IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO. 1 JANUARY 1994 35

PSPICE Simulation of Three-phase Inverters by Means of Switching Functions

L. Salazar, Member, IEEE, and G. J o ~ s , Member, IEEE

Abstract-Static power converters can be analyzed by means of widely available circuit simulation software packages such as PSPICE. However, they are usually modeled as a set of real switches, which results in long execution times and possible convergence problems in the case of complex circuits. This paper proposes macromodels to simulate three-phase power converters on such packages. The proposed macromodels are based on converter switching functions rather than actual circuit config- uration, and they are suited for steady state and large signal transient analysis at system level. In this approach, voltage source inverters (VSI), current source inverters (CSI), and controlled rectifiers (CR) are simulated as multiport networks avoiding the physical nonlinear micromodels of the power switches. Com- puter memory and the run-times required for the simulation are thereby minimized. Complete examples of VSI, CSI and CR, with different PWM techniques, are given with specific reference to the PSPICE software to illustrate the effectiveness of the proposed models.

I. INTRODUCTION HE SPICE circuit simulation program has become an T industry standard. The major advantage of using SPICE in

power electronics is that, with the same software, a particular circuit can be designed and analyzed at different system and subsystem levels, i.e., at the levels of the power switch (de- vice), the converter circuit, and converter systems, including feedback control. However, for higher levels of simulation, simplified models for the switch and the converter must be implemented, in order to minimize convergence problems and reduce the run times.

Recent research work in switching power converters has demonstrated that the transfer or switching function concept [ 1, 2, 31 is a powerful tool in understanding and optimizing the performance of converters such as the voltage source inverter (VSI), current source inverter (CSI) and controlled PWM rectifier (CR). Furthermore, general functional models for three-phase PWM inverter/rectifier converters have also been proposed [3,4]. A functional model produces a dramatic simplification of the total converter circuit. This approach has been shown to be a very useful technique for harmonic analysis and system simulation of different converter topologies on personal computers (PC) with limited memory and speed [4, 5, 81. Moreover, i t allows a converter to be modeled as a multiport circuit that contains only voltage-controlled voltage sources, and current-controlled current sources connected at the input and output ports. The signals controlling these

Manuscript received January 2, 1991; revised September 29, 1993. The authors are with Department of Electrical and Computer Engineering,

IEEE Log Number 9214829. Concordia University, Montreal, P.Q., H3G 1M8 Canada.

sources are two or three-level switching functions defined by the PWM technique and the converter mode of operation (i.e., inverter or rectifier).

The main objective of this paper is to illustrate how the functional definition of a switching converter can be used to model any three-phase voltage source inverter (VSI), current source inverter (CSI) or PWM controlled rectifier (CR) with its respective control circuit on PSPICE, allowing higher levels of analysis and simulation in a low-cost computer environment. Furthermore, a new classification of the CR is introduced. This family includes two members: the voltage-controlled rectifier (VCR), also known as a PWM ac/dc rectifier, and the current-controlled rectifier (CCR), also described as a PWM boost rectifier [6]. Detailed examples of VSI, CSI, VCR, and CCR operation with different PWM schemes are given to illustrate the effectiveness of the proposed models and simulation methods.

11. LEVELS OF SIMULATION IN POWER ELECTRONICS

A . Simulation at the Switch and Converter Circuit Levels

PSPICE micromodels are in common use for simulation analysis at the level of the switch circuit, i.e., power switch, driver or snubber circuit, allowing for example to study switching losses. Models to simulate a variety of semicon- ductor devices on PSPICE have been proposed [7] and they can be adapted, through parameter adjustments, to power applications. These are physical intrinsic micromodels of the appropriate semiconductor technology, usually available from a library of parts provided by various companies. However, these micromodels are not practical at the converter circuit and system levels of simulation for the following reasons:

Diodes, bipolar transistors, and switches in general are described as non-linear controlled source by means of functions that contain exponential terms, resulting in slow execution times, large amounts of generated data (small time steps are required in the numerical integra- tion process), and occasional convergence problems. Accurate modeling of complex converter circuits may lead to a very large number of devices and components and the simulation becomes very time consuming. The same applies when increasing the switching frequency and the simulation time interval. The probability of encountering convergence problems also increases with the number of comDonents.

