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ADVANCED POWER ELECTRONICSADVANCED POWER ELECTRONICS
PWM INVERTERSPWM INVERTERS
Dr. Adel GastliEmail: [email protected]
http://adel.gastli.net
Dr. Adel Gastli PWM Inverters 2
CONTENTSCONTENTSCONTENTS
1. Single-Phase Half-Bridge Inverter
2. Single-Phase Bridge Inverter
3. Three-Phase Inverter4. Three-Phase PWM Inverter5. Sinusoidal PWM6. Modified Sinusoidal PWM7. Sinusoidal PWM 3-Phase
Textbook: Chapter 6Textbook: Chapter 6
8. 60-Degree Modulation 9. Transformer Connection10. Single-Phase Current
Source11. Three-Phase Current
Source12. Variable DC Link Inverter13. AC Filters14. Summary
Dr. Adel Gastli PWM Inverters 3
SingleSingle--Phase HalfPhase Half--Bridge InverterBridge Inverter
2
( ) 0
1( )
2
2 2
20.45
2
S So rms
So rms S
V VV d
VV V
πθ
π
π
⎛ ⎞= =⎜ ⎟⎝ ⎠
= =
∫
1,3,5,.
2( ) sin
0 2,4,..
So
n
Vv t n t
n
for n
ωπ
∞
=
=
= =
∑
Dr. Adel Gastli PWM Inverters 4
Performance ParametersPerformance Parameters
1
2
2,3,..1
2
22,3,..1
21
1
1
1
1
1
3%
onn
o
onno
on
no
onn
o
o
VHF for n
V
THD VV
VDF
V n
VDF for n
V n
LOH V
∞
=
∞
=
= >
=
⎛ ⎞= ⎜ ⎟⎝ ⎠
= >
≥ ×
∑
∑
Harmonic factor of nth harmonic
Total Harmonic Distortion factor
Distortion factor
Distortion factor of nth harmonic
Lowest Order Harmonic- Frequency is closest to fundamental- Amplitude is greater than or equal to 3% the fundamental
Dr. Adel Gastli PWM Inverters 5
Example 6.1 (Homework)
Study the example by yourself.
Simulate the circuit and check the results. (Use any software)
(Life-long learning)
Dr. Adel Gastli PWM Inverters 6
SingleSingle--Phase Bridge InverterPhase Bridge Inverter
2( ) 0
1( )
2
40.90
2
o rms S S
So rms S
V V d V
VV V
πθ
π
π
= =
= =
∫
1,3,5,.
4( ) sin
0 2,4,..
So
n
Vv t n t
n
for n
ωπ
∞
=
=
= =
∑
Dr. Adel Gastli PWM Inverters 7
Example 6.3Example 6.3
2 22 2
1
1,3,5,.
10, 31.5 , 112 , 60 , 220 , 2 377 /
23.6811.87 ,
1 23.6810 11.87
11.87 23.68tan
10 10
4( ) sin
0 2,
o s
L c
n
n
So
n
R L mH C uF f Hz V V f rad s
j jX jn L j n X
n C n
Z R n L nn C n
n
n
Vv t n t
n
for n
ω π
ωω
ωω
θ
ωπ
−
∞
=
= = = = = = =−
= = Ω = = Ω
⎛ ⎞ ⎛ ⎞= + − = + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠
=
= =
∑
21,3,5,.
2
4,..
