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Chapter 7
PWM Techniques
The most widely used control technique
in power electronics
Power
DC/DC
AC/AC
Pulse Width Modulation (PWM)
(Chopping control)
DC/AC
AC/DC
2
Outline
7.1 Basic principles
7.2 Some major PWM techniques in DC/AC inverters
Power
7.3 PWM techniques with feedback control
7.4 PWM rectifiers
3
7.1 Basic principles of PWM
Power
Similar response to different shape of impulse input
The equal-area theorem:
Responses tend to be identical when input signals
have same area and time durations of input impulses
become very small.
4
Basic principles of PWM
Application of the equal-area theorem
Power
This is sinusoidal
PWM (SPWM)
The equal-area
theorem can be applied
to realize any shape of
waveforms
5
A list of PWM techniques
Triangular-wave sampling
– Natural sampling
– Uniform sampling
Calculation
Power
– Calculation based on equal-area criterion
– Selective harmonics elimination
Hysteretic control
Space Vector Modulation (SVM, or SVPWM)
Random PWM
6
7.2 Some major PWM techniques
Natural sampling
Uniform sampling
Selective harmonics elimination
Some practical issues
Power
– Synchronous modulation and asynchronous modulation
– Harmonics in the PWM inverter output voltages
– Ways to improve DC input voltage utilization and reduce
switching frequency
– Connection of multiple PWM inverters
7
Triangular-wave natural sampling
Uni-polar PWM in single-phase VSI
V1
Ud
+
V3
VD1
R
L
uo
V2
V4
Power
VD2
Control
signal
VD3
VD4
ur
Carrier
uc
Mudulation
Carrier
图6-4
Uni-polar sampling is used to
realize uni-polar PWM.
8
Triangular-wave natural sampling
Bi-polar PWM in single-phase VSI
V1
Ud
+
V3
VD1
R
L
uo
V2
V4
Power
VD2
Control
signal
VD3
VD4
ur
Carrier
uc
Mudulation
Carrier
图6-4
Bi-polar sampling is used to
realize bi-polar PWM.
9
Triangular-wave natural sampling
Power
In 3-phase VSI
Three-phase bridge inverter
can only realize bi-bolar PWM
therefore should be controlled
by bipolar sampling.
10
Triangular-wave uniform sampling
Power
Easier to realize
by computercontrol
Modulation factor
11
Power
Selective harmonics elimination
PWM (SHEPWM)
12
Frequency relationship between triangularwave carrier and control signal
Power
Asynchronous Modulation
Synchronous Modulation
13
Harmonics in the PWM inverter
output voltages
Spectrum of 1-phase
bridge PWM inverter
output voltage
1.4
a=1.0
a=0.8
a=0.5
a=0
1.2
No lower order
harmonics
The lowest frequency
harmonics is wc and
adjacent harmonics.
wc has the highest
harmonic content.
Magnitude(%)
Power
1.0
0.8
0.6
0.4
0.2
k 1
n
0
0 +- 2 +- 4 0 +- 1 +- 3 +- 5 0 +- 2 +- 4
1
2
3
(nc +kr)
14
Harmonics in the PWM inverter
output voltages
Power
No lower order
harmonics
No harmonics at c.
The lowest
frequency and
highest content
harmonics are
c2r and 2cr.
1.2
a=1.0
a=0.8
a=0.5
a=0
1.0
Magnitude(%)
Spectrum of 3-phase
bridge PWM inverter
output voltage
0.8
0.6
0.4
0.2
k 1
n
0
0 +- 2 +- 4 0 +- 1 +- 3 +- 5 0 +- 2 +- 4
1
2
3
(nc +kr)
15
Ways to improve utilization of DC input
voltage and reduce switching frequency
Power
Use trapezoidal waveform as modulating signal
instead of sinusoidal
16
Ways to improve utilization of DC input
voltage and reduce switching frequency
Use 3k order harmonics
bias in the modulating
signal
u
1
O
-0.5
t
urU
urV
urW
uc
t
-1
uUN'
ur3
Power
uc
t
O
Ud
2
t
O
urW1
-1
uP
ur1
O

ur1
urV1
O
u
1
u
urU1
t
Ud
2
uVN'
ur uc
u
O
t
uWN'
t
O
O
t
uUV
Ud
图6-18
O
t
-Ud
图6-19
17
Power
Connection of multiple PWM inverters
Purposes
– Expand output power rating
– Reduce harmonics
18
Space Vector PWM (SVPWM or SVM)
Vector Space of 3-phase Line-to-Line Variables
Power
• Phase variables (a, b and c) produce
line-to-line variables (ab, bc and ca) in plane-
• Line-to-line variables (ab, bc and ca) do not have
-component in -coordinate system
c

bc
ca
[1 1 1]T

b
ab

ab

bc
a
ca
19
Line-to-Line Voltage Space Vector
vab 
v
 
v 

T

v   / abc  bc  where
 
vca 
1

1

2 
2
T / abc 

3
3 
0
2

Power
• Space vector

v    e j
bc
  v2  v2
If Vm is the amplitude of balanced,
symmetrical, three-phase line-to-line
voltages, then   3  Vm
2

