methodology for cable modeling and simulation for high-frequency phenomena studies in pwm motor...

744 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 2, MARCH 2008

Methodology for Cable Modeling and Simulationfor High-Frequency PhenomenaStudies in PWM Motor Drives

Helder De Paula, Darizon Alves de Andrade, Member, IEEE, Marcelo Lynce Ribeiro Chaves,Jose Luis Domingos, and Marcos Antônio Arantes de Freitas

Abstract—The analysis of the transient overvoltages in apulsewidth modulation (PWM) motor drive system comprises awide frequency range, which starts with the low values corre-sponding to the motor speed, includes the switching harmonics,which can reach up to few hundreds of kHz, and also the cableresonance frequency, which value can be in the MHz range, de-pending on the cable length. In this context, this work presents atime domain methodology for cable modeling able to represent thecable parameters variation due to skin effect in this broad rangeof frequencies. The proposed technique reproduces accurately thewave propagation and reflection phenomena, thus showing to bevery appropriate to transient overvoltage studies in PWM motordrives. A new alternative to represent the frequency-dependentcable earth-return path is also included, allowing the computationof the zero-sequence currents generated by the common-modevoltage produced by the inverter. Simulations using the proposedmethodology are conducted and the obtained results are comparedwith measurements, showing good agreement.

Index Terms—Cable modeling, common-mode currents,pulsewidth modulation (PWM) motor drive, skin effect, transientovervoltages.

I. INTRODUCTION

PULSEWIDTH modulation voltage source inverter(PWM–VSI) electronic converters make most of the

topologies to drive induction machines. Two main problemsoccur with such systems: 1) transient overvoltages at the motorterminals and 2) high-frequency common-mode currents.The transient overvoltages at the motor terminals appear as aconsequence of the continuous application of voltage pulses,originated from the electronic converter and reflected at thecable endings, while the common-mode currents are due toinherent inverter common-mode voltages. Distinct and un-desirable consequences can emerge from this situation. Theovervoltages will mainly compromise the stator winding in-sulation, and the common-mode currents, flowing to ground

Manuscript received April 23, 2007. This work was supported by theBrazilian agencies CAPES and FAPEMIG. Recommended for publication byAssociate Editor P. Tenti.

H. De Paula is with the Universidade Federal de Minas Gerais (UFMG), BeloHorizonte 31270-010, Brazil (e-mail: [email protected]).

D. A. Andrade and M. L. R. Chaves are with the Faculdade de EngenhariaEletrica (FEELT), Universidade Federal de Uberlandia (UFU), Uberlândia38.400-902, Brazil (e-mail: [email protected]; [email protected]).

J. L. Domingos is with the Centro Federal de Educacao Tecnologica (CEFET/GO), Goiânia 74.055-110, Brazil.

M. A. A. de Freitas is with the Centro Federal de Educacao Tecnologica(CEFET)-UNED/GO, Jataí 75800-000, Brazil.

Digital Object Identifier 10.1109/TPEL.2007.915759

through the capacitive couplings of the motor and feedingcable can lead to electromagnetic interference (EMI) problems,misoperation of ground-fault protection and motor bearingfailures [1]. Solutions for these high-frequency problems aregenerally based on filters. A comprehensive comparison amongdifferent filter topologies for differential mode overvoltagereduction was conducted in [2], which elected the RLC filterat the inverter terminals as the most interesting alternative. ALC filter placed at the inverter output with dc link feedbackwas analyzed in [3] as a very effective solution for reducing thecommon mode phenomena. Particularly from the EMI pointof view, this topology makes possible the use of unshieldedinverter-to-motor cables in industrial facilities, in most cases.

