Magnetics Book

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Magnetics


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Contents

Articles

Saturation (magnetic) 1

Intermodulation 2

Volterra series 7

References

Article Sources and Contributors 11

Image Sources, Licenses and Contributors 12

Article LicensesLicense 13


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Saturation (magnetic)

Magnetization curves of 9 ferromagnetic

materials, showing saturation. 1.Sheet steel,

2.Silicon steel, 3.Cast steel, 4.Tungsten steel,

5.Magnet steel, 6.Cast iron, 7.Nickel, 8.Cobalt,

9.Magnetite

Seen in some magnetic materials, saturation is the state reached when

an increase in applied external magnetic field H cannot increase the

magnetization of the material further, so the total magnetic flux density

B levels off. It is a characteristic particularly of ferromagnetic

materials, such as iron, nickel, cobalt and their alloys.

Description

Saturation is most clearly seen in the magnetization curve (also called

BH curve or hysteresis curve) of a substance, as a bending to the right

of the curve (see graph at right). As the H field increases, the B field

approaches a maximum value asymptotically, the saturation level for

the substance. Technically, above saturation, the B field continues

increasing, but at the paramagnetic rate, which is 3 orders of magnitude

smaller than the ferromagnetic rate seen below saturation.

The relation between the magnetizing fieldH and the magnetic fieldB

can also be expressed as the magnetic permeability: or

the relative permeability , where is the vacuum

permeability. The permeability of ferromagnetic materials is not constant, but depends onH. In saturable materials

the relative permeability increases with H to a maximum, then as it approaches saturation inverts and decreases

toward one.

Different materials have different saturation levels. For example, high permeability iron alloys used in transformers

reach magnetic saturation at 1.6 - 2.2 teslas (T), whereas ferrites saturate at 0.2 - 0.5 T. Some amorphous alloys

saturate at 1.2-1.3 T. Mu metal saturates at around 0.8 T.

Due to saturation, the magnetic permeability fof

a ferromagnetic substance reaches a maximum

and then declines

Explanation

Ferromagnetic materials (like iron) are composed of microscopic

regions called magnetic domains, that act like tiny permanent magnets

that can change their direction of magnetization. Before an external

magnetic field is applied to the material, the domains are oriented in

random directions. Their tiny magnetic fields are oriented in random

directions and cancel each other out, so the material has no significant

magnetic field. When an external magnetizing fieldH is applied to the

material, it penetrates the material and aligns the domains, causing

their tiny magnetic fields to turn and align parallel to the external field,

adding together to create a large magnetic field B which extends out

from the material. This is called magnetization. The stronger the

external magnetic fieldH, the more the domains align yielding a higher magnetic flux densityB. Saturation occurs

when practically all the domains are lined up, so further increases inH can't increase B beyond the increment that

would be caused in a nonmagnetic material, in other words, cannot cause further alignment of the domains.
http://en.wikipedia.org/w/index.php?title=Magnetizationhttp://en.wikipedia.org/w/index.php?title=Magnetic_fieldhttp://en.wikipedia.org/w/index.php?title=Permanent_magnethttp://en.wikipedia.org/w/index.php?title=Magnetic_domainhttp://en.wikipedia.org/w/index.php?title=File%3APermeability_of_ferromagnet_by_Zureks.svghttp://en.wikipedia.org/w/index.php?title=Magnetic_permeabilityhttp://en.wikipedia.org/w/index.php?title=Mu_metalhttp://en.wikipedia.org/w/index.php?title=Amorphous_metalhttp://en.wikipedia.org/w/index.php?title=Ferrite_magnethttp://en.wikipedia.org/w/index.php?title=Tesla_%28unit%29http://en.wikipedia.org/w/index.php?title=Vacuum_permeabilityhttp://en.wikipedia.org/w/index.php?title=Vacuum_permeabilityhttp://en.wikipedia.org/w/index.php?title=Permeability_%28electromagnetism%29http://en.wikipedia.org/w/index.php?title=Magnetic_fieldhttp://en.wikipedia.org/w/index.php?title=Magnetizing_fieldhttp://en.wikipedia.org/w/index.php?title=Orders_of_magnitude_%28magnetic_field%29http://en.wikipedia.org/w/index.php?title=Paramagnetichttp://en.wikipedia.org/w/index.php?title=Asymptotehttp://en.wikipedia.org/w/index.php?title=Hysteresishttp://en.wikipedia.org/w/index.php?title=Cobalthttp://en.wikipedia.org/w/index.php?title=Nickelhttp://en.wikipedia.org/w/index.php?title=Ironhttp://en.wikipedia.org/w/index.php?title=Ferromagnetichttp://en.wikipedia.org/w/index.php?title=Magnetic_flux_densityhttp://en.wikipedia.org/w/index.php?title=Magnetizationhttp://en.wikipedia.org/w/index.php?title=Magnetic_fieldhttp://en.wikipedia.org/w/index.php?title=Magnetichttp://en.wikipedia.org/w/index.php?title=File%3AMagnetization_curves.svg


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Effects and uses

Saturation limits the maximum magnetic fields achievable in ferromagnetic-core electromagnets and transformers to

around 2 T, which puts a limit on the minimum size of their cores. This is one reason why high power motors,

generators, and utility transformers are physically large; because they must have large magnetic cores.

