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New methods for transducer calibration:Free-field reciprocity calibration of condenser

microphones

PhD thesis by:

Salvador Barrera Figueroa

ØrstedCDTU, Acoustic Technology

Technical University of Denmark

The figure on the front page shows the sound field generated between two microphones placed in

front of each other in a free field; one of the microphones is acting as sound source. The upper

figure shows the sound field generated by the transmitter microphone in the absence of the receiver

microphone. The figure in the middle shows the reflected field generated between the microphones

after the radiated field from the transmitter microphone is removed. A “standing” wave between

the microphones can be clearly seen. The lower figure shows the total sound field. The frequency

of the sound is 20 kHz.

Preface

This thesis is submitted in partial fulfilment of the requirements for the Danish Ph.D.

degree. This Ph.D. project has been financially supported by the Consejo Nacional de Ciencia y

Tecnología (CONACYT) of Mexico. The project has also been supported by the Centro Nacional

de Metrología (CENAM) of Mexico.

This work has been carried out under the supervision of Associate Professors Finn Jacobsen

and Knud Rasmussen at ØrstedCDTU, Acoustic Technology of the Technical University of

Denmark, from the 1st of November 1999 to the 31st of January 2003. I would like to thank them

for their constant support and guidance that took the form of fruitful discussions and thorough

revision of the manuscripts that I submitted to them.

The numerical simulations using the axisymmetrical Boundary Element formulation were

made using OpenBEM. This software is a set of formulations of the Boundary Element Method

originated by the thesis work of Peter Møller Juhl (at the Acoustics Department, Technical

University of Denmark), on axisymmetrical BEM and has been programmed mainly by Peter Møller

Juhl, Morten Skaarup Jensen and Vicente Cutanda Henríquez.

Abstract

The unit of sound pressure, the pascal (Pa), is realised by calibrating a condenser microphone

in a closed coupler where the sound pressure is uniformly distributed over the diaphragm. When the

microphone is placed in a free field, the distribution of sound pressure over the diaphragm will

change as a result of the diffraction of the body of the microphone and the load of the radiation

impedance onto the impedance of the microphone diaphragm. Thus, its sensitivity will change. In

the two cases, a technique based on the reciprocity theorem can be applied for obtaining the

absolute sensitivity either under uniform pressure or free-field conditions.

However, some imperfections on the realisation of the free field may invalidate the

theoretical background for the application of the reciprocity technique. Specifically, the walls of the

anechoic chamber will reflect a portion of the incident energy back to the microphones. Also, under

certain conditions the microphones will couple in the free field and the result is a “standing wave”

between the diaphragms of the microphones. Additionally, there are some problems related with the

measurement set-up. One is the electrical effect known as cross talk, and another is the random error

introduced by the finite measurement time.

This thesis describes the application of a time selective technique to the reciprocity

calibration of laboratory standard microphones in free field. This technique is used for removing the

reflections from the walls of an anechoic chamber, the standing wave between microphones, and

electrical noise by manipulating with the frequency response – electric transfer impedance function

– between two microphones and the corresponding impulse response, thus providing a valid

realisation of the free field.

The acoustic centres of the microphones have been determined from the cleaned transfer

impedance values. The complex free-field sensitivities of the microphones have also been calculated.

The resulting complex sensitivities and acoustic centres have been compared to simulated results and

proved to be in good agreement. This confirms the reliability of the time-selective technique, even

in non-anechoic environments.

Resumé

Lydtryksenheden pascal (Pa) realiseres ved at kalibrere en kondensatormikrofon i en kobler

hvor lydtrykket er ligeligt fordelt over mikrofonens membran. Når mikrofonen anbringes i et frit felt,

ændres lydtryksfordelingen over mikrofonens membran. Det skyldes diffraktionen fra mikrofonens

geometri kombineret med virkningen af strålingsimpedansen og membranens akustiske impedans.

Det betyder at mikrofonens følsomhed ændres. Mikrofonens absolutte følsomhed kan bestemmes

med en teknik baseret på reciprocitetsprincippet under fritfeltbetingelser og under

trykfeltsbetingelser.

I praksis er der ufuldkommenheder ved realiseringen af det fri felt. Specielt vil væggene i den

lyddøde rum reflektere en del energien tilbage til mikrofonerne. Desuden dannes en stående bølge

mellem de to mikrofoner membraner. Dertil kommer i praksis to problemer der skyldes

måleapparatet. Det ene problem er en elektrisk effekt kendt som “kryds-tale” . Det andet er den

tilfældig målefejl som skyldes den endelige måletid.

Denne afhandling beskriver anvendelse af en tidsselektiv metode til reciprocitetskalibrering

af referencemikrofoner i frit felt. Metoden fjerner de reflektioner der kommer fra det lyddøde rums

vægge, stående bølge mellem mikrofoner samt elektrisk støj ved at manipulere med

frekvensresponsen – den elektriske overføringsimpedans – mellem mikrofonerne og det tilsvarende

impulssvar, og giver derfra en forbedret realisation af et frit felt.

Mikrofonernes akustiske centrum og komplekse fritfeltsfølsomhed bestemmes af

overføringsimpedansen “renset” for virkninger af refleksioner m. m.. Resultaterne sammenlignes

med simulerede resultater. Der er fundet god overenstemmelse mellem eksperimentele og

simulerede resultater. Det konstateres at den tidsselektive metode er pålidelig, selv hvor

omgivelserne ikke svarer til et frit felt.

Resumen

La unidad de presión acústica, el pascal (Pa), se materializa mediante la calibración de un

micrófono de condensador en un acoplador cerrado, en el que la presión acústica está distribuida

uniformemente sobre el diafragma. Cuando el micrófono se coloca en un campo libre, la

distribución de la presión sobre el diafragma cambia como resultado de la difracción sufrida por la

onda acústica al incidir sobre el cuerpo del micrófono y por la carga que la impedancia de radiación

ejerce sobre el diafragma del micrófono. Por tanto, la sensibilidad del micrófono cambia. La

sensibilidad del micrófono en ambos casos, en condiciones de presión uniforme y de campo libre,

puede determinarse haciendo uso de una técnica basada en el principio de reciprocidad.

Sin embargo, algunas imperfecciones en la materialización del campo libre pueden invalidar

las suposiciones teóricas que dan lugar a la técnica de calibración por reciprocidad. Las paredes de

la cámara anecóica en la que se materializa el campo libre pueden reflejar hacia los micrófonos una

porción de la energía que incide sobre aquellas. Además, en algunas circunstancias, los micrófonos

se acoplarán en el campo libre; el resultado es una onda estacionaria entre los diafragmas de los

micrófonos. También existen algunos problemas relacionados con el sistema de medición. Uno es

de naturaleza eléctrica y se conoce como “cross-talk”. Otro es un error aleatorio causado por el uso

de tiempos de medición finitos.

Esta tesis describe la aplicación de una técnica temporalmente selectiva en la calibración de

micrófonos patrón de laboratorio en el campo libre. Esta técnica es usada para remover las

reflexiones de las paredes de la cámara anecóica, la onda estacionaria entre los micrófonos y el ruido

eléctrico mediante la manipulación de la respuesta en frecuencia, o impedancia eléctrica de

transferencia, entre dos micrófonos y su respuesta impulsiva. Esta manipulación da como resultado

una mejor materialización del campo libre.

Los centros acústicos de los micrófonos han sido determinados usando los valores de la

impedancia eléctrica de transferencia limpia de perturbaciones. Con estos resultados se han calculado

también las sensibilidades complejas de los micrófonos en campo libre. Ambas magnitudes han sido

comparadas con simulaciones numéricas y se ha observado que hay buena coincidencia entre ellos.

Esto es una confirmación de la confiabilidad de la técnica temporalmente selectiva aun en

condiciones no anecóicas.

Table of contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 The measurement unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 The free-field reciprocity calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 The motivation for this project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 The contents of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2. Free-field reciprocity calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Microphone modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 The microphone as a sound source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 The microphone as a receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.3 The transfer function between two microphones in a free field . . . . . . 34

Chapter 3. Removal of the imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Motivation for using the time-frequency transform . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Treatment of the frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 The low frequency patching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Low-pass filtering of the electrical transfer impedance . . . . . . . . . . . . . . 50

3.3 The cleaning procedure using the Fourier transform . . . . . . . . . . . . . . . . . . . . . 52

3.3.1 The impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.2 The time selective window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 The effect of the cleaning procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 4. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 The application of the cleaning technique onto experimental measurements . . 79

4.1.1 Disturbances in the impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1.1.1 Noise – random noise and electrical cross talk . . . . . . . . . . . . . . 91

4.1.1.2. Standing wave between the microphones and reflections from the

walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2 Absolute determination of the free-field sensitivity . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.1 Physical properties of air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 The free-field sensitivity and derived quantities . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3.1 Free-field correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4 Other experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4.1 Calibration of LS2P microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4.1.1 Preamplifier mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4.1.2 The impulse response of the electrical transfer impedance between

two LS2 microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.4.1.3 Determination of the absolute free-field sensitivity of LS2

microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.4.1.3.1 Free-field correction . . . . . . . . . . . . . . . . . . . . . 134

4.4.2 Combination of LS1 and LS2 microphones . . . . . . . . . . . . . . . . . . . . . 141

4.4.2.1 The impulse response of the 4145 – 4180 microphone combination

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 5. Acoustic centres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.1 The determination of the acoustic centres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.1.1. Determination of the acoustic centres based on the modulus of the electrical

transfer impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1.2 Determination of acoustic centre using phase measurements . . . . . . . 153

5.2. Experimental results obtained from modulus measurements . . . . . . . . . . . . . . 155

5.3 Experimental acoustic centres obtained from phase measurements . . . . . . . . . 164

5.4 Acoustic centres of LS2 microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Chapter 6. Conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Appendix A. The measurement system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.1 Measurement procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.2 Measurement instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

A.2.1 Reciprocity apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

A.2.2 Sound analyser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

A.3 Harmonic distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

A.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

A.5 Additional configurations of the measurement set-up . . . . . . . . . . . . . . . . . . . . 193

Chapter 1 Introduction

1It should be mentioned that the definition of the kilogram as the mass of the kilogram prototype may be possiblychanged to a definition based on fundamental or atomic constants. This is one of the resolutions of the 21st GeneralConference of Weights and Measures, CGPM-BIPM, Comptes Rendus de la 21e Conférence Générale des Poids et Mesures (1999),2001.

11

Chapter 1. Introduction

Metrology deals with the realisation of the measurement units. This realisation can be

achieved at levels of different complexity or accuracy. In any case, the unit is realised on a given

device which can represent the unit while showing the desirable metrological characteristic of

stability. For example, there is only one physical unit that is self-contained in an artifact: the

prototype of the kilogram1, and even this is not completely stable: repeated manipulations of the

prototype draw away minuscule particles of it, or perhaps attach foreign particles to it, changing

inevitably the true value of the weight of the kilogram in an unknown way. On the other hand, there

is only one kilogram prototype (and some copies), thus, there should be a way for comparing the

weight of the kilogram prototype to the weight of other devices such as its copies. And there is it:

a weighing balance. By weighing the kilogram prototype on such a balance the operation of

transferring the true value of the kilogram to the balance itself is carried out. The balance, in its turn,

will measure other weights whose weight will express – in one way or other – the true value of the

kilogram. Thus, there should be an uninterrupted chain between the prototype and the balance used

in the grocery store. Starting from the initial measurement of the prototype one can wonder how

good was the measurement itself, or in other words, how close to the true kilogram was the reading

of the first balance. It is obvious that in each transfer of the unit, a degradation occurs: no copy is

better than the original. And it should be expected that this degradation grows as the level of

realisation gets closer and closer to the humble letter one is willing to send trough the post service

with the right stamps. This increased lack of knowledge of the true value of the weight of the letter

is known as the uncertainty of the measurement. Once the unit is realised, the major concern of any

metrologist is to determine and to reduce its uncertainty. This marks a very thin line between the

establishment of the physical fundamentals of the definition and realisation of the unit and the

routine tasks devoted to the determination of stability, repeatability and reproducibility. It is not easy

for the metrologist to separate the two worlds. So, the author of this document being a metrologist

himself is always walking on that thin line.


12

L ppp =

log ,

2

02 (1.1)

But this work is not about the kilogram. For better or for worse, the rest of physical units

are described by definitions based on fundamental laws of physics. This is the case of the unit of

sound pressure: there is a theoretical definition that defines the physical relationships among the

physical units that give coherence to the unit of sound pressure which is the pascal (Pa). However,

the case of the kilogram helps to establish some similarities with the pascal. First, there is a device

through which the unit of sound pressure is realised with the highest achievable stability: a

microphone. Second, there is a technique through which the device is transformed into a standard:

the reciprocity principle applied to electroacoustics. This combination of the device with the

realisation technique supposes a unique and symbiotic relation: there is no standard without the

realisation technique. Furthermore, it can be argued that the technique is the standard itself. Finally,

there is the third element: uncertainties. The realisation of the unit has inevitably an uncertainty. This

uncertainty comes from the different aspects of the process: environmental conditions, limited

accuracy of the measurement instruments, repeatability of the mechanical mounting, etc. The last

is the routine duty of the metrologist. The first two are the subject of this research.

1.1 The measurement unit

Formally speaking, the unit of sound pressure is the pascal (Pa). However, the dynamic range

of the sound pressure has made it necessary to use an expression that compresses that range in such

a way that the comparison of different sound pressures may simplify. Such an expression is the bel

(B), and its widely used sub-multiple, the decibel (dB). The sound pressure level is the base 10 logarithm

of the ratio of the mean square sound pressure to a conventional reference value,

where p0 is the conventional reference of 20 µPa. The unit of this level is the bel (B). This provides

a simple and elegant way for handling the unit of sound pressure. However, it is not the only way.

Another approach is to use the natural logarithm (Napierian) of the ratio. This yields another unit:

the neper (Np). These two units can be used for expressing the same ratio.

The adoption of either one has been the subject of several discussions and papers in the

recent years. See for example references [1], [2], [3], and [4]. The major argument is the physical

coherence of the selected unit with the fundamental units of the SI. It is apparent that the final


2Schottky, W., Tiefempfangsgesetz, Zeits. f. Physik 36, pp. 689-ff, 1926, as quoted in [6].13

accepted recommendation may be a compromise between this coherence and the widespread use

of the decibel instead of the neper. This recommendation reserves the use of the neper to the ratio

of pure sinusoidal functions, and the use of the bel for quantities for which there is no single

frequency. This in facts means that the CGPM-BIPM would accept that the bel and decibel will

continue to be used in acoustic metrology because of the difficulty of achieving pure sinusoidal

functions that are formally defined in the time interval [-4, 4].

1.2 The free-field reciprocity calibration

The calibration of microphones in a free field was originally carried out by making use of

the Rayleigh disk. Ballantine [5] determined the wave response of a spherically shaped microphone by

making use of this apparatus. The free-field reciprocity calibration of microphones was first

discussed by MacLean [6]. Based on the definition of W. Schottky of a microphone as a reversible

transducer2, MacLean proposed the absolute calibration of a microphone under free-field conditions

and under pressure conditions as well. Later, Watten-Dunn [7] described in more detail some

theoretical aspects of the calibration, such as the acoustic centre of the microphones, and suggested

that the coupling of the microphones may occur. Rudnick and Stein [8] presented and experimental

study of the free-field calibration based on the works of MacLean and DiMattia [9]. Further

experimental attempts of free-field calibrations were described by Terry [30] and Niemoller [31] who

introduced the possibility for carrying out calibrations in the time domain.

In the 1980's and the early 1990's, some national laboratories of metrology described the

experimental apparatus used for free-field reciprocity calibrations, and showed some obtained

results. For example, Gibbings and Gibson [10] described the calibration system of the National

Laboratory of Australia, and Burnett and Nedzelnitski [11] reported the calibration system of the

National Bureau of Standards in the United States. Durocher [12] also presented some

considerations for the realisation of free-field calibrations in the National Laboratory of France.

Barham [16] has described the free-field calibration facilities of the National Laboratory of UK.

Furthermore, an international comparison involving some of these laboratories has been carried out

[46] with some mixed and inconclusive results.


14

In any case, these works provide a framework where this project fits. These works have

shown that there are a number of problems associated with the realisation of the free field where

the calibrations are carried out. Specifically, the walls of the anechoic chamber will reflect a portion

of the incident energy back to the microphones. Additionally, under certain conditions the

microphones will couple in the free-field and the result is a “standing wave” between the diaphragms

of the microphones. This standing wave may be considered as a violation of the reciprocity principle

because this coupling is not allowed in the formulation of the reciprocity principle. Finally, there are

some problems related with the measurement set-up. One is an electrical effect known as cross talk,

and other is a random variation or “noise” introduced by the finite measurement time.

1.3 The motivation for this project

A solution to the problems encountered in the realisation of the free-field calibration of

microphones may be the application of a time selective technique, as suggested by Durocher [32],

Blem[33] and Vorländer [34]. Thus, it is the motivation of this project to find an alternative for

realising and improving the realisation of the free field where the unit of sound pressure will be

realised through the calibration of condenser microphones by making use of time selective

techniques.

The need of alternative realisations is supported by unavoidable economical considerations.

The capital investment required to build a suitable anechoic chamber is always large. Large in terms

of money and in terms of time. It is not a trivial task to design and build the chamber. A careful

design of the geometry of the chamber as well as the geometry of the absorbing lining is a

painstaking process. Not less demanding is the realisation of tests for qualifying the room. All these

factors make it difficult to find the sufficient funding for investing in such projects. Especially when

the “only” product is the realisation of a unit, and the offering of a calibration service that will be

required by few customers.

And even if a well designed anechoic chamber is available, the very nature of the practical

implementation of the calibration method and the transducers will introduce conditions alien to the

theory that may invalidate the realisation of the free field in subtle ways as mentioned above.


15

1.4 The contents of this thesis

This thesis describes the theoretical background of a time selective technique based on

application of the Fourier transform applied to the free-field reciprocity calibration of condenser

microphones. This time selective technique is applied to experimental measurements carried out on

laboratory standard microphones, and the results are compared to results from simulated results

obtained using the Boundary Element Method.

This thesis is divided into six chapters, including this introduction, and one appendix.

Chapter 2 deals with the theoretical background that allows the application of the reciprocity

principle to the absolute calibration of condenser microphones. The theoretical basis for the

application of the reciprocity principle to the absolute calibration of condenser microphones is

outlined in this chapter. The fundamental fact that the condenser microphone is a reciprocal

transducer is shown. Afterwards, the electrical transfer impedance is described. The electrical

transfer impedance between two microphones contains the free-field sensitivity of the microphones,

thus providing means for determining it without the need of a sound pressure reference source.

Chapter 3 describes a time selective technique for removing the imperfections of the

electrical transfer impedance that invalidate the assumption of a free field. The practical realisation

of the free field where the reciprocity calibration of microphones is carried out is imperfect.

Reflections from the non-totally absorbent boundaries, acoustical coupling between the

microphones, and electrical and background noise are some of the contaminating agents of the free

field. All these phenomena have a well defined time occurrence. Thus, it is possible under certain

conditions to remove them if a time-frequency transformation is carried out. The equivalence

between frequency response and impulse response can be used for isolating the direct wave between

the microphones from the contaminating phenomena. A condition that must be fulfilled if the time-

frequency transformation is to be applied is that the frequency response should be known at all

frequencies. However, when a reciprocity calibration is carried out, it is not possible to measure the

complex frequency response (electrical transfer impedance between the two microphones) at all

frequencies. Thus, a procedure for completing the frequency response is described and studied in

this chapter. Once the frequency response is complete, the impulse response can be obtained using


16

the inverse Fourier transform. The direct wave between the two microphones can be isolated from

the rest of the impulse response by means of a time selective window. Then, the cleaned frequency

response can be obtained by applying the Fourier transform. The whole procedure involves thus,

the completion of the contaminated frequency response followed by an inverse Fourier

transformation. Once in the time domain a time window is applied and thereafter, a Fourier

transformation returning to the frequency domain where the final product is a clean frequency

response. However, the procedure has an overall effect that is present in the clean frequency

response and may show a deviation from the true value of the frequency response. The cleaning

procedure and its overall effect are studied in this chapter.

Chapter 4 presents the results obtained when the time selective procedure is applied to the

measurements carried out with the experimental set-up. A number of phenomena are described and

analysed with the help of numerical simulations. The objective is to show the different factors that

invalidate the practical realisation of the free field, and how the cleaning technique helps to remove

them, yielding thus a better realisation of the free field. This cleaned function is used for obtaining

the free-field sensitivity of the microphones. The basic parameters on which the sensitivity depends

will be described together with their measurement procedures After this account, the calculation

procedure of the free-field sensitivity is described. Additionally, a derived quantity, the free-field

correction is also defined. This quantity is useful for comparing the obtained results with results

from elsewhere. A valuable source for comparison is the numerical simulation of the problem of the

microphones in the free field using the Boundary Element Method (BEM). The objective of the

comparisons is to show that the cleaning procedure effectively removes any reflections form the

walls, standing waves between the microphones and random noise.

Chapter 5 presents a discussion about the acoustic centres of condenser microphones. An

analysis of the problem of determining the acoustic centres of condenser microphones is carried

out. Procedures for obtaining the acoustic centres from the cleaned electrical transfer impedances

are outlined. The convenience of determining the acoustic centres based on the fulfilment of the

inverse distance law (modulus based) and on the phase of the free-field sensitivity is analysed


17

Chapter 6 contains the most relevant conclusions of the project. It also describes some

proposals for future development and improvement of the free-field calibration of condenser

microphones.

Finally, the appendix contains a brief description of the measurement system and procedure

employed for measuring the electrical transfer impedances, and a description of some practical

problematic of the measurements. A brief description of efforts for minimising the cross talk is

given.


18

Chapter 2 Free-field reciprocity calibration

19

TRANSMITTERMICROPHONE

(SOUND SOURCE)

PROPAGATIONMEDIUM

(FREE FIELD)

RECEIVERMICROPHONE

INPUTQUANTITY

(ELECTRICAL CURRENT)

OUTPUTQUANTITY

(ELECTRICAL VOLTAGE)

Figure 2.1 Schematic representation of the propagation process from a microphone used as sound source to a

microphone used as a passive receiver in free field.

Chapter 2. Free-field reciprocity calibration

Overview

The theoretical basis for the application of the reciprocity principle to the absolute

calibration of condenser microphones is outlined in this chapter. The fundamental fact that the

condenser microphone is a reciprocal transducer is shown. The analytical determination of the

electrical transfer impedance is described. The electrical transfer impedance between two

microphones contains the free-field sensitivity of the microphones, thus providing means for

determining it without the need of a sound pressure reference source.

2.1 Introduction

A condenser microphone can be used as a sound source because of its reciprocal behaviour.

In this situation, the microphone should be excited with a suitable input signal, a sinusoidal electrical

current, for example. This will generate a time varying movement of the microphone’s diaphragm.

This movement is the result of the coupling of the vibrational behaviour of the stretched membrane

and the damping that occurs in the air film between the diaphragm and the backplate of the

microphone. It is difficult to predict the shape of the actual displacement distribution. However,

some assumptions can be made if some limitations are accepted.

The medium that surrounds the source is perturbed, compressed and expanded, by this

movement. In this case, the medium is a gas, more specifically, air. Ideally, the medium is

unbounded. It implies that no energy is reflected back to any of the microphones. The perturbation

propagates as sound pressure. Thus, the pressure perturbation generated by the movement of the

diaphragm will propagate away from the microphone. The spatial distribution of the perturbation

is highly dependent on the displacement distribution. Additionally, if the body of the source


20

microphone is not very small compared with the wavelength, there will be an interference caused

by the propagating wave incident over the body of the microphone and the reflections from it. This

makes it difficult to determine the actual sound pressure at a given point.

If a second microphone is placed in the medium, the incident sound wave will have an effect

on the diaphragm and the finite body of the microphone. Similarly to the case of the source or

transmitter microphone, it will cause an interference that will modify the actual pressure distribution

over the diaphragm, i.e., the sound pressure over the diaphragm will not be identical with the sound

pressure in the absence of the receiver microphone. The sound pressure over the diaphragm will

provoke it to move in a similar manner as the transmitter microphone. However, since the

microphone is acting as a passive receiver, the changes of distance between the diaphragm and the

back plate will cause a change on the capacitance of the microphone that can be detected as an

alternating voltage at the electrical terminals of the microphone.

According to the above exposition, which is visualised in the diagram shown in figure 2.1,

it is needed to solve the wave equation for all the boundary conditions described in order to

determine the actual transfer function between the input to the transmitter microphone to the

output of the receiver microphone. Additionally, the internal behaviour of each microphone must

also be solved. The solution of the external field has been attempted by several researchers, Matsui

[13], Bjørnø [14], Juhl[15], and Barham [16] among others, either numerically or analytically, by

assuming that the microphone is mounted on a semi-infinite rod, which is a condition that should

be normally met if the rod comes from a wall in an anechoic chamber; and that the movement of

the diaphragm has an analytically defined shape, such as a parabolic or a Bessel-like movement, that

is considered a realistic approximation to the actual displacement distribution. Less work has been

done on the solution of the internal field (see, for example, references [17], [18], [19]) by solving a

coupled system that contains the wave equation, the Navier-Stokes equation, and the equation of

state. Bao [18] attempted to couple the internal solution with the external field using the Boundary

Element Method.

Because of the difficulties of solving the complete system, a common approach is that the

system can be analysed when decomposed into three simplified subsystems, namely, the transmitter

microphone, the propagation medium, and the receiver microphone. These subsystems can be


21

p T i Z qae a= + , (2.1)

modelled separately and a coupled solution can be found. This task can be simplified if some

assumptions can be made.

The first is to assume that the wavelength is long compared to the size of the microphones,

i. e., , λ being the wavelength, and a the largest transverse dimension of the microphone,λ >> a

normally its diameter.

The second assumption is to consider that the distance between the microphones, r, is long

compared to the wavelength, i. e., . This indicates that the receiver microphone is locatedλ << r

in the far field of the sound source. Thus, in the free field, the sound pressure should follow the law

of the inverse distance.

Considering that the wavelength is long compared to the dimensions of the microphones,

it is possible to say that the sound pressure is uniform over the sensitive elements of the

microphone, and that the microphones can be substituted by a point, and modelled after its

representation as a two port (four pole) electrical network with lumped parameters.

The medium can be modelled as a boundary free, isotropic medium, taking into account the

propagation losses caused by the fact that air is a non-ideal gas.

2.2 Microphone modelling

The condenser microphone can be considered as a reciprocal transducer. It means that it will

generate an electrical output when subjected to an acoustical perturbation and vice versa. The

behaviour of such devices has been studied and analysed by many authors, see for example

references [20], [21], and [22]. The microphone can be analysed as a two port network from which

a set of equations relating the electrical and acoustical quantities can be obtained. In general, these

equations are called canonical equations of the network and are the result of the application of

Kirchhof’s laws,


22

fQSee=

−2 0ε

, (2.3)

u Z i T qe ea= + , (2.2)

CS

d x=

+ε0 , (2.4)

Le R

e

U0

u

+

+

-

-

uc

d

+

-

i

Diaphragm+x

x

fA

fe

Figure 2.2 Simplified scheme of a condenser microphone, showing the electrical and acoustical

components.

where q is the volume velocity of the diaphragm, p is the sound pressure over the diaphragm, i is the

electrical current flowing through the electrical terminals of the microphone, and u is the voltage

across the electrical terminals, Za is the acoustical impedance, Ze is the electrical impedance, and Tea

and Tae are the transduction coefficients. If a transducer is reciprocal, the transduction coefficients

are equal, .T T Tae ea= =

The next step is to find an appropriate expression for the transduction coefficient of an

electrostatic transducer as the condenser microphone. The elementary expressions for the attraction

force between two oppositely charged plates, fe, and for the capacitance of a parallel plate condenser,

C, are

where Qe is the electrical charge, ε0 is the permittivity of free space, S is the area of the plates, d is

the distance between plates, and x is the deflection from the equilibrium position caused by the

balance between excitation forces. Figure 2.2 shows a simplified diagram of the transducer. It is

apparent from this diagram that the electrostatic force should be in opposite direction to the

restoring force of the diaphragm, therefore the negative sign in equation (2.3).

Then, a balance of forces acting on the diaphragm can be made, that is the sum of the

restoring force of the diaphragm, the acoustic forces and the electrostatic force. This balance is to


23

f f L x R x C x

f L x R x C xQS

A e m mm

A m mm

e

+ = + +

= + + +

&& & ,

&& & ,

1

12

2

0ε

(2.5)

( )

U u L Q R QQC

U u L Q R QQ d x

S

e e e ee

e

e e e ee

0

00

+ = + +

+ = + ++

&& & ,

&& & .ε

(2.6)

provide ultimately the interchange between kinetic and potential energies that will keep the system

in harmonic equilibrium. Thus, the balance can be stated as a differential equation

where fA is the acoustical force. The same can be done in the electrical mesh, where the balance of

voltages is given by the combination of the polarisation and signal voltages and the drop of voltages

around the mesh. The result is

It should be mentioned that in the case of the condenser microphone, the inductance and

resistance can be neglected. However, these quantities are included in the following analysis in order

to keep its generality.

It can be seen in equation (2.5) that the term containing the square of the electrical charge

introduces a non linearity in the equation. The same happens in the electrical equation, where the

product Qex also introduces a non linearity. This non linearity can be disregarded because the

displacement of the diaphragm around the equilibrium position, x, is very small compared to the

distance d, such as . This can also be applied to the charge itself, which is a( )Q d x Q de e+ ≈

function of the capacitance which is in its turn, a function of the changes of distance d, that are given

by x. Thus the capacitance can be approximated by a constant, and the charge also becomes constant

for a given voltage across the capacitor.

This linearisation is enough for obtaining a suitable solution for the system. However, it may

also obscure some characteristics of the system itself, especially those related with the possibility of

the non reciprocal behaviour of the transducer. An alternative approach is to expand these variables


24

( )f t f en

njn t

n=

=−∞

∞

∑ 1η

ω , (2.7)

( )

( )

( )

x t x e

Q t Q e

f t f e f

uUe

Ue U t

nn

jn t

n

en

e njn t

n

An

A njn t

An

j t j t

=

=

= −

= + =

=−∞

∞

=−∞

∞

=−∞

∞

∑

∑

∑

1

1

1

2 2

0

1 11

η

η

η

ω

ω

ω

ω

ω ω

,

,

,

cos .

,

, ,

(2.8)

into a Fourier series of complex coefficients. The mathematical manipulations can be read in detail

in [20]. The highlights of the results will be analysed here.

The Fourier series of complex coefficients of a real, periodic function f(t) are given by (see

reference [23])

where η0 is the Neumann factor, that takes values of 1 if n = 0, and 2 otherwise. The coefficients

fn have the property that f+n = f*-n.