The effects of transient overvoltages were first verified in sys-tems where the motor is fed by inverter through long length ca-bles [4]. The voltage pulses applied by the inverter travel alongthe cable to reach the motor, and their combination with the cor-responding reflected waves can lead to overvoltages at the motorterminals. Transient overvoltages of up to 3 p.u. (1 p.u. dc linkvoltage) have been reported in the literature [5], [6]. With con-verters using modern semiconductors with very small turn ontime, these effects of overvoltages have become observable evenin short length cables. Very fast pulse rising time means veryhigh frequencies; therefore, to accurately predict the possibleovervoltages, both the motor and cable models must be appro-priate for a very wide frequency range, up to MHz level. Induc-tion motor models that incorporate the effects of high frequencyare found in literature [8]–[10], where the basic idea is to use anequivalent circuit whose frequency response approximates themeasured values. An important aspect of these models is the ad-equate representation of the intrinsic capacitances of the motor,which, in high frequencies, represent low impedance paths forthe current generated by the voltage pulses. For the intercon-nection cable, the key point is the correct inclusion of the skineffect in its modeling considering the range of frequencies statedabove, as in a PWM motor drive system the voltage frequencyspectra starts with the low values associated to the motor speed,includes the switching harmonics and go up to the highest fre-quency components associated to the pulse rise time, situated inthe MHz range. Since the skin effect in the earth-return path isvery intense [7], the correct prediction of common-mode cur-rents is also dependent on a model that takes into account thelarge frequency spectra. Besides, the transient overvoltage os-cillations at motor terminals can go from some kHz (as in thecase of submarine systems) up to a few MHz, depending on thecable length. Thus, the model must correctly represent the resis-

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DE PAULA et al.: METHODOLOGY FOR CABLE MODELING AND SIMULATION FOR HIGH-FREQUENCY PHENOMENA STUDIES 745

tance and inductance of the cable in this wide frequency rangein order to assure an accurate reproduction of the transient over-voltages in terms of frequency, peak values, shape and attenua-tion of the voltage oscillations.

Several works attempted to reach this goal, but, in general, allof them present some limitation or disadvantage. Some are ableto correctly represent the cable parameter dependency with fre-quency only in a limited frequency range [11], [12], being thenappropriate for applications with very long leads only, as for ex-ample those found in submarine systems. Others involve com-plex mathematical procedures [13], [14], being of hard compre-hension for the machine and drive engineer. An efficient modelis presented in [8], but it requires measurements to calculate thenecessary parameters and that represents a serious limitation.Fixed-parameter models with resistance and inductance valueschosen at high frequencies (generally the cable resonance fre-quency) were used in [15], [16]. Despite their good represen-tation of the overvoltage amplitude damping, they fail to ac-curately reproduce the attenuation of all other components ex-isting in the inverter output voltage. The simplified modeling ofthe cable just by its lumped capacitance, as in [17], can onlybe applied in drive systems that employ a step-up transformerat the inverter output. This is typically the case of submarineand underground mining long cable motor drives, where themain resonance occurs between the transformer inductance andthe cable capacitance and not within the cable L–C parameters,which makes this strategy inadequate for study of typical in-dustrial drive systems. The method proposed in [18] is not verypractical, since the parameters of the model are calculated ontrial and error basis. In [19], the values of the various elementsof the circuit were obtained by means of a mathematical solvertool and the frequency range analyzed (starting at 100 kHz) isnot suitable for motor drives studies. The difficulty in obtainingthe model parameters was observed in [20], but no solution wasproposed, neither was shown how to determine the values usedin the work.

The methodology presented in the present work overcomesall the limitation and disadvantages mentioned above. It issimple, accurate and fully adequate to the requirements ofmotor drives application. It is named “N-Branch” model andis based in a ladder-type network. Initially developed in [21]for a limited range of frequencies up to fiftieth harmonic orderand used in power quality studies, the modeling techniquehas been improved to consider the skin effect in the requiredfrequency range. The paper also shows a new equivalent circuitfor the prediction of common-mode currents. This equivalentcircuit, which uses the “n-branch” cable model, does not re-quire modal transformation or any other complex mathematicalprocedure, and is therefore simpler than existing techniques.As will be shown, the prediction of both differential voltagesand common-mode currents is achieved in the same simulationusing the proposed circuit.