( ) 4( ) sin( )
1
o So n
nn n
v t Vi t n t
Zn R n L
n C
ω θθ
π ωω
∞
=
= = −∠ ⎛ ⎞+ −⎜ ⎟
⎝ ⎠
∑
Dr. Adel Gastli PWM Inverters 8
a. The instantaneous output voltage
L+×+×+×+×+=
)3779sin(12.31)3777sin(2.40
)3775sin(02.56)3773sin(4.93)377sin(1.280)(0
tt
ttttv
Dividing the output voltage by the load impedance and considering the appropriate delay due to the load impedance angles, we can obtain the instantaneous load current as
L+−×+
−×+−×+
−×++=
)52.843779sin(3.0
)85.823777sin(5.0)63.793775sin(
)17.703773sin(17.3)72.49377sin(1.18)(0
o
oo
o
t
tt
ttti
Dr. Adel Gastli PWM Inverters 9
b. The peak fundamental load current is Im1=18.1A. The rms current at fundamental frequency is I01=12.8A
The rms harmonic load current is
A
IIIIII mmmmmm
41.18
3.05.00.117.31.18 22222
29
27
25
23
21
=++++=
++++=
c. Considering up to 9th harmonic, the peak load current,
AII
I mmh 38.2
2
1.1841.18
2
2221
2
=−
=−
=
%59.181
21
2
=−
=m
mm
I
IITHD
Dr. Adel Gastli PWM Inverters 10
d. The rms load current is AI
I m 02.132
42.18
20 ==≅
The total load power is WRIP 16951002.13 2200 =×==
The fundamental output power is WRIP 4.1638108.12 22
0101 =×==
e. The average supply current AV
PI
ss 7.7
220
16950 ===
f. The peak transistor current AII mp 41.18=≅
The maximum permissible rms transistor current is
AII
I pQ 2.9
2
41.18
220
max ====
Dr. Adel Gastli PWM Inverters 110 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
-30
-20
-10
0
10
20
30
Time, (sec)
Load
Vol
tage
(V
) an
d C
urre
nt (
A)
v0/10
v01
/10
i0i01
Q1,Q2 Q3,Q4D1,D2 D3,D4
g. The waveforms of the output voltage and current and their fundamental components are shown below.
Dr. Adel Gastli PWM Inverters 12
h. The conduction time of each transistor is found approximately from the previous waveforms as
mstt Qo
Q 031.6377180
28.130or 28.13072.49180 =
×==−=
πω
i. The conduction time for each diode is approximately
377180
72.49
302.210)031.6333.8(
10031.6120
1
23
3
×=
=×−=
×−=−=
−
−
ms
tT
t QD
Dr. Adel Gastli PWM Inverters 13
Notes: This example can be repeated for different types of loads (R, RL, RLC) with an appropriate change in load impedance ZL and load angle θn
Gating sequence is as follows:– Generate two square-wave gating signals vg1 and vg2
at an output frequency f0.– The gating signals vg3 and vg4 should be the logic
invert of vg2 and vg1 respectively.– Signals vg1 and vg3 drive Q1 and Q3, respectively,
through gate isolation circuits.– Signals vg2 and vg4 drive Q2 and Q4, respectively,
without any gate isolation circuits.
Dr. Adel Gastli PWM Inverters 14
Three Single-Phase Inverter
Three-phase Bridge Inverter 180o Conduction
120o Conduction
THREETHREE--PHASE BRIDGE PHASE BRIDGE INVERTERINVERTER
Dr. Adel Gastli PWM Inverters 15
Three SingleThree Single--Phase InverterPhase Inverter
12 transistors12 diodes3 transformersRisk of voltage unbalance
Figure 6.4
Transformer secondary windings can be connected in Y or Δ.
Δ connection eliminates triplen harmonics (3, 6, 9,..)
Dr. Adel Gastli PWM Inverters 16
ThreeThree--Phase Bridge InverterPhase Bridge Inverter
Dr. Adel Gastli PWM Inverters 17
3
3
23
scn
sbn
san
Vv
Vv
Vv
=
−=
=
3
3
3
2
scn
sbn
san
Vv
Vv
Vv
−=
−=
=
3
23
3
scn
sbn
san
Vv
Vv
Vv
−=
=
=
180180oo ConductionConduction
Dr. Adel Gastli PWM Inverters 18
∑
∑
∑
∞
=
∞
=
∞
=
⎟⎠⎞
⎜⎝⎛ −=
⎟⎠⎞
⎜⎝⎛ −=
⎟⎠⎞
⎜⎝⎛ +=
K
K
K
,5,3,1
,5,3,1
,5,3,1
6
7sin
6cos
4
2sin
6cos
4
6sin
6cos
4
n
sca
n
sbc
n
sab
tnn
n
Vv
tnn
n
Vv
tnn
n
Vv
πωππ
πωππ
πωππ
Note that for n=3,9,15,21,... vab=vbc=vca=0
Dr. Adel Gastli PWM Inverters 19
( )
ss
sL
VV
tdVv
8165.03
2
2
22/13/2
0
2
==
⎥⎦
⎤⎢⎣
⎡= ∫
π
ωπ
Line-to-line rms voltage
Dr. Adel Gastli PWM Inverters 20
ss
Ls
Ln VV
vn
n
Vv 7797.0
6cos
2
4
6cos
2
41 ==⇒=
ππ
ππ
Line-to-line rms harmonic voltage
ssL
p VVv
v 4714.03
2
3===
Phase rms voltage
Dr. Adel Gastli PWM Inverters 21
Only two transistors remain on at any time.
