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the s ~ g n a l sarc nc& puiodic. Hence conventional discrete FourierAdysis cannot be used I21,[31.> The comparison with the convenhnal sinusoidal P W with fixed switching frequency show thc itnprovemertw with respect inner surface oE the shcor, w h c h genetarex vibrations of the stalm snd c a w s thc magnetic noise. The &comical ru~ulmwe vcriflCd cwperimentaIly- Jn the second pnrt we derive the frequency camponears of rhe pulse width modulartd I F " ) and pulse Eequency modulated witage and of thc magnetic nMst. This is the basrs for d calcwlable spreading a€ the s p e z t " by m m of~pr;M. shown in rhc third put. TIE p~i~pusc is noise Fedudion and t l i m i a a t i h of selected harmonics to waid mwbarlkni s s o m e c . to the mist generation. {m sw FIG. 1. FIG. 1: Inverter-fed induction m d M We stnrt with the dynamic equations 4.t I O of t h induction motor. T h e vultagees, curenis and flw linkaps b c rcprcscmted as rimling phauors. All phawrs we related to the s M o r Tied cowdinate system with the red-axis dong the axis aE the stator windhg la. I n order to makc algebraic mannipuldions w9itx. the equations XIZ 269 written with the differential operator d/dt--+s. The voltage drop across the stator resistance can be neglected in our case. 61 =s$1 0 - i1, (4) = &R2 + (s - jw)i42 (5) - - = il + i2u (8) $1 = i10l L1+ YlmL1, $2 = T2'32L2 ~1 + TI, U : voltage i : current \~r: fluxlinkage o : angular speed (11) (12) (6) l/u-L2, (7) =w~+w w=2mzp L, = q L I + L 1 , L2 =62L2+L2, operation mode between no load W ~ G S = O and nominal load @GS=CJ.l2", results in ULlm =l/U.L2, (1-01)(1-02)=(1-0) (s-iw) -+ ( j w l ~ s - j o ) = j w 2 ~ ~ ~ c s o T 2<c 1 (e.g. 2x.1.4 1Hz~7%~210m~=O.13<<1) (9) (10) (13) (14) (23) 02<0 For the harmonic components * l m ~we s replace the differential operator by s-+jolos, the harmonic angular frequency. The simplifications given in eq(25) is valid as the harmonic angular frequencies are high compared to the angular speed 2 n ~ 1 . of z ~the rotor w 1os>>0=2x.n-zp. R : resistor L : inductance ts : leakage factor ii : rotor-stator winding ratio w i w l w : number of turns of the winding zp : number of pole pairs (s-.i@ -j 6qos -jm)=j9os wzop2T2 >> 1 (e.g. 2rc~lkHz2%~21Oms=26.4>>1) (25) 0>62 n : rotor speed index 1 : stator 2 : rotor m : magnetizing S : differential operator d/dt+s x%b,c : 3-phase instantaneous quantities X : stator related rotating phasor x2 .ii : rotor quantity transformed to the stator X I . l/u : stator quantity transformed to the rotor The superposition of eq(22) for the fundamental and eq(24) for the harmonic components leads to eq(26). It shows the remarkably simple relationship between the magnetizing flux linkage $lm in the air gap and the applied stator voltage GI, both represented as stator related rotating phasors. $1, = +$lmOS The transformation from the three phase quantities Xla,b,c of the stator windings to the stator related rotating phasor ?I is shown in eq( 16). Eq(17) shows the backtransformation. = Xla,b,c 61 = GIGS +GlOS ~IGS/(j~IGS).(1-~1)+ + ~ ~ I o s / ( ~ ~ I(1 os )01) - (26) (27) 2.1d Flux linkage and induction Our next goal must be to show the relationship between the magnetizing flux linkage $1, in the air gap and the air gap induction B. A "snapshot" at any instant t always delivers a sinusoidal distribution of the induction B over the circumference of the air gap even with harmonics in the inverter output voltages! (FIG.1). Therefore we can write: The voltages across the motor windings Ula,b,c differ from the inverter output voltages Ua,b,c by the zero sequence voltage uo. in12 This is shown in eq(2,3) in FIG.l. The transformation from the lqlml= wl.qlm = w1 JB-.cos(cp)-R.l.dv= "real 3-phase world" to the rotating phasor 61 of the stator vol-nlZ tage (according to (eq16)) can be made with the voltages Ula,b,c = 2R .1. ~1 B, across the stator windings or with the inverter output voltages (30) uab,c: The zero sequence voltage uo does not contribute to the R, 1 : inner radius and active lenght of the stator phasor ! ,,B : peak value of the sinusoidally distributed induction B 2 . 