Chapter 16
Line-Commutated Rectifiers
16.1
The single-phase full-wave
rectifier
16.1.1 Continuous conduction
mode
16.1.2 Discontinuous
conduction mode
16.1.3 Behavior when C is
large
16.1.4 Minimizing THD when C
is small
16.2
The three-phase bridge
rectifier
16.2.1 Continuous conduction
mode
16.2.2 Discontinuous
conduction mode
Fundamentals of Power Electronics
1
16.3
Phase control
16.3.1 Inverter mode
16.3.2 Harmonics and power
factor
16.3.3 Commutation
16.4
Harmonic trap filters
16.5
Transformer connections
16.6
Summary
Chapter 16: Line-commutated rectifiers
16.1 The single-phase full-wave rectifier
ig(t)
iL(t)
L
+
D4
D1
Zi
vg(t)
D3
C
v(t)
D2
R
–
Full-wave rectifier with dc-side L-C filter
Two common reasons for including the dc-side L-C filter:
• Obtain good dc output voltage (large C) and acceptable ac line
current waveform (large L)
• Filter conducted EMI generated by dc load (small L and C)
Fundamentals of Power Electronics
2
Chapter 16: Line-commutated rectifiers
16.1.1 Continuous conduction mode
Large L
THD = 29%
vg(t)
Typical ac line
waveforms for
CCM :
ig(t)
10 ms
20 ms
30 ms
40 ms
As L →∞, ac
line current
approaches a
square wave
t
CCM results, for L →∞ :
distortion factor =
1
distortion factor
THD =
Fundamentals of Power Electronics
I 1, rms
= 4 = 90.0%
I rms
π 2
3
2
– 1 = 48.3%
Chapter 16: Line-commutated rectifiers
16.1.2 Discontinuous conduction mode
Small L
vg(t)
Typical ac line
waveforms for
DCM :
As L →0, ac
line current
approaches
impulse
functions
(peak
detection)
THD = 145%
ig(t)
10 ms
20 ms
30 ms
40 ms
t
As the inductance is reduced, the THD rapidly
increases, and the distortion factor decreases.
Typical distortion factor of a full-wave rectifier with no
inductor is in the range 55% to 65%, and is governed
by ac system inductance.
Fundamentals of Power Electronics
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Chapter 16: Line-commutated rectifiers
16.1.3 Behavior when C is large
Solution of the
full-wave
rectifier circuit
for infinite C:
cos (ϕ1 − θ1),
PF, M
Define
K L = 2L
RTL
THD
1.0
β
β
200%
180˚
150%
135˚
100%
90˚
50%
45˚
0
0˚
cos (ϕ1 − θ1)
0.9
PF
0.8
0.7
M
M= V
Vm
0.6
THD
0.5
CCM
DCM
10
1
0.1
0.01
0.001
0.0001
0.4
KL
Fundamentals of Power Electronics
5
Chapter 16: Line-commutated rectifiers
16.1.4 Minimizing THD when C is small
Sometimes the L-C filter is present only to remove high-frequency
conducted EMI generated by the dc load, and is not intended to
modify the ac line current waveform. If L and C are both zero, then the
load resistor is connected directly to the output of the diode bridge,
and the ac line current waveform is purely sinusoidal.
An approximate argument: the L-C filter has negligible effect on the ac
line current waveform provided that the filter input impedance Zi has
zero phase shift at the second harmonic of the ac line frequency, 2 fL.
i (t)
i (t) L
g
L
D4
D1
Zi
vg(t)
D3
Fundamentals of Power Electronics
+
D2
6
C
v(t)
R
–
Chapter 16: Line-commutated rectifiers
Approximate THD
50
THD=30%
1
2π LC
L
R0 =
C
Q= R
R0
f
fp = 1 = 0
2πRC Q
THD=10%
THD=3%
10
THD=1%
Q
f0 =
THD=0.5%
1
0.1
1
10
100
f0 / fL
Fundamentals of Power Electronics
7
Chapter 16: Line-commutated rectifiers
Example
THD = 3.6%
vg(t)
ig(t)
10 ms
20 ms
30 ms
40 ms
t
Typical ac line current and voltage waveforms, near the boundary between continuous
and discontinuous modes and with small dc filter capacitor. f0/fL = 10, Q = 1
Fundamentals of Power Electronics
8
Chapter 16: Line-commutated rectifiers
16.2 The Three-Phase Bridge Rectifier
øa
ia(t)
iL(t)
+
D2
D1
3ø
ac
L
D3
øb
V
C
D4
D5
D6
–
øc
ia(ωt)
Line current
waveform for
infinite L
Fundamentals of Power Electronics
dc load
R
iL
ωt
0
90˚
180˚
270˚
360˚
–iL
9
Chapter 16: Line-commutated rectifiers
16.2.1 Continuous conduction mode
Fourier series:
∞
i a(t) =
Σ
n = 1,5,7,11,...
ia(ωt)
iL
nπ
nπ
4
nπ I L sin 2 sin 3 sin nωt
ωt
0
90˚
•
Similar to square wave, but
missing triplen harmonics
•
THD = 31%
•
Distortion factor = 3/π = 95.5%
•
In comparison with single phase case:
180˚
270˚
360˚
–iL
the missing 60˚ of current improves the distortion factor from 90% to
95%, because the triplen harmonics are removed
Fundamentals of Power Electronics
10
Chapter 16: Line-commutated rectifiers
A typical CCM waveform
THD = 31.9%
ia(t)
vbn(t)
vcn(t)
van(t)
10 ms
20 ms
30 ms
40 ms
t
Inductor current contains sixth harmonic ripple (360 Hz for a 60 Hz ac
system). This ripple is superimposed on the ac line current waveform,
and influences the fifth and seventh harmonic content of ia(t).