The default parameters a i d the device models in the PSPICE program can be adjusted to reduce the above difficulties. Rec- ommended values and guidelines to simulate power converter

0885-8993/94$04.00 0 1994 IEEE

36 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO. 1, JANUARY 1994

TABLE I TYPICAL PARAMETERS A N D GUIDELINES TO SIMULATE POWER CONVERTER CIRCUITS ON PSPICE (10 kW 100 kW)

DIODE D (IS=O.OOl CJ O=O.OOI UF RS=O.OOl) .MODEL BIPOLAR NPN (BF=1000 RB=O.I CJE=0.001 UF)

NMOS (VTO=3 KP=20 CBD=O.OOI CBS=O.OOl) values between 0.01 to 0.05

.OPTIONS ~ ~ ~ o ~ : values between 1 @ A to 1 mA values between 1 p V to 1 mV

Notes:

MOSFET RELTOL:

ITL5=O

1. The switching signal (PULSE) should he defined with a rise and fall time different from zero. 2. The Mosfet switch execute faster than the bipolar switch and presents less convergence problems. 3. Voltage and current controlled switches run slower than mosfet switches and present convergence

problems. 4. Use of controlled source to replace the non-linear active device and integrated analog circuit. 5. Large resistors must be used in parallel with current sources and inductors. 6. Floating nodes are eliminated by using a large resistor to ground (node 0). 7. R-C circuits are useful to minimize ringing oscillation and convergence problems. 8. Use small resistors in series with voltage sources.

DM 3 $"'+, I

ZSWDV + 5 '

1 I 15

Fig. I . Two switch macromodel with voltage controlled source for a three-phase converter leg, subcircuit 2SWDV.

circuits on PSPICE [7, 81 are given in Table I. Moreover, simplified device models using controlled sources have been proposed to speed up the simulation [8, 9, 101. For example, a simple model for a two-switch converter with controlled voltage sources and diodes, shown in Fig. 1 , has been used successfully by the authors [ 101 for simulation at the converter circuit level. Table I1 presents a comparison of the simula- tion run-times using this model and the PSPICE transistor micromodels. Moreover, to evaluate the simulation run-times with a larger number of such switches (Fig. l), the three- phase inverter driven by (1 KHz triangular waveform carrier) shown in Fig. 2 has been simulated using a sinusoidal PWM switching pattern with PSPICE running on a 386,20 MHz P C . The simulation results are shown in Fig. 3 (run time was 12.7 minutes with a step size = 10 ps).

B . Simulation at the System Level

The examples presented in the previous section show that simplified switch models in PSPICE can be used to fully sim- ulate three-phase rectifier-inverter structures with reasonable run times on a low cost computer. However, simulation of multiconverter systems, even with the use of the simplified switch model, results in large computation times. Therefore, further simplifications of the converter model are required to reduce the design cycle.

SPICE VSll SCH OFTIONS RELTOL=O n5 ABSTOL=IGUA ms=n LIB C WELUB CR TRAN IMVS zom n m SOUP mc

XJDP IDP

X3DN 1DN

Fig. 2. A three-phase VSI system with a sinusoidal PWM pattem simulated in PSPICE by using the switch macromodel shown in Fig. 1 (schematic file VSI I .SCH).

Simulation at the system level is concerned with the de- sign verification and performance analysis of the converter as a module, and its interaction with the respective input source, load or other converter if any, in the complete system. Accurate models for the converter switch are not required at this level of simulation. Instead, each converter can be simulated as a multiport network, wherein the time solution of the currents and voltages, in the input and output terminal, constitute the main objectives of the analysis. The transfer function of this multiport network can be easily simulated on PSPICE by using controlled sources in the Analog Behavioral Modeling option (ABM). The models for simulation at the system level using this approach are presented in the next section.