1 ~ Voltuge andflux linkage Our first goal is, to show the dependency of the magnetizing flux = ilmLlm in the air g_ap, on th_e stator voltage GI. linkage For this purpose we eliminate i2 and il in eqs(k15). After some intermediate steps, in which only algebraic conversions are used, we come to the result shown in eq(20). 1 1+ (s - j0)'32T2 YlmLlm= $Im = -ii1(1-'31) S 1+ (s - jo)oTz Eq(30) means: The integration of the air gap induction B over the whole area in which B is positive is the magnetic flux .$ ,l In the same way as the flux linkage, the induction B, too can be represented by a rotating phasor B,. Absolute value IBI and phase position of this rotating phasor B are given by the peak value B, of the sinusoidally distributed induction B and by its position in the air gap (FIG. 1). Using eq(30) we can write: The differential operator s can be used to simplify eq(20) for the fundamental (GS) and for the harmonic (OS) components. 2.1e Induction, magneticforce and noise If we consider a small stator section in position (px (see RG.1) on the circumference of the stator, as a source of noise, we have to know the induction B(t,cp,) as a function of time at this pcsition. B(t,cpx) is given by the projection of the rotating phasor B upon the %-axis: $1, = ~ I ~ ++ilmos G S (21) For the fundamental component G l m ~ofs the magnetizing flux linkage in the air gap, we replace the differential operator by s+jwl ~ sthe , fundamental angular frequency. Applying the simplification given in eq(23), which is valid in the normal B=@lm/(wl.2R.l) B(t,q,) = Re(Be-jqX) 270 (31) (32) The magnetic force density p(t,cpx) (see eq(1) in FIG.l), given by p(t7cp,) = B(t,cpx)2/(2P0) (33) causes vibrations and makes this stator section a source of noise. Magnetic force density p(t,qxj and noise have the same frequency distribution. 2.1f Rtsumt: inverter voltuge and magnetic noise Summing up the results of the previous chapters The projection of the phasor B at a qtator section in position (px leads to the induct~onB(t,cp,) and to the magnetic Force dcnsrty p(t,qx)at t h s po+,itionqxa4 a function of timc t B(t,cp,) = Re(6e-'"X) = : 3-phase inverter output voltages; BOSI.[I fundamental and harmonic components : rotating phasors; inverter=stator voltage; fundamental and harmonic Components : angular frequency of the stator; fundamental and harmonic components stator (1 ;I, rotor (2) and total leakage factor; number of turns of a stator winding; inner radius and active length of the stator; magnetic force density at a stator section as a function of time t and position cp, of the corresponding stator section. leads to a remarkably simple relationship for the magnetic force density p(t,cpx) of a small section of the stator in the position cp, in terms of the fundamental and harmonic components of the inverter output voltages U~b,c=UGSab,c+UOga,b,c. Magnetic force density p(t,cpx) and noise have the same frequency distribution. We will see in the next chapter, that the position of the corresponding stator section has no influence on the frequency distribution of the magnetic force density and the noise. : : : : (44) << BGS Since BOSI,II<<BGS, the products Bos1,(o).Bosr,(n)can be neglected. We expect from eqs(41 and 45), that the inverter output voltages with the frequencies WIGS = 2n.50Hz; WI(~SI = 2n.3.5kHz WIOSII = 2n.4.SkH~ generate radial magnetic forces and magnetic noise with the frequencies 2 0 1 =~2 ~ 1 0 0 H z ; Wlosl 4 0 1 ~ = s 2a.