If the variables u, Qe, fA and x are expanded using the complex Fourier series, the result is

The last equation indicates that the voltage feeding the microphone is the reference phase. Thus, any

phase difference is to be referred to this quantity. With these expressions, it is possible to calculate

the square of the electrical charge and the product of the charge and the displacement present in

equations (2.5) and (2.6); the derivatives of the two quantities can also be calculated. This results in


25

( ) ( )

( ) ( )( )

& ,

&& ,

& ,

&&

,

,

, ,

, ,

,

x jn x e

x n x e jn jn x e

Q jn Q e

Q n Q e jn jn Q e

Q Q Q e

xQ x Q e

nn

j t

nn

j t

nn

j t

en

e nj t

en

e nj t

ne n

j t

en m

e n e mj n m t

en m

n e mj n

=

= − =

=

= − =

=

=

∑

∑ ∑

∑

∑ ∑

∑∑ +

1

1 1

1

1 1

1

1

2 2

2 2

2

ηω

ηω

ηω ω

ηω

η ω η ω ω

η η

η η

ω

ω ω

ω

ω ω

ω

( )+∑∑ m tω .

(2.9)

x xxe

xe

Q QQ

eQ

e

ff

ef

e

u U e U e U t

j t j t

e ee j t e j t

AA j t A j t

j t j t

= + +

= + +

= +

= + =

−

−

−

−

01 1

01 1

1 1

1 1 1

2 2

2 2

2 212

12

ω ω

ω ω

ω ω

ω ω ω

*

,, ,

*

, ,*

*

,

,

,

cos .

(2.10)

These equations provide a whole series expansion for each physical variable. When

equations in (2.8) and (2.9) are substituted in equations (2.5) and (2.6), this will result in an infinite

number of terms having a generic form HNejNwt. If the equations are to be satisfied at any instant,

sets of equations formed with terms containing the same frequency must be also satisfied. Although

there are an infinite number of terms, the series converge rapidly, and few expansions are needed

in order to find a suitable solution. In any case, some further simplifications may be required because

some higher order terms, > N, may be obtained when expanding the equations with the Nth order.

It is done by assuming that the system behaves closer to a linear system, and that the higher order

terms can be safely discarded. This is to be addressed when the case occurs in the coming

development.

If only the first two terms of each series are considered, that is n = 0, and n =1, the

expansion of equations (2.8) becomes


26

( ) ( )

( ) ( )

& ,

&& ,

& ,

&& .

*

*

, ,*

, ,*

x j x e j x e

x j j x e j j x e

Q j Q e j Q e

Q j j Q e j j Q e

j t j t

j t j t

e ej t

ej t

e ej t

ej t

= −

= −

= −

= −

−

−

−

−

12

12

12

12

12

12

12

12

1 1

1 1

1 1

1 1

ω ω

ω ω ω ω

ω ω

ω ω ω ω

ω ω

ω ω

ω ω

ω ω

(2.11)

( ) ( )( )

xQ x Q x Q x Q e x Q x Q e

x Q x Q x Q e x Q e

Q Q Q Q e Q Q e

Q Q

e e e ej t

e ej t

e e ej t

ej t

e e e ej t

e ej t

e e

= + + + + +

+ + +

= + + +

+

−

−

−

0 012 0 1 1 0

12 0 1 1 0

14 1 1 1 1

14 1 1

2 14 1 1

2

20

20 1 0 1

12 1 1

1

, , , ,* *

,

*, ,

*,

*,

*

, , , , ,*

, ,*

,

ω ω

ω ω

ω ω

4 12 2 1

4 12Q e Q ee

j te

j t, ,

*2 .ω ω+ −

(2.12)

( ) ( )[ ]( )( )

[ ]

12 1

12 1

12 1

12 1

12 1

12 1

012 1

12 1

00

20 1 0 1

1

12

f e f e L j j x e j j x e

R j x e j e

C x x e x e

S Q Q Q e Q Q e

j t j tm

j t j t

mj t j t

m

j t j t

e e ej t

e ej t

ω ω ω ω

ω ω

ω ω

ω ω

ω ω ω ω

ω ω

ε

+ = + +

+ +

+ + +

+ +

− −

−

−

−

* *

*

*

, , , , ,* ,

(2.13)

The derivatives resulting in

and the products

The terms in the second row of each expansion in equation (2.12) can be neglected for two reasons.

The terms containing information about the higher harmonics of the fundamental frequency are not

to be used in the analysis of the fundamental frequency. However, they must be included if an

analysis of the harmonic distortion is to be made. This is discussed later in the chapter. The terms

containing products of coefficients whose sum is larger than 1 can be neglected by considering that

this product is negligible when compared with the fundamental frequency coefficients because the

microphone is a quasi-linear system, and thus, the coefficients of the harmonic components are very

small.

Now, the expansions from equations (2.10) and (2.11) and (2.12) can be substituted in (2.5)

and (2.6).


27

( )UQS d x

xC

QS

e

m

e

00

00

0 02

00 2

= +

= +

,

,

,

.

ε

ε

(2.15)

( ) ( )[ ]( )

( )( )

U U e U e L j j Q e j j Q e

R j Q e j Q e

dS Q Q e Q e

x Q x Q x Q e x Q

j t j te

j te

j t

ej t

ej t

e ej t

ej t

e e ej t

012 1

12 1

12 1

12 1

12 1

12 1

00

12 1

12 1

00 0

12 0 1 1 0

12 0

1

+ + = + +

+ +

+ + +

+ + +

− −

−

−

ω ω ω ω

ω ω

ω ω

ω

ω ω ω ω

ω ω

ε

ε

*, ,

*

, ,*

, , ,*

, , , ( )[ ]e ej tx Q e,

* *, .1 1 0+ − ω

(2.14)

The terms can be rearranged according to the frequency information they contain. That is, for each

different multiple of the fundamental frequency, an equation is obtained. The result of this algebraic

manipulation of equations (2.13) and (2.14) is, for the zero frequency terms,

Equation (2.15) indicates the existence of an electric equilibrium when the transducer is

charged. The first equation points to the fact that a change in the polarisation voltage will induce a

proportional change in distance between the plates, which is the operation principle of the

microphone. This is an expected result. On the other hand, the second equation represents the

mechanical equilibrium of the transducer. It suggests the existence of an equilibrium between the

restoring force of the diaphragm and the electrical force. This relation between the static

displacement and the compliance of the microphone combined with the changes in the polarisation

voltage may help to find some cases where the non-linearity of the system may dominate the

mechanical behaviour of the diaphragm.

For extracting and rearranging the terms containing the fundamental frequency in equations

(2.13) and (2.14), it is useful to remember the following definitions that are present in the canonical

equations,


28

Z j L Rj C

Z j L Rj C

CS

d xq j x i j Q

m m mm

e

e

≡ + +

≡ + +

≡+

= =

ωω

ωω

εω ω

1

1

00

01 1 1 1

,

,

, , .,

(2.16)

T TQj Sea ae

e= = , .0

0ωε(2.18)

U Z iQj S

q

fQj S

i Z i

ee

Ae

m

1 10

01

10

01 1

= +

= +

,

,,

,

.

ωε

ωε

(2.17)

Tj C

CS

Qaek

ke

= =1 0

0ωε

, .,

(2.19)

Thus, the equations from the terms containing the fundamental frequency terms are

Comparing these equations with the canonical equations (2.1) and (2.2), it can be seen that the

transduction coefficient is symmetric:

This expression, having a phase quadrature, suggests that the transduction coefficient may be

considered as the impedance of a capacitor, Ck,

This result is very useful, because the equivalent circuit can be drawn as shown in figure 2.3. This

lumped parameter model of the microphone can be used extensively for analysing its behaviour. It

is worth mentioning that more complete models have been described in the literature. For example,

Rasmussen [24] describes a model that includes the lumped elements describing the microphone

diaphragm in terms of a displacement that follows the Bessel function, the air film between the

diaphragm and the back plate, the holes and slit on the back plate, and the back cavity. Although this

representation represents the state of the art concerning the modelling of the microphone, as a first

approximation the simpler model, depicted in figure 2.3, that considers the total acoustical

parameters of the microphone can be used.


29

uj C

qj C

ik c

= − +1 1

ω ω, (2.20)

p Z qj C

iak

= −1

ω, (2.21)

Zj C

j L Raa

a a= + +1

ωω . (2.22)

Ra La Ca Ck

q

p u

i-Ck

Ck Cc

Figure 2.3 Equivalent circuit of the microphone using the global lumped parameter elements.

In figure 2.3, Ca is the acoustic compliance of the microphone, La is the acoustic mass, Ra

is the acoustic resistance (damping), Ck is the compliance due to the electro-acoustical coupling, Cc

is the electrical capacitance of the microphone when the diaphragm is blocked, q is the volume

velocity of the diaphragm, p is the sound pressure over the diaphragm, i is the electrical current

flowing through the electrical terminals of the microphone, and u is the voltage across the electrical

terminals.

In order to find the analytical relation between the different quantities shown in the diagram,

some common tools of electrical engineering can be applied. By making use of the network

equations, or Kirchhoff’s second law, it is possible to determine that the voltage on the electrical

terminals is

and correspondingly, the sound pressure on the microphone diaphragm is

where Za is the acoustical impedance of the diaphragm given as

Using equations (2.20) to (2.22) it is possible to define the different relations between the

acoustic and electrical quantities that describe the behaviour of the condenser microphone. For

example, it is possible to obtain the sensitivity of the microphone. If the sound pressure is uniform

over the surface of the diaphragm, the so-called pressure sensitivity can be obtained.


30

Z j Caa

ω ω→≈

0

1. (2.24)

M j Cj C

CCp

k

a

a

kω ωω

→= − ⋅ = −

0

1 11 , (2.25)

M up

j Cq

Z q j C Zpi

k

a k a

= =−

= −=0

11ω

ω. (2.23)

The sensitivity of the microphone is defined as the ratio of the open circuit voltage in the

electrical terminals to the sound pressure over the diaphragm when the electrical current is equal to

zero. If the diaphragm is exposed to a uniform sound pressure, the lumped parameter model can

be used for obtaining the pressure sensitivity of the microphone. Using equations (2.20) and (2.21)

according to the sensitivity definition yields

Therefore the sensitivity of the microphone is inversely proportional to its acoustical impedance.

This is a logical result because the changes in the polarisation voltage are a function of the

displacement of the diaphragm. Thus, a larger impedance means a smaller velocity and displacement

of the diaphragm for a given pressure, hence a lower sensitivity.

The result given in equation (2.23) makes it possible to obtain the sensitivity of the

microphone as long as the coupling compliance is known. This can be overcome by further

manipulations of the circuit equations of the microphone and its impedance. As the diaphragm is

depicted as a single degree of freedom system, it can be said that one of its properties is that at very

low frequencies, the movement of the diaphragm is controlled by its stiffness. Then, its impedance

approaches that of the compliance of the diaphragm,

Using the above result in equation (2.24), the microphone sensitivity at low frequencies is

which is a real valued quantity. This is in agreement with the behaviour of a single degree of freedom

system, which at frequencies well below its resonance frequency has a phase of nearly zero. By using

this result, the coupling compliance can be defined in terms of the low frequency value of the

sensitivity, and then substituted in equation (2.23) to obtain the pressure sensitivity at any frequency,

i. e.,


31

M j C

M

Zpa

p

a= ⋅ →1 0

ωω . (2.26)

102

103

104

−16

−14

−12

−10

−8

−6

−4

−2

0

2

Frequency (Hz)

Nor

mal

ised

sen

sitiv

ity (

dB)

Figure 2.4 Comparison between measured and calculated pressure sensitivity for a LS1 microphone,

–––– Calculated, – - – - – Measured.

The parameters of the acoustic impedance of the microphone and the low frequency

sensitivity can be obtained from experimental measurements. Figure 2.4 shows a comparison

between the measured and calculated pressure sensitivities of a B&K 4160 microphone. There is a

very good agreement between the experimental and the modelling results at low frequencies, but the

agreement worsens above the resonance frequency where the model breaks down. However, the

results can be used when the limitations are taken into account. A possibility for extending the range

of validity of the model is to use the model proposed by Rasmussen [24]. This will be made in

chapter 3, when a simulation of the transfer function between two microphones is to be developed

and applied for assessing the possibility of applying a time selective technique to the free-field

calibration.


32

RaLa Ca Ck Ck Cc

-Ck

qu

iZa,r p

Figure 2.5 Equivalent circuit of the microphone when used as a sound source. The radiation impedance

appears as a load in series with the acoustical impedance of the microphone.

p Z qa r= − , . (2.27)

u Z i M Z qe p a= + , (2.28)

− = +Z q M Z i Z qa r p a a, . (2.29)

2.2.1 The microphone as a sound source

The condenser microphone is a reciprocal transducer, that is, it can be used as a source or

as a receiver. When the microphone is used as a sound source in a free field, there will be a load over

the diaphragm of the microphone. This load is caused by the radiation impedance. When the

microphone is analysed as a sound source, it can be considered as a high impedance source. Thus,

it will keep its volume velocity whatever the load on it. Then, the load should be connected as an

impedance in series with the microphone impedance. Figure 2.5 shows the equivalent circuit for

such a condition.

In order to obtain the correct relation between the acoustic and electrical quantities in this

new configuration if the equivalent circuit, it is needed to carry out the same analysis as before. It

can be deduced from the network that the sound pressure generated by the microphone is the

product of the radiation impedance, Za,r, and the volume velocity, q,

Considering this, and using the expression for the sensitivity of the microphone at any frequency,

equation (2.23), for extracting the value of the coupling capacitance, the equations of the equivalent

network – equations (2.20) and (2.21) – can be rewritten as

Thus, the volume velocity can be obtained by re-arranging equation (2.29)


33

( )− =+

qM Z

Z Zip a

a a r,

. (2.30)

p p Z qa r= ′ − , . (2.31)

RaLa Ca Ck Ck Cc

-Ck

q

p' u

i

Za,r

p

Figure 2.6 Equivalent circuit of the microphone when used as a receiver. The radiation impedance appears

as a load in series with the acoustical terminals of the microphone

Thus, the volume velocity will depend on the ratio of the acoustical impedance of the microphone

to the radiation impedance. The radiation impedance of the microphone cannot easily be calculated

because the wave equation must be solved for the interior and exterior problems posed by the

microphone. An approximation could be used, for example a baffled piston (see reference [25]). This

indicates that the radiation impedance increases with the frequency. This impedance has also a real

– resistance – and an imaginary – reactance – part.

2.2.2 The microphone as a receiver

When the microphone is located in a free field where a plane wave propagates with a sound

pressure p0, it will disturb the field, and the sound pressure on the diaphragm, p, will not be the

pressure in absence of the microphone. This suggests that the load impedance is connected in series

with the acoustical impedance of the microphone. Figure 2.6 shows the diagram of the circuit.

Za,r is the radiation impedance of the microphone, p’ is the sound pressure when the

diaphragm is blocked, i. e., when q = 0.

From the figure, it can be deduced that the sound pressure incident on the microphone

diaphragm is

The equations of the equivalent network then become:


34

u Z i M Z qe p a= + , (2.32)

p Z q M Z i Z qa r p a a' ,,− = + (2.33)

( )p M Z i Z Z qp a a a r' .,= + + (2.34)

( )pp

S f'

, ,0

= θ (2.35)

Mupf

i

==0 0

. (2.36)

u M Z qp a= , (2.37)

( ) ( )p S f Z Z qa a r0 ⋅ = +, .,θ (2.38)

( )M MZ

Z ZS ff p

a

a a r

= ⋅+

⋅,

, .θ (2.39)

Furthermore, the acoustic pressure p’ is defined when the diaphragm is blocked, i.e., rigid. Thus, the

sound pressure, p’, and the undisturbed sound pressure, p0, can be related by the expression

where S(f, θ) is the scattering factor, which is function of the frequency, f, and the angle of incidence

of the sound wave on the microphone diaphragm, θ. This quantity depends on the geometrical

configuration of the microphone.

The free field sensitivity can be defined (see references [26], and [IEC61094-1]) as the ratio

of the open circuit voltage on the terminals of the microphone, u, to the sound pressure that would

exist at the position of the acoustic centre of the microphone in the absence of the microphone, p0,

Using equation (2.35) and the equations from the equivalent network of the microphone

used as receiver – equations (2.32) and (2.34) – when the current is equal to zero gives,

The free-field sensitivity is then

The result in equation (2.39) shows that the free field microphone sensitivity differs from the

pressure sensitivity not only due to the geometrical configuration of the microphone but also

because of the relation between the acoustic impedance of the microphone and the radiation

impedance. The relation between microphone and load (radiation) impedances may be approximated


35

( )p j cQkd e

j t kd0 4= −ρ

πω , (2.41)

( )Q qS f= , .θ (2.42)

( )q M

ZZ Z

iMS f

ipa

a a r

f= −+

= −, ,

.θ

(2.43)

( ) ( )C M Mf f p= −log log . (2.40)

for some simple cases such as considering the diaphragm as a piston mounted on an infinite wall or

at the end of an unbaffled cylinder. Also, the geometrical diffraction factor could be obtained using

analytical or numerical techniques also under simplifying assumptions.

For practical purposes, the last two factors in equation (2.39) can be combined in one, and

the ratio of the free-field to pressure sensitivity can be obtained. If the decimal logarithm of this ratio

is taken, a new quantity can be defined

This is called the free-field correction. This correction can be considered to be approximately the

same for all microphones of the same type due to the fact that the ratio of the impedance of the

microphone to the radiation impedance will be the approximately the same for all microphones if

they have similar impedance and geometry; this also applies to the geometrical configuration of the

microphones. The use of this correction makes it simpler and easier to obtain the free-field

sensitivity of a microphone from the pressure sensitivity in practical situations, although with

reduced accuracy.

2.2.3 The transfer function between two microphones in a free field

If the microphone is substituted by a point source radiating to the open space, the sound

pressure generated at a point at a radial distance d from the point source is (using the far field

approximation, this is, provided that ka <<1 or kd >>1):

The parameter Q is the source strength. It can be shown that the source strength is related

to the volume velocity as:

From the equations of the equivalent network, the volume velocity is:

Thus, the sound pressure generated by the transmitter microphone is:


36

( )p jfd M i ef

j t kd0 2= ⋅ ⋅ ⋅ −ρ ω . (2.44)

u M prec f rec= ⋅, .0 (2.45)

( )u jfd M M i erec f rec f trans trans

j t kd= ⋅ ⋅ −ρ ω

2 , , , (2.46)

( )Zui

jfdM M etrans rec

rec

transf rec f trans

j t kd, , , .= = ⋅ ⋅ −ρ ω

2(2.47)

When the sound wave generated by the transmitter microphone travels to the position where

the receiver microphone is placed, it will provoke a variation of the voltage on the electrical

terminals of the microphone. The value of this open circuit voltage can be calculated using the very

definition of free field sensitivity

Substituting the sound pressure (2.44) in (2.45):

The ratio of the output voltage to the input current of a passive electrical network is known

as electrical transfer impedance. Rearranging (2.46) it is possible to obtain the electrical transfer

impedance of the two microphone systems:

It can be seen that the electrical transfer impedance is the basis of the reciprocity technique

as it contains information of the free-field sensitivities of the two microphones, thus providing

means for determining their sensitivities.

Chapter 3 Removal of imperfections

37

Chapter 3. Removal of the imperfections

Overview

Any practical realisation of the free field where the reciprocity calibration of microphones

is carried out will imperfect. Reflections from the non-totally absorbent boundaries, acoustical

coupling between the microphones, and electrical noise and background noise are some of the

contaminating agents of the free field. All these phenomena have a well defined time occurrence.

Thus, it is possible under certain conditions to remove these if a time-frequency transformation is

carried out. The mutual equivalence between frequency response and impulse response can be used

for isolating the direct wave between the microphones from the contaminating phenomena. A

condition that must be fulfilled if the time-frequency transformation is to be applied is that the

frequency response should be known at all frequencies. However, when a reciprocity calibration is

carried out, it is not possible to measure the complex frequency response (electrical transfer

impedance between the two microphones) at all frequencies. Thus, a procedure for completing the

frequency response is described and studied in this chapter. Once the frequency response is

complete, the impulse response can be obtained using the inverse Fourier transform. The direct

wave between the two microphones can be isolated from the rest of the impulse response by means

of a time selective window. Then, the cleaned frequency response can be obtained by applying the

Fourier transform. The whole procedure involves thus a completion of the contaminated frequency

response followed by an inverse Fourier transformation. Once in the time domain a time window

is applied, and thereafter a Fourier transformation returning to the frequency domain where the final

product is a clean frequency response. However, the procedure has an overall effect that is present

in the clean frequency response and may show a deviation from the true value of the frequency

response. The cleaning procedure and its overall effect are studied in this chapter.

3.1 Motivation for using the time-frequency transform

In the preceding chapters a general panorama of free-field calibration of microphones has

been described. This ranges from the theoretical background that gives shape to the reciprocity

calibration to the artificial realisation of the free field by means of an anechoic chamber, and the

setting up of a measurement system that is able to measure the required quantities. At this point,

some limitations can be pointed out as well, and it is desirable to bring to the analysis two of them.


38

One is the difficulty of measuring a complete transfer impedance between the two microphones.

The second is the imperfect realisation of the free field.

The difficulties related to the measurement of the transfer impedance function are inherent

to the microphones themselves: poor low-frequency radiation and low-frequency (1/f) noise

(references [27], and [35]), and to the measurement set-up that may have a limited frequency range.

The last can be overcome by using or developing a system that can measure higher frequencies. The

first is unavoidable because it comes together with the very physical realisation of the transducer.

The imperfection of the anechoic chamber where the free field is realised is due to the fact

that the absorption coefficient of the wedges that cover the walls is nearly, but never, one. This

implies that a portion of the acoustic energy will be reflected back to the supposedly freely

propagating wave, invalidating the free field assumption. This “invalidation” is, for the immense

majority of applications, negligible if some criteria are met. However, it may have a significant effect

when the sensitivity of microphones is to be determined. It could be argued that the effect of the

reflections could be minimised by placing the walls far enough so as to have the amplitudes of the

reflections decreased by the inverse distance law and the effect of the air absorption. Although it

seems to be a solution, it would introduce two new problems, one of economical nature: it is

expensive to build very large anechoic rooms; and the other related with the measurement: the

longer the cables, the greater the difficulty of measuring the electrical voltages, especially the output

voltage of the receiver microphone, and the greater the presence of the cross-talk problem.

And there is an “unforeseen” problem caused by the fact that the microphone has

dimensions that may not be negligible when compared with the wavelength of the propagating

sound. It can be argued that the finite size of the condenser microphone is taken into account when

introducing the diffraction factor in the analysis. This is also correct. But the diffraction factor does

not prevent the two microphones to interact with each other, especially when they are front to front

(normal incidence), where the diaphragms are parallel; this interaction results in a standing wave

between the two microphones. The amplitude of this standing wave will depend on the distance

between the microphones, and it is expected that it may be frequency dependent because the

directivity pattern of the microphones becomes more concentrated in the front of the microphone


39

pPde p

PdeA

A

jkdB

B

jkdA B= =− −0 0; . (3.1)

d

A

dA

dB

B

dFW dFW

Source

MirrorSource

dw

Figure 3.1 Schematics of the problem of a spherical wave reflected by a wall behind two transducers

as the frequency increases, or as the wavelength gets shorter compared to the microphone’s

dimensions. This standing wave will have an effect similar to the reflections from the walls.

The result of these unwanted disturbances is to be reflected on the measured electrical

transfer impedance between the microphones, and thus on the free-field sensitivity of the

microphones. This means a high uncertainty of the sensitivity. If a lower uncertainty is to be

achieved, these disturbances must be removed from the measured transfer impedance.

The expected effect of the reflections should be predicted in order to propose a suitable

solution for the problem. Consider two microphones located at a distance d from each other in an

acoustic field where a spherical wave propagates in the positive x direction. Behind microphone B,

there is a perfectly rigid wall. Figure 3.1 shows the schematics of the problem.

The wavefront will first reach microphone A, then microphone B, and then it will reach the

wall where it will be reflected and will go back reaching microphone B, and then microphone A. The

wave at the position of the microphones in the absence of any reflection will have an amplitude

The transfer function between these two points is given by the ratio of pB to pA. Additionally, the

sound pressure pB can be written in terms of the pressure at A, and the distance d, between the two

points,


40

( ) ( )( )

( )H f

dd e

dd e

dd e

B

A

jk d d

B

B

jk d d

A

A

jk d d

B A

B B

A A

=+

′

+′

− −

− ′

− ′

1

1. (3.4)

( )

( )

Hpp

Pd d e

Pd e

dd d e

B

A

A

jk d d

A

jkd

A

A

jkd

A

A

ω = =+

=+

− +

−

0

0. (3.2)

pde

de

pde

de

AA

jkd

A

jkd

BB

jkd

B

jkd

A A

B B

= +′

= +′

− − ′

− − ′

1 1

1 1

,

,(3.3)

It can be seen that the transfer function shows a phase delay between the two microphones. This

phase delay corresponds to the time it takes the wave to travel the separation distance between the

two points. Additionally, there is a scaling factor which is a function of the distance between the

points A and B. It corresponds to a decrease of the pressure as the distance between the two points

increases.

If a reflection from a wall located behind the point B at a distance dw is introduced by

assuming that a mirror source is behind the wall, the sound pressure at each point is

where d’A and d’B are the distances from the mirror source to the points A and B, respectively. The

transfer function between the two points is then

Again, there is a term containing the ratio of pressures of the direct wave, and the delay incurred

when the wave travels from A to B. Additionally, there is a term introduced by the interference

pattern of the mirror source and the direct wave. The ratio of the distances to the primary source

and the mirror source acts only as a scaling factor in the denominator and the numerator of the ratio

in equation (3.4). However, if there is any difference between these distances, an additional phase

delay will be present which will cause the interference pattern itself. Figure (3.2) shows the modulus

of a frequency response obtained using equation (3.4), with distances dA = 5; dA = 6; d’A = 10; and

d’B = 9, for values of k from 0 to 20.


41

0 5 10 15 20−15

−10

−5

0

5

10

15

k

Mod

ulus

(dB

)

Figure 3.2 The transfer function between the sound pressure at two points generated by an point source.

–––– The transfer function when a mirror source is introduced corresponding to a situation where a

reflection form a wall occurs, and – – – – the transfer function when there is no mirror source.

( ) ( )x t X f e dfj ft=−∞

∞

∫ 2π . (3.5)

It can be seen in figure 3.2 that the modulus is not necessarily symmetrical over the transfer

function in reflectionless conditions. This makes it difficult to assess any linear regression or a

similar estimate of the reflectionless frequency response calculated using the interfered transfer

function. On the other hand, a simple calculation of the frequencies of the disturbances should

indicate the delays and the relative positions of the primary and mirror sources as well. This may

complicate if a number of additional sources are present.

Another possibility for analysing the function is to transform it to the time domain. This can

be done by applying the inverse Fourier transform that is defined in any signal analysis textbook (see

reference [37]) as

The impulse response of a complex frequency response is given by


42

( ) ( )( )

( )H f

dd e

dd e

dd e

ABA

B

A

jk d d

B

B

jk d d

A

A

jk d d

B A

B B

A A

=+

′

+′

=++

− −

− ′

− ′

1

1

11' . (3.7)

( )( )( )

11

1 1

1

2 3

2 2 3 3

++

= + − + − +

= + − − + + − − +

BA

B A A A

B A BA A BA A BA

...

... .(3.8)

( ) ( )h t H f e dfj ft=−∞

∞

∫ 2π . (3.6)

( ) ( )H f A B A BA A BA A BA

A A B A A A BA A A A BA A A A BA

= + − − + + − − +

= + − − + + − − +

' ...

' ' ' ' ' ' ' ' ...

1 2 2 3 3

2 2 3 3(3.9)

( )h t a a b a a a b a a a a b a

a a a b a

= ′ + ′∗ − ′∗ − ′∗ ∗ + ′∗ + ′∗ ∗ −

′∗ − ′∗ ∗ +

2 2

3 3 K(3.10)

The calculation of the inverse Fourier transform of the function described in equation(3.4)

can be done using routine procedures that implements equation (3.5). However, a simple analysis

of the problem and a description of the expected results can be made to explain the results. For

example, the transfer function – equation (3.4) – can be simplified and expressed as

The rational term can be expanded as

Thus, the complete frequency response function is

If the inverse Fourier transform is applied, the result will be a sum of convolutions corresponding

to each product in equation (3.8),

This by itself does not provide any information; it is just a consequence of the properties of the

Fourier transform. However, recalling that the functions of frequency represented by the symbols

in equation (3.8) are complex exponentials, the corresponding Fourier transform pair is a delta

function with a given delay, i.e., thus . Considering this, andA en jnkd A= − , ( )a t nd cnA= −δ

substituting it in equation (3.10), the result is


43

( ) ( ) ( ) ( )[ ]( ) ( )[ ]( ) ( )[ ] ( )[ ]( ) ( )[ ]( ) ( )[ ] ( )[ ]

h t K t d c K t d c K t d d c

K t d c K t d d c

K t d c K t d d c K t d d c

K t d c K t d d c

K t d c K t d d c K t d d c

A A B B B

A A A A

A B B B A A A

A A A A

A B B B A A A

= − + − − −

− − − −

− − − − − −

+ − − −

+ − − − − −

' '

'

'

'

'

* '

* '

* ' * '

* '

* ' * ' .

δ δ δ

δ δ

δ δ δ

δ δ

δ δ δ

2

2 2

(3.11)

It can be seen that the impulse response is a series of delta functions that are located at

different time instants corresponding to the differences in the distance travelled by the direct and

reflected waves. In the microphone case the output of a source and the response of the receiver may

be characterised as single degree of freedom systems. The reflections from the wall as well will have

a frequency dependence. Therefore, it is expected to find that the direct wave and the secondary

components will not be delta functions, but the shape of an exponentially decaying sinusoidal

instead.

It is clear that, when there is only a portion of the impulse response that is interesting for

the analysis, it can be isolated or extracted from the whole impulse response by applying a sort of

time selective technique, i.e., a time selective window.

Time selective techniques have been applied extensively in very different fields under

different forms. For example, in the electro magnetics field, the analysis of the performance of

antennas is of interest because there are some similarities with the microphone calibration, especially

the fact that measurements are also made in reflection free chambers and reflections form them may

disturb the measurement of the characteristics of the antenna. An interesting work in this field was

carried out by Williams et al. [28], who developed a method for eliminating the scattered field from

the edges of a finite ground plane. This technique consisted of the application of time gating and

subtracting a theoretically calculated interference pattern from the edges of a finite ground. In this

case, the time gating was successful when used for eliminating the reflections from the walls of the

anechoic chamber, and the noise in the chamber, providing that the bandwidth of the pulse used was

broad enough as to yield a sufficient time resolution. Novotny et al. [29] used a similar technique for

evaluating the free-field response of a broadband absorber.


44

In the past, time selective techniques have also been applied to the free-field calibration of

condenser microphones. Terry and Watson [30] presented a technique where modulated pulses were

used for generating transient signals that were conveniently gated in a non-anechoic environment.