02
2
=
−=
=
cn
sbn
san
v
Vv
Vv
2
02
scn
bn
san
Vv
v
Vv
−=
=
=
2
2
0
scn
sbn
an
Vv
Vv
v
−=
=
=
vca
vbc
vab
π π23/π 3/2π
Note: The waveforms of phase voltages are the same as the waveforms of line voltages with the only difference in the amplitudes (Vs/2 instead of Vs)
120120oo ConductionConduction
Dr. Adel Gastli PWM Inverters 22
∑
∑
∑
∞
=
∞
=
∞
=
⎟⎠⎞
⎜⎝⎛ −=
⎟⎠⎞
⎜⎝⎛ −=
⎟⎠⎞
⎜⎝⎛ +=
K
K
K
,5,3,1
,5,3,1
,5,3,1
6
7sin
6cos
2
2sin
6cos
2
6sin
6cos
2
n
scn
n
sbn
n
san
tnn
n
Vv
tnn
n
Vv
tnn
n
Vv
πωππ
πωππ
πωππ
phline vv 3=
Dr. Adel Gastli PWM Inverters 23
Single-Pulse-Width modulation
Multiple-Pulse-Width Modulation
Sinusoidal-Pulse-Width Modulation
Modified Sinusoidal-Pulse-Width Modulation
Phase Displacement control
Voltage Control of SingleVoltage Control of Single--Phase Phase InvertersInverters
Dr. Adel Gastli PWM Inverters 24
SingleSingle--Pulse Width ModulationPulse Width Modulation
1,3,5,.
4( ) sin sin
2S
on
V nv t n t
n
δ ωπ
∞
=
= ∑
2 1
2s
d t t
TMT M
δω
= = −
= =
Figure 6.11 πδθ
πδπ
δπ ssrms VdVV == ∫+
−
2/)(
2/)(
2)(0 2
2
Modulation index Switching PeriodM=Ar/Ac
Dr. Adel Gastli PWM Inverters 25
Pulse widthPulse width
2)1(1
1sT
Mt −==ωα
2)1(2
2sT
Mt +==ωα
sMTttd =−== 12ωδ
2
TTs = T is the desired period of the output voltage
Prove these two t1 and t2 equations
Dr. Adel Gastli PWM Inverters 26
Harmonic Profile for p =1Harmonic Profile for p =1
The dominant harmonic is the third.
DF increases significantly at a low output voltage (small M).
Figure 6.12
1,3,5,.
4( ) sin sin
2S
on
V nv t n t
n
δ ωπ
∞
=
= ∑
Dr. Adel Gastli PWM Inverters 27
Gating Signals Gating Signals AlgorithmAlgorithm
StartStart
Generate a triangular carrier signal vcr
(Magnitude Vc, Switching Period Ts=T/2)Generate a triangular carrier signal vcr
(Magnitude Vc, Switching Period Ts=T/2)
Compare vcr with a dc reference signal vr
ve=vcr-vr>0 gate signal vg=0ve=vcr-vr<0 gate signal vg=1
Compare vcr with a dc reference signal vr
ve=vcr-vr>0 gate signal vg=0ve=vcr-vr<0 gate signal vg=1
vg should be multiplied by a unity pulsesignal vz with 50% duty cycle at
a period of T vg1=vg*vz
vg should be multiplied by a unity pulsesignal vz with 50% duty cycle at
a period of T vg1=vg*vz
vg2 is obtained by inverting the square signal vz.
vg2 is obtained by inverting the square signal vz.
Change vr to change the modulation index and hence the output voltage rms
Chan
ge f
requ
ency
Dr. Adel Gastli PWM Inverters 28
s
p
p srms
Vp
Vp
V
πδ
πδπ
δπ
=
= ∫+
−
2//
2//
2)(0
MultipleMultiple--Pulse PWMPulse PWM
1
2
m m
s
d t t
TMT M
p
δω += = −
= =
Figure 6.13 (Prove these integral limits)
Dr. Adel Gastli PWM Inverters 29
Harmonic Profile for p =5 Harmonic Profile for p =5
1,3,5,.