II. “N-BRANCH” CABLE MODEL

To include the parameters frequency dependence, the cablecan be represented as an association of infinite shunt-connectedconcentric tubular sub-conductors (Fig. 1), each one with itsown value of inductance and resistance. Considering that the

Fig. 1. Transversal section of a cable sub-divided in “n” concentric tubularconductors.

current in each sub-conductor does not vary along its way, theexpressions of the voltage drop in each sub-conductor, includingmutual coupling, lead to an equivalent circuit with infinite R–Lbranches [21] as shown in Fig. 1. Because infinite number ofbranches is not practical, a simpler representation is in demand.The simplification consists in defining a finite number of sub-conductors (branches) for the model. This circuit, referred toas the “n-branch model,” comprises for each branch, R–L el-ements that do not vary with frequency, but are properly con-nected resulting in an equivalent impedance that represents thecable resistance and inductance frequency variation [Fig. 2(a)].The problem consists in the determination of the n-branch cir-cuit parameters, from a given set of resistance and inductance

cable parameters known for different frequencies.The procedure is summarized as follows.

Considering the particular case of a two-branch model as rep-resented in Fig. 2(b), the model parameters are determined an-alytically [21] as

(1)

(2)

(3)

(4)

(5)

(6)

where:— are the resistance and inductance of the

first and second branches of the model, respectively;— are the input data, comprising the re-

sistance and inductance of the cable known, at frequenciesand respectively. As seen, from (1)–(6), with two

known values of resistance and inductance of the cable forthe two corresponding frequencies, the parameters of thetwo branch model are analytically determined.

A better representation can be obtained if more than two setsof cable parameters are available. If “n” pairs of cable param-eters are known for “n” different frequencies, a cir-cuit with up to “n” branches can be drawn and its parameterscalculated. However, the calculation of R-L circuit parameterswith three or more branches cannot be made in the analytic wayshown above. It is achieved using the equations for two branch


Fig. 2. (a) “N-Branch” circuit and the corresponding equivalent R Limpedance. (b) Two-branch model.

model and an additional procedure called “sequential circuitorder reduction.” It is explained as follows.

1) From the data set consisting of “n” pairs of resistance/in-ductance for the “n” corresponding frequencies, twoof them ( and ), related to thehighest pair of frequencies ( and ), are taken. Then,the parameters of the n-branch model corresponding to theexternal branch are evaluated through (1)–(4).

2) To proceed with the model determination, the n-branchmodel is represented by the equivalent circuit shown inFig. 3. Note that this equivalent circuit is not a two branchmodel. It is composed by the external (nth) branch andan equivalent impedance ( and ) that rep-resents the association of all remaining n-1 branches forall frequencies. That is, the total impedance of the cir-cuit shown in Fig. 3 matches the known data cable in allgiven frequencies. Therefore, it is possible to determine theequivalent parameters of the inner branches association for

Fig. 3. Cable equivalent circuit with “n” branches.

the frequencies , by successivelyapplying (7) and (8) derived from the circuit of Fig. 3. Itcan be interpreted as having the external tubular sub-con-ductor of the n-branch extracted, and calculating the newset of parameters for the equivalent cable represented byn-1 branches

(7)

(8)

3) With the n-1 pairs of and determined,the inner equivalent conductor formed by n-1 branches iscompletely characterized. The n-1 set of parameters of theequivalent conductor can now be used to determine the(n-1)th branch in the same way as done for the n branchesconductor. Steps “2)” and “3)” are repeated until a data setwith only two frequencies is reached, when the inductiveand resistive parameters of the first branch ( and )are evaluated using (5) and (6).

When exercising this methodology, a concern that arisesis related to the optimal number of branches to represent agiven cable configuration. One will find that not always thehighest number of branches is the best solution. In fact, theoptimum solution is the one that requires the smallest numberof branches while delivering an accurate representation of Rand L frequency dependency in the whole range of frequenciesinvolved in the problem. If an excessive number of branchesis used, it can happen that some branch parameters come upwith negative values, even so, the equivalent resistance andinductance of the model are always positive values. Inves-tigations showed that adjusting the frequency range of thedata set and/or reducing the number of branches lead only topositive parameters, which is more meaningful in the analysis.It was verified that a data set given with proportionally spacedfrequencies almostalways lead to a good modeling.

