ifJE!C
ELSEVIER
Harmonic
SYSTErnS
AESECIRCH
Electric Power Systems Research 40 (1997) 45. 49
elimination in pulse-width modulated inverters using
piecewise constant orthogonal functions
J. Nazarzadeh a, M. Razzaghi h,c.*, KY.
Nikravesh a
Received 17 May 1996: accepted 8 July 1996
Abstract
A new method is presented for selective harmonic elimination in pulse-width modulated (PWM) inverter waveforms by the use
of piecewise constant orthogonal
functions. The block-pulse functions are first applied and the relationships between these
functions with Walsh functions and Fourier series are used for harmonics elimination in a PWM inverter. The set of systems of
linear equations obtained replaces the system of nonlinear transcendental equations used in the Fourier analysis approach. As
compared with the Walsh domain technique. the present algorithm reduces the number of combinations for the case where more
than one angle is allowed to vary within a given interval. % 1997 Elsevier Science S.A.
KeJwordJ:
Harmonic; Elimination; Piecewise: Orthogonal: Inverters
1. Introduction
has the restriction that, within a given interval, only
one angle is allowed to vary; moreover, if there exists a
The principles
and applications
of pulse-width
modulation (PWM) techniques have been investigated
in the
literature extensively. There are three methods of PWM
solution
waveform generation: the suboscillation technique [1:2],
distortion minimization [3.4] and harmonic elimination
[5,6]. All three have the common characteristic that the
analysis of PWM waveform generation takes place in
the Fourier domain. In particular, the harmonic elimination approach given in [S] produces a system of
nonlinear transcendental equations that requires the
NewtonRaphson matrix method for its solutions. This
algorithm requires starting values for the angle and
does not always converge to the required solution.
Asumadu and Hoft [7] and Swift and Kamlibries
[8]
used a different approach to the problem of harmonic
elimination for PWM waveform generation. This approach is based on the use of Walsh series expansion of
PWM waveforms rather than of the Fourier series.
More recently Razzaghi and Nazarzadeh [9] applied
Walsh seriesfor optimum PWM patterns in an induction motor. It is pointed out in [8] that their algorithm
* Corresponding author.
0378-7796 97 $17.00 e 1997 Elsevirr Science S.A. All rights reserved
PII SO378-7796(96)01
134-O
that requires
two or more angles to vary in the
same selected interval, then such a solution can not be
detected by the method in [8]. This shortcoming can be
overcome by increasing the number of intervals which
increases
the number
of cases to be considered.
It is
shown in [S] that the problem of harmonic elimination
using Walsh series is converted into a combinatorial
problem.
The goal of this paper is the harmonic elimination
of
PWM waveforms using piecewise orthogonal functions
and reducing the number of combinations given by the
algorithm
in [8]. The present
work
first uses the block-
pulse functions and the relationships between the blockpulse functions and Walsh seriesand Fourier seriesare
given. The set of systems of linear equations obtained
replaces the system of nonlinear transcendental equations used in the Fourier series harmonic elimination
approach.
Algorithms based on block-pulse functions are computationally more efficient and faster than those based
on Fourier
analysis.
Illustrative
examples
are given
to
demonstrate the applicability of the proposed method.
2. Walsh functions
Walsh functions form a complete orthogonal and
orthonormal
set that take only two amplitude values.
+ 1 and - 1, in the time interval [0, 1). There are three
ways of ordering the Walsh functions (dyadic, natural,
and sequence ordering) depending on the method used
to generate them [lo]. Walsh functions have two arguments; the order n and the normalized time I The
function is represented as
Fig. 2. PWM
wavefbrm
for the case of 0 < r < $
Wal(n. I)
Wal(0, I) dt
(2)
‘f(r) Wal(n, t) dt
s0
A = [Ll,). NI...., a,y- ,I’
(3)
Cl,, =
Fig. 1 shows a set of the first eight dyadic-ordered
Walsh functions.
