Equivalent Circuit Modeling and Application of Electro-Thermal and Electro-Mechanical Power Conversion Systems

Thesis submitted in partial fulfillment of the requirements for the degree of

“DOCTOR OF PHILOSOPHY”

by

Simon Lineykin

Submitted to the Senate of Ben-Gurion University of the Negev

March 2006

Beer-Sheva

i

Equivalent Circuit Modeling and Application of Electro-Thermal and Electro-Mechanical Power Conversion Systems

Thesis submitted in partial fulfillment of the requirements for the degree of

“DOCTOR OF PHILOSOPHY”

by

Simon Lineykin

Submitted to the Senate of Ben-Gurion University of the Negev

Approved by the advisor _______________________________________________________________ Approved by the Dean of the Kreitman School of Advanced Graduate Studies ___________________

March 2006

Beer-Sheva

ii

This work was carried out under the supervision of Professor Shmuel (Sam) Ben-Yaakov

In the Department Electrical & Computer Engineering

Faculty of Engineering Sciences

iii

Abstract

In the present work, an approach to application of devices that use direct electro-

thermal and electro-mechanical energy conversion processes (DEC) is presented.

Simplified mathematical models of physical processes were used to create generic

electrical equivalent schemes of the devices. An in-depth analysis of the equivalent circuits

was performed. Several applications of piezoelectric and thermoelectric devices were

proposed.

The present dissertation proposes a user-friendly method for solving steady-state

operational regime of an active cooling system. The method was devised for one-stage

thermoelectric (Peltier) cooler and may be extended to multistage coolers. An active

cooling system is represented as a two-port equivalent circuit (Thevenin's generator).

Instead of using the traditional iteration method, we propose a simple and intuitive

graphical approach to design and analyze an active cooling system.

Another type of model, the tri-port thermo-electrical dynamic model of an active

cooling system and thermoelectric generator is also presented. This model shows very

good results when simulated using PSPICE in DC, TRANSIENT, and AC simulation

types. This tool gives the user the opportunity to design, optimize, and simulate a

thermoelectric system together with electrical drives, loads and close-loop control system,

instead the laboratory examinations. A methodology for extracting the parameters of the

proposed model from manufacturers’ data of TECs is developed. Several examples of

successful applications of the model are presented. Data of many different manufacturers

were examined and the model parameters were extracted. In all cases, the model was found

to accurately reproduce the performance of commercial TECs. The accuracy of the model

was verified also by experiments.

The feasibility of using a piezoelectric transformer (PT) as a galvanic barrier was

examined in this study theoretically and experimentally. The research included the factors

of drive, demodulation, bandwidth, and common mode rejection. In particular, two types

of excitation signals were compared: amplitude modulated signal and frequency modulated

signal. The frequency response of the piezoelectric isolator was studied both

experimentally and by small signal envelope simulation using ORCAD/PSPICE.

The envelope simulation method, developed earlier for large signal simulation in time

domain, (TRAN), was extended to include small signal envelope simulation (AC) and DC

sweep simulation. The model was derived for AM, FM and PM modulation schemes and

was demonstrated on a piezoelectric transformer circuit. The analytical derivations of the

iv

model were verified both experimentally and against full circuit simulations that included a

high frequency carrier. Excellent agreement was found between the simulation and

experimental results.

The problem of maximum power point tracking of high output DC voltage converters

that apply PTs and voltage doublers was studied theoretically and experimentally. It was

shown that the operating frequency of a PT at maximum power is a function of the load.

Hence, under load variations and to overcome parameter instability, there is a need for

some frequency tracking mechanism that will help to lock the operating frequency to the

optimum frequency. The proposed method of frequency tracking is based on phase locked

loop (PLL). By applying this approach, the system’s operation can be made independent of

input voltage, load variations, temperature (within a permitted range), and the spread and

non-linearity of the PT parameters as well as their drift with time.

v

Acknowledgements

The Kreitman school of Advanced Graduate Studies and the Electrical and Computer

Engineering department of Ben-Gurion of the Negev is gratefully acknowledged for

hosting the research and full financial support. A sub-project was supported by THE

ISRAEL SCIENCE FOUNDATION (grant No. 113/02) and by the Paul Ivanier Center for

Robotics and Production management.

I would like to thank Prof. Sam Ben-Yaakov for his invaluable guidance and continued

support during the course of this research work. I would also like to thank Professor

Gregory Ivensky for his help and encouragement. I have enjoyed the working environment

at the Department of Electrical and Computer Engineering and extend my thanks to its

entire staff. During these years, I have had good relationships with my colleagues at the

power electronic laboratory and at the department and I would also like to thank my

colleagues for pleasant times. Special thanks go to all the PEL students and staff for their

help during my graduate career.

I would like to express my profound gratitude to my parents for their constant support

and encouragement in my educational pursuits. Finally yet importantly, I would like to

thank my wife Olga for her love and unlimited patience and my sons, Israel and Alexander

who made life worth struggling during difficult times.

vi

List of Figures

Fig. 1. The flow chart of the approach to applying a DED using an equivalent circuit model. 2

Fig. 2. Rectangular longitude-mode piezoelectric transformer including two pairs of electrodes and the piezoelectric ceramic body. The schematic electrical source and load are connected to the PT at primary and secondary electrode pairs. 4

Fig. 3. The topology of the equivalent circuit of a PT operating close to resonant frequency. Rm, Cr, and LR describe the mechanical resonant system. The capacitors Ci and Co are a result of capacitance between the input and output electrode pairs. The ideal transformers 1:n1 and 1:n2 describe the DEC processes. 5

Fig. 4. Replacement of reactive elements by equivalent circuits for envelope simulation: (a) - Replacing an inductor by an inductor with a dependent voltage source; (b) - Replacing a capacitor by a capacitor with a dependent current source. 9

Fig. 5. Single-stage module construction. 10

Fig. 6. Flow chart of research. The blocks in the flow chart are the milestones of the work (paper publications), the ellipses are the questions and problems that have appeared at every step. (a) - the flow chart of study of the PT and its applications, (b) - the flowchart of the study of thermoelectric modules that has been carried out for the present dissertation. 16

vii

List of Appended Publications

• S. Lineykin and S. Ben-Yaakov, “Feedback isolation by piezoelectric

transformers: a feasibility study,” Power Conversion and Intelligent Motion,

PCIM-2000, 175-181, Nuremberg, Germany, 2000.

• S. Ben-Yaakov and S. Lineykin, “Frequency Tracking to Maximum Power of

Piezoelectric Transformer HV Converters under Load Variations,” IEEE

Transactions on Power Electronics, vol. 21, no. 1, pp. 73 - 78, 2006.

• S. Lineykin and S. Ben-Yaakov, “A Unified SPICE Compatible Model for Large

and Small Signal Envelope Simulation of Linear Circuits Excited by Modulated

Signals,” IEEE Transactions on Industrial Applications, accepted, 2003 - 2005.

• S. Lineykin and S. Ben-Yaakov, “Feedback Isolation by Piezoelectric

Transformers: Comparison of Amplitude to Frequency Modulation,” HAIT

Journal of Science and Engineering, v. 2, Issues 5-6, pp. 830-847, 2005.

• S. Lineykin and S. Ben-Yaakov, “Modeling and analysis of thermoelectric

modules,” IEEE Applied Power Electronics Conference APEC’05, pp.2019 -

2023, Austin, Texas, USA.

• S. Lineykin and S. Ben-Yaakov, "PSPICE-Compatible Equivalent Circuit of

Thermoelectric Coolers," IEEE Power Electronics Specialists

Conference, PESC'05, 608-612, Recife, Brazil, 2005.

• S. Lineykin and S. Ben-Yaakov, “Analysis of thermoelectric coolers by a SPICE-

compatible equivalent circuit model,” IEEE Power Electronics Letters, vol. 3, no.

2, pp. 63 - 66, 2005.

• S. Lineykin and S. Ben-Yaakov, A Simple and Intuitive Graphical Approach to

the Design of Thermoelectric Cooling Systems, International journal of

Refrigeration, submitted, 2006.

viii

Table of Contents

ABSTRACT.............................................................................................................................................III

ACKNOWLEDGEMENTS ......................................................................................................................V

LIST OF FIGURES................................................................................................................................. VI

LIST OF APPENDED PUBLICATIONS.............................................................................................. VII

TABLE OF CONTENTS ......................................................................................................................VIII

1 STATEMENT OF THE PROBLEM..................................................................................................1

2 REVIEW OF THE LITERATURE ON DED MODELING. .............................................................3

2.1 PIEZOELECTRIC TRANSFORMER ................................................................................................4 2.2 THE ENVELOPE ANALYSIS .........................................................................................................7 2.3 THERMOELECTRIC MODULES ..................................................................................................10

2.3.1 The steady state models .................................................................................................11 2.3.2 The dynamic models.......................................................................................................12 2.3.3 Use of electrical equivalent circuit for modeling heat transfer processes.....................12

3 OBJECTIVES ..................................................................................................................................14

4 APPENDED PUBLICATIONS .......................................................................................................15

4.1 EQUIVALENT CIRCUIT MODELING AND APPLICATION OF ELECTRO-MECHANICAL POWER

CONVERSION SYSTEMS ............................................................................................................17 4.1.1 Feedback isolation by piezoelectric transformers: a feasibility study [66]...................17 4.1.2 Frequency Tracking to Maximum Power of Piezoelectric Transformer HV Converters

under Load Variations. [67], [68]. ...............................................................................18 4.1.3 A Unified SPICE Compatible Model for Large and Small Signal Envelope Simulation of

Linear Circuits Excited by Modulated Signals [37], [38]. ...........................................19 4.1.4 Feedback Isolation by Piezoelectric Transformers: Comparison of Amplitude to

Frequency Modulation. [70], [72]................................................................................20 4.2 EQUIVALENT CIRCUIT MODELING AND APPLICATION OF ELECTRO-THERMAL POWER

CONVERSION SYSTEMS ............................................................................................................21 4.2.1 Modeling and Analysis of Thermoelectric Modules [73] ..............................................21 4.2.2 PSPICE-Compatible Equivalent Circuit of Thermoelectric Coolers [76], [77]...........22 4.2.3 Analysis of Thermoelectric Coolers by a SPICE-compatible Equivalent Circuit Model.

[71]................................................................................................................................23 4.2.4 A Simple and Intuitive Graphical Approach to the Design of Thermoelectric Cooling

Systems...........................................................................................................................24 5 CONCLUSIONS AND FUTURE WORK.......................................................................................25

REFERENCES.........................................................................................................................................26

1

1 Statement of the problem

Direct energy conversion (DEC) is the production of electricity from an energy source

without transferring the energy to a working fluid or steam. For example, thermoelectric

generators transform heat directly into electricity. The process is also reversible, so that a

generator can work as a cooler or heater; applying electrical power to thermoelectric

module one gets a temperature difference. Another example is when the electric field

forces a body of piezoelectric material to undergo deformation and vice versa, the piece of

piezoelectric material generates the electric field under mechanical pressure. The DEC is

possible by virtue of variations in thermodynamic properties of charge carriers (electrons,

ions, dipoles in points of lattice etc.).

Devices that rely on the principle of DEC (direct energy conversion devices - DED)

are used in a variety of applications [1], such as electro-mechanical, electro-chemical,

electro-thermal, photoelectrical and other converters. Such devices have been successfully

applied as sensors in measurement equipment. The state-of-the-art technology and novel

materials have also enabled power applications of DEC processes.

There is currently a surge of interest in studying and application of DEC processes.

Miniaturization of electronic and mechanic devices together with high efficiency of

operation have put DED in the forefront of technology. We will find more and more

articles on such processes. Piezoelectric motors, solid-state micro coolers, electro-

acoustical transducers, piezoelectric transformers etc. - a search by these keywords returns

hundreds of papers with varied applications and investigations.

DEC processes have been studied for more than a hundred years. All these processes

have been thoroughly described in mathematical terms, and many software solutions have

been created for simulating these processes. The effectiveness of the devices that use DEC

processes has been rapidly increasing with time. New materials allow direct energy

processes to perform high power operations. It appears that the technology is now ready

for large-scale applications.

In spite of the wide research that has been carried out on possible applications of

DEDs, on the market these devices still remain rare, almost exotic. One reason for this fact,

apparently, is due to mismatch between the vast scientific knowledge of the subject and the

limited number of simple behavior models for consumers - electrical engineers. An

engineer needs an easy-to-understand, universal model, adjustable for specific conditions,

or at least a template for building such a model on his own.

2

The study of DEC processes implies the study of both primary and secondary energy

types as well as corresponding thermodynamic processes, transport phenomena, and nature

of losses, which is a complicated task. The objective of the current research was to develop

a universal approach to the application of electro-thermal and electro-mechanical DEDs

using their equivalent circuit models.

Fig. 1 illustrates the scheme of the approach: the DED has a very complex

mathematical description but under the conditions of specific application, the model can be

significantly simplified and described by means of an electrical equivalent circuit. Each

specific application needs its own methodology to extract the model parameters, and the

topology of the equivalent circuit may be different from application to application.

Nonetheless, the approach for modeling the processes remains similar. When a preliminary

equivalent circuit is available, one can perform all types of network calculations and

simulations of the electronic system with a DED as the element, and estimate the behavior

of the system. This helps to choose the best DED from the manufacturers stock to achieve

the best performance of the application. The specific device needs corresponding

adjustments of the model. The final experimental stage of the thesis estimates the accuracy

of the model and tests the validity of the assumptions.

Fig. 1. The flow chart of the approach to applying a DED using an equivalent circuit model.

3

In the present work, a systemized modeling and approach for application of DEDs is

suggested. All the steps are illustrated by example of electromechanical (electro-

acoustical) and electro-thermal DEDs. Our analysis of the DED as a "regular" electrical

equivalent circuit instead of as a "black box," grants the designer the new perspective on

the problem and enables novel solutions for applying the devices.

2 Review of the literature on DED modeling.

This study was concerned with equivalent circuit modeling and application of electro-

thermal and electro-mechanical power conversion systems for two reasons: because of

their increasing use in power electronics in recent years, and because of the strong

dissimilarity between their available mathematical models. The present study explores

different topologies of the equivalent circuit, fields of application, methods of drive and

loading, and thus the universality of the proposed method is illustrated.

Mathematical modeling is a mathematical tool for solving real world problems

analytically or by simulations [2]. Today we have mathematical models describing almost

anything. DEC processes and DEDs are not an exception. A mathematical model can be

useful for studying physical processes by mathematical calculations. The mathematical

model enables computer simulations of processes. Simulations help to save money and

time. Simulation can help measure immeasurable things (such as worst case or critical

conditions); it also helps to optimize the parameters of a system without rebuilding it.

Mathematical models are informative: it means that the observation by modeling may tell

something about the way one parameter influences the performance of the system.

Among the known differential equations, only a limited number is usable in physics

and engineering. The same mathematical models sometimes describe vastly different

physical processes. For example, the same Fourier equation is used to describe both

conductive heat transfer and electrical fields. This makes models into a universal tool for

an engineer, because the quantity of applications is infinite, while the number of

mathematical models is limited.

There exists a very extensive literature on mathematical models. The author of [1]

gives a comprehensive representation of the physical nature and mathematical description

of main DEC processes. There are many books with detailed analysis of every specific

process and type of device, for example [3] and [4] about thermoelectric devices and [5],

[6], and [7] about piezoelectric devices. For the purposes of the present thesis, we will

concentrate on the survey of literature on modeling and applications of PTs and

thermoelectric modules, with the problems that their modeling involves.

4

2.1 Piezoelectric Transformer

Piezoelectric transformer (PT) is a solid state DED. PTs transform the energy of

alternating electric field on its primary (input) electrodes into electrical energy that may be

picked up by the secondary (output) electrode pair by means of direct conversion of energy

of the electrical field into mechanical vibrations and vice versa. A simple rectangular PT is

shown in Fig. 2. There are various forms and types of PTs [8]. The PTs normally operate

at a frequency close to one of its mechanical resonance frequencies. The PT is a resonant

piezoelectric device.

Fig. 2. Rectangular longitude-mode piezoelectric transformer including two pairs of electrodes

and the piezoelectric ceramic body. The schematic electrical source and load are connected to the PT at primary and secondary electrode pairs.

PTs have a number of merits, including potentially low costs, compact size, high

quality factor (Q) for solid-state devices, high efficiency for solid-state devices, ability to

work at high frequency, good insulation capability, high transformation coefficient, no

windings etc., which make the PT attractive for a range of applications in power

electronics [9]. Among possible applications are galvanic isolation in feedback loops,

electronic ballasts etc. The most recent PTs are able to transform significant values of

energy (tens of watts). A matrix of PTs in parallel operation can become a new generation

of power transformers.

However, PTs have several disadvantages, which seriously hinder their employment.

For example, their implementation demands complex frequency tracking, high modeling

complexity, using a carrier (resonant) frequency for signal transmission. The bandwidth of

a modulated signal is difficult to estimate, but it is normally low because it is conditioned

by a high Q. In addition, over time their main properties, which depend on temperature,

load etc. are obviously unstable.

Publications [5] and [6] present a full mathematical analysis of vibration of

piezoelectric plates and shells with distributed parameters. These studies provide a useful

3D dynamic modeling of PT vibrations. Such models are unique in allowing a cycle-by-

cycle examination of mechanical vibrations, acoustic wave propagation, vibration modes,

resonance conditions etc. for a variety of forms of bars and electrodes. Many works

concerned with design of an optimal PT use finite element model simulation. These models

are very informative and useful for PT design, except that they are practically useless when

5

in comes to designing applications. A PT model with distributed parameters is over-

informative; building and simulation of such models takes a great deal of time and involves

special software. In general, the software for simulating these models is incompatible with

the commonly used software for simulation of electronic components. Moreover, when

electronic components and DEDs are simulated separately, their interrelation is unclear.

One solution is to simplify the model under the specific application conditions and to use

simulation software compatible with traditional electrical circuit simulators, for example

PSPICE. A more appropriate model in this case would be the equivalent circuit model.

Publication [8] gives comprehensive information about resonant piezoelectric devices

such as PTs, filters, oscillators etc. Elements of modeling of piezoelectric devices using the

equivalent electrical circuit are also suggested. The proposed method of analogy is to use

an electrical resonant tank (RLC) to describe a mechanical vibrating system with lumped

parameters. The interrelations between mechanical and electrical parts of the circuit are

achieved by using ideal transformers (see Fig. 3). Additional information on building

electrical equivalent circuits of vibrating plates and bars can be found in [10] and [11].

Fig. 3. The topology of the equivalent circuit of a PT operating close to resonant frequency. Rm,

Cr, and LR describe the mechanical resonant system. The capacitors Ci and Co are a result of capacitance between the input and output electrode pairs. The ideal transformers 1:n1 and 1:n2 describe the DEC processes.

In [12], the topology of Fig. 3 was improved by eliminating the first transformer (1:n1,

see Fig. 3) by reflection of the right hand part onto the input part. The ideal transformer

(1:n2) was replaced by a couple of dependent sources to exclude any DC problems at the

output.

Simplified equivalent circuits of a PT operating close to mechanical resonant

frequency in a specific mode is given also in [13] - [25] among many others. Generally,

the behavior of a PT can be represented by a transmission line with distributed parameters,

which in turn is transferred into a lumped parameters circuit as done in [26], [27], and [7].

The analysis shown in [28] reinforces the notion that under specific conditions the

equivalent circuit model, the finite elements model and the relevant experiment gives

practically the same results. Thus, the topology shown in Fig. 3 is appropriate for modeling

and will be used in the present work for modeling and analysis of the applications.

6

Since a PT is a non-linear element and its parameters are strongly dependent on the

operational conditions such as the electric field, mechanical stress, temperature,

depolarization, aging etc., creating a convenient electrical equivalent circuit that represents

the PT’s behavior is not a simple matter. Such a circuit could estimate the equivalent PT

parameters only under a very narrow range of frequencies and electrical loads.

Several studies have described the non-linear behavior of piezoelectric materials ( [28],

[30], and [31]), but the models they propose are quite complicated. The PT is represented

as an electro-mechanical system that is described by a high order system of differential

equations. This approach is essentially theoretical and cannot obtain a simple electrical

equivalent circuit.

Because of the different vibration modes and mechanical structures, the different

categories of PTs have different mechanical and electrical characteristics. However, for a

specific frequency bandwidth around the corresponding mechanical resonant frequency

and a narrow load range, a single-branch equivalent circuit model can be developed (as

shown in Fig. 3).

The main problem that remains is how to track the frequency of maximum output

voltage (maximum output power). Since the parameters of a PT are dependent on many

factors such as load, amplitude of signal, temperature etc., the frequency of interest is

floating and needs to be tracked.

A PT's model parameters vary with the operational conditions. Therefore one needs to

find a method for finding the parameters of the proposed model directly from experiment

under specific conditions.

There is as yet no clear mathematical description for the transfer function in the

frequency domain of the PT. The one resonant branch model of a PT available is still too

complicated despite the fact that it has been simplified from a system with distributed

parameters. An analysis of the modulations of the carrier wave is needed. An important

question that arises is how to implement the different modulation types and what kind of

modulations is preferable. To determine these parameters, an analysis of the signal on the

PT's output under small deviations of frequency or amplitude of the carrier wave is

required. Since the signals may be transmitted through the PT by means of modulation,

there is a need to study the envelop behavior of the output signal in time and frequency

domain. Some feasible solutions for the problems discussed above are proposed in the

present research.

7

2.2 The envelope analysis

The nature of the Newton-Raphson algorithm, which all Spice-like simulators (HSpice,

PSpice, Spectre, and Eldo) use, is difficult to converge for very high Q circuits. The

transient (time domain) simulation of the devices with a carrier frequency demands a great

deal of time and powerful computer facilities, due to the high frequency component. There

are various power electronics systems, such as resonant converters, motor drives, and

electronic ballasts for fluorescent lamps, that use carriers of relatively high drive frequency

as compared to the control bandwidth. In these cases, the small signal response relevant to

the feedback network design is embedded in the envelope of the signals rather than in the

carrier. The PT is one such system because it employs the resonant frequency as a carrier.

This is why envelope simulations are within the sphere of interest of the present work.

A possible method for envelope simulation is the harmonic balance method [32]. In

the last few years, the harmonic balance (HB) method has gained widespread acceptance

among microwave engineers as a simulation tool for nonlinear circuits. The main

advantages of this approach are its ability to directly address steady-state circuit operation

under single- or multiple- tone excitation, and its full compatibility with the

characterization of the linear sub-network in the frequency domain, which is usually a

prerequisite for high-frequency applications. Nevertheless, the HB method is only for

frequency domain simulations and is non-SPICE-compatible. An alternative is the time-

varied phasor simulation method, which we use in the present research.

The method of time-varying phasor is described in [33]. In [34], [35], [36] the authors

show how one can implement the idea of [33] for large signal steady-state and transient

simulations using the PSPICE electronic circuits simulator. The method in these works is

based on the fact that any analog modulated signal (AM, FM, and PM) can be described by

the following general expression:

( ) ( ) ( ) ( ) ( )tsintUtcostUtu c2c1 ω−ω= (1)

where U1(t) and U2(t) are modulation signals and ωc is the angular frequency of the carrier

signal.

Expression (1) can also be written in a complex form as:

( ) ( ) ]e)t(jU)t(URe[tu tj21

cω−= (2)

or

( ) ( ) ( ) ]eeRe[tUtu tj)t(Uarg cω=rr

(3)

where

8

( ) ( ) ( )tjUtUtU 21 −=r

(4)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−= −

)t(U)t(Utan)t(Uarg

1

21r (5)

)t(U)t(U)t(U 22

21 +=

r (6)

Expression (3) reveals that any modulated signal can be represented by a generalized

phasor with time dependent magnitude and phase.

Response of the elements of the electric circuit (R, L, C) can be described in terms of

Envelope Impedance.

