AN IMPROVED WEIGHTED TOTAL HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES

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1 AN IMPROVED WEIGHTED TOTA HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES Tomas A. IPO University of Wisconsin, 45 Engineering Drive, Madison WI, USA P: -(608)-6-087, Fax: -(608) , Abstract: Te weigted total armonic distortion (WTHD) is a commonly used expression to assess te quality of pulse widt modulated (PWM) inverter waveforms. Te WTHD weigts te voltage armonics inversely wit its frequency. Wile tis is adequate for some inductor type loads, te commonly employed induction motor load as important effects resulting from eddy currents in te rotor bars not incorporated in te WTHD. In tis paper a new distortion index, te IMWTHD, is proposed to more accurately evaluate modulated inverter waveform quality for squirrel-cage induction motor loads. Key words: Harmonic Distortion, Total Harmonic Distortion, Weigted Harmonic Distortion, THD, WTHD, Harmonics, Pulse Widt Modulation.. Introduction Te proliferation of converter modulation algoritms tat ave appeared over te years as created confusion as to te effectiveness of one metod over anoter for comparing various algoritms. One metod of comparing te effectiveness of modulation processes is by comparing te unwanted components, i.e. te distortion, in te output current waveform relative to tat of an ideal sine wave. Te most commonly used distortion index is te weigted total armonic distortion (WTHD) in wic te inverter output voltage armonics are weigted inversely wit te armonic frequency so as to approximate te current distortion in an inductive load. However, in te large majority of practical applications te load is an induction motor wose impedance does not simply vary linearly wit frequency. In tis paper a new performance index is proposed wic quantifies armonic distortion of te ac current waveform wen te inverter supplies a squirrel cage induction motor load. Tis new index can be used as an improved means of comparing various modulation algoritms wen supplying a squirrel cage induction motor load.. Te Harmonic Voltage Distortion Factor Given tat te output voltage of a solid state converter vt () is a periodic function wit period T, te RMS value of te function is, by definition, () 0 Since vt () is periodic wit no dc component, ten it can be represented by te Fourier Series, wereupon T V rms vt () dt T vt () V cos ω t + V cos ω t + V cos ω t + T' () V rms V T V cos( k ω t) cos( kω t) dt 0 0 k 0 () Upon expanding, te integration of terms in wic k become zero, so tat, finally V V rms or, in terms of RMS values of te armonics, (4)

2 V rms V, rms (5) In most practical cases te fundamental component can be considered as te desired output. Te remainder of tis expression is ten considered as te distortion. Factoring out te desired component V rms V, + rms V rms, V, rms (6) Te total armonic distortion (THD) of te voltage can now be defined as THD V rms V, rms V rms, V, rms,, (7) Since te RMS values of bot contain a factor, te THD is equally expressed in terms of peak values, i.e. THD,, V V (8). Harmonic Current Distortion Factor (WTHD) Wile it is te output voltage wic is te quantity tat is controlled in a voltage source/stiff inverter type system, it is te current wic is frequently of most interest since losses, output power etc. typically involve tis quantity rater tan voltage directly. Defining te current armonic distortion in te same manner as for voltage, one can state tat ( I ),, HD - i (9) I Te current waveform is, of course, dependent upon te load impedance and, as suc, can not be predicted or caracterized in advance. However, in some applications, te load can be caracterized by a lossy inductance, tat is, by an inductance in wic te resistance is relatively small. In tis case, te armonic current amplitudes can be approximated by te expression, V I -,,4,... (0) ω were ω is te angular frequency of te fundamental component of te current waveform. If dc components do not exist V HD i () ω Normalizing tis expression to te quantity V ( ω ) te weigted total armonic distortion (WTHD) becomes defined as V WTHD - V () 4. Te Induction Motor oad Te WTHD is superior to te THD as a figure of merit for a non-sinusoidal converter waveform since te WTHD predicts te distortion in te current and subsequent additional losses wic are typically te major issues in te application of suc converters. However, it sould be recalled tat Eq. () was derived assuming tat te load resistance and inductance were constant. Wen te load is passive, tis assumption in generally quite valid. However, wen used wit a motor load special precautions must be taken because of te non-linear nature of te load impedance. Te assumption of constant inductance and resistance is again valid for te stator circuit of any random wound ac macine. However, te sorted bars of te rotor of an induction motor produce special problems due to deep bar effect wic causes te rotor current to rise to te air gap side of te rotor bars. Te overall result is tat te resistance and inductance of te rotor circuit become frequency dependent. 5. Rectangular Squirrel Cage Bar [] Consider te simple rectangular bar placed in an iron slot as sown in Figure. Te total IR drop across te lengt of te bar at any eigt y can be obtained by integrating te electric field over te lengt of te bar.