A. Obtaining the Cable Parameters for Different Frequencies

There are at least four options to determine the cable R–Lparameters at different frequencies to make the input dataset for the modeling: 1) using the “Cable Constants” routineof EMTP platform, 2) by means of finite element analysis


Fig. 4. Resistance (top) and inductance (bottom) variation in relation to fre-quency. Results from models with four, five, and six branches.

[16], [19] (where the proximity effect can also be considered);3) direct measurements and iv) using the method proposedin [22], which uses very simple equations. A remark is to bemade about direct measurements, which are often difficultand/or unavailable due to very high cost of measuring equip-ment, and aside from that, requires special attention to avoidexciting the cable natural frequencies during measurements,which would lead to mistaken parameters. Using the equationsand methodology proposed in [22] it is possible to obtain thecable resistance and inductance matrixes for a given cablearrangement at any desired frequency. For this, the followinginformation is required: a) electrical resistivity and diameter ofthe conductors, b) dielectric permittivity, magnetic permeabilityand thickness of the insulation, c) resistivity of the ground, andd) distance between wires and from them to the ground plane(see Appendix I). If the cable is shielded, the parameters of theshielding must also be provided. The equations of this methodwere incorporated in the “N-branch” model software routine,making it possible to directly obtain the parameters for thedesired “N-Branch” models by entering the cable constructiveand geometric data, along with the desired frequency rangeand the number of branches to be used in the models. Theprogram also outputs the equivalent impedance curves andthe corresponding errors for the various “n-branch” modelsanalyzed, as a function of the frequency.

Figs. 4 and 5 show the results for the resistance and induc-tance representation of a 3 4 mm cable with the proposedmodeling, for four-, five-, and six-branch models, from 20 Hzup to 2 MHz. As can be seen from Fig. 5, the maximum er-rors obtained for the cable resistance and inductance are 4.4%at 2 MHz and 0.25% at 500 kHz, respectively, in the case of asix-branch model.

B. Wave Propagation Phenomena: Lumped versus DistributedParameter Modeling

The computational analysis of the transient overvoltages re-quires models that represent the frequency dependency and alsothe distributed nature of the cable parameters, reproducing the

Fig. 5. Errors obtained for the model equivalent resistance (top) and inductance(bottom). Results from models with four, five, and six branches.

wave propagation and reflection phenomena that occur in thecable endings and the associated voltage/current oscillations.Distributed-parameter models present this feature, but the in-clusion of its dependence with frequency, in the time domain, isvery complex to be implemented [13], [14].

An alternative solution is to use a lumped-parameter modelassociating its cells in a number high enough to “capture” thepropagation phenomenon, keeping in mind that these cellsmust comprise parameter-dependency with frequency. The“n-branch” model fulfills this requirement. The higher thenumber of lumped segments, the more accurately the dis-tributed nature of the cable parameters is reproduced. However,an excessive quantity of “pi” cells leads to a prohibitive com-putational effort, and thus a reasonable number must be found.The key point to define the appropriate number of circuits to becascade-associated is the relation between the wavelength of theelectric field established in the conductor and the length of thecable. The length of each lumped segment should be calculatedin order to make the electric field variation negligible or rea-sonably small in each “pi” circuit. The question becomes howto define the wavelength, since the PWM voltage waveform isa summation of many harmonic components, all contributingto the formation of the electric field. The highest-frequencycomponents of a PWM voltage are associated to the pulse risetime, and a pulse equivalent frequency “ ” can be defined[24]

(9)

The corresponding wavelength “ ” can be calculated by

(10)

(11)

where “ ” is the pulse rise time, “ ” is the wave propagationspeed and “ ” and “ ” are the cable positive-sequenceinductance and capacitance. “ ” was obtained from the cable


Fig. 6. Transient overvoltage simulated results obtained by modeling the motoras a high ohmic value resistor. The cable was modeled with 60 Hz-fixed param-eters (a), 500 kHz-fixed parameters (b) and by the “six-Branch” model (c).

impedance matrixes calculated using [22], while “ ” wasdetermined by the ATP “Cable Constants” routine.

