input current interharmonics of variable-speed drives due to motor current imbalance

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 4, OCTOBER 2010 2797

Input Current Interharmonics of Variable-SpeedDrives due to Motor Current Imbalance

Duro Basic

Abstract—This paper presents theoretical, simulation, andexperimental investigations of input current interharmonics inmodern variable-speed drives based on voltage source invertersand diode input rectifiers that are caused by motor currentimbalance. It investigates how a disturbance in the inverter dcside current created by unbalanced motor currents propagatesfrom the inverter to the rectifier stage and appears as variablefrequency interharmonic distortion in the rectifier input cur-rents. Particular emphasis is given to theoretical analysis of thefrequency transformations created by the inverter and rectifierstages and the magnification of the disturbance current causedby parallel resonance in the drive dc bus circuit. The theoreticalresults are confirmed by simulation and experimental results.They demonstrate that motor current imbalance can be respon-sible for high non-characteristic inter-harmonic distortion in thedrive input currents. A calculation example outlines a procedurefor estimation of the drive input current interharmonic distortionbased on measurements of the motor currents. The paper shouldbe helpful for people investigating the origin of problems causedby variable frequency interharmonic currents.

Index Terms—Harmonic transfer, imbalance, interharmonics,power quality (PQ), switching function, voltage-source inverter.

NOMENCLATURE

, DC bus capacitor capacitance and resistance.

, Power system input frequency.

, Drive output frequency.

Ripple current of inverter and rectifier side.

DC bus currents.

Motor current positive and negative-sequencecurrent components.

Drive input (supply) currents.

DC component of dc bus current.

Inverter and rectifier-side dc bus currents.

, Motor and source-current space vectors.

Drive output (motor) currents.

Current magnification factor.

AC input choke inductance and resistance.

Manuscript received May 22, 2009. First published April 12, 2010; currentversion published September 22, 2010. Paper no. TPWRD-00386-2009.

The author is with the Schneider Electric-Schneider Toshiba Inverter, Pacysur Eure 27120, France (e-mail: duro.basicschneider-electric.com).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRD.2010.2044811

Drive dc choke inductance and resistance.

Power system impedance parameters.

Inverter modulation index.

Diode dynamic resistance.

Rectifier switching functions.

Inverter switching functions.

Inverter dc bus voltage.

Conjugated-complex variables.

I. INTRODUCTION

I NTERHARMONICS are spectral components observed inpower frequency voltages and currents at frequencies which

are not an integer of the fundamental [1], [2]. Subharmonicsare a special case of interharmonics which appear at frequen-cies lower than the power system frequency including dc [1].Interharmonics are usually generated when two ac systems op-erating at different frequencies (input and output fre-quency) are mutually interconnected via some static frequencychanger [1]–[8]. For example, the current ripple harmonicallyrelated to the converter output frequency can be transferred tothe input side where it appears at a linear combination of theinput and output frequencies which is not generally harmoni-cally related to the converter input frequency. As the periodsof interharmonic components are not synchronous with the fun-damental frequency cycle, the amplitude of the converter inputcurrents pulsates (beats) with a ripple running across the tops ofthe current pulses.

Particular concerns in relation to interharmonics are powerfactor correction capacitors and tuned filters [1]–[3]. Capaci-tors normally create a parallel resonance (antiresonance) withthe power system impedance. The result is a potential ampli-fication of current (and voltage) distortion of spectral compo-nents found close to the parallel resonance frequency. Even ifthe filter/capacitor bank is tuned to avoid parallel resonance atinteger harmonic frequencies, the resonance point is not elim-inated, but just shifts typically below the lowest characteristicharmonic (5th). Since the interharmonics usually drift across theharmonic spectrum, changing in amplitude and frequency, theycan align with the antiresonant frequency. In these situations,large-current magnification of the drive current interharmonicdistortion found in this region can be created, causing severevoltage flicker problems [11]. In addition in some situations, atparticular drive output frequencies (motor speeds), the interhar-monic frequency can reduce to zero, causing dc premagnetiza-tion current and overheating of transformers [3].