111. CONVERTER MODELS

A . Transfer Functions

Analytical models for the VSI, CSI, CVR and CCR are derived by using the transfer function concept for a generalized converter [ 3 ] . The converter as a non-linear multiport network has three ports: dc, ac, and control. The dc and ac ports can

SALAZAR AND JO6S: PSPlCE SIMULATION OF THREE-PHASE INVERTERS 37

1.5 Vfc) 0.5

0 0

*) I V v v v v v v v v v v v-v v v v v V I -0.5

I I os 5m9 IomS I5ms 2oms

I I

I I

os 5ms loms Isms 2oms

b) Fig. 4. Typical switching function (SF); (a) two-level SF (S2); (b) three-level SF (5'3).

TABLE II

DC-DC CONVERTER USING DIFFERENT SWITCHES PSPICE SIMULATION TIME OF A HALF BRIDGE

TIME SPEED FACTOR SWITCH Bipolar (*) 25.02 sec 1 .oo

0 Mosfet (8) 17.90 sec 1.39 Controlled Source 13.24 sec 1.89

d)

V k Y ) .OPTIONS RELTOL = 0.01 ITL5 = 0 -400 os 5na IOma l5mS zoms .TIME 2 p s 100 p s 0 2 p s

.MODEL Operating Conditions:

values given in Table I Fig. 3. Simulation results for the VSI system shown in Fig. 2. (a) Sinusoidal PWM generation; (b) two-level switching function (s%. I - ( r ) ) . (c) DC-link capacitor voltage ( I - ( / . 1 1 ) ) and VSI input current ( J ( L / / ) - I ( C i ) ) . (d) Line-to-line output voltage ( I - ( . r . y ) ) and line output current ( I ( L 1 ) ) .

= ohm, L = 200 L l ~ , vin = v Frequency = 2o k ~ z , D~~~ cycle = 50% (*) A dead time of 2 c i s is required

be inputs or outputs depending on the mode of operation. The control port consists only of inputs. The transfer function is the instantaneous relation between the dc input/output and the ac input/output variables. A general converter has a current and a voltage transfer function, both defined by the same switching function applied to the control port.

B. SMYtching Functions

The signals applied to the control input port are two-level or three-level switching functions (SFs). Examples for both are shown in Fig. 4. A two-level SF ( s2 ) is used to generate the ac output leg voltage in a VSI, or the dc output current in a CCR. Similarly, a three-level SF ( s3 ) is used to generate the ac output line current in a CSI, or the dc output voltage in a VCR. Thus, these two types of SF can be considered dual. By using the concept of switching functions, and with the assumptions of no losses and no parasitic reactive elements in the converter, the functional representation of various types of converters is derived, as shown in the next sections.

C. Lbltage Source 1inw.ter (VU) Model

In a VSI, the output voltage is generated by reflecting simul- taneously the instantaneous input voltage to the output, and the instantaneous output line current to the input during intervals defined by the switching function. These two processes can be formulated in analytical form by the following equations,

in vectorial form:

(1) Output (ac) : ij,(t) = w i ( t ) . &(t) Input (dc) : i i ( t ) = &(t)T . &?(t) ( 2 )

where G o ( t ) = [va( t ) ub( t ) vc(t)lT is a vector that contains the instantaneous three-phase output voltages, and i,(t) = [ ia ( t ) ib(t) ic(t)lT is a vector that contains the three in- stantaneous output line currents. Also, v;(t) and i ; ( k ) are the instantaneous input voltage and current respectively (dc quantities), and & ( t ) = [ s a ( t ) sb(t) sc(t)lT is a vector that contains the two-level switching functions for each leg of the converter.