At that time, the measurements of the pulse amplitude were made graphically, with the consequent

lack of accuracy. No consideration of the possible presence of non linearity in the behaviour of the

microphones because of the use of the impulses was made. Later, Niemoeller [31] based on the fact

that the low frequency transfer impedance between two microphones is very difficult to measure,

worked in the time domain instead, and presented a very elaborate technique for determining the

impulse response of a condenser microphone by successive approximations on a measured impulse

response, making use of recursive equations and the least squares method. This approximated

impulse response was then convolved with a similar one, corresponding to a second microphone.

This should be equivalent to the reciprocity calibration in the frequency domain where the

sensitivities of the microphone are multiplied in the transfer impedance. However, the technique

showed some drawbacks caused by the fact that the microphones may not be similar to each other,

leading to large approximation errors.

More recent work has been done by Lambert and Durocher [32], and Blem [33], who

explored the possibility of removing unwanted reflections by applying a time selective window using

a Fourier transform based procedure. In both cases, the complex transfer impedance between the

two microphones was measured using pure tones at regularly spaced frequencies, then the impulse

response was obtained, and a time window was applied onto it for removing possible reflections.

Vorländer [34] also proposed the application of a time selective window to the impulse response,

which was obtained by the use of broad band signals in the time domain by using the Hadamard

transform.

References [28], [29], [30], and [31] have in common the use of impulsive input signals and

the use of time selective windows. While in the antenna’s case, linearity problems may not be of

concern, they are in the case of the condenser microphones. In references [32] [33], and [34] periodic

signals are used, which may help to keep the performance of the microphones in the linear range.


45

Frequency

Measured Complex Frequency Response

0

IFFT⇒

Time

0

Direct Impulse response Reflections

Time

Time selective window

FFT ⇒

Frequency

Cleaned Complex Frequency Response

0

Figure 3.3 Schematics of the procedure for removal of reflections of a complex frequency response

function.

In this chapter, a Fourier transform based solution for removing the imperfections caused

by the reflections from the walls of the anechoic chamber and the standing waves between the

microphones on the electrical transfer impedance is proposed. The proposed procedure consists of

three major stages: a) a treatment of the frequency response, b) the determination of the impulse

response and the application of a time selective window, and c) the calculation of the cleaned

frequency response. It is possible to measure the electrical transfer impedance at equidistantly

placed frequencies in a finite frequency range. If any Fourier transformation is to be applied, the

missing portions of the frequency response must be completed by an adequate procedure. Then, the

inverse Fourier transform is applied to the completed frequency response in order to obtain the

impulse response. It contains information of the direct wave between the microphones as well as

of the reflections and standing waves. These undesired effects can be removed by means of a time

selective window. Finally, the cleaned electrical transfer impedance can be obtained by applying the

Fourier transform. This result can be used for determining the acoustic centres of the microphones

and the free-field sensitivity of the condenser microphone. A schematic representation of the

procedure is shown in figure 3.3


46

The overall effect of the cleaning procedure on the accuracy of the microphone sensitivity

can be studied by making use of a simulated transfer impedance between the two microphones.

These results are used for validating the application of the time selective technique by comparing

them with complementary experimental and numerical results.

3.2 Treatment of the frequency response

The frequency response must be complete in the whole frequency domain before the inverse

Fourier transform is applied. This means that the low-frequency portion [0, f0] and the high

frequency portion [fmax, +4] must be generated in such a way that follows the physical behaviour of

the function but does not modify the measured electrical transfer impedance function. The

procedure is shown schematically in figure 3.4.

However, in practice, it is only possible to measure the electrical transfer impedance in a

limited frequency range. There is a low frequency limit, f0 , below which is impossible to measure the

electrical transfer impedance. This is caused by the microphone’s thermal noise (see references [35],

and [27]). When combined with the microphone’s very low radiation capabilities, this results in an

extremely poor signal-to-noise ratio at low frequencies. On the high frequency side, the limitations

come from the capacity of the data acquisition system, which makes it possible to measure only up

to a frequency fmax where the electrical transfer impedance may not have decayed sufficiently.

If any Fourier transform based post-processing is to be performed on the electrical transfer

impedance, it has to be defined in the whole frequency interval, formally from -4 to +4 as in

equation (3.5), or from 0 to +4 for a one-sided frequency response. Then, a treatment of the missing

portions of the frequency response must be carried out.

3.2.1 The low frequency patching

The missing low frequency values can be completed by patching ideal values to the measured

of the electrical transfer impedance. This is analogous to the time domain solution presented in

reference [31], where the impulse response is approximated using a recursive procedure. The

solution proposed here is based on the fact that at low frequencies, diffraction and radiation


47

Frequency Domain

Frequency Frequency Frequency

Measured Complex Frequency Response Frequency Window Windowed Complex

Frequency Response 1.0

00 0 fmax fmax fmaxfmin fmin

× ⇒ Frequency

Extended Complex Frequency Response

fmaxfmin fmin

⇒

Figure 3.4 Schematics of the procedure for the treatment of an incomplete frequency response prior to the Fourier

analysis.

( )M M S fZ

Z Zf pa

a a r

=+

, .,

θ (3.12)

( ) ( ) ( )Z j fdM S f

ZZ Z

M S fZ

Z Zetr rec p tr tr

a tr

a tr a tr rp rec rec

a rec

a rec a rec r

j t kd, ,

,

, , ,,

,

, , ,

, , .= ⋅+

⋅+

⋅ −ρ θ θ ω

2(3.13)

( ) ( )Z jfd M M S f

ZZ Z etr rec p tr p rec

a

a a r

j t kd, , ,

,, .= ⋅ ⋅ ⋅

+

⋅ −ρ

θ ω

22

2

(3.14)

impedance effects are almost negligible, and the free-field sensitivity becomes constant and almost

equal to the pressure sensitivity.

The free field sensitivity can be expressed in terms of the product of the pressure sensitivity,

the diffraction factor, and the ratio of the diaphragm acoustic impedance to the loaded impedance

(see chapter 2),

Substituting it into the equation of the electrical transfer impedance yields

Considering that the acoustic impedance, the radiation impedance and the geometry are similar,

within close limits when the microphones are of the same type, equation (13) can be rewritten as

At sufficiently low frequencies, where the wavelength becomes longer compared to the size of the

microphone, the geometrical diffraction factor, S(f,θ), becomes very close to one. At the same time,

the radiation impedance becomes very small compared to the microphone impedance; thus the ratio

of the acoustic impedance to the sum of the diaphragm impedance plus the radiation impedance

becomes nearly one. Thus, at these frequencies, the transfer impedance can be expressed as:


48

( ) ( )( )Z f

Z f f fZ f f f fe Ee i

e m, ,

, ,

, , max

,1212 0

12

0=

≤ ≤≤ ≤

0

(3.16)

( )Z jfd M M etr rec k p tr p rec

j t kd, , , .

<<

−≈ ⋅ ⋅ ⋅1 2

ρ ω (3.15)

Thus, a lumped-parameter model (see reference [24]) can be used for generating the ideal electrical

transfer impedance, Ze,12,i(f). The lumped parameters can be determined from actual pressure

sensitivity data. The resulting patched electrical transfer impedance, Ze,12,E(f), is defined as

whereZe,12,m(f) is the measured electrical transfer impedance.

One of the possible shortcomings of this procedure is that the effect of the reflections is

larger at low frequencies. As the procedure uses an ideal, reflection less frequency response for filling

the missing part, a realistic representation of the disturbances may not be obtained, though ultimately

the reflections are to be removed and no information may be extracted from them, except on the

case of the standing wave between the microphones.

Furthermore, the patching of ideal data to measured data may introduce an additional

problem. The value of the electrical transfer impedance at the lowest measured frequency, f0, may

correspond either to a maximum or a minimum of the standing wave pattern. This may introduce

a discontinuity on the slope of the extended electrical transfer impedance function that may be

reflected onto the impulse response. In order to reduce this discontinuity, some further processing

could be performed. For example, a localised average of the ideal and the measured response could

be carried out. Even some recursive patching can be attempted by using “cleaned” frequency

response data, for assessing in a better way the most likely value of the frequency response at the

patching point. In any case, the effect of the slope discontinuity should be assessed.

Another procedure for extending the low frequency portion of the frequency response may

be to extrapolate it. However, this approach suffers from the same drawbacks as the patching

procedure, because the discontinuity of the slope will remain.


49

( ) ( )( )Z f

Z f f fZ f f f fe E Ne i N

e m N, , ,

, , ,

, , , max12

12 0

12

0=

≤ ≤≤ ≤

. 0

(3.18)

( )Z Z A f em E m E Nj f

12 12 00

, , , , , .= θ (3.19)

( ) ( )

( ) ( )

Z fA

A e

Z fA

A e

e mNm

m f f

j

e iNi

i f f

j

m m f f

i i f f

, ,,

, ,,

,

,

,

,

12

12

0

0

0

0

=

=

=

− −

=

− −

=

=

θ θ

θ θ

(3.17)

The next issue to be examined is how the ideal transfer impedance is to be patched to the

measured one. Consider that the measured complex transfer impedance, Z12,m(f) is defined in the

interval [f0,ff], and the ideal transfer impedance function, Z12,i(f) is defined in the interval [0,f0]. This

is to be patched to the measured transfer impedance function at f = f0. It is very likely that there is

a difference between the measured and the ideal transfer impedance functions, thus, a simple

continuation of the measured to the ideal is not possible. In order to solve this problem, a

normalisation of the two functions at a given frequency must be made. Then the ideal function

should be patched to the measured function at the normalisation frequency, where the two

normalised functions have value unity. The patching then should be followed by a further de-

normalisation.

The normalisation of the measured and ideal transfer impedances to the corresponding

complex value of the function at f = f0 is defined as

where the subscript N means normalised, A is the amplitude of the function at a given frequency

and θ is the phase. It can be seen that at f = f0, the normalised functions have the unity value, and

zero phase.

At this point, it is possible to patch the normalised ideal function to the normalised

measured function. Then, the extended or patched normalised measured function is defined as

Then, the de-normalisation is carried out by multiplying the patched function by the complex value

normalisation value Ze,12(f0),


50

( ) ( ) ( )Z f Z f L fe L e E, , , ,12 12= ⋅ , (3.20)

It should be noticed that the complex normalisation eliminates the possibility of a phase

discontinuity on the patched function. This discontinuity may be present if a modulus-only

normalisation is made.

At this point, there is an impedance function that contains the low frequency information.

The treatment of the high frequencies is to be described below.

3.2.2 Low-pass filtering of the electrical transfer impedance

The next step is to develop a procedure for dealing with the high frequency portion of the

frequency response, [fmax, +4]. This procedure should not violate or modify the physical nature of

the microphone-medium-microphone system.

It is well known that the sensitivity of a microphone tends to decay at high frequencies as

a consequence of the mechanical behaviour of the diaphragm. This is explained by the fact that

above the resonance frequency of the diaphragm, its movement is controlled by its mass, and

therefore the sensitivity approaches zero asymptotically. This behaviour is also reflected in the

electrical transfer impedance. This suggests the possibility of accelerating artificially this decay in

the measured frequency range, provided that the upper frequency is well above the resonance

frequency of the microphones. This can be done with a low-pass filter. The low-pass filtered

electrical transfer impedance, Ze,12,L(f) is defined by

where L(f) is the low-pass filter. At first, it may be thought that this filter should not alter the

modulus of the electrical transfer impedance in the frequency range of concern, and therefore,

ideally, it should have a value of unity in the interval [0, ff] and zero in the rest of the whole

frequency range. However, an ideal filter with these characteristics will introduce a non-causality

problem on the impulse response. This may be a problem if any cross-talk is to be removed; cross-

talk will be present at time zero because it is an electrical problem. Therefore, a realistic filter with

linear phase should be used. Such a filter can be developed by means of the techniques described

in several digital signal processing textbooks, for example in reference [36].


51

( ) ( ) ( )h t h t l t' ,= ∗ (3.21)

Causality is not the only effect associated with the use of the filter in the frequency domain.

The shape of the impulse response will also be affected by the filtering. The true impulse response

is obtained when the complex frequency response is obtained as defined in equation (?), this requires

knowing the whole frequency response. It has been explained that the limitations of the

measurement set-up may hinder measuring above a given frequency, and thus the result will be an

incomplete frequency response. If a filter is applied, as suggested in equation (19), the resulting

impulse response will be the convolution of the true impulse response with the impulse response

of the filter,

where h’(t) is the calculated impulse response, and l(t) is the filter impulse response. Thus, the width

of the filter’s impulse response will have an influence on the width of the calculated impulse

response. It is relevant if the impulse response is to be separated from other portions which are not

part of the direct wave between the two microphones. Then, it is important to design the frequency

window in such a way that it minimises the widening of the determined impulse response.

The width of the calculated impulse response is a function of two quantities associated with

the filter impulse response. One is the width of the main lobe, and the second is the attenuation of

the secondary side lobes. These two quantities are not independent but inversely proportional. The

larger the attenuation of the side lobes, the wider the main lobe.

On the other hand, the effective bandwidth of the frequency response is an important

parameter. The effective bandwidth is the frequency interval that contains sufficient information for

characterising the system in terms of resonance frequency, damping, compliance, etcetera. A normal

practice with realisable physical systems is to define the half-power point bandwidth [37]. This frequency

band is centred around the resonance frequency, and it is a function of the damping and resonance

frequency, thus of the mass and stiffness of the system. This can be used as a reference. Thus, the

cut-off parameters of the frequency filter should be chosen considering these limits.

Additionally, some of the contaminating effects of the frequency response may have a

different frequency distribution. For example, the standing wave between the microphones may be

highly concentrated at high frequencies, where the wavelength is comparable with the diameter of


52

( ) ( )z t Z f e dfe ej ft

, , .12 122=

−∞

∞

∫ π (3.23)

( ) ( ) ( ) ( ) ( )Z f z t z t Z feIDFT

ew t

e wDFT

e C, , , , , ,12 12 12 12 → → → . (3.22)

the transducer, thus, if this effect is to be removed effectively, it must be properly represented in the

processing.

Once the frequency response is filtered, the inverse Fourier transform can be applied in

order to obtain its impulse response

3.3 The cleaning procedure using the Fourier transform

The reflections from the walls of the anechoic chamber and the standing waves between

the microphones can be eliminated from the electrical transfer impedance, Ze,12(f), using the

following procedure. The impulse response of the electrical transfer impedance, ze,12(t), can be

calculated by applying the inverse Fourier transform (IFT). In the time domain, the effect of the

reflections appears as scaled and perhaps distorted replicas delayed from the main impulse response.

Thus a time selective window, w(t), may be applied to remove them. Then the “cleaned” electrical

transfer impedance, Ze,12,C(f) can be calculated by applying the Fourier Transform to the windowed

impulse response, ze,12,w(t). The procedure graphically shown in figure 3.3 can be formalised as

As mentioned before, the frequency response must be known at all frequencies. A solution

for this problem has been proposed in section 3.2. Now, the cleaning procedure outlined in equation

(3.22) can be studied.

3.3.1 The impulse response

The first step is the application of the inverse Fourier transform to the electrical transfer

impedance that has already been processed by extending it at low frequencies, and by applying a low-

pass filter. According to equation (3.6), this operation gives as a result the impulse response

As the electrical transfer impedance has been low-pass filtered and artificially extended at low

frequencies – equations (3.16) and (3.20) –, any distortion introduced by such operations can be

observed in the impulse response. The effect of the discontinuity introduced by the operation


53

defined in equation (3.16) is difficult to assess, but provided that the level where the discontinuity

appears is much lower than the maximum value of the frequency response, it may be neglected. On

the other hand, the low-pass filtering of the frequency response corresponds to a convolution of the

impulse responses in the time domain. Thus, it will introduce a modulation of the impulse response.

This modulation will in practice increase the duration of the obtained impulse response.

However, it is difficult to assess the actual effect on the impulse response in a quantitative

manner. The size of the discontinuity introduced by the patching of the low frequency portion will

depend on the change of slope, and the relative size of the discontinuity compared to the maximum

of the frequency response.

Furthermore, a time selective window will be applied to the impulse response for removing

the unwanted reflections from the walls, electrical noise (cross-talk), and the standing wave between

the microphones. This implies a multiplication of the impulse response with a window function. As

the windowed impulse response is to be transformed back to the frequency domain, such a

multiplication corresponds to a convolution in the frequency domain. If the window truncates the

impulse response at an instant where it has not decayed sufficiently, a ripple may be introduced in

the resulting frequency response. This ripple is difficult to quantify. Thus, an alternative procedure

for quantifying the effects is proposed.

This alternative procedure is to apply the cleaning procedure onto a simulated electrical

transfer impedance function. This is to be studied later in this chapter. First, the windowing

procedure is analysed below.

3.3.2 The time selective window

Once the electrical transfer impedance has been processed, the inverse Fourier transform

has been applied and the impulse response, ze,12(t), has been obtained. As deduced from equation

(3.11), this impulse response contains information about the direct wave between the two

microphones, the standing wave between the microphones, the reflections from the walls of the

anechoic room, and electrical noise in the form of cross-talk. In order to remove the two last effects,

a time selective window, w(t), can be applied. This time selective window must not modify the direct


54

( ) ( ) ( )f t f t w tτ τ= ⋅ , (3.24)

( ) ( ) ( ) ( ) ( )F f t w t e dt F W y dywj tω ω ωω

τ

τ

= ⋅ ⇔ −−

− −∞

∞

∫ ∫ . (3.25)

( ) ( )m W d w221

20= = − ′′

−∞

∞

∫πω ω ω , (3.26)

wave between the microphones, and must eliminate completely the reflections, which can be

considered as attenuated, delayed replicas of the direct wave. Therefore, the time selective window

should have value of unity in the time interval where the impulse response contains the direct wave,

and zero everywhere else. The first choice may be a rectangular window. However, as this window

has high side lobes, it is expected that it will introduce some ripple in the frequency response caused

by cutting abruptly the impulse response where it has not decayed completely to zero. This effect

can be reduced by applying a smoothing function on the extremes of the time window. A window

with smooth extremes can be generated by convolving the rectangular window with a smoothing

function. Although the extremes of the window can be considerably smoothed by the convolution,

the height of the first side lobe is approximately the same as that of the rectangular window. The

properties of the time selective window are discussed below.

The time windows are used for determining the Fourier transform of a continuous signal

of infinite duration, f(t), in terms of a finite segment of such signal, fτ(t)

where wτ is the window function, and the subscript τ indicates the finite nature of the corresponding

function. The application of the Fourier transform to the signal segment will yield an approximation,

Fw(w), of the Fourier transform of the infinite signal, F(ω). The difference between these two

quantities can be considered as a windowing error. Furthermore, the windowing operation in the

time domain corresponds to a convolution in the frequency domain,

Thus, it is clear that in order to minimise the difference between the windowed and the exact Fourier

transform, the transform of the window should be very short, and the length of the window large.

This is shown in figure 3.5.

This can be analysed following a development described by Papoulis [38]. He maintains that

minimisation of the second amplitude moment, m2, of the window, w(t),


55

Time Frequency

Time Frequency

h(t)

h(t)

w(t)

w(t)

W(f)

W(f)

H(f)

H(f)

Figure 3.5 Effect of the length of the time window w(t) on the estimated frequency

response H(f). The shorter the window, the wider its lobes. This may increase the

difference between the actual and the estimated frequency responses.

( ) ( ) ( ) ( ) ( ) ( ) ( )g t m f t m f tmf t m F j m F m F= − ′ + ′′ + ⇔ − − +0 1

20 1

222

... ...ω ω ω ω ω (3.27)

leads to a minimisation of the windowing error. In equation (3.26) w=>(0) is the second derivative of

the window function evaluated at t = 0.

The response of a system, g(t), to an arbitrary input, f(t) can be expressed in terms of the

derivatives of f(t) and the moments mn

Equation (3.27) shows the moment expansion in the time and frequency domain. Applying this

expansion to the truncation error , it results in( ) ( )F Fw ω ω−


56

( )( )

11 4

122 2 2 2π

πτ τ

πτ

τπτω

π τ ωτsin cos

cos,t

tt p t+ −

⇔

+

−(3.30)

( ) ( ) ( )F F Fw ω ωπτ

ω− ≈ ′′2

22. (3.31)

( ) ( ) ( ) ( ) ( )( )

F F F y W y dy FF

mw ω ωπ

ω ωω

− = − − ≈′′

−∞

∞

∫1

2 2 2. (3.28)

( ) ( ) ( ) ( )F FF y

y W y dyw ω ωω

π− =

′′ −

−∞

∞

∫0 2

4, (3.29)

This indicates that the error can be minimised if m2 is minimised. However, a non-trivial

minimisation only occurs if F(ω) is a smooth function, and m2 is not zero. The minimisation of m2

leads to a minimum error is a positive window is used. If W(ω) is greater or equal to zero, the

truncation error is

where y0 is a constant of the order 1/τ, the inverse of the window length. Then, for a large τ,

. If these assumptions are fulfilled, the error is minimum if m2 is minimum.( ) ( )′′ − ≈ ′′F y Fω ω0

Based on the above development, Papoulis proposed the use of a window

where pτ(t) is a function having unity value on the whole length of the time window, i.e., a

rectangular window. This window is optimal in the sense that it minimises the error

However, it should be noticed that one of the conditions mentioned above is that the length of the

time window should be large enough to approximate . In the case of isolating( ) ( )′′ − ≈ ′′F y Fω ω0

the direct impulse response between two microphones from the reflections and electrical noise

contained in the whole impulse response, the length of the window may be about few miliseconds.

This may invalidate the solution proposed by Papoulis.

Kwon[39] used Papoulis’ formulation for obtaining a minimum error window by minimising

higher order moments of the window and not only the second moment. The solution he proposes

implicitly contains the condition that the length of the window should be long enough to

approximate . However, one of the conclusions of that work is that the shape( ) ( )′′ − ≈ ′′F y Fω ω0


57

( ) ( ) ( )w t w w t dTukey rect Hann= −∫ τ τ τ . (3.32)

( ) ( )w nn N

rect =≤ ≤ −

1 0 10 elsewhere,

α(3.33)

( )w nn

Nn N

Hanning = −

≤ ≤

12

1 22

0 2

0

cos πα

α

elsewhere. (3.34)

( )

( )( )

( ) ( )w n

n Nn N

NN n NTukey =

≤ ≤

+− +

−

≤ ≤

1 212

11 2

1 22 2

0

0 1+

1+

elsewhere.

αα

απ αcos (3.35)

of the window is like a Tukey window, i. e., it has a flat portion like a rectangular window, with

smoothed extremes like a Hanning window.

Recently, Sjöström[40] considered the use of several windows when time gating impulse

responses for removing noise from a longitudinal beam transfer function. The results show that the

Tukey window is a feasible choice when the bias error is to be minimised.

The Tukey window has been extensively described in the literature (see for example Harris

[41], and Papoulis[38]). It can be thought of as the product of the convolution of a rectangular

window and a Hanning window (a raised cosine)

The smooth ends are introduced by the raised cosine of the Hanning window. The smoothing

portion of the convolved window can be associated with the quantity α that can take values between

0 and 1; at each end of the convolved window the smoothing portion is α/2. Considering that the

length of the window is N, N being a power of two in the discrete time, the expressions of the

rectangular and the cosine lobe windows are, respectively:

Considering that the result is symmetric around the vertical axis, the convolution of these two

functions is:


58

( ) ( )( )

( ) ( )w nnN

nN n NBlackman = +

+

≤ ≤12

0 42 05 22

0 082 2

2 0 2. . cos . cos .π

απ

α α

0 elsewhere

(3.36)

( ) ( )[ ] ( )

w na

nN

a n NKaiser =

−

≤ ≤

I

I

elsewhere,

0

0

πα

πα

12 2

0 2 2

0

2

(3.37)

That is the Tukey window. Figure 3.6(a) presents the shape of this window for different values of

the smoothing portion, α. Figure 3.6(b) shows the modulus of the Fourier transforms of the

windows. All for windows with a length, N, of 64 terms in the discrete time domain.

The convolution procedure can also be applied using other smoothing functions such as

those related with Kaiser or Blackman windows that have a better side lobe attenuation (see

reference [41]). However, the height of the side lobes in the convolved window depends on the

length the flattened portion. This is an important consideration: As one of the requirements for the

window is to have a processing gain almost equal to 1, the flat portion of the window must cover

the most of the impulse response. Another window that can be generated using the same procedure

uses as a smoothing function the Blackman window, which is defined as

Figure 3.7 shows the windows for different values of α, and their corresponding Fourier

transform.

Finally, the last window considered is generated using the Kaiser window. The Kaiser

window is defined in terms of the Bessel functions, and this gives the special advantage of choosing

the main lobe width in terms of the roots of the Bessel Function involved. The function for this

window around zero is:

where I0 is the zero order modified Bessel function of first kind and the product πa is half of the

time-bandwidth product. Figure 3.8 shows the windows for different values of α, and their

corresponding Fourier transforms.


59

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Length

Am

plitu

dea)

0 0.1 0.2 0.3 0.4 0.5−70

−60

−50

−40

−30

−20

−10

0

Frequency in π units

Nor

mal

ised

am

plitu

de (

dB)

b)

Figure 3.6 a) Shape of the Tukey window in the discrete time domain as a function of the smoothing

portion, α, equally divided on the two extremes of the window for:

——— α = 0.25, – – – – α = 0.5, and – . – . – . – α = 0.75.

b) Normalised modulus of the Fourier transform of the Tukey window as a function of the smoothing

portion, α, equally divided on the two extremes of the window for: ——— α = 0.25, – – – – α

= 0.5, and – . – . – . – α = 0.75.


60

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Length

Am

plitu

dea)

0 0.1 0.2 0.3 0.4 0.5−70

−60

−50

−40

−30

−20

−10

0


Nor

mal

ised

am

plitu

de (

dB)

b)

Figure 3.7 a) Shape of the Blackman window convolved with a rectangular window in the

discrete time domain as a function of the smoothing portion, α, equally divided on the two

extremes of the window for: ——— α = 0.25, – – – – α = 0.5, and

– . – . – . – α = 0.75.

b) Normalised modulus of the Fourier transform of the Blackman window convolved with a

rectangular window as a function of the smoothing portion, α, equally divided on the two

extremes of the window for: ——— α = 0.25, – – – – α = 0.5,and – . – . – α = 0.75.


61

10 20 30 40 50 60

0.2

0.4

0.6

0.8

1

1.2

Length

Am

plitu

dea)

0 0.1 0.2 0.3 0.4 0.5−70

−60

−50

−40

−30

−20

−10

0


Am

plitu

de (

dB)

b)

Figure 3.8 a) Shape of the Kaiser window convolved with a rectangular window in the discrete time

domain as a function of the smoothing portion, α, equally divided on the two extremes of the window

for: ——— α = 0.25, – – – – α = 0.5, and – . – . – . – α = 0.75.

b) Normalised modulus of the Fourier transform of the Kaiser window convolved with a rectangular

window as a function of the smoothing portion, a, equally divided on the two extremes of the window

for: ——— α = 0.25, – – – – α = 0.5, and – . – . – α = 0.75.


62

In the preceding figures a common characteristic to all the convolved windows can be

noticed. It is that the height of the side lobes is mainly determined by the portion of the window that

corresponds to the rectangular window, i.e., 1-α. As the smoothing portion grows, the height of the

side lobes tends to be that of the smoothing window function. A linked effect is that the width of

the main lobe will also change as a function of the smoothing proportion as well. The larger the

smoothing portion, the wider the main lobe.

The effect described has a large influence on the selection of the smoothing portion to be

used in the window. It is clear that a large smoothing portion will improve the side lobe attenuation

but also give rise to a wider main lobe. Simultaneously, the enlargement of the smoothing portion

may create a window that modifies significantly the direct impulse response, thus modifying the

modulus of the frequency response in an unwanted manner. On the other hand, if the restrictions

introduced by the duration of the direct impulse response, and its closeness to the disturbances that

should be removed require using a window with a small smoothing portion, the dominance of the

rectangular portion will be such that the selection of the smoothing function will not have a strong

effect on the size of the side lobes nor on the width of the main lobe. This means that using a Tukey

or a Kaiser windows may not introduce a significant difference. However, the smoothed window

will be always better than the rectangular window itself. Figure 3.9 shows the shape of different

windows with the same smoothing portion, and the corresponding rectangular window. The above

effects can be visualised clearly in this figure.

3.4 The effect of the cleaning procedure

The combined effect of the low pass filtering of the frequency response followed by the

application of the time selective window to the impulse response is very difficult to assess, especially

from a quantitative point of view. The characteristics of the low-pass filter and the time selective

window may be well known, but their interaction with the actual impulse response cannot be

determined analytically for all cases. However, a heuristic determination can be made by means of

the application of the cleaning procedure to a simulated frequency response function that resembles

closely the electrical transfer impedance between the two microphones in a free field. The elements

that constitute such a function have been already determined in chapter two, and these can be used

for the purpose of the simulation.


63

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Window length

Am

plitu

dea)

0 0.2 0.4 0.6 0.8 1−70

−60

−50

−40

−30

−20

−10

0

Nor

mal

ised

mod

ulus

(dB

)

b)


Figure 3.9 Comparison of several windows having a smoothing portion α = 0.25. Figure a) shows

the different windows in the discrete time: ——— Tukey, — – — Chebyshev, – – – – Blackman,

and — . — Kaiser. Figure b) shows the normalised modulus of the Fourier transform of the above

windows, plus – * – Rectangular window of the same length.


64

Recalling that the electrical transfer impedance is defined when two microphones located

at an acoustical distance d12 from each other in a free field are considered. The electrical transfer

impedance between the microphones at the frequency f is defined (see for example reference [26])

as the ratio of the open-circuit voltage, u2, on the electrical terminals of the receiver microphone to

the electrical current, i1, through the electrical terminals of the second microphone acting as sound

source,

Zui

jfd

M M ee f fj d

, , ,122

1 121 22

12= = −ρ γ , (3.38)

where Mf,1 and Mf,2 are the free-field sensitivities of the microphones, ρ is the density of the medium

and γ is the complex propagation coefficient, which includes the effect of the air absorption.

The electrical transfer impedance is the basis of the reciprocity technique as it contains

information of the free-field sensitivities of the two microphones, thus providing means for

determining their sensitivities without the need of any reference but the electrical quantities

measured by a suitable technique.

Some interesting information regarding the shape of its impulse response can be extracted

from the factors in equation (3.38). It is possible to divide the electrical transfer impedance into

three different factors. The first is the frequency multiplied by a constant, the second is the product

of sensitivities and the third is the complex exponential of the product of acoustical distance and the

complex propagation coefficient. The shape of the impulse response of the electrical transfer

impedance is the convolution of the impulse responses of each factor.

The first term may be considered as a differentiation that emphasises high frequencies. The

second term can be considered as a multiplication of two systems of single degree of freedom. The

last term can be interpreted as a time delay. This indicates that the time varying characteristics of the

impulse response should mostly be associated with the variations in the parameters of the single

degree of freedom systems associated with the free-field sensitivities. This indicates that the

parameters of the time selective procedure should be selected taking such information as a basis.


