SPWM

Power Electronics and Drives (Version 2): Dr.

Zainal Salam, 2002

1

4.4.1 SPWM

• Natural sampling– Amplitudes of the triangular wave

(carrier) and sine wave (modulating) are compared to obtain PWM waveform

Modulating Waveform Carrier waveform

1M1+

1−

0

2dcV

2dcV−

00t 1t 2t 3t 4t 5t


Zainal Salam, 2002

2

SPWM (2)

– Implementation example�Analog comparator chip that

compares the 2 waveforms

�Generation of the carrier signal


Zainal Salam, 2002

3

SPWM (3)

�Generation of the modulating signal


Zainal Salam, 2002

4

SPWM (4)

• Regular sampling– Asymmetric and symmetric

T

samplepoint

tM mωsin11+

1−

4T

43T

45T

4π

2dcV

2dcV

−

0t 1t 2t 3tt

asymmetric sampling

symmetricsampling

t

Generating of PWM waveform regular sampling


Zainal Salam, 2002

5

SPWM (5)

( )

(1,2,3...)integer an is and signal modulating theoffrequency theis where

M

:at locatednormally are harmonics The .frequency" harmonic" the torelated is M

waveformmodulating theofFrequency veformcarrier wa theofFrequency M

)(MRATIO MODULATION

ly.respective voltage,(DC)input and voltageoutput theof lfundamenta are , where

M

:holds iprelationshlinear the1, M0 If

versa. viceandhigh isoutput wavesine the thenhigh, isM If magnitude. tageoutput vol

wave)(sine lfundamenta the torelated is M

veformcarrier wa theof Amplitude waveformmodulating theof AmplitudeM

:MINDEX MODULATION

R

R

R

R

1

I1

I

II

I

I

kf

fkf

p

p

VV

VV

m

m

in

in

=

==

==−−−−−−−−−−−−−−−−−−−−−−−−−−−−

=

<<

=

=


Zainal Salam, 2002

6

SPWM (6)

• Bipolar switching– Pulse width relationships

k1δk2δ

kα

∆4∆=δ

π π2

carrierwaveform

modulatingwaveform

pulsekth

π π2


Zainal Salam, 2002

7

SPWM (7)

– Characterisation of PWM pulses for bipolar switching

∆

0δ 0δ 0δ 0δ

k1δk2δ

2SV+

2SV−

kα


Zainal Salam, 2002

8

SPWM (8)

– Determination of switching angles for kth PWM pulse

v Vmsin θ( )

Ap2Ap1

2dcV+

2dcV

−

AS2

AS1

22

11

second,-volt theEquating

ps

ps

AAAA

=

=


Zainal Salam, 2002

9

SPWM (9)

[ ]

)sin(sin2

cos)2cos(sin

sinusoid, by the supplied second- voltThe

where; 2

Similarly,

where

22

2)2(

2

:asgiven is pulse PWM theof cyclehalfeach during voltageaverage The

21

2222

11

11

111

okom

kokmms

o

okk

dckk

o

okk

sk

o

okdc

o

kokdck

V

VdVA

VV

VV

VV

k

ok

δαδ

αδαθθ

δδδββ

δδδβ

βδ

δδδ

δδδ

α

δα

−=

−−==

−=

=

−=

=

−

=

−−

=

∫−


Zainal Salam, 2002

10

SPWM (11)

)sin()2(

)sin(222

edge leading for the Hence,

;strategy, modulation thederive To

22

;22

, waveformsPWM theof seconds- voltThe

)sin(2Similarly,

)sin(2, smallfor sin

Since,

1

1

2211

21211

2

1

okdc

mk

okmoodc

k

spsp

odc

kpodc

kp

okmos

okmosooo

VV

VV

AAAA

VAVA

VA

VA

δαβ

δαδδβ

δβδβ

δαδ

δαδδδδ

−=⇒

−=

==

=

=

+=

−=→


Zainal Salam, 2002

11

SPWM (12)

[ ]