D. Current Source Inverter (CSI) Model

by applying the duality principle. Hence, The CSI transfer functions can be obtained from (1) and (2)

Output (ac) : ;,(t) = i ; ( t ) . i 3 ( t ) (3) Input (dc) : v; ( t ) = 60(t)T . .+3(t) (4)

where i 3 ( t ) is a vector that contains the three-level switching function for each line of the converter.

E . Voltage Controlled Rectifrer (VCR) Model

The transfer functions are derived from (3) and (4) by applying the inversion principle. Thus, the output becomes

38 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO, I , JANUARY 1994

ii(t)

il(t) = ia(t)*SZa(t) iZ(1) = ib(t)*SZb(t) i3(t) = ic(l)*SZc(I)

ea@) = vi(t)*SZa(t) eqt) = vi(t)*S2b(t) ec(1) = vi(t)*S2C(l)

VSl

I el(t) = va(t)*S3a(t) eZ(t) = vb(t)*S3b(t) e3(1) = vc(t)*S3c(t)

ia(t) = ii(t)*S3a(t) ib(t) = ii(t)*S3b(t) ic(t) = ii(t)*S3c(t)

CSI

il(t) = ia(t)*SZa(t) 12(t) = Ib(t)*SZb(t) e2(t) = vb(I)*S3b(t)

e3(t) = vc(t)*S3c(t) a ( t ) = lC(t)*S2C(t)

ia(t) = io(l)*S3a(t) ib(t) = io(t)*S3b(t) ic(t) = io(t)*S3c(t)

ea@) = vi(t)*SZa(t) eqt) = vl(t)*S2b(t) =(I) = VI(t)*S2C(t)

VCR CCR

Fig. 5. Proposed macromodels of three-phase converters

an input, and the input becomes an output. Hence,

Input (ac) : i z ( t ) = i o ( t ) . i 3 ( t ) ( 5 ) Output (dc) : vo( t ) 1 C,(t)' . i 3 ( t ) (6)

where ; ( t ) and G l ( t ) are vectors that contain the three-phase instantaneous input current and voltage respectively.

F . Current Controlled Rectifier (CCR) Model

The transfer functions for a CCR can be obtained by applying the duality principle to ( 5 ) and (6), or the inversion principle to (1) and (2). Hence,

Input (ac) : ~ ~ ( t ) = 6,(t)' . i&t) (7) Output (dc) : io(t) = . & ( t ) (8)

The respective circuit macromodels derived from (1) to (8) for the three-phase VSI's, CSI's, VCR's, and CCR's in terms of the controlled voltage and current sources are shown in Fig. S .

G. PSPICE Implementation

Using the ABM option in PSPICE, the implementation of the converter macromodels shown in Fig. 5 (see Appendix 11) can be done directly. In this option the controlling function of the controlled Fource, E and G, can be defined as a mathemati- cal expression in standard notation using the keyword VALUE. The mathematical expression can contain the variable TIME and any mixture of voltages and current. The standard SPICE polynomial function operator (POLY) can also be employed. However. the syntax for specifying the polynomial is quite difficult to use. Also, many other transfer functions are not well represented by polynomials.

H . Switching Function Implementation

The PWM subcircuits control modules designed to generate the switching functions are: 3PWM for the sinusoidal PWM method, CPH for the hysteresis control method, and SHES7 for a PWM switching pattern method that eliminates harmonics 5 and 7. The PSPICE input file for each of these is presented in Appendix 11. The sinusoidal and hysteresis control methods require a comparator circuit to generate the switching func- tion. The macromodel for a comparator (subcircuit CPNF) is implemented by means of a voltage source E, representing the output of a differential amplifier with a clamped output. This non-linear amplifier is defined as a lookup table (TABLE) as follows:

E034 TABLE{G*(V(l) - V ( 2 ) ) ) = (-1; -10)(1, 10)

The ideal amplifier has a high gain G(G > 10 K) and the input to the table is the difference between V(1) and V(2). The output is interpolated for inputs between -1 and 1, and clamped to (-10, 10) when the range of the table is exceeded.