(3.5kHz-S0Ha) Wlos1 +WIGS = 2~.(3.5kHz+5OH~) WI OSII-WIGS = 2n.(4.5kHz-50Hz) WIOSII+WIGS = ~ K . ( ~ . S ~ H Z + ~ ~ H Z ) The measured results in FIG.4 fully confirm the theoretical expectations - except for the low frequency w=2n.l50Hz, which was caused by a different source of noise in the same room. 2.2 Experimental verification The relation in eqs(34) between magnetic force density and inverter output voltages, deduced in chapter 2.1, has been verified experimentally with an (original 15kW elevator motor shown in FIG.2. For this purpose, the stator of the motor was connected to the 3phase 50Hz a.c. system (wl~s=2xSOHz).In addition two 3-phase harmonic voltages, one with ~ 1 0 ~ 1 = 2 n . 3 . 5 k Hand z one with Wlosl1=2X.4.5kHZ were injected by means of power amplifiers (see FIG 3). These stator voltages (eq 41) transformed to the stator related rqtating phasor representation (eq 34,16-+42) generate the phasor B of the air gap induction (according to eq(34,26,31-+43)). FIG.2: 15kW elevator motor and sound intensity measurement equipment 27 1 50Hz a.c. system - The right side is valid for 3-phase sinusoidal PWM with the PWM-frequency f l and sinusoidal PFM with the PFM-frequency fF, which modulates the switching frequency F around Fo Eq(4) in TABLE I bottom presents the result: The 3-phase inverter output voltages with 3-phase sinusoidal pulse width modulation (PWM)and sinusoidal pulse frequency modulation (PFh4). We obtain eq(4) immediately by inserting (b/T)&b,c (eq 2) and R (eq 3) in eq( 1) for the inverter output voltage U&b,c. Unfortunately the harmonic components in eq(4) consist of rather complicated terms, which do not disclose the individual frequency components. The next goal therefore must be to decompose the PWM and PFM inverter output voltage in individual frequency components. C: coupling capacitor L: decoupling inductance FIG.3: Test circuit with 50Hz a.c. supply and injection of two predefined 3-phase harmonics with flos1=3,5kHz, flos11=4,5kHz 3.2 Frequency components of the inverter voltage In eq(4, 'TABLE I) we derived the inverter output voltages Uab,c(t) as a function of time, which now have to be decomposed into fundamental and harmonic frequency components. At first glance, this seems to be difficult: The functions cos(wlt..) and COS(wpt..) occur in the argument of a sine and a cosine function as sin( ...cos(w1t..)...) and cos{ ...cos(m~t..)..}.But on looking at it again it can be shown that the decomposition can be achieved with the help of the mathematical eqs(l,2, TABLE TI). Key for the decomposition is the introduction of bessel functions J,(x): FIG.4: Experimental result 3. PWM and PFM inverter, magnetic force and noise 3.1 Formulation of the PWM and PFM inverter voltage as a function of time soidal pulse width modulation (PWM) and sinusoidal pulse frequency modulation (PFM) will be formulated as a function of the time using TABLE I. The middle part of TABLE I1 shows all the intermediate steps of the decomposition and leads directly to eq(4, TABLE 11). This equation still contains the products COS(..Rgt..).COS(..wFt..). cos(..wlt..). But the decomposition is easy to do with the help of eq(3, TABLE XI). A three phase inverter consists of six turn-off elements - two per phase. In TABLE I each phase is represented by a change over-switch, the output of which can be connected either to the positive or negative pole of the d.c. supply voltage. At each intersection of a 3-phase sinusoidal (USta,b,c) and a triangular (UH)control signal, the inverter is changing the polarity of its output voltages u,,b,c. The modulation index ao and the frequency fl of the 3-phase signal USt&b,c determine amplitude aoUd/2 and frequency fl of the fundamental component of