Envelope Impedance of an Inductor. The state space equation of an inductor is given

by:

dt

diLu L

L = (7)

Applying (2), we get

( ) ( )( ) ( ) ( )( ) tj21

tj21

cc etjItIdtdLetjUtU ω−ω− −=− (8)

Assuming that the current is also of the form

( ) ( ) ( )( ) tj21

cetjItIti ω−−= (9)

we find that

( ) ( )( ) ( ) ( ) ( ) ( ) tj1c22c1

tj21

cc etILtIdtdLjtILtI

dtdLetjUtU ω−ω−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ω++⎟

⎠⎞

⎜⎝⎛ ω−=− (10)

By dividing out the exponential term from both sides of the equation, we get a complex

expression of time-varied phasor:

( ) ( ) ( ) ( ) ( ) ( )⎟⎠⎞

⎜⎝⎛ ω++⎟

⎠⎞

⎜⎝⎛ ω−=− tILtI

dtdLjtILtI

dtdLtjUtU 1c22c121 , (11)

and by separating out the equation into the real part and imaginary part

( ) ( ) ( )

( ) ( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ω+=

ω−=

tILtIdtdLtU:Im

tILtIdtdLtU:Re

1c22

2c11 (12)

A two-port network (Fig. 4(a)) excited by the real and imaginary portions of the

modulating signal (1) can represent these two cross-coupled equations.

The Z matrix of the network is defined by

( )( )

( )( )⎥⎦⎤

⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡tItI

ZtUtU

2

1LE

2

1 (13)

where:

9

(14) ⎥⎦

⎤⎢⎣

⎡ω

ω−=

sLLLsL

Zc

cLE

Envelope Admittance of a Capacitor. Following the above approach, we find the

following for a capacitor:

(15) ⎥⎦

⎤⎢⎣

⎡ω−

ω=

sCCCsC

Yc

cCE

( )( )

( )( )⎥⎦⎤

⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡tUtU

YtItI

2

1CE

2

1 (16)

As demonstrated in [34], [35], and [36], the SPICE compatible envelope simulation

circuit can be developed according to the following stages:

• Duplicating the circuit to create the real part and the imaginary parts.

• Replacing reactive elements (L, C) in the real and imaginary components of the

circuit by the equivalent ones, as shown in Fig. 4.

• Introducing two excitation sources for real and imaginary parts (U1(t) and U2(t))

but excluding the carrier.

• Adding a behavioral element for calculating the square root of the sum of squares

of real and imaginary components of the output signals.

Now the signal is transparent for the carrier frequency, which is present in the

equivalent circuit as a DC element. As Fig. 5 shows, the simulation of such signal takes

significantly less time.

(a) (b)

Fig. 4. Replacement of reactive elements by equivalent circuits for envelope simulation: (a) - Replacing an inductor by an inductor with a dependent voltage source; (b) - Replacing a capacitor by a capacitor with a dependent current source.

This method helps to study PT responses to large signals and to analyze its steady state

and transient behavior. The innovation of the present research introduced in [37] and [38]

helps to analyze also small signal responses and transfer functions in the frequency domain

of a PT excited by a modulated signal.

10

Fig. 5. Example of transient and envelope simulation results for FM modulation. Upper curve

represents the modulating signal. Middle curves represent frequency modulated input carrier signal (gray) and envelope of the input signal (black line). The lower portion plots the PT’s output signal (gray curve) and its envelope (black curve) obtained by the envelope simulation model. [38].

2.3 Thermoelectric modules

A thermoelectric module (TEM) is a solid-state DED. It normally consists of an array

of pellets from dissimilar semiconductor materials (p and n type), which are joined,

thermally in parallel and electrically in series (see Fig. 5). A TEM can be used for cooling,

heating, and energy generation [4], [1]. As a thermoelectric cooler (TEC), the TEM has

already been used for thermal management and control of microelectronic devices such as

diode lasers and CPUs. As a thermoelectric generator (TEG), the TEM can be used to

produce electric power in remote locations when temperature gradients are available.

Fig. 5. Single-stage module construction.

A thermoelectric device has several distinct advantages over other technologies. It has

no moving parts and, therefore, needs substantially less maintenance. Life tests have

shown the capability of thermoelectric devices to exceed 100,000 hours of steady state

operation. Thermoelectric coolers contain no chlorofluorocarbons or other materials that

may require periodic replenishment ( [39]). Temperature control to within fractions of a

degree can be maintained using thermoelectric devices and the appropriate support

circuitry [4]. Thermoelectric devices function in environments that are too severe, too

sensitive or too small for conventional operation. Thermoelectric devices are not position-

11

dependent. The direction of heat pumping in a thermoelectric system is fully reversible.

Changing the polarity of the DC power supply causes heat to be pumped in the opposite

direction, e.g. a cooler can then become a heater. Thermoelectric module can be of very

small size and can be implemented directly into the integrated circuit ( [40], [56]).

Summing up the situation with thermoelectric processes today, one can say that on the

part of research, much knowledge has been gained about thermoelectric properties of

semiconductors, and thermoelectric generators and refrigerators of various types have been

developed. On the part of the market, the need for active cooling devices and

unconventional power sources has grown. However, in spite of these apparently favorable

conditions, the number of applications of thermoelectric modules has not increased since

the 1960s [39]. It is postulated that by proposing an engineer-friendly universal equivalent

circuit model, one can help realizing the great potential of thermoelectric devices.

The models that are proposed in the present work have been thoroughly grounded by

previous works. A detailed description of all the processes that take place in a

thermoelectric module can be found in [4], [1], and [41]. The analysis (especially that of

the dynamics) of the TEM is presented in a system of equations based on overall energy

balance, junction energy balance, and differential equation of the temperature distribution

[59]. The main assumption is that thermoelectric DEC processes are time independent.

Thus, the dynamics of a TEM is the dynamics of the temperature redistribution in the TEC

with a thermal load and heat exchanger.

The solution of such problem can be achieved analytically [43], as well as by

numerical simulations using finite-element simulation model, as in [44] and [45]. Both

types of analysis are labor intensive. In the literature, the problem of modeling, simulation,

and calculation is divided into two main parts: Steady-state models and dynamic models.

2.3.1 The steady state models

The steady -state models are employed for optimization of the heat pump [46] or

generator [48] - [51], for choosing the cheapest heat sink [52], or for finding more

effective conditions of operation [53], [54]. It was found out that the steady -state

operational conditions is nontrivial to solve. This arises from the variety of parameters

taking part in equations and from the nonlinearity of the system.

In [55] authors give the full steady-state analysis of the multistage TEC with reference

to the Thomson effect, normally omitted because of its fairly low influence. The approach

is universal and precise. The matrix form interpretation of the expressions enables

solutions that make use of common software programs. Nevertheless, the expressions are

12

sophisticated. Among the ways of solution, one can find the equivalent electric circuit for

one- or two- dimension heat flow analysis ( [40], [56], and others).

2.3.2 The dynamic models

All direct energy conversion processes are independent of time, whereas heat-transfer

processes are time-dependent. Thus, in order to perform dynamic analysis of TEMs'

behavior, one needs to introduce the concepts of thermal masses of elements of TEM and

its environment (thermal load and heatsink). The dynamic model of a TEM helps us to

calculate the controller necessary for using the TEM in a close loop application

(temperature control for example), ( [57], [57], [59]).

The analytical linear dynamic model of TEC-based cooler with heat sink and the

cooling load was derived using small-signal linearization method in [57]. The linear

dynamic model is shown there to vary with operational conditions. This analysis is

interesting when the operational conditions are known (Bios-point).

Another way of finding the close-loop system parameters is analyzing the transient

response of the TEM's output amplitude as a function of the frequency of a small-

amplitude sinusoidal excitation. The unsteady state analysis of the transient behavior of

TEM is also possible only with dynamic models. The analytical models of such kind one

can find in [56], [58], and [59].

2.3.3 Use of electrical equivalent circuit for modeling heat transfer processes

All authors of the literature, referenced above, use the assumption that DEC processes

of Peltier heat pumping and Seebeck power generation are time independent. Most authors

assume junctions of very small size and the mass of a junction to be zero. These

assumptions permit to compose an equivalent circuit of the TEC. The energy conversion

processes are represented as dependent on heat, temperature, current or voltage sources.

The differential equation of the temperature distribution in solids may be solved using the

electrical equivalent thermal RC low-pass circuit [60], [61], [62], and [63] (Cauer

network).

The full three port topology of the equivalent circuit of Peltier module (TEC) is shown

in [64]. The model includes two thermal ports (hot and cold sides) and an electrical port.

The electrical and thermal parts interrelate with one another in accordance with steady-

state equations of TEC operation. The simplified Cauer (CRC) low-pass network

(dumbbell-like, with thermal mass divided by two between hot and cold sides), express the

dynamical behavior of the TEC. All external thermal elements such as thermal load and

13

heatsink are shown also as elements of equivalent electrical network by thermal resistances

and thermal capacities [65].

The topology proposed in [64] was used in the present work as a starting point. There

were proposed some improvements in the topology, the model used for thermoelectric

coolers was employed also for thermoelectric generators. The way of extracting the

parameters directly from the manufacturer specification has been developed. The proposed

model was used for all types of simulations: time domain (transient), steady-state (DC-

sweep) and small signal in frequency domain (AC-sweep). Their parametric analysis was

also used for finding the optimal module for specific application.

14

3 Objectives

The objective of the current work was to point out the universal approach of modeling

by the equivalent circuit approach to application of electro-thermal and electro-mechanical

DED using their equivalent circuit model.

When the equivalent circuit is constructed, one can perform all types of network

calculations and simulations of electronic systems with DED as an element. This analysis

helps to choose the best DED from the manufacturer's stock to meet the best performance

of application.

Sub-task was to develop simple models of direct energy conversion devises such as

piezoelectric transformer (PT) thermoelectric cooler (TEC) and thermoelectric generator

(TEG) that can be used as a vehicle to study the inherent characteristics of the devices in

power conversion applications. Special effort was made to create equivalent circuit type

models that can be easily analyzed by computer software such as MATLAB (Mathworks

Co.) and MATHEMATICA (Wolfram Research Co.). The resulting equivalent circuit

needs to be translated to circuit simulators environment such as PSPICE (Microsim Co.) to

provide an engineering analysis and design tools for the designated devices. These tools

will be useful not only to study the devices themselves but also to examine the interaction

between the devices, drivers, and loads.

The usefulness of models needs to be proven analytically, by simulations, and

experimentally. The models that were developed are to be demonstrated through the design

of some concrete applications.

The experimental validation of every suggestion is required. Experiments with test

DEDs estimate the precision of the models and show the relevancy of the assumptions.

15

4 Appended Publications

In this chapter, a number of related papers are presented. All the papers were published

in IEEE magazines or presented at IEEE conferences during the period of my doctoral

studies. The order of the papers reflects the sequential connection between different parts

of the research. The papers are presented in a logical order rather than in a chronological

one. Every paper is accompanied by a brief introduction that includes the description of the

problem, tools of approach, main results, conclusions, and, as the problems come to be

better understood, the direction for future research.

For the sake of clarity, flow chart of the research is presented (see

Fig. 6). There are two different diagrams for electro-mechanical and electro-thermal

energy conversion systems. The "stations" in the diagrams are the published materials; the

arrows are showing the directions of the research. Both of the diagrams are built using the

same principle: the first application is shown at the head of the diagram. This paper

crystallizes the ways of investigations for the papers in lower levels of the diagram. The

papers in medium levels propose the solutions of these problems. The papers shown at the

lowest level in the diagram are concluding the researches.

16

(a) (b)

Fig. 6. Flow chart of research. The blocks in the flow chart are the milestones of the work (paper publications), the ellipses are the questions and problems that have appeared at every step. (a) - the flow chart of study of the PT and its applications, (b) - the flowchart of the study of thermoelectric modules that has been carried out for the present dissertation.

17

4.1 Equivalent circuit modeling and application of electro-mechanical

power conversion systems

4.1.1 Feedback isolation by piezoelectric transformers: a feasibility study [66].

The paper was presented at Power Conversion and Intelligent Motion conference,

PCIM-2000, in Nuremberg. The paper represents the first step in my studies of

piezoelectric transformers. The described application of PT uses it as a galvanic barrier

for feedback signal isolation, as an alternative to the traditional use of the optocoupler

or magnetic transformer.

The application is based on physical principle: the body of PT starts to vibrate when

excited by voltage with a frequency close to the mechanical resonant one of the PT. The

vibrating PT creates a secondary (output) voltage on the output electrodes. The ceramic

body of the PT has to provide a good galvanic insulation between primary and

secondary electrodes. The signal is transferred through the PT by modulation (FM or

AM) of the excitation voltage wave. The "voltage doubler" peak detector is used as an

envelope detector.

The proposed scheme was analyzed using the equivalent circuit of PT. The

experimental results are in good agreement with the theory as presented in the paper

[66].

A detailed understanding of the behavior of the equivalent electrical circuit of the

vibrating piezoelectric bar has guided the way to improvement of the frequency tracking

system. This subject together with detailed study of the modulation method, and the

analysis of the BW of the PT-based signal transmitting system as a function of carrier

frequency and the mechanical resonance quality factor was studied in the further stages

of research. Another subject for additional study was proposing the methods for

extraction of the equivalent circuit parameters.

The solutions of all these problems are discussed in later sections.

Feedback Isolation by Piezoelectric Transformers:

A Feasibility Study

Simon Lineykine and Sam Ben-Yaakov

Power Electronics Laboratory Department of Electrical and Computer Engineering

Ben-Gurion University of the Negev P. O. Box 653, Beer-Sheva 84105

ISRAEL

Tel: +972-7-646-1561; Fax: +972-7-647-2949; Email: sby@ee.bgu.ac.il; Website: http://www.ee.bgu.ac.il/~pel

Abstract - Signal isolation is needed in

power electronics systems that include separate primary and secondary 'grounds'. The feasibility of using a piezoelectric transformer as a galvanic barrier was investigated in this study theoretically and experimentally. The research included the issues of drive, demodulation, bandwidth and common mode rejection. It was found that a small size transformer (2x3mm) could be used effectively to transfer feedback signals from the isolated output of a DC-DC converter to the primary side with a bandwidth of 7kHz when a carrier of 350kHz is used. Wider signal bands can be achieved by applying higher carrier frequencies.

I. INTRODUCTION

Signal isolation is needed in power electronics systems that include separate primary and secondary 'grounds' [1-4]. The general representation of a typical power converter with primary to secondary isolation includes a power stage, a modulator, and an isolation barrier for the feedback signal (Fig. 1a). To ease the accuracy and stability requirements of the signal isolator, the configuration of Fig. 1b is normally preferred. In this case the isolator has to carry the error signal and not the output signal. Comparing the transfer functions of the two configurations we obtain:

)()()(1)()(

)1.(ωωω

ωω

isopm

pm

ref

out

HHHKHH

aFigVV

⋅⋅⋅+⋅

= (1)

(a)

(b)

Hm(ω)Modulator

Hp(ω)Power Stage

Voltagedivider

Isolation

K

Vref

Vout

Hiso(ω)

Hm(ω)Modulator

Hp(ω)Power Stage

Voltagedivider

Isolation

K

Vref

Vout

Hiso(ω)

Fig. 1. Alternative feedback loop configurations

in an isolated converter. (a) Isolating of output signal. (b) Isolating of error signal.

)()()(1)()()(

)1.(ωωω

ωωω

isopm

isopm

ref

out

HHHKHHH

bFigVV

⋅⋅⋅+⋅⋅

= (2)

where: )(ωpH - transfer function of power stage )(ωmH - transfer function of modulator

17 - 1

)(ωisoH - transfer function of the isolator K – voltage divider

refV – reference voltage

outV – output voltage

For a large Loop Gain (LG), equations (1, 2)

reduce to:

)(1)1.( LG ωisoref

out

HKaFig

VV

⋅≈∞→ (3)

KbFig

VV

ref

out 1)1.( LG ≈∞→ (4)

It is thus evident that by choosing the configuration of Fig. 1b (including the voltage reference and error amplifier at the secondary) the system becomes practically independent of the transfer function of the isolator )(ωisoH – provided that the LG is kept high.

Filter++

- -

+

-

Rectifier Inverter Rectifier DC

DriverSecondaryRegulator

AC

In Out

Fig. 2. Conventional feedback isolation in

power converters.

Conventional solutions to the signal isolation problem include galvanic isolation by opto-coupler or transformers [1-6] (Fig. 2). Other solutions (such as capacitor coupling [2,3]) are possible, but not all are compatible with interfering signals of high dtdV normally associated with the power conversion environment. A typical practical design for a flyback converter is shown in Fig. 3. In this case the reference voltage and error amplifier are realized by TL431 (Texas Instruments Inc.), whereas the actual isolation (of the error signal) is carried out by the opto-coupler 4N15.

145n 220u

20k

200

1k

3.3

Cout

1k330p

UC38427

6

3

5

2

1

8

4

1u220

1.2k

5.6n0.1u

10k

Vout=20V

1k

6.8k

1k

1n

1k

TL431

RLoad

4

5

6 1

2

3

4N15

Fig. 3. Flyback converter with opto-coupler

isolation.

In this study we explored the feasibility of using a ceramic Piezoelectric Transformer (PT) [7-9] as the barrier element. That is, the possibility of replacing the isolation element (e.g. 4N15 in Fig. 3) by the PT. Since the PT can not transfer DC signal, modulation techniques need to be applied. Namely, the error signal is first used to modulate a carrier and then recovered by demodulating the output signal past the isolation barrier. This is similar to the method used when an electromagnetic transformer is used to realize the isolation barrier [2,3]. The investigation probed into the theoretical aspects of such coupling and included experimental examination of a specific design that uses FM modulation.

VoutVin

(a)

PT Ro

VinVout

(b) Fig. 4 PT (a) and its connection in the circuit

(b).

17 - 2

Cin Rm Cr Lr Co

Vin Voutir +

_

Vout/n ir/n1:n

Ro

Fig. 5. Equivalent circuit of a piezoelectric

transformer (PT) loaded by Ro.

A typical PT is shown pictorially in Fig. 4. The equivalent circuit of a PT (Fig. 5) includes a resonant network (Lr, Cr, Rm) that emulates the effect of the mechanical vibration and dependent sources that express the gain [7-9]. The model also comprises the physical dielectric capacitors (Cin, Co) that are formed by the input and output electrodes. Since the network is highly selective it will pass, with reasonable gain, only frequencies that are in the vicinity of the resonant frequency. The equivalent circuit of Fig. 5 represents one vibration mode but in reality, many such circuits are in fact connected in parallel since the PT can vibrate in various modes. That is, a typical transfer function of a PT will look like the plot of Fig. 6. For a typical application one resonant peak is selected and the design is done for that specific carrier frequency [7].

0

0.5

1

1.5

2

2.5

3

3.5

4

100 150 200 250 300 350 400

Vout/Vin

frequency,kHz

Fig. 6. Gain versus frequency of experimental PT.

II. MODULATION SCHEMES

In this study we explored two modulation alternatives: AM modulation and FM modulation. In the AM case, a constant frequency carrier is used and by varying the amplitude of the carrier the error signal is transmitted. In the FM case, the error signal is coded into a frequency shift that translates at the output of the PT as an amplitude shift. This is illustrated in Fig. 7 that assumes an operating point above the resonant frequency.

0

0.5

1

1.5

2

2.5

3

3.5

4

345 355 365

V out /V in

frequency,kHz t

t

Amplitude ofoutput

Inputfrequency

Fig. 7. PT’s response to an FM modulated

signal.

AM Modulation

Considering the high quality factor (Q) of practical PTs the carrier frequency has to be around the resonant frequency of the device. To explore the effect of modulation it would be desirable to simplify the electrical equivalent circuit of the PT. This can be done by first reflecting the load Ro and Co (Fig. 8a) to the primary and then converting the parallel Ro’Co’ to a series network (Fig. 8b).

The transformation equations are:

2222

0

2

20

22

2222

2222

2

2

11

)1(

)1(

oooo

oo

ooo

oo

oo

oo

oo

RCnR

CRZ

RCnRC

C

RCnRR

nRR

nCC

ωω

ωω

ω

+=

′′

+′′=″

+=″

+=″

=′⋅=′

(5)

17 - 3

where: n is transformation ratio

Cin

Rm CrVin Ro' Vout'Co'

(a)

Lr

VinCin Rm Cr

Lr

Ro''

Vout'

Co''

(b)

Fig. 8. The stages in simplifying the equivalent circuit of a PT. (a) - reflecting the RoCo network to the primary. (b)-translating the parallel Ro’Co’ network into a series network Ro”Co”.

The series elements can now be lumped into an equivalent capacitor eqC and equivalent resistor eqR :

″+

″=

″+=

or

oreq

omeq

CC

CCC

RRR

(6)

Thus the PT equivalent circuit is reduced into a series RLC network (Fig. 9).

Req Lr

Ceqiin

Vin

t

Fig. 9. The simplified equivalent circuit of a PT

circuit fed by an AM modulated signal.

Fig. 9 implies that the response of the PT to an AM modulated signal is in fact a response of a series resonant network. An AM signal will be composed of a carrier ( 0f ) plus two side-bands located at sff +0 and sff −0 where sf is the

modulating frequency. If the carrier is set at the resonant frequency of the PT then the width of the resonant curve ( f∆ ) will be:

eQf

f 0=∆ (7)

where: eQ is the quality factor of the series resonant

circuit eq

re R

LfQ 02π

= and

0f is the resonant frequency of the network

eqrCLf

ππω

21

20

0 ==

The bandwidth of a resonant circuit is BWf∆ corresponding to the cut off frequency of the modulated signal ( sof )

soBW ff 2=∆ (8)

from which

eso Q

ff

20= (9)

Hence, by adjusting eQ (loading properly) one can control the useful signal bandwidth of the PT.

The simplified analysis given above is supported by a rigorous analysis carried out in this study. It was found that the response of a PT to a step function in the carrier can be expressed as:

)sin())1((1

)sin())1((1

)(

222

2222

tVeVRCn

tVeVRCn

tV

on

t

mooo

onQ

t

mooo

out

e

o

ωωη

ωωη

τ

ω

+−+

=+−+

=

−

−

(10)

where:

o

eQ

ωτ

2= - is the aparent time constant of the

system

om

m

RRR

′′+=η - is the efficiency

mV is the amplitude of the modulation step

nV is the amplitude of the carrier And the bandwidth is:

17 - 4

eQof

eQBW o

⋅=

⋅⋅⋅=

⋅⋅=

22221

πω

τπ (11)

FM Modulation

Following the same intuitive reasoning as above one can find the useful signal bandwidth in the case of FM modulation. Assuming that a constant frequency shift per unit amplitude of the modulating signal is mf∆ , the useful bandpass will be mf∆ . That is, most of transmitted energy is locked within mff ∆+0 and mff ∆−0 . In this respect the behavior is similar to the AM case except that the energy is distributed within the range (rather than having only two side bands as in the case of AM). Consequently, the break point of the signal transfer function is reached in this case when:

em Q

ff

20=∆ (12)

Furthermore, the signal bandwidth sof will also be limited to mf∆ since modulating frequencies higher than mf∆ will be highly attenuated. It is thus clear that in this case as in the case of AM modulation the useful bandwidth can be controlled by adjusting eQ .

III. COMMON MODE REJECTION

A major concern in isolated feedback signal of power systems is the injection of common mode signal via the isolator [1-3]. Such an injection will increase the EMI signal at the output section and will require further filtering to satisfy common standards.

PT

Vin Ro

Vout

Fig. 10. Setup for measuring common mode

transfer ratio.

We have therefore tested the experimental PT for both differential mode gain (Fig. 4b) and common mode gain (Fig. 10). The results (Fig. 11) suggest that the peak at 350 kHz is a good choice since it is a conveniently high frequency and has a large common mode rejection ratio (the ratio of differential gain to common mode gain).

0

0.5

1

1.5

2

2.5

3

3.5

4

100 150 200 250 300 350 400

CMDM

Vout/Vin

Frequency,kHz

Fig. 11. Differential Mode (DM) and Common Mode (CM) transfer ratios of experimental piezoelectric transformer.

IV. EXPERIMENTAL

The experiment circuit included a small (2x3mm) PT transformer made of PXE43 material (Philips).

PT

R1

R2

SW

Vin

Vout

Fig. 12. Setup for measuring BW.

We have used the circuit of Fig. 12 to verify the analysis of signal bandwidth in AM modulation. The element was driven by a 350kHz carrier and the amplitude was changed

17 - 5

stepwise by a divider that was controlled by a switch. The results confirm the estimates presented above. The block diagram of Fig. 13 describes the system that was used the PT isolator concept applying FM modulation. The circuit was built around a commercial resonant controller (MC34066). The output was rectified and filtered to recover the error signal. A complete circuit diagram is given in Fig. 14. The design was made compatible to the flyback converter of Fig. 3. That is, the PT isolation system can replace the TL431 and 4N15. The input was designed to be connected to the output voltage while the output of the PT isolator is compatible with the UC3842. During testing, the PT isolation block was fed by a DC signal on which a low frequency AC signal was superimposed.