3 () 0 were l denotes te lengt of te rotor bar in te laminations. It is sown in [] tat te voltage drop reduces to were, Ṽ R Ṽ R l Ẽ dl γρi l m b cos( γy) - sin( γd) (4) jω µ γ - b o (5) ρ ω b is te angular frequency of te EMF impressed on te rotor bar, µ o is te permeability of air and ρ is te resistivity of te conducting bar. Note tat te resistive drop is a maximum at te top of te slot were te parameter y d. µ o I m l φ m ( y) cos( γd) cos( γy) (7) γb sin( s γd) Te total IX drop across te lengt of te bar at any eigt y is terefore l d Ṽ t B m dydz 0 y or in complex form (8) Ṽ jω b φ m (9) wereupon Ṽ jω µ - b o m l cos( - γd) cos( γy) γ b sin( s γd) (0) Note tat te reactive drop is zero at te top of te slot and a maximum at te bottom of te slot, just te reverse of te resistive drop. Te total voltage drop along te bar at an arbitrary eigt y is te sum of te inductive and resistive drop or Ṽ bar Ṽ R + Ṽ () l γρi l m b cos( γy) - sin( γd) + jω µ - b o - I m l γ b s cos( γd) cos( γy) sin( γd) Te total flux crossing te slot above te eigt y wic terefore links te current below y is or b d y Figure Rotor bar embedded in laminations l d φ m ( y) B m dydz 0 y (6) () wic can also be written as [48], γρi l m cos( Ṽ bar γd) - () b sin( γd) Note tat te total drop down te lengt of te bar is constant, wic is to be expected since all of te current filaments are connected in parallel. Te current distribution in te bar may be tougt of as te superposition of a uniform (average) current and a circulating current flowing additively at te top of te bar and negatively at te bottom, directed in suc a way as to oppose te time rate of cange of slot leakage flux. Te skin effect forces current to flow in te part of te conductor area wic lies at te top of te slot and te effective resistance is many times (usually tree to four times) as large as te round bar or sallow bar rotor. In te case of te double bar rotor, te resistance of te top bar usually as a iger resistance tan te bottom bar making te effective resistance at starting even larger if desired.