( ) sino nn
v t B n tω∞
=
= ∑
2
1
4 3sin sin sin
4 4 4
pS
n m mm
V nB n n
n
δ δ δα π απ=
⎡ ⎤⎛ ⎞ ⎛ ⎞= + − + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∑
(See textbook for detailed calculation of Bn)
Note that the harmonics’variation as a function of output voltage has decreased.
Dr. Adel Gastli PWM Inverters 30
Sinusoidal PWMSinusoidal PWM
2
( )1
pm
o rms Sm
V Vδπ=
= ∑
1,3,5,.
( ) sino nn
v t B n tω∞
=
= ∑
1m
m m md t tδω += = −
LOH = 2p-1 p: number of pulses per half a cycle
More practical
vo=Vs(g1-g4)
Dr. Adel Gastli PWM Inverters 31
Harmonic Profile for p =5 Harmonic Profile for p =5
1,3,5,.
( ) sino nn
v t B n tω∞
=
= ∑
2
1
4 3sin sin sin
4 4 4
pS m m m
n m mm
V nB n n
n
δ δ δα π απ=
⎡ ⎤⎛ ⎞ ⎛ ⎞= + − + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∑
LOH = 2p-1=9
Significant decrease in DF and harmonics content.
Commonly used in industrial applications
Dr. Adel Gastli PWM Inverters 32
Peak Fundamental versus M Peak Fundamental versus M
For M<1 the maximum output voltage over the input voltage ratio varies linearly with M.
For M>1, the inverter operation is called overmodulation.
Overmodulation leads to basically square waveform and add more harmonics. (Not recommended)
Dr. Adel Gastli PWM Inverters 33
Modified Sinusoidal PWMModified Sinusoidal PWM
1m
m m md t tδω += = −
Carrier signal is modified
Because of the nature of sine waveform, the width of pulses does not change much with the modulation index near the peak of the sine.
Dr. Adel Gastli PWM Inverters 34
Less number of switching of power devices between 60o and 120o
Reduction of switching lossesIncrease of fundamental component. Harmonic characteristics are improved.
Harmonic Profile for p =5 Harmonic Profile for p =5
Dr. Adel Gastli PWM Inverters 35
Phase DisplacementPhase Displacement
Full-bridge is equivalent to summation of two half-bridge inverters where vbo is shifted 180o from vao.
To vary the output voltage amplitude, the phase shift of 180o can be varied from 0o to 180o.
00 baab vvv −=
Dr. Adel Gastli PWM Inverters 36
Phase DisplacementPhase Displacement
Fundamental rms is a function of the phase displacement angle α.
ππα
2
400 01
sVV ≤≤⇒≤≤
( )
⎟⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛=
−=
=
=
∑
∑
∑
∞
=
∞
=
∞
=
2sin
2
4
2cos
2sin
4
sin2
sin2
01
,5,3,1
,5,3,10
,5,3,10
)(0
απ
αωαπ
αωπ
ωπ
πα
s
n
sab
n
sb
n
sa
srms
VV
tnn
n
Vv
tnn
Vv
tnn
Vv
VV
L
L
L
Dr. Adel Gastli PWM Inverters 37
Phase DisplacementPhase Displacement
Vs/2
-Vs/2
Vs/2
-Vs/2
Vs
-Vs
180o
180o+α180o-α
α
van
vao
vbo
To obtain a quarter-wave symmetry at 90o it is possible to shift the gate signal g1 by α and g3 by 180o-α.
( )
( )
( ) ( )
( )απ
ωαπ
απωπ
αωπ
cos2
4
sincos4
sin2
sin2
01
,5,3,1
,5,3,10
,5,3,10
s
n
sab
n
sb
n
sa
VV
tnnn
Vv
tnn
Vv
tnn
Vv
=
=
+−=
−=
∑
∑
∑
∞
=
∞
=
∞
=
L
L
L
Dr. Adel Gastli PWM Inverters 38
Sinusoidal Pulse-Width Modulation
60o PWM
Third-Harmonic PWM
Space Vector modulation
Voltage Control of ThreeVoltage Control of Three--Phase Phase InvertersInverters
Will not be Will not be coveredcovered
Dr. Adel Gastli PWM Inverters 39
Sinusoidal PWM 3Sinusoidal PWM 3--Phase Phase
It is similar to single-phase SPWM but with 3-reference sine waveforms shifted by 120o each.
vab=Vs(g1-g3)
o
cf f
fm =
Frequency modulation ratio should be odd multiple of 3.