+

-

Vout

Vref

E/A

V/F PT Rectifier

Vin

Gndout

Fig. 13. Block diagram of the experimental PT

isolator.

1k

6.8

1k

100p

22k

56k

1.8k

1.5k

3.4V

10k

5k

6

7

4

E/Ainverting

input

E/Aoutput

GND

12

14

Drive A

Drive B

PT 10n220k 2.2k

15V

MC34066Vout

PWMMOD

Fig. 14. View of feedback path the whole

(practical).

The measured static transfer-function of the isolator reflects the high gain of the error amplifier in MC34066 (Fig. 15). In actual use, the output signal will be forced to approach the reference signal by the global feedback action.

A network analyzer was used to test the signal bandwidth of the FM approach. The measured bandwidth of the experimental circuit was found to be 7KHz (Fig. 16).

15 2520

6

5

4

Vin(input toPWM), V

Vout(input toFeedback), V

Fig. 15. Static transfer function of the

experimental PT isolator.

Gain1

Gain2

Phase1

Phase2

Gain,[db]

Phase,[o]

Frequency,[Hz]1K 10K

0

-5

-10

-100o

100o

0o

-200o

Fig. 16.Transfer function of the experimental PT

isolator at two carrier frequencies (gain 1: 358kHz; gain 2: 357.6kHz).

V. DISCUSSIONS AND CONCLUSIONS

The main advantages of the PT in the proposed application are the small size and the very high isolation breakdown voltage that can be achieved [7-9]. This is due to the good isolation of the ceramic material. The main disadvantage is the common mode stray capacitance between primary and secondary sides of the PT. Common mode rejection can be improved by ensuring that the harmonics of the switching frequency of the converter do not coincide with he common mode peaks of the PT.

To fully exploit the engineering benefit of the proposed isolation approach, there would be a need to develop a dedicated IC that will be compatible with the PT. A desirable feature of such an IC would be a self oscillation and locking

17 - 6

to the preferred resonance frequency of the device. In the case of AM modulation, locking to the resonant frequency is preferred. In the case of FM, a phase lock loop can be used to stabilize the operating frequency.

REFERENCES

[1] T. Fleming, "Isolation amplifiers break ground loops and achieve high CMRR," EDN, vol. 24, no. 26, pp. 97-102, Dec. 1987.

[2] M. F. Zirngast, "Capacitive isolation expands analogue design options-Part 1," Electronic-Engineering- (London), vol. 61, no. 748, pp. 37-38,40, Apr 1989.

[3] M. F. Zirngast, "Capacitive isolation expands analogue design options-Part 2," Electronic-Engineering- (London), vol. 61, no. 749, pp. 33, 36,38,40,43,45, May 1989.

[4] S. Howard, "Isolator measures 12-bit signals across ±3500V barriers," EDN,

vol. 32 no. 19, pp. 203-210, Sep 17 1987.

[5] J. B. Simöes, R. M. C. Siva, A. M. L. S. Morgado, C. M. Correia, "The optical coupling of analog signals," IEEE Transactions on Nuclear Science, pt. 2, vol. 43, no. 3, pp. 1672-1674, Jun 1996.

[6] H. Whittington, B. Flynn, and D. Macpherson, Switched mode power supplies design and construction, New York: John Wiley & sons, 1992.

[7] R. Holland and E. P. Eernisse, Design of resonant piezoelectric devices, Cambridge: The M.I.T. Press, 1966.

[8] C. Y. Lin and F. C. Lee, "Piezoelectric transformer and its applications," Proc. of VPEC Seminar, pp. 129-136, Sept. 1995.

[9] N. Volkert, "DC-DC converter with very high insulation capability," EPE'99, pp.1-8, 1999.

17 - 7

18

4.1.2 Frequency Tracking to Maximum Power of Piezoelectric Transformer HV

Converters under Load Variations. [67], [68].

The paper was presented in IEEE Power Electronics Specialists Conference, PESC-

2002 [67] and was accepted with small revisions to IEEE Transactions on Power

Electronics. [68]. Some new experiment results were included. The measurements were

reconstructed using modern high precision measuring equipment. Here is presented only

the last, revised version.

The problem of maximum power point tracking of high output DC voltage

converters that apply Piezoelectric Transformers (PT) and voltage doublers was studied

theoretically and experimentally. It was shown that the operating frequency of the PT, at

which maximum voltage gain is reached, is a function of the load. Hence, under load

variations, and to overcome parameters instability, there is a need for some mechanism

of frequency tracking that will help to lock the operating frequency to the optimum one.

The proposed frequency tracking idea is based on the analysis of rectifiers that one

can use with the PTs, made by Professor Ivensky in [69]. My part in the paper was the

analysis of optimal frequency, analytical proving the method, and providing the

experiments.

The main idea is to synchronize the phase of the input signal with the phase of the

vibration velocity of the piezoelectric bar of the PT. Analysis of the equivalent circuit

shows that the sample of the current, which is equivalent to the mechanical velocity,

may be measured as electrical current on one of the diodes of the rectifier circuit.

The proposed method to achieve frequency tracking is based on a Phase Locked

Loop (PLL). The PLL inputs are the phase of the input voltage driving the PT and the

phase of the current flowing through one of the voltage doubler diodes.

Theoretical analysis, verified by experiments, shows that when the phase shift of the

diode current relative the phase of the input voltage is zero, the voltage gain of the

system is at its maximum point. By applying this approach, the system operation can be

made independent on input voltage, load variations, temperature (within permitted

range), and the spread and non-linearity of the PT parameters, as well their drift with

time.

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006 73

Maximum Power Tracking of PiezoelectricTransformer HV Converters

Under Load VariationsShmuel (Sam) Ben-Yaakov, Member, IEEE, and Simon Lineykin

Abstract—The problem of maximum power point trackingof high output dc voltage converters that apply piezoelectrictransformers (PT) and voltage doublers was studied theoreticallyand experimentally. It was shown that the operating frequency ofthe PT, at which maximum power is reached, is a function of theload. Hence, under load variations, and to overcome parameterinstability, there is a need for some frequency tracking mechanismthat will help to lock the operating frequency to the optimum one.The proposed method to achieve frequency tracking is based ona phase locked loop (PLL). The PLL inputs are the phase of theinput voltage driving the PT and the phase of the current flowingthrough one of the voltage doubler diodes. Theoretical analysis,verified by experiments, shows that when the phase shift of thediode current relative the phase of the input voltage is zero, thevoltage gain of the system is at its maximum. By applying thisapproach, the system’s operation can be made independent ofinput voltage, load variations, temperature (within a permittedrange), and the spread and nonlinearity of the PT parameters, aswell as their drift with time.

Index Terms—Phase locked loop (PLL), piezoelectric trans-formers (PTs).

I. INTRODUCTION

THE main advantages of piezoelectric transformers (PTs)are potential low cost, small size, low profile, good in-

sulation capability, and the absense of windings, and hence,magnetic fields. In some specific applications, PTs are superiorto electromagnetic transformers, making the PT a good designchoice. Among these is the generation of a low power, high dcvoltage (HV). Rosen type (see Fig. 1) PTs have a high gain ratiothat, when combined with excellent insulation properties of thePT, make it a good candidate for the construction of compactHV converters—up to few kilovolts. Since the PT is a resonantelement, its output to input voltage gain is strongly dependenton the operating frequency [1].

For any given load, the problem of maximum power trackingtranslates into searching and locking to the frequency that pro-vides a maximum voltage gain. Hence, to maintain maximumoutput voltage under variable operating conditions (load varia-tion, temperature changes and components tolerances) it is nec-essary to lock the operating frequency to the one that will ensurethe highest possible output voltage for a given load.

Manuscript received February 11, 2004; revised May 11, 2005. This workwas supported by The Israel Science Foundation under Grant 113/02 and by thePaul Ivanier Center for Robotics and Production Management.

The authors are with the Power Electronics Laboratory, Department of Elec-trical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (e-mail: sby@ee.bgu.ac.il).

Digital Object Identifier 10.1109/TPEL.2005.861125

Fig. 1. Rosen type PT.

Fig. 2. Equivalent circuits of a PT: (a) original equivalent circuit and(b) simplified equivalent circuit reflecting secondary to the primary side.Values are for PXE43, Philips, operating around 73 kHz.

As a prerequisite for solving the frequency-tracking problem,one needs first to find a parameter that can be used as a mea-sure of the deviation from the desired frequency. Differentapproaches have been suggested for frequency tracking of PTdrivers. In [2], the phase angle between input voltage and inputcurrent was used as a criterion while in [3], the phase anglebetween input and output voltage was used as a measure of thedeviation from the frequency of maximum gain. Unfortunatelythese criteria are load dependent and could be used only overa narrow load resistance range.

The difficulty in locating the optimum tracking parameter canbe appreciated by considering the equivalent circuit of a typicalPT [Fig. 2(a)] and its series resonance representation [Fig. 2(b)].

As can be easily observed from Fig. 2(a) [1], the frequencyof the maximum output to input voltage ratio will be between

for high resistance loads (close to the open circuit situation,load is negligible) and for low resistance loads, when ispractically shorted

(1)

(2)

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Thus, the frequency giving the maximum output to inputvoltage ratio is a strong function of the load resistance. How-ever, since the series resonant branch [Fig. 2(b)] is responsiblefor the input to output power transfer, it stands to reason thatmaximum voltage gain will be obtained when the operatingfrequency is locked to this series resonance. In this studywe explored this possible criterion for frequency locking tomaximum power and propose a novel method for obtaining areliable bipolar signal that is a measure of the deviation fromthe optimal frequency. It is then demonstrated how this signalcan be used to lock the frequency to the optimal one.

II. ROSEN TYPE PT IN HV APPLICATIONS

The Rosen-type piezoelectric transformer 48 12 2 mm(Phillips [4]) was chosen for the present research (see Fig. 1).This unit has a large output to input voltage ratio at the fre-quency of the maximum output voltage, relatively high power(up to 5 W), and high input to output insulation (tens of kilo-volts). The equivalent circuit of a PT operating near its reso-nance point [Fig. 2(a)] includes a resonant network ( , ,

) that emulates the effect of the mechanical vibration anddependent sources that express the mechanical to electrical en-ergy transformation [1]. The model also comprises the phys-ical dielectric capacitors ( , ) that are formed by the inputand output electrodes. Since the network is highly selective itwill pass, with reasonable gain, only frequencies that are in thevicinity of the resonant frequency.

To simplify the analysis some equivalent transformations ofthe equivalent circuit can be applied. First, the output part [right-hand side of Fig. 2(a)] of the equivalent circuit is reflected to theinput side. and of the equivalent circuit are transformedto and . Output voltage V is transformed to V

(3)

(4)

VV

(5)

Second, the parallel network , is transformed to a seriesfrequency dependent network , [Fig. 2(b)], where

(6)

(7)

The voltage transfer function of a PT (in terms of equivalentcircuit parameters) is

VV

(8)

Fig. 3. Output stage of PT with voltage-doubler rectifier: (a) topology and(b) voltage-doubler waveforms versus angle # = !t.

As one can see, the system is of third order. An analysis of thissystem is given in [1]. In terms of angular frequency 2 ,the transfer function will be (9), shown at the bottom of the page.

In a typical HV application, one would use a voltage doublerto boost the output voltage. Hence, the output section will beconstructed as shown in Fig. 3(a), where is the output filtercapacitor and is the load resistance.

Assuming that the quality factor ( ) of the PT is high, thecurrent through the primary series resonant circuit will be si-nusoidal. This current, after being transferred to the secondary,charges and discharges capacitor of the output section[Fig. 3(b)].

As shown in [5], the voltage doubler and load section can berepresented as an equivalent reactive load (a resistor in parallelto a capacitor). Consequently the gain function (8) is valid forthis case too, except that the load resistance and output ca-pacitance in Fig. 2(a) need to be replaced by an equivalentresistor and capacitor, respectively.

III. CRITERIA OF MAXIMUM POWER POINT FREQUENCY

One can obtain the analytical expression for the frequency ofthe maximum output to input voltage ratio by taking the deriva-tive of (9) with respect to and equating it to zero (10). This re-sults in a third-order equation, which can be solved, for example,by using Cardan’s method. Since the analytical expression wasfound to be too cumbersome, we have solved (10) numerically

(10)

VV

(9)

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BEN-YAAKOV AND LINEYKIN: MAXIMUM POWER OF PIEZOELECTRIC TRANSFORMER HV CONVERTERS 75

where: 2 , —frequency of the maximum ofPT’s transfer function.

The proposed criterion for maximum voltage gain is zerophase-shift between the input voltage V and the virtual current

of the resonant branch of the equivalent circuit [Fig. 2(b)].The basis of this hypothesis is that maximum output powerwill be obtained when the PT is driven at the series resonancefrequency. This is further supported by the observation madein [1] that the behavior of transfer function (9) is like that ofa second order – – band-pass filter, because within thenarrow band around the resonant frequency, andchange slowly.

Based on the simplified equivalent circuit of Fig. 2 the transferfunction V j is

V(11)

where and are expressed in (12) and (13) as afunction of the parameters shown in Fig. 2

(12)

(13)

From (11), one can see that zero phase-shift between andV occurs when the imaginary part is zero, i.e.

(14)

or

(15)

Applying (6), (7), (12), (13), and (15) the frequency of zerophase-shift gives (16), shown at the bottom of the page.

This frequency of series resonance of the equivalent circuit isnot identical to but is very close to it. For the experimentalhigh voltage PT used in this study, the maximum difference be-tween and is less than 50 Hz (less than 0.1% of the op-erating frequency) and the phase-shift of at is less than1.2 (Fig. 4). Fig. 5 shows the ratio of the PT’s output power at

to the output power at . These numerical values clearlypoint out to the fact that is a very good approximation of

. Hence, the phase shift of can be used for all practicalpurposes as a sensing signal for the deviation from . Theobjective of the tracking system will thus to be zero this phaseshift.

IV. PROPOSED TRACKING METHOD

The results of Section III suggest that the maximum gain isreached when the phase angle between the input voltage V and

is zero.Unfortunately, there is no direct way to measure this current

or its phase since there is no physical access to the (virtual) se-

Fig. 4. Phase of i when f = f (dashed-line), and difference betweenf and f (solid line).

Fig. 5. The ratio between the output power of PT driven at f to the outputpower at f as a function of R .

ries branch. This is overcome in the proposed method by an indi-rect measurement that makes use of the observation that the cur-rent of diode is in fact a sample of the current [Fig. 3(b)].Note, in particular, that will stop conducting when the po-larity of the current is reversed. Hence, this polarity reversalinstant can be used as an indicator for the phase of the series cur-rent. This proposed sensing method is demonstrated in Fig. 6.Comp1 and Comp2 are used to generate two square waves; oneis synchronized to V while the second one is synchronizedwith . The phase angle between V and can thus be mea-sured by feeding these two signals to a phase detector.

V. EXPERIMENTAL RESULTS

The experimental circuit (Fig. 7) included a phase lockedloop (PLL) fed by the two phase signals. The digital frequency

(16)

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Fig. 6. Proposed method for extracting the phase signals of V and i .

Fig. 7. Experimental setup of the proposed frequency tracking to maximumoutput voltage.

Fig. 8. V (upper trace) and V (lower trace) (see Fig. 6) at the series resonantfrequency.

phase detector of the PLL (CD4046A) compares these rect-angular waveforms and feeds the VCO with the phase errorsignal. The PLL was designed to have a lock-in range

2 that covers the frequency range to (1),(2). Typical experimental waveforms are shown in Figs. 8–10. Abias network (BN) was used to slightly shift the zero point suchthat frequency locking is obtained when there is a small phaseshift between the Comp1 and Comp2 signal. This was foundnecessary for the compensation of the phase shift caused by theparasitic capacitances of the diodes used as clamps (Fig. 7). Thisphase offset was found to be constant and independent of loadresistance. Practical designs of this tracking system should at-tempt to minimize this parasitic effect by choosing low capaci-tance diodes.

The objective of the experimental work was to validate theability of the proposed method to track the frequency of max-imum output voltage and to ascertain that the operation is stable.These goals were achieved by conducting two different exper-iments. In the first one, a comparison was made between the

Fig. 9. V (upper trace) and V (lower trace) at the series resonant frequency.(See Fig. 6).

Fig. 10. V (upper trace) and V (lower trace) of the comparators outputs (seeFig. 6).

output voltage measured with and without frequency tracking,while the PT was subjected to a temperature rise. In the secondset of experiments the output voltage was measured, under openand closed loop conditions, while the load was switched fromone resistance value to another.

A. Frequency Tracking While the PT is Exposed to aTemperature Rise

The output voltage of the system was measured while the PTwas heated. Since the parameters of the PT are temperature de-pendent, the frequency of maximum output voltage is expectedto change as the temperature of the PT is varied. Fig. 11 summa-rized the measured output voltage when the PT was exposed to atemperature change of 30 C to 65 C. With frequency trackingthe voltage drop at 65 C was only 2% while with no trackingthe output voltage dropped by 34%. It should be pointed outthat the small 2% variation could be due to less efficient oper-ation at the elevated temperature and not necessarily improperfrequency tracking.

B. Response to a Load Step

The frequency tracking system was also tested under staticand dynamic conditions. In the dynamic response tests, thesystem was subjected to a load variation by an auxiliary switch(see Fig. 12). First, the open loop output voltage as a func-tion of the drive frequency was measured for each load (seeFig. 13). The plot clearly shows that maximum output voltageis obtained at different frequencies and , correspondingto 1.2 M and 1.76 M , respectively. Three

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BEN-YAAKOV AND LINEYKIN: MAXIMUM POWER OF PIEZOELECTRIC TRANSFORMER HV CONVERTERS 77

Fig. 11. Output voltage versus PT temperature. (a) Constant frequencyoperation. (b) With proposed frequency tracking scheme.

Fig. 12. Variable load.

Fig. 13. Steady state output voltage as a function of the operation frequencyf , for two values of the load resistance R . (V = 20.3 V).

experiments were conducted: (a) operating the system underclosed loop conditions, (b) operating the system under openloop condition with drive frequency , and (c) operating thesystem under open loop condition with drive frequency . Itis expected that under a switched load situation, each one ofthese conditions will produce a different output voltage stepper Fig. 13. In case (a) the step should be (from 150.5 to178.5 V); in case (b) it should be and in case (c) it should be

. Fig. 14 shows the operation under closed loop conditionsas per case (a). The output voltage variations as well as thefrequency hopping matched the expected ones ( in Fig. 13).

Fig. 15 shows the results for case (b) while Fig. 16 depictsthe results for case (c). The amplitudes of the output voltagesteps were found to match to the expected ones: and(see Fig. 13), respectively. The results of these tests suggest thatthe tracking circuitry of the experimental setup is functioning asexpected.

Fig. 14. PLL response under closed loop conditions. Amplitude of V isequal toA of Fig. 13. V is the output voltage of the loop filter, (VCO input,Fig. 7).

Fig. 15. Open loop response when the drive frequency is f —the frequency ofresonance when R = 1.2 M. Amplitude of V is equal to A of Fig. 13.V is constant voltage to the VCO input (Fig. 7).

Fig. 16. System without PLL control. Drive frequency is f —the frequency ofresonance with R = 1.76 M. Amplitude of V is equal to A of Fig. 13.V is constant voltage to the VCO input (Fig. 7).

VI. CONCLUSION

The proposed tracking method offers a way to lock to the fre-quency that provides the maximum output power for any load.This could be useful in various applications that need to gen-erate high output voltage (e.g., ionization equipment, sparkersand the like). By applying the proposed approach, the system’soperation can be made independent of input voltage, load vari-ations, temperature (within the permitted range), and the spreadand nonlinearity of the PT parameters, as well their drift withtime.

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78 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 21, NO. 1, JANUARY 2006

The phase detection method proposed here can also be usedin cases that call for output voltage regulation. In such cases, asimple feedback loop via a voltage controlled oscillator (VCO)would be ambiguous. If, say, the output voltage is too low,should the frequency be increased or decreased? This ambi-guity can be resolved by applying the phase detection methodproposed here that generates a clear unequivocal bipolar signal.

The proposed method was verified experimentally and it wasdemonstrated that the control circuitry needed for the imple-mentation is simple and can be easily constructed from off-the-shelf components.

REFERENCES

[1] G. Ivensky, I. Zafrany, and S. Ben-Yaakov, “Generic operational char-acteristics of piezoelectric transformers,” IEEE Trans. Power Electron.,vol. 17, no. 6, pp. 1049–1057, Nov. 2002.

[2] N. Volkert, “DC-DC converter with very high insulation capability,” inProc. Eur. Conf. Power Electronics Application (EPE’99), Sep. 1999,pp. 1–8.

[3] S. Nakashima, H. Ogasawara, T. Ninomiya, and H. Kakehashi, “Piezo-electric-transformer inverter with maximum efficiency tracking anddimming control,” in Proc. IEEE APEC’02, vol. 1, Mar. 2002, pp.918–924.

[4] Philips Components, “Application Note Phillips Magnetic Products:Piezoelectric Transformers,” Philips Components, Nijmegen, TheNetherlands, 1997.

[5] G. Ivensky, M. Shvartsas, and S. Ben-Yaakov, “Analysis and modelingof a piezoelectric transformer in high output voltage applications,” IEEETrans. Power Electron., vol. 19, no. 2, pp. 542–549, Mar. 2004.

Shmuel (Sam) Ben-Yaakov (M’87) received theB.Sc. degree in electrical engineering from theTechnion, Haifa, Israel, in 1961 and the M.S. andPh.D. degrees in engineering from the Universityof California, Los Angeles, in 1967 and 1970,respectively.

He is presently a Professor with the Department ofElectrical and Computer Engineering, Ben-GurionUniversity of the Negev, Beer-Sheva, Israel, andheads the Power Electronics Group there. His currentresearch interests include power electronics, circuits

and systems, electronic instrumentation, and engineering education. He alsoserves as a consultant to commercial companies in the areas of analog andpower electronics.

Simon Lineykin received the B.Sc. degree inmechanical engineering and the M.S. degree in elec-trical engineering from Ben-Gurion University ofthe Negev, Beer-Sheva, Israel, where he is currentlypursuing the Ph.D. degree in electrical engineering.

His research interests are modeling and emulationof the physical processes and active cooling systemsusing the Peltier effect.

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4.1.3 A Unified SPICE Compatible Model for Large and Small Signal Envelope

Simulation of Linear Circuits Excited by Modulated Signals [37], [38].

This paper was presented at the IEEE International Conference PESC-2003,

Acapulco, Mexico [37], and was accepted for publication in IEEE Transactions on

Industrial Applications [38] with small revisions. Some new experimental results were

added.

The method proposed in this paper was used for investigating the BW of a

piezoelectric transformer excited by modulated waves [66] - [72]

[66]

. It was previously

shown in that BW depends on several parameters such as the mechanical

resonance quality factor, the frequency of the carrier wave, etc. The method proposed in

the present paper provides designers with a simple simulation tool to analyze the

transfer function of piezoelectric transformers.

The envelope simulation method developed earlier ( [33] - [36]) for large signal

simulation (time domain - TRAN) is extended to include small signal envelope

simulation (AC) and DC Sweep simulation (steady state for a range of carrier

frequencies). The model is relevant for AM, FM and PM modulation schemes and is

demonstrated on a piezoelectric transformer circuit. The analytical derivations of the

model were verified against full circuit simulations that include a high frequency carrier

and also experimentally. Excellent agreement was found between the results of

simulation according to the new unified envelope model, full simulation and

experimental results.

The proposed approach to simulation also gives a preliminary understanding of

envelope impedance, which opens a new wide field for research. For example, we can

better understand the problem of instability of the envelope of voltage in fluorescent

lamp, notwithstanding the fact that the system is stable for carrier frequency.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 3, JUNE 2006 745

Unified SPICE Compatible Model for Large andSmall-Signal Envelope Simulation of Linear

Circuits Excited by Modulated SignalsSimon Lineykin and Shmuel Ben-Yaakov, Member, IEEE

Abstract—The envelope-simulation method, developed earlierfor large-signal simulation [time domain (TRAN)] is extended toinclude small-signal envelope simulation (ac) and dc sweep simu-lation (steady state for a range of carrier frequencies). The modelis derived for amplitude modulation (AM), frequency modulation(FM), and phase modulation (PM) modulation schemes and isdemonstrated on a piezoelectric transformer circuit. The modelis based on the equivalent circuit approach and can be run on anymodern electronic circuit simulator. An excellent agreement wasfound between the simulation results according to the new unifiedenvelope model, full simulation, and experimental results.

