Chapter 18
Pulse-Width Modulated Rectifiers
18.1
Properties of the ideal rectifier
18.2
Realization of a near-ideal rectifier
18.3
Control of the current waveform
18.4
Single-phase converter systems employing ideal rectifiers
18.5
RMS values of rectifier waveforms
18.6
Modeling losses and efficiency in CCM high-quality rectifiers
18.7
Ideal three-phase rectifiers
Fundamentals of Power Electronics
1
Chapter 18: PWM Rectifiers
18.1 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
vac(t)
i ac(t) =
Re
Re is called the emulated resistance
iac(t)
+
vac(t)
Re
–
Fundamentals of Power Electronics
2
Chapter 18: PWM Rectifiers
Control of power throughput
V 2ac,rms
Pav =
Re(vcontrol)
iac(t)
+
vac(t)
Power apparently “consumed” by Re
is actually transferred to rectifier dc
output port. To control the amount
of output power, it must be possible
to adjust the value of Re.
Fundamentals of Power Electronics
3
Re(vcontrol)
–
vcontrol
Chapter 18: PWM Rectifiers
Output port model
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.
Fundamentals of Power Electronics
Ideal rectifier (LFR)
iac(t)
+
2
p(t) = vac /Re
vac(t)
Re(vcontrol)
i(t)
+
v(t)
–
–
ac
input
dc
output
vcontrol
v 2ac(t)
p(t) =
Re(vcontrol(t))
4
v 2ac(t)
v(t)i(t) = p(t) =
Re
Chapter 18: PWM Rectifiers
The dependent power source
i(t)
i(t)
p(t)
v(t)i(t) = p(t)
i(t)
+
+
v(t)
v(t)
–
–
power
source
p(t)
v(t)
power
sink
i-v characteristic
Fundamentals of Power Electronics
5
Chapter 18: PWM Rectifiers
Equations of the ideal rectifier / LFR
When connected to a
resistive load of value R, the
input and output rms voltages
and currents are related as
follows:
Defining equations of the
ideal rectifier:
i ac(t) =
vac(t)
Re(vcontrol)
v(t)i(t) = p(t)
v 2ac(t)
p(t) =
Re(vcontrol(t))
Fundamentals of Power Electronics
6
Vrms
=
Vac,rms
R
Re
I ac,rms
=
I rms
R
Re
Chapter 18: PWM Rectifiers
18.2 Realization of a near-ideal rectifier
Control the duty cycle of a dc-dc
converter, such that the input current
is proportional to the input voltage:
dc–dc converter
ig(t)
1 : M(d(t))
+
iac(t)
vac(t)
i(t)
+
vg(t)
v(t)
–
–
C
R
d(t)
ig
Controller
vg
Fundamentals of Power Electronics
7
Chapter 18: PWM Rectifiers
Waveforms
vac(t)
ig(t)
VM
t
v(t)
iac(t)
V
VM /Re
M(t)
t
Mmin
vg(t)
VM
Fundamentals of Power Electronics
vac(t) = VM sin (ωt)
M(d(t)) =
vg(t) = VM sin (ωt)
M min = V
VM
8
v(t)
V
=
vg(t) VM sin (ωt)
Chapter 18: PWM Rectifiers
Output-side current
Averaged
vg(t)i g(t) v 2g(t)
i(t) =
=
over
V
VRe
switching
V 2M
i(t) =
sin 2 (ωt)
period
VRe
V 2M
1 – cos (2ωt)
=
2VRe
Averaged
over ac line I = i(t)
period
V 2M
P=
1 : M(d(t))
+
vac(t)
2Re
dc–dc converter
ig(t)
iac(t)
V 2M
=
TL
2VRe
i(t)
+
vg(t)
v(t)
–
–
C
R
d(t)
ig
Controller
vg
Fundamentals of Power Electronics
9
Chapter 18: PWM Rectifiers
Choice of converter
M(d(t)) =
v(t)
V
=
vg(t) VM sin (ωt)
M(t)
Mmin
• 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
Fundamentals of Power Electronics
10
Chapter 18: PWM Rectifiers
18.2.1 CCM Boost converter
with controller to cause input current to follow input voltage
Boost converter
i(t)
ig(t)
+
iac(t)
vac(t)
+
L
D1
vg(t)
Q1
v(t)
R
–
–
vg(t)
C
ig(t)
d(t)
Controller
• DC output voltage ≥ peak AC input voltage
• Controller varies duty cycle as necessary to make ig(t) proportional
to vg(t)
Fundamentals of Power Electronics
11
Chapter 18: PWM Rectifiers
Variation of duty cycle in boost rectifier
M(d(t)) =
v(t)
V
=
vg(t) VM sin (ωt)
Since M ≥ 1 in the boost converter, it is required that V ≥ VM
If the converter operates in CCM, then
M(d(t)) =
1
1 – d(t)
The duty ratio should therefore follow
vg(t)
d(t) = 1 –
V
Fundamentals of Power Electronics
12
in CCM
Chapter 18: PWM Rectifiers
CCM/DCM boundary, boost rectifier
Inductor current ripple is
vg(t)d(t)Ts
∆i g(t) =
2L
Low-frequency (average) component of inductor current waveform is
i g(t)
Ts
=
vg(t)
Re
The converter operates in CCM when
i g(t)
2L
>
∆i
(t)
⇒
d(t)
<
g
Ts
ReTs
Substitute CCM expression for d(t):
Re <
2L
vg(t)
Ts 1 –
V
Fundamentals of Power Electronics
for CCM
13
Chapter 18: PWM Rectifiers
CCM/DCM boundary
Re <
2L
vg(t)
Ts 1 –
V
for CCM
Note that vg(t) varies with time, between 0 and VM. 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
Re < 2L
Ts
The converter always operates in DCM provided that
Re >
2L
VM
Ts 1 –
V
For Re between these limits, the converter operates in DCM when vg(t)
is near zero, and in CCM when vg(t) approaches VM.