The hysteresis comparator (subcircuit CPH) is designed by adding positive feedback to the comparator CPNF with another controlled source (EVH) applied through a capacitor:

CERR 1620.001 UF VH 50{ VHYST} RVH 5010 XCPN 16260 CPNF EVH 31 VALUE = {V(5,0)*V(6,0)*0.05}

The hysteresis band is defined by the voltage source VH and it can be modified with the keyword PARAMS: VHYST = (value).

The subcircuit 3PWM uses three comparators (subcircuits XCP1, XCP2, XCP3) and a three-phase sinusoidal reference ( E l , E2, E3). The amplitude (modulation index) and the frequency of this sinusoidal reference are modified externally with two dc sources (or regulator error signals).

IV. SIMULATION ENVIRONMENT The PSPICE software, with the ABM option, version 4.03,

running on a 386, 20 MHz PC, has been used to validate the proposed macro-models. The schematic capture program ORCAD, version 3.22 [ 1 11 is employed as an effective front- end to PSPICE. It allows modification of all parameters and components values directly on the circuit schematic. The macromodels for the converter and PWM switching function generators are implemented as subcircuit modules and stored in a separate library (CIR.LIB). Symbols for each subcircuit have been implemented by using ORCAD. The symbols are stored in a schematic dedicated library file (SCH.LIB). The symbols are listed in Appendix I, and the respective subcircuits list are presented in Appendix 11. The ORCAD program generates the input netlist for PSPICE from the schematic circuit diagram files. The programs are linked by means of a batch file. The PSPICE post-processor PROBE is used as a calculator, harmonic analyzer and for waveform display. The overall system is a very flexible, low cost and complete

+ Fig. 6. Example 1 , input file VSIZSCH.

SPICE vsw SCH OPnoNS RELTOL-0 05 lTL5=0 ABSTot=IuA LIB c \pEL\LIB CIR ?RAN lous Zomsoms IOurulC PROBE

4

Fig. 7 . Example 2, input file VSIH2.SCH

simulated laboratory for analysis and design of three-phase power converters.

v. APPLICATION EXAMPLES AND SIMULATION RESULTS In order to establish the effectiveness of the proposed

models and simulation methods, the performance of standard three-phase converter systems (VSI, CSI, VCR, CCR) oper- ating with different PWM control methods is evaluated. The respective input schematic diagrams for PSPICE are shown in Figs. 6, 7, 8, 9, and 10, as they would appear on the screen of a PC computer with the ORCAD software. Resistances are incorporated in the circuits to account for the losses. Notice that the only non-linear physical elements used in the simulation are the diodes in the front-end three-phase rectifier of the VSI and CSI systems. Moreover, by using a functional definition for a rectifier in terms of absolute value, the three- phase rectifier in the example 2 and 3 has been replaced with one equivalent voltage source E defined as follows

E 46 VALUE = { [ ABS(V(1) - V ( 2 ) )

+ ABS(V(2) - V ( 3 ) )

+ ABS(V(3) - V ( 1 ) ) I P l

where V(1). V(2). V ( 3 ) are the phase voltages. This source is defined as subcircuit REC in Appendix 11. One diode is still used to account for the unidirectional current in the dc bus.

Fig. 8. Example 3, input file CSI2.SCH.

2mh ic- 34.28 c 1 m K-48.26 2mh ic - -115.4 U5OM ic = -125.9 2mh ic= 81.14 C 3 M ( M k = 190.8

Fig. 9. Example 4, input file VCR2.SCH

I .LIBC\PEL\LIB.cIR

c

I l l MSIR

3HYs PARMI:

39

Fig. 10. Example 5, input file CCR2.SCH.

The same applies to VCR and CCR for discontinuous current analysis in the dc output terminal. The three-phase rectifier could also have been replaced by ideal voltage and current sources,respectively, as a first approximation.