Index Terms—Envelope simulation, piezoelectric transformers,resonant inverters, simulation program with integrated circuitemphasis (SPICE).

I. INTRODUCTION

VARIOUS power electronics systems such as resonant con-verters, electronic ballasts for discharge lamps, piezoelec-

tric transformers, and others, are based on resonant networksthat are often exposed to modulated signals. For example,the conventional method of setting the light outputs of lampspowered by electronic ballasts is to control the drive frequencyof the ballasts. When such systems operate in closed loop(Fig. 1), the feedback signal of interest is normally the envelopeof the sensed signal. In this case, the error signal is translatedinto a frequency-modulated signal which, in turn, is affectingthe envelope of the signal that is sensed at the output (Fig. 1).Consequently, the analysis and simulation of the small-signaltransfer functions, needed for controller design and stabilityanalysis, is rather complicated. Earlier studies attempted totackle the problem of small-signal analysis of carrier-drivensystem by one of the following approaches: 1) signal pertur-bation and linearization of the state-space equations [1]–[4]or 2) signal perturbation and linearization of the system’sequations after phasor transformation [5]–[12]. The commondenominator of all these methods is the need for derivinganalytically the small-signal expressions before calculation or

Manuscript received July 30, 2003; revised October 19, 2005. Abstractpublished on the Internet March 18, 2006. An earlier version of this paperwas presented at the 2003 IEEE Power Electronics Specialist Conference(PESC’03) Acapulco, Mexico, June 15–19. This work was supported by TheIsrael Science Foundation under Grant 113/02 and by the Paul Ivanier Centerfor Robotics and Production Management.

The authors are with the Power Electronics Laboratory, Department ofElectrical and Computer Engineering, Ben-Gurion University of the Negev,Beer-Sheva 84105, Israel (e-mail: sby@ee.bgu.ac.il).

Digital Object Identifier 10.1109/TIE.2006.874421

Fig. 1. Resonant converter under closed-loop control.

simulation can be carried out. This normally requires rathertedious manual work, which has to be repeated for each sys-tem. Furthermore, the resulting expressions are exceedinglycomplex to the point that they might be too involved for thenonanalytical expert such as the common design engineer.

As demonstrated earlier [6]–[9], envelope simulation by asimulation program with integrated circuit emphasis (SPICE)compatible model could be used to simplify the extractionof the large-signal response of the carrier-driven linear powerelectronics systems. The method can also be used to simulatenonlinear systems by applying equivalent linear networks toemulate the behavior of the nonlinear parts of the system(such as the rectifier in a resonant dc–dc converter [13]–[17]).Obviously, linearization is required in any small-signal analysismethod (e.g., [1] and [3]) that attempts to handle nonlinearsystems.

The earlier SPICE compatible envelope-simulation methodwas confined to large-signal (time domain) simulation. Small-signal responses can be extracted from the envelopes of thetime-domain-simulation runs by repeating the simulation fora range of modulating signal [8]. This, of course, is a tediousprocess since each frequency domain point requires a lengthytime-domain-simulation run.

The objective of this study is to develop a unified modelthat can be used to run both large-signal (time domain, TRAN)envelope simulation as well as small-signal (frequency domain,ac) simulation by applying the same model, that is, to develop amodel that can be used as is, without the need for an analyticalperturbation and linearization effort, for TRAN, AC, as well asDC (steady-state sweep) analysis. Since the method hinges onthe earlier SPICE compatible envelope-simulation model, wepresent the essentials of that model by a way of an example:a piezoelectric transformer driven by a modulated signal [9],[18]–[20]. The details of the basic envelope-simulation modelare given in [7] and an example of its application in [8].

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It should be noticed that the attribute “SPICE compatible”used here is meant to imply that the proposed model is basedon the equivalent circuit approach and, as such, can be runon any electronic circuit simulator. Furthermore, by rewritingthe state space equations of the equivalent circuits, one canrun the simulation on MATLAB as was demonstrated earlier inconnection with a behavioral model of pulsewidth-modulation(PWM) converters [21].

II. LARGE-SIGNAL ENVELOPE SIMULATION

Any analog-modulated signal (AM, FM, and PM) can bedescribed by the following general expression:

u(t) = U1(t) cos(ωct) − U2(t) sin(ωct) (1)

where U1(t) and U2(t) are modulation signals, and ωc is theangular frequency of the carrier signal.

Equation (1) can also be written in complex form as

u(t) = Re[(U1(t) − jU2(t)) ejωct

](2)

or as

u(t) =∣∣∣U(t)

∣∣∣ Re[earg(U(t))ejωc(t)

](3)

where

U(t) = U1(t) − jU2(t) (4)

and

arg(

U(t))

= tan−1

(−U2(t)

U1(t)

)(5)

∣∣∣U(t)∣∣∣ =

√U2

1 (t) + U22 (t). (6)

Equation (3) reveals that any modulated signal can be repre-sented by a generalized phasor with time-dependent magnitudeand phase.

As demonstrated earlier in [7] and [8], the SPICE compatibleenvelope-simulation circuit can be developed by means of thefollowing stages:

1) duplicating the circuit to create the real part and theimaginary part;

2) replacing reactive elements (L, C), as shown in Fig. 2, intothe real and imaginary sections of the circuit;

3) placing two excitation sources for real and imaginaryparts (U1(t) and U2(t)) but excluding the carrier;

4) adding a behavioral element for calculating the squareroot of the sum of squares of real and imaginary com-ponents of the output signals.

The expressions for U1 and U2 for various modulationschemes are given below for the case of a single-tone modu-lation with a modulating signal m(t) and a carrier c(t)

m(t) =Am sin(2πfmt) (7)

c(t) =Ac cos(2πfct) (8)

Fig. 2. Replacement of reactive elements by equivalent circuits for envelopesimulation. (a) Replacing an inductor by an inductor and dependent voltagesource. (b) Replacing a capacitor by a capacitor and dependent current source.

Fig. 3. Equivalent circuit of a piezoelectric transformer loaded by a resistorRo and driven by a modulated signal (AM, FM, or PM).

where Am and Ac are the amplitudes of the modulatingsignal m(t) and carrier signal c(t), respectively, fc is thefrequency of the carrier, and fm is the frequency of the mod-ulating signal.

The amplitude modulation (AM) signal is described by

u(t) = (1 + kam(t)) c(t)

= Ac (1 + kaAm sin(2πfmt)) cos(2πfct). (9)

A frequency modulation (FM) signal for any modulatingsignal m(t)

u(t) = Ac cos(

2πfct + 2πkf

∫m(t)dt

)(10)

and in the single-tone case

u(t) = Ac cos(

2πfct − kfAm

fmcos(2πfmt)

). (11)

The phase modulation (PM) signal is expressed as

u(t) = Ac cos (2πfct + kpm(t))= Ac cos (2πfct + kpAm sin(2πfmt)) (12)

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LINEYKIN AND BEN-YAAKOV: MODEL FOR LARGE AND SMALL-SIGNAL ENVELOPE SIMULATION OF CIRCUITS 747

Fig. 4. Schematics of envelope-simulation model of the piezoelectric transformer circuit (Fig. 3) excited by a frequency-modulated signal (PSPICE/OrCADevaluation Version 9.2).

where “k” is the modulation coefficient and the subscript in-dexes “a,” “p,” and “f” stand for AM, PM, and FM, respectively.

The decomposed signal sources (U1, U2) for AM are

U1 =Ac + kaAmAc sin(2πfmt) (13)

U2 =0. (14)

For FM

U1 =Ac cos (β cos(2πfmt)) (15)

U2 =Ac sin (β cos(2πfmt)) (16)

where β = Amkf/fm.For PM

U1 =Ac cos (kpAm sin(2πfmt)) (17)

U2 = − Ac sin (kpAm sin(2πfmt)) . (18)

Based on the above, the piezoelectric transformer circuit ofFig. 3, driven by an FM-modulated source, can be representedby the SPICE circuit of Fig. 4 (shown for OrCAD, V 9.2).The circuit is split into two sections representing the realand imaginary parts and includes two excitation ports “inre”and “inim,” which are driven by the U1 and U2 sources ofthe FM case. The behavioral source “abs_out” carries out thecalculation of the square root of the sum of squares of real andimaginary components of the output voltage.

Typical simulation results that compare the traditional tran-sient simulation of Fig. 3, as is, to the results of envelopesimulation by the model of Fig. 4 are depicted in Fig. 5. Thetwo sets of results are identical and not merely “similar” sincethe envelope-simulation method is based on an exact analyticalrepresentation of the circuit.

III. UNIFIED MODEL FOR ENVELOPE SIMULATION

The envelope-simulation model will now be extended toinclude not only the case of large-signal (time domain, TRAN)analysis, but also the small-signal analysis case (frequencydomain, AC) and the DC sweep case. The latter is a steady-state analysis carried out for a range of carrier frequencies (fc).Since the large-signal SPICE-compatible circuit of the phasor-transformed model is linear, the circuit itself is applicable asis for all three types of analysis. The difference will be in theexcitation signals, that is, the expression for the real part U1 andthe imaginary part U2. The required excitation signals will be,in general, different for each modulation scheme and for eachtype of analysis (TRAN, DC, and AC).

A. TRAN Analysis Cases

The excitations of the time-domain analyses follow (13) and(14) for AM, (15) and (16) for FM, and (17) and (18) for PM.

B. DC Analysis Cases

In this steady-state analysis, the simulation is repeated fora number of carrier frequencies within a specified range. Thatis, the amplitude of the carrier frequency (Ac) is constant; theamplitude of the modulation frequency (Am) is zero; and the“dc” sweep is over the carrier frequency (fc). Under theseconditions, the excitations for all modulation types (AM, FM,PM) are found from (13)–(18) to be

U1 = Ac

U2 = 0.(19)

C. AC Analysis Case

Small-signal analysis is carried out after linearization of thecircuit and excitation sources. Since the circuit is linear, it will

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748 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 3, JUNE 2006

Fig. 5. Transient and envelope-simulation results for FM modulation. Upper curve: modulating signal; middle curves: frequency-modulated input carrier signal(gray) and envelope of the input signal (black line); and lower plot: PT’s output signal (gray curve) and its envelope (black curve) obtained by the envelope-simulation model of Fig. 4.

be left as is by the simulator when running the AC analysis.The excitation sources, however, need to be modified. This canbe accomplished by: 1) reducing the large-signal expressionsto the small-signal case (i.e., narrowband modulation) and2) replacing the time-dependent representation of the TRANsources by phasors. This will be exemplified next by consider-ing the case of FM modulation. For small signal Am → 0, and(15) and (16) reduce to

U1 = Ac

U2 = Ackffm

Am cos(2πfmt). (20)

Hence, in ac analysis, U1 needs to be represented by a DCsource of magnitude Ac and U2 by

U2 = 2πAckf

∫Amdt (21)

where Am is a phasor of magnitude Am.Hence, for AC analysis, the port “inre” needs to be fed by a

dc source of magnitude Ac and the port “inim” by an ac sourceof magnitude Am followed by an integrator (a standard behav-ioral model) and multiplied by 2πAckf . A proposed implemen-tation in OrCAD Version 9.2 is shown in Fig. 6(a). Similartransformations produce the AC sources for other modulationtypes: AM [Fig. 6(b)] and PM [Fig. 6(c)].

A summary of the excitation sources and conditions for eachtype of modulation and analysis is given in Table I. For all

Fig. 6. PSPICE implementation of the decomposed sources for AC-sweepenvelope simulation. Real and imaginary components of the source for small-signal envelope simulation for (a) FM, (b) AM, and (c) PM. Ac is the amplitudeof the carrier wave, Am is the amplitude of the modulating signal, kf isthe coefficient of frequency modulation, ka is the coefficient of amplitudemodulation, and kp is the coefficient of phase modulation.

modulation schemes, the variables that are swept in eachanalysis are as follows: “time” for TRAN analysis, carrier fre-quency (fc) for dc analysis, and “frequency” for AC analysis.Each type of analysis calls for a unique real excitation (inre)and imaginary excitation (inim), while the circuit itself is left asis. The excitation sources used in each type of analysis shouldbe consistent with the analysis. That is, in TRAN analysis, DCand time-dependent sources are used; in AC analysis, dc andac sources are used, while in DC analysis, only dc sourcesare used.

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LINEYKIN AND BEN-YAAKOV: MODEL FOR LARGE AND SMALL-SIGNAL ENVELOPE SIMULATION OF CIRCUITS 749

TABLE IREAL (INRE) AND IMAGINARY (INIM) SIGNALS REQUIRED FOR CARRYING OUT SMALL SIGNAL,

LARGE SIGNAL, AND DC SIMULATION IN THE AM, FM, AND PM CASES

Fig. 7. Steady-state output voltage as a function of excitation (carrier) fre-quency of the simulation model (Fig. 4) for different loads Ro obtained byapplying dc-envelope analysis.

IV. SIMULATION RESULTS

The proposed unified envelope-simulation method was testedby comparing the envelope-simulation results obtained by theproposed model to the results of full simulation (which includesthe carrier) of the piezoelectric circuit of Fig. 3. The equiv-alent circuit of the experimental PT includes: Ci = 200 pF,Co = 225 pF, Lr = 22.6 mH, Cr = 9.83 pF, Rm = 1.121 kΩ,and N = 0.647. Typical simulation results are given inFigs. 7 and 8.

The “dc” simulation results of Fig. 7 were obtained by apply-ing the source, as shown in Table I and sweeping the parameterfc, that is, the carrier frequency, over the frequency range of340–370 kHz. The simulation was repeated for different loadsby applying the “parametric sweep” option. The simulationresults of Fig. 7 represent the frequency dependence of theoutput voltage of a piezoelectric transformer for different load-resistance values.

Fig. 8 compares the results of ac-envelope simulation,according to the proposed method, to the results of conventionalsimulation of the original circuit as is. The ac small-signalresponse was obtained from the full-circuit TRAN simulationby running the simulation for many (time domain) FM modu-lated signals and measuring the resulting steady-state envelopes[6]–[8]. Fig. 8 shows the results for the conventional simulationruns (phase shift and amplitude) and the results of the ac-envelope simulation for the case of FM. Fig. 8 confirms thatthe results obtained by the two methods are identical. However,while the results of the ac–envelope simulation were obtainedin a few seconds, running the set of transient simulations andextracting the results took hours.

Fig. 8. Small-signal (ac) simulation results of the PT phasor model Fig. 4excited by an FM signal (lines) compared to the results of multiple runs oftransient analysis (symbols) for different carrier frequencies fc.

V. EXPERIMENTAL

The small-signal transfer function of the PT under study wasmeasured by applying the experimental setup of Fig. 9. The PTwas driven by a modulated signal; the output was buffered (tocontrol loading), rectified by a voltage doubler, and the rectifiedsignal was buffered again.

Typical results of the experimental measurements (dashedlines) and simulations (heavy lines) are shown in Figs. 10and 11. The small disagreement between the experimental dataand the simulation results is probably due to the fact that theparameters of the PT model are in slight error. This is dueto experimental limitations of practical parameter-extractionprocedures and the nonlinearity of the PT. Notwithstandingthe slight experimental errors, the good agreement betweenthe experimental and simulation results supports the conjec-ture that small-signal envelope simulation is a viable tool toexplore the transfer function of a PT under various mod-ulating conditions. Further information on PT excitation bymodulated signals and their envelope simulations can be foundin [15] and [16].

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750 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 3, JUNE 2006

Fig. 9. Experimental setup. The network analyzer measures the ratio betweenthe signals (A)—envelope of the output signal, and the reference terminal (R)which is connected to the excitation port (RF ).

Fig. 10. Small-signal transfer function of the experimental PT under AM exci-tation. Solid line: small-signal envelope simulation; dashed line: experimental;and carrier frequency: 353 kHz.

Fig. 11. Small-signal transfer function of the experimental PT under FM exci-tations. Solid line: small-signal envelope simulation; dashed line: experimental;carrier frequency. (a) fc = 346.1 kHz. (b) fc = 357.7 kHz.

VI. DISCUSSION AND CONCLUSION

The major contribution of this paper is the novel-modelingmethod that facilitates an SPICE compatible ac-envelope sim-ulation by the equivalent circuit approach. The two majoradvantages of the method are: 1) the use of the AC-analysiscapability of SPICE rather than the tedious extraction of thesmall-signal response from sets of TRAN analysis and 2) thesimplicity of the excitation sources that can be used to run as isand without further analytical derivation (such as small-signalperturbation), TRAN, DC, and AC analyses. A good agree-ment was found between the simulation results according tothe proposed method, full circuit simulation, and experimentalresults.

As simulation tools are developed, power electronics willfollow the trend of other electronic areas, in which a major partof the engineering-design work is carried out by simulation.The proposed modeling method could be useful to the engineerand to the researcher since it provides the tool to exploresystems that are very difficult to be examined analytically. Inparticular, the method can be advantageously used to extractthe small-signal responses needed for the design of the controlloops in feedback systems. The simple and unified approachand the short simulation time are making the method easy touse and user friendly.

REFERENCES

[1] A. F. Witulski, A. F. Hernandez, and R. Erickson, “Small signal equivalentcircuit modeling of resonant converters,” IEEE Trans. Power Electron.,vol. 6, no. 1, pp. 11–27, Jan. 1991.

[2] V. Vorperian, “Approximate small-signal analysis of the series and theparallel resonant converters,” IEEE Trans. Power Electron., vol. 4, no. 1,pp. 15–24, Jan. 1989.

[3] J.-H. Cheng, A. F. Witulski, and J. U. Vollin, “A small-signal modelutilizing amplitude modulation for the class-d converter at fixed fre-quency,” IEEE Trans. Power Electron., vol. 15, no. 6, pp. 1204–1211,Nov. 2000.

[4] E. X. Yang, F. C. Lee, and M. M. Jovanovic, “Small-signal modelingof series and parallel resonant converters,” in Proc. IEEE APEC, 1992,pp. 786–792.

[5] C. Rim and G. Cho, “Phasor transformation and its application tothe DC/AC analyses of frequency phase-controlled series resonant con-verters,” IEEE Trans. Power Electron., vol. 5, no. 2, pp. 201–211,Apr. 1990.

[6] S. Ben-Yaakov, S. Glozman, and R. Rabinovici, “Envelope simula-tion by SPICE-compatible models of electric circuits driven by modu-lated signals,” IEEE Trans. Ind. Electron., vol. 47, no. 1, pp. 222–225,Feb. 2000.

[7] ——, “Envelope simulation by SPICE-compatible models of linear elec-tric circuits driven by modulated signals,” IEEE Trans. Ind. Appl., vol. 37,no. 2, pp. 527–533, Mar./Apr. 2001.

[8] S. Glozman and S. Ben-Yaakov, “Dynamic interaction analysis of HFballast and fluorescent lamps based on envelope simulation,” IEEETrans. Ind. Appl., vol. 37, no. 5, pp. 1531–1536, Sep./Oct. 2001.

[9] S. Lineykin and S. Ben-Yaakov, “A unified SPICE compatible model forlarge and small signal envelope simulation of linear circuits excited bymodulated signals,” in Proc. IEEE PESC, Acapulco, Mexico, Jun. 2003,pp. 1205–1209.

[10] Y. Yin, R. Zane, R. Erickson, and J. Glaser, “Dynamic analysis offrequency-controlled electronic ballasts,” in Conf. Rec. 37th IEEE-IASAnnu. Meeting, Oct. 2002, pp. 685–691.

[11] G. Spiazzi and S. Buso, “Small-signal analysis of cold cathode fluores-cent lamp ballasts,” in Proc. IEEE PESC, Recife, Brazil, Jun. 2005,pp. 2783–2789.

[12] Y. Yin, R. Zane, R. Erickson, and J. Glaser, “Direct modeling of enve-lope dynamics in resonant inverters,” in Proc. IEEE PESC, 2003, vol. 3,pp. 1313–1318.

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LINEYKIN AND BEN-YAAKOV: MODEL FOR LARGE AND SMALL-SIGNAL ENVELOPE SIMULATION OF CIRCUITS 751

[13] R. Steigerwald, “Analysis of a resonant dc–dc converter with capacitiveoutput filter,” IEEE Trans. Ind. Electron., vol. IE-32, no. 2, pp. 439–444,Nov. 1985.

[14] G. Ivensky, M. Shvartsas, and S. Ben-Yaakov, “Analysis and modelingof a voltage doubler rectifier fed by a piezoelectric transformer,” IEEETrans. Power Electron., vol. 19, no. 2, pp. 542–549, Mar. 2004.

[15] S. Lineykin and S. Ben-Yaakov, “Feedback isolation by piezoelectrictransformers: Comparison of amplitude to frequency modulation,” HAITJ. Sci. Eng., vol. 2, no. 5/6, pp. 830–847, Dec. 2005.

[16] ——, “Feedback isolation by piezoelectric transformers: Comparisonof amplitude to frequency modulation,” in Proc. IEEE PESC, Aachen,Germany, Jun. 2004, pp. 1834–1840.

[17] J. A. Oliver, C. Fernandez, R. Prieto, and J. A. Cobos, “Circuit orientedmodel of rectifiers for large signal envelope simulation,” in Proc. IEEEPESC, Recife, Brazil, Jun. 2005, pp. 2771–2776.

[18] S. Lineykin and S. Ben-Yaakov, “Feedback isolation by piezo-electric transformers: A feasibility study,” in Proc. PCIM, Jun. 2000,pp. 175–181.

[19] S. Ben-Yaakov and S. Lineykin, “Frequency tracking to maximum powerof piezoelectric transformer HV converters under load variations,” inProc. IEEE PESC, Jun. 2002, pp. 657–662.

[20] G. Ivensky, I. Zafrany, and S. Ben-Yaakov, “Generic operational char-acteristics of piezoelectric transformers,” IEEE Trans. Power Electron.,vol. 17, no. 6, pp. 1049–1057, Nov. 2002.

[21] S. Ben-Yaakov and D. Adar (Edry), “Average models as tools for studyingthe dynamics of switch mode DC–DC converters,” in Proc. IEEE PESC,Taipei, Taiwan, R.O.C., 1994, pp. 1369–1376.

Simon Lineykin received the B.Sc. degree in me-chanical engineering in 1997 and the M.S. degreein electrical engineering in 2000 from Ben-GurionUniversity of the Negev, Beer-Sheva, Israel, wherehe is currently working toward the Ph.D. degree inelectrical engineering.

His research interests are modeling and emulationof the physical processes and active cooling systemsusing Peltier effect.

Shmuel (Sam) Ben-Yaakov (M’87) received theB.Sc. degree in electrical engineering from the Tech-nion, Haifa, Israel, in 1961, and the M.S. and Ph.D.degrees in engineering from the University of Cal-ifornia, Los Angeles (UCLA), in 1967 and 1970,respectively.

He is presently a Professor in the Department ofElectrical and Computer Engineering, Ben-GurionUniversity of the Negev, Beer-Sheva, Israel, andheads the Power Electronics Group there. His currentresearch interests include power electronics, circuits

and systems, electronic instrumentation, and engineering education. He alsoserves as a Consultant to commercial companies in the areas of analog andpower electronics.

19 - 7

20

4.1.4 Feedback Isolation by Piezoelectric Transformers: Comparison of

Amplitude to Frequency Modulation. [70], [72]

This paper was presented at the IEEE international Power Electronics Specialists

Conference PESC’04, June 2004, Aachen, Germany [70] and revised version was

published in the HAIT Journal of Science and Engineering, [72].

Signal isolation is required in power electronics systems with separate primary and

secondary "grounds." The feasibility of using a PT as a galvanic barrier was

investigated theoretically and experimentally. The research addressed the issues of

drive, demodulation, bandwidth, and common mode rejection. In particular, two types

of excitation signals were compared: amplitude-modulated (AM) excitation signals and

frequency modulated (FM) excitation signals. The frequency response of the

piezoelectric isolator was studied by small signal envelope simulation using

ORCAD/PSPICE and also experimentally. We concluded that the FM approach offers a

larger bandwidth but the AM approach seems to be easier to implement.