65

τ =2mRa

a

, (3.39)

( )p j cQ kd ereflectmirror

reflect

j t kdreflect=−ρ

πω

4 , (3.41)

( )p j cQkd e

j t kd0 4= −ρ

πω , (3.40)

A quantity used for determining the time it takes the impulse response of a system of a single

degree of freedom to decay to an arbitrary level with respect to its maximum amplitude is the time

constant, t, which is given by

where ma is the acoustical mass and Ra is the acoustical resistance from the system of a single degree

of freedom.

Equation (3.38) is deduced considering that the two microphones are located in a free field,

i.e., an environment where the waves propagate freely away from the source. However, as a free field

cannot be perfectly realised, it is expected that the boundaries may reflect some energy back to the

microphones. These reflections can be considered to be coherent with the transmitter microphone.

This gives the possibility for considering them as being generated by additional mirror sources

located at each reflecting boundary, i.e., each wall.

If the space where the free field is realised is a rectangular room, it is possible to consider

six mirror sources, one for each wall. Thus the effect of these reflections can be obtained as a sum

of coherent mirror sources. The arrangement of the mirror sources associated with each reflection

from the walls is presented in figure 3.10.

The sound pressure, p0, generated in the free field by a microphone considered as a simple

source of strength, Q, at a distance d is (recalling equation (2.41):

where the source strength is defined in terms of the volume velocity through the diaphragm, q, and

the scattering factor S, as This can be extended to the sound pressure preflect generated( )Q qS f= , .θ

by a mirror source located at a distance dmirror from the primary source (see figure 3.11). This yields

where the distance dreflect is defined by .d d dreflect mirror2 2 2= +


66

dA

dB'

dB

dC'

dD'

dE'

dD

dC

dE

Figure 3.10 Arrangement of mirror sources inside the anechoic chamber when two microphones are

placed inside it. Microphone A acts as a sound source, then it is the origin of the mirror sources. There

are two additional mirror sources because the ceiling and the floor of the chamber are also considered.

d

dreflect

dB

dB'dmirror

Figure 3.11 Schematics of the arrangement of the transmitter microphone, the receiver

microphone and a mirror source.


67

pj ck Q

de

Qd

e ejkd mirror

reflect

jkd j treflect0 4

= +

− −ρπ

ω . (3.42)

p p jf Q

d e emm

m

m

jkd

m

j tm0 2= =

∑ ∑ −ρ ω . (3.43)

u M p M jf Q

d e ereceiver f receiver mm

f receiverm

m

jkd

m

j tm= =

∑ ∑ −

, , .ρ ω

2(3.44)

Considering that the sources are coherent, the total sound pressure at the point O is given

by the sum of the two sound sources

This can be generalised for a number, m, of sources as follows

When this sound pressure acts on the diaphragm of the receiver microphone, the open circuit

voltage on the electrical terminals is

This equation can be used for determining the electrical transfer impedance between the two

microphones when reflections from the boundaries are present.

It should be mentioned that Delany and Bazley [42] have pointed out that the total field at

any point inside the enclosure should be calculated by summing the six waves reflected from the

walls and the direct wave from the source with due regard of the relative phase of these waves, and

that their relative phase may change as a function of the impedance of the walls. This should give

a better agreement between the calculated and measured interference patterns. However, as it is not

intended to find the spatial interference pattern, but to use an approximation, the simple sum of the

coherent mirror sources described by equation (3.44) will be used.

It should also be noticed that reflections from the walls may not be the only effect that can

disturb the realisation of a free field. The coupling of the microphones and electrical noise may be

reflection-like disturbances; the standing wave between the microphones is clearly another reflection.

The cross talk and the background noise should be present at time zero. The first because it is

electrical in nature, thus travelling much faster than the sound wave, should be occurring at an

instant very close to zero, and perhaps having a duration that is function of the signal to noise ratio

in the frequency domain. There is a “noise” caused by random fluctuations introduced by the use


68

of a finite measurement time; the effect of this “noise” will be present at zero time if it is randomly

distributed all over the frequency response.

It should be recalled that the free field sensitivity of the microphone has a dependence on

the angle of the incident waves because of the diffraction factor. Therefore, if the mirror sources

have incidence angles other than the normal incidence (0°) the corresponding change of sensitivity

should be considered. This change would imply that, in equation (3.44), instead of using a single

free field sensitivity for the receiver microphone, the sound pressure coming form one of the mirror

sources with a given angle of incidence should be sensed by the receiver microphone with the

sensitivity corresponding to such angle of incidence, this means that there should be used as many

different sensitivities as mirror sources are present. On the other hand, the walls of the anechoic

chamber are formed by a number of wedges that will certainly modify the shape of the reflected

wave in an unpredictable manner. Thus, the phenomena may become very complicated. However,

in this case it is not very important to reproduce the exact shape of the reflections, but to indicate

the instants when they occur. Then, the simple expression in equation (3.44) is to be used in the

simulation of an electrical transfer impedance contaminated by reflections from boundaries of the

anechoic space.

A computer model based on equations (3.38) and (3.44) has been developed using

MATLAB. The product of sensitivities in the electrical transfer impedance is generated by making

use of the lumped-parameter model of the microphones and a typical free-field correction [43]. The

reflections from the imperfect anechoic chamber and the standing wave between the microphones

are introduced using the concept of image sources as described above. The cleaning procedure is

applied onto this simulated electrical transfer impedance in order to evaluate the effect of the

procedure as well as of the signal processing parameters used to generate it. The selection of such

parameters is described below.

The first aspect to be analysed is that of the duration of the direct wave between the

microphones. This can be estimated by making use of the lumped-parameter model of the

microphones and the time constant defined in equation (3.39).


69

LS1 (B&K 4160) LS2 (B&K 4180)

acoustic mass, ma 345 kgAm-4 750 kgAm-4

acoustic compliance, Ca 1.19H10-12 m3APa-1 0.068H10-12 m3APa-1

acoustic resistance, Ra 20H10-6 PaAsAm-3 120H10-6 PaAsAm-3

Table 3-1. Values of the lumped parameters for condenser microphones LS1P and LS2P

The microphones to be analysed are laboratory standard microphones: one-inch laboratory

standard microphones, LS1P. The above type correspond to the microphone Brüel & Kjær models

4160. The results from this study can be extended to other types of microphones as for instance the

half-inch laboratory standard, LS2P. Typical values given by the manufacturer for the lumped-

parameter model of the pressure sensitivity are shown in table 3.1.

The impulse response will have a peak at the instant corresponding approximately to the

time it takes by the sound wave to travel from the transmitter microphone to the receiver. After this

peak it will decay exponentially according to the time constant defined in equation (3.39). This

constant provides a measure of how rapidly the amplitude of an impulse response of a system of a

single degree of freedom decays following an exponential behaviour, , where A(t)( ) ( )A t A e t= −max

1 τ

is the instantaneous amplitude, Amax is the maximum amplitude, τ is the time constant and t is the

time. This equation can be used to predict the time when a given decay is expected to( )A t Amax

occur. Thus, for the parameters given in table 3.1, the time constant is 0.0345 ms. This value

indicates a fast decay of the impulse response. It should be noticed that equation (3.38) contains a

product of two systems of a single degree of freedom. Then, the resulting convolution has a time

constant that is two times the time constant of one system. Thus, using the calculated time constant,

the direct wave should have decayed to one thousandth of its maximum amplitude about 0.5 ms

after the peak has been reached. However, it is worth mentioning that it is expected that the actual

values of the lumped-parameter model may change under free-field conditions, due to the presence

of a radiation impedance. Such changes may not modify the above expectations in a significant

manner, though.


70

With the above result it is possible to determine whether the reflections form the walls and

the expected standing wave between the microphones will be far enough from the direct wave as

to avoid truncating it when applying the time selective window. Consider that the microphones are

located at a distance of 30 cm from each other in the middle of an anechoic room having free space

dimensions of 120 H 80 H 175 cm. At these conditions, the direct wave between the microphones

will have its peak about 0.87 ms, and the amplitude of the impulse response should decay to a one

thousandth of its maximum value at 1.37 ms.

On the other hand, the closest disturbance to the direct wave is the standing wave between

the microphones. The time instant where it should be appearing in the impulse response is about

2.6 ms. The rest of the reflections will come at a later time: first from the lateral walls and then from

the ceiling and the floor of the chamber. These two last come at a time of about 5.1 ms, the actual

reflection point on the walls is unknown, and therefore the times given above may change in the

actual measurements..

In order to obtain a realistic impulse response, the frequency response should be sampled

using parameters that are in full accordance with the physical characteristics of the microphones.

The resonance frequency and the quality factor, Q, of the microphones, give information about the

lowest higher limit frequency, fmax, that must be measured or simulated in order to determine a

realistic impulse response. For LS1P microphones, the resonance frequency is about 8.5 kHz, and

the quality factor is nearly unity. This implies that the system has a broad resonance. And at a

frequency 1.5 times the resonance frequency, the average power has dropped 50% or 3 dB the from

the value at resonance frequency; if a realistic decay of the impulse response is to be achieved, the

maximum frequency should not be less than the above limit. It must be also noticed that the

equation (3.38) contains the frequency as a factor, which constitutes a differentiation that clearly

emphasises high frequencies. As it is desirable to have a frequency response whose high frequency

values should have decayed sufficiently, a proper minimum value for fmax should be estimated.

As the maximum frequency defines the size of the time step in the corresponding impulse

response, the number of sampled frequency points should be large enough to a) describe properly

the irregularities caused by the standing wave and reflections from the walls, and b) be large enough

as include all the primary reflections, i.e., to give a time record that includes such reflections.


71

The first reflection appears with a frequency periodicity of about 380 Hz, and the latest at

about 200 Hz. This implies that in order to sample the disturbances appropriately, the size of the

frequency step should not be larger than 100 Hz and smaller if possible. These figures may change

slightly if the distance between the microphones is increased, as it is in practical situations where it

can take values between 20 and 40 cm. This is a minor variation however, because the time where

the reflections appear do not change as in a significant manner when the distance between the

microphones is between the given interval.

The number of frequency samples is given by dividing fmax by the frequency step. The

number of frequency points as well as the sampling frequency can be increased in order to obtain

a sampling of the shape of the disturbances.

These calculated time parameters have been used as a basis for a number of simulations of

the cleaning procedure applied to a simulated electrical transfer impedance that contains reflections

from the walls.

There should be a suitable value of fmax so as to generate a realistic impulse response that

contains all the relevant information required for a proper separation of the direct wave between the

microphones. For this purpose, an electrical transfer impedance function between two LS1P

microphones located 30 cm from each other has been generated at four different frequency ranges

with maximum frequencies, fmax, of 1.5Afres, 2Afres 3Afres and 4Afres. The frequency interval has been

divided into 1024 points in all four cases. A low-pass frequency window has been applied onto the

simulated electrical transfer impedance in order to make it converge to zero at the high frequency

limit. Figure 3.12 shows the Hilbert envelope of the corresponding impulse responses of the four

cases.

As expected, the direct wave between the microphones rises to a maximum after about 0.9

ms. It can be seen that in the case where fmax, = 1.5fres, the impulse response rises and decays slower

than under the other three conditions. This has a very important consequence if the direct wave is

to be isolated from the reflections. Additionally, a small time shift of a fraction of a millisecond can

be observed between the peaks of the four cases; the difference tends to converge as the value of

fmax increases. This is basically a consequence of the major high frequency information contained in


72

0 0.5 1 1.5 2

x 10−3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nor

mal

ised

Am

plitu

de

Time (s)

Figure 3.12 Hilbert envelope of the Impulse response from four electrical transfer impedance functions

with maximum frequencies: –––– 4Afres; - - - - 3fres; . . . . .2fres, and – . – 1.5fres.

the frequency responses that has a higher fmax and has no important consequence on the overall

procedure.

The second parameter to be evaluated is the size of the frequency step. The intention is to

find a suitable step size that will represent accurately the effect of the reflections on the impulse

response. The simulated electrical transfer impedance corresponds to the case of two LS1P

microphones located at a distance of 30 cm from each other. The upper frequency fmax is chosen to

be two times the resonance frequency of the microphones. Artificial reflections have been added by

introducing mirror sources. Three cases have been studied, dividing the frequency interval into: a)

64, b) 128 and c) 256 frequency steps that corresponds to frequency steps of 264 Hz, 132 Hz and

66 Hz respectively. The impulse responses corresponding to the three cases are shown in figure 3.13.

It can be seen that the step size of the frequency response is an important parameter. As

expected, the large size of the frequency step in case a) leads to a sub-sampling of the disturbances


73

caused by the latest of the reflections. This can be seen in figure 3.13(a) where the corresponding

impulse response is not long enough to include a portion of the reflections and even at the zero time

a portion of the last reflection is present. This may be caused by a aliasing effect. Case b) describes

a situation that is on the limit of the aliasing, but it contains all the reflections. Finally case c) is a

frequency interval short enough as to avoid any aliasing. However, it must be added that a larger

number of samples may emphasise the effect of the random noise that may be present in the actual

measurements.

From the above results it can be concluded that it is possible to carry out a simulation of the

cleaning procedure of the electrical transfer impedance describing accurately its impulse response.

It is intended to assess experimental measurements with the simulation results, therefore, sampling

parameters derived from the above methodology that are compatible with those used in the

experimental set-up are to be used in the following. The electrical transfer impedance between two

LS1P microphones has been generated by using 993 frequency steps in the frequency range from

900 to 30690 Hz, i.e., a frequency step of 30 Hz. Artificial reflections have been introduced. The

“noise” caused by the finite measurement time has been introduced by adding a random variation

of similar level to that observed in the measurement set-up to the simulated electrical transfer

impedance. The missing portion of the contaminated electrical transfer impedance is filled with

values of a reflectionless electrical transfer impedance. Then, a realistic low-pass filter is applied to

the frequency response. Thereafter, the impulse response is obtained and a Tukey time window

(equation (3.35)) is applied. These are shown in figure 3.14. The peak of the impulse response is

expected to occur at 0.87 ms, however, the realistic low-pass filter introduces the additional delay

that is observed in figure 3.14.

The most important outcome of this simulation study is the effect caused by the application

of the cleaning procedure described in equation (3.22). For this purpose, the cleaned electrical

transfer impedance is compared with an ideal reflectionless electrical transfer impedance; the overall

effect of the procedure is defined as the difference between the two quantities. This result is to be

used for assessing the accuracy improvements caused by the removal of the reflections and any

deviation introduced by the procedure. Figure 3.15 shows the difference between the cleaned

electrical transfer impedance and the reflectionless electrical transfer impedance. It can be seen that

major differences are present at the extremes of the residual function. These differences are


74

apparently caused by the truncation of the impulse response at an instant when it has not decayed

sufficiently. The ripple on the residual is explained by the fact that the time selective window has a

short duration compared with the total length of the impulse response. Thus, its Fourier transform

has high and wide side lobes; it can be seen from the residual function that the frequency of the

distortions at the extremes coincides with that present in the Fourier transform of the time window

whose modulus is also shown in figure 3.15.

Another effect that can be observed is that at the patching frequency, f0, there is a

discontinuity of amplitude and slope between the ideal and the contaminated electrical transfer

impedance functions. This is caused by the fact that the ideal value of the frequency response at that

frequency may coincide with a maximum or a minimum of the contaminated frequency response.

Though this may introduce an additional harmonic distortion, this may not be significant because

the patching level is normally 40 dB lower than the maximum value of the frequency response.


75

0 1 2 3 4 5 6

x 10−3

−1

−0.5

0

0.5

1

Nor

mal

ised

am

plitu

de

(a)

0 1 2 3 4 5 6

x 10−3

−1

−0.5

0

0.5

1(b)

Nor

mal

ised

am

plitu

de

0 1 2 3 4 5 6

x 10−3

−1

−0.5

0

0.5

1(c)

Time (s)

Nor

mal

ised

am

plitu

de

Figure 3.13 . Impulse response of the electrical transfer impedance function for different

sizes of the frequency step: a) 264 Hz, b) 132 Hz, and c) 66 Hz.


76

0 1 2 3 4 5 6 7 8

x 10−3

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Nor

mal

ised

Am

plitu

de

Time (s)

Figure 3.14 . Impulse response of the electrical transfer impedance function and the time

selective window: ––– impulse response, and – . – . – time selective window.

It can be concluded that the most significant deviations introduced by the cleaning

procedure are caused by the time selective window. The frequency range where the time window

introduces the maximum levels of distortion can be determined from its frequency spectrum by

evaluating the width end height of the main and secondary lobes. Thus, the accuracy remains

unchanged in the frequency range where the distortion caused by the time selective window can be

considered as negligible. For metrological purposes, this can be set to be compared with a given

accuracy level.


77

0 0.5 1 1.5 2 2.5 3

x 104

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Mod

ulus

(dB

)

Figure 3.15 . Difference between the cleaned and an ideal electrical transfer impedance

function as a measure of the global effect of the cleaning procedure on the accuracy of the

electrical transfer impedance. –––– residual; - - - - spectrum of the time selective window

Summary

A time selective procedure has been developed and tested using computer simulations.

The procedure has proved to remove the imperfections on the electrical transfer impedance

caused by reflections effectively. The major limitation of the procedure is introduced by the

length of the time selective window because it cuts the impulse response at points where the

impulse response has values that are larger than zero. This introduces a ripple with a frequency

that depends on the length of the time selective window. The amplitude of this ripple sets the

accuracy limitations of the application of the cleaning technique.


78

Chapter 4 Experimental results

79

Chapter 4. Experimental results

Overview

In this chapter, an account of the application of the cleaning technique described in Chapter

3 onto experimental data will be given. The objective is to show the different factors that invalidate

the practical realisation of the free field, and how the cleaning technique helps to remove them,

yielding thus a better realisation of the free field. After this account, the calculation procedure of the

free-field sensitivity is to be described. Once the electrical transfer impedance has been measured,

the cleaning procedure can be applied, and a clean function is obtained. This cleaned function is

used for obtaining the free-field sensitivity of the microphones. The basic parameters on which the

sensitivity depends will be described together with their measurement procedures. Additionally, a

derived quantity, the free-field correction, is also defined. This quantity is useful for comparing the

obtained results with results from elsewhere. A valuable source for comparison is the numerical

simulation of the problem of the microphones in the free field using the Boundary Element Method

(BEM). The BEM will be extensively used in the development of this chapter. The objective of the

comparisons is to show that the cleaning procedure effectively removes any reflections form the

walls, standing waves between the microphones, and random as well as electrical noise.

4.1 The application of the cleaning technique onto experimental measurements

In the following, the cleaning procedure described in chapter 3 will be applied to a number

of measurements of the electrical transfer impedance between two microphones. The different

measurements correspond to different configurations that contain valuable information about the

realisation of the free field. A brief description of the measurement set-up is given below; a more

detailed description of is given in appendix A. The experimental set up consists of two “one-inch”

LS1P standard microphones B&K 4160 placed in a small anechoic room with free space dimensions

of 120x80x175 cm. The measurement of the frequency response is made using the so-called steady

state response mode of the analyser B&K 2012. The analyser is connected to a reciprocity apparatus

that measures the voltage on the terminals of the receiver microphone and the current through the

terminals of the transmitter microphone. The frequency range goes from 900 Hz to 30720 Hz.


80

101

102

103

104

105

−20

−10

0

10

20

30

40

50a)

101

102

103

104

105

−20

−10

0

10

20

30

40

50b)

Mod

ulus

(dB

)

101

102

103

104

105

−20

−10

0

10

20

30

40

50c)

Frequency (Hz)

Figure 4.1 Modulus of the electrical transfer impedance or frequency response between two

microphones located at a distance of 28 cm in the free field. a) measured, b) extended, and c)

low-pass filtered.


81

The first case to be analysed corresponds to an electrical transfer impedance between the two

microphones located at a distance of 28 cm. Figure 4.1 shows the measured, the extended and the

low-pass filtered frequency responses. It can be noticed in figure 4.1a that the dynamic range in the

frequency range where the electrical transfer impedance was measured is nearly 40 dB. This may

indicate that the impulse response obtained from this frequency response will be very similar to the

actual impulse response. This dynamic range could be increased by enlarging the frequency range.

It should be expected that well below the resonance frequency the slope of the frequency response

is 6 dB/octave, thus if a further increase of the dynamic range is wanted by extending the

measurement rage at low frequencies, the above slope is to be taken into account. It should be

mentioned that extending the frequency range of the measurement at low frequencies may result in

an longer measurement time. It should also be recalled that the signal to noise ratio is very poor at

these low frequencies. At high frequencies, the decay of the frequency response is more uncertain.

If the microphones would act as single degree of freedom system, the decaying rate should be at

least as rapidly as at low frequencies. However, as it can be seen after the resonance frequency, the

frequency response decays smoothly, but about 15-16 kHz, a small jump is present. This could be

the first resonance of the back cavity of the microphones (see Rasmussen [24]). After this

disturbance, the function decays again, but around 26 kHz a peak appears. This frequency

corresponds to the second radial resonance of the diaphragm of the microphone. After this peak,

the frequency response appears to decay even more rapidly. It can be expected that any higher

resonance in the microphone will be damped by the behaviour of the air film between the diaphragm

and the backplate of the microphone.

In figure 4.1b the frequency response extended at low frequencies is shown. In this case,

although the size of the disturbances is small, a discontinuity in the slope is observed. As this

patching occurs at a frequency about 40 dB from the maximum if the frequency response, the effect

of the discontinuity may be very small. In any case, this extended portion serves only as a tool for

completing the frequency response, and it cannot be considered as a part of the measurement itself,

thus it can be disregarded afterwards. As the patched portion is calculated from ideal data, it does

not contain any information of reflections. This may imply that the actual characteristics of the

reflected waves may not be exactly represented in the time domain. However, as the objective of the

procedure is to eliminate the effect of them on the frequency response, it is not important that their

characteristics are fully known.


82

Finally, in figure 4.1c the low-pass filtered frequency response is shown. It repeats the

features already described, but it also shows the accelerated decay of the low pass filtering. It is

important to notice that this decay has not modified the modulus of the frequency response

significantly except above the roll-off frequency.

After the low-pass filtering, an inverse Fourier transform is applied to the frequency

response. In figure 4.2 the normalised impulse response is shown. Its features are discussed in the

following. First, the duration and decaying time of the impulse response. Second, the time instant

where the reflections and the standing wave between the two microphones occur in the impulse

response. The disturbances will be analysed in the following.

If the distance between the microphones is 0.28 m disregarding the acoustic centres, it is

expected that the direct wave coming from the transmitter microphone will reach the receiver

microphone at the instant it takes the sound to travel such distance. Considering that the sound

speed, c, under standard environmental conditions (23 C temperature, 101325 Pa static pressure, and

50% relative humidity) is about 345.86 m/s, the time it takes the wave to travel the above mentioned

distance is 0.82 ms. It is possible to see in figure 4.2 that the impulse response effectively starts at

that instant. Then, it reaches its peak at about 0.92 ms. Thereafter, it decays slower that it rises. Here

it is necessary to recall the analysis about the expected properties of the impulse response carried out

in Section 3.4. The decay of the impulse response should be defined in terms of the time constant

of the system – see equation (40) –. If the same parameters of the lumped parameter of the

microphones given in table 3.1 are considered, the same decay rate defined in Section 3.4 should be

expected. Thus, the impulse response should have decayed to one thousandth of its maximum value

after 0.5 ms. It can be observed in figure 4.2b. Thus, the start of the impulse response and the

decaying time calculated from the lumped parameter model can be used as a valuable guide when

the decision of where to place the time selective window will be made.

The previous considerations prepare the way for applying the time selective window for

separating the direct wave from the disturbances. The window to be applied is a Tukey window as

described in equation (3.35). The window’s length is about 2 ms, and the smoothing portion on the

extremes of the window is 30%. Figure 4.3 shows the impulse response and the window as well as

the residuals of the impulse response after the application of the window. It can be seen that the


83

time selective window effectively separates the disturbances from the direct wave between the two

microphones, and that it does not modify any part of it significantly while it makes a smooth

transition that will decrease any distortion caused by the short length of the time selective window.

The frequency at which the disturbances occur in the frequency response can be predicted

by its position on the impulse response. Then, the standing wave has a peak at about 2.5 ms, this

corresponds to a frequency of 400 Hz. Later reflections from the walls appear at 3.5 ms, 4 ms, and

perhaps at 6.5 ms; this corresponds to 285 Hz, 250 Hz, and 153 Hz. This could not be noticed easily

on the frequency response because of the large dynamic range (Figure 4.1). However, once the

inverse Fourier transform of the windowed impulse response is obtained, the clean frequency

response is determined and the difference between the cleaned and the uncleaned (raw) response

can be calculated. This difference is shown in figure 4.4. It can be seen that it is very difficult to

assess the frequency of the disturbances. It is expected that they will modulate themselves creating

a very complex pattern. However, three different regions can be observed. A low frequency region

– from 900 Hz to 6 kHz –, a mid frequency range – 6 kHz to 15 kHz –, and a high frequency range

– 15 kHz to 30 kHz –. In these regions, the difference between the cleaned and the raw frequency

response is different.

At low frequencies, the frequency of the pattern is about 200 Hz to 250 Hz. Besides, the

amplitude of the disturbances seems to decrease as the frequency increases in this frequency range.

These two characteristics observed in the difference may suggest that it could be caused by the

reflections from the walls. It is expected that the absorption coefficient of the absorbent walls is low

at low frequencies, improving as the frequency increases – see reference [44] – . Thus, the amplitude

of the reflections should change as the frequency increases.

The mid frequency range could be considered as a transition range. The first disturbances

seem to die out while other disturbances with different frequency appear. It is in this region where

the amplitude of the disturbances is the smallest in the whole frequency range.

The characteristics of the emerging disturbances can be better analysed in the high frequency

region. There, the amplitude of the disturbances grows slowly. The sudden increase at about 28 -

30 kHz may be due to the effect of the time selective window, as shown in figure 3.15. The


84

frequency of the disturbances is less clear. There could be a modulation of two disturbances, one

with a frequency of about 1250 Hz which carries another of about 450 Hz, the fact that the second

is almost a integer multiple of the first. The second one is due to the standing wave between the

microphones, while the first is caused by the cross talk effect. Their frequencies are necessarily

related by an integer because of the fact that the cross talk occurs at zero time, and the direct wave

between the microphones occurs at an instant when the sound has travelled the distance between

the microphones, and the standing wave at an instant when the sound has travelled a distance which

is three times the distance between the microphones. Thus, according to the above, the frequency

to which the standing wave should occur is 1220 Hz, and the standing wave at about 410 Hz. The

increase of the amplitude is also in agreement with the fact that the standing wave appears when the

wavelength becomes comparable with the size of the microphone. The effect of the size of the cross

talk is less evident. But it may be related with the signal to noise ratio. Thus, it can be significant at

the extremes of the frequency range.

A way for showing the effect of the reflections on the frequency response is to apply a time

selective window that does not remove the cross talk. Thus, the difference between the cleaned and

the raw frequency responses should be only due to the reflections and the standing wave. Figure

4.6 shows the impulse response, and the difference between the cleaned and the raw frequency

responses. The residuals in figure 4.6a show that the cross talk has not been removed from the direct

impulse response between the microphones. Therefore, only the reflections from the walls are

removed from the frequency response, and the difference between the cleaned and raw frequency

responses should only contain the disturbances caused by the reflections. It can be seen in figure

4.6b that the high frequency region is filled with disturbances having a frequency about 500 Hz. This

can be related to the standing wave between the microphones.

The effect of the cross talk can also be illustrated using the same procedure. Then, a

different time selective window is applied. It does not remove the reflections form the walls nor the

standing wave between the microphones. It only removes the cross talk. Figure 4.5 shows the

residuals after the time windowing, and the difference between the cross talk cleaned and the raw

frequency response. It can be seen in figure 4.5a that the time selective window has removed only

the cross talk, leaving the reflections on the impulse response. Figure 4.5b shows the difference

between the cross talk cleaned and the raw frequency responses. The difference has also an


85

oscillating pattern with frequency about 1200 Hz; this corresponds to the period of time it takes the

direct wave to reach the receiver microphone . Three different regions can be observed as well. A

low frequency region – up to 5 kHz –, a mid frequency region – 5 kHz to 12 kHz –, and a high

frequency region – 12 kHz and above –. In the low frequency region the amplitude decreases as the

frequency increases. In the mid frequency range, the amplitude is about zero; the small variations

may be caused by the random noise that is present at the same time instant as the cross talk. In the

high frequency range the amplitude increases again. However, this tendency seems to stop at about

25 kHz and then increase again. If the cross talk depends on the signal level, the differences

observed may be related to the signal to noise ratio at the different frequency regions though it is

difficult to find a qualitative relation between them.


86

0 1 2 3 4 5 6 7 8

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Nor

mal

ised

Am

plitu

de

0 1 2 3 4 5 6 7 8

x 10−3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3

Time (s)Figure 4.2 Impulse response obtained from experimental data. (a) is the normalised amplitude and

the time window: ––––– Impulse response, and – – – – – Hilbert envelope of the ideal impulse

response. (b) ––––– Hilbert envelope of the measured response, and – – – – – Hilbert envelope

of the ideal impulse response.


87

0 1 2 3 4 5 6 7 8

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1N

orm

alis

ed A

mpl

itude

0 1 2 3 4 5 6 7 8

x 10−3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3

Nor

mal

ised

Am

plitu

de

Time (s)Figure 4.3 Impulse response of the electrical transfer impedance and the time selective window:

a) ——— normalised impulse response, and —C—C— time selective window.

b) ——— residuals after the application of the time selective window, C C C C C C C Hilbert envelope

of the impulse response, and —C—C— time selective window.


88

0.5 1 1.5 2 2.5 3

x 104

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Frequency (Hz)

Mod

ulus

(dB

)

Figure 4.4 Difference between the cleaned and raw electrical transfer impedances.


89

−1 0 1 2 3 4 5 6 7 8 9

x 10−3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 a)

Time (s)

Nor

mal

ised

am

plitu

de

0 0.5 1 1.5 2 2.5 3

x 104

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5b)

Frequency (Hz)

Diff

eren

ce (

dB)

Figure 4.5 a) Impulse response of a frequency response when the two microphones are located at 28

cm from each other. The time selective window does not remove the reflections.

b) Difference between the cleaned and raw frequency responses. The “cleaned” version still includes

the effect of the reflections.


90

−1 0 1 2 3 4 5 6 7 8

x 10−3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 a)

Time (s)

Nor

mal

ised

am

plitu

de

0 0.5 1 1.5 2 2.5 3

x 104

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5b)

Frequency (Hz)

Diff

eren

ce (

dB)

Figure 4.6 a) Impulse response of a frequency response when the two microphones are located at 28

cm from each other. The time selective window does not remove the cross talk.

b) Difference between the cleaned and raw frequency responses. The “cleaned” version still includes

the effect of the cross talk.


91

4.1.1 Disturbances in the impulse response

The direct wave between the two microphones is not the only noticeable feature of the

determined impulse response. It can be seen in figure 4.3b that a small disturbance appears at instant

zero, and another slightly larger disturbance followed by a more or less confuse set of smaller

disturbances appear after the direct wave.