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2798 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 4, OCTOBER 2010

Fig. 1. Equivalent circuit of a variable-speed drive for system analysis.

A major source of interharmonics is a direct frequency changeror cycloconverter. The cycloconverter output voltage is formedfromselectedsegments of the input three-phasevoltagesandcon-sequently, the input current in each phase consists of the corre-sponding segments of the output currents. The harmonic contentof the cycloconverter input current can be derived by expressingtheconverterswitchingfunctionsasaphase-modulatedharmonicseries.Thispowerfulconcept isexplainedindetail inPelly’sbook[4] forvariouscycloconverterconfigurations(pulsenumber)withcosine-wave crossing control.

Various two-stage converters with separate ac/dc and dc/acconverters interconnected via a dc link (current and voltagesource converters) are also sources of interharmonics becausethe two ac systems are not perfectly decoupled by the dclink [1]–[10]. Among them, variable-speed drives based onpulse-width-modulated (PWM) voltage-source inverters areincreasingly being recognized as potential sources of inter-harmonics [5]–[11]. The disturbance current injected by theinverter into the dc link is generally low in well-designed PWMinverters. The inverter disturbance current and related rectifierinput current interharmonics are typically generated if theinverter is operated in overmodulation or if the inverter loadis unbalanced [8]. For the frequency transformation analysisthrough ac/dc/ac converters, the switching function concept isa commonly used tool [6], [7].

A particular problem with PWM drives with large dc bus ca-pacitance is that propagation of the inverter disturbance cur-rent through the converter is affected by parallel resonance cre-ated by the interaction of the bus capacitor and power systemand/or filter inductances [6], [7]. The dc bus parallel resonancecan considerably amplify the disturbance current on the rectifierside, causing excessive noncharacteristic distortion in the recti-fier input currents.

In this paper, we systematically investigate interharmonics inthe input currents of PWM drives caused exclusively by motorcurrent imbalance. First, we examine the interharmonic gener-ation by using the conventional switching function concept andtheoretically derive the frequency transformations caused by theinverter and rectifier stages. The convolution concept in the fre-quency domain is also introduced which can be easily appliedfor fast predictions of harmonic spectra of relevant three-phasespace vectors and/or dc (scalar) variables. Further, the dc-bus

circuit parallel resonance and corresponding current magnifica-tion of the inverter disturbance current during its propagationfrom the inverter to rectifier stage are analyzed. Related to this,an approximate approach for taking the effect of the rectifierac-side inductors into account on the resonance damping is used.The theoretical analysis is illustrated and supported by severalsimulation and experimental results. Finally, a practical proce-dure based on the motor current measurements for the estima-tion of the drive-input current interharmonics is outlined.

II. HARMONIC TRANSFER THROUGH INVERTER

In this section, we will investigate the transfer of the motorcurrent imbalance through the inverter stage. The system modeland major parameters used in this study are shown in Fig. 1(an inductor of 0.125 mH was added to emulate motor-windingimbalance).

The inverter is used for synthesizing symmetrical sinusoidalthree-phase voltages between the output terminals atsome arbitrary output frequency from the constant dc busvoltage by using some PWM modulation techniques. If thePWM switching harmonics in the inverter output voltages areomitted from this consideration (only low-frequency compo-nents are of interest in this study), the inverter output to—busaverage (low frequency) voltages are

(1)

where is the modulation index defined as(dc bus voltage ripple is neglected) and are the so-calledinverter switching functions defined as

(2)

BASIC: INPUT CURRENT INTERHARMONICS OF VARIABLE-SPEED DRIVES 2799

The switching function values are restricted to between 0 and1 and represent average relative time within the PWM periodwhen the outputs are connected to the positive dc bus rail (inthe rest of period they are connected to the negative dc bus rail).