HAIT Journal of Science and Engineering B, Volume 2, Issue x, pp. xxx-xxxCopyright C° 2005 Holon Academic Institute of Technology

Feedback isolation by piezoelectrictransformers: comparison of amplitude

and frequency modulationSimon Lineykin∗ and Sam Ben-Yaakov

Power Electronics Laboratory,Department of Electrical and Computer Engineering,

Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel∗Corresponding author: sby@ee.bgu.ac.il

Received 1 April 2005, accepted 10 July 2005

Abstract

Signal isolation is needed in power electronics systems that includeseparate primary and secondary ’grounds’. The feasibility of using apiezoelectric transformer (PT) as a galvanic barrier was investigatedin this study theoretically and experimentally. The research includedthe issues of drive, demodulation, bandwidth, and common mode re-jection. In particular, two types of excitation signals were compared:amplitude modulated and frequency modulated. The frequency re-sponse of the piezoelectric isolator was studied both by small signalenvelope simulation using ORCAD/PSPICE and experimentally. Itis concluded that the FM approach offers a larger BW but the AMscheme seems to be easier to implement.

Keywords: Acoustoelectric devices, isolators, piezoelectric devices,transformers, AM, FM.

1 Introduction

Signal isolation is needed in power electronics systems that include separateprimary and secondary ’grounds’ [1,2]. Conventional solutions to the signalisolation problem include galvanic isolation by opto-couplers, capacitors, ortransformers (Fig. 1). In this study, we explored the feasibility of using a

2

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ceramic Piezoelectric Transformer (PT) as the barrier element, that is, thepossibility of replacing the isolation element by the PT [3]. Since the PTcannot transfer low frequency and DC signal, modulation techniques need tobe applied. Namely, the error signal is first used to modulate a carrier andthen recovered by demodulating the output signal past the isolation barrier.This is similar to the method used when an electromagnetic transformeris applied to realize the isolation barrier. The primary objective of thisinvestigation was to compare the PT isolator performance in terms of thesmall signal bandwidth of the PT transfer function, when modulated by AMor FM signals (Fig. 2).

Figure 1: Conventional feedback isolation in power converters.

Figure 2: Block-diagram of a PT-Isolator with FM or AM excitation.

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2 The piezoelectric transformer

The PT is a solid-state device that transforms the electrical energy at itsinput port to a secondary electrical energy at the output port using acousticwaves as a medium. The PT operates at frequencies that are close to itsmechanical resonance so the PT is, in fact, a selective resonant system.The equivalent circuit of a Fig. 3 describes the PT for a given resonantmode [3,5,6]. It includes a resonant network (Lr, Cr, Rm) that emulates theeffect of the mechanical vibration and dependent sources that express thegain. The model also comprises the physical dielectric capacitors (Cin, Co),which are formed by the input and output electrodes. Also included in thismodel are stray capacitances Cnm(n,m = 1, 2) between input and outputelectrodes. These capacitances play an important role in deteriorating theCommon Mode Rejection Ratio (CMRR) of the device, that is, the ratiobetween the differential and the common mode transfers. High commonmode signal penetration will not only deteriorate the signal to noise ratio atthe output but also increase the EMI signal at the output section and willtherefore require extra filtering to satisfy common standards.

Figure 3: Equivalent circuit of a piezoelectric transformer (PT)loaded by Ro. For experimental PT: Cin=200pF, Co=220pF,Lr=16.58mH, Cr=12.97pF, Rm=910Ω, and N=1.2 (Qm=49.3, fr=342.9kHz,fs=354.8kHz). Cnm are parasitic capacitive couplings between input andoutput electrodes.

In this study, we have used a small size PT (2 × 2 × 3mm) made ofLead Zirconate Titanate material EC65. The transfer functions of the ex-perimental PT are shown in Fig. 4. The plots suggest that the peak at

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Figure 4: Differential Mode (DM) and Common Mode (CM) transfer ratiosof experimental PT.

Figure 5: Gain versus frequency dependence in the experimental PT (solidlines) and transfer function of equivalent circuit (dashed lines) of Fig. 3 forthe loads: 1.2MΩ, 100kΩ, 56kΩ, 22kΩ, 10kΩ, and 5kΩ.

350 kHz is a good choice since it is a conveniently high frequency and has thebest CMRR. A closed look at the differential mode transfer function of thePT (Fig. 5) reveals that the peak around the chosen operating frequency(350kHz) is adjacent to another peaks in the vicinity. In the following,though, it is assumed that the PT can be represented to reasonable degreeby a single resonant network.

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3 Extracting PT parameters

The common method for studying the behavior of systems that includedifferent energy types is to build an electrical equivalent circuit, which emu-lates the behavior of the complete system. The advantage of the equivalentcircuit is that it can be studied by means of traditional techniques of elec-tronic circuits analysis. Several methods have been proposed for extractingthe equivalent circuit parameters of a PT [5—7]. The proposed method usedin this study was motivated by the need to get the parameters of a PT ofmoderately low mechanical quality factor Qm using minimum number ofmeasurements and simple equipment.

The method applies the measurement of the maximum of output to inputvoltage ratio for unloaded PT and when loaded with a known arbitrary load,and the frequencies of corresponding maxima.

The calculation of the equivalent circuit elements was based on the fol-lowing assumptions:

1. There is only one significant resonant mode around the frequency ofoperation. The influence of other frequencies is negligible.

2. The input resistance of the measuring equipment is large enough sothat the loading effect can be neglected.

3. ESRs of input and output capacitors are small and may be neglected.

4. Output capacitance Co of the secondary electrode pair is known.

5. There are four unknown elements in the equivalent circuit: Cr, Lr, Rm,and N (see Fig. 3).

For the equivalent circuit of Fig. 3, the output to input voltage ratio isexpressed as

A (Ro, f) =VoVin

=

rC2rNR2oω

2

F(1)

where:

F =¡N2 − CrLrN

2ω2 − CoCrN2RmRoω

2¢2

(2)

+¡CrN

2Rmω + CrRoω + CoN2Roω − CoCrLrN

2Roω3¢2,

ω = 2πf, (3)

and f is frequency.

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The frequency at which the transfer ratio reaches a maximum, fmax, canbe derived by differentiating (1) with respect to f and solving for the rootsof the resulting equation

2C2rN6R2o − 4C2oC4rL2rN6R4ox

3 (4)

−µ2C4rL

2rN

6R2o + 4CoC4rLrN

4R4o+4C2oC

3rLrN

6R4o − 2C2oC4rN6R2mR4o

¶x2 = 0

wherex = (2πfmax)

2. (5)

Figure 6: Simplification of the equivalent circuit of the open-circuited PT.

The simplified equivalent circuit of the open-circuited PT (Fig. 6) wasderived from the previous expression by setting the resistance of the loadresistor Ro to infinity. The transfer function was found to be:

A (∞, f) =VoVin

¯Ro→∞

=NCrq

C2rC‘2o R

2mω

2 + (Cr + C 0o (1− CrLrω2))2

(6)

whereC 0o = Co ·N2. (7)

Output voltage of Fig. 6 is:

V 0o =VoN. (8)

By differentiating expression (6) with respect to ω and equating theresulting expression to zero, the angular frequency of the maximum of thetransfer function of open circuited PT, fpk was obtained as:

ωpk =

sC0o + Cr

C 0oCrLr− R2m2L2r

=

sC 0o +Cr

C 0oCrLr

µ1− 1

2Q2

¶≈s

C 0o + Cr

C 0oCrLr(9)

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where

Q =1

Rm

rLr

Cr

s1 +

Cr

C 0o(10)

andfpk = ωpk/2π, (11)

is the frequency of maximum input to output ratio of the unloaded PT. Q isa quality factor of the open-circuited PT, usually larger than five. Hence,the frequency of the peak of the transfer function of the open-circuited PTis very close to the series resonance frequency fs (ωs = 2πfs)

ωpk ≈ ωs =

sCo + CrN2

LrCrCo=1

ωr

s1 +

Cr

C 0o(12)

whereωr = 1/

pLrCr, (13)

is the angular frequency of the mechanical resonant circuit (ωr = 2πfr).Consequently, the output to input voltage ratio at the frequency of max-

imum is

A (∞, fpk) =N

Rm

2Lr

√Crp

C 0o (4Lr (C 0 + Cr)− CrC 0oR2m)≈ N

Rm

sCrLr

C 0o (C 0o + Cr).

(14)Solving together equations (6) - (14), one obtains:

Rm =Q³f2pk − f2r

´2π CoA (∞, fpk)

2 f3pk, (15)

Cr =A (∞, fpk)

2 Co f4pk³

f2pk − f2r

´f2r Q

2, (16)

N =A (∞, fpk) f

2pk³

f2pk − f2r

´Q, (17)

Lr =

³f2pk − f2r

´Q2

4π2 f4pk CoA (∞, fpk)2 . (18)

The above solutions contain the two measured values of the open-circuitedPT: A(∞, fpk) and fpk, and assume that Q and fr are known.

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The exact numerical values of parameters can be evaluated by an iter-ation procedure. The iteration process starts after assuming some initialvalues for Q and fr and plugging them in (15) - (18) for further evaluation.The results are then used to estimate fmax and A(Ro, fmax) and these arecompared to the measured values. In a case of mismatch, Q and fr areincremented and the calculation is repeated.

Figure 7: Block diagram of the algorithm for estimating the parameters ofa PT equivalent circuit.

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The block diagram of the iteration algorithm is shown in Fig. 7. Once theprocedure converges to the solutions, one needs to check the validity of the apriori assumptions that Q is larger than five and that the input resistance ofthe measuring equipment Rin is equal or larger than the output impedanceof the PT:

Rin ≥ 50 +p2500−N44π2f2rC

2oR

2m

N44π2f2rC2oRm

. (19)

Applying the above procedure to the experimental PT, the followingparameters were extracted: Lr = 16.58mH, Cr = 12.97pF, Rm = 910Ω,and N = 1.2 (Qm = 49.3, fr = 342.9kHz, fs = 354.8kHz). Input andoutput capacitances were measured as: Cin = 200pF, Co = 220pF. Thetransfer function of the remodeled equivalent circuit is shown in Fig. 5 bydashed lines for different loads and is found to be in good agreement withthe experimental data (solid lines).

4 Modulation schemes

In this study, we explored two modulation alternatives: amplitude modula-tion (AM) and frequency modulation (FM) (see Fig. 2). In the AM case,a constant frequency carrier is used and by varying the amplitude of thecarrier, the error signal is transmitted. In the FM case, the error signal iscoded into a frequency deviation that is translated at the output of the PTinto an amplitude change.

4.1 Frequency modulation

For a single-tone harmonic modulating signal m(t)

m (t) =M cosωmt (20)

where ωm is angular frequency of the modulating signal ωm = 2πfm, andM is its amplitude, one can express the modulated carrier wave v(t):

v (t) = Ac cos(ωct+K

tZ0

m (t) dt) (21)

where ωc and Ac are angular frequency and amplitude of the carrier signaland K is the modulation coefficient. Or, after integration:

v (t) = Ac cos(ωct+

µKM

ωm

¶sinωmt), (22)

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KM

ωm= β. (23)

The spectrum of a frequency-modulated signal includes the fundamen-tal term at the frequency of the carrier signal and an infinite set of side-frequencies located symmetrically on either side of the carrier at frequencyseparations of fm, 2fm, 3fm, etc. For a modulating signal of small amplitude(β ¿ 1) (narrow band frequency modulation - NBFM), only two sidebands(fc+fm) and (fc−fm) have significant values. Thus, the required bandwidth(BW) to transfer a modulating signal of frequency fm is:

BW = 2fm. (24)

4.2 Amplitude modulation

For the single-tone amplitude modulation m(t) of (22)

v (t) = (1 + kam (t)) Ac cos (ωct) (25)

where ka is the modulation index. Or:

v(t) = Ac cos (ωct) + kaAcM cos (ωmt) cos (ωct) (26)

and after some trigonometric transformations:

v (t) = Ac cos (ωct) +kaAcM

2cos ((ωc − ωm) t) +

kaAcM

2cos ((ωc + ωm) t) .

(27)Hence, the spectrum of an AM signal will include the carrier (fc) plus

two side bands located at fc + fm and fc − fm where fm is the modulatingfrequency. Thus, like in the case of narrow band FM, one can express therequired bandwidth for passing m(t):

BW = 2fm. (28)

It can thus be concluded that for small signal analysis, both the AMand FM scheme require the same bandwidth for a given modulating signalfm. However, since the PT is a resonant element, its gain is frequencydependent. As a result, the carrier and side bands of the modulated signalwill be attenuated differently and this may affect the practical BW that canbe obtained via the PT. It could further be expected that the location ofthe carrier frequency in relation to the PT’s maximum gain frequency willalso have an impact on the BW. These questions were studied by envelopesimulation and then verified experimentally.

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5 Small signal envelop simulation

The question of small signal bandwidth of a PT when passing a modulatedsignal can conveniently be studied by small signal envelope simulation [8].In this technique, the original signal is broken into two coupled parts, a realand imaginary network.

Figure 8: Replacement of reactive elements by equivalent circuits for enve-lope simulation: (a) replacing an inductor by an inductor and dependentvoltage source; (b) replacing a capacitor by a capacitor and dependent cur-rent source.

As demonstrated in [8], the SPICE compatible small-signal envelope sim-ulation circuit can be developed by means of the following stages:

1. Duplicating the circuit to create the real and imaginary parts.

2. In the two sections: replacing reactive elements (L, C) into the realand imaginary sections of the circuit, as shown in Fig. 8.

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3. Placing two excitation sources for real and imaginary parts accordingto the type of modulation as shown in Fig. 9.

4. Adding a behavioral element for calculating the square root of the sumof squares of real and imaginary components of the output signals.

Figure 9: Real and imaginary components of the source for small-signalenvelope simulation, representing excitation by FM (a) and AM (b). Ac -amplitude of the carrier wave, Am - amplitude of the modulating signal, kf— coefficient of frequency modulation, ka.

Using this technique, the equivalent circuit of Fig. 3 was transformedinto the SPICE compatible circuit of Fig. 10. This circuit is valid for AMas well as FM excitation. The only difference is the source element for FM(Fig. 9(a)) and for AM (Fig. 9(b)). This circuit was than used to exploreby simulation the transfer of a modulated signal via the PT.

6 Simulation and experimental results

The small signal transfer function of the PT under study was measured bythe experimental setup of Fig. 11. The PT was driven by a modulatedsignal, the output was buffered (to control loading), rectified by a voltagedoubler and the rectified signal was buffered again.

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Figure 11: The experimental setup.

Figure 12: Small signal transfer function of the experimental PT in AMscheme. Solid line: small signal envelope simulation. Dashed line: expe-rimental. Carrier frequency: 353kHz.

Typical results of the experimental measurements (dashed lines) and sim-ulations (solid lines) are shown in Fig. 12 and Fig. 13. The good agreementbetween the experimental and simulation results support the conjecture thatsmall signal envelope simulation is a viable tool to explore the BW of a PTunder various modulating conditions.

The experimental and simulation results demonstrate that the tested PThas a BW of about 5 - 7 kHz both in AM and FM excitation. However, theresults show that the location of the carrier frequency in relation to the reso-nant frequency affects the transfer function of the PT. In particular, whereasAM works well if the carrier and resonant frequencies coincide, better resultsare obtained for FM when the carrier is off the resonant frequency. To test

15

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this point, we have run additional experiments and simulations. The results(Fig. 14) show that around the peak, the BW is the smallest and that largerBW is obtained off peak.

Figure 13: Small signal transfer function of the experimental PT in FMscheme. Solid line: small signal envelope simulation. Dashed line: ex-perimental. Carrier frequency: (a) fc = 346.1kHz, (b) fc = 357.7 kHz.

This can be explained by the fact that the high frequency side bands arehighly attenuated when the carrier frequency is at the peak of the transferfunction. Results of small signal envelop simulation for both excitationtypes coincide. However, in practice, it was not possible to measure the

16

20 - 15

transfer function of FM excitation when the carrier frequency was too closeto the resonant frequency while the AM results were inconsistent when thecarrier frequency was far from the resonance frequency peak. Fig. 14 showsthe combined results of the simulation and experiments for the relevantexcitation carrier frequencies. Experiments were also run to determine theCommon Mode Rejection (CMRR), that is, the ratio of differential modegain (DM) to the common mode gain (CM), of the PT over the chosenfrequency of operation. The results (Fig. 15) suggest that the CMRR forthis PT is in the range of 10db.

Figure 14: Bandwidth (BW) of the PT transfer function for AM and FMmodulated signal versus frequency of the carrier signal. Simulation: line.Experimental measurements: dots.

Figure 15: Differential-mode (DM) signal and CMRR of the experimentalPT for different loads (5kΩ to 1.2MΩ) in the frequency range close to re-sonance.

17

20 - 16

7 Discussion and conclusions

The main advantages of the PT in the proposed application are the small sizeand the very high isolation breakdown voltage that can be achieved. This isdue to the good isolation of the ceramic material. The main disadvantage isthe common mode stray capacitance between primary and secondary sidesof the PT. The measured CMRR of about 10db (Fig. 15) may be too highin the noisy environment of a switch mode converter, if there is a significantcommon mode signal across the isolator. The possible interference of thecommon mode signal can be minimized by proper design, for example byconnecting the primary of the isolator to a quite ‘ground’ with respect tothe ground of the output. The common mode rejection could be furtherimproved by ensuring that the harmonics of the switching frequency of theconverter do not coincide with the common mode peaks of the PT.

The present study suggests that both AM and FM can be used to passfeedback signals via an isolation PT. It was found that the small signalbandwidth of both excitation cases is practically the same. However, theexperiments that were carried out with practical finite signals, contrary toinfinitesimally small modulating signal assumed in small signal envelop sim-ulation, reveal a difference between the AM and FM schemes. It was foundthat in the AM option, best results were obtained when the carrier fre-quency was close to the resonant frequency (Fig. 12). When the carrier wasoff the resonance frequency of the PT, the demodulated signal was highlydistorted. The inverse was found in the case of FM. In this case, best resultswere obtained when the carrier frequency was off the peak of the resonance(Fig. 13). Since a higher BW is obtained off resonance, the FM approachis better in this respect. However, the AM scheme might have a practicaladvantage since it is possibly simpler to realize by self-oscillating circuit thatlocks to a frequency that is close to the resonant frequency. Another pos-sible frequency tracking technique is described in [9]. In the FM case, onewould need to stay off resonance, which might pose a problem consideringthe spread and temperature dependence of the PT parameters. It can thusbe concluded that the FM approach offers a larger BW but the AM seemsto be easier to implement.

This research was supported by the Israel Science Foundation (grantNo. 113/02) and by the Paul Ivanier Center for Robotics and Productionmanagement.

18

20 - 17

References

[1] M. Zirngast, Electronic Engineering (London) 61, no. 748, 37 (Apr.1989).

[2] M. Zirngast, Electronic Engineering (London) 61, no. 749, 33 (May1989).

[3] G. Ivensky, I. Zafrany, and S. Ben-Yaakov, IEEE Trans. on Power Elec-tronics 17, 1049 (2002).

[4] S. Lineykin and S. Ben-Yaakov, Power Conversion and Intelligent Mo-tion, PCIM’00, p.175, Nürnberg, Germany (2000).

[5] R. Holland and E. Eernisse, Design of Resonant Piezoelectric Devices(The MIT Press, Cambridge, MA, 1966).

[6] An American National Standard, IEEE Standard on Piezoelectricity,ANSI: IEEE Std. 176 (1978).

[7] J. Merhaut, Theory of Electroacoustics (McGraw-Hill, N.Y., 1981).

[8] S. Lineykin and S. Ben-Yaakov, IEEE Power Electronics Specialists Con-ference, PESC’03, p.1205, Acapulco, Mexico (2003).

[9] S. Ben-Yaakov and S. Lineykin, IEEE Power Electronics Specialists Con-ference, PESC’02, p.657, Cairns, Australia (2002).

19

20 - 18

21

4.2 Equivalent circuit modeling and application of electro-thermal

power conversion systems

4.2.1 Modeling and Analysis of Thermoelectric Modules [73]

This paper was presented at the Applied Power Electronics Conference APEC’05,

Austin, Texas, USA. This is a preliminary approach to the study of thermoelectric

modules. The objective of this work was to develop a SPICE compatible equivalent

circuit of a thermoelectric module. The analysis was built on the basis of theory of

semiconductor thermoelectric cooler and thermoelectric generator given in [4] and [1]

[64]

.

The approach to the modeling of the module was based on one given in . The

resulting equivalent circuit was modified to be compatible with any one-stage

thermoelectric module, universal and user friendly.

The study shows how a manufacturer’s data for a TEC [74] or a TEG [75] can be

used to extract the parameters of the proposed model. The model is helpful for

analyzing the drive requirements of TECs and loading effects of TEGs. Another

important application of the proposed model is for analyzing performance of a TEM

under specific conditions such as heat leakage, non-ideal thermal insulation, etc.

Furthermore, with this model one can also construct an optimal module for a specific

problem.

The present model is compatible with PSPICE or other electric circuit simulators

for DC, AC, and TRANSIENT simulation types.

Several examples of successful utilization of the model are presented. The paper is

based on data of many different manufacturers that were used to accurately reproduce

the performance of commercial TEMs.

Modeling and Analysis of Thermoelectric Modules Simon Lineykin and Sam Ben-Yaakov'

Power Electronics Laboratory Department of Electrical and Computer Engineering

Ben-Gurion University of the Negev P. 0. Box 653, Beer-Sheva 84105, ISRAEL, Phone: +972-8-646-1561, Fax: +972-8-647-2949

Email: sby@ee.bgu.ac.il, Website: www.ee.bgu.ac.iU-pel

Abstract - The objective of this work was to develop a SPICE compatible equivalent circuit of B thermoelectric module (TEM). A methodology is developed for extracting the parameters of the proposed model from manufacturers' data of Thermoelectric Coolers (TEC) and Thermoelectric Generators (TEG). The model could be helpful for analyzing the drive requirements of TECs and loading effects of TEGs. The present model is compatible with PSPICE or other electric circuit simulators. An importsnt feature of the model is its ability to generate small signal transfer functions that c m be used to design feedback networks for temperature control applications.

I. INTRODUCT~ON

A thermoelectric module (TEM) is a solid-state energy converter. It normally consists of an array of pellets from dissimilar semiconductor material (p and n type), which are joined, thermally in parallel and electrically in series. The TEM can be used for cooling, heating, and energy generation [ 11 - [3]. As a thermoelectric cooler (TEC), the TEM already found applications in thermal management and control of microelectronic devices such as diode lasers and CPUs. As thennoelectric generator (TEG), the TEM could be used to produce electric power in remote locations when temperature gradients are available [2].

The objective of this work was to develop a SPICE compatible equivalent circuit of a TEM. Equivalent circuit is a convenient tool for electronic engineers. It helps to present the problem in electronic circuit terms, helps to understand its functionality, and facilitates the solving of cooling or power- generation problems without the need for expertise in thermal engineering.

If. PRINCIPLES OF OPERATION

There are five main physical processes taking place in thermoelectric module: Thermal convection - the phenomenon named by Fourier process, described by physical constant k (WKm), which is determined by thermal conductivity and geometry of the pellet. 0 (WW) i s a thermal resistance of the couple

l h @ = - - k A

T = @q (2) where MA is geometry factor, h - height of the pellet (m), A - cross-section area (m'), T - temperature (K), and q - heat (W). Joule heating i s the physical process of heat dissipation on the resistive elements. The electrical resistance R of a couple of pellets is:

h A

R = p - (3)

q j =12R (4) where p - resistivity of the material (Q m), Q - Joule heating (W), I - electric current (A). Peltier cooling/heating - the phenomenon of absorption (or dissipation) of heat by a junction between two dissimilar materials when ekctrical current flows throw the junction. The heat q,, absorbeddissipated by the junction is:

where x (V) is a temperature dependent Peltier coefficient corresponding to a specific pair of materials. Seebeck power generation is a process by which heating (or cooling) of the junction of two dissimilar materials generates an electrical potential of the junction:

where a (VK] is a Seebeck coefficient.

be:

where xde is a potential of absorbing/emitting junctions and Tde is a temperature of absorbing/emitting junctions.

The additional thermoeIectric phenomena - Thompson phenomenon, which is described by the Thompson coefficient T = d d d T (V/Kz). The effect of this phenomenon is small [ 11, [3] and is therefore neglected in this work.