4.1.1.1 Noise – random noise and electrical cross talk

The early disturbance may be associated either with the random variation caused by the finite

averaging time or with the cross talk because they appear at the same instant of time. If the noise

is randomly distributed al over the whole frequency range, it should appear in the impulse response

at as a delta function zero time if the frequency interval was infinite, or with a given effective

duration if the frequency range is finite, as it is the case. On the other hand, cross talk is an electrical

problem with a frequency dependence as the value of the cross talk depends on the signal to noise

ratio. As it is electrical in nature, it will travel faster than the sound wave, appearing thus at zero time.

Because its value depends on the signal to noise ratio at any frequency, it should also have a finite

duration. Thus, it is very difficult to predict which effect is the one present in the impulse response.

However, further considerations of the actual nature of the noise present in the measured frequency

response should help to solve the dilemma. The frequency response is measured using the 2012

sound analyser. The working principle of this analyser is described in Appendix A and it will not be

repeated here, but a short description will serve the purpose of helping to understand the cause of

the early disturbance. The steady state values of the input of the analyser are measured by applying

the so-called adaptive scan algorithm. For each frequency, blocks of data are collected and

processed in order to obtain the average and standard deviation. Any transient behaviour is avoided

by introducing a settling time. Once the system is settled, the detector starts the data collecting

procedure. Then, it calculates the average and standard deviation of the sampled block. These

results are compared with a set of user defined requirements for the standard deviation. When these

requirements are met, the final result is recorded. If the requirements are not met, the data

acquisition and processing continues adding block after block of data, working actually as a filter that

adaptively narrows the bandwidth until the variations disappear. However, another user defined

parameter is the maximum measurement time that overrules the accuracy parameter when a given


92

time is reached and the wanted accuracy is not achieved. It has a special relevance because the signal

to noise ratio at low frequencies is very poor. Thus, it is expected that the noise distribution along

the frequency response is not constant. Furthermore, the accuracy parameter is not defined in

absolute terms but relative to the measured value, as a percentage of it. It means that the amplitude

of the “noise” is not constant, but having a value related to the standard deviation multiplied by the

modulus of the measured frequency response. This means that the noise may have an exponential

decay at low frequencies, and as the frequency increases, it will replicate the shape of the frequency

response. When this is transformed into the time domain, it shows up as the combination of the

exponentially decaying function convolved with the scaled down replica of the impulse response of

the frequency response. This means that the time transformation of the noise may show an

oscillatory behaviour that decays exponentially in time. It should be mentioned that the noise levels

should be high for observing this behaviour. Observing the impulse response, there are some later

oscillations with the same period as the main impulse response. On the other hand, although cross

talk may also have a frequency dependence perhaps inversely proportional to the signal to noise

ratio, its characteristics are less evident. However, it can be seen in figure 4.3b that the early

disturbance is a sharp peak that decays very rapidly, thus implying that it is not random noise.

Additionally, it is expected that the amplitude of this disturbance will increase as the distance

increases because the signal to noise ratio becomes poorer. In either case, the instant when such a

disturbance appears may indicate the position of the time selective window if the disturbance is to

be removed.

4.1.1.2. Standing wave between the microphones and reflections from the walls

The next disturbances to be analysed are these coming after the direct wave between the

microphones. There are two different perturbations. One is the standing wave between the

microphones, and the second is the set of reflections from the walls. In figure 4.3b it is possible to

see a well defined peak at about 2.5 ms, and later, a set of disturbances that rises above the floor

noise. The instant where the first peak appears corresponds roughly to 3 times the distance between

the microphones. This is the distance that the sound wave shall travel in order to form the standing

wave between the two microphones. It is expected that the size of the standing wave decreases as

the distance increases.


93

−1 0 1 2 3 4 5 6

x 10−3

−5

0

5x 10

−3 a)

−1 0 1 2 3 4 5 6

x 10−3

−5

0

5x 10

−3 b)

−1 0 1 2 3 4 5 6

x 10−3

−5

0

5x 10

−3 c)

Nor

mal

ised

am

plitu

de

−1 0 1 2 3 4 5 6

x 10−3

−5

0

5x 10

−3 d)

−1 0 1 2 3 4 5 6

x 10−3

−5

0

5x 10

−3 e)

Time (s)Figure 4.7 Impulse response of the electrical transfer impedance measured at several distances

between the microphones. A) 180 mm, b) 280 mm, c) 350 mm, d) 420 mm, and e) 490 mm.


94

RddO R− =

= −20

39 54log . dB. (4.1)

The latter disturbances appear at instants corresponding to reflections from the walls. It is

expected that their magnitudes increase as the distance increases. In any of the two cases, it is

difficult to assess the frequency characteristics because of the complicated path they follow.

However, it is possible to examine the measurements made at several distances in order to describe

the behaviour of the disturbances. Figure 4.7 shows the impulse response of electrical transfer

impedances measured at several distances between the microphones: 180 mm, 280 mm, 350 mm,

420 mm, and 490 mm. The standing wave between the microphones may easily be identified –

especially at short distances – because its occurrence is well defined as a function of the distance

between the microphones. Consider the five cases shown in figure 4.7. At these measurement

distances the standing wave happens at the instants of 1.56 ms, 2.43 ms, 3.04 ms, 3.65 ms, and 4.26

ms. At the distances of 180 mm and 280 mm, the standing wave is clearly separated from any other

disturbance. It is at a distance of 350 mm and longer distances where it seems to be confused with

the first reflections from the walls.

The behaviour of the standing wave can be analysed using a simple approach. Consider a

point source of strength Q located on an infinite baffle, and a second parallel baffle located at a

distance d in front of the baffled source. The wavefronts emitted by the baffled source will travel

the distance d between the baffles, and will be reflected back. The reflected wavefront can be

thought to be generated by a mirror source located at a distance d behind the reflecting baffle. These

reflected wavefronts will travel to the baffle with the original source and will be reflected back again.

This second reflection can be though to be caused by a second mirror source located behind the

original point source at a distance 2d behind the baffle. Thus, when this second reflection reaches

the baffle in front of the point source by a second time, it should have travelled a distance equal to

3 times the distance between the baffles, d. It is important to notice that the energy carried by the

original and the reflected wavefronts is the same because the energy is fully reflected on the infinite

baffles. Thus, the logarithmic ratio of the amplitude of the direct wave hitting the baffle to the

second reflection reaching the baffle is simply

The relation between the amplitudes of reflected waves at two different distances between

the baffles can be expressed as the logarithmic ratio between the distances


95

RDdO R− =

20

16

2

2log . (4.5)

Rddd d1 2

1

220− =

log . (4.2)

′ ∝Q QDd

, (4.3)

′′ ∝ ′ ∝Q QDd

QDd2 2

2

2 . (4.4)

The case of the microphones in a free field can be analysed in a similar way, considering that

the subsequent reflections are caused by mirror sources behind the microphones. However, some

considerations should be taken into account. One is that the dimensions of the microphone are

finite. Thus, in the frequency range where the measurements are made, the microphone cannot be

considered as very large compared to the wavelength, and thus the specular mechanism for

explaining the reflections does not fully occur. It can be thought that the reflection mechanism at

lower frequencies is more or less diffuse. This means that the reflected energy will be only a fraction

of the incident because of the finite dimensions of the microphone. It could also be argued that it

is a function of the solid angle formed by the reflecting microphone and seen by the incident

wavefront. This solid angle is proportional to the square of the ratio of the transverse dimension,

the diameter in this case, to the distance between sources. Thus, the source strength of the mirror

source behind the reflecting microphone, Q’, should be a fraction of the original source strength,

Q,

where D is the microphone diameter. Therefore, the source strength of the secondary mirror source

generating the second reflection, Q=>, is proportional to the source strength of the first mirror source

and the ratio of the diameter to the distance between the mirror source and the microphone

Then, the ratio of amplitudes between the direct wave and the second reflection is a function of the

distance and of the source strength of the second mirror source. The first part is given by equation

(4.1), the second one is a function of the distance.

The logarithmic ratio between the amplitude of the secondary reflections at two different

distances, similar to equation (4.2), can be calculated using


96

Rdd

ddd d1 2

13

23

1

220 60− =

=

log log . (4.6)

In figure 4.8 the logarithmic Hilbert envelope of two impulse responses obtained from

measurements at different distances, namely 200 mm and 300 mm, is shown. According equation

(4.5), it should be expected that the ratio of amplitudes is about -52 dB and -59 dB respectively. If

we consider the peak values, we can see that the difference is about -48 and -56 dB respectively. It

indicates that the observed values are smaller than the expected. However, it is important to recall

that in the frequency range at which the microphones are analysed, there is neither an entirely

specular reflection phenomenon nor an entirely diffuse reflection phenomenon but a combination

of the two. Thus, a sort of transition value should be expected. In this case, it seems that the diffuse

reflection seems to be dominant.

Thus, a very important consequence is that the amplitude of the standing wave is heavily

dependent on the square ratio of the diameter to the distance. Then, it is expected that the amplitude

of the standing wave will decrease very rapidly, it decreases 4 times if the distance is doubled. It

implies that the amplitude of the standing wave can be reduced by increasing the distance

sufficiently. However, this may not be a practical solution because the logarithmic signal-to-noise

ratio becomes poorer as the distance increases, and the reflections from the walls start to become

dominant.

If the ratio of amplitudes is calculated using equation (4.6), a -10 dB difference is expected;

the observed difference is slightly larger. The differences between the expected and observed values

may be caused by the fact that the frequency content of the second reflection will be different

because of the difference in distances. The larger the distance, the more dominant the high

frequencies become; this makes a sharper impulse response.


3It seems to be more appropriate here to talk about a plane where the reflection from the wall occurs because the wallis composed by absorbing wedges of a given length. This makes it very difficult to define the place where the actualreflection at a given frequency occurs.

97

0 0.5 1 1.5 2 2.5 3

x 10−3

−40

−30

−20

−10

0

10

20

30M

odul

us (

dB)

Time (s)

Figure 4.8 Logarithmic modulus of the Hilbert envelope of the impulse response obtained from

measurements at two different distances between microphones ——— 200 mm, and — C — C —

300 mm.

The reflections from the walls have positions more or less fixed because the distance they

should travel depends on the quadratic sum of two times the distance to the plane of the wall3 where

the reflection comes from and the distance between microphones. This total distance will not change

significantly as the distance between microphones increases. Furthermore, the reflections from the

walls behind the microphones (ceiling and floor in this case) are always at the same position because

the distance the wave must travel remains constant independently of the distance between

microphones. The reflections are more easily identified at the shortest distance, because the noise

floor is at a lower level. It can be seen in figure 4.7a that after 3 ms there is a group of disturbances

that rise above the noise floor. These are the reflections from the walls. As the distance increases,


98

( )[ ]M fd dd

Z ZZ d d dfe e

e,

, ,

,exp .1

2 12 13

23

12 13

2312 13 23

2= + −

ργ (4.7)

the instant where the disturbances occur does not seem to change but, a small increase on their

amplitude may be observed at the longer distances. The same effect may be observed with

measurements where the absorbent material is not so efficient as in this case. This implies that

measurements could be made in rooms with some absorbing material on the walls, but not

necessarily high performance wedged walls.

4.2 Absolute determination of the free-field sensitivity

In chapter 2, an expression for the electrical transfer impedance was deduced –

equation(2.47) –. It contains the free-field sensitivity of the two microphones coupled in the free

field. It is an equation with two unknowns that cannot be solved by itself. However, if a third

microphone is coupled successively to the other two microphones, a set of three equations with

three unknowns is obtained. Then, the sensitivity of each microphone can be obtained in absolute

terms by solving the simultaneous system of equations. Solving the system of equations, the free-

field sensitivity of one of the microphones here labelled as microphone 1 is

Similar expressions can be obtained for the other two microphones. This equation can be used for

calculating the free-field sensitivity after cleaning the measured electrical transfer impedances.A

number of parameters must be determined. These parameters are the effective distance between the

two microphones, and the physical properties of the air inside the chamber such as density, and air

absorption. Although the acoustic centres are extensively used in the determination of the free-field

sensitivity and in the analysis of the problem of the air absorption, its determination is described in

detail in chapter 5 in order to restrict the remaining of this chapter to the analysis of the free-field

sensitivity and the free-field correction.

4.2.1 Physical properties of air

The physical properties of air are of great importance when the free-field sensitivity is

calculated. As it happens with any other factor in equation (98), if a better estimate of Mf is wanted,

to use an accurate value of the physical properties of air is a must.


99

The standard [26] provides a set of expressions for the calculation of the variables involved

in the calculation of the sensitivity. However, there are other sources of expressions for such a

calculation. Rasmussen[45] presents a compendium of expressions for calculating the physical

properties of the air. It is without doubt recommendable to follow the references given in the report.

However, no matter how accurate the used expression is, it is useless when the air inside the

anechoic chamber does not have the composition of the so-called “standard” air. The following is

a description of a particular case study that took place during the development of the project.

During an international comparison of free-field calibration of condenser microphones[46],

deviations of the acoustic centres obtained by the laboratories from the expected values given in the

standard [26] were observed. These deviations consisted in a large spread of the calculated acoustic

centres at high frequencies. The reported acoustic centres, and the standard values are shown in

figure ?.

It can be seen that there are some differences in the low frequency range; there seems to be

a constant difference in the whole frequency range. However, it can also be seen that two of the

participating laboratories deviated considerably from the expected values at frequencies above 6.3

kHz. The third laboratory showed only a small difference at 25 kHz. It was suggested in the report

that the cause of the deviations in the measurements of the Acoustics Laboratory at DTH were

caused by an anomalous composition of the air inside the chamber. This would show up as an

anomalous air absorption coefficient of 2 to 2.5 times larger than the value of the “standard” air. It

was suggested that this anomaly would have been caused by some chemical compounds released

from the absorptive wedges. This was also “supported” by the peculiar smell inside the chamber!

The measurements of this project were carried out in the same anechoic chamber. Thus, it

was expected to obtain similar results of the acoustic centres. Figure 4.9 shows the results of acoustic

centres from the first series of measurements. The results are obtained from the cleaned

measurements using the calculation procedure based on the linear regression of the electrical transfer

impedance measured at several distances described in section 5.1.1. The values from [46] are also

shown.


100

103

104

−5

0

5

10

15

Frequency (Hz)

Aco

ustic

cen

ter

posi

tion

(mm

)

Figure 4.9 Acoustic centres measured in the small anechoic chamber during the first attempts of

measuring the electrical transfer impedance: ———— Calculated using simple Linear regression

procedures, — + — + — DTH meas [46], and —~—~— IEC standard[26]

It can be seen that the recently measured acoustic centres showed a behaviour similar to the

old values. However, the new values show a slight difference moving towards the standardised

values. It must be recalled here that this approximation is based on the fact that the air absorption

coefficient is known. Therefore, in order to verify that the air absorption was indeed the cause of

the problem, the acoustic centres were calculated using the procedure that determines simultaneously

the acoustic centres and the air absorption; this is described in section 5.1.1.

Figure 4.10 shows the acoustic centres calculated after the determination of the quadratic

coefficients of equation (5.5)


101

103

104

−5

0

5

10

15

Frequency (Hz)

Aco

ustic

cen

ter

posi

tion

(mm

)

Figure 4.10 Acoustic centres measured in the small anechoic chamber during the first attempts of

measuring the electrical transfer impedance: ———— Calculated using the quadratic curve fitting,

— + — + — DTH meas [46], and —~—~— IEC standard[26]

However, another confirmation was needed in order to proceed with modifications of the

anechoic chamber itself. An experiment was conducted where the door of the anechoic chamber was

open during the measurement. It was expected that the measurements would be contaminated by

the increased background noise, and reflections from the hard surfaces in the laboratory. At same

time, it was expected that the air inside the chamber would be replaced by fresh air or at least non-

contaminated air. Figure 4.11 shows the acoustic centres obtained from the experiment that were

calculated using the simple linear regression procedure.


102

103

104

−5

0

5

10

15

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 4.11 Acoustic centres measured in the small anechoic chamber calculated from measurements

of the electrical transfer impedance inside the anechoic chamber with the door open:

———— Calculated using a linear regression procedure, — + — + — DTH meas [46],

and —~—~— IEC standard[26]

After the confirmation of the fact that the air inside the chamber was contaminated by some

compounds released from the absorbing material of the walls, a solution for the anomalous air inside

the chamber was then needed. The simplest solution was to introduce a ventilation system which

was implemented shortly after the experiments.

This experiment also confirms the importance of providing that the air where the

measurements are made is not contaminated, or at least its composition is close to that of the

standard air.


103

4.3 The free-field sensitivity and derived quantities

Once the acoustic centres have been determined according to the procedures described in

chapter 5 – either by the experimental procedures or by assuming that the theoretical considerations

are correct in the frequency range of interest –, and when the physical properties of air have been

calculated – using the appropriate procedures –, the free-field sensitivity can be calculated using

theexpression given in equation (4.7).


104

103

104

−40

−35

−30

−25

−20

−15M

odul

us (

dB)

103

104

−350

−300

−250

−200

−150

−100

−50

0

Frequency (Hz)

Pha

se (

°)

b)

Figure 4.12 Free-field sensitivities of three B&K 4160 condenser microphones calculated using

the experimental acoustic centres: a) Modulus in dB re 1 V/Pa, and b) Phase angle. In both

figures ———— Microphone s/n 1453784, — — — microphone s/n 1453798,

and C C C C C C microphone s/n 1453804


105

The calculation procedure is very simple indeed. Equation (4.7) provides a closed and direct

expression for the calculation of the complex sensitivity. This equation was used thus for calculating

the free-field sensitivity of a number of LS1P microphones. In the following, the results are to be

analysed. In all cases, it is assumed that the physical properties of air are described accurately by the

equations referred in section 4.2.2. It means that the composition of the air inside the chamber is

close to the so-called standard air, and that the anomalies in such composition shown in section 4.2.2

are no longer present.

4.3.1 Free-field correction

The free-field sensitivity shown in figure 4.12 cannot be compared with any other reference

of the same accuracy level except when the sensitivity of the same device is determined

experimentally in another calibration system. This is not very practical if the accuracy of the

calibration is to be disseminated to other metrological levels. However, in chapter 2 another quantity

was defined. It is the free-field correction – equation (2.40) –. The determination of this quantity

requires precise knowledge of the sensitivity of the same microphone under pressure conditions, i.e.,

when a uniform sound pressure is applied onto the diaphragm of the microphone. The

determination of the sensitivity of the microphone under uniform pressure conditions is more

widespread than under free-field conditions, and it is not unusual that the pressure calibration set-up

is set long before any attempt of free-field calibration is made. Thus, it was possible to obtain

pressure calibration data of a number of microphones, and these were used for determining the free-

field correction of the microphones. In the following, the results of the free-field correction of these

microphones is discussed. This is calculated only up to 16 kHz because of the frequency limitations

of the pressure calibration. The free-field sensitivity of each microphone was calculated using the

experimental values of the acoustic centres.

Figure 4.13 shows the experimental acoustic centres of all the microphones. It can be seen

that there is a large spread in the determined values of this quantity. The deviation at the lowest

frequency – 1 kHz – is caused by the effect of the time selective window. The spread is about 3 mm

in the whole frequency range. This spread may be partly due to some of the limitations of the

measurement set-up in terms of repeatability. This spread should be also reflected in the calculated

free-field sensitivity, and thus in the free-field correction.


106

103

104

−5

0

5

10

15

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 4.13 Acoustic centres of a number of condenser microphones. The full lines are the individual

values, and —~—~—~— is the IEC standard [26]

Figure 4.14 shows the difference between the complex free-field and pressure sensitivities.

The spread in the modulus is about 0.2 dB at low and high frequencies, while it is about 0.1 dB in

the mid frequency range. The spread of the phase is also large. As mentioned above, it may be a

consequence of the large spread of the calculated acoustic centres.

This fact may contradict a statement given on page 160, chapter 5 in which the use of the

experimental acoustic centres is recommended instead of the standardised values. However, that

statement is valid because the use of the experimentally determined acoustic centres actually results

in a little spread of the sensitivities at high frequencies, above 20 kHz. However, it also seems that

the large spread of the experimental acoustic centres has also a significant effect on the phase

difference. This is not unexpected.


107

Figure 4.15 shows the free-field correction obtained from the free-field sensitivities when

these are calculated with the standardised acoustic centres [26] of all the microphones. It can be seen

that the spread in the modulus of the free-field correction remains unchanged. However, there is

a dramatic improvement of the phase difference. This is because in this calculation the spread of the

experimental acoustic centres has been removed.

Physically, the acoustic centres of the microphones do not vary very much. The geometry

of the microphones is the same within close tolerances, and the diaphragms of the microphones

will probably move in the same way. Thus, the observed variations may be associated with

instabilities of the measurement system itself.

Figure 4.16 shows the average of the free field correction of the microphones when the free-

field sensitivities are calculated using the standardised and the experimental acoustic centres.

It can be seen that the difference between the modulus values is not large, about a 0.02 dB,

at low frequencies – up to 10 kHz –, and decreases to 0.01 dB above 10 kHz. Below 2 kHz there

are remaining effects of the time windowing. Although the modulus difference is not large, the

phase difference shows a dramatic difference, about 5 degrees. This large spread is caused by the fact

that the experimental acoustic centres have a large spread that is reflected in the average as well.

The modulus of the free-field correction can be compared with the results reported in the

international comparison [46] or with a recent polynomial approximation of the free-field correction

that has been circulating as an internal draft among the members of a working group of the IEC

[47], or with simulated data from the BEM formulation. As the IEC data are based on the average

values either from calculations or experiments provided by several laboratories around the world,

the experimental results obtained here are compared with this reference. The results are also

compared with the results of the BEM formulation with the frequency range extended to frequencies

above the resonance frequency.


108

0

2

4

6

8

10F

ree−

field

cor

rect

ion

(dB

)

103

104

−30

−20

−10

0

10

20

30

Frequency (Hz)

Pha

se d

iffer

ence

( °

)

Figure 4.14 Free field correction form a number of microphones. The free-field sensitivities were

calculated using the experimental acoustic centres: a) Modulus, and b) Phase


109

0

2

4

6

8

10a)

Mod

ulus

(dB

)

103

104

−30

−20

−10

0

10

20

30b)

Frequency (Hz)

Pha

se (

° )

Figure 4.15 Free-field correction for a number of microphones. The free-field sensitivity was

calculated using the standardised acoustic centres [26]: a) Modulus of the free-field correction, and

b) phase angle.


110

103

104

0

5

10a)

Fre

e−fie

ld c

orre

ctio

n (d

B)

103

104

−0.2

−0.1

0

0.1

0.2

Diff

eren

ce (

dB)

103

104

−20

−10

0

10

20

Pha

se (

°)

103

104

−20

−10

0

10

20b)

Frequency (Hz)

Diff

eren

ce (

°)

Figure 4.16 Difference between the free-field corrections when the free-field sensitivity is

calculated using the standardised and experimental values: a) modulus, and b) phase. In the two

figures: ———— Difference, — C — C — Experimental acoustic centres, and – – – –

standardised acoustic centres.


4 It is known that the BEM solution for exterior problems can be contaminated with spurious results coming fromfictitious eigenfrequencies of the internal domain of the geometry. This is also known as the non-uniqueness problem.According to Juhl, Peter, M., et al, On the non-uniqueness problem in a 2-D half-space BEM formulation, Proceedings of theNinth International Congress on Sound and Vibration, Orlando, Fla., USA, 2002 (CD-ROM), the non-uniquenessproblem is reflected numerically as an ill-conditioning of the matrix containing geometrical constants related with thesolid angle seen by the elements on the surface when measuring the field point P. The matrix also contains terms relatedto the integrals of the Green’s function. In the problem described here, the non-uniqueness was tested by checking theill-conditioning of the matrix. It was found that the frequencies where the larger condition numbers were presentshowed no significant deviations from the expected results. This was more evident when analysing the acoustic centresof the microphones. This is described in the following chapter.

111

23.77

18.6

1.95

600

11.88

600

12.7

13.3

9.3

12.7

6.35

LS1 Geometry LS2 GeometryFigure 4.17 Geometry of LS1 and LS2 microphones used for the BEM simulations.

The BEM simulation is carried out using the geometry4 shown in figure 4.17. The free-field

correction is calculated from equation (2.40) by multiplying the diffraction factor, S(f,θ), and the load

of the radiation impedance on the acoustical impedance of the microphone. The calculation of these

quantities is described below.

The radiation impedance is calculated by forcing the diaphragm of the microphone to move

as a Bessel function of zero order, as described in equation (5.11). Afterwards, the radiation

impedance is calculated as


112

( )

( )Z

pq

S p r rdr

u r rdra r

a

a, ,= =∫

∫

2

2

0

0

π

π

(4.8)

( )( )

( )

( )

( )S f

pp

S p r rdr

S p r rdr

p r rdr

p r rdr

T

I

T

a

I

a

T

a

I

a, ,θ

π

πθ == = =

∫

∫

∫

∫0

0

0

0

0

2

2(4.9)

where S is the area of the circular diaphragm. Once the radiation impedance has been determined,

the acoustical impedance of the microphone is obtained using the equivalent circuit described by

Rasmussen in [24]. Thus, the load of the radiation impedance over the acoustic impedance can be

calculated.

The diffraction factor is determined in a similar fashion. However, in this case the diaphragm

as well as the body of the microphone and the mounting rod are supposed to be rigid. The rigid

body is introduced in a sound field where a plane wave propagates in a direction parallel to the

longitudinal axis of the geometry. The diffraction factor is calculated as

where pI is the pressure of the incident plane wave at the position where the diaphragm is later

introduced, and pT is the total pressure on the diaphragm.

Figure 4.18 shows the comparison of the experimental and simulated modulus of the free-

field correction for LS1 microphones. It can be seen that at low frequencies and up to 7 kHz the

agreement between the measurements, the standardised curve, and the simulation results is very

good. However above 7 kHz, the results spread.

The polynomial approximation has a peculiar behaviour between 8 kHz and 10 kHz

following an almost straight line between these frequencies. This can be explained by the fact that

traditionally, laboratories make measurements at the central frequencies of third octave bands, thus

leaving an empty space in the middle of the frequency range where the slope of the free-field

correction may change rapidly; this is the case of the frequency interval between 8 and 10 kHz where

no intermediate frequency has been used for the fitting of polynomial coefficients.


113

1 2 3 4 5 10 200

1

2

3

4

5

6

7

8

9

10

Frequency (kHz)

Fre

e−fie

ld c

orre

ctio

n (d

B)

Figure 4.18 Comparison of the modulus of the free-field correction for LS1 microphones.

——— Average of experimental measurements, — — — BEM simulations,

and — C — C — ref. [47].

However, the spread among the corrections above 7 kHz cannot be easily explained. At

these frequencies many things occur simultaneously:

1. The simulation starts to break down although the equivalent circuit of the microphone can be

used above the resonance frequency [24] and the assumption of the movement of the diaphragm

as a Bessel function may still be valid.

2. The repeatability of the free-field measurements becomes comparable to the observed

difference. This is clear when the spread of sensitivities determined from measurements at

different distances is calculated – see figure (5.4) – .


114

3. The pressure calibrations also suffer from some problems. The radial movement of the

wave inside the calibration couplers becomes significant, and some particular resonances

inside the couplers may begin to have some significant effects. This is reflected in a

degradation of the repeatability of the sensitivity of a microphone obtained in plane wave

couplers of different length.

All these factors obscure the analysis making it very difficult to draw any conclusions. The

last two can be assigned to the realisation of the measurements. A better repeatability should help

to clarify some of the problem. The first point can be analysed based on the experimental results and

the simulation results. Figure 4.19 shows the calculated and the experimental free-field corrections,

and the calculated contributions of the diffraction factor and the load of the radiation impedance

to the total free-field correction.

It can be seen that, as expected, the largest contributor to the free-field correction is the

diffraction of the plane wave caused by the body of the microphone. The diffraction factor depends

on the geometry of the microphone and the mounting rod, and last but not least, on the spatial

distribution of the incident sound field.

On the other hand, the load of the radiation impedance over the microphone impedance,

which is very small at low frequencies, becomes significant at the resonance frequency of the

microphone. The load of the radiation impedance depends on the acoustic impedance of the

microphone; furthermore, it depends on the actual movement of the diaphragm, and the spatial

configuration of the set-up.

In the two cases the geometry of the microphone and the mounting rod is an important

factor. However, it remains invariant within close tolerances, of the order of few hundredths of a

millimetre. Furthermore, the expected variations due to the tolerances are very small. Thus, the

variations in the geometry can be neglected.


115

103

104

−2

0

2

4

6

8

10

12

Frequency (Hz)

(dB

)

Figure 4.19 Contributions of the diffraction factor and the load of the radiation impedance to the

modulus of the free-field correction: — — — diffraction factor, — C — C — load of the radiation

impedance, and ——— total free-field correction.

The movement of the diaphragm is an important factor when calculating the radiation

impedance. It is widely accepted that at frequencies below the resonance frequency of the

microphone, the movement of the diaphragm can be approximated by a parabolic function which

is very similar to a Bessel function of zero order. It is known that the Bessel function is a solution

of the differential equation that describes the displacement of a stretched membrane in vacuum [48].

Thus, a natural assumption is that the diaphragm has a displacement distribution that is a Bessel

function of zero order. However, the microphone is a strongly coupled system, and its internal

behaviour is strongly controlled by the damping of the air film between the backplate and the


116

diaphragm. This coupling may modify the actual movement of the diaphragm around the resonance

in an unknown manner. Different attempts to solve this problem ([13], [14], [17], [18], and [19]) have

not yielded a satisfactory solution. This lack of knowledge may lead to some deviations of the

estimated load of the radiation impedance with respect to the actual load.

It can be seen in figure 4.21 that the difference between the uniform and the Bessel function

displacement distribution is very small up to 8 kHz. It increases afterwards up to a maximum about

15 kHz, and then decreases again. The interval where the difference is maximum does not coincide

with the interval where the difference between the experimental and calculated is larger.

The last factor to consider is the shape of the sound field in which the microphone is

immersed. The shape of the sound field may have a significant effect on the diffraction factor.

Although the definition of the free-field sensitivity [26] is based on the assumption that the

microphone is subjected to a plane wave, in the practical realisation some deviations from the plane

wave may be observed. This is implicitly introduced in the concept of the acoustic centre, which

corresponds to a point source that substitutes the microphone. As the measurements may be made

at a distance where the shape of the spherical front may not be yet that of a plane wave, a deviation

of the diffraction factor from the plane wave case may be observed. This can be verified by means

of the BEM simulation of the diffraction problem under two conditions: when the incident wave

is a plane wave and when it is a spherical wave. Figure 4.20 shows the diffraction factor obtained

under the two conditions, and the difference between them.

It can be seen in figure 4.20 that the diffraction factors have almost the same value in the frequency

range considered. Thus, the difference between the diffraction factors (secondary axis) is very small

and almost negligible in the most of the frequency range. This indicates that the chosen distances

are long enough as to assure that at the microphones diaphragm the wavefronts are approximately

plane. Thus, it cannot be considered the cause of large deviations in the free-field correction, at least

not in the frequency range considered.