In electronic circuits, transformers and inductors with ferromagnetic cores operate nonlinearly when the current

through them is large enough to drive their core materials into saturation. This means that their inductance and other

properties vary with changes in drive current. In linear circuits this is usually considered an unwanted departure from

ideal behavior. When AC signals are applied, this nonlinearity can cause the generation of harmonics and

intermodulation distortion. To prevent this, the level of signals applied to iron core inductors must be limited so they

don't saturate. To lower its effects, an air gap is created in some kinds of transformer cores. The saturation current,

the current through the winding required to saturate the magnetic core, is given by manufacturers in the

specifications for many inductors and transformers.

On the other hand, saturation is exploited in some electronic devices. Saturation is employed to limit current in

saturable-core transformers, used in arc welding, and ferroresonant transformers which serve as voltage regulators.

When the primary current exceeds a certain value, the core is pushed into its saturation region, limiting further

increases in secondary current. In a more sophisticated application, saturable core inductors and magnetic amplifiers

use a DC current through a separate winding to control an inductor's impedance. Varying the current in the control

winding moves the operating point up and down in the saturation curve, controlling the AC current through the

inductor. These are used in variable fluorescent light ballasts, and power control systems.

References

Intermodulation

A frequency spectrum plot showing intermodulation between two injected

signals at 270 and 275 MHz (the large spikes). Visible intermodulation products

are seen as small spurs at 280 MHz and 265 MHz.

Intermodulation or intermodulation

distortion (IMD) is the amplitude modulation

of signals containing two or more different

frequencies in a system with nonlinearities.

The intermodulation between each frequency

component will form additional signals at

frequencies that are not just at harmonic

frequencies (integer multiples) of either, but

also at the sum and difference frequencies of

the original frequencies and at multiples of

those sum and difference frequencies.

Intermodulation is caused by non-linear

behaviour of the signal processing being used.

The theoretical outcome of these

non-linearities can be calculated by generating

a Volterra series of the characteristic, while the

usual approximation of those non-linearities is

obtained by generating a Taylor series.

Intermodulation is rarely desirable in radio or audio processing, as it creates unwanted spurious emissions, often in

the form of sidebands. For radio transmissions this increases the occupied bandwidth, leading to adjacent channel
http://en.wikipedia.org/w/index.php?title=Taylor_serieshttp://en.wikipedia.org/w/index.php?title=Spurious_emissionhttp://en.wikipedia.org/w/index.php?title=Sidebandshttp://en.wikipedia.org/w/index.php?title=Spurious_emissionhttp://en.wikipedia.org/w/index.php?title=Sidebandshttp://en.wikipedia.org/w/index.php?title=Sidebandshttp://en.wikipedia.org/w/index.php?title=Spurious_emissionhttp://en.wikipedia.org/w/index.php?title=Taylor_serieshttp://en.wikipedia.org/w/index.php?title=Approximationhttp://en.wikipedia.org/w/index.php?title=Calculatehttp://en.wikipedia.org/w/index.php?title=Theoreticalhttp://en.wikipedia.org/w/index.php?title=Signal_processinghttp://en.wikipedia.org/w/index.php?title=Non-linearhttp://en.wikipedia.org/w/index.php?title=Multiple_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Integerhttp://en.wikipedia.org/w/index.php?title=Harmonichttp://en.wikipedia.org/w/index.php?title=Frequencieshttp://en.wikipedia.org/w/index.php?title=Signal_%28electrical_engineering%29http://en.wikipedia.org/w/index.php?title=Amplitude_modulationhttp://en.wikipedia.org/w/index.php?title=File%3ARF_Intermodulation_at_280_MHz.jpghttp://en.wikipedia.org/w/index.php?title=Frequency_spectrumhttp://en.wikipedia.org/w/index.php?title=Ballast_%28electrical%29http://en.wikipedia.org/w/index.php?title=Fluorescent_lighthttp://en.wikipedia.org/w/index.php?title=Electrical_impedancehttp://en.wikipedia.org/w/index.php?title=Magnetic_amplifierhttp://en.wikipedia.org/w/index.php?title=Saturable_core_inductorhttp://en.wikipedia.org/w/index.php?title=Voltage_regulatorhttp://en.wikipedia.org/w/index.php?title=Arc_weldinghttp://en.wikipedia.org/w/index.php?title=Intermodulation_distortionhttp://en.wikipedia.org/w/index.php?title=Harmonichttp://en.wikipedia.org/w/index.php?title=Signal_%28circuit_theory%29http://en.wikipedia.org/w/index.php?title=Alternating_currenthttp://en.wikipedia.org/w/index.php?title=Linear_circuithttp://en.wikipedia.org/w/index.php?title=Inductancehttp://en.wikipedia.org/w/index.php?title=Linear_circuithttp://en.wikipedia.org/w/index.php?title=Inductorhttp://en.wikipedia.org/w/index.php?title=Electronic_circuithttp://en.wikipedia.org/w/index.php?title=Electric_utilityhttp://en.wikipedia.org/w/index.php?title=Transformerhttp://en.wikipedia.org/w/index.php?title=Electromagnet