4 Equation () can be written in terms of te dc resistance of te bar if it is recalled tat for a rectangular bar ρl R - DC (4) bd Hence, te effective impedance of te bar is Ṽ bar Z bar (5) I m ( γd) cos( γd) R DC (6) sin( γd) Te real portion of te impedance representing te ac resistance of te bar can be readily evaluated as sin( αd) + sin( αd) R AC αdr DC cos( αd) cos( αd) were (7) α ( ω µ ) ( ρ) b o (8) Te imaginary component of te impedance represents te reactance of te bar and is sin( αd) sin( αd) X AC αdr DC (9) cos( αd) cos( αd) Te inductance of a rectangular bar in a slot is [48], dl µ - DC o (0) b so tat (9) can also be written as, X ω DC sin( - αd) sin( αd) AC αd cos( αd) cos( αd) Te ac inductance is terefore, () DC sin - ( αd) sin( αd) AC () αd cos( αd) cos( αd) If αd is large Eqs. (7) and (9) can be approximated by R AC αdr DC () and DC - AC (4) αd Equation () can, by use of Eq. (4), be expressed in te form R AC ρl - ( αd) bd l r ρ b( α) (5) Hence, for sufficiently ig frequencies, te ac resistance can be calculated in te same manner as for te dc resistance if one replaces te actual dept of te bar d by an equivalent dept /α. Te quantity /α is called te skin dept and te fact tat te current distributes itself unevenly over a conductor due to sinusoidal excitation is called te skin effect. A plot of R AC and AC normalized wit respect to teir dc values is given in Figure. R ac /R dc and ac / dc Figure (R ac /R dc ) approx ( ac / dc ) actual (R ac /R dc ) actual ( ac / dc ) approx α d (ω b / )0.5 Sketc of normalized resistance and inductance of solid bar in a slot as a function of αd and teir approximations One can write Eqs. () and (4) directly in terms of frequency if it is noted tat αd ( ω µ ) ( ρ) b o d ω µ b o d ( ρ) (6) Defining a bar or break frequency ρ ω b0 (7) µ o d one can also write R AC and AC for large αd (ig frequency) as, R AC ω - b R ω DC 0 b (8) 4

5 DC - AC (9) ω ω b 0 b If te dc value of R AC and AC are assumed below ω ω b 0b and 9/4 respectively and ig frequency approximations are used above tese values te dased lines of Figure are obtained. A good approximation is obtained wic can be used to estimate te armonic distortion caused by te frequency dependant rotor bar parameters. 5. Non-Rectangular Rotor Bars It is well known tat te bar sape of a practical induction motor design is rarely a simple rectangular structure as considered in Section 5. Indeed bar sapes ave been devised so as to minimize te losses due to inverter armonics. Since te analysis of te previous section was limited to tis case, it is appropriate to question te validity of te results for nonrectangular bars. Wile a variety of sapes are possible, it appears tat essentially all bar configurations possess a bar break frequency of te type identified as Eq. (7). For example Figure sows te frequency dependence of an inverted coffin saped bar []. ac / dc r ac /r dc α d Figure b b Rotor Surface Variation of bar resistance and inductance for inverted coffin saped bar wit αd were d is te bar dept, parameter is b/b b /b 6 4 b /b Te frequency beavior of te rotor resistance and leakage inductance can typically be approximated as a constant below tis break point and as proportional (i.e. R AC ) or inversely proportional ( AC ) to ω b beyond tis point. Te bar sapes can be sown to mainly affect te slope of te curves beyond te break point. However, te analysis tat as been performed can always be modified to accommodate a different slope beyond te break point. For example, let K b denote te multiplier wic canges te slope from a unity value described in Eqs. (8) and (9) obtained for example by finite element analysis. Above te break frequency te AC resistance for te modified case can ten be expressed as Similarly ω R AC ' K b b - + R ω b0 DC AC ' K R + ( K )R b AC b DC DC - ω b K b - + ω b0 (40) (4) (4) ω b - ω AC b0 - (4) ω b K b - + ω b0 Hence, wen te AC resistance and inductance for a particular bar armonic as been calculated for a simple rectangular bar, te equivalent result can be readily translated to a non-rectangular bar by Eqs. (4) and (4). 