Similar to the Fourier
series representation,
the
Walsh series representation
of any periodic function
f‘(t) defined over [O, 7’j is given by
v-1
J’(t) = a,, Wal(0, f) + 1 u,, Wal(n, t) = ATW(t)
I, = I
-
k
I0
‘fir)
u,, = f
(4)
and
(1)
IV(t) = [Wal(O, r). Wal(1, t) . . . . . Wal(N-
1, t)]’
(5)
Eqs. (l)-(3)
form a Walsh transform that applies to
a function defined in the time interval ]O, I). For the
piecewise continuous function, the integral of Eq. (3)
can result in a different answer. For example, for a
PWM waveform with one switching angle x, 0 < K < $,
using Fig. 2 and considering the coefficient u, we have
where
1
u;
J(r) Wal(7, 1)4[=4[l
s0
= -4x
=
* df +i’k
-df]
(6)
and using Fig. 3 for the case f < 2 <i
the coefficient u, we have
and considermg
14
- dt = 43: - 1
(7)
I1
Thus, we get two different values for ~1~.
It is shown in [8] that, by using a Walsh series, 41 664
different cases should be considered for N = 64 in the
first quarter period and with 3 switching
angles. TO
reduce the number of different cases we use the blockpulse functions and the result will be converted to a
Walsh series using a conversion matrix.
Ll7 =
I
I
G
0
Fig.
I The first e&It
8,
0.5
dyadic-ordered
,
4
,t
1
Walsh
functions
Fig. 3 PWM
wdveform
for the case of $ < Y < $
where R is the block-pulse-Walsh
The inverse relation is also valid.
A
+
---------...
+1
-
f
C= R
conversion
‘A
matrix.
(15)
- -
for N = 4, we have [I l]
f-
”
N
Fig. 4. ,f
is the incoming
3. Block-pulse
1
4
b
a
t
“+I
N
and f + is the outgoing
R=waveform.
functions
Since the proposed method is essentially based on
block-pulse functions, a short review of the relationship
between Walsh functions and block-pulse functions is
given as follows. If N = 2d, where d is a positive integer,
both N Walsh functions Wal(n, t) and block-pulse functions b(n, t), n = 0, l,..., N - 1. can be defined on the
interval te[O, 1) [l 11. The set of block-pulse functions is
a complete orthogonal set and is defined as
1 for t~[niN,
0 elsewhere
b(n, t) =
(n + 1);N)
(II+ I) .\
C=[c,,
f(t)b(n.
t)dt,
n=O,
l,.,.. N-
(‘I,..‘. C.,7LJT
1
(10)
(11)
and
B(t) = [b(O, t), b(1, t) ,.... b(N-
-1
-1
1
-1
1
1
-1
-1
-1
1
1
(16)
When the Walsh series representation of a time signal is
required to be converted to the more familiar Fourier
series representation, then the Fourier transforms of the
Walsh functions are needed in the conversion equations. A recursive formula by Blachman [8] is used to
evaluate the Walsh transforms of sinusoids. As a result,
the following expression holds
HA = v
(17)
where V is the vector containing the coefficients of the
Fourier series expansion of the same time function and
H is the Fourier-Walsh
conversion matrix.
4. Harmonic
where
i I, ,v
1
1
(8)
The block-pulse
representation
of a periodic function
f(t) defined over [0, 1) is given by a finite series as
.v ~ I
f(t) = 1 Nc,,b(n. t) = NCTB(r)
(9)
,I = 0
c,, =
;
1
1
1
1
1, r)]’
(12)
For a piecewise continuous waveform
the integral of
Eq. (10) in the interval [n/N, (n + 1)/N) is given by
elimination
in PWM
inverters
The proposed scheme first uses block-pulse functions
and then Walsh functions and eventually Fourier series
to eliminate a given harmonic in a PWM waveform. In
this scheme we first choose a fixed number for blockpulse functions, say N,, and some starting intervals for
switching
angles x,, x7,. , x,. We then eliminate the
required harmonics and find zj, i = 1, 2 ,..., j. If the
calculated angles satisfy the starting intervals for all CI,,
i= 1. 2...., ,j then we obtain satisfactory results. Otherwise we change the starting intervals for the same N,
until we get the required r,, i = l,.... j. To increase our
accuracy we increase N, to 2N, and the same procedure
will be carried out until the required precisions are
obtained. It should be noted that the intervals for the
switching angles in the second stage are obtained from
the results calculated in the first stage.
(13)
where .f’- and f”
are the incoming and outgoing
waveforms
respectively (see Fig. 4). In the case of a
PWM waveform,
from zero to one we get f - = 0 and
f’ = 1 and from one to zero we have f- = 1 and
f'=O.