In (2), the conventional sinusoidal PWM modulation is as-sumed. Despite this, the results obtained are universal and valideven when different modulation methods are used. It should benoted that the effect of any constant offset in the inverter outputto—bus voltage or common-mode voltages deliberately intro-duced in the modulator to modify the PWM switching patternis cancelled out in the line-line voltages so that they will not beaffected (in average) by the modulation method.

According to the average model (neglecting the PWMswitching ripple), the inverter currents will be sinusoidal. How-ever in the general case, if the motor connected on the inverterterminals draws unbalanced phase currents , the motorcurrents will have, in addition to the positive-sequence currentcomponent , a certain amount of the negative-sequencecurrent

(3)

The current components have some arbitrary phase shifts (and ). For the positive-sequence current , the powerfactor at nominal load is between 0.8–0.85 while is notthat important because, theoretically, there is not any negative-sequence voltage generated and, thus, no active power transferis possible by the negative-sequence system (just oscillations ofpower).

All three-phase currents will be combined at the bus andbus terminals in a common inverter-side dc bus current. As theswitching functions define the average time when the switchesare connected for the bus, the average bus current can befound as a sum of the averaged phase currents

(4)

If we combine (2) and (3) in (4), it is easy to show that the re-sulting rail inverter-side current is (5), shown at the bottomof the page. It has two distinctive components as follows.

Fig. 2. Equivalent circuit for the analysis of disturbance current transfer fromthe inverter to rectifier side.

1) The dc component in (5) is linked to the fundamentalpositive-sequence current which is responsible for activepower transfer

(6)

2) The ac component is related to the presence of themotor current unbalance. It does not transfer any activeenergy but creates fluctuations of the energy at twice theinverter frequency . In other words, it appears as adisturbance current agitating the inverter dc bus subsystem

(7)

Both current components depend linearly on the modulationindex . Thus, for a motor working with constant torque (cur-rent) and constant current imbalance, the inverter current distur-bance will be lower at lower motor speeds (lower modulationindices).

III. HARMONIC TRANSFER THROUGH THE DC BUS SUBSYSTEM

The current transfer from the inverter dc side to the rectifier dcside can be approximately modeled with the equivalent circuit inFig. 2 [8]. The inverter dc current component directly transfersto the rectifier dc side, as the capacitor dc current componentis equal to zero. The disturbance component in (5) at twice theinverter frequency excites the parallel (antiresonant) LC circuitin Fig. 2.

In first approximation, we can examine the current transferwith equivalent dc-link inductance and damping resistance

, assuming that the rectifier operates in continuous conduc-tion mode (from Fig. 1, diode dynamic resistance is also takeninto account). The last term in (9) is introduced to account forthe voltage drop caused by diode commutations (proportional tothe source reactance and dc current )

(8)

(9)

(5)


Fig. 3. Current magnification factor due to dc bus parallel resonance.

The rectifier dc-side current can be found from the current divi-sion rule

(10)

where

(11)

is defined as the current magnification factor and is thephase shift introduced by the dc link. Our simulations show thatthis model predicts unrealistically low damping and very highcurrent magnifications near the parallel resonance point if onlyohmic resistances are taken into account in and the diodecommutation effects are neglected [Fig. 3, curve (a)]. Very goodagreement with simulation results is obtained if the commuta-tion voltage drop is accounted for by adding an equivalent com-mutation resistance in the damping resistance [last item in(9)]. This approach is justified if the dc bus current fluctuatesat the frequency which is much smaller than the commutationfrequency. Thus, as the averaged commutation voltage drop isproportional to the instantaneous dc bus current, it can be mod-eled as an additional equivalent resistance introduced in the dcbus from the supply side. With this correction frequency plot ofthe current magnification factor calculated from (11) and shownin Fig. 3 [curve (b)], it is realistic even near the resonance fre-quency. It also agrees well with the simulation results (diamondmarkers) presented later in Section VII.