Dr. Adel Gastli PWM Inverters 40
Comments:Comments:
All phase voltages are identical but 120o
out of phase without even harmonics.
Harmonics multiple of 3 are identical in amplitude and phase in all the 3-phases.
Thus, the ac output line voltages do not contain the harmonics multiple of 3.
Dr. Adel Gastli PWM Inverters 41
6060--Degree Modulation Degree Modulation
Less switching lossesUtilizes more available dc voltageHigher fundamental in both phase and line voltagesAll triplen harmonics are absent in three-phase voltages.
Similar to the modified PWM seen earlier.
Flat top between 60o and 120o
Dr. Adel Gastli PWM Inverters 42
Harmonic ReductionHarmonic Reduction
Phase displacement control
Bipolar output voltage notches
Unipolar output voltage notches
60-Degree modulation
Transformer connections
Dr. Adel Gastli PWM Inverters 43
Phase Displacement Control
It was seen that the nth harmonic can be eliminated by a proper choice of displacement angle α if:
( )
n
n
o90
or
0cos
=
=
α
α
Thus, the 3rd harmonic can be eliminated if: o30=α
Dr. Adel Gastli PWM Inverters 44
Bipolar Notches Bipolar Notches A pair of unwanted harmonics at the output of single-phase inverters can be eliminated by introducing a pair of symmetrically placed bipolar voltage notches as shown below.
)()( 00 πθθ +−= vv
Half-wave symmetry
only odd harmonics (i.e. n=1,3,5,…)
)()( θθ −−= oo vv
Point symmetry
01 =→nA
Dr. Adel Gastli PWM Inverters 45
1,3,5,.
( ) sino nn
v t B n tω∞
=
= ∑
[ ]oo
s
sn
BB
nnn
V
dndndnV
B
3.33 and 62.230For
cos2cos214
)sin()sin()sin(4
2153
21
0
2/1
2
2
1
==⇒==
+−=
⎥⎦⎤
⎢⎣⎡ +−= ∫ ∫∫
αα
ααπ
θθθθθθπ
α π
α
α
α
These type of equations can be solved iteratively or using specialized program such as MathCAD or MATLAB Symbolic Toolbox.
Dr. Adel Gastli PWM Inverters 46
clear, syms a1 a2equ1 ='1-2*cos(3*as1)+2*cos(3*as2)';equ2 ='1-2*cos(5*as1)+2*cos(5*as2)';[as1,as2] = solve(equ1, equ2);
a1=double(as1)*180/pi;a2=double(as2)*180/pi;
for n=1:length(a1)if(a1(n)+a2(n)<=90)
n1=n;break
endend
a=[a1(n1) a2(n1)]
Example of Matlab program for solving such equations
Dr. Adel Gastli PWM Inverters 47
The previous equation of Bn can be extended to mnotches as follows:
( ) ( ) ,...5,3,1for cos1214
1
=⎥⎦
⎤⎢⎣
⎡−+= ∑
=
nnn
VB
m
kk
ksn α
π
221
πααα <<<< kL
where
Dr. Adel Gastli PWM Inverters 48
Example 6.4Example 6.4
Figure 6.38
A single-phase full-wave inverter uses multiple notches to give bipolar voltage as shown in Figure 6.38 and is required to eliminate the fifth, seventh, eleventh, and thirteenth harmonics from the output wave. Determine the number of notches and their angles.
Dr. Adel Gastli PWM Inverters 49
SolutionSolutionFor elimination of the fifth, seventh, eleventh and thirteenth harmonics we should have:
0131175 ==== BBBB
That is m=4 notches per half wave.