6. Per Pase Equivalent Circuit Altoug te impedance as been determined for a single bar, all of te bars of an induction motor experience te same impressed flux (only wit different pase relationsips) so tat te motor equivalent circuit rotor impedance maintains te same frequency dependence. Altoug a portion of te rotor circuit is comprised of te end ring portion, again te same frequency dependence exists. Since te rotor can, for practical purposes, be considered as rotating near syncronous speed, differing only by a small slip, it can be assumed tat te bar frequency is related to an arbitrary armonic impressed on te stator by ω (44) b ω ± ω were te plus sign is taken wen te stator armonic rotates in te negative direction, (i.e,5,8,,...) and te minus sign is used wen te armonic rotates in te positive direction (i.e.,4,7,0,,...). Te fre- 5

6 quency dependant rotor resistance and inductance can ten be well approximated by, ω ± ω R AC R (45) ω DC 0 b and DC - AC (46) ( ω ± ω ) ω 0 b Te resulting per pase equivalent circuit for an induction motor wit frequency dependent parameters is sown in Figure 4(a). + ~ V' + _ ~ V' _ r' jω ' jω b r' jω ' jω b ~ I (a) (b) r r ( ω ± ω ) ω 0b ( ) [( ω ± ω ) ω ] 0b m m r ' ' r Ṽ + m + ' m Figure 4 r b /S r b m Ṽ + m (a) Frequency dependant induction motor equivalent circuit for non-triplen armonic component and (b) approximation wit stator side referred to te rotor Wen ig frequency armonics are superimposed on te fundamental, superposition principles can be applied assuming tat saturation is not too severe. Wen te frequency is ig te slip frequency corresponding to an arbitrary non-triplen armonic is ( ± ω ) ω r S - (47) ( ± ω ) were ω r is te rotor speed in electrical radians per second and eiter te plus or minus sign applies depending upon weter te armonic is a positive or negative sequence respectively. If ω is sufficiently ig tis expression approaces unity regardless of te polarity of ω. Because te rotor parameters are frequency dependant it is useful to refer te stator side of te circuit to te rotor rater tan vice versa. Using Tevenin s Teorem, te voltage observed at te air gap from te rotor side is jω m V ' V (48) r + jω ( + ) m For sufficiently ig frequencies, Eq. (48) can be conveniently approximated as m V ' V (49) + m Te Tevenin driving point impedance is obtained by sorting te voltage source in wic case ( r + jω )jω m Z ' (50) r + jω ( + ) m Again, if te angular frequency ω is sufficiently ig, m Z ' r m + jω (5) + m + m It is clear tat te frequency dependence of te rotor parameters must reflect te fact tat te rotor is rotating so te fift armonic stator voltage depending upon its direction of rotation, impresses a fourt or sixt armonic voltage on te rotor due to te fact tat it is positive or negatively rotating wit respect to te rotor. In general, for armonics related to te fundamental component, te tree pase voltage waveforms remain balanced. Tat is, te tree voltages ave identical wavesapes but are only pase sifted in time by / of a complete cycle. Te rotation sequence can be determined by dividing te armonic number by and examining te numerator of te resulting fraction (if any). If te numerator is one te sequence is positive and te minus sign in applies in Eq. (47) wile if te numerator of te fraction is two te sequence is negative (te plus sign applies). If te value of is divisible by tree te component corre- 6

7 sponds to zero sequence term and sould be omitted from te summation since tis component does not link te rotor. 