In the case of the 3 4 mm cable utilized in this section,the propagation speed is around 1.3 10 m/s and the corre-sponding “ ”, for a pulse rise time of 80 ns, is about 33 me-ters. Let “ ” be defined as the relation between the length ofthe lumped cells and the electric field wavelength. By means ofseveral preliminary simulations, it was verified that a reasonablenumber for “ ” would be around 0.01 or 0.02, leading to “pi”circuits between 33 and 66 cm. It was then decided to use twolumped cells to represent one meter of cable length.

III. SIMULATIONS AND MEASUREMENTS OF

THE TRANSIENT OVERVOLTAGES

In order to validate the proposed modeling technique, simula-tions were carried out using ATP platform. The system analyzedcomprises a PWM inverter, a 3 4 mm , 95 meter interconnec-tion cable and a 2 HP three-phase induction motor. The PWMvoltage pulse is approximated as a trapezoidal shape. The cableis represented by a cascade-connection of “pi” circuits, whichseries impedance is the six-Branch model shown in Section II,whose parameters are shown in Table I (see Appendix II). Each“pi” circuit represents 0.5 m of cable length; therefore, the so-lution of the problem involves 190 “pi” cells. The cable shuntcapacitance is 102,0 pF/m.

In order to observe the attenuation of the oscillations dueonly to the effect of the cable resistance, the motor was re-placed by a resistor of high ohmic value, resulting in a reflec-tion coefficient near to 1. Fig. 6(a) and (b) show results whenthe cable is represented by fixed parameters. In Fig. 6(a) the pa-rameters of the cable at 60 Hz were chosen, while in Fig. 6(b)the resistance and inductance refer to the cable resonant fre-quency, around 500 kHz. Fig. 6(c) shows the result obtainedby using the proposed “n-branch” model with 6 branches. Itis observed that the modeling with 60 Hz-fixed parameters re-sulted in very low attenuation of the transient overvoltage, faraway from the expected result, represented by the 500 kHz-fixedparameter model. On the other hand, the “six-Branch” model

Fig. 7. Simulation results obtained with (a) fixed-parameter model at cable res-onance frequency and (b) “N-Branch” model.

produced a voltage amplitude attenuation very close to that ob-tained with 500 kHz-fixed parameter model. Additionally, theproposed model was the only one that could reproduce the pro-gressive “rounding” in the pulse edge during its propagationsthrough the cable, as can be seen in Fig. 7, showing that themodel is able to provide the correct attenuation correspondingto each one of the harmonics that compose the PWM pulse. Forthe same reason, the undesired spurious oscillations overlappedto the main oscillation due to parameter lumpiness [7] were alsoeliminated.

Fig. 8 shows the simulation and experimental results of thetransient overvoltage. Curves marked “A” and “B” correspondto the simulated and measured voltage at the inverter terminals,respectively. Curve marked “C” shows the simulated voltageat the cable terminals when the load is a resistor, whose ohmicvalue corresponds to the motor characteristic impedance at500 kHz. Curves marked “D” (measured) and “E” (simulated)refer to the voltage at the cable terminals when the load isthe 2 HP induction motor. The simulation shows that thecascade association of “N-Branch”-“pi” circuits led to a goodrepresentation of the transient overvoltage, showing goodagreement between simulated and experimental results. Themotor model [8] used in this simulation takes into account


Fig. 8. Voltage at the inverter terminals: simulated (curve A) and experimental(curve B). Voltage at motor terminals: simulated representing the motor as aresistor (C), simulated with a high-frequency model for the motor (E) and ex-perimentally obtained (D).