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )⎪
⎪⎩
⎪⎪⎨
⎧
=+−+−=+−+−
=+−+−=+−+−
013cos213cos213cos213cos21
011cos211cos211cos211cos21
07cos27cos27cos27cos21
05cos25cos25cos25cos21
4321
4321
4321
4321
αααααααα
αααααααα
oooo 87.32 91.30 09.16 55.10 4321 ==== αααα
Dr. Adel Gastli PWM Inverters 50
Unipolar Voltage Notches Unipolar Voltage Notches
1,3,5,.
( ) sino nn
v t B n tω∞
=
= ∑ [ ]oo
s
sn
BB
nnn
V
dndnV
B
93.37 and 83.170For
coscos14
)sin()sin(4
2153
21
0
2/1
2
==⇒==
+−=
⎥⎦⎤
⎢⎣⎡ += ∫ ∫
αα
ααπ
θθθθπ
α π
α
Similarly to bipolar notches symmetrical unipolar notches can also be introduced.
( ) ( ) ,...5,3,1for cos114
1
=⎥⎦
⎤⎢⎣
⎡−+= ∑
=
nnn
VB
m
kk
ksn α
π
Dr. Adel Gastli PWM Inverters 51
Transformer ConnectionTransformer ConnectionOutput voltages of two or more inverters may be connected in series through a transformer to reduce or eliminate certain unwanted harmonics.
Phase shifted by 60o.
Dr. Adel Gastli PWM Inverters 52
Transformer ConnectionTransformer Connection
1 1 3 5
2 1 3 5
1 2 1 5
( ) sin sin 3 sin 5 ...
( ) sin( ) sin 3( ) sin 5( ) ...3 3 3
3 sin( ) sin 5( ) ..6 6
o
o
o o o
v t A t A t A t
v t A t A t A t
v v v A t A t
ω ω ωπ π πω ω ω
π πω ω
= + + +
= − + − + − +
⎡ ⎤= + = − + − +⎢ ⎥⎣ ⎦
Elimination of third (an all triplen) harmonics
π/3
Dr. Adel Gastli PWM Inverters 53
Current Source InverterCurrent Source Inverter
Voltage Source Inverter
AC Load
Voltage control
Current varies with load impedance
Vdc
Current Source Inverter
AC Load
Current control
voltage varies with load impedance
Vdc
Dr. Adel Gastli PWM Inverters 54
SingleSingle--Phase Current Source (ContPhase Current Source (Cont’’d)d)
For continuous current flow, 2 switches must always conduct.
Dr. Adel Gastli PWM Inverters 55
SingleSingle--Phase Current Source (ContPhase Current Source (Cont’’d)d)
0Q1, Q4 , D1 , D4
-ILQ3, Q4 , D3, D4
0Q3, Q2 , D3 , D2
ILQ1, Q2 , D1 , D2
ioConducting Switches
Dr. Adel Gastli PWM Inverters 56
SingleSingle--Phase Current Source (ContPhase Current Source (Cont’’d)d)
δ
See Eq. (6.28) p.249
( )∑∞
=⎟⎠⎞
⎜⎝⎛=
,..5,3,1
sin2
sin4
)(n
Lo tn
n
n
Iti ωδ
π
⎟⎠⎞
⎜⎝⎛=
2sin
2
4)(1
δπL
rmso
II
Dr. Adel Gastli PWM Inverters 57
ThreeThree--Phase Current SourcePhase Current Source
Similar to voltage waveform for 180o
conduction (p. 239)
Dr. Adel Gastli PWM Inverters 58
ThreeThree--Phase Current Source (ContPhase Current Source (Cont’’d)d)
∑∞
=⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛=
,..5,3,1 6sin
3sin
4)(
n
La tn
n
n
Iti
πωππ
YY--Load ConnectionLoad Connection
( )∑∞
=⎟⎠⎞
⎜⎝⎛=
,..5,3,1
sin3
sin4
)(n
La tn
n
n
Iti ωπ
π
ΔΔ--Load ConnectionLoad Connection
⎟⎠⎞
⎜⎝⎛=
3sin
2
4)(1
ππL
rmsa
II
From Eq. (6.16a)
From Eq. (6.21a)
Dr. Adel Gastli PWM Inverters 59
Alternative ConfigurationAlternative Configuration
Dr. Adel Gastli PWM Inverters 60
Current Control TechniquesCurrent Control Techniques
PWM, SPWM, MSPWM, and other techniques can be applied to vary the load current and improve the quality of its waveform.