7. WTHD for Frequency Dependant Rotor Resistance A very simple figure of merit for te effect of impressing a non-sinusoidal voltage waveform on te rotor branc can be developed if one considers only te frequency dependence of te rotor resistance and assumes tat te rotor inductance remains constant. In tis case, referring to Eq. (8) and Figure, one can approximate te variation of r wit frequency as r r wen ω ω b 0b and ω r r - b wen ω > ω. ω b 0b 0b In te first case te loss in te rotor resistance due to a single armonic is simply V ' P I r r (5) ω ( ' + ) if stator resistance can be neglected. Wen normalized to te loss occurring during te starting period (due to in-rus current), V ' r P ω ( ' + ) (5) P, inrus V ' r ω ( ' + ) V ' - ω (54) V ' ω wic reduces finally to P (55) P, V inrus V If it can be assumed tat te stator resistance drop is negligible compared to te drop across te stator leakage inductance at te frequencies of interest. Wen ω b >, P V ' ω ± ω ω ( ' + ) r (56) P P, inrus V ' ω ± ω ω ( ' + ) r (57) V ' r ω ( ' + ) V ' - ω ω ± ω V ' ω V ' - ω ω ω ω - ± - V ' ω ω ω V - ± - (58) V One can now define a weigted armonic distortion factor wic includes two sets of terms suc tat WTHD bar 0 V V ω V - ± + - ω V 0b + 0 (59) were 0 denotes te igest armonic for wic ω < ω b 0b. It is apparent tat te geometry dependent parameter ω as an important effect on te WTHD since te second group of terms are weigted more eavily wit respect to frequency tan te first group. In most practical cases were pulse widt modulation is used, te lower frequency armonics are suppressed and all of te armonics are greater tan in wic case one can define ω 4 WTHD - bar V ± - V (60) Te quantity ω can be considered as a bar factor related to te design of te macine over wic te inverter specialist as little control wereas te remaining portion is most relevant for te purpose of selecting a PWM algoritm. For 60 Hz operation te value of ω takes on values ranging from 0.5 for small macines (5 HP) to 4.0 for large macines (500 HP). 7

8 8. WTHD Also Including Effect of Frequency Dependant Rotor eakage Inductance Tus far, only te effects of frequency dependant rotor resistance as been considered. Wen te variation in rotor leakage inductance is also taken into account, te procedure is similar. In tis case tree regions can be identified, r r ; wen ω ω b 0b (6) (6) (6) Te solution for te contributions to te WTHD from te first two ranges of ω is te same previous work in Section 7.. A typical loss term for a armonic belonging to te tird region is, or, ω ± ω r r ; ω ± ω r r ; P P 9 wen < ω ω b 4 0b 9 wen ω > ω b 4 0b ( ) - ( ω ± ω ) V ' ω ± ω - r ω ω ' +.5-0b ( ω ± ω ) (64) V ' ω ± ω r ω ' ( ω ± ω ) +.5- (65) and wen normalized to te power dissipated during te current in-rus interval (assuming constant parameters under tis condition), P P, inrus V ' - + V ' ' ( ω ± ω ) ω +.5 0b ' ω ± ω (66) Some simplification is possible if it is assumed tat te primed stator and rotor leakage inductances are equal, a standard assumption in induction macine analysis. Again neglecting te effects of stator resistance on te voltage ratio V ' V ', Eq. (66) becomes, P V P, inrus V 4 ω - ( ± ) ω - ( ± ) +.5 (67) A weigted THD factor, termed WTHD, can now be developed as te square root of tree sets of terms in wic Eq. (54) is used in te first region were ω ω b 0b, Eq. (58) applies for armonics in wic < ω ( 9 4)ω b 0b and Eq. (67) is applicable for values of armonics for wic ω > ( 9 4)ω b 0b. Again, te WTHD must be presented as function of te macine dependant parameter ω rater tan as a single number. Te resulting function is plotted in Figure 5 for te case of te quasi-square wave inverter. Te complicated nature of IMWTHD prompts one to consider acceptable approximations. In most cases te armonic content of te lower armonics are negligible, indeed tis result is te goal of most pulse widt modulation algoritms. In suc cases it can be assumed tat te armonic components of flux rotate in te air gap at a sufficiently ig rate of rotation tat te rotor appears as essentially stationary. Hence, one can, in effect replace te bar frequency b ± terms in simply by. Te final result for IMWTHD is were IMWTHD W + W + W W 0b V - V dc k ± k,,,... b W 0b + ω V - - ω V 0b ( ) (68) (69) (70) 8