The disturbance current transferred on the rectifier side canbe expressed now as

(12)

Its rms value and phase are

(13)

(14)

where is the current magnification factor at the frequency.

Obviously at the frequencies when the denominator in (11)high magnifications of the inverter disturbance

Fig. 4. Rectifier switching functions.

current are created by the dc bus and large currents can appearon the rectifier side. This critical frequency is the dc-bus parallelresonance frequency

(15)

We can conclude that due to the frequency-dependent currenttransfer from the inverter side to the rectifier side, when the driveis operated at a variable frequency ranging from ,the rectifier-side current can be considerably increased if the dcbus resonance frequency is found somewhere within the

range.

IV. HARMONIC TRANSFER THROUGH THE RECTIFIER

To explain harmonic current transfer from the rectifier dcside to the input ac side, we have to use the so-called rectifierswitching functions . Assuming that the rectifier worksin continuous conduction mode and that the commutation ef-fects are neglected, every diode/thyristor conducts withinperiod distributing the dc current evenly among all three inputphases. The rectifier input current is created by combining twosegments of the rectifier dc current shifted by (one from thecathode and one from anode group). The segments can be de-fined by multiplying the rectifier dc current with a square-waveswitching function as shown in Fig. 4.

By using the rectifier switching function concept, the rectifierinput currents are

(16)

The rectifier switching functions are nonsinusoidal and can besplit into Fourier’s series as follows:

(17)


where

(18)

The most important term in (18) is the fundamental harmonicwith an amplitude of

(19)

To simplify the analysis, we will further approximate theswitching functions with their fundamental harmonics only

(20)

With this simplification only, the principal interharmonic familywith two sidebands around the power system fundamental com-ponent can be derived. Higher order interharmonic families cre-ated as sidebands around the characteristic rectifier harmonicscan be derived in similar way by using higher order terms ofthe rectifier switching functions (5th and 7th). These interhar-monics are much lower in magnitude (illustrated in Fig. 10) asthey decrease inverse proportionally to the characteristic har-monic order (19). However, in some situations, they may havethe potential to excite parallel resonance in resonant power sys-tems and, thus, cannot be always completely neglected.

From (12) and (16), the rectifier input current is

(21)

where

(22)

(23)

From (21), we can conclude that the disturbance current presentin the rectifier dc-side current will be transferred to the rectifierinput side as two symmetrical spectral components atwhose amplitudes are 55% of the original disturbance current.

For the fixed supply frequency , these spectral compo-nents can appear at arbitrary frequencies that are determined bythe drive output frequency . In the general case, these fre-quencies are not harmonically related to the fundamental power-supply frequency and, thus, they will appear as interharmonicsin the rectifier input current. For example, for 50 Hz and

50 Hz, the interharmonic familycovers the frequency range from 50 Hz to 150 Hz. It is inter-esting that for 50 Hz, this interharmonic appears as the

third harmonic (150 Hz) in the rectifier input current which isspecific for rectifier unbalanced operation. In other words, it ap-pears that the motor current imbalance reflects directly as therectifier input current imbalance.

In the same circumstances, the interharmonic familycovers the frequency range from 0 Hz to 50 Hz. Thus, this

interharmonic family appears as a subharmonic component. For25 Hz, its frequency becomes zero and it appears as a

dc offset in the rectifier currents. For 25 Hz, it changesthe sequence order and at 50 Hz, this interharmonicbecomes a negative-sequence fundamental (~50 Hz) frequencycomponent.