Fig. 1 shows a section of a TEM, which operates in the mode of thermoelectric cooler, when the power supply is connected to eIectric port and heat is pumped from the cold side to the hot side.

q p =nI (5)

?l=CY"T (6)

The potential difference the two junctions of the pellet wilI

U = X , - ~ ~ = U ( T , - T , ) = U A T (7)

* Corresponding author

CX7803-8975-1/05/$u1.00 Q2005 IEEE. 2019

21 - 1

I ,SouiCe,T. " " .

Fig. 1. Energy equilibrium for thermoelectric cooler.

Following the first law of thermodynamics, one can express the energy equilibrium at both sides of the thermoelectric module that are defined as the absorbing (a) and emitting (e) junctions. For absorbing side, one can write:

q, = c + a m T , I - - AT 12R, 2

and for the emitting side: AT I'R,

qe =-+a,T,I+- @nl 2

a, =aN R, = R N 0, = O / N

where q, is a heat absorbing at a-side, qc - heat emitting at e- side, N - number of couples, T, and T, - temperatures of (a-) and (e-) sides in K, and AT=T,-T,.

The electrical part cif the module is described as electrical resistance R, and an electrical potential difference U:

U = a,T, - a,T, ;= a,AT (13) It is a common practice in one-dimensional heat transfer

problems [4] to apply an equivalent electrical circuit scheme. This approach was adopted in this study to describe the TEM system in which severa1 energy types exist. All non-electrical processes are described in terms of electrical analogies, and transformers (or dependent sources) represent their interconnections. By this, the equivalent circuit of thermo- electrical system of TBM can built as a pure electrical circuit. The proposed equivalent circuit topology of the model (Fig. 2) is based on equations (8), (9), and (13) for a- and e-junctions PI.

111. CALCULATION OF'THE PARAMETERS OF THE MODEL FROM THE MANUFACTURER'S DATASHEETS

Manufacturers of TECs ([6], [7], [8], and others) use the foIlowing parameters to specify their product:

AT,,, - is the largesi temperature differential (IC) that can be obtained between the hot and cold ceramic plates of a TEM for the given level of hot-side temperature Th,

I,,, is the input current (A) which will produce the maximum possible AT across a TEC, and

I I f Te

Fig. 2. Proposed equivalent circuit of a thermoelectric module for steady state. Bold lines are used for the thermal parts,

U,,, is the DC voltage (V) that will deliver the maximum possible AT at the suppiied lmw

Applying (8), (9), and (1 3) one can use the set of data: Th, AT-, U,,, I,, for calculating the parameters of proposed model:

Using the relations (8), (9), and (13), one can derive the characteristic parameters:

(14) (1 - J-)

ATma Th + Z

utnax = QmTh (16) where Z is figure of merit of the TEM. Z = akO,/R, .

Applying (14) - (16), one can use the set of data: Th, AT, U,,, I,, for calculating the parameters of the proposed model:

- urn, a, -- Th

(19) 0, = 2Th

I-umax Gi7ZiJ Manufacturers of TEGs normally specify the electrical

properties rather than the thermai ones. The quoted parameters include electric power, open-circuit voltage, maximum power (for matched load), efficiency for matched load, maximum efficiency etc. The temperature range of power generator is usually grater than that of a cooling module and therefore one cannot neglect the dependence of the parameters on temperature. That is why some manufacturers provide data for different temperatures.

The datasheet of TEGs often includes power at matched load W, (load is matched to internal resistance), load voltage at matched load U,, open circuited voltage U,, and maximum efficiency qOpt.

Using this data, one can calculate the parameters of the equivalent circuit directly from datasheets:

2020

21 - 2

R,=- uf, wnl 2U,

a, =- AT

IV. EXAMPLES

In this chapter, several examples of model application are presented:

I . The TB-127-1.4-1.2 - is one of thermoelectric cooling modules available from Kryotherm [ 6 ] . From manufacturer's datashects: AT,,,= 70 K, 1,,,=7.6 A, U,,=15.9 V, under condition that Th=300 K,

Applying (16) - (IS) one can calculate the model parameters: a,=0.053 V K ,

Fig. 3 shows the result of application of the DC-sweep simulation that reconstructs the performance plot of the TEM TB-127-1.4-1.2. Dashed line is simulation result. Original perfonnance plot was copied from the s o h a r e , pfaced on the Internet by Kryotherm. 2. Thc HZ-20 is manufactured by Hi-Z Technology [9]. The data from datasheets: for hot side temperature Th = 230°C and cold side temperature T, = 30 "C, W, = 19 W, U, = 2.38 V, qop, = 4.5%. From (20) - (22): a, = 0.0238 V/K, 0, = 0.589 WW, R, = 0.298 !2.

Fig. 4 shows the DC-sweep simulation of the HZ-20 module. The plot looks identical to the one given in the datasheet of the module.

Cl, 0,=1.5 K N .

v. COMPARISON OF THE EXPERIMENTAL DATA WITH SIMULATION USING THE MODEL

The laboratory measurements of physical TEM were compared with computer simulations to be certain that the model permits to simulate the processes taking place in TEM. The experiment was carried out using the TEM TB-127-1.4-1.2 (Kryotherm) that had the dimensions of 40mmx40mm. The module was inserted between two massive aluminum plates (40mmx40mmx5mm) with implemented thermocouples for temperature measurement. Both plates were insulated themally from the ambient air. The TEM was first used as a cooler and a DC voltage was applied to it (Fig. 5). When the temperature difference between the plates reached a ptedetemined value, the supply voltage was tumed off. From that point, TEM was continuing its operation as a generator and its output voltage U measured for different loads.

To simulate the experimental conditions by proposed model, one needs first to calculate:

I. The parameters of the model (16) - (18). 2. Lumped heat capacity of the aluminum plates. This can

be calculated by:

where c - specific heat (kJ/kg K), p - density (kg/m3), and V - c,, = cpv (23)

Fig 3. Performance ofTEM TB-127-1.4-1.2. Dashed line is simulation resuIt obtained by proposed model, solid line is the published performance plot by manufacturer

20

16

12

8

4

0 0 2 4 6 8 LO 12 14 16

Fig. 4. DC-sweep simulation of performance of the Thermoelectric Generator HZ-20 made by Hi-Z Technology, Inc.

V

C

Fig. 5. The experimental setup.

volume (m3). In this specific case, the Ca is about 19 JK. 3. Thermal insulation. Even though the system is thermally

insulated, small heat leakage stiIl exists. The value of the thermal resistance can be calculated from steady state measurements when applying a low input power.

In steady state (in our case after about five hrs), the thermal resistance Oilo can be calculated from Th, T, and T,,, by:

(24) R. = a:Ok(T, +Th +2.273)2(Tc -Tmom) 190 (T, -Th)2(2Rm+u~0, (Tc+Th+2-273))

202 1

21 - 3

Fig. 6. PSPICE model ufTEM TB-127-1.2-1.4.

4. The thermal resistance of a contact between the TEM and the plate can be estimated from datasheets of the thermal interface material (sillcon grease in our case).

The scheme for PSPICE simulation is shown on Fig. 6 and Fig. 7.

4

3

2

f

0 a 20 40 60 80

(4

Fig, 7. PSPICE model for simulating the experiment. R,, is a thermal resistance of the thermal insulation, C,I - lhermal capacity of the aluminum plates. - thermal rehistance of the thema1 contact between TEM and the plate.

Fig. 8 shows the experimental results together with computer simulation. The good agreement clearly shows that the model is valid not only for steady state conditions (DC) but also for simulating dynamic behavior,

40

35

30

25

20

35 U 50 700

@I

35 32

27 30

25 22

20 77

12 0 50 $00 75

0 50 TOO

Fig. 6, The behavior of the TEM under sequence of powering and loading on electric pori. Experimental data shown in gray line, simulation results are black line. (aj voltage on the electrjc reminal. (b), (c), and (d) - Temperatures of absorbing (TJ and emitting [T.) sides o f TEM, for open- circuited electric terminal, loaded by 2 fl resistor, and loaded with 4.5 R resistor respectiveIy.

2022

21 - 4

VI. CONCLUSIONS

In this study, equivalent circuits are used to describe TEM systems in which several energy types exist, when all non- electrical processes are emdated by electrical analogies, and their interconnections are represented by transformers or dependent sources. The model on Fig 2 is a two-port electrical system when one of the ports is an equivalent circuit of the thermal part. Consequently, the model can be implemented as a block in any electrical scheme.

The study shows how the manufacturer’s data for Thermoelectric Cooler (TEC) as well as for Thermoelectric Generator (TEG) can be used to extract the parameters of the proposed model.

The model could be helpful for analyzing the drive requirements of TECs and loading effects of TEGs. Another. important application of proposed. model is when the performance of the TEM needs to be analyzed under specific conditions such as heat leakage, non-ideal thermai insulation etc. Using the model one can analyze not only existing modules, but also specify an optimal module for a specific problem.

The present model is compatible with PSPICE or other electric circuit simulators for DC, AC, end TRANSIENT simulation types and will thus be an excellent tool for solving

problems of temperature control. Several examples of successful utilization of the model are

presented. The paper is based on data of many different manufacturers that were used to reproduce accurately the performance of commercial TEMs.

An important feature of the model is its ability to generate small signal transfer functions that can be used to design feedback network in temperature control applications.

REFERENCES

A. F. loffe, Semiconductors thermoelments and thermoelectric cooling, London: Infoserch Iimited, 1957. S. Noll, Pedier Device hfOrYRUtiOR Direcrory, online. Available: http://www.geltier-info.com. S. I,. Soo, Direci energy conversion, London: Prentrce-Hall, 1968. J. P. Holman, Heoi rrunsfer, 7th ed., London: McGraw-Hill, 1992, pp25-

J. Chavez, J. Ortega, J. Salazar, A. Turo, and J. Garcia, “Spice model of thermoelectric elements including lhermal erects,” Proceedings o j the Instrumenlation and Meearurement Technology Cofiference, 2000, pp. 1019 - 1023. Kryotherm Co., products, online. Available: http://www.kryotherrn.ru Beijing Huimao Cooling Equipment Co., products online, Available: htrp://mvw.huimao.com Marlow Industries, products, online. Available: http://www.marlow.com Hi-2 Technology, products, online. Available: http://www.hi-z,com

56, and 137-143.

2023

21 - 5

22

4.2.2 PSPICE-Compatible Equivalent Circuit of Thermoelectric Coolers [76]

[77]

,

This paper was presented at the IEEEI’04 Conference in Tel-Aviv, Israel. It deals

with thermoelectric coolers (chillers). The technique for extracting the model

parameters proposed in [73] was improved so as to be compatible with the data sheets

of all known manufacturers of the thermoelectric coolers. The improved version of the

present paper was presented on IEEE Power Electronics Specialists Conference,

PESC'05, that took place in Recife, Brazil, 2005. This latest version of the paper is

presented here.

The study shows how a manufacturer’s data for thermoelectric coolers can be used

to extract the parameters of the model proposed in this study. The model is helpful for

analyzing the drive requirements of TECs. Another important application of the

proposed model is to analyze the performance of a TEC under specific conditions, such

as thermal leakage, non-ideal thermal insulation, etc. Using the model one can analyze

not only existing modules, but also specify an optimal TEC for a specific problem. The

present model is compatible with PSPICE or other electric circuit simulators for DC,

AC, and TRANSIENT simulation types and thus should prove to be an excellent tool

for solving problems relating to temperature control. Several examples of successful

utilization of the model are presented. This paper is based on data from many different

manufacturers that were used to reproduce accurately the performance of commercial

TECs. An important feature of the model is its ability to generate small-signal transfer

functions that can be used to design a feedback network in temperature control

applications.

PSPICE-Compatible Equivalent Circuit of Thermoelectric Coolers

Simon Lineykin and Sam Ben-Yaakov*

Power Electronics Laboratory Department of Electrical and Computer Engineering

Ben-Gurion University of the Negev P. O. Box 653,Beer-Sheva 84105, ISRAEL, Phone: +972-8-646-1561,Fax: +972-8-647-2949

Email: sby@ee.bgu.ac.il, Website: www.ee.bgu.ac.il/~pel

Abstract-the objective of this work was to develop a PSPICE-compatible equivalent circuit of a thermoelectric cooler (TEC). Equivalent circuits are convenient tools for power electronics engineers since they help in presenting a problem in electronic circuit terms and can assist in the design of power stages and the control circuitry and algorithms. A methodology is developed for extracting the parameters of the proposed model from manufacturers’ data of TECs. The present model is compatible with PSPICE or other electronic circuit simulators. An important feature of the model is its ability to generate small-signal transfer functions that can be used to design feedback networks for temperature control applications. Several examples of successful utilization of the model are presented. Data of many different manufacturers were examined and the model parameters were extracted. In all cases, the model was found to reproduce accurately the performance of commercial TECs. The accuracy of the model was also verified by experiments.

I. INTRODUCTION A thermoelectric cooler (chiller) (TEC) is a solid-state

energy converter (Fig. 1). It normally consists of an array of pellets from dissimilar semiconductor material (p and n type), which are thermally joined in parallel and electrically in series. The thermoelectric module (TEM) can be used for cooling, heating, and energy generation [1] - [3]. The objective of this work was to develop a SPICE-compatible equivalent circuit of a TEC. An equivalent circuit is a convenient tool for electronic engineers. It helps in presenting the problem in electronic circuit terms and understanding its functionality, and it facilitates the solution of cooling or power-generation problems without the need for expertise in thermal engineering. A SPICE-compatible model is especially useful when dealing with a non-linear devices such as a TEC and incorporating it in a closed-loop system. In such cases a SPICE-compatible model can help in obtaining the transfer functions needed to design feedback circuitry.

II. PRINCIPLES OF OPERATION Five energy-conversion processes take place in a

thermoelectric module: conductive heat transfer, Joule heating, Peltier cooling/heating, Seebeck power generation and the Thompson phenomenon. All these processes account for the interrelations between thermal and electrical energies. Following the first law of thermodynamics, one

can express the energy equilibrium at both sides of the thermoelectric module that are defined as the absorbing (a) and emitting (e) junctions. For the absorbing side:

2RIITTq m

2

amm

a −α+ΘΔ

= (1)

For the emitting side:

2RIITTq m

2

emm

e +α+ΘΔ

= (2)

Nm α=α (3)

RNR m = (4)

N/m Θ=Θ (5)

where qa is heat absorbed at the a-side, qe heat emitted at the e-side, N number of couples, Ta and Te temperatures of (a-) and (e-) sides in K, Θ thermal resistance of the couple in the direction of the heat flow, R electrical resistance of the couple, α Seebeck coefficient, and ΔT=(Te-Ta).

It is conventional to leave out the effect of the Thompson phenomena because it is negligibly small.

The electrical part of the module is described as an electrical resistance Rm and an electrical potential difference V:

TTTV mamem Δα=α−α= (6) * Corresponding author

Fig. 1 Single-stage thermoelectric module construction

6080-7803-9033-4/05/$20.00 ©2005 IEEE.

22 - 1

III. EQUIVALENT SCHEME It is common practice in one-dimensional heat transfer

problems to apply an equivalent electrical circuit scheme [4]. This approach was adopted in this study to describe the TEM system in which several energy types exist. All non-electrical processes are described in terms of electrical analogies, and transformers (or dependent sources) represent their interconnections. In this way, the equivalent circuit of the thermo-electrical system of a TEC can be built as a pure electrical circuit.

Table 1 shows the physical parameters of the thermal system and corresponding parameters of the equivalent electric circuit.

This system of analogies permits the equivalent circuit of the thermo-electrical system of the TEC to be constructed as an electrical network. Fig. 2 shows the equivalent circuit of the TEC using the analogies from Table 1, which are based on equations (1), (2), and (6) for a- and e-junctions [5].

The scheme consists of the Cauer (C-Θm-C) network, which is normally used in equivalent circuits to represent conductive heat transfer in solids [6], supplemented by current sources. The sources shows Joule heating of the TEC, qj, Peltier cooling on the heat-absorbing side of the TEC, qpa, and Peltier heating on the heat-emitting side of the TEC, qpe. The electrical part consists of the voltage source Vs and electrical resistance Rm. All capacitors have the initial charge IC = Tamb.

A modified equivalent circuit topology of the model, based on the circuit of Fig. 2, is shown in Fig. 3 with two dependent sources instead of three and lumped parameters instead of distributed ones. This new representation is clearly closer to the intuitive understanding of active cooling.

IV. CALCULATION OF THE PARAMETERS OF THE MODEL FROM MANUFACTURERS’ DATASHEETS

Manufacturers of TECs (Kryotherm [7], Hui Mao [8], Marlow [9], and others) use the following parameters to specify their product: ΔTmax is the largest temperature differential (K) that can be obtained between the hot and cold ceramic plates of a TEC for a given level of Th (temperature of the hot side), Imax is the input current (A) which will produce the maximum possible ΔT across a TEM, Vmax is the dc voltage (V) that will deliver the

maximum possible ΔT at the supplied Imax, Qmax is the maximum amount of heat (W) that can be absorbed at the TEC’s cold plate at Imax and at a ΔT equal to 0. Note that Qmax is not the maximum possible amount of heat that can be handled by the TEC, rather the heat flow corresponding to the current Imax. Qopt is the maximum amount of heat that can be absorbed at the TEC’s cold plate for a ΔT equal to 0. Qopt is larger than Qmax. Some manufacturers apply the notation Qmax instead of Qopt, so one needs to carefully read the description given in the datasheets.

Using the relations (1), (2), and (6), the characteristic parameters of the TEC can be derived:

( )Z

ZT211TT h

hmax+−

+=Δ (7)

mm

hmax

1ZT21I

Θα−+

= (8)

TABLE I Thermal to electrical analogy

Thermal quantities Units Analogous Electrical Quantities Units

Heat, q W Current, I A

Temperature, T K Voltage, V V

Thermal Resistance, Θ K/W Resistance, R Ω

Heat capacity, C J/K Capacity, C F Absolute zero temperature 0 K Ground 0 V

Fig. 2. The equivalent circuit of the TEC. The scheme is based on a Cauer-type network for describing heat transfer in a solid with internal heat sources (qJ). qpa and qpe (Peltier cooling and heating). The Vs voltage source describes Seebeck power generation. IC is the initial temperature of the device, Tamb.

Fig. 3. Modified proposed equivalent circuit of a thermoelectric module.

609

22 - 2

hmmax TV α= (9)

( )

Z21ZT21ZT21

Qm

2hh

max Θ−++

= (10)

Rm

ThI mopt

α= (11)

Rm2ThQ

22m

optα

= (12)

where Z is a figure of merit of the TEC, Z=αmθm/Rm. Applying (7) - (12), one can now use the set of data: Th,

ΔT, Vmax, Imax for calculating the parameters of the proposed model:

( ) [ ]ΩΔ−

=h

maxh

max

maxm T

TTIVR (13)

( ) ⎥⎦⎤

⎢⎣⎡

Δ−Δ

=ΘWK

TTT2

VIT

maxh

h

maxmax

maxm (14)

⎥⎦⎤

⎢⎣⎡=α

KV

TV

h

maxm (15)

V. STEADY-STATE ANALYSIS The TB-127-1.4-1.2 is one of the thermoelectric cooling

modules available from Kryotherm [7]. From the manufacturer’s datasheets: Under the Th=300K condition ΔTmax= 70 K, Imax=7.6 A, Vmax=15.9 V, and Qmax=75 W. Applying (13) – (15), one can calculate the model parameters: αm=0.053 V/K, Rm=1.6 Ω, Θm=1.5 K/W. Fig. 4

shows the result of the application of the dc-sweep simulation that reconstructs the performance plot of the TEM TB-127-1.4-1.2. The dashed line is the simulation result. The original performance plot was copied from the software tool placed by Kryotherm on the Internet. The results obtained by the two methods are in close agreement.

VI. COMPARISON OF EXPERIMENTAL TO MODEL TIME-DOMAIN RESPONSE

The laboratory measurements of a physical TEC were compared with computer simulations that apply to the proposed model. The experiment was carried out using the TEC TB-127-1.4-1.2 (Kryotherm) TEC with dimensions of 40mmx40mm and ceramic plates 1mm thick on both sides. The module was thermally insulated. In the first phase of the experiment, a constant voltage was applied for several seconds to the electrical port. As a result, the temperature difference between the absorbing and emitting sides of the TEC was established. Then the time domain relaxation of the temperature difference to zero was observed by measuring the voltage of the open electrical port.

The capacitors C of the equivalent scheme of Fig. 3, determine the dynamic behavior of the model. The capacitors represent the lumped heat capacitance of the alumina ceramic plates and pellets of the TEC. The lumped heat capacitance of the TEC is CTEC = 0.35 J/K and lamped heat capacitance of each one of the ceramic plates is Cc = 5.33 J/K. Data on thermal volumetric capacity are taken from [11], and the volumes of the ceramic plates and pellets from datasheets [7]. Thus C = Cc+CTEC/2 = 5.68 F.

Fig. 5 compares the results of the experimental TEC time response and the PSPICE transient simulation of the equivalent circuit. The figure shows a good fit of the simulation to the experimental data.

Fig. 4. Performance plot of TEM: TB-127-1.4-1.2. Temperature of the a-side (cold) vs. current under conditions: Th=300, cooling power 20W. The dashed line is the simulation result obtained by the proposed model; the solid line is the performance plot published by the manufacturer.

Fig. 5. Time response of the TEC. The gray thick line shows the result of the experiment on the physical module. The black dashed line is the result of the computer simulation using PSPICE (black line).

610

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VII. SMALL-SIGNAL TRANSFER FUNCTION GENERATIONY USING EQUIVALENT CIRCUIT OF THE TEC

For a better controller design, one has to know the system frequency response (transfer function). An analytical method for calculating the poles and zeros of the transfer function of the TEC-based system is given in [12]. However, since the TEC is a non-linear system, the transfer function will be different for each operating (bias) point. The analytical derivation of the transfer function for all conditions of operation is thus a cumbersome process. The proposed model provides a simple way to get the transfer functions of the system directly from the large signal model by just carrying out a small-signal (ac) simulation of the cooling system by an electronic circuit simulator such as PSPICE.

Fig. 6(a) shows the experimental system of a TEC with a thermal load (two massive aluminum plates). The system is thermally insulated. There are two thermocouples inserted into the thermal load for temperature measurement. The setup permits the measurement of a response of the system to a sine wave voltage input. By making the measurements at different frequencies, one can get the frequency response of the system (the transfer function). Fig. 6(b) shows the equivalent circuit for the experiment simulation. The results of the simulations using the proposed equivalent circuit model are shown on Fig. 7. As one can see, the results of the small-signal (ac) simulation are in good agreement with those of the transient cycle-by-cycle simulation as well as with experimental results.

VIII. CONCLUSIONS The study shows how the manufacturer’s data for the

thermoelectric cooler can be used to extract the parameters of the proposed model. The model could be helpful for analyzing the drive requirements of the TEC. Another important application of the proposed model is to analyze the performance of the TEC under specific conditions such as thermal leakage, non-ideal thermal insulation, etc. Using the model one can analyze not only existing modules, but also specify an optimal TEC for a specific problem. The present model is compatible with PSPICE or other electric circuit simulators for dc, ac, and transient simulation types and will thus be an excellent tool for solving problems of temperature control.

Several examples of successful utilization of the model are presented. The paper is based on data given by many different manufacturers that were used to reproduce accurately the performance of commercial TEMs. An important feature of the model is its ability to generate small-signal transfer functions that can be used to design a feedback network in temperature control applications.

(a)

(b)

Fig. 6. Measurement of the small-signal transfer function of the system: TEC sandwiched between two massive aluminum plates with built-in thermocouples. (a) Experimental setup. (b) PSPICE/OrCAD simulation scheme. The model of the TEC is the one shown in Fig. 3 with parameters αm=0.053 V/K, Rm=1.6 Ω, Θm=1.5 K/W calculated above, 1: sine voltage source, 2: thermal interface material (TIM), 3: aluminum plates.

Fig. 7. Transfer function of the system of Fig. 6. Input variable is the input voltage and output is the temperature difference between the aluminum plates on both sides of the TEC. Dashed line is the result of cycle-by-cycle transient simulation of the system with sine input voltage (amplitude 1.3 V); Solid line is the small-signal (ac) simulation result, and points are data of the experimental measurements.

611

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REFERENCES [1] A. F. Ioffe, Semiconductors thermoelments and

thermoelectric cooling, London: Infoserch limited, 1957. [2] S. Noll, Peltier Device Information Directory, online.