[ ])sin(1 and

)sin(1

width,-pulse for the solve tongSubstituti

)sin(

:derived becan edge trailing themethod,similar Using

)sin(

Thus,

1. to0 from It varies depth.or index

modulation asknown is 2

ratio, voltageThe

2

1

11

2

1

okIok

okIoko

okk

okIk

okIk

dc

mI

M

M

M

M

)(VVM

δαδδ

δαδδδ

δδβ

δαβ

δαβ

++=

−+=⇒

−=

−=

−=

=


Zainal Salam, 2002

12

SPWM (13)

[ ]kIok

kk

kk

M αδδ

δδδδδ

δα

δα

sin1

,Modulation SymmetricFor different. are andi.e ,Modulation

Asymmetricfor validisequation above The

:edge Trailing

:edge Leading

:is pulsekth theofanglesswitching theThus

k 2k 1k 2k 1k

1

1

+=⇒

==

+

−

– ExampleFor the PWM waveform shown, calculate the switching angles for all the pulses.


Zainal Salam, 2002

13

SPWM (14)

V5.1V2

π π2

1 2 3 4 5 6 7 8 9

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12t13

t14t15

t16t17

t18 π2π

1α

carrierwaveform

modulatingwaveform

– Harmonics of bipolar PWM waveform


Zainal Salam, 2002

14

SPWM (15)

{

})2(cos)(cos )(cos)(cos )(cos)2(cos

: toreduced becan Which

sin2

2

sin2

2

sin2

2

sin)(12

:as computed becan pulse PWM (kth)each ofcontent harmonicsymmetry,

wave-halfiswaveformPWM theAssuming

212

1

2

2

0

2

2

1

1

okkkkkkk

kkokdc

nk

dc

dc

dc

T

nk

nnnnnn

nV b

dnV

dnV

dnV

dnvfb

ok

kk

kk

kk

kk

ok

δαδαδαδαδαδα

π

θθπ

θθπ

θθπ

θθπ

δα

δα

δα

δα

δα

δα

+−++−−++−−−−=

−+

+

−=

=

∫

∫

∫

∫

+

+

+

−

−

−


Zainal Salam, 2002

15

SPWM (16)

[]

equation. thisofn computatio

theshows pagenext on the slide The

:i.e. period, oneover pulses for the of sum isthe waveformPWM

for the coefficentFourier ly.Theproductive simplified becannot equation This

2coscos2 )2(cos)(cos2

Yeilding,

1

11

∑=

=

+−−−=

p

knkn

nk

ok

kkkkdc

nk

bb

pb

nnnn

nVb

δααδα

π


Zainal Salam, 2002

16

SPWM (17)– Harmonics spectra

p p2 p3 p40.1=M

8.0=M

6.0=M

4.0=M

2.0=M

Amplitude

Fundamental

0

2.0

4.0

6.0

8.0

0.1

NORMALISED HARMONIC AMPLITUDES FORSINUSOIDAL PULSE-WITDH MODULATION

Depth ofModulation


Zainal Salam, 2002

17

SPWM (18)– Spectra observations�Amplitude of fundamental

decreases/increases linearly in proportion to the depth of modulation (modulation index). Relationship given as: V1= MIVin

�Harmonics appear in “clusters” with main components at frequencies of : f = kp (fm) k=1,2,3.... where fm : frequency of the modulation signal

� “Side-bands” exist around main harmonic frequencies


Zainal Salam, 2002

18

SPWM (19)

�Amplitude of the harmonics changes with MI. Its incidence (location on spectra) does not

�When p>10, or so, the harmonics can be normalised (as shown in Figure). For lower values of p, the side-bands clusters overlap, and the normalised results no longer apply


Zainal Salam, 2002

19

SPWM (20)– Normalized Fourier coefficients

h MI

0.2 0.4 0.6 0.8 1.0

1 0.2 0.4 0.6 0.8 1.0

MR 1.242 1.15 1.006 0.818 0.601

MR +2 0.016 0.061 0.131 0.220 0.318

MR +4 0.018

2MR +1 0.190 0.326 0.370 0.314 0.181

2MR +3 0.024 0.071 0.139 0.212

2MR +5 0.013 0.033

3MR 0.335 0.123 0.083 0.171 0.113

3MR +2 0.044 0.139 0.203 0.716 0.062

3MR +4 0.012 0.047 0.104 0.157

3MR +6 0.016 0.044

4MR +1 0.163 0.157 0.008 0.105 0.068

4MR +3 0.012 0.070 0.132 0.115 0.009

4MR+5 0.034 0.084 0.1194MR +7 0.017 0.050


Zainal Salam, 2002

20

SPWM (21)– Example

( )

harmonics.dominant theof some and voltagefrequency -lfundamenta theof valuestheCalculate 47Hz. is lfrequencyfundamenta