V. HARMONIC TRANSFER BY USING SPACE VECTOR

SPECTRA AND CONVOLUTION APPROACH

The space vector approach is a very elegant and compact wayto analyze three-phase systems because it allows us to representall three-phase quantities with a single space vector. We willstart from the inverter/motor current space vector defined as

(24)

For unbalanced currents (replacing (3) in (24)), the motor cur-rent space vector can be found to have two vectors rotating inopposite directions

(25)

where

(26)

A particular advantage of using the space vector approach isthat the frequency spectrum of three-phase currents can be ex-pressed in terms of frequency and sequence. We can attributea sign to the frequency to indicate the sequence order of par-ticular spectral components (positive or negative) as shown inFig. 5 where the space vector current spectrum has two distinc-tive components at and corresponding to the posi-tive- and negative-sequence components in (25). Compared tothe well-known two-sided spectrum of real signals with positiveand negative sides that are conjugated-complex, the harmonicspectrum of a space vector is not symmetrical nor conjugated-complex. In other words, the harmonic spectrum of an ideallybalanced motor current space vector has only a single spectralline at the positive-sequence fundamental output frequency. Thepresence of a spectral component at the negative-sequence fun-damental output frequency with unbalanced motor currents canbe considered in a generalized sense as an unwanted distortioncomponent in the motor current space vector spectrum whosepresence will be eventually reflected to the rectifier input stage.

Now, we will see how the motor current space vector spec-trum is transformed into the scalar dc current spectrum. For this,we will start from (4) which, is for convenience, repeated here

(27)


Fig. 5. Frequency transformations during harmonic propagation from the in-verter ac side to dc side.

After the corresponding cosinusoidal switching functions inare replaced by

(28)

we can obtain the following expression for the inverter dc-sidecurrent (the constant offset in is lost because motor cur-rents do not have a zero-sequence component)

(29)

The result in (29) illustrates a general rule for the mappingof the inverter output current space vector spectrum to thedouble-sided dc current spectrum. The resulting dc currentspectrum is obtained by scaling the inverter output currentspace vector spectrum by 1/2, frequency shifting for(multiplication by ) and adding a complementaryconjugated-complex spectrum. In this way, the resulting spec-trum gets the properties of the classical two-sided spectrumof real signals (dc current is scalar) that is symmetrical andconjugated-complex.

The result in (29) is not surprising. The inverter acts as a fre-quency changer (providing frequency shift from ) andthis fact is reflected in the frequency transformations in (29).When analyzing the harmonic propagation in the opposite di-rection, from the inverter ac to dc side, a frequency shift in theopposite direction has to be made . It can be under-stood as a demodulation process.

The frequency transformations in the inverter are illustratedin Fig. 5. We start from the motor current space vector spec-trum that contains a negative-sequence current component cor-responding to the motor current imbalance (25). This spectrumshould be shifted by . By adding the complementary com-plex-conjugated spectrum, the inverter dc-side current spectrum

is obtained (29). This frequency shift can be explained as theconvolution in the frequency domain of the motor current andinverter switching function space vector spectra which reflectsthe time-domain multiplications of the motor currents and in-verter switching functions (in this case, a sinusoid when PWMoperates in the linear region) in (27). The use of the convolu-tion concept is very useful when dealing with nonsinusoidalswitching functions, for example, in overmodulation. In thiscase, the motor current spectrum will be replicated around everyharmonic of the inverter switching function.

The harmonic current propagation from the rectifier dc sideto its input (ac) side can be analyzed in a similar manner. Wewill start with the rectifier input current space vector defined as

(30)

As we have mentioned in Section II, the rectifier input currentscan be obtained by multiplying the rectifier dc current by the rec-tifier switching functions. By combining (30) and (17), it is easyto show that the resulting rectifier input current space vector iscomposed of a series of the rectifier switching function spacevectors (rotating at positive or negative direction at respectivefrequencies) where the amplitudes are modulated by the recti-fier-side dc current

(31)

where .Due to time-domain multiplication in (31), we can conclude

that the resulting rectifier input current space vector spectrumcan be obtained as the convolution in frequency domain of therectifier switching function space vector spectrum and rectifierdc current spectrum.