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Intermodulation 3

interference, which can reduce audio clarity or increase spectrum usage. It should not be confused with harmonic

distortion (which has common musical applications), nor with intentional modulation (such as a frequency mixer in

superheterodyne receivers) where signals to be modulated are presented to an intentional nonlinear element

(multiplied) (see non-linear mixers such as mixer diodes and even single-transistor oscillator-mixer circuits). In

audio, the intermodulation products are nonharmonically related to the input frequencies and therefore "off-key"

with respect to the common Western musical scale.

Causes of intermodulation

A linear system cannot produce intermodulation. If the input of a linear time-invariant system is a signal of a single

frequency, then the output is a signal of the same frequency; only the amplitude and phase can differ from the input

signal. However, non-linear systems generate harmonics, meaning that if the input of a non-linear system is a signal

of a single frequency, then the output is a signal which includes a number of integer multiples of the input

frequency; (i.e. some of ).

Intermodulation occurs when the input to a non-linear system is composed of two or more frequencies. Consider an

input signal that contains three frequency components at , , and ; which may be expressed as

where the and are the amplitudes and phases of the three components, respectively.

We obtain our output signal, , by passing our input through a non-linear function:

will contain the three frequencies of the input signal, , , and (which are known as thefundamental

frequencies), as well as a number of linear combinations of the fundamental frequencies, each of the form

where , , and are arbitrary integers which can assume positive or negative values. These are the

intermodulation products (or IMPs).

In general, each of these frequency components will have a different amplitude and phase, which depends on the

specific non-linear function being used, and also on the amplitudes and phases of the original input components.

More generally, given an input signal containing an arbitrary number of frequency components

, the output signal will contain a number of frequency components, each of which may be described by

where the coefficients are arbitrary integer values.
http://en.wikipedia.org/w/index.php?title=Linear_combinationhttp://en.wikipedia.org/w/index.php?title=Harmonichttp://en.wikipedia.org/w/index.php?title=Phase_%28waves%29http://en.wikipedia.org/w/index.php?title=Amplitudehttp://en.wikipedia.org/w/index.php?title=LTI_system_theoryhttp://en.wikipedia.org/w/index.php?title=Transistorhttp://en.wikipedia.org/w/index.php?title=Diodehttp://en.wikipedia.org/w/index.php?title=Electronic_mixer%23Product_mixershttp://en.wikipedia.org/w/index.php?title=Non-linearhttp://en.wikipedia.org/w/index.php?title=Analog_multiplierhttp://en.wikipedia.org/w/index.php?title=Superheterodyne_receiverhttp://en.wikipedia.org/w/index.php?title=Frequency_mixerhttp://en.wikipedia.org/w/index.php?title=Distortion_%28music%29http://en.wikipedia.org/w/index.php?title=Harmonic_distortionhttp://en.wikipedia.org/w/index.php?title=Harmonic_distortionhttp://en.wikipedia.org/w/index.php?title=Interference_%28communication%29


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Intermodulation 4

Intermodulation order

Distribution of third-order intermodulations: in blue the position of the fundamental

carriers, in red the position of dominant IMPs, in green the position of specific IMPs.

The order of a given

intermodulation product is the sum of

the absolute values of the coefficients,

For example, in our original example

above, third-order intermodulation

products (IMPs) occur where

:

In many radio and audio applications,

odd-order IMPs are of most interest, as

they fall within the vicinity of the

original frequency components, and

may therefore interfere with the

desired behaviour.

Passive Intermodulation

(PIM)

As explained in a previous section, intermodulation can only occur in non-linear systems. Non-linear systems are

generally composed of active components, meaning that the components must be biased with an external power

source which is not the input signal (i.e. the active components must be "turned on"). Passive intermodulation (PIM),

however, occurs in passive devices (which may include cables, antennas etc.) that are subjected to two or more high

power tones. The PIM product is the result of the two (or more) high power tones mixing at device nonlinearities

such as junctions of dissimilar metals, metal-oxide junctions and even loose connectors. The higher the signal

amplitudes, the more pronounced the effect of the nonlinearities, and the more prominent the intermodulation that

occurs - even though upon initial inspection, the system would appear to be linear and unable to generate

intermodulation.

Sources of PIM

Ferromagnetic materials are the most common materials to avoid and include ferrites, nickel, (including nickel

plating) and steels (including some stainless steels.) These materials exhibit hysteresis when exposed to reversingmagnetic fields resulting in PIM generation.

PIM can also be generated in components with manufacturing or workmanship defects, such as cold or cracked

solder joints or poorly made mechanical contacts. If these defects are exposed to high RF currents, PIM can be

generated. As a result, RF equipment manufacturers perform factory PIM tests on components, to eliminate PIM

caused by these design and manufacturing defects.