the voltages ua,b,c. Induction motors are normally operated with linear voltage frequency characteristic ag-fl. The frequency F of the triangular control signal (UH)determines the inverters switching frequency F. We obtain the final result in eq(5, TABLE XI) with the frequency fl for the fundamental component and the frequency combinations nF& mf#vfl for the harmonic components. n, m and v are ordinal numbers. F=Fo+AF-sin(2xf~t+(p~) is the pulse frequency, Fo the basic pulse frequency, AF the amplitude of the pulse frequency deviation, fF the frequency of the PFM (i.e. the frequency which modulates the pulse frequency) and fl the frequency of the PWM (i.e. the frequency which modulates the pulse width and delivers the fundamental frequency). Now, after this decomposition of the PWM and PFM inverter voltages, the next step is to derive the frequency components of the density of the radial magnetic forces, which cause the magnetic noise. In the first step, the inverter output voltages Ua,b,c(t) with sinu- TABLE I shows in the upper part the line diagrams for the phase a and in the lower part the mathematicalformulation for - the modulation signals uSta (resp. uSab,c) and the carrier signal UH (only line diagram), - the inverter output voltages Ua (resp. Ua,b,c as Fourier series for a given pulseyidth b&bF and pulse frequency F ( = N ~ A ) ) , - the pulse width ba (resp. bab,c) and - the pulse frequency F. Line diagrams and equations are arranged in TABLE I as follows: - The left side is valid for constant pulse width b and constant pulse frequency F - The middle part is valid for 3-phase sinusoidal PWM with the PWM-frequency fi, and constant pulse frequency F 3.3 Frequency components of magnetic force and noise In order to derive the frequency components of the density of the radial magnetic force at the inner surface of the stator, we have to proceed according to eqs(34). This is shown in TABLE I11 with the following intermediate steps: - In a first step the 3-phase PWM and PFM inverter voltages ua,b,c in eq(5, TABLE 11) have to be represented as stator related rotating phasor 61 according to eq(34,16). Eq(1, TABLE 111) is the transformation algorithm and eq(2, TABLE 111) the mathematical equation used for the transformation. Eq(3, TABLE 111) shows the result: ii, = i i l ~ (upper s part of eq.3) + 610s (lower part of eq.3). 27 2 TABLE I Illustration and derivation of the equation for the pulse width modulated (PWM) and pulse frequency modulated (PFM) 3-phase inverter output voltages Ua,b,c(t) as a function of the time. ' "dD UH modulator & ~ I modulation signal uSta, camer signal uH modulation sianal uSta, carrier sianal uH IiI inverter 1% motor modulation sianal uSta. camer sianal uH A 4JH +UH O 0 h A un UH inverter outout vdtaae ua inverter output voltage ua inverter wtwt voltam ua 1 I 0 0 I I pulse width (phase a) pulse width (phase a) 1 M M a5 05 05 0 0 0 pulse frequency (all three phases) pulse frequency (all three phases) pulse frequency (all three phases) I .5 I I FIFO F/FO I 0.6 3-phase sinusoidal modulation signals wl USb,b,c = const %ta,b,c = 'st cos(olt+60 +Dl = 2rcfl; fl: PWM-frequency fundamental frequency D=OO; -120"; +I200 in phase I ua,b,c = Ud[2(b/T)a,b,c -l] + I G$c 4 u I: n = 1,2,3,.. -1s i n [ n x ( b ~ T ) a , b , ~ ] c o s [ n [ ~ ~ d t + ~ ~ ]!:=]p%z fEqUenCy n 2 (1) inverter switching freq. mean value, fundamental (inter-) harmonics 3-phase sinusoidal pulse width modulation 1 (b/T),*,, = { +5 1 UH cos(wlt + 6, + D)} 2 a0 (2) a0 = dation mod 0 + 1 index constant pulse frequency n = 2xF Q, = 2xFO Ai2 = 2xAF na,b,c = Ra,b,c = S2 = a, = const ; AS2 = 0 = R o + A Q sin(oFt+qF) (3) 3-phase inverter output voltages with 3-phase sinusoidal pulse width modulation (PWM) and sinusoidal pulse frequency modulation (PFM) fundamental = + Ud/2. a. cos(w,t+ 6o + D) + + 4/~.