v
1  v 
  tan  
 v 
1 
2 

3

2 




v
v
ab

ca
20
Switching States for 3-phase
Voltage Source Inverter
p
ia
ib
ic
va
vb
vc
idc
sa
sb
Vdc
sc
Power
n
sa
sb
sc
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
1
0
1
1
Switching state
nnn
nnp
npn
npp
pnn
pnp
ppn
ppp
idc
vab
vbc
vca
0
ic
0
0
-Vdc
0
Vdc
Vdc
0
Vdc
ib
0
-Vdc
ib+ic
-Vdc
0
ia
Vdc
-Vdc
ia+ic
Vdc
0
-Vdc
ia+ib
0
Vdc
-Vdc
ia+ib+ic
0
0
0
0
21
Switching State Vector [pnn]
V pnn
 v 
 
v  pnn
1

vab 
1

2 
2


 T / abc   vbc  

3 
3
 vca  pnn
0

2

1 
V
   dc  
2 
 0   
 
3 
 Vdc  


2 


3
Vdc 
2


1
Vdc 
2


Power


V pnn  V1    e j
bc
v
  2  Vdc


v 
  tan 1     30
 v 

V1
v
ab, 
ca
22
Switching State Vector [ppn]
1

vab 
1


v 
2 


2
V ppn     T / abc   vbc  

3
3 
v  ppn
0
vca  ppn

2

Power


V ppn  V2    e j
bc
v
1   0 
2  V    0 

3   dc   2  Vdc 

 Vdc 

2 


V2

  2  Vdc

v 
  tan 1     90
 v 
ab, 
ca
23
Switching State Vector [ppp]
1

vab 
1


v 
2 


2
V ppp   
 T / abc  vbc 


3
3 
v  ppp
0
vca  ppp

2
1  0 
2    0   0 

3    0

0

2 


Power
bc


V ppp  V0  0

V0
ab, 
ca
24
Switching State Vectors

Power


V1[ pnn]

V2 [ ppn]

V3 [npn ]

V4 [npp ]

V5 [nnp ]

V6 [ pnp ]

V0 [ ppp ]

V0 [nnn ]
30
90
2  Vdc
bc
 (°)

V3 [npn ]
150
-150
-90
II
0
Sector I
VI
III

V4 [npp ]
IV
-30
0
0

V2 [ ppn]
ca
V

V1[ pnn]
ab, 

V6 [ pnp ]

V5 [nnp ]

V0  [ ppp ]  [nnn ] at center point
25
Reference Voltage Vector, Vref
vab 
 Vm  cost  
 


Assume vbc   Vm  cost  120
vca  ref Vm  cost  120
Power

v 
Vref       e j
 v  ref

bc

V3 [npn]

where   v2  v2  3  Vm
2
v 
  tan     t
 v 
1
 In general,

3
Vref (t ) 
 Vm (t )  e j(t )
2

V2 [ ppn]

V
v
ref

v

V4 [npp]
ca

V1[ pnn]
ab, 

V6 [ pnp]

V5 [nnp]

V0  [ ppp]  [nnn] at center point
26
Definition of High Frequency Synthesis

 Ti  
0 Vref dt  i  0 Vi dt ,
TS

T  T
i
S
i
T1
T1 T2



Vref dt  V1 dt   V2 dt 
TS
For example
0
0
T1
TS

V0 dt
T1 T2
v
Power
V1()
V2()
Vref ()
t
T1
T2
T0
TS
Total area of
= Area of
27
Synthesis of Vref using Switching State Vectors
p
idc
1
ia va
sa
a
1
0
ib vb
sb
b 0
ic vc
1

Vdc
c 0sc

V3 [npn]
Power
n

V0

II

III

V4 [npp]

Vref

I

d1  V1
v

Vref


V2

d 2  V2

V2 [ ppn]
bc

V1
ca
v
IV

V1[ pnn]
ab, 
VI
V

V6 [ pnp]

V5 [nnp]

V0  [ ppp]  [nnn]
28
Duty Ratio of Switching State Vectors in SVPWM
T
T T



From HF synthesis definition,  Vref dt   V1 dt   V2 dt 
TS
1
0
0
1
2
T1




Assume Vref is constant in TS , Vref  TS  V1  T1  V2  T2
Power
cos 
 
  TS  V1
sin



1
    T1  V2
0 
cos60

  T2
sin
60



T2
2 
 d2 

 sin 
TS
3 V2
d0  1  d 1  d 2

d 2  V2

V0
T1 T2

V2
where     30
T1
2 
 d1 

 sin(60  )
TS
3 V1

 V0 dt
TS


Vref


d1  V1

V1
29
7.3 PWM techniques with
feedback control
Current hysteretic control
Power
Voltage hysteretic control
Triangular-wave comparison (sampling) with
feedback control
30
Current hysteretic control
Power
In Single-phase VSI
31
Current hysteretic control
Power
In 3-phase VSI
32
Voltage hysteretic control
Power
Ud
2
Filter
Ud
2
+ u*
u
u
图6-26
33
Power
Triangular-wave comparison (sampling)
with feedback control
34
7.4 PWM rectifiers
Operation Principles
Power
a) Rectification mode
c) Reactive power
compensation mode
b) Inversion mode
d) Current leading by 
35
PWM rectifiers
Power
Three-phase circuit
36
PWM rectifiers
Indirect current control
Triangular-wave
Power
u*d +
u-d
PI
id
uR +
+
R
- uA,B,C
sin(t+2k/3)
(k=0,1,2) uL
XL
cos(t+2k/3)
(k=0,1,2)
R
L
ua,ub,uc
ud +
Load
图6-31
37
PWM rectifiers
Direct current control
ia,b,c
u*
d
+
u-
PI
id
i*a,b,c
R
ua,ub,uc
d
Power
L
sin(t+2k/3)
(k=0,1,2)
ud
+
Load
图6-32
38