Fundamentals of Power Electronics
14
Chapter 18: PWM Rectifiers
Static input characteristics
of the boost converter
A plot of input current ig(t) vs input voltage vg(t), for various duty cycles
d(t). In CCM, the boost converter equilibrium equation is
vg(t)
= 1 – d(t)
V
The input characteristic in DCM is found by solution of the averaged
DCM model (Fig. 11.12(b)):
p(t)
V – vg(t)
ig(t)
Solve for input current:
+
vg(t)
+
–
2L
d 2T s
p(t)
+
–
i g(t) =
V
–
Beware! This DCM Re(d) from
Chapter 11 is not the same as the
rectifier emulated resistance Re = vg/ig
Fundamentals of Power Electronics
15
with
vg(t)
2L
d 2T s
+
p(t)
V – vg(t)
p(t) =
v 2g(t)
2L
d 2T s
Chapter 18: PWM Rectifiers
Static input characteristics
of the boost converter
Now simplify DCM current expression, to obtain
2L i (t) 1 – vg(t) = d 2(t) vg(t)
VTs g
V
V
CCM/DCM mode boundary, in terms of vg(t) and ig(t):
2L i (t) > vg(t)
V
VTs g
Fundamentals of Power Electronics
1–
vg(t)
V
16
Chapter 18: PWM Rectifiers
Boost input characteristics
d=0
d = 0.2
d = 0.4
d = 0.6
d=1
1
d = 0.8
with superimposed resistive characteristic
jg(t) = 2L i g(t)
VT s
0.75
CCM:
vg(t)
= 1 – d(t)
V
DCM:
0.5
)
i g(t
=
t)
v g(
2L i (t) 1 – vg(t) = d 2(t) vg(t)
VTs g
V
V
/R e
CCM when
CCM
0.25
2L i (t) > vg(t)
VTs g
V
DCM
1–
vg(t)
V
0
0
0.25
0.5
m g(t) =
Fundamentals of Power Electronics
0.75
1
vg(t)
V
17
Chapter 18: PWM Rectifiers
Open-loop DCM approach
We found in Chapter 11 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 EMI
Like other DCM applications, this approach is usually restricted to low
power (< 200W).
The boost converter can also be operated in DCM as a low harmonic
rectifier. Input characteristic is
vg(t)
v 2g(t)
i g(t) T =
+
Re
s
R v(t) – v (t)
e
g
Input current contains harmonics. If v is sufficiently greater than vg,
then harmonics are small.
Fundamentals of Power Electronics
18
Chapter 18: PWM Rectifiers
Other similar approaches
• Use of other converters (in CCM) that are capable of increasing the
voltage:
SEPIC, Cuk, buck-boost
Flyback, isolated versions of boost, SEPIC, Cuk, etc.
• Boundary or critical conduction mode: operation of boost or other
converter at the boundary between CCM and DCM
• Buck converter: distortion occurs but stresses are low
• Resonant converter such as parallel resonant converter or some
quasi-resonant converters
• Converters that combine the functions of rectification, energy
storage, and dc-dc conversion
Fundamentals of Power Electronics
19
Chapter 18: PWM Rectifiers
18.2.2 DCM flyback converter
Flyback converter
EMI filter
ig(t)
iac(t)
n:1
D1
+
L
vac(t)
i(t)
+
v(t)
C
R
vg(t)
Q1
–
–
D
Operation in DCM: we found in Chapter 11 that the converter input
port obeys Ohm’s law with effective resistance Re = 2n2L/D2Ts.
