4.4.2 OPTIMAL PWM /
HARMONIC ELIMINATION
PWM
− Terminology
Optimised PWM switching strategies
Harmonic elimination PWM technique
(HEPWM)
Programmed PWM techniques
− Optimal PWM switching strategies
Designed to optimise some specific
performance criteria
harmonic voltage elimination
harmonic current minimisation
harmonic torque and rotor speed ripple
minimisation
harmonic loss minimisation
Harmonic energy tend to be distributed
NAA-2002
1
OPWM/HEPWM (2)
− Harmonic elimination PWM
switching strategies
Subset of OPWM
− Programmed PWM techniques
Derived from practical implementation
aspect of the OPWM switching
strategies
Large number of OPWM switching
angles are usually programmed off-line
into an EPROM or a microprocessor’s
memory
− General concept
NAA-2002
2
OPWM/HEPWM (3)
Consider a particular performance
criteria : elimination of several lowerorder harmonics in the inverter output
Consider generalised quarter-wave
symmetric PWM waveform
Line to line PWM waveform (three-level
switching)
Line to neutral PWM waveform (two-level
switching)
Extended to various schemes based on
single or three-phase inverter configuration
NAA-2002
3
OPWM/HEPWM (4)
Determine Fourier coefficients of
generalised PWM waveform
Fourier coefficient equation in terms of
N variables (N : number of switching
angles per quarter cycle)
Equate N-1 harmonics to 0 and assign
specific value of amplitude of the
fundamental of inverter output voltage in
per unit value (ap1)
Equations are non-linear and
transcendental → multiple solutions are
possible
NAA-2002
4
OPWM/HEPWM (5)
A set of solutions for switching angles
satisfying criterion
α1 < α2 < α3 < ------ < αN < π/2
have to be obtained for each increment in
ap1 for voltage control with simultaneous
elimination of harmonics
Example of generalised quarter-wave
symmetric PWM waveform
0
180
α1
360
α2
NAA-2002
5
OPWM/HEPWM (6)
Only odd harmonics exist, Fourier
coefficients given by:
The non-linear equations to eliminate N-1
lower-order harmonics such as 3, 5, 7 etc.
are in the form of
NAA-2002
6
OPWM/HEPWM (7)
Non-linear equations have to be solved
using suitable numerical methods i.e.
standard math library for PC
environment – IMSL, NAG (Matlab)
Example of results obtained for
varying ap1 values ( N = 8)
Angles (°)
α1
α2
α3
α4
α5
α6
α7
α8
0
17.1289
17.1289
38.2281
38.2281
57.7423
57.7423
79.8438
79.8438
0.1
19.6472
20.3303
39.3403
40.6252
59.1186
60.8510
79.0085
80.9795
0.2
19.2740
20.6352
38.6476
41.2126
58.2048
61.6715
78.0003
81.9507
0.3
18.8821
20.9106
37.9216
41.7569
57.2542
62.4590
76.9690
82.9167
0.4
18.4723
21.1513
37.1605
42.2496
56.2598
63.2076
75.9055
83.8804
ap1
0.5
18.0443
21.3503
36.3605
42.6774
55.2099
63.9060
74.7955
84.8447
0.6
17.5968
21.4976
35.5147
43.0188
54.0862
64.5320
73.6150
85.8124
0.7
17.1261
21.5780
34.6105
43.2374
52.8571
65.0403
72.3174
86.7875
0.8
16.6234
21.5650
33.6237
43.2645
51.4623
65.3223
70.7964
87.7750
0.9
16.0634
21.3982
32.4922
42.9384
49.7584
65.0385
68.7283
88.7851
Switching angles solutions trajectories
for N=8
NAA-2002
7
1
15.2985
20.7779
30.8777
41.5097
47.0563
62.2443
64.3129
89.8477
OPWM/HEPWM (8)
α8
90
Angles (degree)
80
70
α7
60
α6
50
α5
40
α4
30
α3
20
α2
10
α1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
.
Amplitude of the fundamental of each
MSMI module in p.u. (ap1- m)
For N = 8 (7 harmonics eliminated)
Inverter switching frequency
fsw = (N + 1)f
Order of 1st and 2nd significant harmonic
component
D1 = 2N + 1
D2 = 2N + 3
NAA-2002
8
OPWM/HEPWM (9)
PWM waveform and harmonic
spectrum of a single-phase bridge
inverter output voltage for ap1 = 0.8
250
200
150
)
s
tl
o
V
(
o
V
e
g
a
tl
o
v
t
u
p
t
u
O
100
50
0
-50
-100
-150
-200
-250
0. 005
0. 01
0. 015
0. 02
0. 025
0. 03
0. 035
t (s e c . )
0. 04
0. 045
0. 05
0. 055
2
1. 8
1. 6
s
t
n
e
i
c
fi
f
e
o
c
r
e
ri
u
o
F
d
e
s
il
a
m
r
o
N
1. 4
1. 2
1
0. 8
0. 6
0. 4
0. 2
0
0
5
10
15
20
25
Ha rm onic orde r (n)
NAA-2002
30
35
40
9
OPWM/HEPWM (10)
Disadvantages
Computational difficulties (lower-output
frequency range, large number of PWM
switching instants)
Only local minimum obtained
Advantages over SPWM
About 50% reduction in inverter switching
frequency
Higher voltage gain due to over modulation
– higher utilization of power conversion
process
High quality of output voltage and current
– small ripple in DC link current –
reduction in size of DC link filter
components
Reduction in switching frequency –
reduction in switching losses - high power
applications
NAA-2002
10
OPWM/HEPWM (11)
Elimination of lower-order harmonics
causes no harmonic interference such as
resonance with external line filtering
networks typically employed in inverter
power supplies
Traction application – power frequency
signaling components can be avoided over
entire frequency range of drive
Precalculated PWM switching patterns
avoids online computations
References
Enjeti, P N., Ziogas, P. D. and Lindsay, J.
F. (1990). “Programmed PWM Techniques
to Eliminate Harmonics: A Critical
Evaluation.” IEEE Transactions on
Industry Applications. 26 No. 2. 302-316
H. S. Patel and R. G. Hoft (1973).
“Generalized Techniques of Harmonic
Elimination and Voltage Control in
Thyristor Inverters: Part I-Harmonic
Elimination.” IEEE Transactions on
Industry Applications. IA-9 No. 3. 310-317
NAA-2002
11
OPWM/HEPWM (12)
Performance parameter suitable for
drive application
Harmonic loss minimization instead of
eliminating individual harmonics
Harmonic loss in machine is dictated by rms
ripple current
NAA-2002
12
OPWM/HEPWM (13)
Harmonic copper loss
PL = 3I2rippleR
R : effective per phase resistance of the
machine
Iripple is a function of α because Vn is a
function of α
The values for α must be solved for so as to
minimize Iripple for a certain desired
fundamental magnitude
Typical implementation
Microprocessor, lookup table of angles
Down-counters to generate pulse widths in
time domain
NAA-2002
13
OPWM/HEPWM (14)
Counters clocked at fck = Kf. If K = 360,
waveform generated has 1° resolution
Lower fundamental frequency, switching
angles can be increased, higher numbers of
significant harmonics can be eliminated −
larger lookup table
NAA-2002
14