In the field, PIM can be caused by components that were damaged in transit to the cell site, installation workmanship

issues and by external PIM sources. Some of these include:

Contaminated surfaces or contacts due to dirt, dust, moisture or oxidation.

Loose mechanical junctions due to inadequate torque, poor alignment or poorly prepared contact surfaces. Loose mechanical junctions caused during transportation, shock or vibration.

Metal flakes or shavings inside RF connections.
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Intermodulation 5

Inconsistent metal-to-metal contact between RF connector surfaces caused by any of the following:

Trapped dielectric materials (adhesives, foam, etc.), cracks or distortions at the end of the outer conductor of coaxial

cables, often caused by overtightening the back nut during installation, solid inner conductors distorted in the

preparation process, hollow inner conductors excessively enlarged or made oval during the preparation process.

PIM can also occur in connectors, or when conductors made of two galvanically unmatched metals come in

contact with each other. Nearby metallic objects in the direct beam and side lobes of the transmit antenna including rusty bolts, roof

flashing, vent pipes, guy wires, etc.

PIM Testing

IEC 62037 is the international standard for PIM testing and gives specific details as to PIM measurement setups. The

standard specifies the use of two +43 dBm (20W) tones for the test signals for PIM testing. This power level has

been used by RF equipment manufacturers for more than a decade to establish PASS / FAIL specifications for RF

components.

Intermodulation in electronic circuits

Intermodulation is caused by nonlinearity or parameter limitations in an amplifier system. This nonlinearity can be

characterized in many ways, including the slew rate, crossover distortion, reduced transistor current gain, or

saturation of collector-emitter junctions near clipping. Slew-induced distortion (SID) can produce intermodulation

distortion (IMD) when the first signal is slewing (changing voltage) at the limit of the amplifier's power bandwidth

product. This induces an effective reduction in gain, partially amplitude-modulating the second signal. If SID only

occurs for a portion of the signal, it is called "transient" intermodulation distortion.[1]

This usually occurs due to soft

clipping of the signal peaks.[2]

Intermodulation in audio applications

Any system that produces harmonic distortion also produces intermodulation distortion. The difference between the

two types of distortion is the signal used to test it, not the device. Only sine waves produce harmonic distortion;

music signals always produce intermodulation.

Audio engineers usually strive to avoid intermodulation, as for anything other than extremely simple input

waveforms, it introduces frequency components that are not harmonically related, which tends to sound unmusical

and unpleasant. However, certain audio effects rely on amplitude modulation; these include tremolo and ring

modulation. One way to generate such effects is through deliberate intermodulation in a non-linear device, but may

also be achieved without intermodulation by an analog multiplier.Wikipedia:Disputed statement Transient

intermodulation distortion, or TIM, occurs in amplifiers that employ negative feedback when signal delays make theamplifier incapable of correcting distortion when exposed to fast, transient signals.

Harmonic distortion occurs when non-linearity (in an amplifier or loudspeaker, for instance) only creates new

frequencies that are harmonically related to the original sine wave signal. Intermodulation distortion occurs when a

different typeWikipedia:Disputed statement of non-linearity can create new frequencies that are not harmonically

related to the original signal. All audio devices give rise to distortion to some extent; harmonic distortion and

intermodulation distortion tests highlight different aspects of imperfections, and one type of distortion may be

inaudibly low while the other is significantly high for some equipment under certain conditions.Wikipedia:Disputed

statement[citation needed]

In analog recording, wow and flutter are forms of intermodulation distortion caused by speed variations in the

medium (usually tape). When the flutter rate is above a certain point, typically about 20Hz, the modulation products

impressed into the musical signal no longer present as an audibly obvious flutter, yet continue to interfere with the
http://en.wikipedia.org/w/index.php?title=Magnetic_tapehttp://en.wikipedia.org/w/index.php?title=Audio_frequencyhttp://en.wikipedia.org/w/index.php?title=Magnetic_tapehttp://en.wikipedia.org/w/index.php?title=Audio_frequencyhttp://en.wikipedia.org/w/index.php?title=Magnetic_tapehttp://en.wikipedia.org/w/index.php?title=Audio_frequencyhttp://en.wikipedia.org/w/index.php?title=Audio_frequencyhttp://en.wikipedia.org/w/index.php?title=Magnetic_tapehttp://en.wikipedia.org/w/index.php?title=Flutter_%28electronics_and_communication%29http://en.wikipedia.org/w/index.php?title=Wow_%28recording%29http://en.wikipedia.org/wiki/Citation_neededhttp://en.wikipedia.org/wiki/Disputed_statementhttp://en.wikipedia.org/wiki/Disputed_statementhttp://en.wikipedia.org/w/index.php?title=Audio_system_measurementshttp://en.wikipedia.org/w/index.php?title=Distortionhttp://en.wikipedia.org/wiki/Disputed_statementhttp://en.wikipedia.org/w/index.php?title=Loudspeakerhttp://en.wikipedia.org/w/index.php?title=Amplifierhttp://en.wikipedia.org/w/index.php?title=Harmonic_distortionhttp://en.wikipedia.org/wiki/Disputed_statementhttp://en.wikipedia.org/w/index.php?title=Analog_multiplierhttp://en.wikipedia.org/w/index.php?title=Ring_modulationhttp://en.wikipedia.org/w/index.php?title=Ring_modulationhttp://en.wikipedia.org/w/index.php?title=Tremolohttp://en.wikipedia.org/w/index.php?title=Amplitude_modulationhttp://en.wikipedia.org/w/index.php?title=Soft_clippinghttp://en.wikipedia.org/w/index.php?title=Soft_clippinghttp://en.wikipedia.org/w/index.php?title=Amplitude_Modulationhttp://en.wikipedia.org/w/index.php?title=Power_bandwidthhttp://en.wikipedia.org/w/index.php?title=Slew-induced_distortionhttp://en.wikipedia.org/w/index.php?title=Current_gainhttp://en.wikipedia.org/w/index.php?title=Crossover_distortionhttp://en.wikipedia.org/w/index.php?title=Slew_ratehttp://en.wikipedia.org/w/index.php?title=Galvanization