Hence, simply connect input port to AC line.
Fundamentals of Power Electronics
20
Chapter 18: PWM Rectifiers
Averaged large-signal model
EMI filter
⟨ ig(t) ⟩T
Averaged model
⟨ i(t) ⟩T
+
iac(t)
vac(t)
vg(t)
+
2n 2L
D 2T s
⟨ p(t) ⟩T
–
⟨v(t)⟩T
C
R
–
D
• Under steady-state conditions, operate with constant D
• Adjust D to control average power drawn from AC line
Fundamentals of Power Electronics
21
Chapter 18: PWM Rectifiers
Converter design
Select L small enough that DCM
operation occurs throughout AC line
cycle. DCM occurs provided that
d3 > 0, or
d 2(t) < 1 – D
But
vg(t)
d 2(t) = D
nV
Substitute and solve for D:
D<
1
vg(t)
1+
nV
Area q1
ipk
i 1(t)
d1Ts
Ts
d2Ts
d3Ts
t
Ts
To obtain DCM at all points on
input AC sinusoid: worst case is at
maximum vg(t) = VM :
Converter operates in DCM in
every switching period where
above inequality is satisfied.
Fundamentals of Power Electronics
i1(t)
D<
22
1
V
1+ M
nV
Chapter 18: PWM Rectifiers
Choice of L
to obtain DCM everywhere along AC sinusoid
We have:
D<
1
V
1+ M
nV
with
Substitute expression for Re to obtain
D = 2nV
VM
L
RT s
Solve for L:
L < L crit =
RT s
4 1 + nV
VM
2
Vrms
=
Vac,rms
R
Re
Worst-case design
For variations in load
resistance and ac input
voltage, the worst case
occurs at maximum load
power and minimum ac input
voltage. The inductance
should be chosen as follows:
L < L crit-min =
Fundamentals of Power Electronics
23
Rmin T s
4 1 + nV
V M-min
2
Chapter 18: PWM Rectifiers
18.3 Control of the Current Waveform
18.3.1
Average current control
Feedforward
18.3.2
Current programmed control
18.3.3
Critical conduction mode and hysteretic control
18.3.4
Nonlinear carrier control
Fundamentals of Power Electronics
24
Chapter 18: PWM Rectifiers
18.3.1 Average current control
ig(t)
Boost example
vg(t)
+
–
v(t)
–
+
–
Low frequency
(average) component
of input current is
controlled to follow
input voltage
+
L
Gate
driver
Pulse width
modulator
va(t)
≈ Rs ⟨ ig(t)⟩T
Gc(s) Compensator
s
–
Current v (t)
reference r
Fundamentals of Power Electronics
25
+
Chapter 18: PWM Rectifiers
Block diagram
Boost converter
• Current
reference
derived
from input
voltage
waveform
i(t)
ig(t)
+
iac(t)
vac(t)
vg(t)
vg(t)
Multiplier
Q1
X
C
v(t)
–
ig(t)
Rs
PWM
va(t)
v (t)
+– err
Gc(s)
vr (t)
= kx vg(t) vcontrol (t)
• Compensation of
current loop
Fundamentals of Power Electronics
+
D1
–
vcontrol (t)
• Multiplier allows
control of emulated
resistance value
L
Compensator
Controller
26
Chapter 18: PWM Rectifiers
R
The emulated resistance
• Current sensor has
gain Rs :
va(t) = Rs i g(t)
Ts
Boost converter
i(t)
ig(t)
+
iac(t)
vg(t)
vac(t)
• If loop is well
designed, then:
L
+
D1
C
Q1
–
vcontrol (t)
va(t) ≈ vr (t)
vg(t)
Multiplier
X
v(t)
–
ig(t)
Rs
PWM
va(t)
v (t)
+– err
Gc(s)
• Multiplier:
vr(t) = k x vg(t) vcontrol(t)
vr (t)
= kx vg(t) vcontrol (t)
Compensator
Controller
• Hence the emulated resistance is:
Re =
vg(t)
=
i g(t)
vr (t)
k xvcontrol (t)
which can be simplified to
Re vcontrol(t) =
va(t)
Rs
Fundamentals of Power Electronics
27
Rs
k xvcontrol(t)
Chapter 18: PWM Rectifiers
R
System model using LFR
Average current control
Ideal rectifier (LFR)
⟨ig(t)⟩Ts
iac(t)
vac(t)
+
⟨vg(t)⟩T
⟨i(t)⟩T
+
⟨ p(t)⟩Ts
s
Re
C
–
⟨v(t)⟩T
s
–
Re(t)
Re(t) =
Fundamentals of Power Electronics
s
28
Rs
vcontrol (t)
k x vcontrol(t)
Chapter 18: PWM Rectifiers
R
Use of multiplier to control average power
An analog multiplier
introduces the
dependence of Re
on v(t).