117

100

101

0

2

4

6

8

10

12D

iffra

ctio

n fa

ctor

(dB

)

100

101

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Frequency (kHz)

Diff

eren

ce (

dB)

Figure 4.20 Modulus of the diffraction factor determined when the incident wave is: ——— a plane

wave, and — — — a point source located at 36 cm from the diaphragm. The two curves overlap each

other in the whole frequency range. The difference between the two is represented by the curve with the

dash-dotted line (—C—C—), the values of the difference can be read in the secondary axis, to the right.

The above results indicate that the most probable cause of differences in the free-field

correction may be caused by the radiation impedance. This is a function of the actual displacement

distribution of the diaphragm, which is unknown.

On the other hand, the phase of the free-field correction has not been studied in detail.

Some estimations based on measurements on scale models of the microphone [49] are available.

However, this study does not contain enough detail to address these results to a particular type of

microphone. In any case, these results are a reference for further analysis.


118

100

101

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Frequency (kHz)

Mod

ulus

(dB

)

Figure 4.21 Modulus of the load of the radiation impedance when the diaphragm of the microphone

is assumed to have a velocity distribution of: ——— Zero order Bessel function, and — — —

uniform velocity distribution.

As there are no other experimental measurements of the phase of LS1 microphones than

the ones shown here, the phase of the experimental free-field correction can be compared with the

phase of the free-field correction obtained from the BEM simulation. This comparison is not

straightforward because the use of the acoustic centres modifies the actual phase of the correction.

Thus, for a direct comparison, either the phase of free-field correction calculated from the BEM

should be modified by the acoustic centre, or the experimental phase should be obtained using no

acoustic centres but considering the distance between the diaphragms only. The second option is

considered here. Thus, the free-field correction is recalculated using the physical distance between

microphones. Figure 4.22 shows the comparison between the calculated and measured phase for

LS1 microphones. The measured phase is the average of all the microphones measured.


119

1000 5000 10000−10

−5

0

5

10

15

20

25

30

35

40

Frequency (Hz)

Pha

se (

°)

Figure 4.22 The phase of the free-field correction for LS1 microphones: ——— average from

experimental measurements, — C — C — calculated using the BEM, — — — contribution from

the load of the radiation impedance, and C C C C C C C contribution form the diffraction factor.

The phase seems to have a value which is of the same order of magnitude as the data shown

in reference [49]. The agreement between the calculated and measured phase confirms that the

measurement and the subsequent cleaning procedure effectively removes the disturbances without

modifying the electrical transfer impedance except at low frequencies. It can be seen that the

agreement between the calculated and experimental phases is actually good up to 7 kHz. The

difference increases above that frequency. This is in correspondence with the differences observed

in the modulus of the free-field correction, and they may be explained with the same arguments.

Again, the most important contributor is the diffraction factor, although at the frequencies around

the resonance frequency of the microphone the effect of the load of the radiation impedance may

be slightly different, explaining thus such differences.


120

4.4 Other experimental results

Measurements of LS1 microphones was not the only aspect of calibration studied. If the

calibration of LS1 microphones is implemented and functional, it seems quite natural to move

forward to the calibration of LS2 microphones.

Though the same experimental set-up can be used with very small changes in the mechanical

mounting, some problems may arise in the electrical side of the measurements. The reason is more

or less obvious: the sensitivity of a LS2 microphone is about 12 dB lower than the LS1's. This

immediately suggests that for the same level of input voltage, the output of the receiver microphone

could be 24 dB lower when compared with the LS1 case. Thus, the signal to noise ratio becomes

lower, and the cross talk becomes more significant. This is not a trivial problem, and huge efforts

have to be applied in order to solve it. In the following, an account of the measurement results

obtained of LS2 microphones is given.

Some of the solutions implied the use of microphones of different dimensions whose signal

level could help to minimise the cross talk. Thus, combinations of LS1 and LS2 microphones were

tried. The results are presented below.

4.4.1 Calibration of LS2P microphones

4.4.1.1 Preamplifier mounting

In normal applications, the effect of the mounting jig used for placing the transducers in the

interior of the anechoic chamber may be regarded as negligible. This may not be the case when the

reciprocity calibration is performed. However, the standard [50] does not give a clear indication of

the mounting jig. Furthermore, this standard gives a definition of the free-field sensitivity only in

terms of the microphone:

“For a sinusoidal plane progressive wave of given frequency, for a specified direction of sound incidence, and

for given environmental conditions, the quotient of the open-circuit voltage of the microphone by the sound

pressure that would exist at the position of the acoustic centre of the microphone in the absence of the


121

microphone. This quotient is a complex quantity, but when phase information is of no interest, the free-field

sensitivity may denote its modulus only.”

The standard [26] is more explicit giving some indications about the mounting itself. The

microphone shall be attached to a cylinder having the same diameter as the nominal diameter.

Additionally, the length of the cylinder shall be long compared to the diameter of the microphone,

at least 10 times as a practical limit, and then gradually tapered. In practice this configuration has

been realised by mounting the preamplifier on a long rod coming from the walls of the chamber;

this can be assumed to be a semi-infinite rod. However, the mounting of the microphone takes

significant relevance when the absolute realisation is to be transferred to other transducers by means

of comparison techniques. A study case is described below.

During the establishment of the measurement set up, a new model of preamplifier was used.

Geometrically, the main difference with older models is the change in the shape: from cylindrical

to tapered. The mounting of the preamplifier on the rod is shown in figure 4.23. Measurements were

made using the same experimental instrumentation but using two Brüel & Kjær 4180 condenser

microphones of the type LS2P. Figure 4.24 shows the Hilbert envelope of the impulse response

obtained from measurements of the electrical transfer impedance made at different distances.

Figure 4.24 shows some of the features already known. There is the cross talk at the

beginning of the impulse response. It can be noticed that it is larger than in the case of LS1

microphones. This is expected because the sensitivity of an LS2 microphone is 20 dB smaller than

the LS1 sensitivity. This means an even lower signal to noise ratio. This is also reflected in the fact

that the amplitude of the cross talk increases as the distance increases.

After the direct wave between the microphones, two disturbances appear to be dominant.

Analysing the impulse response when the distance between the microphones is 160 mm – figure

4.24a – , it can be seen that the latest disturbance appears at t = 1.4 ms. This corresponds to the time

instant where the standing wave should appear. This behaviour repeats at the other distances; it can

also be observed that the amplitude of this disturbance decreases as the distance increases, as it

should be. Therefore, it is possible to conclude that this particular disturbance is indeed the standing

wave between the two microphones.


122

600

12.7

13

.3

9.3

12.7

6.35

LS2 Geometry with new preamplifier

12.7

10

.2

82

1

Figure 4.23 Geometry of the LS2 mounted on the new preamplifier. This is used for

a BEM simulation.

The early disturbance appears at t = 1.1 ms when the distance is 160 mm, at t = 1.2 ms when

d = 200 mm, and t = 1.35 ms when d = 250 mm. This corresponds in fact to the distance travelled

by a reflection from the step of the preamplifier mounting shown in figure 4.23. This is a surprising

effect because the size of the step seems to be small. It is also interesting to note that the amplitude

of the disturbance seems to increase as the distance grows.

In order to verify that the reflection effectively comes form the step of the mounting, the

step was covered by tape in such a way that the mounting did not have such a step. Figure 4.24

shows the taped impulse response, and the untaped impulse response of an electrical transfer

impedance measured at d = 250 mm. It can be seen that the disturbance suspected to be associated

with the step of the preamplifier mounting is not present in the impulse response of the taped case.

This is a confirmation that such disturbance was caused by the step of the mounting. Therefore, it

becomes very important to provide a mounting rod that does not present any sudden change of

section. Even small changes may induce disturbances that have amplitudes which are even more

significant than the standing wave between the microphones or the reflections from the walls.


123

0 1 2 3 4 5 6 7 8

x 10−3

−2

0

2

4

6

8

10x 10

−3 c)

Time (s)

0 1 2 3 4 5 6 7 8

x 10−3

−2

0

2

4

6

8

10x 10

−3 a)

0 1 2 3 4 5 6 7 8

x 10−3

−2

0

2

4

6

8

10x 10

−3 b)

Nor

mal

ised

am

plitu

de

Figure 4.24 Hilbert envelope of the impulse response of the electrical transfer impedance measured

at a distance a) 160 mm, b) 200 mm, and 250 mm.


124

0 1 2 3 4 5 6

x 10−3

−2

0

2

4

6

8

10x 10

−3

Nor

mal

ised

Am

plitu

de

Time (s)

Figure 4.25 Hilbert envelope of the impulse response of an electrical transfer impedance measured at

a distance d = 250 mm. –——— Taped mounting step, and C C C C C C untaped mounting.

It is difficult to find the actual shape of the effect of this disturbance on the frequency

response. This is due to the fact that the cross talk has a large effect and all disturbances are

superimposed on the cross talk. It is also difficult to separate the disturbance of the direct wave

because they are very close indeed. It should be expected that the amplitude of the reflection from

the step should be very small at low frequencies where the wavelength is large compared to the size

of the step. As the wavelength becomes comparable with the size of the step, the amplitude of the

reflection should increase. In order to prove the above, a simulation exercise was carried out.


125

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Nor

mal

ised

mod

ulus

(dB

)

Frequency (Hz)

Figure 4.26 Modulus of the sound pressure obtained from the BEM simulation using the geometries

shown in figures 4.17 and 4.23 of the mounting set-up. The sound pressure has been corrected for the

frequency and the diffraction factor associated with the geometry: ——— without the step on the

preamplifier mounting, and — . — . — with the step on the mounting.

The mounting set-up was simulated using the axisymmetric BEM. The movement of the

diaphragm was supposed to have a uniform velocity distribution. The sound pressure is obtained

using the geometries shown in figures 4.17 (without the step on the mounting) and 4.23 at a given

distance is shown in figure 4.26. The sound pressure has been corrected for the frequency and for

the diffraction factor of the corresponding geometry.

It can be seen that a disturbance is present in the case of the sound pressure calculated using

the geometry with the step on the mounting of the preamplifier all over the frequency range; this

disturbance thus will be present at the actual measurements of the sound pressure as an additional

reflection with an effect that can be comparable with the reflections from the walls. This proves the


126

importance of providing a mounting that has a constant section of equal diameter as the

microphone. Additionally, it also proves the convenience of handling the information of the

measurements in the time domain.

4.4.1.2 The impulse response of the electrical transfer impedance between two LS2 microphones

The impulse response of the electrical transfer impedance between two LS2 microphone is

slightly different from that of the LS1 microphone – it is shorter and has a resonance frequency

about two times the LS1 resonance frequency –. On the other hand, it is not easier to separate the

direct wave between microphones from the cross-talk and the standing wave between diaphragms

and wall reflections. The main reason is that the sensitivity of LS2 microphones is about 12 dB lower

than the sensitivity the LS1 microphones. This implies that the electrical transfer impedance is about

24 dB lower than the LS1 case at the same distance. Furthermore, the input voltage of the

transmitter microphone is about the same level in the two cases, but the output voltage of the

receiver will be 24 dB lower. This aggravates the cross-talk problem. On the other hand, the

measurements are made at shorter distances in order to increase the signal to noise ratio. The result

of these two facts is that a) the cross talk is larger, and it occurs closer to the direct wave between

the microphones, and b) the standing waves and reflections from the walls are closer to the direct

wave.

Measurements of the electrical transfer impedance between two LS2 microphones were

carried out using the same experimental set-up as with the LS1 microphones. The major difference

is that the frequency range is now from 1.8 kHz to 40 kHz and the frequency step is 40 Hz instead

of 30 Hz. The measurement distances were also different. The measurements were made at four

distances: 160 mm, 200 mm, 250 mm, and 320 mm.

The analysis is very similar to the case of LS1 microphones. First, a measurement of the

electrical transfer impedance measured when the microphones are located at 160 mm from each

other is analysed. Figure 4.27 shows the procedure followed for completing the frequency response

function.

As in the LS1 case, it can be seen in figure 4.27a that the dynamic range is about 40 dB,


127

though the value of the electrical transfer impedance at the high frequency extreme is only about 20

dB lower than its maximum value. This is because the resonance frequency of the LS2 microphones

is about 20 kHz, and thus, the maximum frequency is less than two times the resonance frequency.

This may introduce some modulation of the impulse response with the low-pass filter. A

consequence of this modulation is that the estimated impulse response has a slightly longer duration.

The electrical transfer impedance is smoother than in the LS1 case. This could be caused by the fact

that the higher resonances of the diaphragm of the microphone have not been reached at the

measured frequency range. The second radial resonance of the diaphragm occurs above the highest

measured frequency.


128

102

103

104

−40

−20

0

20

40a)

Mod

ulus

(dB

)

102

103

104

−40

−20

0

20

40b)

102

103

104

−40

−20

0

20

40c)

Frequency (Hz)Figure 4.27 Modulus of the electrical transfer impedance or frequency response between two

microphones located at a distance of 28 cm in the free field. a) measured, b) extended, and c)

low-pass filtered.


129

0 1 2 3 4 5 6

x 10−3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3

Nor

mal

ised

Am

plitu

de

Time (s)

b)

0 1 2 3 4 5 6

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1a)

Figure 4.28 Impulse response obtained from experimental data. (a) is the normalised amplitude

and the time window, (b) is the same as (a) but in a amplitude range that allows to see the

disturbances. In the two figures: ––––– Impulse response, and – – – – – Hilbert envelope of the

ideal impulse response.


130

In figure 4.27b the extended frequency response function is shown. As in the case of the

LS1, there may be a discontinuity at the patching frequency. But the patching also is made at a

frequency where the modulus of the electrical transfer impedance is 40 dB lower than its maximum

value. Thus, the discontinuity may not have a large effect on the impulse response.

In figure 4.27c the low-pass filtered frequency response is shown. Although in this case the

roll-off frequency of the filter is above the resonance frequency, it may not be enough for avoiding

widening the impulse response. Thus, it is convenient to measure the frequency response at higher

frequencies. Unfortunately, the analyser used in the measurements has a high frequency limit of 40

kHz, and therefore it was not possible to measure frequencies above this limit.

The inverse Fourier transform is then applied to the completed frequency response in order

to obtain the impulse response. The normalised impulse response is shown in figure 4.28.

Considering that the distance between diaphragms is 160 mm (disregarding the acoustic centres),

and that the speed of sound is about 345 m/s, the time it takes the wave to travel the above distance

is about 0.46 ms, which is about the instant where the impulse response between the microphones

appears to rise. Actually, the rising of the impulse response starts a little earlier, but it may be a

consequence of the fact that the roll-off frequency of the low-pass filter is not above 2 times the

resonance frequency, as described in section 3.4. Furthermore, the impulse response reaches its

maximum at about 0.5 ms, and then it decays at 1/1000th of its maximum at 0.8 ms. According to

section 3.4, the decaying time should be given by the time constant which for a LS2 microphone is

about three times the time constant of a LS1 microphone. For the parameters given in table 3.1, the

time constant for a LS2 microphone is 1.25E-5 s. Thus, the impulse response should have decayed

to one thousandth of its maximum amplitude about 0.18 ms after the peak. It is apparent that in this

case, such reduction is reached at 0.8 ms, slightly later. It can be a consequence of the low roll-off

frequency of the low-pass filter. However, the time constant provides a reliable criterion for

determining the length of the time selective window.

The cross-talk, standing wave and reflections from the walls can easily be identified in the

impulse response. Cross-talk occurs at zero time, as expected. However, the size of the disturbance

is substantially larger than in the LS1 case (see figure 3.14). This is an expected result. Additionally,

the first disturbance after the direct wave between the microphones is the standing wave between


131

the microphones. It appears at the expected instant, about 1.4 ms, which corresponds to the time

it takes to travel 3 times the distance between microphones. Reflections from the walls come at a

later instant and have a smaller amplitude. In the figure it is clear that the dominant contaminating

agent is the cross-talk. The standing wave between the microphones and the reflections from the

walls are much more smaller. Thus, it is expected that the contribution of the first is larger. This can

be seen after a time selective window is applied.

If the disturbances are to be removed, a time selective window can be applied to the impulse

response. The time window applied is 0.4 ms long, and it is centred around the maximum of the

impulse response. Figure 4.29 shows the impulse response and the time selective window.

As well as in the LS1 case, the frequency interval can be divided into three different regions

according the effect of the different perturbations, although it is evident that the cross-talk is the

dominant effect all over the frequency range. It is then of the utmost importance to eliminate it from

the measurement system, especially if LS2 and less sensitive microphones are to be calibrated using

the reciprocity technique.

4.4.1.3 Determination of the absolute free-field sensitivity of LS2 microphones

Once the imperfections caused by the cross-talk, standing wave between microphones, and

the reflections from the walls have been removed from the electrical transfer impedance, it is

possible to determine the acoustic centres of the microphones and the free-field sensitivity.

Although it was recognised that the cross-talk problem posed a significant problem for the accurate

calibration of the microphones, measurements over a number of LS2 microphones of the type 4180

manufactured by Brüel & Kjær were carried out in order to obtain their free-field sensitivities and

the free-field correction. The results are described below.


132

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01Nor

mal

ised

Am

plitu

de

Time (s)

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 4.29 Impulse response of the electrical transfer impedance and the time selective window:

a) ——— normalised impulse response, and —C—C— time selective window.

b) ——— residuals after the application of the time selective window, C C C C C C C Hilbert envelope

of the impulse response, and —C—C— time selective window.


133

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Mod

ulus

(dB

)

Figure 4.30 Difference between the cleaned and raw electrical transfer impedances.

4.4.1.3.1 Free-field correction

Similarly to the case of LS1 microphones, the information about the pressure sensitivity of

the LS2 microphones calibrated under free field conditions was available, thus it was possible to

7determine the free-field correction. Figure 4.31 shows the free-field correction of six microphones

Brüel & Kjær 4180. The free-field sensitivities were calculated using the acoustic centres shown in

figure 5.9.

First, it should be noticed that the effect of the time selective window is evident at

frequencies below 3 kHz. This is caused by the fact that the window is just 0.4 ms long. This means

that the first lobe is about 2.5 kHz wide. And, as shown in chapter 3, the width of the main and

second lobes is the major limitation when the time selective window is applied.


134

The spread of results in the resulting free-field corrections observed in figure 4.31b has a

maximum value of 0.2 dB in the mid frequency range. This spread decreases slowly as the frequency

decreases. In figure 4.31b can also be seen that the difference between the standardised value of the

free-field correction [47] follows about the same trend. On the other hand, as the frequency

increases, the difference between the measured and standardised correction increases up to a

maximum of 0.2 dB at 12 kHz. This difference is similar to that observed in the case of LS1

microphones. This difference may be caused by the fact that the standardised free-field correction

is a polynomial approximation based on the average of free-field corrections -- provided by a

number of laboratories around the world – at a number of frequencies, specifically at the central

frequencies of third octave bands contained in the frequency interval 1 kHz to 25 kHz. Thus, at low

frequencies, the frequencies are more or less evenly distributed, but the frequencies scatter as they

increase, for example, between 1 and 10 kHz, there are 9 frequency steps, but between 10 and 25

kHz there are only 5 frequency steps. This may explain the large difference observed in the

frequency range from 10 to 25 kHz.

In any case, it is a very interesting difference although there are no means for validating the

obtained results so-far. Traditionally, laboratories make measurements at the central frequencies of

one-third octave bands, thus leaving an empty space in the middle of some frequencies where the

slope of the free-field correction may change rapidly; this is the case of the frequency interval

between 10 and 20 kHz.

A possibility for validating the results is the use of the BEM formulation, as in the case of

the LS1 microphones.

The next feature to analyse is the free-field correction. The free-field correction is obtained

using the same procedure as the LS1 microphones. Thus, it will not be repeated here. Figure 4.33

shows the modulus of the free-field correction of experimental data together with standard data [47],

and data calculated with the axisymmetrical BEM formulation.


135

2 5 10 200

2

4

6

8

10M

odul

us (

dB)

2 5 10 20 30−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency (kHz)

Diff

eren

ce (

dB)

Figure 4.31 Free-field correction of a number of LS2 microphones. The sensitivity was calculated using

the acoustic centres shown in figure 5.9: a) modulus of the calculated free-field corrections, and b)

Differences with respect to the average. In figures a) and b): ——— Experimental measurements, and

— C — C — Data from [47].

It can be seen that at frequencies below 18 kHz the agreement between the simulation and

the average value from the measurements is better than the agreement between the IEC values [47]

and the measurements. However, the simulation appears to break down above that frequency. It is

apparent that the contribution of the load of the radiation impedance becomes significant above

these frequencies. Figure 4.32 shows the contributions of the diffraction and the load of the

radiation impedance for the LS2 case.


136

103

104

−2

0

2

4

6

8

10

12

Frequency (Hz)

Mod

ulus

(dB

)

Figure 4.32 Contributions to the modulus of the free-field correction of the load of the radiation

impedance and the diffraction factor to the free-field correction: ——— total free-field correction,

— C — C — load of the radiation impedance, and — — — diffraction factor.

As mentioned above, the load of the radiation impedance has a very similar shape, but the

LS2 case is shifted towards a higher frequency. This can be explained by the fact that the impedance

of the LS2 microphone is not scaled in the same proportion as the geometry when compared to the

LS1 microphone. The resonance frequency of a LS1 microphone is about 8.5 kHz while the

resonance of the LS2 is about 23 kHz. This explains why the radiation load has its maximum

contribution at such higher frequency in the LS2 case. This larger effect occurs at the frequency

range where the diffraction factor reaches its maximum, thus flattening the total free-field

correction. This explains partially why the free-field correction of the LS2 microphone is not as large


137

2000 5000 10000 200000

1

2

3

4

5

6

7

8

9

10

Frequency (Hz)

Mod

ulus

(dB

)

Figure 4.33 Modulus of the free-field correction for LS2 microphones: ——— Average from

experimental measurements, — C — C — Polynomial approximation from [47], and — — —

calculated using the axisymmetrical BEM.

as the LS1 correction.

Another difference is that between the diffraction factors also observed in figure 4.34. This

difference is not only in the maximum value reached by the modulus but also at the frequency where

the maxima occur. It is known that geometrically the LS2 microphones are more or less scaled

versions of the LS1 microphones except in one feature, which is the cavity depth: 1.95 mm for LS1

and 0.5 mm for the LS2. This may be the cause of such difference. Experimentally, the only

possibility for proving this is to manufacture either a LS1 microphone with a cavity depth of 1 mm,

or a LS2 microphone with a cavity depth of 1 mm. As it may be difficult to convince the


5Formally speaking, these modified microphones would not belong to the category of laboratory standard microphones,labelled LS, because the standard [47] specifies that LS1 microphones shall have a nominal cavity depth of 1.95 mm,while for LS2 this is 0.5 mm.

138

0.2 0.5 1 2 3−2

0

2

4

6

8

10

12

ka

Mod

ulus

(dB

)

Figure 4.34 Comparison between the contributions of the load of the radiation impedance and the

diffraction factor to the free-field correction for LS1 and LS2 microphones: ——— LS1 microphones,

and — — — LS2 microphones.

manufacturers to do so5, in order to prove this difference, a simulation using the BEM formulation

was made considering the geometry of a LS1 microphone but modifying the cavity depth to 1 mm.

The diffraction factor and the load of the radiation impedance of the LS2 case are compared with

those of the modified LS1 microphone in figure 4.35.

It can be seen in figure 4.35 that by making this small change of the cavity depth, the

diffraction factor changes dramatically in modulus, and in the frequency where the maximum occurs.

The difference between the maximum with the standard cavity and the modified cavity is 1.2 dB.


139

0.2 0.5 1 2 3−2

0

2

4

6

8

10

12

ka

Mod

ulus

(dB

)

Figure 4.35 Comparison of the contributions of the load of the radiation impedance and the diffraction

factor for the LS2 and the modified LS1 with a shorter front cavity: — C — C — standard LS1

microphone, — — — LS2 microphone, and ——— modified LS1 microphone.

And the frequency where the maximum of the modified version shifts to a higher frequency, and

it seems to coincide with the diffraction factor of the LS2 microphone. The difference between the

modulus of the diffraction factor of the LS2 and the modified LS1 is of the order of 0.4 dB. This

can be explained by the fact that the diameter of the LS2 is not exactly half of the diameter of the

LS1 but slightly more: 23.77/12.7 = 0.534. Additionally, the actual geometry of the LS2 has a small

change of the diameter about 10 mm behind the front ring.

These changes may be explained in terms of the modal behaviour inside the cavity. The

change in the length of the cavity will shift the eigen frequencies of the longitudinal, axisymmetric

and three-dimensional modes of the cavity to higher frequencies, thus modifying the actual

contribution of such modes to the global effect of the diffraction.


140

It is also observed a small change in the modulus of the radiation impedance about the

resonance frequency of the LS1. This is related with the changes of the diffraction factor as well and

it could be explained in a similar manner.

So far the results of the modulus of the free-field correction have been considered. In the

following the phase of the free-field correction is analysed. As well as in the case of the LS1

microphones, to the author’s knowledge no results exist other than the guide given in [49]. Thus,

the results of the BEM axisymmetrical formulation will be used for assessing the phase of the free-

field correction. For a direct comparison the experimental phase is calculated without using acoustic

centres but considering the distance between the diaphragms only. Figure 4.36 shows a comparison

between the calculated and measured phase for LS2 microphones. The measured phase is the

average of all the measured microphones.

The dominant contributor to the phase of the free-field correction is the diffraction factor,

but the load of the radiation impedance plays an important role around the resonance frequency of

the microphone. The agreement between the calculated and experimental phases is between 0.4E

in the most of the frequency range, though it reaches a maximum of 0.6E around 10 kHz. As well

as in the case of the LS1, the agreement between the calculated and measured phase confirms that

the measurement and the subsequent cleaning procedure effectively removes the disturbances

without modifying the electrical transfer impedance except at low frequencies. It is interesting to

note that the agreement in the phase does not exactly correspond to the modulus, where the

deviations from the BEM results are larger above 20 kHz. It may indicate that the modulus of the

load of the radiation impedance is slightly overestimated although the relation between real and

imaginary part remains constant.

4.4.2 Combination of LS1 and LS2 microphones

As the cross talk poses a significant obstacle for measurement of the LS2 microphones, a

solution that minimises the problem, or ideally removes it, is needed. One possibility is to combine

LS1 and LS2 microphones. This solution was studied, and some preliminary results are shown

below. It is expected that the characteristics of the impulse response are different from a

combination of two LS2 or LS1 microphones because the difference in resonance frequency and


141

103

104

−20

−10

0

10

20

30

40

Frequency (Hz)

Pha

se (

°)

Figure 4.36 Phase of the free-field correction of LS2 microphones: ——— Average of the calibrated

microphones, — C — C — results from the BEM calculations, — — — contribution of the load

of the radiation impedance, and C C C C C contribution of the diffraction factor.

time constant between the two types of microphone. Thus, the resulting impulse response as a result

of the convolution of the two impulse responses should have a modulated frequency. The

modulation should, however, be of short duration because of the combination of the two highly

damped systems.

For the experiments a triad of microphones composed of a one inch free-field microphone

Brüel & Kjær model 4145 and two LS2 microphones Brüel & Kjær model 4180 was used. The

microphone 4145 was used solely as transmitter because it has a flat frequency response that reaches

higher frequencies. The frequency interval and frequency steps used are the same as in the case of

only LS2 microphones. In the following, only features that deviate significantly from those

presented in the preceding sections will be described.


142

4.4.2.1 The impulse response of the 4145 – 4180 microphone combination

Once the incomplete electrical transfer impedance has been processed as described in

chapter 3, the inverse Fourier transform can be applied for obtaining the impulse response. This

impulse response is shown in figure 4.38.

The impulse response can be compared with that shown in figure 34 that corresponds to a

pair of LS2 microphones located at the same distance. It can be seen that the cross-talk is effectively

reduced by about 6 dB. And the standing wave between the microphones is significantly larger. This

is caused by the fact that the microphone 4145 presents a larger surface for the reflections than the

4180 microphone. It should be of smaller amplitude if two 4145 microphones were located at the

same distance however. As the length of the impulse response and the location of the disturbances

is very similar to the case of two LS2 microphones, the same time selective window is used for

isolating the direct impulse response from the disturbances. This time selective window has already

been shown in figure 4.29. Figure 4.37 shows the difference between the cleaned and the raw

electrical transfer impedance between the 4145 and 4180 microphones. This figure can be compared

with figure 4.30, which shows the same difference for a combination of two LS2 microphones.

As expected, the contribution of the cross-talk is significantly smaller than in the LS2-to-LS2

combination, although it has still a large value at low frequencies. The contribution of the standing

wave is considerable at high frequencies; its maximum is comparable with the contribution of the

cross talk at low frequencies. However, it appears that the two disturbances have been removed

effectively from the electrical transfer impedance. Thus, it is possible to use the cleaned functions

for calculating the free-field sensitivity of the microphones.

The first result obtained from the electrical transfer impedances measured at several

distances is the value of the acoustic centres. These are shown in figure 4.39. It can be seen that the

acoustic centres of the 4180 microphones are in agreement with the standard values. The value of

the 4145 is more or less consistent with some experimental data presented in reference [52]. It

should be expected that the fact of having a better signal to noise ratio may help to obtain a better

estimation of the acoustic centres of the 4180 microphones. However, as there was only one


143

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Mod

ulus

(dB

)

Figure 4.37 Difference between cleaned and raw electrical transfer impedances between a one inch

microphone Brüel & Kjær 4145 and a half inch microphone Brüel & Kjær 4180.

measurement made with this configuration, it cannot be with any certainty concluded that the

combination of 4145 and 4180 helped to get a better estimate.

Finally, the free-field sensitivity is calculated using equation (4.7) and the free-field correction

determined from the difference between free-field and pressure sensitivity. The result is shown in


144

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−1

−0.5

0

0.5

1

Nor

mal

ised

Am

plitu

de

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−5

0

5

x 10−3

Time (s)

Figure 4.38 Normalised impulse response of the electrical transfer impedance between a Brüel & Kjær

4145 free-field microphone and a Brüel & Kjær 4180 microphone: a) the impulse response shown in

full scale, b) the normalised impulse response shown in a fraction of the full scale that allows to detect

the disturbances caused by cross talk, standing waves and reflections from the walls. In the two figures:

— C — C — Hilbert envelope of the impulse response, and ——— impulse response.

figure 4.40. It can be seen that the agreement between the free-field correction of each microphone

and the average estimated from LS2-to-LS2 calibrations is very good in the whole frequency range.

The maximum difference is 0.03 dB and it occurs at low frequencies. This is an important indication

that the combination of microphones of different dimensions yields similar results as combinations

of microphones of the same type. Evidently, more research must be done in this particular

application. But this is encouraging if free-field calibration of quarter inch microphones is to be tried

in the future


145

5000 10000 20000 30000 40000−5

0

5

10

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 4.39 Acoustic centres determined from the modulus of the electrical transfer impedance between

one-inch and half-inch microphones measured at several distances: — C — C — acoustic centres of the

LS2 microphones, ——— acoustic centre of the 4145 microphone, and — ~ — ~ — acoustic centres

from [26].


