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The Ideal Rectifier Fundamentals of Power Electronics
1
Chapter 17: The Ideal Rectifier
Chapter 17
The Ideal Rectifier
17.1
Properties of the ideal rectifier
17.2
Realization of a near-ideal rectifier
17.3
Single-phase converter systems employing ideal rectifiers
17.4
RMS values of rectifier waveforms
17.5
Ideal three-phase rectifiers Fundamentals of Power Electronics
2
Chapter 17: The Ideal Rectifier
17.1 Properties of the ideal rectifier
It is desired that the rectifier present a resistive load to the ac power
system. This leads to
unity power factor
ac line current has same waveshape as voltage
i
ac
(t) = v
ac
(t)
R
e
R
e
is called the
emulated resistance
R
e
+ v
ac
(t)
i
ac
(t) Fundamentals of Power Electronics
3
Chapter 17: The Ideal Rectifier
Control of power throughput
R
e
(v
control
)
+ v
ac
(t)
i
ac
(t)
v
control
P
av
=
V
ac,rms
2
R
e
(v
control
)
Power apparently consumed by
R
e
is actually transferred to rectifier dc
output port. To control the amount
of output power, it must be possible
to adjust the value of
R
e
. Fundamentals of Power Electronics
4
Chapter 17: The Ideal Rectifier
Output port model
R
e
(v
control
)
+ v
ac
(t)
i
ac
(t)
v
control
v(t)
i(t)
+ p(t) = v
ac
2
/R
e
Ideal rectifier (LFR)
ac
input
dc
output
The ideal rectifier is
lossless and contains
no internal energy
storage. Hence, the
instantaneous input
power equals the
instantaneous output
power. Since the
instantaneous power is
independent of the dc
load characteristics, the
output port obeys a
power source
characteristic.
p(t) =
v
ac
2
(t)
R
e
(v
control
(t))
v(t)i(t) = p(t) = v
ac
2
(t)
R
e Fundamentals of Power Electronics
5
Chapter 17: The Ideal Rectifier
The dependent power source
p(t)
+
v(t) i(t)
p(t)
+
v(t) i(t)
v(t)i(t) = p(t)
v(t)
i(t)
power
source
power
sink
i-v characteristic Fundamentals of Power Electronics
6
Chapter 17: The Ideal Rectifier
Equations of the ideal rectifier / LFR
i
ac
(t) =
v
ac
(t)
R
e
(v
control
)
v(t)i(t) = p(t)
p(t) =
v
ac
2
(t)
R
e
(v
control
(t))
V
rms
V
ac,rms
=
R
R
e
I
ac,rms
I
rms
=
R
R
e
Defining equations of the
ideal rectifier:
When connected to a
resistive load of value
R
, the
input and output rms voltages
and currents are related as
follows: Fundamentals of Power Electronics
7
Chapter 17: The Ideal Rectifier
17.2 Realization of a near-ideal rectifier
1 : M(d(t))
dcdc converter
Controller
d(t)
R
v
ac
(t)
i
ac
(t)
+
v
g
(t) i
g
(t)
i
g
v
g
+
v(t) i(t)
C
Control the duty cycle of a dc-dc
converter, such that the input current
is proportional to the input voltage: Fundamentals of Power Electronics
8
Chapter 17: The Ideal Rectifier
Waveforms
t
v
ac
(t)
t
i
ac
(t)
V
M
v
g
(t)
V
M
V
M
/R
e
v(t)
i
g
(t)
V
M(t)
M
min
v
ac
(t) = V
M
sin ( t)
v
g
(t) = V
M
sin ( t)
M(d(t)) = v(t)
v
g
(t) =
V
V
M
sin ( t)
M
min
= V
V
M Fundamentals of Power Electronics
9
Chapter 17: The Ideal Rectifier
Choice of converter
M(d(t)) = v(t)
v
g
(t) =
V
V
M
sin ( t)
To avoid distortion near line voltage zero crossings, converter should
be capable of producing
M(d(t))
approaching infinity
Above expression neglects converter dynamics
Boost, buck-boost, Cuk, SEPIC, and other converters with similar
conversion ratios are suitable
We will see that the boost converter exhibits lowest transistor
stresses. For this reason, it is most often chosen
M(t)
M
min Fundamentals of Power Electronics
10
Chapter 17: The Ideal Rectifier
Boost converter
with controller to cause input current to follow input voltage
Boost converter
Controller
R
v
ac
(t)
i
ac
(t)
+
v
g
(t) i
g
(t)
i
g
(t)
v
g
(t)
+
v(t) i(t)
Q
1
L
C
D
1
v
control
(t)
Multiplier
X
+
v
ref
(t)
= k
x
v
g
(t) v
control
(t)
R
s
v
a
(t)
G
c
(s)
PWM
Compensator
v
err
(t) Fundamentals of Power Electronics
11
Chapter 17: The Ideal Rectifier
Variation of duty cycle in boost rectifier
M(d(t)) = v(t)
v
g
(t) =
V
V
M
sin ( t)
Since
M 1
in the boost converter, it is required that
V V
M
If the converter operates in CCM, then