Available: www.peltier-info.com [3] S. L. Soo, Direct energy conversion, London: Prentice-Hall,

1968. [4] J. P. Holman, Heat transfer, 7th ed., London: McGraw-Hill,

1992, pp25-56, and 137-143. [5] J. Chavez, J. Ortega, J. Salazar, A. Turo, and J. Garcia,

“Spice model of thermoelectric elements including thermal effects,” Proceedings of the Instrumentation and Measurement Technology Conference, 2000, pp. 1019 - 1023.

[6] P. Bagnoli, C. Casarosa, M. Ciampi, and E. Dallago, “Thermal resistance analysis by induced transient (TRIAT) method for power electronic devices thermal characterization – part I: fundamentals and theory,” IEEE transactions on power electronics, vol. 13, n. 6, pp 1208-1218, November 1998.

[7] Kryotherm Co., products, online. Available: www.kryotherm.ru.

[8] Beijing Huimao Cooling Equipment Co., products, online. Available: www.huimao.com.

[9] Marlow Industries, products, online. Available: www.marlow.com.

[10] Hi-Z Technology, products, online. Available: www.hi-z.com.

[11] E. De Baetselier, W. Goedertier, and G. De Mey, “A survey of the thermal stability of an active heat sink,” Microelectronic Reliability, v. 37, n. 12, pp. 1805-1812, 1997.

[12] B. Huang and C. Duang, “System dynamic model and temperature control of a thermoelectric cooler,” International Journal of Refrigeration, n. 23, pp. 197-207, 2000.

Shmuel (Sam) Ben-Yaakov received the BSc degree in Electrical Engineering from the Technion, Haifa Israel, in 1961 and the MS and PhD degrees in Engineering from the UCLA, in 1967 and 1970 respectively.

He is presently a Professor at the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev,

Beer-Sheva, Israel, and heads the Power Electronics Group there. His current research interests include power electronics, circuits and systems, electronic instrumentation and engineering education. Professor Ben-Yaakov also serves as a consultant to commercial companies in the areas of analog and power electronics.

Simon Lineykin received the BSc degree in Mechanical engineering and MS degree in Electrical Engineering from Ben-Gurion University of the Negev, Israel. He is currently working toward his PhD degree in electrical engineering at Ben-Gurion University of the Negev. His research interests are modeling and

emulation of the physical processes and active cooling systems using Peltier effect.

612

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23

4.2.3 Analysis of Thermoelectric Coolers by a SPICE-compatible Equivalent

Circuit Model. [71]

This paper was published in the IEEE Power Electronic Letters journal. It is a

continuation of a previous one, "PSPICE-Compatible Equivalent Circuit of

Thermoelectric Coolers" [77]. The experimental part was improved by measuring the

sine response of the cooling system. As shown in the paper, the results of simulation are

very close to the measured ones. In addition to the applications described earlier in [73]

and [77], another important application of the model was proposed, which analyzes the

performance of the TEC under specific conditions such as thermal leakage, non-ideal

thermal insulation, etc. Using the model one can analyze not only existing modules, but

also specify an optimal TEC for a specific problem. The present model is compatible

with PSPICE or other electric circuit simulators for DC, AC, and TRANSIENT

simulation types and will thus serve as an excellent tool for solving temperature control

problems.

Several examples of successful utilization of the model are presented. The paper is

based on data from many different manufacturers that were the basis for accurately

reproducing the performance of commercial TEMs.

An important feature of the model is its ability to generate small signal transfer

functions that can be used to design a feedback network in temperature control

applications.

IEEE POWER ELECTRONICS LETTERS, VOL. 3, NO. 2, JUNE 2005 63

Analysis of Thermoelectric Coolers by aSpice-Compatible Equivalent-Circuit Model

Simon Lineykin and Sam Ben-Yaakov, Member, IEEE

Abstract—The objective of this work was to develop aPSPICE-compatible equivalent circuit of a thermoelectriccooler (TEC). Equivalent circuits are convenient tools for powerelectronics engineers since they help in presenting a problem inelectronic circuit terms and can assist in the design of power stagesand the control circuitry and algorithms. A methodology is devel-oped for extracting the parameters of the proposed model frommanufacturers’ data of TECs. The present model is compatiblewith PSPICE or other electronic circuit simulators. An importantfeature of the model is its ability to generate small-signal transferfunctions that can be used to design feedback networks for tem-perature-control applications. Several examples of successfulutilization of the model are presented. Data of many differentmanufacturers were examined and the model parameters were ex-tracted. In all cases, the model was found to reproduce accuratelythe performance of commercial TECs. The accuracy of the modelwas also verified by experiments.

Index Terms—Active cooling, equivalent circuits, modeling, sim-ulation, temperature control, thermoelectric devices, thermoelec-tricity.

I. INTRODUCTION

ATHERMOELECTRIC cooler (chiller) (TEC) is a solid-state energy converter (Fig. 1). It normally consists of an

array of pellets from dissimilar semiconductor material (p andn type), which are thermally joined in parallel and electricallyin series. The TEM can be used for cooling, heating, and energygeneration [1]–[3]. The objective of this work was to developa SPICE-compatible equivalent circuit of a TEM. An equiva-lent circuit is a convenient tool for electronic engineers. It helpsin presenting the problem in electronic circuit terms and under-standing its functionality and facilitates the solution of coolingor power-generation problems without the need for expertisein thermal engineering. A SPICE-compatible model is espe-cially useful when dealing with a nonlinear device such as aTEC and incorporating it in a closed-loop system. In such casesa SPICE-compatible model can help in obtaining the transferfunctions needed to design feedback circuitry.

II. PRINCIPLES OF OPERATION

Five energy-conversion processes take place in a thermo-electric module: conductive heat transfer, Joule heating, Peltiercooling/heating, Seebeck power generation and the Thompson

Manuscript received November 7, 2004; revised November 19, 2004. Rec-ommended by Associate Editor D. J. Perreault.

The authors are with the Power Electronics Laboratory, Department of Elec-trical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (e-mail: sby@ee.bgu.ac.il).

Digital Object Identifier 10.1109/LPEL.2005.846822

Fig. 1. Single-stage module construction.

phenomenon. All these processes account for the interrelationsbetween thermal and electrical energies. Following the first lawof thermodynamics, one can express the energy equilibrium atboth sides of the thermoelectric module that are defined as theabsorbing (a) and emitting (e) junctions. For the absorbing side

(1)

For the emitting side

(2)

(3)

(4)

(5)

where is heat absorbed at the a-side, heat emitted at thee-side, N is the number of couples, and are the temper-atures of the (a-) and (e-) sides in K, is the thermal resis-tance of the couple in the direction of the heat flow, R is theelectrical resistance of the couple, Seebeck coefficient, and

. It is conventional to leave out the effect ofthe Thompson phenomena because it is negligibly small. Theelectrical part of the module is described as an electrical resis-tance and an electrical potential difference V, as follows:

(6)

III. EQUIVALENT SCHEME

It is common practice in one-dimensional heat transfer prob-lems to apply an equivalent electrical circuit scheme [4]. Thisapproach was adopted in this study to describe the TEM systemin which several energy types exist. All nonelectrical processesare described in terms of electrical analogies, and transformers(or dependent sources) represent their interconnections. In thisway, the equivalent circuit of the thermo-electrical system of aTEC can be built as a pure electrical circuit.

1540-7985/$20.00 © 2005 IEEE

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64 IEEE POWER ELECTRONICS LETTERS, VOL. 3, NO. 2, JUNE 2005

TABLE ITHERMAL TO ELECTRICAL ANALOGY

Fig. 2. Equivalent circuit of the TEC. The scheme is based on a Cauer-typenetwork for describing heat transfer in a solid with internal heat sources (q ).q and q (Peltier cooling and heating). The V voltage source describesSeebeck power generation. IC is the initial temperature of the device, T .

Table I shows the physical parameters of the thermal systemand corresponding parameters of the equivalent electric circuit.

This system of analogies permits the equivalent circuit of thethermo-electrical system of the TEC to be constructed as anelectrical network. Fig. 2 shows the equivalent circuit of theTEC using the analogies from Table I, which are based on (1),(2), and (6) for a- and e-junctions [5].

The scheme consists of the Cauer network,which is normally used in equivalent circuits to representconductive heat transfer in solids [6], supplemented by currentsources. The sources shows Joule heating of the TEC, , Peltiercooling on the heat-absorbing side of the TEC, , and Peltierheating on the heat-emitting side of the TEC, . The electricalpart consists of the voltage source and electrical resistance

. All capacitors have the initial charge .A modified equivalent circuit topology of the model, based

on the circuit of Fig. 2, is shown in Fig. 3 with two dependentsources instead of three and lumped parameters instead of dis-tributed ones. This new representation is clearly closer to theintuitive understanding of active cooling.

IV. CALCULATION OF THE PARAMETERS OF THE MODEL

FROM MANUFACTURERS’ DATASHEETS

Manufacturers of TECs (Kryotherm [7], Hui Mao [8],Marlow [9], and others) use the following parameters to specifytheir product: is the largest temperature differential(K) that can be obtained between the hot and cold ceramicplates of a TEC for a given level of (temperature of thehot side), is the input current (A) which will produce themaximum possible across a TEM, is the dc voltage(V) that will deliver the maximum possible at the supplied

is the maximum amount of heat (W) that can be

Fig. 3. Modified proposed equivalent circuit of a thermoelectric module.

absorbed at the TEC’s cold plate at and at a equalto 0. Note that is not the maximum possible amount ofheat that can be handled by the TEC, rather the heat flow cor-responding to the current is the maximum amountof heat that can be absorbed at the TEC’s cold plate for aequal to 0. is larger than . Some manufacturers applythe notation instead of , so one needs to carefullyread the description given in the datasheets.

Using the relations (1), (2), and (6), the characteristic param-eters of the TEC can be derived as follows:

(7)

(8)

(9)

(10)

(11)

(12)

where is a figure of merit of the TEC,Applying (7)–(12), one can now use the set of data

for calculating the parameters of theproposed model

(13)

(14)

(15)

23 - 2

LINEYKIN AND BEN-YAAKOV: ANALYSIS OF THERMOELECTRIC COOLERS 65

Fig. 4. Performance plot of TEM: TB-127-1.4-1.2. Temperature of the a-side (cold) versus current under conditions:T = 300, cooling power 20 W. The dashedline is the simulation result obtained by the proposed model; the solid line is the performance plot published by the manufacturer.

V. STEADY-STATE ANALYSIS

The TB-127-1.4-1.2 is one of the thermoelectric coolingmodules available from Kryotherm [7]. From the manu-facturer’s datasheets: Under the K condition

K, A, V, andW. Applying (13)–(15), one can calculate the

model parameters: V/K,K/W. Fig. 4 shows the result of the application of the dc-sweepsimulation that reconstructs the performance plot of the TEMTB-127-1.4-1.2. The dashed line is the simulation result. Theoriginal performance plot was copied from the software toolplaced by Kryotherm on the Internet. The results obtained bythe two methods are in close agreement.

VI. COMPARISON OF EXPERIMENTAL TO MODEL

TIME-DOMAIN RESPONSE

The laboratory measurements of a physical TEC were com-pared with computer simulations that apply to the proposedmodel. The experiment was carried out using the TEC TB-127-1.4-1.2 (Kryotherm) TEC with dimensions of 40 mm 40 mmand ceramic plates 1 mm thick on both sides. The module wasthermally insulated. In the first phase of the experiment, a con-stant voltage was applied for several seconds to the electricalport. As a result, the temperature difference between the ab-sorbing and emitting sides of the TEC was established. Thenthe time-domain relaxation of the temperature difference to zerowas observed by measuring the voltage of the open electricalport.

The capacitors C of the equivalent scheme of Fig. 3 deter-mine the dynamic behavior of the model. The capacitors repre-sent the lumped heat capacitance of the alumina ceramic platesand pellets of the TEC. The lumped heat capacitance of theTEC is J/K and lamped heat capacitance of

Fig. 5. Time response of the TEC. The gray thick line shows the result of theexperiment on the physical module. The black dashed line is the result of thecomputer simulation using PSPICE (black line).

each one of the ceramic plates is J/K. Data onthermal volumetric capacity are taken from [11], and the vol-umes of the ceramic plates and pellets from datasheets [7]. Thus

F.Fig. 5 compares the results of the experimental TEC time re-

sponse and the PSPICE transient simulation of the equivalentcircuit. The figure shows a good fit of the simulation to the ex-perimental data.

VII. SMALL-SIGNAL TRANSFER FUNCTION GENERATION

USING EQUIVALENT CIRCUIT OF THE TEC

For a better controller design, one has to know the system fre-quency response (transfer function). An analytical method forcalculating the poles and zeros of the transfer function of theTEC-based system is given in [12]. However, since the TEC is anonlinear system, the transfer function will be different for eachoperating (bias) point. The analytical derivation of the transferfunction for all conditions of operation is thus a cumbersomeprocess. The proposed model provides a simple way to get thetransfer functions of the system directly from the large signalmodel by just carrying out a small-signal (ac) simulation ofthe cooling system by an electronic circuit simulator such asPSPICE.

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66 IEEE POWER ELECTRONICS LETTERS, VOL. 3, NO. 2, JUNE 2005

Fig. 6. Measurement of the small-signal transfer function of the system: TECsandwiched between two massive aluminum plates with built-in thermocouples.(a) Experimental setup. (b) PSPICE/OrCAD simulation scheme. The model ofthe TEC is the one shown in Fig. 3 with parameters = 0:053 V/K, R =

1:6 ;m = 1:5 K/W calculated above, 1: sine voltage source, 2: thermalinterface material (TIM), 3: aluminum plates.

Fig. 7. Transfer function of the system of Fig. 6. Input variable is the inputvoltage and output is the temperature difference between the aluminum plateson both sides of the TEC. Dashed line is the result of cycle-by-cycle transientsimulation of the system with sine input voltage (amplitude 1.3 V), solid linethe small-signal (ac) simulation result, and points are data of the experimentalmeasurements.

Fig. 6(a) shows the experimental system of a TEC with athermal load (two massive aluminum plates). The system is ther-mally insulated. There are two thermocouples inserted into thethermal load for temperature measurement. The setup permitsthe measurement of a response of the system to a sine wave

voltage input. By making the measurements at different fre-quencies, one can get the frequency response of the system (thetransfer function). Fig. 6(b) shows the equivalent circuit for theexperiment simulation. The results of the simulations using theproposed equivalent circuit model are shown on Fig. 7. As onecan see, the results of the small-signal (ac) simulation are ingood agreement with those of the transient cycle-by-cycle sim-ulation as well as with experimental results.

VIII. CONCLUSION

The study shows how the manufacturer’s data for the thermo-electric cooler can be used to extract the parameters of the pro-posed model. The model could be helpful for analyzing the driverequirements of the TEC. Another important application of theproposed model is to analyze the performance of the TEC underspecific conditions such as thermal leakage, nonideal thermal in-sulation, etc. Using the model, one can analyze not only existingmodules, but also specify an optimal TEC for a specific problem.The present model is compatible with PSPICE or other electriccircuit simulators for dc, ac, and transient simulation types andwill thus be an excellent tool for solving problems of tempera-ture control.

Several examples of successful utilization of the model arepresented. The paper is based on data given by many differentmanufacturers that were used to reproduce accurately the perfor-mance of commercial TEMs. An important feature of the modelis its ability to generate small-signal transfer functions that canbe used to design a feedback network in temperature-control ap-plications.

REFERENCES

[1] A. F. Ioffe, Semiconductors Thermoelments and ThermoelectricCooling. London, U.K.: Infoserch Ltd., 1957.

[2] S. Noll, Peltier Device Information Directory [Online]. Available:http://www.peltier-info.com

[3] S. L. Soo, Direct Energy Conversion. London, U.K.: Prentice-Hall,1968.

[4] J. P. Holman, Heat Transfer, 7th ed. London, U.K.: McGraw-Hill,1992, pp. 25–56, 137–143.

[5] J. Chavez, J. Ortega, J. Salazar, A. Turo, and J. Garcia, “Spice model ofthermoelectric elements including thermal effects,” in Proc. Instrumen-tation and Measurement Technology Conf., 2000, pp. 1019–1023.

[6] P. Bagnoli, C. Casarosa, M. Ciampi, and E. Dallago, “Thermal resistanceanalysis by induced transient (TRIAT) method for power electronic de-vices thermal characterization—Part I: Fundamentals and theory,” IEEETrans. Power Electron., vol. 13, no. 6, pp. 1208–1218, Nov. 1998.

[7] Kryotherm Co., Products [Online]. Available: http://www.kryotherm.ru[8] Beijing Huimao Cooling Equipment Co., Products [Online]. Available:

http://www.huimao.com[9] Marlow Industries, Products [Online]. Available: http://www.marlow.

com[10] Hi-Z Technology, Products [Online]. Available: http://www.hi-z.com[11] E. De Baetselier, W. Goedertier, and G. De Mey, “A survey of the

thermal stability of an active heat sink,” Microelect. Reliab., vol. 37, no.12, pp. 1805–1812, 1997.

[12] B. Huang and C. Duang, “System dynamic model and temperaturecontrol of a thermoelectric cooler,” Int. J. Refrigeration, no. 23, pp.197–207, 2000.

23 - 4

24

4.2.4 A Simple and Intuitive Graphical Approach to the Design of

Thermoelectric Cooling Systems [78], [79].

This paper was presented on international conference IEEE PESC'06, in Korea

[78]. Extended and revised version of the paper was accepted for publication in

International Journal of Refrigeration [79].

In various applications, thermoelectric active cooling systems can help maintain

electronic devices at a desired temperature condition better than passive coolers.

Thermoelectric Coolers (TEC) are especially useful when the temperature of a device

needs to be precisely controlled. This study proposes a user-friendly graphical method

for calculating the steady-state operational point of a TEC based active cooling system,

including the heatsink role. The method is simple and intuitive and provides

comprehensive information about the cooling system such as its feasibility, required

heatsink, the TEC current, temperatures of the cold and hot sides, coefficient of

performance (COP) and others. The method could help designers to examine and

choose a thermoelectric module from catalogues to meet a specific cooling problem. To

start using the method, designers need only the experimental TEC data provided by

practically all manufacturers of such devices. The experimental results of this study

verify the high accuracy of the proposed model and graphical approach.

User-friendly and Intuitive Graphical Approach to the Design of Thermoelectric Cooling Systems

Simon Lineykin*, Sam Ben-Yaakov

Department of Electrical and Computer Engineering, Power Electronics Laboratory,

Ben-Gurion University of the Negev P. O. Box 653, Beer-Sheva 84105, ISRAEL, Phone: +972-8-6479811,

Fax: +972-8-647-2949, Email: simonl@ee.bgu.ac.il, Website: www.ee.bgu.ac.il/~pel

Abstract

This study proposes a user-friendly graphical method for calculating the steady-

state operational point of a thermoelectric cooler (TEC) based active cooling system,

including the heatsink role. The method is simple and intuitive and provides

comprehensive information about the cooling system such as its feasibility, required

heatsink, the TEC current, temperatures of the cold and hot sides, coefficient of

performance (COP) and others. The method could help designers to examine and

choose a thermoelectric module from catalogues to meet a specific cooling problem. To

start using the method, designers need only the experimental TEC data provided by

practically all manufacturers of such devices. The experimental results of this study

verify the high accuracy of the proposed model and graphical approach.

Keywords: Chiller; Thermoelectricity; Modeling; Thermoelectric cooling;

Thermoelectric cooler design.

________

* Corresponding author. Tel: +972-8-647-9811; fax: +972-8-647-2949.

E-mail addresses: simonl@ee.bgu.ac.il (S. Lineykin)

sby@ee.bgu.ac.il (S. Ben-Yaakov)

24 - 1

Nomenclature

a, b Segments of the y-axis of the plot giving

information on COP (K)

COP coefficient of performance, COP =

cooling capability/electric power

I electrical current (A)

Imax data sheet parameter. The current that

provides a temperature difference of

ΔTmax under a specific Th and heat flux

qc = 0 (A)

N number of thermoelectric couples in the

TEC

qc amount of heat dissipated by thermal

load = amount of heat, pumped out TEC

(W)

qcr required qc in a specific application (W)

qh heat dissipated by heatsink (W)

Qmax amount of heat that can be pumped by

the TEC for I = Imax and Th=Tc (W)

Qopt maximum possible amount of heat that

can be pumped by the TEC for = Th=Tc,

Qopt>Qmax (W)

R electrical resistance of the couple (Ω)

Rm electrical resistance of the TEC (Ω)

Tamb temperature of the ambient air (K)

Tc temperature of the cooling surface =

temperature of the cold side of the TEC

(K)

Tca =Tc-Tamb (K)

Tcar required Tca in specific application (K)

TEC thermoelectric cooler (Chiller)

Th temperature of the hot side of the TEC

(K)

Tha =Th-Tamb (K)

Umax data sheet parameter. The voltage drop

across the TECs’ terminals,

corresponding to current Imax and the

temperature difference ΔTmax (V)

V voltage (V)

x, y labels of the coordinate axes, argument

and function correspondingly.

α Seebeck coefficient (V K-1)

αm energy conversion coefficient of the

TEC (V K-1)

ΔTkm(I) apparent temperature source of the TEC

with heatsink as a function of current (K

or oC)

ΔThkm(I) apparent temperature source of the TEC

with heatsink as a function of current for

hot side (K or oC)

ΔTmax data sheet parameter. The largest

temperature differential that can be

obtained between the hot and cold

ceramic plates of a TEC for the given

level of Th and qc = 0 (K or oC)

Θ thermal resistance of the couple (K W-1)

Θk thermal resistance of the heatsink

(K W-1)

Θm thermal resistance of the TEC (K W-1)

Θkm(I) apparent thermal resistance of the TEC

with heatsink as a function of current (K

W-1)

Θkmh(I) apparent thermal resistance of the TEC

with a heatsink as a function of current,

for a hot side (K W-1)

1. Introduction

The demand for small-size active cooling equipment has increased in recent years,

since the traditional passive cooling systems (heatsink and fan) are not powerful enough

to cope with the task of cooling a variety of modern electronic devices. One potential

alternative solution is active cooling [1] - [3]. In various applications, thermoelectric

24 - 2

active cooling systems can help maintain electronic devices at a desired temperature

condition better than passive coolers. Thermoelectric coolers (TEC) are especially

useful when the temperature of a device needs to be precisely controlled. The difference

between passive and active cooling is depicted in Fig. 1. The passive cooling system

(Fig. 1a) includes a heatsink, possibly with a fan, with thermal resistance Θk, (K/W).

The active cooling system (Fig. 1b) uses an energy conversion process to absorb the

thermal energy from the surface needed to be cooled and to pump this energy out.

Active cooling can be realized by applying a thermoelectric (Peltier) cooler (TEC) and a

heatsink of thermal resistance Θk.

The main problem of designing thermoelectric active cooling systems is the fact

that the system depends on a large number of parameters. The parameters involved are

thermal resistance of the heatsink, temperature of the ambient air, the many parameters

of the TEC, and the electrical current through it.

The objective of this work has been to simplify the treatment of TEC based cooling

systems by applying a unified model. The paper proposes a universal graphical method

for the design of TEC-based active cooling systems.

2. The unified TEC model

The behavior of a thermoelectric couple is determined by three fundamental

parameters: Θ – the thermal resistance of the couple in the direction of the heat flow, R

– the electrical resistance of the couple, and α - the Seebeck coefficient. Commercial

TECs include N couples. Assuming that all couples are identical, and that the heat flow

is unidirectional, the lumped parameters of the TEC αm, Θm, and Rm will be:

(1) Nm α=α

(2) RNR m =

(3) N/m Θ=Θ

24 - 3

Practically all TEC manufacturers ( [4], [5], [6] and others) use the following

parameters to specify their products: ΔTmax - is the largest temperature difference (K)

that can be obtained between the hot and cold ceramic plates of a TEC for the given

level of hot-side temperature Th, Imax is the input current (A) which will produce the

maximum possible temperature drop ΔTmax across a TEC, and Umax is the DC voltage

(V) that will deliver the maximum possible temperature drop ΔTmax at the supplied Imax.