146

2000 5000 10000 200000

1

2

3

4

5

6

7

8

9

10

Frequency (Hz)

Mod

ulus

(dB

)

Figure 4.40 Comparison of the modulus of the free-field correction of LS2 microphones calculated from

measurements where LS2 microphones were combined with a one inch Brüel & Kjær 4145 microphone:

——— LS2 microphones, — — — BEM calculations, and — C — C — standard data [47].

Summary

In this chapter a number of experiments have been described. First, the cleaning technique

has been applied to an electrical transfer impedance measured at a given distance. The consequences

of the technique have been described as well as the characteristics of the observed disturbances such

as the electrical cross talk, the random noise due to a finite measurement time, the standing wave

between the microphones, and the reflections from the walls.


147

Then, a number of measurements with one inch laboratory standard microphones (LS1) have

been carried out, and the cleaning procedure applied to them. With these cleaned functions the

complex free-field sensitivity, and the free-field correction have been determined. The results were

compared with data obtained from simulations using the BEM axisymmetrical formulation, and with

data proposed by an international standard. Traditionally, the free-field correction includes only

modulus information, and it seems to be in agreement with the measurements. The phase is also

compared with the BEM simulations, and good agreement has been found.

The calibration of half inch laboratory standard microphones (LS2) has also been studied. As

in the case of the LS1 microphones, the free-field sensitivity and free-field correction have been

determined. The experimental results where compared to the simulated data and standardised data.

Good agreement was found. An explanation for some observed differences between the diffraction

factor of LS1 and LS2 microphones have been examined by making use of the BEM formulation.

As the cross talk poses a significant problem in the calibration of LS2 microphones, some

ways of improvement have been tried. This is the combination of LS1 and LS2 microphones. The

results are in good agreement with values obtained when only LS2 microphones are used. This may

be a partial solution to the cross talk problem in the case of the calibration of LS2 microphones.

The agreement between experimental and simulated data indicates that the cleaning

technique effectively removes the disturbances while simultaneously does not modify the measured

functions except at low frequencies, in agreement with the simulations described in chapter 3.


148

Chapter 5 Acoustic centres

149

Chapter 5. Acoustic centres

Overview

An analysis of the problem of determining the acoustic centres of condenser microphones

is described in this chapter. Procedures for obtaining the acoustic centres from the modulus of

cleaned electrical transfer impedances and the phase of cleaned free-field sensitivity are outlined. The

convenience of determining the acoustic centres based on the fulfilment of the inverse distance law

(modulus based) and on the phase of the free-field sensitivity is analysed.

5.1 The determination of the acoustic centres

It can be seen that in equation (4.7) there is a parameter with great influence on the final

free-field sensitivity. This parameter is the true acoustic distance between the microphones labelled

as x and y, dxy. This is a function of the physical distance between the microphones and their acoustic

centres , where dx and dy are the acoustic centres of microphone x and yd d d dxy x y= + +

respectively.

The acoustic centre of a microphone is defined in [26] as follows:

“For a sound emitting transducer, for a sinusoidal signal of given frequency and for a specified direction and

distance, the point from which the approximately spherical wavefronts, as observed in a small region around

the observation point, appear to diverge”

In the case of a reciprocal transducer, the acoustic centre when used as receiver is the same

as when used as transmitter. This equivalence is explained by the fact that the diffracted field is

equivalent to the radiated field when the radiating/diffracting object is acting in either condition (see

reference [51]).

Although the concept of acoustic centre has been closely linked to the conceptual

development and the realisation of the free-field calibration of microphones, there are only few

attempts for predicting the behaviour of this quantity in the literature. In an early work Rasmussen


150

[52] tried to explain the expected behaviour of the acoustic centres departing from the determination

of different sources that generate a sound field that can be known analytically. This theoretical

speculation was supplemented with experimental measurements.

The prediction of the acoustic centre position requires a precise knowledge of the sound

field generated by the microphone. This in turn demands a precise knowledge of the displacement

distribution of the diaphragm of the microphone and its geometry. There have been some attempts

for solving the sound field when a microphone is introduced in the propagation path of a plane

wave (see references [13], [14], [15], [16], [18], and [53]). However, all these works have been

focussed on obtaining the free-field correction of the microphone, except reference [15] that

provided values of the acoustic centres of LS1 and LS2 microphones.

5.1.1. Determination of the acoustic centres based on the modulus of the electrical transfer

impedance

The experimental determination of the acoustic centre can be carried out using simple linear

regression techniques if measurements of the electrical transfer impedance are made at several

distances. This procedure can be used when there is confidence that the characteristics of the air in

the environment where the measurements are made can be calculated using well defined or standard

procedures.

According to equation (2.44), the sound pressure generated by a transmitter microphone

decreases as a function of distance. If an ideal receiver that does not disturb the propagating wave

is located at a given distance, the output voltage will be inversely proportional to the true distance,

. It can also be stated vice versa, . This proportionality can be converted to anu dt∝ 1 1 u d t∝

equality by introducing a proportionality constant, m, in the right side . Furthermore, the1 u mdt=

true distance is the sum of the physical distance, d, and the acoustic centre, dt, of the transducer,

. This linear equation can be solved if a sufficient output( )1 u m d d md md md bx x= + = + = +

voltages are measured at given distances using the least squares method. Thus, the value of the

acoustic centre can be determined by dividing the independent term, b, by the slope, m.


151

Z eAde

d

t, ,12

α ≈ (5.2)

1

12Z edA

ed

t

,

,α

≈ (5.3)

( )1 1

121 2Z e A

d d md be

d,

.α

≈ + = ++ (5.4)

ZAdee

t

dt, ,12 = −α (5.1)

A realistic implementation of the above procedure must take into account the air absorption

and the fact that the receiver will also have an acoustic centre. Furthermore, the electrical transfer

impedance can be measured at several distances and be used for the calculations.

The modulus of the electrical transfer impedance can be expressed as

where A is a constant, dt is the true acoustic distance, and α is the air absorption. The true distance

is the sum of the physical distance between the microphones, d, and the sum of the acoustic centres

of the microphones, d1+2. It can be assumed that the d is large compared with d1+2, in such a way that

d is approximately equal to dt. Thus, the exponential factor in equation (5.1) can be eliminated from

the right side of the equation by multiplying both sides of the equation by eαd. This yields

The inverse dependence of the distance can be reverted if the inverse of both sides is calculated

Substituting the true distance by its two components, and rearranging terms

Thus, the sum of the acoustic centres can be obtained by dividing the independent term by the

slope. The simplicity of the procedure is based on three assumptions. One is that the observation

distance is long compared to the sum of the acoustic centres. Under normal conditions, distances

range from 250 mm to 500 mm. According to values given in the standard [26], the acoustic centres

are about 9 mm at low frequencies. Thus, this assumption can be considered to be fulfilled. The

second assumption is that the physical properties of the air inside the anechoic chamber can be

calculated. The third assumption is that in the range of observation distances where the acoustic

centres are determined, the acoustic centre is independent of the observation distance [52].

An alternative approach was suggested by Hruska and Koidan [54]. There, a procedure for


152

( ) ( ) ( )ln ln ln .,Z d A d ddde x yx y

12 1= − + − −

+

+α (5.5)

( ) ( )[ ]ln ln .,Z d d A ddde x yx y

12 = − + − +++

α α (5.6)

( ) ( )[ ]ln ln .,Z d d d A d d de x y x y122= − + − ++ +α α (5.7)

obtaining simultaneously the acoustic centres was described. Instead of inverting equation (5.1), the

two sides of the equation are multiplied by the physical distance, the natural logarithm is applied

If the physical distance is large compared to the sum of the acoustic centres of the microphones,

d>dx+y the third term on the right can be expanded into a Taylor series

( )ln ...12 3

2 3

+ = − + −z z z z

Normally, d is much larger that dx+y, thus the higher order terms can be neglected, and the expansion

can be substituted in equation (5.5). Substituting and rearranging

In order to eliminate the inverse dependence on d equation (5.6) is multiplied by d

The right side of equation (5.7) is a quadratic expression whose coefficients can be determined by

a curve fitting technique. Thus, the coefficient of the quadratic term is the acoustic air absorption,

and the independent term is the sum of the acoustic centres of the microphones. This method for

obtaining the two quantities can be quite useful when the air attenuation is suspected of being

anomalous.

5.1.2 Determination of acoustic centre using phase measurements

The definition of the acoustic centre states that this point in space is the origin of the

spherical source that substitutes the microphone. It implies that the phase at the acoustic centre

should be zero. Thus a fully coherent acoustic centre is that point generating spherical waves with

zero delay.

Except for Rasmussen’s report [52] and Vorländer’s paper [34], there is no mention of the

use of phase measurements for the determination of the acoustic centre of the microphone in the

literature, although Trott [55] and Ando [56], [57] have used the phase for determining the acoustic


153

( )[ ]τω

θ ωgdd

= − , (5.8)

( )τ

θ ωωp = − . (5.9)

centre for other devices. Furthermore, if the determination of the acoustic centre is based on

modulus measurements, it is necessary to carry out the measurement of the electrical transfer

impedance at several distances in order to obtain a better estimate of the acoustic centres. Vorländer

suggested that the acoustic centres could be calculated from phase measurements making use of

measurements at just one distance. The method is based on the calculation of the group delay, τg.

This quantity is defined (see reference [38]) as the negative derivative of the phase response of a

frequency response

where θ(ω) is the phase of the frequency response. Thus, a group delay equal to zero indicates a

constant phase response, while a constant group delay indicates a linear phase response. Thus, if

there is any distortion or deviation from these recognisable behaviours in the phase response, the

group delay will indicate it.

Vorländer [34] calculated the group delay of the phase response of the absolute free-field

sensitivity and showed some results relating directly the acoustic centre with the group delay. Such

relation seems to be a direct multiplication of the group delay in time units with the sound speed.

However, this direct relationship between the group delay and the acoustic centre is difficult to

prove. Unfortunately, Vorländer does not describe his procedure with sufficient detail, but it is

apparent that he calculated the complex free-field sensitivity using the distance between the

diaphragms of the microphones, and then, the group delay of its phase. And this group delay was

related directly with the position of the acoustic centre of the microphone; no comparison with

values measured with other techniques is given.

However, a closer examination of the problem may indicate that it is the phase delay [38]

that is appropriate for describing the problem. The phase delay is given by

The response of a system when the envelope function of an input varies( )y t ( ) ( )f t y t t= cosω0

slowly, is [38]


154

( ) ( ) ( ) ( )g t A y t tg p= − −ω τ ω τ0 0cos . (5.10)

Thus, the phase delay expresses the phase response as a time delay, that is, the phase delay will give

the time delay that a sinusoidal component of the signal will experience when passing through the

system. On the other hand, the group delay can be interpreted as a time delay of the envelope

function of a narrow band signal centred at a given frequency. If the phase is proportional to the

frequency, these two quantities are identical. When the phase is not proportional to the frequency,

the phase delay can still be considered as a time delay, but the group delay will represent a phase

distortion. This distortion can be related to dispersive processes, where the propagation velocity is

different at different frequencies. This is not the case of sound propagation in homogeneous media.

Thus, as the acoustic centre is a static point in front of the microphone diaphragm that will delay

the sinusoidal component of the signal, it is correct to use the phase delay instead of the group delay

as a mean for determining the acoustic centre of the microphone based on phase measurements.

5.2. Experimental results obtained from modulus measurements

First the calculation of three microphones is to be analysed. The electrical transfer

impedance of the three pairs of coupled microphones in the free field have been measured at four

different distances. The environmental conditions are measured during the measurement of the

electrical transfer impedances, and a record is kept together with the measurement file. Once the

measured electrical transfer impedance has been cleaned, the acoustic centres are determined using

the linear regression procedure. Figure 5.1 shows the acoustic centres of the three microphones.

It can be seen that the acoustic centres follow the shape of the values of the IEC standard

in most of the frequency range, but above 20 kHz the acoustic centres diverge to a positive value.

The acoustic centre reaches a maximum about 26 kHz, and then they decrease again. As the

frequency increases, the behaviour of the acoustic centres becomes more difficult to assess. This

behaviour was observed also in measurements of the PTB laboratory in the intercomparison [46,]

although measurements were made only up to 25 kHz.. A difference of 5 mm in a measurement

distance of 500 mm would indicate a difference of about 0.08 dB in the modulus of the calculated

sensitivity. Therefore, whether the calculated value is correct is important.


155

103

104

−5

0

5

10

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 5.1 Acoustic centres of a set of condenser microphones: ———— Microphone s/n 1453784,

— — — microphone s/n 1453798, C C C C C C microphone s/n 1792671, and — ~—~— IEC

standard [26].

The frequency where the maximum occurs coincides with the frequency of the second radial

resonance of the diaphragm. This suggests that the deviation may be related to the actual

displacement of the diaphragm.

The use of a modal sum of Bessel functions for describing the movement of the diaphragm

has been used extensively ([13], [14], [18], [16], and [17]). This has yielded some good

approximations in the calculation of the free-field correction, but no calculation of the acoustic

centres based on these results have been made. It was mentioned above that Juhl [15] presented

some results of the acoustic centres numerically calculated using the BEM. However, his low

frequency approximation was based on the widely accepted assumption that at low frequencies –

below the resonance of the microphone – the movement of the diaphragm can be approximated by


156

( ) ( )( )

η ηrJ KrJ Ka

= −

0

001 , (5.11)

a parabolic function; just as the shape of the Bessel function of zero order at its first zero crossing,

or the first radial resonance of the diaphragm. Under this assumption he calculated the acoustic

centres up to 10 kHz, which is certainly a low frequency for comparing with the experimental results.

Obviously, this assumption would break down at higher frequencies where the experimental results

show significant deviations from the theoretical values.

It was decided to used the same BEM formulation as Juhl [15], already used for calculating

the free-field correction in chapter 4, and assuming that the diaphragm displacement, η, under

uniform pressure conditions is

where J0 is the Bessel function of zero order, η0 is a constant that defines the amplitude of the

movement, K is the wave number of the diaphragm, a is the radius of the diaphragm, and r is the

radial coordinate. It is expected that this displacement distribution may represent the actual

distribution, thus yielding a higher frequency approximation. Additionally, a constant displacement

distribution was also considered. The geometry used in the simulation is shown in figure 4.17. The

semi-infinite rod was simulated having a length of 60 cm. This would introduce a small disturbance

in the simulated results because of the reflections from the back of the rod. However, it is expected

that the amplitude of such disturbances is small. The frequency range used in the calculations is 1

kHz - 32 kHz. The size of the smallest element in the axisymmetric mesh is 2.5 mm. Thus, there will

be at least 4 elements per wavelength at the highest calculation frequency. The results were

calculated using the linear regression calculation procedure in the distance range from 30 cm to 60

cm, which corresponds to the experimental measurement distances. Figure 5.2 shows the calculated

results.

It can be seen that at low frequencies the three assumed shapes of the displacement follow

the standardised value, although the uniform distribution yields a slightly smaller value. The Bessel-

like and the parabolic distribution yield the same value, as expected. Above 15 kHz the two

predictions diverge. At this frequency, it is very likely that the assumed parabolic displacement

distribution deviates from the actual distribution, thus it will yield results that do not correspond to

the actual ones. Therefore, the parabolic distribution will not be used for further comparisons. The


6 The sharp peaks observed in the acoustic centres calculated using the Bessel displacement distribution could beconsidered to be contaminated by the non-uniqueness problem – see footnote 4, page 111 –. However, this is not thecase because the peaks do not appear at the same frequencies where the condition number is larger. Furthermore,varying the parameters that define the Bessel displacement (such as the stiffness of the diaphragm) causes the peaks tomove to different frequencies, as it should be, bearing no relation to the condition number of the matrix.

157

103

104

−5

0

5

10

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 5.2 Acoustic centres obtained using the axisymmetric BEM formulation for three displacement

distributions —*—*— Piston, –C–C– Parabol, —~—~—~— Bessel, and ——— IEC

standard..

comparison of experimental and calculated results is shown in figure 5.3

It can be seen in figure 5.3 that the high frequency behaviour of the experimental acoustic

centres has some coincidences with the simulated results. First it can be seen that the three curves

have a maximum about the same frequency, which is the second radial resonance of the diaphragm.

It is interesting to observe that at the above frequency, it is actually the uniform distribution

that shows a better agreement with the experimental values6. It may be caused by the fact that at


158

103

104

−5

0

5

10

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 5.3 Comparison among acoustic centres obtained using the axisymmetric BEM formulation and

experimental values: ——— average of 3 microphones, —o—o— Piston, —~—~— Bessel, and

—∗—∗—∗— IEC standard..

such high frequencies, the effect of the air film between the diaphragm and the back plate of the

microphone begins to damp the amplitude of the movement of the diaphragm, perhaps heavily in

the middle of the diaphragm, flattening thus the displacement distribution and making it look more

like the uniform distribution. Another explanation could be that the uniform distribution can be

expressed as a sum of the modes of a Bessel function of zero order; as the movement of the

diaphragm is considered to be axisymmetrical, the Bessel functions of higher order Bessel functions

may be neglected. At frequencies below the resonance frequency this would replicate the sound field

generated by the first mode (zero) of the Bessel function, J0, – with small additions from the higher

modes of J0. These contributions may become significant at frequencies higher than the resonance

frequency. Thus, when the second zero of J0 occurs, the contribution of the higher modes may be

significant enough as to flatten the displacement of the diaphragm. At this point it should be very

interesting to have either numerically calculated or measured figures of the actual velocity


159

distribution of the diaphragm. In this context an investigation that involves the measurement of the

diaphragm displacement by a laser Doppler vibrometer by Behler and Vorländer [58] is promising.

The next consideration is to use the experimental acoustic centres for the calculation of the

free-field sensitivity. Figure 4.12 shows the free-field sensitivities of three microphones calculated

using the experimental acoustic centres. The shown sensitivities are the average of the free-field

sensitivities obtained from the measurements at different distances.

The spread of the sensitivities obtained at different distances may yield additional

information. Figure 5.4 shows the difference between the sensitivities obtained at different distances

for the microphone 1453784 It can be seen that the spread of sensitivities is reduced at high

frequencies when the acoustic centres determined from experimental results is used. It seems to be

a natural consequence of the difference observed at such frequencies between the theoretical and

experimental acoustic centres. At low frequencies, where the difference between theoretical and

experimental acoustic centres is not that large, the spread is almost the same in the two cases

although it appears to be slightly lower when the experimental acoustic centres are used. This may

be an indication that the use of the experimental instead of the theoretical acoustic centres is

recommendable, especially at high frequencies.

The spread of the phase of the microphone sensitivity obtained at different distances is

shown in figure 5.5. It can be seen that the spread of the phase is small, about K0.1E at low

frequencies, while it increases monotonously up to K2E as the frequency increases. This high

frequency spread can be caused either by a small error in the repeated positioning of the

microphones, or by small changes in the temperature that causes a change in the speed of sound.

In both cases, it is expected that the error grows as the wavelength becomes smaller. However, if

the error is caused by a temperature change, it can be calculated according to measurements of the

temperature. If the positioning error is random, it cannot be predicted. In the measurements made

for this project, no systematic behaviour related to the temperature was observed, therefore, it was

assumed that the deviations were caused by small positioning errors. Thus, it is necessary to


160

introduce a positioning aid that may allow to “calibrate” the measurement distance by using a simple

length standard, such as a rod with a given length. It can be also observed that there is no difference

between the spread of the phase of the sensitivity calculated with and without acoustic centres,

although its absolute value changes accordingly.


161

103

104

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1a)

103

104

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)

Diff

eren

ce (

dB)

b)

Diff

eren

ce (

dB)

Figure 5.4 Deviations of the modulus of the sensitivities calculated at several distances from the average

modulus of the sensitivity for microphone Brüel & Kjær 4160, s/n 1453784: a) using the calculated

acoustic centres, and b) using the standardised acoustic centres. – C – C – 250 mm,

— — — 320 mm, ———— 400 mm, and C C C C C C 500 mm.


162

103

104

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2a)

103

104

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Pha

se d

iffer

ence

( °

)

b)

Frequency (Hz)Figure 5.5 Deviations of the phase of the sensitivities calculated at several distances from the average

phase sensitivity for microphone Brüel & Kjær 4160, s/n 1453784: a) using the calculated acoustic

centres, and b) using the standardised acoustic centres. —~—~— 250 mm, — — — 320 mm,

———— 400 mm, and C C C C C C 500 mm.


163

5.3 Experimental acoustic centres obtained from phase measurements

According to section 5.1.2, it is possible to determine the acoustic centre from phase

measurements, or more accurately, from the phase response of the microphone in the free field.

Thus, the first step is to calculate the phase of the free-field sensitivity using equation (4.7). In this

calculation, the distance between the two diaphragms is used. This will give the phase response of

the microphone referred to these surfaces. Then, the phase delay is calculated from the phase

response, and multiplied by the speed of sound. The resulting distance is the acoustic centre of the

microphone. The group delay and its corresponding acoustic centre as suggested by Vorländer[34]

are also calculated. The results are shown in figure 5.6.

The difference between the estimations of the acoustic centre based on the phase delay, and

on the group delay are evident. All but the low frequencies show a disagreement. The low frequency

agreement happens because at these frequencies, the phase may be proportional to the frequency,

a condition that is lost above 1 kHz.

The appropriateness of using the phase delay based acoustic centre instead of the group

delay based acoustic centre as proposed by Vorländer can easily be proved. Whatever choice is

correct, the resulting phase of the free-field sensitivity calculated using the calculated acoustic centres

should be zero. Figure 5.7 shows the phase of the free-field sensitivity when the two approximations

are used; the scale difference should be noticed. From analysing the figures, it is clear that when the

estimate of the acoustic centre based on the phase delay is used for the calculations, it actually yields

a free-field sensitivity the phase of which is equal to zero; the small residual observed in the figure

can be neglected as a sub-product of the inaccuracies of the calculation procedure. On the other

hand, the estimate of the acoustic centre based on the group delay gives a phase that deviates largely

from zero, varying between -180 and 100 degrees. This is a proof that the group delay cannot be

used for determining the acoustic centre after an expected phase of the free-field sensitivity.


164

103

104

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

20a)

Pha

se (

° )

103

104

−20

−15

−10

−5

0

5

10b)

Frequency (Hz)

Aco

ustic

cen

tre

(mm

)

Figure 5.6 Determination of the acoustic centres based on phase measurements a) Phase of the free-field

sensitivity calculated with equation (1) considering the physical distance between microphones, b)

Acoustic centre ——— Calculated from the phase delay, C C C C C C calculated from the group

delay, – – – – Calculated from modulus measurements, and —~—~— IEC values


165

103

104

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1a)

103

104

−150

−100

−50

0

50

100

150

b)

Frequency (Hz)

Pha

se a

ngle

(°)

Figure 5.7 Phase of the free-field sensitivity calculated using the acoustic centre determined based on

measurements of the phase response of a microphone: a) from phase delay, and b) from group delay.


166

( )∠ = +Z d kd Ce 2 2 . (5.13)

( )∠ = + + +

= +

Z d kd kd

kd Ce xy M Mθ θ

1 2

,(5.12)

Furthermore, it can be seen in figure 5.6 that the value of the phase based acoustic centre

calculated from the phase delay and group delay is completely different from the acoustic centre

determined from modulus measurements. This seems to be in contradiction to the concept of

acoustic centre.

The contradiction can be explained by analysing the phase of the electrical transfer

impedance. The definition of the acoustic centre states that this point in space is the origin of the

spherical source that replaces the microphone. Thus, the change of phase in the electrical transfer

impedance measured at different distances should indicate the position of the acoustic centre. The

phase angle of the electrical transfer impedance at a given distance is

where C is the sum of the phase of the free-field sensitivities, and the phase introduced by the

acoustic centre. This is constant and does not change as a function of distance. If the measurement

is made at another distance, d2, the phase angle is

The constant C is, again, the sum of the phase of the sensitivities and the phase delay of the acoustic

centre.

But the phase of the free-field sensitivity is a function of the acoustic centre. Then, in order

to calculate it, it is necessary to take a given value for the acoustic centre. This value can be arbitrarily

selected or calculated, for example, with the procedure based on the calculation of the phase delay.

This procedure yields the phase of the free-field sensitivity which is zero in the whole frequency

range. However, the resulting acoustic centre is in contradiction with the acoustic centre calculated

from modulus measurements.

Therefore, a decision must be taken in order to have full compatibility between

measurements. To have two different acoustic centres defined according to the quantity which they

are based upon may introduce confusion about which one to use, and which is the reference.


167

102

103

104

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)

Sen

sitiv

ity d

iffer

ence

(dB

)

Figure 5.8 Deviations of the modulus of the sensitivities calculated at several distances from the average

modulus of the sensitivity for microphone Brüel & Kjær 4160, s/n 1453784 using the acoustic centres

calculated using the phase delay. – C – C – C – 240 mm, — — — 320 mm,

———— 400 mm, and C C C C C C 500 mm.

An important consequence of the use of the acoustic centre determined from modulus

measurements is that the resulting modulus of the free-field sensitivity becomes independent of the

calibration distance while the phase is not zero except at very low frequencies. This is not fulfilled

when the acoustic centre calculated using the phase delay is used for the calculations. Figure 5.8

shows the deviations of the modulus of the free-field sensitivity at different distances when the

acoustic centres based on the phase delay are used for the calculation of the sensitivity. It can be

seen that there are differences in the sensitivities calculated at different distances. A systematic

difference occurs as the higher sensitivity is associated with the shorter distance, and the lower

sensitivity with the longer distance.


168

5000 10000 20000−5

0

5

Frequency (Hz)

Aco

ustic

cen

ter

posi

tion

(mm

)

Figure 5.9 Experimental acoustic centres of LS2 condenser microphones compared with data from [15]

and [59]: ——— Experimental values, — ~ — ~ — Juhl [15], and — * — * — Wagner [59].

This is a strong argument in favour of calculating the free-field sensitivity using the acoustic

centres determined on the assumption that the modulus of the electrical transfer impedance in the

free-field must fulfill the inverse distance law. Thus, the resulting sensitivity will be independent of

the distance and it will have a phase that will be the sum of the phase of the pressure sensitivity, the

radiation load factor, the diffraction factor, and the phase introduced by the chosen acoustic centre.

5.4 Acoustic centres of LS2 microphones

The acoustic centres of six Brüel & Kjær 4180 microphones were determined from modulus

measurements using the simple linear regression procedure described in section 5.1.1. Figure 5.9

shows the acoustic centres of the microphones. The results are compared with data contained in the

reference [15], and with data obtained by Wagner et al [59].


169

2 10 20 30−4

−2

0

2

4

6

8

10

Frequency (kHz)

Aco

ustic

Cen

ter

(mm

)

Figure 5.10 Acoustic centres of LS2 microphones: ——— Average of the experimental acoustic

centres, — * — * — data from [59], —~—~— calculated with BEM using a displacement

distribution of a zero order Bessel function, and —Î—Î— using a uniform displacement distribution.

It can be seen that the spread of the results of the acoustic centre is significant. At

frequencies below 20 kHz, such spread is about 4 mm, while above 20 kHz it decreases to 1.5 mm.

This is the same spread observed in the case of LS1 microphones (see figure 4.13) though the

average value may follow closely the results given for comparison. It could be argued that the

cleaning technique may introduce such a deviation. However, by examining the acoustic centres

calculated from raw measurements, it can be seen that the same spread is observed, thus this spread

could be an expression of the repeatability of the measurement set-up.

First, the acoustic centres are calculated using the axisymmetrical BEM formulation

assuming that the displacement distribution of the diaphragm is uniform or a Bessel function of zero

order. The simulation results and a comparison with the average of the measurements are shown

in figure 5.10


170

It can be seen that the calculated values seem to coincide with the average of the

experimental acoustic centres, also with the values presented by Wagner[59]. The trend is

approximately the same at low frequencies. But above 20 kHz the spread increases. However, it is

worth noticing that the Bessel approximation seems to coincide better with data from [59].

However, it should be remembered that the experimental data shown here is an average of a number

of measurements. As it has been shown, the spread of the experimental values is large, and thus, it

may be possible that an improvement of the measurement repeatability may help to clarify the

results.

Summary

The problem of determining the acoustic centres has been addressed in this chapter. The

acoustic centres of the microphones have been determined both from considering that the modulus

of the electrical transfer impedance must fulfil the inverse distance law, and from considerations

about the characteristics of the phase of the free-field sensitivity. It has been found that the acoustic

centres determined in these two ways are inconsistent. However, by selecting the acoustic centre

based on modulus measurements, the sensitivity of the microphone is independent of the distance

at which the calibration was made. Thus, the modulus based acoustic centres are recommended for

determining the free-field sensitivity of the microphone. The determined acoustic centres are in good

agreement with the acoustic centres obtained by simulating the microphone with an axisymmetrical

BEM formulation assuming different velocity distributions of the diaphragm.

Additionally, the acoustic centres of LS2 microphones have also been analysed and

compared with simulated results.

Conclusions

171

Chapter 6. Conclusions and future research

6.1 Conclusions

A time selective procedure has been developed and tested using computer simulations. The

procedure has proved to remove imperfections of the electrical transfer impedance caused by

reflections effectively.

The cleaned results are considered to be reliable, i.e. not modified by the cleaning procedure,

in a frequency range that goes from a low frequency that is a function of the length of the time

selective window to the highest measured frequency minus the same low frequency limit. This is

because the time selective window may cut the impulse response at an instant where the impulse

response has not decayed completely to zero. This introduces a ripple with a frequency that depends

on the length of the time selective window. The amplitude of this ripple sets the accuracy limitations

of the application of the cleaning technique.

The cleaning technique has been applied to an electrical transfer impedance experimentally

measured at a given distance. The characteristics of the observed disturbances such as the electrical

cross talk, the random noise due to a finite measurement time, the standing wave between the

microphones, and the reflections from the walls have been described and discussed.

A number of measurements with one inch laboratory standard microphones (LS1) have been

carried out, and the cleaning procedure applied to them. With these cleaned functions, the acoustic

centres of the microphones have been properly determined both from considering that the modulus

of the electrical transfer impedance must fulfil the inverse distance law, and from considerations

about the characteristics of the phase of the free-field sensitivity. It has been found that the acoustic

centres determined in these two ways are inconsistent. However, by selecting the acoustic centre

based on modulus measurements the sensitivity of the microphone is independent of the distance

at which the calibration was made. Thus, the modulus based acoustic centres are recommended for

determining the free-field sensitivity of the microphone. The estimated acoustic centres are in good

Conclusions

172

agreement with the acoustic centres obtained by simulating the microphone with an axisymmetrical

BEM formulation assuming different velocity distributions of the diaphragm.

The complex free-field sensitivity and the free-field correction have been determined as well.

The results were compared with data obtained from simulations using the BEM axisymmetrical

formulation, and with data proposed by an international standard. Traditionally, the free-field

correction includes only modulus information, and it seems to be in agreement with the

measurements. The phase is also compared with the BEM simulations, and good agreement has

been found.