The frequency transformations that occur in the rectifier stageare illustrated in Fig. 6. We start from the double-sided rectifierdc current spectrum (note only dc and disturbance current com-ponents are shown, the rectifier current ripple at is omittedfor simplicity). Further, we convolve this spectrum with the rec-tifier switching function space vector spectrum. Only the funda-mental and fifth harmonic components of the switching functionspace vector spectrum are shown in Fig. 6 (note that the fifth har-monic of the switching vector has negative sequence). The re-sulting rectifier input current space vector spectrum shows thatas the result of the motor current imbalance, around every char-acteristic harmonic of the rectifier input current, two interhar-monic families shifted by are created. This result is ba-sically the same as the result obtained in Section II and givesgood insight into the frequency transformation mechanism.

VI. SIMULATION RESULTS

A series of simulations has been carried out in PSIM by usinga simulation model of Fig. 1. A 75-kW four-pole motor wassupposed controlled by using simple law. Themotor current imbalance was created by introducing an addi-tional inductor in series with the motor phase . The motor anddrive currents have been found at different drive-output frequen-cies ranging from 20 Hz to 45 Hz. In the simulations,the motor load torque was maintained nearly constant so that


Fig. 6. Frequency transformations during harmonic propagation from the rec-tifier dc side to ac side.

Fig. 7. DC bus inverter �� , rectifier side �� , and rectifier input current�� waveforms at � � 40 Hz without and with 5.5% motor current im-balance.

motor current imbalance remained nearly thesame (5%-6%) at different motor speeds.

Typical current waveforms illustrated in Fig. 7 are obtainedwith 5.5% imbalance for 50 Hz and 40 Hz. Theinverter injects the disturbance current at 80 Hz which is closeto the dc bus parallel resonance frequency so the rectifier-sidedc current and, consequently, the rectifier input currents exhibithigh fluctuations.

The results in Figs. 8–10 show harmonic spectra of the dcbus inverter and rectifier side currents and driveinput currents at the drive output frequencies

Fig. 8. DC bus inverter-side current spectrum �� at � � �� Hz�� Hzwith an approximate constant motor current imbalance of 5.5%.

Hz Hz for approximately a constant motor current im-balance of 5.5%. The inverter-side dc bus current has a domi-nant dc component and disturbance current at with an am-plitude that is linearly varying with the drive-output frequency(modulation index ).

The rectifier-side dc bus current spectrum in Fig. 9 illustratesthe disturbance current magnification that gets higher when thedisturbance current frequency approaches the dc busparallel resonance frequency (in our example, approximately100 Hz). Also, conventional ripple at (300 Hz) created bythe rectified voltage ripple is visible.

Finally, Fig. 10 shows the drive input current spectra forHz Hz. Apart from characteristic integer

harmonics and two distinctive major interharmonic families at, several minor interharmonic sidebands around

the 5th and 7th harmonics are also visible (exactly as illustratedin Fig. 6) but their magnitudes are much smaller.

The current magnification factors are also found in thesimulations with an expanded inverter output frequency rangeof Hz Hz and shown in Fig. 3 (diamond markers).These simulation results compare well with the theoretical anal-ysis results (continuous line). However, the dc bus antiresonantpeak appears at a somewhat higher frequency than that pre-dicted theoretically in (15). It can be explained by lower ef-fective inductance seen by the dc bus [8] as the result of diodecommutations.

VII. EXPERIMENTAL RESULTS

Several experimental measurements have been made on asetup with parameters roughly corresponding to the system pa-rameters in Fig. 1 with a 75-kW motor that was recently serviced(rewound). The motor currents were measured ( 131.9 A,


Fig. 9. DC bus rectifier-side current �� spectrum at � � ��Hz��Hzwith an approximate constant motor current imbalance of 5.5%.

Fig. 10. Rectifier input current �� spectrum at � � �� Hz � �� Hzwith an approximate constant motor current imbalance of 5.5%.