The block-pulse series representation of a time signal
can be converted to the Walsh series representation by
a conversion matrix given in [ll]. Using Eqs. (4) and
(11) the following expression holds:
RC=A
(14)
5. Illustrative
examples
5.1. E~~unple 1 (eliminution
half-bridge incerter)
of’ the jifth
hurmonic in u
Let L’,,, denote the nzth harmonics of the voltage
supply and let 2, and rz define the two angles used to
eliminate the fifth harmonic cg and control the fundamental component L‘, to a value of 0.8391 in a halfbridge inverter. Fig. 5 shows the waveform
of the
inverter output in the first quarter period.
We choose N, = 16 block-pulse
functions
starting intervals for switching angles as
oat;<&,
and the
+ei:<&
(18)
By using Eq. (13). we first find the coefficients. For cg
we have fP = 1 and f’ = - 1 and for C, we have
f - = - 1 and f + = 1. Thus we get
c(,=[X)-o]-[+rj]=2rf-&
(19)
Fig. 6. Full-bridge
inverter.
and
c,=
-[x:-&]+[+cf;]=
-2%Af&
(20)
Furthermore,
C~ = cj = & and in a similar manner ci,
i = 5, 6,..., 15 can be calculated. Using Eq. (14) we get
the following nonzero coefficients for the Walsh functions:
=
Cl*,
=
-
8x; + 8nf
4 -
u25=U3,=~-sa~-8X~
c, and L’~ are given by:
z’, = 1.691816+4.63710+
-8~;
and using Eq. (17) we obtain the following
for the Fourier series:
harmonics
us = - 0.28432 + 11.267 1Oa ; - 2.64364~;
11.1981ocr; -9.49319cc;
&<x;<&.
Since the fundamental voltage is to be controlled
the fifth harmonic is to be eliminated we set
cx) = 0.05044
and
and
solutions
x; = 0.10742
(21)
&<2;<&
In this case the nonzero
functions are given by
(23)
coefficients
(23)
for the Walsh
u, = i + 8~; - 8x;
(24)
In this case, we get the following
solutions
( = 22.499”)
and
for ,xi and CC;:
To increase our accuracy, since the solutions in Eq. (21)
are in the starting intervals, we increase N, to 32 and
choose starting intervals for x: and xf as
j+:+.
Y; = 0.11238
&<Z:<&
2; = 0.062497
rs = 0
This yields the following
and
The solutions in Eq. (23) are in the starting intervals
given in Eq. (22); hence to obtain better precision we
increase N, to 64 and choose the starting intervals for
2: and xl as
u3 = 0.825461 + 8.38391a; - 14.80070~;
and
By setting c, = 0.8391 and L’~= 0 we get the following
solutions:
XT = 0.06033
u, = 1.63141 + 3.10143~; - 8.83212~;
c’,= - 1.11131+
10.13411~;
t15= - 1.42381 + 15.29121~; + 4.46027~~:
all = a13 = -1+Zrf+8r;
L+ = 0.8391
u,lJ
Furthermore,
cl, = 1 + 8%; -8x?
u,=8ci;
u11= a13= - 1 + 81: + 8a;
2; = 0.113999
(= 41.039”)
5.2. E.vumple 2 (&.ninution oj’ the fjih
hurmonics in u fill-bridge
inL)erter)
and seventh
Let x,, x, and xi define the three angles that eliminate L-‘~and C, and give the value of the fundamental
component L', = $, Fig. 6 shows the waveform
of the
inverter output in the first quarter period. Let NI = 16
and
1
,,<x
t <z.
2
2
~iIyI‘<16’
3
3
ik<“:<G
4
(25)
A
+1
-
0
-1
The coefficients for the block-pulse
first quarter period are given by
“1
O12
a1
0.5
1
b
I
C” =[o.+x;,a:-&,~-“:]
Using Eq. (14) we get the following
for the Walsh functions:
-
Fg
5. Half-bridge
inverter
u, = 1-4x;
+4x;-4%:
functions
in the
(26)
nonzero coefficients
J. ~Va~arzudeh
er al. /Electric
Polcer
UT= -4x;-4x:+4x:
all = -2f4cr;
a13
= - 1+4a)
+4cr;+4cr:
-4cr;-4cI:
and using Eq. (17) we obtain the following
for the Fourier series:
harmonics
u, = 1.67486 - 4.41606 + 3.0~; + 6.60911~;
- 7.79596~~;
u3 = - 0.827386 - 7.40034n; + 1.47202~: + 6.27371~;
u5 = 1.93694 - 1.32182~; - 6.64524cr; - 3.76422~;
c’,=
- 1.26821 f4.74662~;
Since the fundamental
the fifth and seventh
we set
u, = 0.8391
and
1>(5
= u, = 0
(27)
solutions
a: = 0.148996
for x t, Y: and 2:
and
$<a$<&,
r$ = 0.20997
(28)
&<@:<J$
(29)
Using Eq. (27), the solutions obtained in this case are
in the starting intervals given in Eq. (29) thus to
increase our precision we increase N, to 64 and
choose the starting intervals for a; and X: and LY: as
&aY:<g,
g<,:<g,
g<&,;<g
In this case, using Eq. (27) we get the following
tions
ai’ = 0.146675
( = 52.803”)
a; = 0.179182
( = 64.505”)
(30)
solu-
and
u33 = 0.214895
40 (I 997) 45-49
49
verters. An analysis of PWM waveforms
in this way
shows that linear solutions can be obtained relating
angles and voltage fundamentals.