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Intermodulation 6

signal as extraneous frequency modulation, and the resulting sideband products manifest as distortion. This

distortion results in a thicker, grainier texture due to the excess non-musical sum and difference components riding

above and below the harmonic content of the material.

Measurement

Intermodulation distortion in audio is usually specified as the Root Mean Square (RMS) value of the various

sum-and-difference signals as a percentage of the original signal's RMS voltage, although it may be specified in

terms of individual component strengths, in decibels, as is common with RF work. Audio IMD standard tests include

SMPTE standard RP120-1994 where two signals (at 60 Hz and 7 kHz, with 4:1 amplitude ratios) are used for the

test; many other standards (such as DIN, CCIF) use other frequencies and amplitude ratios. Opinion varies over the

ideal ratio of test frequencies (e.g. 3:4,[3]

or almost -but not exactly - 3:1 for example).

After feeding the equipment under test with low distortion input sinewaves, the output distortion can be measured by

using an electronic filter to remove the original frequencies, or spectral analysis may be made using Fourier

Transformations in software or a dedicated spectrum analyser, or when determining intermodulation effects in

communications equipment, may be made using the receiver under test itself.

Using a modern network analyzer with two internal RF sources and sensitive RF detectors simplifies the

measurement setup and also provides a sensitivity level comparable to spectrum analyzers. Furthermore, a calibrated

VNA setup also removes mismatch errors from measurements which otherwise would be present in spectrum

analyzer measurements. Meanwhile error-corrected IM measurement systems are available. These system support

frequency converting vector-measurements of S-parameters.[4]

The user can locate IM-sources and perform a vector

or time-domain fitting or modelling of the IM-signals and components.

External links

Lloyd Butler (1997). "Intermodulation Performance and Measurement of Intermodulation Components"[5]

.

VK5BR. "Amateur Radio," August 1997. Retrieved 30 January 2012.

References

[1] Rane Pro Audio Reference for IM (http://www.rane.com/par-i.html#IM)

[2] http://waltjung.org/PDFs/SID_TIM_TAA77_P1.pdf Slewing Induced Distortion in Audio Amplifiers, Part 1 by Walter Jung in The Audio

Amateur Issue 1/1977

[3] http://www.leonaudio.com. au/3-4.ratio.distortion.measurement.pdf Graeme John Cohen: 3-4 Ratio; A method of measuring distortion

products

[4] Thalayasingam, K. and Heuermann, H.,Novel Vector Non-Linear Measurement System for Intermodulation Measurements, European

Microwave Conference, Rom, Italy, IEEE, 2009, Available online (http://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&

queryText=thalayasingam&x=32&y=12=no)

[5] http://users.tpg.com.au/ldbutler/Intermodulation.htm

This article incorporates public domain material from the General Services Administration document "Federal

Standard 1037C" (http://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm) (in support of MIL-STD-188).
http://en.wikipedia.org/w/index.php?title=MIL-STD-188http://www.its.bldrdoc.gov/fs-1037/fs-1037c.htmhttp://en.wikipedia.org/w/index.php?title=General_Services_Administrationhttp://en.wikipedia.org/w/index.php?title=Copyright_status_of_work_by_the_U.S._governmenthttp://en.wikipedia.org/w/index.php?title=File:PD-icon.svghttp://users.tpg.com.au/ldbutler/Intermodulation.htmhttp://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=thalayasingam&x=32&y=12=nohttp://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=thalayasingam&x=32&y=12=nohttp://www.leonaudio.com.au/3-4.ratio.distortion.measurement.pdfhttp://waltjung.org/PDFs/SID_TIM_TAA77_P1.pdfhttp://www.rane.com/par-i.html#IMhttp://users.tpg.com.au/ldbutler/Intermodulation.htmhttp://en.wikipedia.org/w/index.php?title=Network_analyzer_%28electrical%29http://en.wikipedia.org/w/index.php?title=Spectrum_analyserhttp://en.wikipedia.org/w/index.php?title=Fast_Fourier_transformhttp://en.wikipedia.org/w/index.php?title=Fast_Fourier_transformhttp://en.wikipedia.org/w/index.php?title=Electronic_filterhttp://en.wikipedia.org/w/index.php?title=Audio_system_measurementshttp://en.wikipedia.org/w/index.php?title=Radio_frequencyhttp://en.wikipedia.org/w/index.php?title=Decibelhttp://en.wikipedia.org/w/index.php?title=Root_Mean_Squarehttp://en.wikipedia.org/w/index.php?title=Frequency_modulation