Fig. 9. (a) Zero sequence and (b) positive sequence representation of the cable“pi” circuit.

the winding-to-ground and winding turn-to-turn capacitivecouplings, which, combined to the machine inductance, definethe frequency response of the motor input impedance.

Thus, the characteristic impedance of the motor is not con-stant, but frequency-dependent, and so is the reflection coeffi-cient at motor terminals. Besides, these intrinsic capacitances ofthe motor play an important role on the dynamics of the pulsereflection phenomena, affecting the shape and the frequency ofthe voltage oscillations, as observed in Fig. 8. It is thus recom-mended that an appropriate model for the motor be also used tobring additional accuracy to the simulation of the transient overvoltage phenomenon.

IV. INCLUSION OF THE COMMON MODE CURRENT PATH

The determination of differential and common mode quan-tities of an electric system is usually performed by means ofseparated circuits, where the three-phase voltage is decom-posed in its differential and common-mode components andthen applied to the corresponding circuits. The modal quantitiesare thus obtained and latter recombined for the calculation ofthe phase quantities. This methodology, based on the modalanalysis, uses a transformation matrix, which becomes fre-quency-dependent in the case of the phenomena studied here.

Fig. 10. Proposed circuit for common and differential mode voltage and currentsimultaneous calculation.

Some methods overcome this drawback [13], [14], but theyare quite complex. In this context, this section shows a newmethod, that makes use of the “N-Branch” cable model, for thesimultaneous determination of the differential mode (transientovervoltages) and common mode (“earth” currents) high-fre-quency phenomena in a PWM motor drive system, by meansof a single circuit, with no complex mathematical procedures.

The cable series impedance and shunt capacitance matrixescan be written as

(12)

(13)

where the on-diagonal elements in (12)refer to the series self-impedance of the loop formed by theconductor “ ” and the ground return. The off-diagonal ele-ments correspond to the series mutualimpedance between conductors “ ” and “ ”, and determine thelongitudinally induced voltage in conductor “ ” if a currentflows in conductor “ ”, and vice-versa. The resistive terms inmutual coupling are introduced by the presence of the ground.

Reporting to (13), the on-diagonal elements “ ” repre-sent the sum of the conductor “ ” capacitance to ground and toall other conductors, while the off-diagonal “ ” refers to themutual capacitance between conductors “ ” and “ ,” with a neg-ative sign.

Considering a continuously transposed cable, all the on-di-agonal elements in (12) present the same value and will be de-noted by “ ” . Likewise, the off-diagonals will bewritten as “ ” . In the case of the capacitances,“ ” will represent the conductor capacitance to ground and“ ” the capacitance between conductors . The


Fig. 11. Experimental setup for common mode current measurement.

positive and zero-sequence values for the cable resistance, in-ductance and capacitance can be written as

(14)

(15)

(16)

(17)

(18)

(19)

where the subscripts “pos” and “zero” refer to positive and zerosequence.

From (14)–(19), the “pi” circuits representing the positiveand zero-sequence parameters of the cable can be presentedas shown in Fig. 9. In a three-phase cable system, the posi-tive and zero-sequence circuits correspond to the differentialand common-mode circulation paths, respectively. Consideringa single “pi” circuit, the voltage drop for the positive and zerosequence, according to Fig. 9, are

(20)

(21)

Replacing (14)–(19) in (20) and (21)

(22)

(23)

Comparing expressions (22) and (23), it is observed that thepositive and zero-sequence currents, circulating in the corre-sponding circuits, flow through different impedances. Both ofthem circulate by “ ” and “ ;” but “ ” does not flowthrough “ .” Besides, the voltage drop on “ ” producedby “ ” and “ ” are multiplied by different factors. Theidea is to elaborate a single special circuit, for the phase cur-rent circulation, where its differential and common mode com-ponents (“ ” and “ ,” respectively) flow through only therespective impedances, correctly producing the voltage dropsaccording to (22) and (23). To accomplish this, it is necessary

to include in (22) an impedance equal to and a capaci-tance that cancels the term

(24)

The proposed circuit configuration is shown in Fig. 10, where,by means of a single decoupled three-phase circuit, it is pos-sible to directly determinate the common and differential modevoltage and currents, simultaneously.