~,,/2.x;=~ vn (4) sin[ nx/2.{ l+ao cos(wlt + a0 +D)}] (inter-) harmonics 273 cos[n{ Rot + yo - ~ n / w F'cos(wFt+ (pF I}] TABLE I1 Decompositionof the PWM and PFM inverter output voltages ua,b,=(t)in eq(4, TABLE I) to fundamental and (inter-) harmonic components. Mathematical equations used for the decomposition cos(p + q) = cos( p). cos(q) - sin(p). sin( q) sin(p+ q) = sin(p). cos(q) +cos(p). sin(q) ; [ sin(x. cosy) = C~=,sin(zn/2). 2 - ($)I. J, (x) . cos(zy) ; cos(x . cos y) = Cy==, cos(z n/2).[ 2 -(Lo)]. (1) J, (x) . cos(zy) (2) - 4.cos( h) cos( i) cos( k) = cos( h + i+ k)+cos( h - i+ k)+ cos( h+ i k)+ cos( h - i- k) (3) Decomposition of the expressions sin(...cos...) and cos(...cos...) in eq(4, TABLE I) P q r sin[nn/2+nn/2.aocos(wlt+?50 +D)1 - P = P = -- 9 Y X C;='=, sin(nn/2).cos(v.x/2) . [2-(:)] +cos(nn/2). sin(yn/2) . . Jv(nn/2~ao)~cos[v(wlt+60 +D)l+ . J~(n7c/2~ao)~cos[y(olt+60+D)1 = [2-(:)] z zc- 2 - y - sin[(n+?n/2. = q sin(nn/2).cos[n.x/2.ao cos(w,t + 6 , +D)]+ +cos(nn/2).sin[nn/2-aocos(wlt + 6 , +D)] = . Y . Jv(nn/2~ao)~cos[v(olt+60 +D)] [2-(:)] P+9 P q b P r cos[n(aot + yo) -nAn/(l$ . cos(wFt ( p ~ ) ] = - P q __1 COS[n(not yo)]. COS(m X/2).[ 2 - = & q -- Cos[n(not + yo)]. COS[nAn/wF ' cos(wFt + ( p ~ )+] +sin[n(Qot+yo)]. sin[nL\n/oF-cos(oFt+(pF)] = Y X (i)] . Jm(nA n / w ~ )COs[m(oft . + (PF)] Result: fundamental and (inter-)harmonics of the PWM and PFM inverter output voltages Ua,b,c U,,b,c =Ud/2.ao COS(2nfl t+60 +D)+ + cos[2n(nFo + mfF+ vf, )t m - where: ,U , ( nyo - m n/2 + mcpF + ~ 6+ vD]+ ~ ) ( nyo - m n/2 + mqF- ~6~) - vD] + +COS[2k(nFo-mf~+vf1)t + (nyo-mn//2-m(P~+V60)+VD]+ n=l m O v=O + cos[ 2 X ( nFo- mfF- vfl )t + ( nyo - m n/2 - mqF- vij0) - vD] [ ( ;)I. [2 -(:)I. = sin[(n + v) n/2] .l/n. ud/2. l/n. 2 L____ for m=O for v=o 1 for m>O 1 for v>O 2 AF/fF).Jv (n n/2 .ao) see below (6) see below Un : d.c. SUDD~Yvoltane : PWMGdkx I Bessel functions 0 J,(n L - +1 for n+v=1.5,9. .. -1 for n+v=3,7,1l,.. 0 for n+v=0,24.6,. In + + cos[2n(nF0+ mfF- vfl ) t + m 4 6 8 10 12 14 274 16 18 fl : PWM-frequency, fundamental frequency D=O"(a); -120°(b); +12Oo(c); F=Fo+AF-sin(b f F t + ( p ~ ) n~n/oF F : switching frequency Fo : basic pulse frequency nn'2.ao AF : amplitude of the pulse frequency deviation fF : PFM-frequency 20 TABLE III Derivation of the density of the radial magnetic force from the inverter voltages uqbc in eq(5, TABLE 11). The derivation is based on eq(34). We start with the inverter output voltage Ua,b,c in eq(5,TABLE U): Transformation algorithmus: 3-phase (Ua,b,c) . . + stator related rotating voltage phasor (iil) U,eJoo + ube+J1200 + uce-jlzOo = 312. fi1 .' (1) Stator related rotating voltage phasor Mathematical equation used for the transformation: - . - 7 zcos(x+vD)-e-JD =3/2.e*Jx forv= 1 * * 4 D=0".-120",+120" =3/2.eTjx forv= 2 5 8 =o forv=O 3 6 = GIGS (upper part of eq.3) (2) +Glos (lower part of eq.3): U1 -- +ud/2. ao. ,+j(ait+b) + Stator related rotating phasor B of the air gap induction B: jj= -j. (1-01) .%.Ud.,+j(alt+Ww1.2R.1 01 2 (4) Projection of the rotating induction phasor B upon the axis of the stator winding l a (cpx=O"): B(t,q,) =Re B(t.9, =0) = (5) (~I1- o. 2l )R~. ~1~1. ~ . 2s i n ( o l t + 6 0 ) + + oz(l-01) o'wl.2R-' +(l/Wn+m+v)'sin(On+m+vt+qn+m+v) +(1/~n-m+v).sin(an-m+vt+ 'Pn-m+v) + n=l m=O v=1,2,4, 5,7,8,.. Density p(t,cpx=0) of the radial magnetic force as a function of the time at the position q,=O of the inner stator surface: - + + - - - O , aO ud unmv (l-O1I2 2 ~ 0 ( w l . 2 R . l )2IKfl ~ 2 2R'fn+m+v - + + - ag : PWMindex++l : PWM-frequency, fundamental frequency F=Fo+AF&i(2nfFt+q~) F : switchingfrequency Fo : basicpulsefrequency AF : amplitude of the pulse frequency deviation fF : PFM-frequency f1 U, -.-.A.