Dr. Adel Gastli PWM Inverters 61
Advantages of the CSIAdvantages of the CSI
The advantages of the CSI are:– Since Idc is controlled and limited, misfiring of
switches, or short-circuit, would not be a serious problem.
– The peak current of power devices is limited.– The commutation circuits for thyristors are
simpler.– It has the ability to handle reactive or
regenerative load without freewheeling diodes.
Dr. Adel Gastli PWM Inverters 62
Disadvantages of the CSIDisadvantages of the CSI
A CSI requires a relatively large reactor to exhibit current-source characteristics and an extra converter stage to control the current.The dynamic response is slower than that of the VSI.Due to current transfer from one pair of switches to another, an output filter is required to suppress the output voltage spikes.
Dr. Adel Gastli PWM Inverters 63
Variable DC Link InverterVariable DC Link Inverter
Varying the modulation index (or pulse width) and maintaining the dc input voltage constant has shown that a range of harmonics would be present on the output voltage.
The pulse width can be fixed to eliminate or reduce certain harmonics and the output voltage can be controlled by varying the level of the dc input voltage.
Dr. Adel Gastli PWM Inverters 64
Variable DC Link Inverter (ContVariable DC Link Inverter (Cont’’d)d)
DrawbacksDrawbacks: – Requires additional converter.
– Power cannot be fed-back to the source.
Dr. Adel Gastli PWM Inverters 65
AC FiltersAC Filters
Output of the inverter is “chopped AC voltage with zero DC component”. In some applications such as UPS, “high purity” sine wave output is required.
An LC section low-pass filter is normally fitted at the inverter output to reduce the high frequency harmonics.
In some applications such as AC motor drive, filtering is not required.
Dr. Adel Gastli PWM Inverters 66
AC Filters (ContAC Filters (Cont’’d)d)
C
L
voFLOADvoi
+ +
voFvoi
LOW PASS FILTERLOW PASS FILTER
Dr. Adel Gastli PWM Inverters 67
Commonly used output filtersCommonly used output filters
C filter is very simple but draws more reactive power.
C filter is very simple but draws more reactive power.
LC tuned filter can eliminates only one frequency.
LC tuned filter can eliminates only one frequency.
CLC filter is more effective in reducing harmonics of wide bandwidth and draws less reactive power.
CLC filter is more effective in reducing harmonics of wide bandwidth and draws less reactive power.
Dr. Adel Gastli PWM Inverters 68
AC Filters (ContAC Filters (Cont’’d)d)
Usually the nth and higher order harmonics would be reduced significantly if the filter impedance Zfn is much smaller than that of the load ZLn, and a ratio 1:10 is normally adequate in most of the cases.
(Study example 6.7 p. 294)
10Ln
fn
ZZ ≤
Dr. Adel Gastli PWM Inverters 69
AC Filters (ContAC Filters (Cont’’d)d)
No control in harmonics and output voltage magnitude
Square waveform
Cut-off frequency of the low-pass filter is somewhat fixed
1,3,5,.
4( ) sin
0 2, 4,..
So
n
Vv t n t
n
for n
ωπ
∞
=
=
= =
∑
The filter size is dictated by the VA ratings of the inverter.
Dr. Adel Gastli PWM Inverters 70
AC Filters (ContAC Filters (Cont’’d)d)
2
( )1
pm
o rms Sm
V Vδπ=
= ∑
1,3,5,.
( ) sino nn
v t B n tω∞
=
= ∑
LOH = 2p-1 p: number of pulses per half a cycle
PWM waveformHarmonics are “pushed”to higher frequencies.
Cut-off frequency of the filter is increased
Hence the filter components (i.e. L and C) sizes are reduced.
Trade off for this flexibility is complexity in the switching waveforms.
Dr. Adel Gastli PWM Inverters 71
SummarySummary
An inverter can convert a fixed dc voltage to a variable or fixed ac voltage and/or frequency.
Various modulation techniques can be used to vary the output voltage. With appropriate choice of switching angles, specific harmonics can be eliminated.
Dr. Adel Gastli PWM Inverters 72
Summary (ContSummary (Cont’’d)d)
The current source inverter is most suited for application requiring well defined controllable current.
A CSI is a dual of a VSI.
In a VSI, the load current depends on load impedance, whereas the load voltage in a CSI depends on the load impedance.