It is to be observed that the value found in (24) must bemultiplied by three in order to make the voltage drop in “ ”equal to “ ”, and not three times this value. For thesame reason, the other impedance that is included is “ ”,and not its triple, assuring a voltage drop of “ ”. Ana-lyzing the proposed circuit, it is seen that the positive and zerosequence currents, flowing through it, produce voltage dropsequivalent to those expressed in (22) and (23). The mutual termsof the cable impedance matrix (“ ”) refer to the earth re-turn path, in which the skin effect is very intense. Thus, bothcable “ ” and “ ” impedances must be represented by“n-branch” models. Following the same methodology describedin the previous section, a new simulation was conducted, nowfocusing on the determination of the common-mode current.The cable was modeled by a cascade-association of the circuitpresented in Fig. 10, using the same methodology described inSection II. The connection between the converter and the motoris made by a 100 m, 4 4 mm cable, where the fourth wireworks as the earth return path, as shown in Fig. 11. The fre-quency converter is supplied by a modified 15 kVA distributiontransformer, Y-Y, 220/220 V, being the fourth wire of the cableconnected to the neutral point of the secondary winding, wherethe current clamp was connected. Fig. 11 illustrates the capac-itive couplings to ground existing in the system under investi-gation. In the converter, the path is established by the capac-itance between the power switches and the heat sink, while inthe motor the couplings are through the stator windings to frameand to the rotor axis. Since both the motor and the converter arenot connected to the earth conductor, only the current flowingto ground through the cable distributed capacitances was mea-sured. Regarding the current measurements, 1 V corresponds to0.61 A.

Fig. 11 shows that the common mode path comprises thezero-sequence impedance of the cable and also the impedanceof the transformer, which must be included in the modeling. InFig. 12, measurements show a common mode current of 90 kHzwith peak values within 1.4 and 1.85 A. In the simulation, whose


Fig. 12. Common mode voltage (upper) and the corresponding current(bottom). Experimental results (1 V! 0.61 A).

Fig. 13. Common mode voltage (upper curve) and the corresponding current(bottom). Simulated results.

results are shown in Fig. 13, the transformer impedance was rep-resented by a 0.1 mH inductance.

As seen, the peak value of the common-mode current oscil-lations situated in the 1.2–1.75 A range, while its frequency isaround 120 kHz, which can be considered quite satisfactory re-sults. Using a more sophisticated model for the transformer, ap-propriate for high frequencies, better results can be expected.

V. CONCLUSION

This work presented an efficient methodology for cable mod-eling and simulation, suitable for the study of the high frequencyphenomena present in PWM motor drive systems. The modelis able to correctly represent the cable parameter variation in awide frequency range, up to MHz level, thus being appropriatefor this application. It was observed that the cascade-associated“N-Branch”-“pi” circuits reproduced quite well the wave prop-agation and reflection phenomena. Besides, the simulation suc-cessfully represented the amplitude, shape and attenuation ofthe voltage oscillations, characterizing well the transient differ-ential overvoltages present in PWM drive systems. A new equiv-alent circuit for determining the common-mode currents wasproposed. The methodology is simple and efficient and does not

TABLE IPARAMETERS OF THE SIX-BRANCH MODEL OF THE CABLE USED IN SECTION III

require complex mathematical transformations. The results ob-tained from the proposed circuit showed good agreement whencompared to measurements.

APPENDIX I

In addition to the geometric characteristics of the cablesystem under analysis, some more data is needed for the cableparameter determination, as mentioned in Section II. Someinformation is given bellow:

• Relative permittivity : for PVC insulation layers,; for PE, ; for XLPE, ; for EPR,

.• Relative permeability : since the insulation materials

are considered diamagnetic, .• Resistivity of the conductor , at 20 C: for copper,

.m; for aluminum, .m.• Resistivity of the ground : depends strongly on the soil

characteristics, ranging from 1 .m (wet soil) to about 10k .m (rock). The resistivity of sea water lies between 0.1and 1.0 .m [23]. Information about this parameter canbe obtained from the project of the facility ground grid;if this is not available, it is recommended to use

.m, an average value obtained from a great numberof measurements [7].