9 W V - V dc 4 ω - ( ) ω ω + 0b b (7) In tese equations 0b represents te igest armonic for wic ω < ω b 0b and b denotes te igest armonic for wic ω < ( 9 4)ω b ob. Included in tis result, also is te fact tat tird armonic components (zero sequence components) of stator current do not link te rotor if te stator winding is an ideal sinusoidally distributed winding. Hence, only values of,4,5,7,8,0, etc. are included in te summation. Since te parameter ω can not be isolated from te remainder of te expression, te WTHD is best portrayed as a function of ω or specified for particular values of ω. Te exact and approximate functions are plotted in Figure 5 for te case of te quasi-square wave inverter. Since it can be recalled tat te quasi-square wave inverter contains a full measure of all non-triplen odd armonics, te approximation can be considered as sufficiently accurate for use as a figure of merit. Note tat at most only two armonics can exist in te middle region and could easily be approximated by relegating te lower of te two armonics to te first region and te upper one to region tree. Also, if all of te armonics are sufficiently iger tan ten only values in te tird region apply and furtermore if ω - ( ± )».5 Eq. (67) simplifies to P ω V 4 - ± - (7) P, inrus V wic is simply four times tat of Eq. (58). Again making te assumption tat te bar frequencies are sufficiently ig tat b, te WTHD for tis condition is ω 4 V IMWTHD( f) - ω 0b V / k ± k,,... (7) Tis result is to be expected since Eq. (7) effectively assumes tat te rotor leakage inductance is zero and, since it as been assumed tat ', te current for all armonics will be twice te value obtained wen is assumed as a constant equal to '. It must be mentioned ere tat te assumption of ideal sinusoidally distributed windings is a convenient matematical artifice wic is never realized exactly in practice. In reality, te stator winding distribution is non-ideal meaning te iger armonic fields are set up in te air gap, tat is, fields wit sets of poles wic are an odd multiple of te number of poles set up by te fundamental component. In most practical cases a small tird armonic spatial component will terefore exist in te air gap wen tird armonic stator currents are allowed to flow, so tat a field is produced wic does link te rotor bars. Tis field can be sown to be single pase in nature, i.e. stationary and pulsating in space. Since tis effect is generally small, it is neglected ere. 9. WTHD for Stator Copper osses Te equivalent circuit of Figure 4 can also be used to obtain a suitable figure of merit for stator losses. Te ratio of stator resistive losses due to armonic as a per unit of power dissipated during te in-rus period wit fundamental frequency voltage applied is, V ' r P ω ( ' + ) - (74) P, inrus V ' r ω ( ' + ) Wen ω < and also wen ω ( 9 4) bot te stator resistance and te rotor leakage inductance are constant and tis expression reduces to P V ' - ω (75) P, inrus V ' ω Hence, for armonics, 0 ω ( 9 4)ω b b0, te WTHD for te stator is te same as previously obtained for te rotor wen r and are constant, namely V IMWTHD (76) V For operation above ( 9 4), 9

10 P P, inrus V ' - r ω ' +.5- ( ω ± ω ) (77) V ' r ω ( ' + ) wic, after some manipulation and te usual assumptions becomes P V P, inrus V 4 ω - ( ± ) ω - ( ± ) +.5 (78) Again if it is assumed tat te bar frequency for eac armonic is sufficiently close to te corresponding stator armonic frequency, ten P V P, inrus V 4 ω - - ω - ( ) +.5 (79) A suitable expression for te stator WTHD is consequently o V IMWTHD V k ± o V + (80) V k ± were 0 is te igest armonic for wic ω < ( 9 4)ω b b0. Finally, for sufficiently ig armonic content (above ( 9 4) ) ω ( ± ) -».5 (8) so tat, as a final figure of merit one can reasonably approximate te stator WTHD by means of te simple expression V IMWTHD( f) (8) V k ± wic is simply twice te WTHD obtained by ignoring te frequency dependence of te motor parameters. Te factor of two is obtained by neglecting, in effect, te rotor leakage inductance. If te inverter waveform as only ig frequency armonic content, tis result demonstrates tat WTHD, as defined by () remains a reasonable figure of merit for te stator losses of an induction motor but not for te rotor losses. Figure 5 sows ow te two WTHD terms vary as a function of ω for te case of a tree pase quasi-square wave inverter. Also sown for reference is te WTHD wic does not include te effect of frequency dependant parameters (Eq. ()). It is Figure 5 IMWTHD (exact) IMWTHD (approx) IMWTHD (exact) IMWTHD (approx) Bar Factor ω /ω b0 Effect of frequency dependant rotor parameters on te stator and rotor weigted total armonic distortion apparent tat te deep bar effect as an important effect on te losses, particularly wen te fundamental component is substantially greater tan te caracteristic bar frequency. Te effect on te loss is particularly dominant in te rotor since te resistance corresponding to te rotor bar armonics increase wit te bar factor wile te stator resistance remains independent of frequency. 0