The method substitutes linear algebraic equations for the nonlinear
equations required in Fourier series harmonic elimination. Compared with the Walsh method the present
method reduces the number of combinations
for the
case where, within a give interval, more than one
angle is allowed to vary (see [S]). The method is
efficient and faster than those based on Fourier analysis. The given numerical examples support
these
claims.
References
111J. Wilson
Since the solutions in Eq. (28) are in the starting
intervals given in Eq. (25) the accuracy is increased
by increasing N, to 32 and starting intervals zt, X:
and G?: are chosen as
+ai;<g,
Reaetrrc/~
1.11371sr;
voltage is to be controlled and
harmonics is to be eliminated
This yields the following
a; = 1.118359,
f3.17157rwi-t
Systems
( = 77.362”)
6. Conclusions
This paper is based on the use of block-pulse functions for selective harmonic elimination in PWM in-
and .I. Yearnans,
Intrinsic
harmonics
of idealized
inverter
PWM systems. Proc. IEEE Indusq
Applications
Societ>’ Annu. Meet.. 1976, pp. 961-913.
A high performance
pulsePI B.K. Bose and H.A. Sutherland,
width modulator
for an inverter-fed
drive system using a microcomputer,
Proc. IEEE Industry
Applications
Society Annu.
Meet.. 1982. pp. 8477853.
PWM
waveforms
of a mi[31 J. Caste1 and R. Hoft, Optimum
croprocessor
controlled
inverter,
IEEE Power Electronics
Specialist Conj: Rec.. 1978, pp. 2433250.
M. Rostami
and K.Y. Nikravesh,
@rimurn
[41 .l. Nazarzadeh,
PWM
Pattern fix Torque Distortion
Minimirufion
in Indurtion
Motors.
ACEMP’92.
May 1992. pp. 27 -29.
techniques
of harmonic
151H. Pate1 and R. Hoft. Generalized
elimination
and voltage control in thyristor
inverters:
Part Iharmonic
elimination,
IEEE Trans. Industry
Applications,
IA-9
(1973) 310~317.
techniques
of harmonic
161H. Pate1 and R. Hoft, Generalized
elimination
and voltage control
in thyrstor
inverters:
Part 2Voltage control
technique.
IEEE Trans. Industry
Applicarions,
IA-10 (1974) 666-673.
and R.G.
Hoft.
Microprocessor
based sinu[71 J.A. Asumadu
soidal waveform
synthesis using Walsh and related orthogonal
functions,
IEEE Trans. Power Electron..
4 (1989) 2344241.
181F. Swift and A. Kamberis. A new Walsh domain technique of
harmonic
elimination
and voltage control
in pulse-width
modulated inv,erters.
IEEE Trans. Power Elecrron.,
8 (April
1993)
170&185.
and J. Nazarzadeh.
Optimum
pulse-width
modu[91 M. Razzaghi
lated pattern in induction
motor using Walsh functions,
Electr.
Po\ter S,,.\r. Rrs., 35 (1995) 87791.
Chen
and W.K.
Leung.
Algorithms
for converting
[lOI C.F.
sequency-dyadic.
and
Hadamard-ordered
Walsh
functions,
Mathematics
and Computers
in Simulation.
27 (5-6)
(1985)
471-478.
the inverse of a ma[I 11 Z.H. Jiang, New method for computing
trix whose elements
are linear combinations
of Walsh
functions. Int. J. Sysrems Sci., 20 (1989) 2335-2340.