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Volterra series 7

Volterra series

The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series

in its ability to capture 'memory' effects. The Taylor series can be used to approximate the response of a nonlinear

system to a given input if the output of this system depends strictly on the input at that particular time. In the

Volterra series the output of the nonlinear system depends on the input to the system at all other times. This provides

the ability to capture the 'memory' effect of devices such as capacitors and inductors.

It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also

used in electrical engineering to model intermodulation distortion in many devices including power amplifiers and

frequency mixers. Its main advantage lies in its generality: it can represent a wide range of systems. It is therefore

sometimes referred to as a non-parametric model.

In mathematics, a Volterra series denotes a functional expansion of a dynamic, nonlinear, time-invariant functional.

Volterra series are frequently used in system identification. The Volterra series, which is used to prove the Volterra

theorem, is a series of infinite sum of multidimensional convolutional integrals.

History

Volterra series is a modernized version of the theory of analytic functionals due to the Italian mathematician Vito

Volterra in work dating from 1887.[1]

Norbert Wiener became interested in this theory in the 1920s from contact

with Volterra's student Paul Lvy. He applied his theory of the Brownian motion to the integration of Volterra

analytic functionals. The use of Volterra series for system analysis originated from a restricted 1942 wartime

report[2]

of Wiener, then professor of mathematics at MIT. It used the series to make an approximate analysis of the

effect of radar noise in a nonlinear receiver circuit. The report became public after the war.[3]

As a general method of

analysis of nonlinear systems, Volterra series came into use after about 1957 as the result of a series of reports, at

first privately circulated, from MIT and elsewhere.[4]

The name Volterra series came into use a few years later.

Mathematical theory

The theory of Volterra series can be viewed from two different perspectives: either one considers an operator

mapping between two real (or complex) function spaces or a functional mapping from a real (or complex) function

space into the real (or complex) numbers. The latter, functional perspective is in more frequent use, due to the

assumed time-invariance of the system.

Continuous time

A continuous time-invariant system withx(t) as input andy(t) as output can be expanded in Volterra series as:

where and .

, are called n-th order Volterra kernel which can be regarded as a higher-order impulse

response of the system.

IfN is finite, the series operator is said truncated.

If a,b andN are finite, the series operator is called doubly-finite Volterra series.
http://en.wikipedia.org/w/index.php?title=Integral_kernelhttp://en.wikipedia.org/w/index.php?title=Massachusetts_Institute_of_Technologyhttp://en.wikipedia.org/w/index.php?title=Brownian_motionhttp://en.wikipedia.org/w/index.php?title=Paul_L%C3%A9vy_%28mathematician%29http://en.wikipedia.org/w/index.php?title=Norbert_Wienerhttp://en.wikipedia.org/w/index.php?title=Vito_Volterrahttp://en.wikipedia.org/w/index.php?title=Vito_Volterrahttp://en.wikipedia.org/w/index.php?title=System_identificationhttp://en.wikipedia.org/w/index.php?title=Functional_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Nonlinearhttp://en.wikipedia.org/w/index.php?title=Mathematicshttp://en.wikipedia.org/w/index.php?title=Non-parametrichttp://en.wikipedia.org/w/index.php?title=Frequency_mixerhttp://en.wikipedia.org/w/index.php?title=Taylor_series


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Volterra series 8

Sometimes the n-th order term is divided by n!, a convention which is convenient when considering the combination

of Volterra systems by placing one after the other ('cascading').

The causality condition: Since in any physically realizable system the output can only depend on previous values of

the input, the kernels will be zero if any of the variables are negative. The

integrals may then be written over the half range from zero to infinity. So if the operator is causal, .

Frchet's approximation theorem: The use of the Volterra series to represent a time-invariant functional relation isoften justified by appealing to a theorem due to Frchet. This theorem states that such a system can be approximated

uniformly and to an arbitrary degree of precision by a sufficiently high finite order Volterra series. The input set over

which this approximation holds must be compact. This is usually taken to be the set of equicontinuous, uniformly

bounded functions which is compact by the ArzelAscoli theorem. In many physical situations this assumption

about the input set is a reasonable one. The theorem however gives no indication as to how many terms are needed

for a good approximation which is the important question in applications.

Discrete time

where and .

, are called Volterra kernels.

IfP is finite, the series operator is said truncated.

If a,b andP are finite the series operator is called doubly-finite Volterra series.