A . Example 1: Rectifier-VSI with Sinusoidal PWM

In this example, a 1 kHz sinusoidal PWM modulator (SUB- CIRCUIT 3PWM) is employed to generate a three-phase, two-level switching function. The modulation index and the frequency are varied by changing the values of the voltage sources Vm (magnitude) and Vf (frequency). The schematic input file is shown in Fig. 6. Table I11 compares the run times

40 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO. I, JANUARY 1994

350 -

200 7

400, I

600, I

0

.. . . ..

os 5ms loznS 15ms 2omS

r I

400

0

I I

(b)

Fig. 1 I . Simulation results for Example 1 (see Fig. 6).

0

I I u l u u u u u u u u l U U \ I

(I(Li)-l(Ci))*2 4 I

200

0

-100 I I

(b)

os 5ms l h I5ms 2oms

Fig. 12. Simulation results for Example 2 (see Fig. 7).

for the VSI systems shown in Figs. 2 and 6. The first result is for the simulation of the VSI with the simplified switch model shown in Fig. 1, and the second corresponds to the proposed converter macromodel of Fig. 5 (a). Notice that an important reduction in the run time is achieved by using the converter macromodels. The time step can also be increased to speed up the simulation. Moreover, Fig. 11 displays input and output waveforms, and confirms that the converter macromodel gives the same results as those shown in Fig. 3.

B . Esample 2 : Rectifier-VSI with Hysteresis

The schematic input file for this example is shown in Fig. 7. A current feedback loop and hysteresis control with a sinu- soidal reference (SUBCIRCUIT 3HYS) is used in this example to generate sinusoidal currents at the output terminal of the inverter. The amplitude of the reference and the frequency are modified on the schematic diagram by changing the value of the voltage sources VA and VF. The range of the hysteresis band is modified by using the statement “PARAMS:” on the schematic diagram. In the input file, the three-phase rectifier

os 5mS lb l5ms

(b)

Fig. 13. Simulation results for Example 3 (see Fig. 8).

300 I I

400

200

0 os Sms lomS l5ms 2oms

(b)

Fig. 14. Simulation results for Example 4 (see Fig. 9).

TABLE I11 SIMULATION TIME OF A THREE PHASE VsI BY

USING THE PROPOSED MODELS ON PSPICE

.TIME Statement Model 1 p s 20 ms 10 ps 100 ps 20 ms 100 ps

761.10 s 436.38 s

240.79 s 59.54 s

Controlled Source Fig. I , Fig. 2 Converter Model Fig. 5(a), Fig. 6

and the VSI are now incorporated as subcircuits (REC, VSI). Fig. 12 shows the results obtained with the proposed methods.

C . Example 3: Rectifier-CSI with Selective Harmonic Elimination Technique

In this example, the control module SHE57 is used to control a CSI. This module generates a three-phase, three-level PWM switching function that eliminates the 5th and 7th harmonics in the output current of the inverter. The schematic input file for PSPICE is illustrated in Fig. 8, and Fig. 13 shows the simulation results.

SALAZAR AND JObS: PSPICE SIMULATION OF THREE-PHASE INVERTERS 41

2 0 0 , I These models are also valid when simulating the behavior of converter systems with feedback control loops.

0

450 , I

0

(b)

Fig. 15. Simulation results for Example 5 (see Fig. IO).

D. E.\-aniple 4 : VCR MVth Selective Husmonic Eliniiriatiori Technique

The modulation technique (SUBCIRCUIT SHE57) of Ex- ample 3 is utilized here to minimize harmonics in the input line current of a VCR system. The schematic input file for this example is shown in Fig. 9. The simulation results are shown in Fig. 14.

E. E-vaniple 5: CCR Mith Hystesesis Control

This example simulates the performance of a CCR that pro- vides near unity input power factor, and almost sinusoidal input line currents by using the same hysteresis control module (3HYS) as in Example 2. The schematic input file is shown in Fig. IO. The simulation results are shown in Fig. 15.