Fundamentals of Power Electronics
11
Chapter 16: Line-commutated rectifiers
16.2.2 Discontinuous conduction mode
THD = 99.3%
ia(t)
van(t)
10 ms
vbn(t)
vcn(t)
20 ms
30 ms
40 ms
t
Phase a current contains pulses at the positive and negative peaks of the
line-to-line voltages vab(t) and vac(t). Distortion factor and THD are increased.
Distortion factor of the typical waveform illustrated above is 71%.
Fundamentals of Power Electronics
12
Chapter 16: Line-commutated rectifiers
16.3 Phase control
Replace diodes with SCRs:
øa
ia(t)
α
L
iL(t)
+
+
Q1
3ø
ac
Phase control waveforms:
Q2
Q5
iL
Q6
øc
ωt
0
vd(t)
Q4
ia(t)
Q3
øb
van(t) = Vm sin (ωt)
C
–
dc load
R
V
–
0˚
180˚
90˚
– vbc
– vca
vab
vbc
270˚
– iL
vca
– vab
vd (t)
Average (dc) output voltage:
3
V=π
90˚+α
3 Vm sin(θ + 30˚)dθ
30˚+α
= 3 π2 VL-L, rms cos α
Fundamentals of Power Electronics
13
Upper thyristor:
Q3
Q1
Q1
Q2
Q2
Q3
Lower thyristor:
Q5
Q5
Q6
Q6
Q4
Q4
Chapter 16: Line-commutated rectifiers
Dc output voltage vs. delay angle α
V
1.5
V L–L, rms
Rectification
Inversion
3
V=π
1
90˚+α
3 Vm sin(θ + 30˚)dθ
30˚+α
= 3 π2 VL-L, rms cos α
0.5
0
–0.5
–1
–1.5
0
30
60
90
120
150
180
α, degrees
Fundamentals of Power Electronics
14
Chapter 16: Line-commutated rectifiers
16.3.1 Inverter mode
IL
øa
3ø
ac
L
+
øb
V
+
–
–
øc
If the load is capable of supplying power, then the direction of power
flow can be reversed by reversal of the dc output voltage V. The delay
angle α must be greater than 90˚. The current direction is unchanged.
Fundamentals of Power Electronics
15
Chapter 16: Line-commutated rectifiers
16.3.2 Harmonics and power factor
Fourier series of ac line current waveform, for large dc-side inductance:
∞
i a(t) =
Σ
n = 1,5,7,11,...
nπ
nπ
4
nπ I L sin 2 sin 3 sin (nωt – nα)
Same as uncontrolled rectifier case, except that waveform is delayed
by the angle α. This causes the current to lag, and decreases the
displacement factor. The power factor becomes:
power factor = 0.955 cos (α)
When the dc output voltage is small, then the delay angle α is close to
90˚ and the power factor becomes quite small. The rectifier apparently
consumes reactive power, as follows:
Q = 3 I a, rmsVL-L, rms sin α = I L 3 π2 VL-L, rms sin α
Fundamentals of Power Electronics
16
Chapter 16: Line-commutated rectifiers
Real and reactive power in controlled rectifier
at fundamental frequency
Q
|| S || sin α
ms
2 V
3 π
S
=
Lr
L–
IL
α
|| S || cos α
P
P = I L 3 π2 VL-L, rms cos α
Q = 3 I a, rmsVL-L, rms sin α = I L 3 π2 VL-L, rms sin α
Fundamentals of Power Electronics
17
Chapter 16: Line-commutated rectifiers
16.4 Harmonic trap filters
A passive filter, having resonant zeroes tuned to the harmonic frequencies
Zs
is
Z1
ac source
model
Fundamentals of Power Electronics
Z2
Z3
Harmonic traps
(series resonant networks)
18
...
ir
Rectifier
model
Chapter 16: Line-commutated rectifiers
Harmonic trap
Ac source:
model with
Thevenin-equiv
voltage source
and impedance
Zs’(s). Filter often
contains series
inductor sLs’.