As shown earlier [7], [8], the following expressions can be used to calculate the

fundamental TEC’s parameters from the set of data given by manufacturers (Th, ΔTmax ,

Umax, Imax ):

h

maxm T

U=α (4)

( (5) )h

maxh

max

maxm T

TTIUR Δ−

=

( )maxh

h

maxmax

maxm TT

T2UI

TΔ−

Δ=Θ (6)

Following the first law of thermodynamics, one can express the energy equilibrium

at each side of the thermoelectric module, defined as the cold (c) and hot (h) junctions.

Note that all parameters taken, in first order approximation are time invariable and

temperature independent, and that the contribution of the Thomson effect is neglected,

as suggested in [1], [3].

For the absorbing (cold) side one can write:

2RITITq m

2

mcmc −

ΘΔ

−α= (7)

and for the emitting (hot) side:

2RITITq m

2

mhmh +

ΘΔ

−α= (8)

where qc is the heat absorbed at the cold side of the TEC, qh – heat dissipated at the hot

side, Tc and Th – temperatures of cold and hot sides respectively in K, and ΔT=Th-Tc.

24 - 4

The electrical section of the module is described as an electrical resistance Rm in

series with an emf-source:

(9) mm IRTV +Δα=

Finally, the temperature of the hot side of the TEC can be expressed as a function

of the heat transferred from that side to the passive heat removal (heatsink).

khambh qTT Θ+= (10)

Applying the set of equation (7) - (10) one can eliminate the variables qh, Th, and V

and get an expression for Tc, the temperature of the absorbing side of the TEC. The

solution can then be used to develop an equivalent circuit type model of the TEC system

(Fig. 1b) for which the temperature difference between the cooled side (Tc) and the

ambient (Tamb), (Tca), is expressed as:

( ) ( )( ) ( )ITIqTTT kmkmkcambcca Δ+Θ+Θ⋅=−= (11)

where

( ) ( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

Θα−Θα+ΘΘα−

=Θkmmm

m2

kmkm I1I1

I1I (12)

is the apparent thermal resistance of the TEC (Fig. 1b) , and

( )( )

( )kmmm

kmmambmm

2

km2

km I1I1

I1TI2RiRi

ITΘα−Θα+

Θα−Θ⎟⎟⎠

⎞⎜⎜⎝

⎛α++Θ

=Δ (13)

is the apparent temperature source (Fig. 1b). This representation provides an intuitive

understanding of the cooling process of a TEC. It shows that the TEC introduces a

temperature difference ΔTkm(I) and an apparent thermal resistance Θkm(I) both of which

are dependent on the TEC parameters and the current I that drives it. In the event of

negative values of ΔTkm(I), TEC operates as a cooling device.

Although the above expressions seem complex, they include only the fundamental

parameters of the TEC that can be calculated from the manufacturers’ data by (4)-(6).

All expressions can of course be easily evaluated by using commonly available

software.

24 - 5

3. Graphical method

Since a direct application of the analytical solution of (11) for the design of TEC

based systems could be rather involved, we propose here a simple and intuitive

graphical approach. It is based on a parametric representation of current dependent

values of (11) denoted here as the S(I) curve. The x and y values of the S(I) (current

dependent) curve are defined as:

(14) ( ) ( )( )⎩

⎨⎧

Δ=Θ+Θ=

=ITy

IxIS

km

kmk

It should be noted that in this presentation, the x-axis variable is thermal resistance,

whereas the y-axis variable is temperature. A typical S(I) plot is shown in Fig. 2. Notice

that each point of the S(I) curve corresponds to a specific TEC current. The graphical

solution for a given active cooling system is facilitated by drawing on same plot a line,

denoted as the d line that represents the objective thermal problem on hand, defined as:

(15) xqTy crcar ⋅−=

where y- and x- axis are as above. The slope of this line qcr corresponds to the power

needed to be dissipated by the cooled unit, whereas Tcar is the required surface

temperature of the cold side of the TEC (assumed to be equal to the surface of the heat

source), referred to the ambient temperature. The intersection(s) between the S(I) curve

and the d line (if existing) satisfies both equations and is thus the solution of the given

TEC cooling system (11).

By way of illustration, consider an active heat removal case that applies a TB-127-

1.2-1.4 [4] module and a heatsink with Θk = 1 K/W, Tamb = 300 K. The data supplied by

manufacturer for the TEC is: ΔTmax = 70 K, Imax=7.6 A, and Vmax = 15.9 V, under the

condition that the temperature of the hot side Th = 300 K. The module has dimensions

of 40mm by 40mm, and is enclosed between two ceramic plates. It is assumed that the

power dissipated by the surface to be cooled is 10W and that the required surface

temperature is (Tamb-10K = 290 K).

24 - 6

Applying the transformation equations (4)-(6) we can first calculate the

fundamental parameters of the TEC. They are found to be: Θm = 1.51 K W-1, αm = 53

mV K-1, Rm = 1.6 Ω. Then, by using (12) and (13), one can plot the S(I)-curve, see

Table 1. The d-line is constructed according to the thermal data, as shown in Fig. 2. The

intersections between the S(I) curve and the d line, are the solutions of the problem. The

currents of the S(I) at the intersection points are the currents needed to drive the TEC in

order to obtain the desired cooling effect. Between the two solutions that are normally

obtained, one would obviously choose the lower current. The coordinates of the

intersection points between the S and d curves correspond to the apparent thermal

resistance and temperature source of the equivalent circuit (Fig. 1b). In this example, the

apparent thermal resistance (for I = 2.37A) is found to be Θkm = 2 KW-1 and the value of

the temperature source ΔTkm = –30K. It should be noted that since the S(I) curve

includes the information of the heatsink used, a different S(I) curve needs to be drawn

for each thermal resistance of the alternative heatsinks as shown in Fig. 3.

4. Proposed active cooling design procedure

The proposed graphical design procedure will be illustrated by way of an example

in which a commercial TEC (TB-127-1.4-1.2 [4]) is considered. It is assumed that the

surface of the unit to be cooled dissipates qcr = 40 W and that it needs to be maintained

at temperature of less than 50°C at an ambient temperature of Tamb = 300 K (27°C).

That is, Tcar ≈ 20 K. Fig. 3 shows a family S(I)-curves, each corresponding to a specific

value of Θk. The solid d-line, starts at point (0, 20) and drops downward with angle

coefficient of 40W. As evident from the graphical representation of the problem (Fig.

3), there are no real solutions for Θk≥0.8 K/W. For the case of Θk = 0.6 K/W, the d-line

crosses at two points: I ≈ 2A, and I ≈ 4A. Operation at the lower current is obviously

more desirable.

24 - 7

5. The temperature of hot side and the Coefficient of Performance (COP)

An important issue is the temperature of the hot side of the TEC, especially since

semiconductor TECs are limited to low temperature operation (normally lower than

125°C). Thus, one needs to make sure that the hot side is not overheating. Applying the

set of equation (7) - (10) one can eliminate the variables qc, Tc, and V and get an

expression for the temperature of the TEC's hot-end above the ambient, given by

( ) ( )ITIqTTT kmhkmhcambhha +Θ⋅=−= (16)

where

( ) ( )kmmm

kkmh I1I1

IΘα−Θα+

Θ=Θ (17)

( ) ( ) kkmmm

ambm

m2m

mmm

2

kmh I1I1

TR

12I

2RIIT Θ⋅

Θα−Θα+

⎟⎟⎠

⎞⎜⎜⎝

⎛ Θα++Θα

⋅= (18)

Applying (16), one can obtain the temperature of the hot side of the TEC using the

same method as for the temperature of its cold side. The curve Sh is described

parametrically as:

(19) ( ) ( )( )⎩

⎨⎧

Τ=Θ=

=IyIx

ISkmh

kmhh

The dh - line is constructed by starting it from a point of the S(I) curve

corresponding to the current of operation (I), that was found in Fig. 2, and continuing

the line with angular coefficient of qc (see the dotted line in Fig. 4). In the illustrated

case the dh line intersects the Y-axis at about 23K which implies that, at steady state, the

hot side of the TEC will be 23K+Tamb = 50°C.

It should be noticed that the apparent temperature source has zero value when the

current is zero, thus the Sh curve minimum point corresponds to I = 0, and that this

minimum value touches the X-axis, at Y=Tamb, the reference temperature. This is of

course expected since at I = 0, the TEC is in conductive heat transfer operation mode.

24 - 8

The value of X-axis that the minimum point of the Sh touches corresponds to the

thermal resistance of the heatsink (Θk)

The coefficient of performance (COP) represents the balance between injected

energy and the pumped energy. Applying the definition of the COP given in [3]:

ch

c

qqqCOP−

= (20)

and multiplying the nominator and the denominator of (20) by Θk we find:

ba

aTa

qTqCOP

hahcha

hc =−

=Θ−

Θ= (21)

where and kcqa Θ= aTb ha −=

The ratio a/b can be obtained graphically as demonstrated in Fig. 5 for the

illustrated example. Since "a" is the heatsink surface temperature above ambient

temperature when electrical power is off, it can be estimated by drawing a dh line of

slope of qc from the Sh minimum (which corresponds to Y=Tamb and Θkmh = Θk). The

intersection of this line with the Y-axis is the solution of a = qcΘk (Fig. 5). Since

b = Tha - a, it can be evaluated as the difference between the TEC hot side temperature

Tha found earlier and "a," as shown in Fig. 5.

6. Experimental

The graphical analysis was verified by two different methods: a) laboratory

experiments that were carried out on a specific TEC, and b) calculations according to

the proposed approach and comparison to experimental data given by manufacturers for

different TECs.

The experimental setup (Fig. 6) included a TEC, a heat source, a heatsink with fan

and aluminum plate with an embedded thermocouple for temperature measurements. A

series of measurements of the steady-state difference of temperatures of T1 (which is

close to Tc) and Tamb under different heat dissipation conditions, and for different TEC

currents, were carried out. The results of the measurements are shown in Table 2. Fig. 7

24 - 9

depicts the S(I)-curve that corresponds to the data of the experimental TEC (TB-127-

1.4-1.2) and the Θk = 0.67 K/W of the heatsink and fan, at Tamb = 294.8K. The slope of

the d-lines corresponds to the heat generated by the heat source. Each line starts at an

S(I) point that corresponds to each of the experimental TEC currents. The point of

intersection between the d line and the "y" - axis is the estimated value of Tca - the

temperature of the cold side above the ambient temperature. The values of these model

estimates are summarized in Table 2. Good agreement was found between the measured

temperature differences and the ones estimated by the proposed graphical method

(Table 2).

Applying the experimental data given by manufacturers, we have used the unified

model to recalculate the value of Qmax also given in the commercial data sheets. Qmax is

defined as the amount of heat (W) that can be pumped by the TEC for I = Imax and ΔT

equal to zero. To evaluate this case by proposed method, one needs to construct the

S(I)-curve using (14) and substituting Tamb = Th=300K (standard temperature of the

experiment) and Θk=0. This substitution means that the temperature of the hot side of

the TEC is constant and equal to the temperature of the cold side (Fig. 8). Now, the

straight line from the origin to the point on the S-curve that corresponds to Imax has a

slope of Qmax. The tangential line from the origin to the S-curve (Fig. 8) has an angle of

Qopt - the maximum possible amount of heat for the experimental condition, which is

also quoted by some manufacturers.

Fig. 9 shows the distribution of the error in the recalculated values of Qmax for 54

commercial TEC devices as compared to the independent data given by the

manufacturers. In about 90% of the cases, the error in the recalculated Qmax by the

proposed model, relative to the values given in the data sheets, was found to be less than

5%.

24 - 10

7. Discussion and Conclusions

In some applications, thermoelectric active cooling systems can help maintain

electronic devices at desired temperature conditions better than passive coolers. Active

cooling is especially useful when the temperature of a device needs to be precisely

controlled. This study proposes a user-friendly graphical method for calculating the

steady-state operational point of a TEC-based active cooling system. The method is

simple and intuitive, and provides comprehensive information about the cooling system

such as its feasibility, required heatsink, the TEC current, temperatures of the cold and

hot sides, coefficient of performance (COP) and others. The method could help

designers to examine and choose a thermoelectric module from catalogues to meet a

specific cooling problem. To start using the method, designers need only the

experimental TEC data provided by practically all manufacturers of such devices.

The proposed method is ‘graphical’ in the sense that it provides a clear graphical

representation and hence an intuitive understanding of the cooling process by a TEC. It

is based on a very basic equivalent circuit of a generic cooling system (Fig. 1b) that

includes a thermal source, an apparent thermal resistance and a temperature source. The

strength of the graphical approach is that it can help the designers to examine and

compare different TEC and heatsinks combinations in a very user-friendly way. In fact,

once having defined a given cooling problem by a 'd' line and drawing a family of 'S'

curves for a range of heatsinks, one can immediately identify the best design options.

After a design is selected, accurate calculation of the operating conditions of the system

can be carried out by applying the equations developed in this work. For example, the

exact value of the required TEC drive current for a given case can be obtained by

solving numerically (11) and (15) for current I. Indeed, practically all the numerical

results quoted in this paper were obtained by numerical solutions of the corresponding

equations.

24 - 11

The proposed ‘graphical’ approach was verified against experimental results

collected during this study and independent data given by manufacturers. The excellent

agreement that was observed attests to the high accuracy of the proposed model and

graphical approach. This good agreement also justifies the approximations that were

done in the development of the proposed model (e.g. neglecting the Thompson effect,

variations of the parameters values with temperature, etc.).

References

[1] A.F. Ioffe, Semiconductors thermoelments and thermoelectric cooling, Infosearch limited,

London, 1957.

[2] S. Noll, Peltier Device Information Directory, online, available: http://www.peltier-

info.com.

[3] S.L. Soo, Direct energy conversion, Prentice-Hall, London, 1968.

[4] Kryotherm Co., products, online. Available: http://www.kryotherm.ru.

[5] Beijing Huimao Cooling Equipment Co., products online, available:

http://www.huimao.com.

[6] Marlow Industries, products, online. Available: http://www.marlow.com.

[7] S. Lineykin, S. Ben-Yaakov, Modeling and analysis of thermoelectric modules, Applied

Power Electronics Conference APEC’05, p. 2019 - 2023, Austin, Texas, USA, 2005.

[8] S. Lineykin, S. Ben-Yaakov, Analysis of thermoelectric coolers by a SPICE-compatible

equivalent circuit model, Power Electronics Letters, vol. 3, no. 2, p. 63 - 66, 2005.

24 - 12

Figures

Fig. 1 Schematic representation of a passive (a) and an active (b) cooling systems, shown under similar

system terms. Tc - temperature of the surface of interest, qc - heat dissipated by the thermal load, Tamb - ambient temperature, Tca - temperature difference between the surface and the ambient, Θkm(I) is apparent thermal resistance, and ΔTkm(I) is apparent temperature source.

Fig. 2. Geometrical solution for the system: TEC (TB-127-1.4-1.2 [5]) with heatsink (Θk=1K/W).

Tamb=300K and a specific thermal problem: Tcar= - 10K (10K below the Tamb), qcr=10W. The thermal requirement is met when the d line intersects the S-curve (at I = 2.37A and I = 4 A).

Fig. 3. The S-curve for TB-127-1,2-1,4 with different heatsinks (Θk from 0.2 to 2 K/W) for Tamb = 300 K.

Solid line specifies the requirements imposed on active cooling system (Tcar = 20K, qcr = 40 W).

24 - 13

Fig. 4. Geometrical solution for the temperature of the hot side of the TEC (Th = Tha + Tamb). The value of

the current is obtained from solution for the cold side (Fig. 2).

Fig. 5. Geometrical solution for the COP: The ratio of lengths of "a" and "b."

Fig. 6. Experimental setup: 1 - TEC, 2 - massive aluminum plate, 3 - heat source (qc, W), 4 -

thermocouple, measuring temperature T1, (K), 5 - thermal insulation, 6 - heatsink and a fan with thermal resistance Θk = 0.67 K/W. The temperature of the ambient air during the experiment was Tamb = 294.8 K.

24 - 14

Fig. 7. The S(I) curve of the experimental system of TEC and heatsink, showing the points that

correspond to specific TEC drive currents, and graphical representations of the experimental conditions (d lines). The slope of each d line corresponds to the power of thermal load, (qc, in W) of each experiment. The intersections of the d lines with the vertical axis (y) are the estimates of the temperature of the TEC cold side above the ambient temperature in K, for each experimental I.

Fig. 8. Verification of the graphical method against experimental data given by manufacturers. The S-

curve shown is for TB-127-1,2-1,4 with Tamb=Th=300K, i.e. Θk=0. The parameter Qmax is the slope of the straight line from Tamb=300 on the Y-axis, to the point of I=Imax on the S-curve (the minimum of S-curve). The slope of the tangential line from the origin is Qopt.

Fig. 9. Comparison of the recalculated Qmax by proposed model to the data given by TEC manufacturers’

for 54 commercial TEC units.

24 - 15

TABLE 1

Building the parametric curve S(I) for TB-127-1,2-1,4, Tamb

I, A X from (14), K/W Y from (14), K -1 2.829918 30.77659 0 2.511161 0 1 2.25968 −18.5922 2 2.056482 −28.1765 3 1.889145 −30.7909 4 1.74921 −27.7789 5 1.630722 −20.036 6 1.529381 −8.15179 7 1.442007 7.505923 8 1.366212 26.74094 9 1.300181 49.50119

TABLE 2.

Results of the experimental measurements by reference to results Estimated using

proposed Graphical method

Measured values

qc, W I, A Tca, K

Estimated Tca from Fig. 7

24.75 1 30 30 25.2 2.9 7 7 24.5 4 1 0.9 25.2 5 2 2

24 - 16

25

5 Conclusions and future work

In the course of this study, we proposed various ready-to-use equivalent circuit

models of PTs and TEMs. The models are simple to understand and to implement, and

can be reconstructed by a designer for every specific case by using the given topology

and the script for parameters extracting. The proposed equivalent circuit models are also

simple and ready for simulation using any available electric circuit simulation software

(for example, PSPICE).

The resonant equivalent circuit model of the PTs allows emulating the signal

response of the PT in the time domain as well as in the frequency domain. The model is

adjustable for any type of PT. The model is helpful in designing and analyzing drive

circuits, choosing the optimal load circuitry and close loop control design.

The equivalent circuit model of PTs for envelope simulations allows very time-

saving simulation of the envelope behavior of the signal in time domain as well as small

signal AC simulations in the frequency domain of the PT excited by a modulated signal.

This simulation is impossible using standard PSPICE tools with a regular equivalent

circuit.

A graphical method of the steady-state analysis of the thermoelectric active cooling

system was proposed and described. The method is based on the two-port equivalent

representation of the active cooling system, where one port is electrical and the other is

thermal. The method is easy and intuitive, it could help designers to examine and

choose the elements of the system from catalogues to meet a specific cooling problem.

A tri-port dynamic model of the thermoelectric module helps to analyze the transient

response of the cooling system by a variation of one of the parameters such as the

ambient temperature or power of the heat source. The model estimates the small signal

transfer function of the system in frequency domain, which is very helpful for close

loop control design.

Several applications of the PT and TEM have been proposed, including a system for

signal and power transfer through the PT, a system for maximum power frequency

tracking of the PT, and using the PT as galvanic barrier in feedback signal loop. A

method of estimation of the BW of the PT excited by modulated signal is presented,

along with the method and design guidelines for design of active cooling system. All the

applications were verified experimentally with very good results.

In conclusion to this work, I want to mention some of the possible directions for

future study. One issue could be to find ways of adjusting the proposed models to

26

achieve the best accuracy, taking into account the instability of the DED parameters

with temperature (in TEMs as well as PTs), extending the method of analysis of one-

stage TEM to multistage TEMs, extending the 1D analysis to 2D analysis of the TEM

by taking into account the distribution of heat and temperature in the direction

perpendicular to the heat flow. Another task would be make in-depth studies of new

applications of thermoelectric coolers and thermoelectric generators using proposed

models. Since the simultaneous operation of a PT array in parallel operation using the

technique of frequency tracking is now possible, we can start an additional research of

the possible applications of this method. The last would be the issue of developing a

universal method for finding the parameters of the PT’s equivalent circuit directly from

the manufacturer's data. In order to be more universal, the method of envelope analysis

must be extended so that it can analyze systems with several modulating signals.

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תקצירמכאניים -בעזרת מעגלי תמורה של מערכות להמרת אנרגיה אלקטרוהדמיה בעבודה הנוכחית מוצגת גישה ל

לשם בניית מעגלי תמורה אלקטרוניים בסיסיים לתיאור רכיבים להמרת אנרגיה ישירה . ויישומםתרמיים -ואלקטרו

(DED)לאחר ניתוח מעמיק של מעגלי תמורה. ם השתמשנו במודלים מתמטיים מפושטים של תהליכים פיזיקאליי ,

.מספר יישומים של מערכות פייזואלקטריות ותרמואלקטריותהוצעו

השיטה . עבודת דוקטורט זו מציעה שיטה ידידותית למשתמש של אנליזת מצב מתמיד במערכות קירור אקטיבי

מערכת קירור אקטיבי . דרגתיים-דרגתי וניתנת ליישום גם במקררים רב- חד(Peltier)פותחה עבור מקרר פלטייה

במקום להשתמש בשיטת איטרציות המקובלת . (Thevenin's generator)ורה בטופולוגיית זוגיים מוצגת כמעגל תמ

. אנו מציעים גישה גראפית פשוטה ואינטואיטיבית לעיצוב וניתוח מערכות קירור אקטיבי

עם שלושה יסוג נוסף של מודל מערכת קירור אקטיבי וגנרטור תרמואלקטרי שאנו מציעים הוא מודל דינאמ

. PSPICE (DC, transient, AC)מודל זה נותן תוצאות מדויקות מאוד בכל סוגי האנליזה הזמינים בתוכנת . קיםהד

, כגון דוחף(כלי עבודה זה מאפשר למשתמש לעצב ולייעל מערכות תרמואלקטריות בצירוף התקנים אלקטרוניים

פיתחנו מתודולוגיה לקבלת . במקום לערוך ניסויי מעבדה, על ידי הדמיה ממוחשבת) ומערכת בקרה בחוג סגור, עומס

בחנו נתונים של מספר יצרנים וקיבלנו את . פרמטרים של המודל המוצע מתוך נתוני היצרן לגבי כל מקרר ספציפי

ובכל המקרים המודל נמצא , שחזרנו ניסויים של היצרנים בהדמיה. הפרמטרים של המודל עבור כל התקן והתקן

. דיוק הביצועים של המודל נבדק גם בניסויי מעבדה שביצענו. דויק את הביצועים של מקררים מסחרייםכמשחזר במ

המחקר לקח . וניסוייתתמעשיות היישום של שנאי פייזואלקטרי בתור מבודד גלווני נבדקה במחקרנו תיאורטי

. FM- וAM: בין שני סוגי אפנוןערכנו השוואה. והנחתת אות משותף, רוחב פס, דמודולציה, בחשבון גורמי דחיפה

. PSPICEתגובת מבודד פייזואלקטרי במישור תדר נבחנה בדרך ניסויית וגם על ידי הדמיית אות קטן בתוכנת

ידינו באופציה -שיטת הדמיית המעטפת שפותחה בעבר בשביל הדמיית אות גדול במישור הזמן הורחבה על

, AMל בחנו שנאי פייזואלקטרי באפנוני "בעזרת השיטה הנ. DCת להדמיית אות קטן במישור תדר ובאופציית סריק

FMו -PM .נמצאה . פיתוחים אנליטיים של המודל אומתו ניסויית וכנגד הדמיות קונבנציונאליות הכוללות תדר נושא

. התאמה מלאה בין תוצאות הניסויים לתוצאות ההדמיה

למתח גבוה הכוללים שנאי פייזואלקטרי ומיישר DC-DC בממירי יבעיית המעקב אחרי תדר ההספק המקסימאל

הוכחנו כי תדר העבודה של שנאי פייזואלקטרי בנקודת הספק מקסימאלי . מכפיל מתח נחקרה אנאליטית וניסויית

. יש צורך במנגנון מעקב וייצוב התדר, יציבות הפרמטרים של המודל-בתנאיי עומס משתנה ואי, לכן. תלוי בעומס

, ב וייצוב התדר מבוססת על חוג נעילת פאזה ומבטיחה עבודה תקינה ללא תלות בשינויי מתחהשיטה המוצעת למעק

.יציבות בזמן של הפרמטרים של השנאי הפייזואלקטרי-ליניאריות או אי-וגם אי) בתחום סביר(טמפרטורה , עומס

יעקוב-בןהעבודה נעשתה בהדרכת פרופסור שמואל

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