-.F-(')]. sin[(n+v)E] 2 = +I for n+v=1,5.9.. -I for n+v=3,7.1 I,.. 0 for n+v=0.2.4.6,.. (7) 1u x [2-(:)] 1 2 n. ."', -, m=O 2 for 1 for m>O 2 for v=O I for v>o Ud : d.c. Supply voltage 61.2 : stator (1) and rotor (2) leakage factor : totalleakagefactor wl : number of turns of the stator winding R : inner radius of the stator 1 : active length of the stator 0 275 A F R .J,(n-).Jv(n-ao) fF 2 , s e TABLE 1' (8) - In the next step the rotating phasor B of the air gap induction B has to be derived from rotating voltage phasor ii1 in eq(3, TABLE 111) according to eq(34,26,31). We obtain eq(4, TABLE In). - Considering a small stator section in position cpx (FIG.1)as a source of noise we have to project the rotating phasor B in eq(4, TABLE 111) on this stator section. For this step we apply the transformation eq(34,32) and use the mathematical equation: Re(-je+J") = Re(+je-jx) = sin(x) Since the position cpx of this section has no influence on amplitude and frequency of the air gap induction B(t,cp,) (see eq(45)), we choose qx=O for the sake of simplicity. cpx=O is the axis of the stator winding la. The result B(t,cpx=O) is represented in eq(5, TABLE 111). B(t,cp,) has still the same frequency components as 61 in eq(3, TABLE 111). - The last step is to square the induction B(t,cpx)+B(t,cpx)2 according to eq(34,33) in order to obtain the density p(t,cpx) of the radial magnetic force at the stator sector, which we regard as a source of noise. For this step we use the mathematical equation 2sinx-siny=cos(x-y)-cos(x+y). In addition we take into account that the harmonic components are much smaller than the fundamental component. The noise generated due to the vibrations of the stator caused by the radial magnetic force has the same frequencies as p(t,qx): The magnetic force density p is a pressure, which generates a sound pressure. - AF, or more precisely dF/fF, determines the amplitudes of the frequency components (see Jm(nhF/fF) in eq(5,6,TABLE 11)). Higher the AF/fF, greater is the number of side band frequency components of higher order m and hence wider is the spread in spectrum. FIGSb shows an example with AF/f~=O.45kHz/0.5kHzS . 9 . FIG.% shows an example with AFVf~3.2kHz/0.4& =8. loo ! I "." nlv " I .V bl 'O" c' The final result in eq(6, TABLE 111) shows that each frequency component of the stator voltage generates two noise frequencies, one which is higher and one which is lower by the fundamental PWM frequency fl. 4. Spreading of the frequency spectrum PWM inverters with constant switching frequency generate voltages with dominant amplitudes at only a few single frequencies around the multiples of the switching frequency. The goal of an additional PFM is to spread the frequencies of the inverter voltage more evenly over the whole spectrum, thus generating noise which is more agreeable. This is demonstrated - for PWM in FIG.5a and - for PWM with PFM in FIG.Sb,c according to eq(5, TABLE 11). In FIGSb the parameters A F and fF are set for optimal illustration of the basic principle. In F I G 3 the parameters AF and fF are set for a more evenly spread of the frequencies. 4.1 Influence of the parameter settings Noise reduction for drives is especially relevant when the switching frequency of the inverter is below 1SkHz. This is the case for drives with IGBT-inverters between some kW and some 1OOkW. Let us assume that the drive has to be operated with the fundamental frequency fl=0 to 50Hz (PWM-index ao=fl/SOHz) and that the upper limit of the basic switching frequency is F~5kHz. The parameters fF (PFM-frequency) and AF (amplitude of the PF deviation), which can be used to "design" the spreading of the spectrum of the inverter output voltage (eq5.6, TABLE 11) have the following influence: - fF directly determines the distance kmfldvfl of the side band frequencies from the multiples nFoof the basic switching frequency. FIG.5b shows an example: n~SkHi&~n.O.SkH&.40Hz I 10'' lo-' 0 2 4 6 8 10 12 14 16 18kHz 2 4 6 8 10 12 14 16 l8kHz II 0 FIGS: Spectrum of the inverter output voltage (eq5,6, TABLEII) with flGS=fl for the fundamental and fn*&v=nFokmf+fl for the harmonic components. nF0 kmfF fV'f1 a) PWM : n.5kHz f v 4 0 H z ; AFd ; e . 