Lump into
effective
impedance Zs(s):
Z s(s) = Z s'(s) + sL s'
Zs
is
Z1
ac source
model
Fundamentals of Power Electronics
Z2
Z3
Harmonic traps
(series resonant networks)
19
...
ir
Rectifier
model
Chapter 16: Line-commutated rectifiers
Filter transfer function
H(s) =
Z 1 || Z 2 ||
i s(s)
=
i R(s) Z s + Z 1 || Z 2 ||
or
H(s) =
i s(s) Z s || Z 1 || Z 2 ||
=
i R(s)
Zs
Zs
is
Z1
ac source
model
Fundamentals of Power Electronics
Z2
Z3
Harmonic traps
(series resonant networks)
20
...
ir
Rectifier
model
Chapter 16: Line-commutated rectifiers
Simple example
Ls
Qp ≈
is
R1
L1
R0p ≈
Z1 || Zs
1
fp ≈
2π L sC 1
ir
ωL
C1
s
Zs
R0p
R1
ωC 1
1
f1 =
Ls
C1
ωL
1
2π L 1C 1
Z1
R01 =
Q1 =
Fifth-harmonic
trap Z1
Fundamentals of Power Electronics
1
L1
C1
R01
R1
R1
21
Chapter 16: Line-commutated rectifiers
Simple example: transfer function
• Series resonance: fifth
harmonic trap
• Parallel resonance: C1
and Ls
Qp
1
fp
– 40 dB/decade
L1
L1 + Ls
f1
Q1
Fundamentals of Power Electronics
22
• Parallel resonance
tends to increase
amplitude of third
harmonic
• Q of parallel
resonance is larger
than Q of series
resonance
Chapter 16: Line-commutated rectifiers
Example 2
Ls
is
R1
R2
R3
L1
L2
L3
C1
C2
C3
5th harmonic
trap Z1
7th harmonic
trap Z2
Fundamentals of Power Electronics
23
ir
11th harmonic
trap Z3
Chapter 16: Line-commutated rectifiers
Approximate impedance asymptotes
Zs || Z1 || Z2 || Z3
ωC 1
1
s
ωL
f1
L1
ω
ωC 1
2
Q1
R2
24
ωL
3
f3
Q3
Q2
R1
Fundamentals of Power Electronics
f2
2
ωL
ωC 1
3
R3
Chapter 16: Line-commutated rectifiers
Transfer function asymptotes
1
f1
Q1
f2
Q2
f3
Q3
Fundamentals of Power Electronics
25
Chapter 16: Line-commutated rectifiers
Bypass resistor
ωL
Z1 || Zs
Rn
Rn
Rbp
Ln
Ln
Cn
Cn
ωL
1
Rbp
1
f1
Z1
Zs
Cb
fbp
ωC 1
fp
Rbp
s
1
fp
– 40 dB/decade
f1
fbp
– 20 dB/decade
Fundamentals of Power Electronics
26
Chapter 16: Line-commutated rectifiers
Harmonic trap filter with high-frequency roll-off
Ls
R5
R7
Rbp
L5
L7
Cb
C5
C7
Fifth-harmonic Seventh-harmonic
trap
trap with highfrequency rolloff
Fundamentals of Power Electronics
27
Chapter 16: Line-commutated rectifiers
16.5 Transformer connections
Three-phase transformer connections can be used to shift the phase of the
voltages and currents
This shifted phase can be used to cancel out the low-order harmonics
Three-phase delta-wye transformer connection shifts phase by 30˚:
ωt
a'
a
a
a'
3 :n
30˚
T1
b
T1
T2
T3
c
T3
n'
b'
T1
T2
c'
c
c'
T2
T3
T1
n'
T2
T3
b
b'
Primary voltages
Fundamentals of Power Electronics
28
Secondary voltages
Chapter 16: Line-commutated rectifiers
Twelve-pulse rectifier
a
ia(t)
1:n
ia1(t)
T1
3øac
source
T1
b
T2
L
IL
+
T3
n'
T3
T2
c
vd (t)
ia2(t)
dc
load
3 :n
T4
T4
T5
T6
n'
T6
Fundamentals of Power Electronics
T5
29
–
Chapter 16: Line-commutated rectifiers
Waveforms of 12 pulse rectifier
ia1(t)
nIL
ωt
90˚
180˚
270˚
360˚
– nIL
• Ac line current contains
1st, 11th, 13th, 23rd, 25th,
etc. These harmonic
amplitudes vary as 1/n
• 5th, 7th, 17th, 19th, etc.
harmonics are eliminated
ia2(t)
nI L 1 + 2 3
3
ia(t)
nI L
3
nI L 1 + 3
3
Fundamentals of Power Electronics
30
Chapter 16: Line-commutated rectifiers
Rectifiers with high pulse number
Eighteen-pulse rectifier:
• Use three six-pulse rectifiers
• Transformer connections shift phase by 0˚, +20˚, and –20˚
• No 5th, 7th, 11th, 13th harmonics
Twenty-four-pulse rectifier
• Use four six-pulse rectifiers
• Transformer connections shift phase by 0˚, 15˚, –15˚, and 30˚
• No 5th, 7th, 11th, 13th, 17th, or 19th harmonics
If p is pulse number, then rectifier produces line current harmonics of
number n = pk ± 1, with k = 0, 1, 2, ...
Fundamentals of Power Electronics
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Chapter 16: Line-commutated rectifiers