ig(t)
+
vg(t)
+
–
C
Gate
driver
Pulse width
modulator
va(t)
Gc(s) Compensator
Multiplier
x
y
v(t)
–
+
–
As discussed in Chapter
17, an output voltage
feedback loop adjusts the
emulated resistance Re
such that the rectifier
power equals the dc load
power:
V 2g,rms
Pav =
= Pload
vg(t)
Re
kx xy
vref1(t)
–
+
verr(t)
v(t)
vcontrol(t)
Gcv(s)
–
+
vref2(t)
Voltage reference
Fundamentals of Power Electronics
29
Chapter 18: PWM Rectifiers
Feedforward
Feedforward is sometimes
used to cancel out
disturbances in the input
voltage vg(t).
ig(t)
+
vg(t)
+
–
v(t)
To maintain a given power
throughput Pav, the reference
voltage vref1(t) should be
+
–
Pavvg(t)Rs
vref 1(t) =
V 2g,rms
–
Pulse width
modulator
vg(t)
multiplier
Peak
detector V
M
x
z
y
Gate
driver
kv
xy
z2
va(t)
vref1(t)
vcontrol(t)
Gc(s) Compensator
–
+
–
Gcv(s)
+
vref2(t)
Voltage reference
Fundamentals of Power Electronics
30
Chapter 18: PWM Rectifiers
Feedforward, continued
ig(t)
Controller with feedforward
produces the following reference:
+
k vvcontrol(t)vg(t)
vref 1(t) =
V 2M
vg(t)
+
–
v(t)
–
Pav =
k vvcontrol(t)
2Rs
+
–
The average power is then
given by
Pulse width
modulator
vg(t)
multiplier
Peak
detector V
M
x
z
y
Gate
driver
kv
xy
z2
va(t)
vref1(t)
vcontrol(t)
Gc(s) Compensator
–
+
–
Gcv(s)
+
vref2(t)
Voltage reference
Fundamentals of Power Electronics
31
Chapter 18: PWM Rectifiers
Modeling the inner wide-bandwidth
average current controller
Averaged (but not linearized) boost converter model:
L
⟨i1(t)⟩T
⟨i(t)⟩T
⟨vg(t)⟩Ts
+
+
s
s
+
–
⟨v1(t)⟩Ts
+
–
⟨i2(t)⟩Ts
⟨v2(t)⟩Ts
–
C
R
⟨v(t)⟩T
s
–
Averaged switch network
In Chapter 7,
we perturbed
and linearized
using the
assumptions
vg(t)
Ts
= Vg + vg(t)
d(t) = D + d(t) ⇒ d'(t) = D' – d(t)
i(t) T = i 1(t) T = I + i(t)
s
v(t)
v1(t)
i 2(t)
Fundamentals of Power Electronics
Ts
Ts
Ts
s
= v2(t)
Ts
= V + v(t)
= V1 + v1(t)
= I 2 + i 2(t)
32
Problem: variations in vg,
i1 , and d are not small.
So we are faced with the
design of a control
system that exhibits
significant nonlinear
time-varying behavior.
Chapter 18: PWM Rectifiers
Linearizing the equations of the boost rectifier
When the rectifier operates near steady-state, it is true that
v(t)
Ts
= V + v(t)
with
v(t) << V
In the special case of the boost rectifier, this is sufficient to linearize
the equations of the average current controller.
The boost converter average inductor voltage is
d i g(t)
L
Ts
dt
= vg(t)
Ts
– d'(t)V – d'(t)v(t)
substitute:
d i g(t)
L
dt
Fundamentals of Power Electronics
Ts
= vg(t)
Ts
33
– d'(t)V – d'(t)v(t)
Chapter 18: PWM Rectifiers
Linearized boost rectifier model
d i g(t)
L
Ts
dt
= vg(t)
Ts
– d'(t)V – d'(t)v(t)
The nonlinear term is much smaller than the linear ac term. Hence, it
can be discarded to obtain
d i g(t)
L
dt
Ts
= vg(t)
Ts
– d'(t)V
L
Equivalent circuit:
i g(t)
i g(s) V
=
d(s) sL
Fundamentals of Power Electronics
vg(t)
Ts
+
–
34
Ts
+
–
d'(t)V
Chapter 18: PWM Rectifiers
The quasi-static approximation
The above approach is not sufficient to linearize the equations needed to
design the rectifier averaged current controllers of buck-boost, Cuk,
SEPIC, and other converter topologies. These are truly nonlinear timevarying systems.