8 b) PWM and PFM: n5kHz+m.500H&v40Hz; AFd.451CHz; wd.8 c) PWM and PFM: n 5 k H z h n 4 0 0 H z i v 4 ; AF=3.21rHz ; w . 8 where: F=Fo+AF.sin(%f~t+) F : PF fF : PFM-frequency Fo : basic PF fl : PWM-frequency, fundamental AF : amplitude of the PF deviation 4.2 Selected harmonics elimination The fact that the harmonic frequencies of the described PWM and PFM inverter are calculable, has the great advantage that the generated frequencies can be "designed. That means for example that frequencies which would exite a mechanical resonance of the stator, can be avoided or eliminated. The F I G 6 illustrates an example for a selected harmonics elimination in a small frequency band of the evenly spreaded frequency spectrum. FIG.6a shows a detail (0...5kHz) from FIG.%. Assuming a mechanical resonance frequency of e.g. 3kHz, FIG.6b illustrates how harmonics which would excite this resonance can be eliminated. The frequencies to be eliminated can be determined by the relation AF/fF. As shown in eq(5,6, TABLE II), the amplitude U,, of the interharmonics with the frequencies nF&mf+vfl are proportional to the bessel function Jm( nAF/fF). By an adequate setting of the parameters AF and fF, the bessel function J,(nAF/fF) can be made zero for predetermined ordinal numbers we get n and m. In our example with AF/f~=3.SlcHz/0.4MIz=8.75 276 J,(nAF/fF)=O for n=l and m=5. That means, the amplitudes UnmpUyjvof the harmonics with the frequencies fn-mfv=fl-5fv =51rHz-5~400Hztv.40Hz =3lrHztv.4OHz are zero (UIS,=O). This selected frequency elimination is independent on the modulation index a0 and the fundamental frequency fl and therefore independent on the operating point ( a p f l ) of the motor! a' course, can be done in both ways with carrier modulators or with space vector modulators with the same result. The implementation of the described PWM and PFM on the basis of a space vector modulator is currently under progress. 5. Conclusions The theoretical results and the experimental verification presented in this paper are the basis for the practical implementation with an IGBT-inverter-fed a.c. motor which will be ready for tests in early summer 1994. The final optimization of the PWM and PFM will be done with the feedback of the practical results. I1 4 wz Acknowledgment 5 The authors wish to express their thanks to Schindler AG at Ebikon in Switzerland for their generous hardware support for experiments. 10.' References 10' 1 2 3 4 wz r 11 Kirlin R.L., 5 FIG.6: Spreaded spectrum of the inverter output voltage a) A detail (0...5kHz) from FIG& b) Selected harmonics elimination at 3kHz 4.3 Carrier modulation -space vector modulation The derivations in the previous chapters were based on carrier modulation. The reason is, that the carrier modulation is better suited than space vector modulation to visualize the PWM and PFM as functions of the time. The practical implementation, of 277 Kwok S., Legowski S., Trzynadlowski A.M.: Power Spectra of a PWM Inverter with Randomized Pulse Position (IEEE PESC Conf. Rec.. pp.1041-1047, 1993) P I Schonung A., Stemmler H.: Static frequency changers with subharmonic control in conjunction with reversible variable speed ac drives (Brown Boveri Rev., AugfSept. 1964) [31 Stemmler H.:Ein- und mehrpulsige Unterschwingungswechselrichter, Steuerverfahren, Strom- und Spannungsverhidtnisse (Control in Power Electronics and Electrical Drives, IFAC, Volume 1,1974)