If the operator is causal.

We can always consider, without loss of the generality, the kernel as symmetrical. In fact, for the

commutativity of the multiplication it is always possible to symmetrize it without changing .

So for a causal system with symmetrical kernels we can write

Methods to estimate the kernel coefficients

Estimating the Volterra coefficients individually is complicated since the basis functionals of the Volterra series are

correlated. This leads to the problem of simultaneously solving a set of integral-equations for the coefficients. Hence,

estimation of Volterra coefficients is generally performed by estimating the coefficients of an orthogonalized series,

e.g. the Wiener series, and then recomputing the coefficients of the original Volterra series. The Volterra series main

appeal over the orthogonalized series lies in its intuitive, canonical structure, i.e. all interactions of the input have

one fixed degree. The orthogonalized basis functionals will generally be quite complicated.

An important aspect, with respect to which the following methods differ is whether the orthogonalization of the basis

functionals is to be performed over the idealized specification of the input signal (e.g. gaussian, white noise) or over

the actual realization of the input (i.e. the pseudo-random, bounded, almost-white version of gaussian white noise, or

any other stimulus). The latter methods, despite their lack of mathematical elegance, have been shown to be more

flexible (as arbitrary inputs can be easily accommodated) and precise (due to the effect that the idealized version of

the input signal is not always realizable).
http://en.wikipedia.org/w/index.php?title=Wiener_serieshttp://en.wikipedia.org/w/index.php?title=Arzel%C3%A0%E2%80%93Ascoli_theoremhttp://en.wikipedia.org/w/index.php?title=Uniformly_boundedhttp://en.wikipedia.org/w/index.php?title=Uniformly_boundedhttp://en.wikipedia.org/w/index.php?title=Equicontinuoushttp://en.wikipedia.org/w/index.php?title=Compact_spacehttp://en.wikipedia.org/w/index.php?title=Maurice_Ren%C3%A9_Fr%C3%A9chet


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Volterra series 9

Crosscorrelation method

This method, developed by Lee & Schetzen, orthogonalizes with respect to the actual mathematical description of

the signal, i.e. the projection onto the new basis functionals is based on the knowledge of the moments of the random

signal.

To allow identification orthogonalization, Volterra series must be rearranged in terms of orthogonal

non-homogeneous G operators (Wiener series):

The G operators can be defined by the following

whenever is arbitrary omogeneous Volterra,x(n) is a Stationary white noise with zero mean and variance

A.

Recalling that every Volterra functional is orthogonal to all Wiener functional of greater order, and considering the

following Volterra functional

we can write

Ifx is SWN, and by letting , we have:

So if we exclude the diagonal elements, , it is

If we want to consider the diagonal points, the solution proposed by Lee and Schetzen is:

Efficient formulas and references for diagonal kernel point estimation can be found in

[5]

and .

[6]

Exact orthogonal algorithm

This method and its more efficient version (Fast Orthogonal Algorithm) were invented by Korenberg. In this method

the orthogonalization is performed empirically over the actual input. It has been shown to perform more precisely

than the Crosscorrelation method. Another advantage is that arbitrary inputs can be used for the orthogonalization

and that fewer data-points suffice to reach a desired level of accuracy. Also, estimation can be performed

incrementally until some criterion is fulfilled.
http://en.wikipedia.org/w/index.php?title=Wiener_series


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Volterra series 10

Linear regression

Linear regression is a standard tool from linear analysis. Hence, one of its main advantages is the widespread

existence of standard tools for solving linear regressions efficiently. It has some educational value, since it highlights

the basic property of Volterra series: linear combination of non-linear basis-functionals. For estimation the order of

the original should be known, since the volterra basis-functionals are not orthogonal and estimation can thus not be

performed incrementally.

Kernel method

This method was invented by Franz & Schlkopf and is based on statistical learning theory. Consequently, this

approach is also based on minimizing the empirical error (often called empirical risk minimization). Franz and

Schlkopf proposed that the kernel method could essentially replace the Volterra series representation, although

noting that the latter is more intuitive.

Differential sampling

This method was developed by van Hemmen and coworkers and utilizes Dirac delta functions to sample the Volterra

coefficients.

References

[1][1] Vito Volterra. Theory of Functionals and of Integrals and Integro-Differential Equations. New York: Dover Publications, 1959.

[2] Wiener N:Response of a nonlinear device to noise. Radiation Lab MIT 1942, restricted. report V-16, no 129 (112 pp). Declassified Jul 1946,

Published as rep. no. PB-1-58087, U.S. Dept. Commerce. URL: http://www.dtic.mil/dtic/tr/fulltext/u2/a800212. pdf

[3] Ikehara S:A method of Wiener in a nonlinear circuit. MIT Dec 10 1951, tech. rep. no 217, Res. Lab. Electron.

[4][4] Early MIT reports by Brilliant, Zames, George, Hause, Chesler can be found on dspace.mit.edu.