An approximate approach that is sometimes used in these cases: the
quasi-static approximation
Assume that the ac line variations are much slower than the converter
dynamics, so that the rectifier always operates near equilibrium. The
quiescent operating point changes slowly along the input sinusoid, and we
can find the slowly-varying “equilibrium” duty ratio as in Section 18.2.1.
The converter small-signal transfer functions derived in Chapters 7 and 8
are evaluated, using the time-varying operating point. The poles, zeroes,
and gains vary slowly as the operating point varies. An average current
controller is designed, that has a positive phase margin at each operating
point.
Fundamentals of Power Electronics
35
Chapter 18: PWM Rectifiers
Quasi-static approximation: discussion
• In the literature, several authors have reported success using this
method
• Should be valid provided that the converter dynamics are suffieiently
fast, such that the converter always operates near the assumed
operating points
• No good condition on system parameters, which can justify the
approximation, is presently known for the basic converter topologies
• It is well-understood in the field of control systems that, when the
converter dynamics are not sufficiently fast, then the quasi-static
approximation yields neither necessary nor sufficient conditions for
stability. Such behavior can be observed in rectifier systems. Worstcase analysis to prove stability should employ simulations.
Fundamentals of Power Electronics
36
Chapter 18: PWM Rectifiers
18.3.2 Current programmed control
Current programmed
control is a natural
approach to obtain input
resistor emulation:
Boost converter
vg(t) +
–
+
C
Q1
v(t)
–
vg(t)
is(t)
ma
Clock
++
Fundamentals of Power Electronics
i2(t)
D1
Peak transistor current is
programmed to follow
input voltage.
Peak transistor current
differs from average
inductor current,
because of inductor
current ripple and
artificial ramp. This leads
to significant input
current waveform
distortion.
L
ig(t)
ia(t)
0
vcontrol(t)
Ts
S Q
Multiplier
X
+
–
R
ic(t)
Comparator
= kx vg(t) vcontrol(t)
Latch
Current-programmed controller
37
Chapter 18: PWM Rectifiers
R
CPM boost converter: Static input characteristics
vg(t)
T s in CCM
L
1
Fundamentals of Power Electronics
e
e
=2
R
bas
e
bas
e
R
bas
Re = 0
.2R
Re = 0.1R
CCM
b as
e
e
DCM
10
R
0.2
vg(t)
i c(t) =
Re
Minimum slope compensation:
0.4
R
It is desired that
vg(t)
1–
V
0.6
=4
TsV m a L vg(t)
i c(t) >
+
L
V
V
Ts
vg(t)
TsV vg(t)
1–
V
2L V
R
or,
Ts
>
jg(t) = i g(t)
i g(t)
0.8
Rbase
V
Mode boundary: CCM occurs when
ma = V
2L
Rbase = 2L
Ts
R =
e
R
ma +
ase
vg(t)
V
base
i c(t) – 1 –
Re =
0.5R
b
V – vg(t) vg(t) + m a L
ase
Ts
=
in DCM
Re =
0.33R
b
i g(t)
Static input characteristics of
CPM boost, with minimum
slope compensation:
base
vg(t)
Li 2c (t) fs
Re =
0
0.0
ma = V
2L
38
0.2
0.4
0.6
0.8
vg(t)
V
Chapter 18: PWM Rectifiers
1.0
Input current waveforms
with current mode control
i g(t)
Peak i g
• Substantial
distortion can
occur
1.0
ma = V
2L
Rbase = 2L
Ts
0.2
ase
Re =
0.33
Rb
R =
e
0.1R
bas
e
0.4
Re =
2R
b
id
uso
Sin
0.6
• Can meet
harmonic limits
if the range of
operating points
is not too large
ase
0.8
0.0
Fundamentals of Power Electronics
ωt
39
• Difficult to meet
harmonic limits
in a universal
input supply
Chapter 18: PWM Rectifiers
18.3.3 Critical conduction mode
and hysteretic control
Variable switching
frequency schemes
ig(t)
Hysteretic
control
• Hysteretic control
• Critical conduction
mode (boundary
between CCM and
DCM)
ωt
ig(t)
Critical
conduction
mode
ton
Fundamentals of Power Electronics
40
ωt
Chapter 18: PWM Rectifiers
An implementation of critical conduction control
Boost converter
EMI filter
ig(t)
i(t)
+
iac(t)
vac(t)
L
+
D1
vg(t)
C
Q1
–
vcontrol (t)
v(t)
–
vg(t)
Multiplier
X
Zero current
ig detector
Rs
va(t)
vr (t)
= kx vg(t) vcontrol (t)
+
–
Comparator
S Q
R
Latch
Controller
Fundamentals of Power Electronics
41
Chapter 18: PWM Rectifiers
R
Pros and cons of critical conduction control
• Simple, low-cost controller ICs
• Low-frequency harmonics are very small, with constant transistor
on-time (for boost converter)
• Small inductor
• Increased peak current
• Increased conduction loss, reduced switching loss
• Requires larger input filter
• Variable switching frequency smears out the current EMI spectrum
• Cannot synchronize converter switching frequencies
Fundamentals of Power Electronics
42
Chapter 18: PWM Rectifiers
Analysis
ig(t)
Transistor is on for fixed time ton