132.9A, and 127.2 A) and they indicated an im-balance of about 3% (32). The drive input current has exhibitedhighest fluctuations at 46.5 Hz (dc bus resonance pointappears to be at a bit lower frequency with respect to the simula-tion model which can be explained by higher source impedancecompared to that assumed in the simulations). The rectifier dcbus and input current waveforms clearly indicate interharmonicdistortion (Fig. 11).

Fig. 11. Rectifier-side dc bus �� and input �� currents at � � 46.5Hz.

Fig. 12 shows the rectifier-side dc current and its spectrumcaptured in a 1-s window (1-Hz frequency resolution). Distor-tion component at 93 Hz is the result of the motor cur-rent distortion. Next to it is the second harmonic (100 Hz) pro-duced by the input voltage imbalance that always existsto some extent in real-life systems. Please note that all experi-mental harmonic spectra are given in [Arms] while the simula-tion spectra are given in [Apeak].

The rectifier input current spectrum in Fig. 13 confirms thatthe interharmonic distortion is present as two equal spectralcomponents at 43 Hz and 143 Hz .Their amplitudes are equal approximately to 50% of the dc buscurrent interharmonic distortion and in our case, they reachabout 4% of the fundamental input current. The third harmonicin Fig. 13 is related to power system voltage imbalance.

Two special cases are illustrated in the results of Fig. 14. Theupper trace shows the rectifier input at 25 Hz. In thiscase, the interharmonic becomes the second in-teger harmonic while the interharmonic frequencyfalls to zero, causing dc current to flow into the power system.The current waveform clearly indicates the presence of even har-monics as its half-wave symmetry is lost.

Finally, the bottom trace in Fig. 14 is obtained at50 Hz when the interharmonic becomes the thirdinteger harmonic and the interharmonic becomesthe negative-sequence fundamental current. Both componentsare typically present in unbalanced rectifier operation, and it isnot possible to distinguish the contributions of the motor currentand input voltage imbalances.

VIII. CALCULATION EXAMPLE

In this section, we will give a sample calculation that illus-trates the procedure for the estimation of the drive interhar-monics based on the motor current measurements (Table I).


Fig. 12. Rectifier-side dc current �� waveform and spectrum at � �46.5 Hz.

We will start from the motor current measurements (in theexample, we will use the simulation result obtained for40 Hz). First, we have to calculate the negative-sequence motorcurrent. For that, we can use an approximate formula (IEC61800-3 Ed. 2)

(32)This motor current imbalance corresponds to about 5.5%. Thus,the corresponding inverter dc-side disturbance current can beestimated from (7)

(33)

From Fig. 8, the current magnification of the dc link at 80 Hzis about 2

Hz (34)

Thus, the rectifier rms dc-side disturbance current is

(35)

Finally, the rectifier input current interharmonics have the fol-lowing rms value each

(36)

(This is the same as in Fig. 10. Note that in Fig. 10, the peakharmonic current is indicated.) Both interharmonic magnitudesare about 4.6% of the nominal current.

Fig. 13. Rectifier input current �� waveform and spectrum at � � 46.5Hz.

These interharmonics are very difficult to mitigate. The bestoption is to suppress their source (i.e., motor current imbalance).For example, current controllers placed in the reference framesynchronous with the negative-sequence motor current vectorcan be used to suppress the motor current imbalance. Also, theinterharmonic magnification can be reduced by the system de-sign. From the analysis presented in this paper, it appears that(for the same equivalent dc-link inductance) dc-link systemswith higher ac side inductance will create better damping andlower interharmonic magnification than dc links with predomi-nant dc-side inductance.

IX. CONCLUSION

In this paper, we have analyzed the mechanisms of a genera-tion of interharmonics in variable-speed drives caused by motorcurrent imbalance. It was shown theoretically and in simulationsthat a significant amount of interharmonics in the drive inputcurrents can be generated when motor currents are unbalanced.These interharmonics can appear at any frequency between thedc and the third harmonic depending on the drive output fre-quency. This presents a major risk because these interharmoniccurrents can excite various resonance modes in a supply system.