The calibration of half inch laboratory standard microphones (LS2) has also been studied. As

in the case of the LS1 microphones, the acoustic centres, free-field sensitivity and free-field

correction have been determined. The experimental results where compared to the simulated data

and standardised data. Good agreement was found. Additionally, an explanation for some observed

differences between the diffraction factor of LS1 and LS2 microphones have been found by making

use of the BEM formulation.

The combination of LS1 and LS2 microphones has also been tried with the objective of

partially solving the cross talk problem by increasing the signal to noise ratio using a one-inch

microphone. The results are in good agreement with values obtained when only LS2 microphones

are used. This may be at least a partial solution to the cross talk problem in the case of the

calibration of LS2 microphones.

The agreement between experimental and simulated data indicates that the cleaning

technique effectively removes the disturbances while simultaneously does not modify the measured

functions except at low frequencies, as predicted in the simulations. On the other hand, the removal

of the standing wave between the microphones is an improvement to the realisation of the free-field

which cannot be achieved by other means.

The results described in this thesis imply that the cleaning technique allows to carry out free-

field calibrations even in non-anechoic environments like a small room furnished with absorbent

material on the walls.

Conclusions

173

6.2 Future research

The work described in this thesis is a contribution to the advancement of the state of the art

of free field reciprocity calibration of condenser microphones. However, there are still subjects that

should be addressed in future work.

Improvement of the time selective technique: The major limitation of the time selective technique

is the fact that the time window cuts the impulse response where it has not decayed sufficiently. It

is obvious that similar windowing procedures will have a similar effect. An example of such

windowing is the “liftering” of the complex cepstrum. During the development of the project, the

same windowing was applied to a wavelets transformation of the impulse response, with similar

results. However, a common tool for “denoising” a given signal using the wavelet transform consists

of thresholding the coefficients in such a way that coefficients below this threshold are eliminated,

and the signal is reconstructed with the remaining coefficients. It could be possible to carry out a

thresholding of the impulse response in such a way that the coefficients of the direct wave between

the microphones remain unmodified while the coefficients of the reflections, standing waves and

cross-talk are subject of a thresholding if their value is larger than the random variations present in

the impulse response. Colloquially, it would mean a “noisification” of the coefficients of the

disturbances that should be removed. This may avoid introducing the characteristic ripple of the

time selective windowing.

The environmental coefficients of the free-field sensitivity: The sensitivity of the microphones is

affected by changes in the environmental conditions. The static pressure and temperature

coefficients of the pressure sensitivity have been studied, and it seems natural to combine them with

the estimated effect of the environmental variables on the diffraction factor and the radiation

impedance of the microphone in order to obtain the environmental coefficients of the free-field

sensitivity. The diffraction factor being a wavelength related phenomenon, it is greatly affected by

changes in temperature. This dependence has been indirectly addressed for calculating the free-field

correction from a polynomial approximation. However, no explicit coefficients have been given for

the free-field sensitivity as such.

Conclusions

174

The determination of the actual velocity distribution of the diaphragm: The numerical calculations of

the quantities related to the free-field sensitivity of condenser microphones such as the acoustic

centre and the free-field correction depend heavily on the correct assumption of the velocity

distribution of the diaphragm of the microphone. There have been advances in solving the problem

of the interior field of a condenser microphone using numerical techniques. The combination of

these numerical models with experimental measurements of the velocity distribution of the

diaphragm, such as optical measurements, may help to confirm the behaviour of the quantities

related to the free-field calibration.

The reduction of electrical cross talk and improvement of the repeatability of the measurements: Further

improvements of the measurement set-up include the elimination of the cross-talk, or at least its

minimisation to a bearable level while simultaneously the stability and day-to-day repeatability of the

measurements are improved. Another possible improvement is an extension of the measurement

range in the high frequency range. Addressing the two problems is of fundamental importance if

smaller transducers with lower sensitivity are to be calibrated.

Calibration in a diffuse field: The absolute calibration of condenser microphones in a diffuse

field continues to be an open question. The requirements for creating a diffuse field with the sound

generated by a microphone are not trivial. The possibility of the simultaneous calibration of free-

field and diffuse field should be extensively studied.

References

175

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[2] Mills, I. M., Taylor, B. N. and Thor A. J., Definitions of the units radian, neper, bel and decibel ,

Metrologia 38(4), pp. 353-61, 2001.

[3] Emerson, W. H., A reply to “Definitions of the units radian, neper, bel and decibel” by I. M. Mills et al.,

Metrologia 39(1), pp. 101, 2002.

[4] Valdés, J., The unit one, the neper, the bel and the future of the SI, Metrologia 39(6), pp. 543-9, 2002.

[5] Ballantine, Stuart, Technique of microphone calibration, Journal of the Acoustical Society of America,

pp. 319-58, 1932.

[6] MacLean, W. R., Absolute measurement of sound without a primary standard, Journal of the Acoustical

Society of America 12, pp. 140-6, 1940.

[7] Wathen-Dunn, Weiant, On the reciprocity free-field calibration of microphones, Journal of the Acoustical

Society of America 21(5), pp. 542-6, 1949.

[8] Rudnick, I., and Stein, M.N., Reciprocity free field calibration of microphones to 100 Kc in air, Journal of

the Acoustical Society of America 20(6), pp. 818-25, 1948.

[9] DiMattia, A. L., and Wiener, F. M., On the absolute pressure calibration of condenser microphones by the

reciprocity method, Journal of the Acoustical Society of America 18(2), pp. 341-4, 1946.

[10] Gibbings, D. L. H., and Gibson, A. V., Free-field reciprocity calibration of capacitor microphones at

frequencies from 19.95 kHz to 316.2 Hz, Metrologia 20, pp. 85-94, 1984.

[11] Burnett, Edwin W., and Nedzelnitsky, Victor, Free-field reciprocity calibration of microphones, Journal

of Research of the National Bureau of Standards 92(2), pp. 129-151, 1987.

[12] Durocher, Jean-Nöel, Etalonnage des microphones à condensateur en champ libre, Journal d’ Acoustique

2, pp. 431-6, 1989.

[13] Matsui, E., Free-field correction for laboratory standard microphones mounted on a semiinfinite rod, Journal

of the Acoustical Society of America, 49 (5 Part.2), pp 1475-83, 1971.

[14] Bjørnø, L., Holst-Jenson, O., Lolk, S., Free-field corrections for condenser microphones, Acustica, 33 (3),

pp 166-73, 1975

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[15] Juhl, Peter M., A numerical investigation of standard condenser microphones, Journal of Sound and

Vibration 177 (4), pp 433-446, 1994.

[16] Barham, Richard, Free-field reciprocity calibration of laboratory standard microphones, Ph.D Thesis, ISVR,

University of Southampton, 1995.

[17] Zuckerwar, Allan, J., Theoretical response of condenser microphones, Journal of the Acoustical Society

of America 64 (5), pp 1278-85, 1978.

[18] Bao, X., and Kagawa, Y., A simulation of condenser microphones in free field by boundary element approach,

Journal of Sound and Vibration, 119 (2), pp 327-37, 1987.

[19] Cutanda, Vicente, Numerical transducer modeling, PhD Thesis, Technical University of Denmark,

Ørsted-DTU, Acoustic Technology, 2002.

[20] Hunt, Frederick V., Electroacoustics. The analysis of transduction and its historical background, The

Acoustical Society of America, 1982.

[21] Beranek, Leo, Acoustic measurements, 1962.

[22] Kinsler, L. E., Fundamentals of acoustics, New York, N. Y., USA, Wiley, 2000.

[23] Kreyzig, Erwin, Advanced engineering mathematics, New York, N.Y, USA, Wiley, 1993.

[24] Rasmussen, K., The static pressure and temperature coefficients of standard laboratory microphones,

Metrologia 36 (1999), pp 265-273

[25] Morse, Phillip M., and Ingaard, Uno, Theoretical acoustics, McGraw Hill, 1968.

[26] IEC International Standard 61094-3, Measurement microphones, Part 3: Primary method for free-field

calibration of laboratory standard microphones by the reciprocity technique, 1995.

[27] Zuckerwar, Allan J, and Ngo, Kim Chi Thi, Measured 1/f noise in the membrane motion of condenser

microphones, Journal of the Acoustical Society of America 95(3), pp. 1419-25, 1994.

[28] Williams, Jeffery T., Delgado, Heriberto J., and Long, Stuart A., An antenna pattern measurement

technique for eliminating the fields scattered from the edges of a finite ground plane, IEEE Transactions on

Antennas and Propagation 38 (9), 1990.

[29] Novotny, D. R., Johnk, R. T., and Ondrejka, A., Analyzing broadband, free-field, absorber

measurements, Proceedings of 2001 International Symposium on Electromagnetic Compatibility (EMC

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[30] Terry, R. L., and Watson, R. B., Pulse technique for the reciprocity calibration of microphones, Journal of

the Acoustical Society of America, 23 (6), pp. 684-5, 1951.

[31] Niemoeller, Arthur F., Reciprocity calibration of condenser microphones in the time domain, Journal of the

Acoustical Society of America, 33 (12), pp. 1712-19, 1961.

[32] Lambert, J. M., Durocher, J. N., Analyse des perturbations acoustiques lors de l’etalonnage en champ libre

des microphones etalons a condesateurs dits d’un pouce par la technique re la reciprocité, Laboratoire National

D’Essais, (1989)

[33] Blem, J. S., Tidsselektiv fritfeltskalibriering af kondensatormikrofoner, Master Thesis, The Acoustics

Laboratory, Technical University of Denmark, (1992)

[34] Vorländer, M., and Bietz, H., Novel broad band reciprocity technique for simultaneous free-field and diffuse-

field microphone calibration, Acustica 80, 365-377 (1994)

[35] Tarnow V., Thermal noise in microphones and preamplifiers, Brüel & Kjær Technical Review 1972(3),

3-14.

[36] Proakis, J. G., and Manolakis D. G., Digital signal processing, principles, algorithms and applications,3rd

edition, Prentice-Hall, 1996.

[37] Bendat, Julius S., and Piersol, Allan G., Random data. Analysis and measurement procedures, Second

edition, John Wiley and Sons, 1986.

[38] Papoulis, Athanasios., Signal Analysis, McGraw Hill, 1977.

[39] Kwon, Hyu-Sang, and Kimm Yang-Hann, Minimization of bias error due to windows in planar acoustic

holography using a minimum error window, Journal of the Acoustical Society of America 98 (4), pp.

2104-11, 1995.

[40] Sjöström, M., Properties of smoothing with time gating, 2000 IEEE International Symposium on

Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat

No.00CH36353) Vol.1, pp. 707-10, 2000 .

[41] Harris, F. J., The use of windows for harmonic analysis with the discrete Fourier Transform, Proc IEEE

66(1), 51-83, (1978)

[42] Delany, M. E., and Bazley, E. N., A note on the sound field due to a point source inside and absorbent-lined

enclosure, Journal of Sound and Vibration 14(2), pp. 151-7, 1971.

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[43] IEC International Standard 655 (1979), Values for the difference between free-field and pressure sensitivity

levels for one-inch standard condenser microphones.

[44] Yong-Sheng, Ding, The design of a small anechoic chamber for free-field calibration of standard condenser

microphones, The Acoustics Laboratory, Technical University of Denmark, Internal report 20, 1984.

[45] Rasmussen, K., Calculation methods for the physical properties of air used in the calibration of microphones,

Technical Report, Department of Acoustic Technology, Technical University of Denmark, Report

PL-11a, 1996.

[46] Rasmussen, K., and Sanderman Olsen, E., Intercomparison on free-field calibration of microphones. Final

version, Technical Report, Department of Acoustic Technology, Technical University of Denmark,

Report PL-07, 1993.

[47] IEC Committee draft IEC/CD 61094-7, Measurement microphones - Part 7: Values for the difference

between free-field and pressure sensitivity levels of laboratory standard microphones, 2002.

[48] Skvor, Zdenek, Vibrating systems and their equivalent circuits, Elsevier, ISBN 0-444-98806-8, 1991.

[49] Brüel & Kjær, Data Handbook: Condenser microphones and microphone preamplifiers for acoustic

measurements, 1982.

[50] IEC Publication 61094-1, Measurement microphones Part 1: Specifications for laboratory standard

microphones, 1992.

[51] Wiener, F. M., On the relation between the sound fields radiated and diffracted by plane obstacles, Journal

of the Acoustical Society of America 23(6), pp. 697-700, 1951.

[52] Rasmussen, Knud, Acoustic centres of condenser microphones, The Acoustics Laboratory, Technical

University of Denmark, Report 5, 1971.

[53] Koidan, Walter, and Siegel, David S., Free-field correction for condenser microphones, Journal of the

Acoustical Society of America 36(?), pp. 2233-4, 1964.

[54] Hruska, G. R., and Koidan, W., Free-field method for sound-attenuation measurements, Journal of the

Acoustical Society of America 58 (2), pp 507-9, 1975.

[55] Trott, W. James, Effective acoustic center redefined, Journal of the Acoustical Society of America

62(2), pp. 468-9, 1977.

[56] Ando, Y., On the sound radiation from semi-infinite circular pipe of certain wall thickness, Acustica 22, pp.

219-25, 1969.

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[57] Ando, Y., Experimental study of the pressure directivity and the acoustic centre of the “circular pipe horn

loudspeaker”, Acustica 20, pp. 366-9, 1968.

[58] Behler, G. K., and Vorländer, M., Method and application of an optical measuring and calibration

technique for microphones, Proceedings in CD of the 17th International Congress in Acoustics (ICA),

Rome, 2001.

[59] Wagner, Randall P., and Nedzelnitski, Victor, Determination of acoustic center correction values for type

LS2aP microphones at normal incidence, Journal of the Acoustical Society of America 104(1), pp. 192-203,

1998.

References

180

Appendix A The measurement system

181

Appendix A. The measurement system

In this appendix the measurement set-up and some aspects of the measurement procedure

are described. Some of the most important elements of the measurement system, especially the

reciprocity apparatus and the sound analyser, are already described in references [A1], [A2] and [A4].

However, it is pertinent to describe the measurement set-up because one of the most important

activities of the project described in the thesis was to process the complex frequency response

measured by the calibration set-up in order to improve the estimation of the free-field sensitivity.

However, during the course of the project, several different configurations of the measurement set-

up were tried.

A.1 Measurement procedure

In order to determine the open-circuit electrical transfer impedance between two

microphones, four voltages should be measured:

A. The output voltage of the receiver microphone (sound pressure)

B. The voltage at the terminals of the reference impedance in series with the transmitter

microphone (electrical current)

C. The insert voltage of the receiver microphone channel, and

D. The insert voltage of the transmitter microphone channel.

By making the appropriate set of calculations with the above measured quantities, it is

possible to determine the open circuit electrical transfer impedance (see chapter 2). The insert

voltage technique has been widely used for obtaining the open circuit output voltage of a transducer

in the calibration of microphones, especially a variety known as the “substitution” technique (see

reference [A3]). As described in chapter 2, the open circuit output voltage occurs when the current

through the electrical terminals is equal to zero. However, although the input impedance of the

microphone preamplifier is large, it is still finite. Then, the load effect will introduce a current flow

through the microphone. This is taken into account by measuring the insert voltage, and the open

circuit voltage can be obtained by determining the ratio of the output voltage to the insert voltage.


182

ZU UU U

Ze e ref, ,''

,122 1

1 2

= (A.1)

B&K 2012 Anakyser

ReciprocityApparatus

PC20 dBAmplifier

20 dBAmplifier

Receiver +2673 preamo +20dB Transmitter +

Transmitter unit B&K

IEEEBus

Signal inputSignal output

Termometer

Barometer

Figure A-1. Schematic representation of the measurement set-up.

The electrical transfer impedance is calculated using the four measured voltages described

above using

where U1 is the voltage at the terminals of the reference impedance connected in series to the

transmitter microphone, U2 is the output voltage of the receiver microphone, U’1 is the insert voltage

of the transmitter microphone, U’2 is the insert voltage of the receiver microphone, and Ze,ref is the

reference impedance connected in series to the transmitter microphone.

The environmental conditions, static pressure, temperature and relative humidity are

measured and recorded during the measurements.

A.2 Measurement instruments

The measurement set-

up used is very similar to that

described in reference [A2].

However, a major change is the

substitution of the signal

synthetiser and the digital

multimeter by a sound analyser,

namely the Brüel & Kjær 2012.

The use of this analyser has

some advantages, among these,

the possibility for making faster

measurements. Figure A.1

shows a block diagram of the

complete measurement set-up.

The reciprocity apparatus and

the sound analyser are described

below.


183

MEASUREMENTDEVICE

HIGH PASS FILTER22/220 Hz

POLARISATIONVOLTAGE

0 - 20 dB

20 dB

RECEIVERMICROPHONE

TRANSMITTERMICROPHONE

B&K2645

400 Hz

0/30 dB

0-30 dB

CHANNEL A

CHANNEL BHIGH PASS FILTER22/220 Hz

0 - 20 dB

20 dB

SIGNAL

INSERT

Figure A-2. Block diagram of the reciprocity apparatus

A.2.1 Reciprocity apparatus

The reciprocity apparatus is designed to make the appropriate switching for allowing the

measurement of the voltages needed for calculating the electrical transfer impedance. It also provides

amplification and low-pass filtering. A signal can be fed into the apparatus, and the output can be

measured by a voltmeter, an analyser, etc. The input signal can be provided by a signal generator or

an analyser itself. Figure A.2 shows a simplified block diagram of the reciprocity apparatus, including

the connections to the microphones.

A.2.2 Sound analyser

The following information can be supplemented from the technical manual of the analyser

[A4]. The B&K 2012 sound analyser contains several modules which handle the different functions

of the analyser: measurement module, memory module, etc.

In this case, the module of interest is the measurement module, which has three different

built-in modes, time selective response (TSR), steady state response (SSR), and FFT spectrum (FFT).

The steady state response module is used because the complex frequency response will be

measured at discrete frequencies spaced equidistantly. When this mode is selected, the analyser


184

( ) ( )( )

H fU fU frel = 2

1

, (A.2)

( ) ( )H f U fabs = 2 , (A.3)

Figure A-3. Block diagram of the sound analyser Brüel

& Kjæer 2012. It shows the operations made in order to

obtain the frequency response in the Steady State Response

(SSR) mode.

measures either a relative frequency response or an absolute frequency response. Each is defined

as

where, U1(f) is the excitation signal which is provided by the analyser itself, and U2(f) is the output

of the system under test which correspondingly is the analyser input.

The SSR mode measures the complex frequency response function under steady state

conditions using a stepped sine excitation. This means that pure frequencies are used at specified

frequency steps that can be linearly or logarithmically separated.

An adaptive scan algorithm can be set up to measure the frequency response under the SSR

mode to a user specified accuracy in a minimum possible time. This is explained below.

In figure A.3, a block diagram of the measurement process, extracted from the analyser

operation manual, is shown.

In order to determine the frequency

response of the device under test, its

response is multiplied by the complex

conjugate of the analytic excitation signal. This

gives a frequency shift to DC. Subsequent

low pass filtering provides the real and

imaginary parts of the response.

The analytical procedure is described

below. The system is excited with a

sinusoidal function of a given frequency.

This signal is provided by the analyser


185

( ) ( ) ( )f t A ft A te = =cos cos .2π ω (A.4)

( ) ( ) ( )[ ] ( ) ( )[ ]f t A t j t A t j ta = + − = +cos cos cos sin .ω ω π ω ω2 (A.5)

( ) ( ) ( )[ ]f t A t j ta* cos sin .= −ω ω (A.6)

( ) ( )f t B ti = +cos .ω ϕ (A.7)

( ) ( ){ } ( ) ( ) ( )[ ]f t f t B t A tAB

ti a⋅ = + ⋅ = + +Re cos cos cos cos .* ω ϕ ω ϕ ω ϕ2 2 (A.8)

( ) ( ){ } ( ) ( ) ( )[ ]f t f t B t A tAB

ti a⋅ = − + = − +Im cos sin sin sin .* ω ϕ ω ϕ ω ϕ2 2 (A.9)

( ){ }Re cos ,f tAB

i = 2 ϕ (A.10)

( ){ }Im sin .f tAB

i = 2 ϕ (A.11)

internal generator. It can be expressed as

The analytic excitation signal is defined as the sum of the original cosine and its Hilbert Transform,

recalling that the Hilbert Transform of a cosine function is a 90° delayed cosine or a sine function

The complex conjugate of the analytical excitation function is

The output of the device under test (analyser input), fi, is, in general, an amplified (or

attenuated) and delayed version of the excitation function, that can be expressed as

Then the multiplication of the analytic excitation function with the response of the device

can be carried out in two parts, one with the real part of the analytic excitation function and the

other with the imaginary part of the analytic excitation function. Multiplication with the real part

gives

Multiplication with the imaginary part yields

A low-pass filter implemented in the analyser eliminates the higher frequency components,

and then the real part of the measurement is

and the imaginary part


186

The next stage in the measurement process is the application of the so-called adaptive scan

algorithm. For each excitation level, blocks of data are collected and processed in order to obtain

its average and standard deviation. However, any transient behaviour should be avoided. This can

be done by introducing a settling time. Once the system has settled, the detector starts the data

collecting procedure. Then, it calculates the average and standard deviation of the sampled block.

These results are compared with a set of defined requirements. If the requirements are met, the final

result is then shown. If the requirements are not met, the data acquisition and processing continues.

This means that the higher the accuracy, the longer the required time to achieve it.

The low-pass filter in figure (A.3) effectively implements a band-pass filter centred at the

frequency of analysis. Thus, the suppression of random variations appears to be the higher the

longer the measurement time. Also, the data collected is complex, and this allows averaging

complex data, which is equivalent to the FFT analysis. Due to this fact, the increased averaging

time has the same effect as narrowing the filter bandwidth, reducing the effect of background noise.

A.3 Harmonic distortion

As described in chapter 2, the working principle of the microphone implies that under

certain conditions, distortion may appear. It is especially important when the microphone is driven

with a voltage that is too high. Unfortunately, when measuring the electrical transfer impedance

between two microphones in a free field, a high excitation signal is needed in order to improve the

poor signal to noise ratio. On the other hand, as described above, the measurement principle of the

analyser actually removes harmonic distortion from the measurements, thus making difficult to

assess the actual level of distortion. However, if an oscilloscope is used for monitoring the linearity

of the voltages being measured, the distortion may be detected.

As the excitation signal is fed to the transmitter microphone, it will generate a sound field

that contains distortion products. These distortion products will be sensed by the receiver

microphone, but the analyser will filter them out. On the other hand, the voltage fed to the

transmitter microphone is measured on the terminals of the reference impedance, and considering

that the feedback of the diaphragm to the electrical terminals of the diaphragm is negligible, this

voltage will correspond to the full excitation level. This implies that if the distortion products are


187

γ = −+

= +

xd x

EE

xd

EE

EE

1

0

0

1

1 12

02

0

1

2 2, (A.13)

( )( )vv

EE

ZZm

m

2

1

1

0

1

241 1 3=

+ +,

,

,γ γ (A.12)

large the ratio of voltages calculated using equation (A.1) will not correspond to the actual electrical

transfer impedance; it will be smaller.

Furthermore, it is possible to assess the expected distortion of a electrostatic transducer (see

reference [A5]). The ratio of the fundamental to the second harmonic of the velocity of an

electrostatic transducer is given by

where v1 and v2 are the volume velocities of the transducer at the fundamental and the second

harmonic frequencies, E0 and E1 are the polarisation and excitation voltages, Zm,1 and Zm,2 are the

mechanical impedances at the fundamental and first harmonic frequencies. The parameter γ is

defined by

where x1 and x0 are the static displacement and the displacement generated by the at the excitation

voltage at the fundamental frequency, and d is the distance between the diaphragm and the backplate

of the transducer.

Equations (A.12) and (A.13) contain very interesting information about the behaviour of the

transducer. The harmonic distortion will be a function of the square of the ratio of polarisation to

excitation voltages. It means that changes in the excitation voltage may have a significant effect on

the distortion products. On the other hand, the ratio of impedances at the fundamental and

harmonic frequencies gives a frequency dependence to the expected distortion that is particular to

the mechanical impedance of the transducer. Furthermore, in the case of the microphone, it implies

that the distortion will have a maximum at a frequency which is half the resonance frequency of the

microphone. Figure A.3 shows the harmonic distortion of the sound pressure measurement when

different excitation voltages are used.


188

103

104

−50

−40

−30

−20

−10

0

10a)

103

104

−80

−70

−60

−50

−40

−30

−20b)

8 V6 V5 V

Mod

ulus

(dB

)M

odul

us (

dB)

Frequency (Hz)

Figure A-3. Measurements of the fundamental and the harmonic distortion of the sound pressure

measured at different excitation voltages: a) Sound pressure, and b) First harmonic distortion. In the

two figures the excitation voltages are: ——— 8 V, —C—C— 6 V, and —B—B— 5 V.

It can be seen that the harmonic distortion actually has a maximum value about half the

resonance frequency of the microphones involved in the measurement. This is because of the fact

that at the resonance frequency the impedance of the transducer is minimal, thus at a frequency

which is half of the resonance frequency of the microphone, the ratio will have a maximum value.

It can also be seen that changes in the excitation voltage are enlarged significantly in the distortion

product. This is a consequence of the dependence on the square – and higher products -- of the ratio


189

of polarisation to excitation voltage.

Thus, in order to minimise the harmonic distortion, two decisions were taken. One was to

use a low excitation voltage. This would decrease the harmonic distortion evenly in the whole

frequency range. Another action was to “shape” the excitation signal. Instead of having a uniform

level in the whole frequency range, a weighting function was used. This function has a unity value

at the lowest frequency, and it decreases at 6 dB per octave as the frequency increases until it reaches

an attenuation of 20 dB. Thereafter, it becomes constant until it reaches the highest measurement

frequency. This shaping also has the purpose of decreasing the dynamic range of the measured

sound pressure.

A.4 Coherence

Another problem related to the measurements is the signal to noise ratio that is especially

poor at low frequencies. An appropriate indicator of the behaviour of the signal to noise ratio is the

coherence function. Figure A.4 shows the coherence when the sound pressure generated by a LS1

microphone is measured by another LS1 microphone in a free field; the frequency response

corresponds to the electrical transfer impedance. The microphones are located 240 mm and 500 mm

from each other. The coherence was measured using a two-channel FFT analyser (Brüel & Kjær

2035) using random noise as excitation signal.

It can be seen that in the two cases the coherence is nearly one at frequencies above 2 kHz.

Below that frequency, the coherence rolls of very rapidly, and below 1 kHz, it is practically zero. This

indicates that making measurements below 1 kHz is very difficult, and very time consuming. The

difference between the two curves is explained by the fact that the sound pressure measured at the

largest distance is lower, thus decreasing the signal to noise ratio. A solution could be to increase the

excitation signal, but this would aggravate the harmonic distortion, as described in section A.3.


190

102

103

104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Coh

eren

ce

Figure A-4. Coherence of the transfer function between two LS1 microphones in a free field. The

function was measured using the same excitation level, but different distances: ——— 240 mm, and

— — — 500 mm.

A.5 Additional configurations of the measurement set-up

The cross talk problem was addressed using different approaches. One solution is that

described in chapter 4, where a combination of LS1 and LS2 microphones was tried. This helped

to reduce the cross talk that was present when two LS2 microphones were used, but apparently this

was not good enough, thus, further attempts for reducing or eliminating the cross talk were made.

In this section, a short description of some attempts made in order to eliminate, or at least to

minimise, the cross talk is given.

First, after the first set of measurements was made with the original configuration inside the

reciprocity apparatus, it was found that the cross talk had a significant influence. Thus, a thorough

revision of the grounding of the circuitry inside the reciprocity apparatus was carried out. The


191

B&K 2012 Anakyser

ReciprocityApparatus

PC20 dBAmplifier

20 dBAmplifier

Receiver +2673 preamp +20dB Transmitter +

Transmitter unit B&K

IEEEBus

Signal inputSignal output

Termometer

Barometer

ManualSwitch

Brûel & KjærReciprocity App.

5998

Figure A-5. Schematic diagram of the modified measurement set-up. The

significant feature of this set-up is the complete separation of the

transmitter and receiver microphone channels.

objective of this was to avoid any ground loop inside the reciprocity apparatus. This set of

operations did neither change the internal configuration of the reciprocity apparatus nor the

measurement set-up as shown in figures A.1 and A.2. It did not change the measurement procedure

either.

Another possibility for removing the cross talk was the complete separation of the channels

of the transmitter and the receiver microphones. This means that there should be an instrument that

is able to measure the output voltage and the insert voltage of the receiver microphone. There

should also be another instrument that is able to measure the voltage on the terminals of the

reference impedance and the insert voltage of the transmitter microphone. This was achieved by

introducing a second reciprocity apparatus, a Brüel and Kjær 5998 and a manual switch in the

configuration described in figure A.2. The modified measurement set-up is shown in figure A.5.

The main disadvantage

of this set-up is the need to

operate manually the switching

between the two reciprocity

apparatuses.

In order to operate this

set-up, it was necessary to

modify the measurement

procedure in several respects.

Apart form modifying the

software, the most significant

change was to measure the

reference voltage used for

measuring the insert voltage in

each apparatus.

The improvements achieved by using the modified configurations are best observed in the

impulse response of the electrical transfer impedance. Figure A.6 shows the impulse response of a


192

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−2

0

2

4

6

8

10x 10

−3 a)

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−2

0

2

4

6

8

10x 10

−3 c)

Time (s)

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

−2

0

2

4

6

8

10x 10

−3 b)

Nor

mal

ised

am

plitu

de

Figure A-6. Hilbert envelope of the impulse responses determined from different configuration of the

measurement set-up: a) original measurement set-up, b) with modifications in the internal grounding of

the reciprocity apparatus, and c) with the transmitter and receiver channels fully separated.

microphone combination measured with the three measurement set-ups at the same distance.


193

The cross-talk occurs at zero time, before the impulse response between the microphones

appears. It can be seen that the internal modification of the reciprocity apparatus actually reduced

the maximum value of the cross-talk about 5 dB. Furthermore, the complete separation between the

transmitter and receiver channels eliminates completely the cross-talk. However, channel separation

is not free of problems. The major problem that was found when implementing this solution is the

lack of repeatability from day-to-day measurements. However the results obtained by means of the

channel separation are promising, and this solution is still being investigated including the

consideration of using galvanic transformers in the set-up that uses only one reciprocity apparatus.

References

[A1] Rasmussen, K., Recent developments in instrumentation for reciprocity calibration of condenser

microphones, Proceedings of the 7th International Congress on Acoustics, pp. 529-32, 1971.

[A2] Rasmussen, K., and Sanderman Olsen, E., Intercomparison on free-field calibration of microphones.

Final version, Technical Report, Department of Acoustic Technology, Technical University

of Denmark, Report PL-07, 1993.

[A3] Rasmussen, K., Recent developments in instrumentation for reciprocity calibration of condenser

microphones, Proceedings of the 7th International Congress on Acoustics, pp. 529-32, 1971.

[A4] Brüel & Kjær Technical Documentation, Sound analyser 2012.

[A5] Hunt, Frederick V., Electroacoustics. The analysis of transduction and its historical background, The

Acoustical Society of America, 1982.

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