Transistor off-time ends when
inductor current reaches zero
Ratio of vg(t) to ⟨ig(t)⟩ is
Re = 2L
t on
Inductor volt-second balance:
On time, as a function of load
power and line voltage:
vg t on + vg – V t off = 0
t on = 4LP
V 2M
Solve for toff:
t off = t on
Fundamentals of Power Electronics
ωt
ton
43
vg
V – vg
Chapter 18: PWM Rectifiers
Switching frequency variations
Solve for how the controller varies the switching frequency over the ac
line period:
1
T s = t off + t on
T s = 4LP
V 2M
vg(t)
1–
V
For sinusoidal line voltage variations, the switching frequency will
therefore vary as follows:
V 2M
V
1
fs = =
1 – M sin (ωt)
V
T s 4LP
Minimum and maximum limits on switching frequency:
V 2M
V
min fs =
1– M
V
4LP
V 2M
max fs =
4LP
These equations can be used to select the value of the inductance L.
Fundamentals of Power Electronics
44
Chapter 18: PWM Rectifiers
18.3.4 Nonlinear carrier control
• Can attain simple control of input current waveform without sensing
the ac input voltage, and with operation in continuous conduction
mode
• The integral of the sensed switch current (charge) is compared to a
nonlinear carrier waveform (i.e., a nonlinear ramp), on a cycle-bycycle basis
• Carrier waveform depends on converter topology
• Very low harmonics in CCM. Waveform distortion occurs in DCM.
• Peak current mode control is also possible, with a different carrier
Fundamentals of Power Electronics
45
Chapter 18: PWM Rectifiers
Controller block diagram
Nonlinear carrier charge control of boost converter
Boost converter
D1
L
ig(t)
is(t)
+
is(t)
v(t)
C
:1
vg(t) +
–
R
n
Q1
–
is /n
vi (t)
Comparator
Ci
– vi (t) +
R Q
vc(t)
Nonlinear carrier
generator
vcontrol (t)
vc(t)
Latch
+
–
vi (t)
S Q
0
0
Ts
dTs
Ts
Clock
Nonlinear-carrier charge controller
Fundamentals of Power Electronics
46
Chapter 18: PWM Rectifiers
Derivation of NLC approach
The average switch current is
t + Ts
1
i s(t) T =
i s(τ)dτ
Ts t
s
We could make the controller regulate the average switch current by
• Integrating the monitored switch current
• Resetting the integrator to zero at the beginning of each
switching period
• Turning off the transistor when the integrator reaches a
reference value
In the controller diagram, the integrator follows this equation:
vi(t) = 1
Ci
i.e.,
vi(dT s) =
Fundamentals of Power Electronics
dT s
0
is
i s(τ)
n dτ
Ts
nC i fs
for 0 < t < dT s
for interval 0 < t < T s
47
Chapter 18: PWM Rectifiers
How to control the average switch current
Input resistor emulation:
vg(t)
i g(t)
=
Ts
Ts
Re(vcontrol)
Relate average switch current to input current (assuming CCM):
i s(t)
Ts
= d(t) i g(t)
Ts
Relate input voltage to output voltage (assuming CCM):
vg(t)
Ts
= d′(t) v(t)
Ts
Substitute above equations to find how average switch current should
be controlled:
v(t)
i s(t)
Ts
= d(t) 1 – d(t)
Fundamentals of Power Electronics
Ts
Re(vcontrol)
48
Chapter 18: PWM Rectifiers
Implementation using nonlinear carrier
v(t)
Desired control, from previous slide:
i s(t)
Ts
= d(t) 1 – d(t)
Ts
Re(vcontrol)
Generate carrier waveform as follows (replace d by t/Ts ):
vc(t) = vcontrol t 1 – t
for 0 ≤ t ≤ T s
Ts
Ts
vc(t + T s) = vc(t)
The controller switches the transistor off when the integrator voltage
equals the carrier waveform. This leads to:
vi(dT s) = vc(dT s) = vcontrol(t) d(t) 1 – d(t)
i s(t)
Ts
nC i f s
= vcontrol(t) d(t) 1 – d(t)
v(t)
Re(vcontrol) = d(t) 1 – d(t)
Fundamentals of Power Electronics
49
i s(t)
Ts
Ts
v(t)
=
Ts
nC i fsvcontrol(t)
Chapter 18: PWM Rectifiers
Generating the parabolic carrier
Removal of dc
component
vcontrol (t)
vc(t)
+–
Integrator with reset
Integrator with reset
Clock
(one approach, suitable for discrete circuitry)
Note that no separate multiplier circuit is needed
Fundamentals of Power Electronics
50
Chapter 18: PWM Rectifiers
18.4 Single-phase converter systems
containing ideal rectifiers
•
It is usually desired that the output voltage v(t) be regulated with
high accuracy, using a wide-bandwidth feedback loop
•
For a given constant load characteristic, the instantaneous load
current and power are then also constant:
pload (t) = v(t)i(t) = VI
•
The instantaneous input power of a single-phase ideal rectifier is
not constant:
pac(t) = vg(t)i g(t)
with
so
vg(t) = VM sin (ωt)
vg(t)
i g(t) =
Re
V 2M
V 2M
2
pac(t) =
sin ωt =
1 – cos 2ωt
Re
2Re
Fundamentals of Power Electronics
51
Chapter 18: PWM Rectifiers
Power flow in single-phase ideal rectifier system
•
Ideal rectifier is lossless, and contains no internal energy storage.