F . Perforniaizce Analysis

Different circuit parameters have been computed for the previous examples from the data generated in the simulation by using the post-processor (PROBE). These parameters include the average, rms and peak values, the harmonic contents, distortion factor and power factor. These values were checked against the results obtained with exact modeling of the con- verter system and with experimental set-ups. Close agreement was found, confirming the validity of the macromodels pro- posed in this paper.

VI. CONCLUSION In order to optimize the run times and the amount of mem-

ory required, and to reduce convergence problems in computer simulations of three-phase converter circuits using SPICE, appropriate modeling must be used. This includes the physical micromodel at the switch level, the switch macromodel at the single converter level, and the converter macromodel at the system level. In this paper macromodels which simulate standard three-phase VSI's. CSI's, VCRs and CCR's at the system level have been proposed and evaluated for use on the PSPICE simulation package. These models greatly reduce the computational times without affecting the accuracy of the results obtained for converter input and output variables.

APPENDIX I SCHEMATIC LIBRARY FILE USED IN ORCAD (PEL.LIB)

U XPWM 3PWM

X3H 3HYS

APPENDIX I1 SUBCIRCUITS LIBRARY USED IN PSPICE ( L I B . C I R )

I

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO. I , JANUARY 1994

REFERENCES Lautaro D. Salazar (S’86-M’91) received the B S degree in electrical engineering from Concepcidn University, Concepci6n, Chile, in 1980.

He specialized in switch-mode power supply de- sign, simulation and control method during the Ph.D. program in Concordia UniverWy, Montredl,

[ I ] P Wood. Theot? of Suijching Power Converter, New York Van Nomand-Reinhold, 198 l

[Z ] P D Ziogas. “Synthesis of optimum gain functions for static power converters,” IEEE Trans Itid A p p ( , vol IA-19, pp. 401408, May/June 1983. Quebec, Cdndda

[3] E. P Wiechmann, P. D Ziogas and V. R. Stefanovic, “Generalized functional model for three phase PWM inverter/rectifier converters,” Cotif Rec IEEE /AS’ 86, pp. 984993, 1986.

141 P. D Ziogas, E P. Wiechman and V. R. Stefanovic, “A computer aided analysis and design approach for static voltage \ource inverters,”/EEE Tiatis I d Appl , vol IA-21, pp. 1234-1240, Sept./Oct. 1985

[5] D Vincenti, M Boost, P D. Ziogas dnd R. V Patel, “A novel simulation progrdm for power electronics equipment,” Can J Elecr Comp Eng , vol 14. DD. 89-97. 1989. B. T. Odi: J. C. Salmon, J. W. Dixon and A. B. Kulkami, “A three- phase controlled current PWM converter with leading power factor,” IEEE Trans. Ind. Appl.. vol. IA-23, pp. 78-84, Jan./Feb. 1987. PSPICE Manual, CA: MicroSim Corp. V. Bello, “Computer-modelling the pulse with modulated (PWM) in- verter,“ Pou<ercon 7. Mar. 24-27, 1980. B. Epler. ‘‘Spice2 application notes for dependent sources,” IEEE Circ,rtit.r Devices Mag.. pp. 3 4 4 4 , Sept. 1987. L. D. Salazar, P. D. Ziogas and D. Vincenti, “Simple models for spice assist power electronics circuit simulation on PC’s,” Conf. Rec. /AS’ 88, pp. 1063-1068, 1988. ORCAD/SDT 111 Addendum, Hillsboro, OR: ORCAD Systems Corp.

G. Joos (M’79-SM’89) received the M.Eng. and Ph.D. degrees from Mc Gill University, Montreal, Quebec, Canada, in 1974 and 1987, respectively.

Since 1988, he has been with the Department of Electrical and Computer Engineering of Concordia University, Montreal, where he is involved in teach- ing and research in the areas of power converters and electrical drives. From 1975 to 1978, he was a design engineer with Brown Boveri Canada, and from 1978 to 1988, a professor at the Ecole de Technologie Supbrieure, in Montreal.