[5] M. Pirani, S. Orcioni, and C. Turchetti, ``Diagonal kernel point estimation of n-th order discrete Volterra-Wiener systems,EURASIP Journal

on Applied Signal Processing, vol. 2004, no. 12, pp. 1807--1816, Sept. 2004.

[6] S. Orcioni, M. Pirani, and C. Turchetti, ``Advances in Lee-Schetzen method for Volterra filter identification,Multidimensional Systems and

Signal Processing, vol. 16, no. 3, pp. 265--284, 2005.

Further reading

Barrett J.F:Bibliography of Volterra series, Hermite functional expansions, and related subjects. Dept. Electr.

Engrg, Univ.Tech. Eindhoven, NL 1977, T-H report 77-E-71. (Chronological listing of early papers to 1977)

URL: http://alexandria.tue.nl/extra1/erap/publichtml/7704263.pdf

Bussgang, J.J.; Ehrman, L.; Graham, J.W:Analysis of nonlinear systems with multiple inputs,Proc. IEEE, vol.62,

no.8, pp. 10881119, Aug. 1974

Giannakis G.B & Serpendin E:A bibliography on nonlinear system identification. Signal Processing, 81 2001

533580. (Alphabetic listing to 2001) www.elsevier.nl/locate/sigpro Korenberg M.J. Hunter I.W: The Identification of Nonlinear Biological Systems: Volterra Kernel Approaches,

Annals Biomedical Engineering (1996), Volume 24, Number 2.

Kuo Y L:Frequency-domain analysis of weakly nonlinear networks, IEEE Trans. Circuits & Systems,

vol.CS-11(4) Aug 1977; vol.CS-11(5) Oct 1977 26.

Rugh W J:Nonlinear System Theory: The VolterraWiener Approach. Baltimore 1981 (Johns Hopkins Univ

Press) Many online versions, e.g. www.ece.jhu.edu/~rugh/volterra/book.pdf

Schetzen M: The Volterra and Wiener Theories of Nonlinear Systems, New York: Wiley, 1980.
http://en.wikipedia.org/w/index.php?title=Georgios_B._Giannakishttp://alexandria.tue.nl/extra1/erap/publichtml/7704263.pdfhttp://www.dtic.mil/dtic/tr/fulltext/u2/a800212.pdfhttp://en.wikipedia.org/w/index.php?title=Dirac_delta_functionhttp://en.wikipedia.org/w/index.php?title=Statistical_learning_theory


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Article Sources and Contributors 11

Article Sources and ContributorsSaturation (magnetic) Source: http://en.wikipedia.org/w/index.php?oldid=593004054 Contributors: A13ean, ARTE, Avoided, BZegarski, BenFrantzDale, Chetvorno, Dcshank, Francisco

Quiumento, Gene Nygaard, JPFen, Jim.henderson, KoenDelaere, Mnmngb, Red Act, Rjwilmsi, RockMagnetist, Rtdrury, Shaddack, StAnselm, Steve Quinn, Teaktl17, The Anome, TomyDuby,

Tone, UAwiki, Zhangzhe0101, 23 anonymous edits

Intermodulation Source: http://en.wikipedia.org/w/index.php?oldid=596334297 Contributors: A. B., Azathoth101, Bllix, Brownout, Cadmium, Chris the speller, ChrisGualtieri, DexDor,

Dlrohrer2003, Eastlaw, Edcolins, Ettrig, Hooperbloob, Howdybob, Jamiehowarth, Jimijames, John of Reading, Keenan Pepper, Kevinpurcell, Kle0012, LCS check, Lampglow, Larrymcp,

LittleWink, Lusum, Maitchy, Misza13, Mr. PIM, Nigelj, Nimur, Ninly, Oleg Alexandrov, Oli Filth, Omegatron, Pfeldman, Pjacobi, Plamka, R'n'B, Radagast83, Radiodef, Radiojon, Ronrkey,Salsb, Scanmap, Serrano24, Skapur, Temblast, The Anome, TheParanoidOne, TiQuA, Toffile, Vegaswikian, 59 anonymous edits

Volterra series Source: http://en.wikipedia.org/w/index.php?oldid=595265387 Contributors: Alsee, Belizefan, Benglitz, Brews ohare, Christaj, DragonflySixtyseven, EffeX2, Foxcob2, Giftlite,

GirasoleDE, JFB80, Jitse Niesen, Jlataire, Konstable, Kuru, Mdd, Memming, Michael Hardy, Myasuda, Oleg Alexandrov, Omphaloscope, Orcioni, Rjwilmsi, Squids and Chips, Sawomir Biay,

TedPavlic, VK35, 19 anonymous edits


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Image Sources, Licenses and ContributorsFile:Magnetization curves.svg Source: http://en.wikipedia.org/w/index.php?title=File:Magnetization_curves.svg License: Public Domain Contributors: Charles Proteus Steinmetz

File:Permeability of ferromagnet by Zureks.svg Source: http://en.wikipedia.org/w/index.php?title=File:Permeability_of_ferromagnet_by_Zureks.svg License: Public Domain Contributors:

Zureks

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Contributors: Alvin-cs, Nimur

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