•
Hence instantaneous input and output powers must be equal
•
An energy storage element must be added
•
Capacitor energy storage: instantaneous power flowing into
capacitor is equal to difference between input and output powers:
pC(t) =
d EC(t)
=
dt
d
1
2
Cv 2C(t)
dt
= pac(t) – pload(t)
Energy storage capacitor voltage must be allowed to vary, in
accordance with this equation
Fundamentals of Power Electronics
52
Chapter 18: PWM Rectifiers
Capacitor energy storage in 1¿ system
pac(t)
Pload
vc(t)
d
=
1
2
Cv 2C(t)
dt
= pac(t) – pload(t)
t
Fundamentals of Power Electronics
53
Chapter 18: PWM Rectifiers
Single-phase system with internal energy storage
⟨ pac(t)⟩T
+
iac(t)
vac(t)
pload(t) = VI = Pload
Ideal rectifier (LFR) i (t)
2
ig(t)
vg(t)
+
+
s
Re
C
–
vC(t)
Dc–dc
converter
i(t)
v(t)
load
–
–
Energy storage
capacitor
Energy storage capacitor
voltage vC(t) must be
independent of input and
output voltage waveforms, so
that it can vary according to
d
=
1
2
Cv 2C(t)
dt
This system is capable of
= pac(t) – pload(t)
Fundamentals of Power Electronics
54
•
Wide-bandwidth control of
output voltage
•
Wide-bandwidth control of
input current waveform
•
Internal independent energy
storage
Chapter 18: PWM Rectifiers
Hold up time
Internal energy storage allows the system to function in other
situations where the instantaneous input and output powers differ.
A common example: continue to supply load power in spite of failure
of ac line for short periods of time.
Hold up time: the duration which the dc output voltage v(t) remains
regulated after vac(t) has become zero
A typical hold-up time requirement: supply load for one complete
missing ac line cycle, or 20 msec in a 50 Hz system
During the hold-up time, the load power is supplied entirely by the
energy storage capacitor
Fundamentals of Power Electronics
55
Chapter 18: PWM Rectifiers
Energy storage element
Instead of a capacitor, and inductor or higher-order LC network could
store the necessary energy.
But, inductors are not good energy-storage elements
Example
100 V 100 µF capacitor
100 A 100 µH inductor
each store 1 Joule of energy
But the capacitor is considerably smaller, lighter, and less
expensive
So a single big capacitor is the best solution
Fundamentals of Power Electronics
56
Chapter 18: PWM Rectifiers
Inrush current
A problem caused by the large energy storage capacitor: the large
inrush current observed during system startup, necessary to charge
the capacitor to its equilibrium value.
Boost converter is not capable of controlling this inrush current.
Even with d = 0, a large current flows through the boost converter
diode to the capacitor, as long as v(t) < vg(t).
Additional circuitry is needed to limit the magnitude of this inrush
current.
Converters having buck-boost characteristics are capable of
controlling the inrush current. Unfortunately, these converters exhibit
higher transistor stresses.
Fundamentals of Power Electronics
57
Chapter 18: PWM Rectifiers
Universal input
The capability to operate from the ac line voltages and frequencies
found everywhere in the world:
50Hz and 60Hz
Nominal rms line voltages of 100V to 260V:
100V, 110V, 115V, 120V, 132V, 200V, 220V, 230V, 240V, 260V
Regardless of the input voltage and frequency, the near-ideal rectifier
produces a constant nominal dc output voltage. With a boost
converter, this voltage is 380 or 400V.