A major factor in the interharmonic transfer is the currentmagnification occurring in the drive dc bus circuit due to itsparallel resonance. Our results suggest that resonance dampingis significantly affected by the drive input inductance via thevoltage drop associated with the rectifier valve commutations.In other words, the presence of the ac-side (commutation) in-ductance can be modeled as an additional equivalent resistance


Fig. 14. Rectifier input current �� waveforms and spectra in two special conditions. (a) At � � �� Hz and (b) at � � �� Hz.

TABLE IINPUT DATA FOR THE CALCULATION PROCEDURE

introduced in the dc bus from the supply side that improves thesystem damping.

A calculation procedure is presented for the estimation of thedrive input current interharmonics that is based on measure-ments of the motor currents. The experimental results and sim-ulation results presented are in good agreement and completelysupport the theoretical analysis.

REFERENCES

[1] IEEE Interharmonic Task Force, “ Interharmonics: theory and mod-eling,” IEEE Trans. Power Del., vol. 22, no. 4, pp. 2355–2348, Oct.2007.

[2] R. Yacamini, “Power system harmonics, part 4 interharmonics,” PowerEng. J., pp. 185–193, Aug. 1996, Tutorial.

[3] A. D. Graham, “Line interharmonic currents in frequency changers,” inProc. 8th Int. Conf. Quality Power, Athens, Greece, Oct. 14–16, 1998,pp. 749–754.

[4] B. R. Pelly, Thyristor Phase-Controlled Converters and Cyclocon-verters. New York: Wiley, 1971.

[5] F. De Rosa, R. Langella, A. Sollazzo, and A. Testa, “On the interhar-monic components generated by adjustable speed drives,” IEEE Trans.Power Del., vol. 20, no. 4, pp. 2535–2543, Oct. 2005.

[6] Y. Jiang and A. Ekstrom, “General analysis of harmonic transferthrough converters,” IEEE Trans. Power Electron., vol. 12, no. 2, pp.287–293, Mar. 1997.

[7] R. Carbone, F. De Rosa, R. Langella, A. Sollazzo, and A. Testa, “Mod-elling of AC/DC/AC conversion systems with PWM inverter,” in Proc.IEEE Power Eng. Soc. Summer Meeting, 2002, vol. 2, pp. 1004–1009.

[8] M. B. Rifai, T. H. Ortmeyer, and W. J. McQuillan, “Evaluation of cur-rent interharmonics from AC drives,” IEEE Trans. Power Del. , vol. 15,no. 3, pp. 1094–1098, Jul. 2000.

[9] A. Testa, “Issues related to interharmonics,” in Proc. IEEE Power Eng.Soc. General Meet., Jun. 6–10, 2004, vol. 1, pp. 777–782.

[10] D. E. Rice, “A detailed analysis of six-pulse converter harmonic cur-rents,” IEEE Trans. Ind. Appl., vol. 30, no. 2, pp. 294–304, Mar./Apr.1994.

[11] T. Tayjasanant and W. Xu, “A case study of flicker/interharmonic prob-lems caused by a variable frequency drive,” in Proc. 11th Int. Conf.Harmonics Quality Power, 2004, pp. 72–76.

Duro Basic received the Dipl.Eng. degree in elec-trical engineering from the University of NoviSad, Serbia, in 1981, the M.E. degree in electricalengineering from the University of Belgrade, Serbia,in 1993, and the Ph.D. degree in electrical engi-neering from the University of Technology, Sydney,Australia, in 2001.

In 2002, he joined Schneider-Electric. Cur-rently, he is a Senior Motor Control Engineer withSchneider Toshiba Inverter, France, a joint venturebetween Schneider-Electric and Toshiba, where he

is mostly involved in the development of variable-speed drive systems. Hisresearch interests are control of electric drives, power-electronic converters,active filters, and power quality.

input current interharmonics of variable-speed drives due to motor current imbalance

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