APPENDIX II

See Table I.

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Helder de Paula was born in Uberlândia, Brazil, onDecember 27, 1975. He received the B.Sc., M.Sc.,and Ph.D. degrees in electrical engineering from theUniversidade Federal de Uberlândia, Brazil, in 1998,2001, and 2005, respectively.

In 2006, he joined the Electrical EngineeringDepartment, Universidade Federal de Minas Gerais(UFMG), Minas Gerais, Brazil, as a Lecturer andmember of the Industry Applications Laboratory.He has worked in R&D projects on cable and linemodeling for high frequency studies. His main

interests are motor drives, electromagnetic compatibility and quality of power.

Darizon Alves de Andrade (M’87) was born inMonte Alegre de Minas, Brazil, on June 23, 1956.He received the B.Sc. and M.Sc. degrees fromthe Universidade Federal de Uberlândia (UFU),Uberlândia, Brazil, in 1980 and 1987, respectively,and the Ph.D. degree from the University of Leeds,Leeds, U.K., in 1994, all in electrical engineering.

During 2000, he was a Visiting Scholar with theMotion Control Group, Bradley Department of Elec-trical Engineering, Virginia Polytechnic Institute andState University, Blacksburg, where he carried out re-

search on new strategies for modeling SRMs. From 1980 to 1985, he was a Lec-turer with the Faculdade de Engenharia de Ituiutaba, Ituiutaba, Brazil. In 1985,he joined the Faculdade de Engenharia Elétrica, UFU, where he is currently aSenior Lecturer. His teaching, research, and consulting interests and activitiesare related to design, simulation, and control techniques associated with motioncontrol of electromechanical energy converter devices and new developments inquality of power. He has authored and coauthored several papers in these areas.

Marcelo Lynce Ribeiro Chaves was born in Itu-iutaba, Brazil, on October 03, 1951. He receivedthe B.Sc. and M.Sc. degrees from the UniversidadeFederal de Uberlândia (UFU), Brazil, in 1975 and1985, respectively, and the Ph.D. degree from theUniversidade Estadual de Campinas (UNICAMP),Brazil, in 1995, all in electrical engineering.

He is a Senior Lecturer with the Faculdadede Engenharia Elétrica, Universidade Federal deUberlândia. His main interests are electromagnetictransients in power systems, insulation coordination,

motor drives, and quality of power.

José Luis Domingos received the M.Sc. and Ph.D.degrees in electrical engineering from the Univer-sidade Federal de Uberlândia, Brazil, in 1998 and2003, respectively.

In 1992, he became a Lecturer with the CentroFederal de Educação Tecnológica de Goiás wherehe teaches electronics related subjects. Since 2003,he has been an Assistant Lecturer at the School ofElectrical Engineering, Universidade Católica deGoiás. Presently, he is actively involved in switchedreluctance machines and drive system research, and

his research interests include solid-state power conditioning and motor drivedevelopment to automation processes.

Marcos Antônio Arantes de Freitas was born inMonte Alegre de Minas, Brazil, in 1970. He receivedthe M.Sc. and Ph.D. degrees in electrical engi-neering from The Federal University of Uberlandia(UFU), Uberlandia (MG), Brazil, in 1998 and 2002,respectively.

He is currently with the Industry Division,Federal Center of Technological Education ofGoiás/UnED-Jataí, where he has been working toestablish research and education activities in industryapplication of power electronics converters and high

power drives. His research interests include high-frequency power conversion,active power-factor correction techniques, motor drives, multipulse rectifiers,and clean power applications.

methodology for cable modeling and simulation for high-frequency phenomena studies in pwm motor...

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