M(d(t)) =
1
1 d(t)
The duty ratio should therefore follow
d(t) = 1 v
g
(t)
V
in CCM Fundamentals of Power Electronics
12
Chapter 17: The Ideal Rectifier
CCM/DCM boundary, boost rectifier
Inductor current ripple is i
g
(t) = v
g
(t)d(t)T
s
2<i>L
i
g
(t)
T
s
= v
g
(t)
R
e
Low-frequency (average) component of inductor current waveform is
The converter operates in CCM when
i
g
(t)
T
s
> i
g
(t) d(t) < 2<i>L
R
e
T
s
Substitute CCM expression for
d(t)
:
R
e
<
2<i>L
T
s
1 v
g
(t)
V
for CCM Fundamentals of Power Electronics
13
Chapter 17: The Ideal Rectifier
CCM/DCM boundary
R
e
<
2<i>L
T
s
1 v
g
(t)
V
for CCM
Note that
v
g
(t)
varies with time, between
0
and
V
M
. Hence, this
equation may be satisfied at some points on the ac line cycle, and not
at others. The converter always operates in CCM provided that
R
e
< 2<i>L
T
s
The converter always operates in DCM provided that
R
e
>
2<i>L
T
s
1 V
M
V
For
R
e
between these limits, the converter operates in DCM when
v
g
(t)
is near zero, and in CCM when
v
g
(t)
approaches
V
M
. Fundamentals of Power Electronics
14
Chapter 17: The Ideal Rectifier
Static input characteristics
of the boost converter
A plot of input current
i
g
(t)
vs input voltage
v
g
(t)
, for various duty cycles
d(t)
. In CCM, the boost converter equilibrium equation is
v
g
(t)
V
= 1 d(t)
The input characteristic in DCM is found by solution of the averaged
DCM model (Fig. 10.12(b)):
Beware! This DCM
R
e
(d)
from
Chapter 10 is not the same as the
rectifier emulated resistance
R
e

= v
g
/i
g
Solve for input current:
p
R
e
(d(t))
+ R
+
V v
g
(t)
i
g
(t)
i
g
(t) =
v
g
(t)
R
e
(d(t)) +
p(t)
V v
g
(t)
with
p(t) =
v
g
2
(t)
R
e
(d(t))
R
e
(d(t)) =
2<i>L
d
2
(t)T
s Fundamentals of Power Electronics
15
Chapter 17: The Ideal Rectifier
Static input characteristics
of the boost converter
Now simplify DCM current expression, to obtain
2<i>L
VT
s
i
g
(t) 1 v
g
(t)
V
= d
2
(t) v
g
(t)
V
CCM/DCM mode boundary, in terms of
v
g
(t)
and
i
g
(t)
:
2<i>L
VT
s
i
g
(t) > v
g
(t)
V
1 v
g
(t)
V Fundamentals of Power Electronics
16
Chapter 17: The Ideal Rectifier
Boost input characteristics
with superimposed resistive characteristic
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
d
= 0
d
= 0.2
d
= 0.4
d
= 0.6
d
= 0.8
d
= 1
CCM
DCM
j
g
(
t
)=
2
L
VT
s
i
g
(
t
)
m
g
(t) = v
g
(t)
V
i
g
(t) =
v
g
(t)/R
e
2<i>L
VT
s
i
g
(t) 1 v
g
(t)
V
= d
2
(t) v
g
(t)
V
v
g
(t)
V
= 1 d(t)
2<i>L
VT
s
i
g
(t) > v
g
(t)
V
1 v
g
(t)
V
CCM:
DCM:
CCM when Fundamentals of Power Electronics
17
Chapter 17: The Ideal Rectifier
R
e
of the multiplying (average current) controller
Boost converter
Controller
R
v
ac
(t)
i
ac
(t)
+
v
g
(t) i
g
(t)
i
g
(t)
v
g
(t)
+
v(t) i(t)
Q
1
L
C
D
1
v
control
(t)
Multiplier
X
+
v
ref
(t)
= k
x
v
g
(t) v
control
(t)
R
s
v
a
(t)
G
c
(s)
PWM
Compensator
v
err
(t)
v
a
(t) = i
g
(t)R
s
v
a
(t) v
ref
(t)
v
ref
(t) = k
x
v
g
(t)v
control
(t)
R
e
= v
g
(t)
i
g
(t) =
v
ref
(t)
k
x
v
control
(t)
v
a
(t)
R
s
R
e
(v
control
(t)) =
R
s
k
x
v
control
(t)
Solve circuit to find
R
e
:
Current sensor gain
when the error signal is small,
multiplier equation
then
R
e
is
simplify: Fundamentals of Power Electronics
18
Chapter 17: The Ideal Rectifier
Low frequency system model
R
v
ac
(t)
i
ac
(t)
R
e
+ v
control
(t) v(t) T
s
+ Ideal rectifier (LFR)
C
R
e
(t)
R
e
(t) =
R
s
k
x
v
control
(t) i(t) T
s i
g
(t) T
s p(t) T
s v
g
(t) T
s
R
e
(v
control
(t)) =
R
s
k
x
v
control
(t)
This model also applies to other
converters that are controlled in the
same manner, including buck-boost,
Cuk, and SEPIC. Fundamentals of Power Electronics
19
Chapter 17: The Ideal Rectifier
Open-loop DCM approach
We found in Chapter 10 that the buck-boost, SEPIC, and Cuk
converters, when operated open-loop in DCM, inherently behave as
loss-free resistors. This suggests that they could also be used as
near-ideal rectifiers, without need for a multiplying controller.
Advantage: simple control
Disadvantages: higher peak currents, larger input current E