11 0. Example Calculation of Harmonic osses Te following induction motor is assumed to be powered by a quasi-square wave voltage source inverter; HP 50 HP V ll,rms 0 V ω 77 r/s r 0.0 Ω r 0.04 Ω X 0.4 Ω X 0.76 Ω X m.5 Ω For te condition of interest, te inverter operates in te field weakening mode at two per unit frequency (based on te rating of te motor). Te bar caracteristic frequency is determined to be ω b0 00 r/s. Te task is to determine (rougly) te motor losses due to te armonic content of te inverter. Solution: Since te angular frequency corresponding to two per unit is ω 77 ( ) 754 te caracteristic bar factor is ω ω 00 bo Te corresponding WTHDs from Figure 5 are WTHD WTHD 0.5 By definition, P P, inrus ( WTHD) Tus ( V ') P P inrus r [ ω ( ' + )] Now, V ' X m.5 V X + m X m ω ' 754 ( 0.4).5 + m Tus, ( V ') ( 6.5 P r ) [ ω ( ' + )] Watts/Pase. Similarly, P P, inrus ( WTHD) Tus 0.08( V ') P 0.08P inrus r [ ω ( ' + )] and, P 0.08( 6.5 ) Watts/Pase Te total copper loss due to armonic content in te quasi-square wave inverter is P P + P 87 Watts wic represents rougly 0.5% of te motor 50 HP rating.. THD Weigting wit PWM Inverter Supply In cases were te fundamental component remains constant or wen te dc link voltage is proportional to te fundamental ac component as is te case for constant volts/ertz drives, te normalization of te WTHD by te fundamental ac component is a suitable coice. In particular, note te THD and WTHD of a quasi square wave inverter types does not cange wit frequency provided tat te dc voltage decreases in direct proportion. In tis case te fundamental component as well as all of te armonics cange by te same value making te THD and te WTHDs independent of frequency. In te case of a pulse-widt modulated drive owever, te dc voltage remains constant wile te fundamental component varies. On te oter and, te armonic components cange relatively little, assuming te same ratio of switcing to output frequency, making te THD and

12 WTHDs vary widely. In fact, it is easy to see tat wen te fundamental component approaces zero (by virtue of te modulation index M approacing zero), te distortion factors approac infinity, clearly an unsatisfactory situation. Te problem can be avoided by simply coosing an normalization factor wic is invariant as frequency canges. For te case of te alf bridge inverter tis quantity is conveniently cosen as te value of ac fundamental ac voltage existing wen te modulation index M, defined as V were V dc dc equals one-alf te pole to pole dc bus voltage. Using te approximation b, te WTHDs normalized to te inverter dc link voltage become ( V ) - IMWTHD0 - V M V - V dc - (8) desired te WTHD is readily obtained from tese expressions by taking, IMWTHD IMWTHD V dc IMWTHD0- V IMWTHD0 - M IMWTHD0 IMWTHD - M IMWTHD0 M (89) (90) (9). Conclusion Tis paper as developed a new figure of merit for te comparison of te performance inverter pulse widt modulation algoritms wen feeding a squirrel cage induction motor load. Suc a quantifiable figure of merit sould prove useful in assessing ow various PWM algoritms influence motor copper losses. IMWTHD0 0b V - V dc 4 ω V - + ω - 0b V dc ω k ± k ± ω k,,,...k 0b k k 0b +,... 0b (84) References were W WTHD0 W + W + W 0b V - V dc k ± k,,,... (85) (86) [] P. Alger, Te Nature of Induction Macines (book) Gordon and Breac Publisers, New York. [] T.A. ipo, Introduction to AC Macine Design, Vol. (book), Univ. of Wisconsin/WEMPEC, 996. W b 0b + ω V - - ω V 0b dc (87) W V - V dc / 4 ω - ω ω + 0b b (88) In tese equations 0b again represents te igest armonic for wic ω < ω b 0b wile b denotes te igest armonic for wic ω < ( 9 4)ω b ob. If