The 39.M 0.8,M 100V,V inverter, PWM phase single bridge-full In the

:Example

M offunction a as2

ˆ:from computed are harmonics The

2

PWM,bipolar phase-singlebridge fullfor :Note

RIDC

I

',

===

=−==

DCnRG

RGGRRGRRo

VV

vvvvv


Zainal Salam, 2002

21

SPWM (22)– Three-phase inverters

�Effect of odd triplens� For three-phase inverters, there is

significant advantage if p is chosen to be:odd and multiple of three (triplens)(e.g. 3,9,15,21, 27..)

� With odd p, the line voltage shape looks more “sinusoidal”

� Even harmonics are absent in the phase voltage (pole switching waveform) for podd

�Spectra observations� The absence of harmonics no. 21 & 63

in the inverter line voltage due to p as a multiple of three


Zainal Salam, 2002

22

SPWM (23)� Overall, spectra of the line voltage is

more “clean” (lower THD, line voltage is more sinusoidal)

� More concern with the line voltage

� Phase voltage amplitude is 0.8 (normalised) for modulation index =0.8

� Line voltage amplitude is square root three of phase voltage due to the three-phase relationship

�Waveform


Zainal Salam, 2002

23

SPWM (24)2dcV

2dcV

−

2dcV

2dcV

−

2dcV

−

2dcV

−

2dcV

2dcV

dcV

dcV

dcV−

dcV−

π π2

RGV

RGV

RYV

RYV

YGV

YGV

6.0,8 == Mp

6.0,9 == MpILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIOTHAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER


Zainal Salam, 2002

24

SPWM (25)

0

2.0

4.0

6.0

8.0

0.1

2.1

4.1

6.1

8.1

Amplitude

voltage)line to(Line 38.0

Fundamental

41 4339

3745

472319

21 63

6159

5765

6769 77

798183 85

8789

91

19 2343

4741

3761

5965

6783

7985

89

COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE(B) HARMONIC (P=21, M=0.8)

A

B

Harmonic Order

�Harmonics


Zainal Salam, 2002

25

SPWM (26)− Overview

� It is desirable to push p to as large as possible. When p is high, the harmonics will be at higher frequencies based on : f = kp(fm), where fm is the frequency of the modulating signal

�Although the voltage THD improvement is not significant, but the current THD will improve greatly because the load normally has some current filtering effect

� If a low pass filter is to be fitted at the inverter output to improve voltage THD, higher harmonic frequencies is desirable because it makes smaller filter component.


Zainal Salam, 2002

26

SPWM (27)− Example

The amplitudes os the pole switching waveform harmonics of the red phase of a three-phase inverter is shown in the following Table. The inverter uses a symmetric regular sampling PWM scheme. The carrier frequency is 1050Hz and the modulating frequency is 50Hz. The modulation index is 0.8. Calculate the harmonic amplitudes of the line-to-voltage(i.e. red to blue phase) and complete the Table.


Zainal Salam, 2002

27

SPWM (28)

Harmonic number

Amplitude (pole switching waveform)

Amplitude (line-to line voltage)

1 1

19 0.3

21 0.8

23 0.3

37 0.1

39 0.2

41 0.25

43 0.25

45 0.2

47 0.1

57 0.05

59 0.1

61 0.15

63 0.2

65 0.15

67 0.1

69 0.05


Zainal Salam, 2002

28

SPWM (29)

• Unipolar switching– 2 pair of switches operating at carrier

frequency


Zainal Salam, 2002

29

SPWM (30)

– Frequency spectrum, MI = 1

– Normalized Fourier coefficients(Vn/VDC)


Zainal Salam, 2002

30

SPWM (31)

– 2 pair of switches operating at carrier frequency, other pair at reference frequency

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SPWM