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)