Fundamentals of Power Electronics
58
Chapter 18: PWM Rectifiers
Low-frequency model of dc-dc converter
Dc-dc converter produces well-regulated dc load voltage V.
Load therefore draws constant current I.
Load power is therefore the constant value Pload = VI.
To the extent that dc-dc converter losses can be neglected, then dc-dc
converter input power is Pload , regardless of capacitor voltage vc(t).
Dc-dc converter input port behaves as a power sink. A low frequency
converter model is
p (t) = VI = P
i (t)
load
2
+
C
vC(t)
+
Pload V +
–
–
Energy storage
capacitor
Fundamentals of Power Electronics
59
load
i(t)
v(t) load
–
Dc-dc
converter
Chapter 18: PWM Rectifiers
Low-frequency energy storage process, 1¿ system
A complete low-frequency system model:
iac(t)
vac(t)
pload(t) = VI = Pload
i2(t)
ig(t)
⟨ pac(t)⟩Ts
+
vg(t)
Re
+
C
vC(t)
+
Pload V +
–
–
–
Ideal rectifier (LFR)
Energy storage
capacitor
i(t)
v(t) load
–
Dc-dc
converter
•
Difference between rectifier output power and dc-dc converter
input power flows into capacitor
•
In equilibrium, average rectifier and load powers must be equal
•
But the system contains no mechanism to accomplish this
•
An additional feeback loop is necessary, to adjust Re such that the
rectifier average power is equal to the load power
Fundamentals of Power Electronics
60
Chapter 18: PWM Rectifiers
Obtaining average power balance
iac(t)
vac(t)
pload(t) = VI = Pload
i2(t)
ig(t)
⟨ pac(t)⟩Ts
+
vg(t)
Re
+
C
vC(t)
+
Pload V +
–
v(t) load
–
–
Ideal rectifier (LFR)
Energy storage
capacitor
i(t)
–
Dc-dc
converter
If the load power exceeds the average rectifier power, then there is a
net discharge in capacitor energy and voltage over one ac line cycle.
There is a net increase in capacitor charge when the reverse is true.
This suggests that rectifier and load powers can be balanced by
regulating the energy storage capacitor voltage.
Fundamentals of Power Electronics
61
Chapter 18: PWM Rectifiers
A complete 1¿ system
containing three feedback loops
Boost converter
i2(t)
+
ig(t)
+
iac(t)
vac(t)
L
vg(t)
Q1
vC(t)
–
vcontrol(t)
vg(t)
Multiplier
X
i(t)
D1
DC–DC
Converter
C
Load v(t)
–
–
ig(t)
Rs
d(t)
PWM
v(t)
va(t)
vref1(t)
= kxvg(t)vcontrol(t)
+
v (t)
+– err
Gc(s)
Compensator
and modulator
–+ vref3
Compensator
Wide-bandwidth output voltage controller
Wide-bandwidth input current controller
vC(t)
Compensator
–+ vref2
Low-bandwidth energy-storage capacitor voltage controller
Fundamentals of Power Electronics
62
Chapter 18: PWM Rectifiers
Bandwidth of capacitor voltage loop
•
The energy-storage-capacitor voltage feedback loop causes the
dc component of vc(t) to be equal to some reference value
•
Average rectifier power is controlled by variation of Re.
•
Re must not vary too quickly; otherwise, ac line current harmonics
are generated
•
Extreme limit: loop has infinite bandwidth, and vc(t) is perfectly
regulated to be equal to a constant reference value
• Energy storage capacitor voltage then does not change, and
this capacitor does not store or release energy
• Instantaneous load and ac line powers are then equal
• Input current becomes
i ac(t) =
Fundamentals of Power Electronics
pac(t)
p (t)
Pload
= load
=
vac(t)
vac(t)
VM sin ωt
63
Chapter 18: PWM Rectifiers
Input current waveform, extreme limit
i ac(t) =
pac(t)
p (t)
Pload
= load
=
vac(t)
vac(t)
VM sin ωt
THD → ∞
Power factor → 0
vac(t)
iac(t)
t
Fundamentals of Power Electronics
64
So bandwidth of
capacitor voltage
loop must be
limited, and THD
increases rapidly
with increasing
bandwidth
Chapter 18: PWM Rectifiers
18.4.2 Modeling the outer low-bandwidth
control system
This loop maintains power balance, stabilizing the rectifier output
voltage against variations in load power, ac line voltage, and
component values
The loop must be slow, to avoid introducing variations in Re at the
harmonics of the ac line frequency
Objective of our modeling efforts: low-frequency small-signal model
that predicts transfer functions at frequencies below the ac line
frequency
Fundamentals of Power Electronics
65
Chapter 18: PWM Rectifiers
Large signal model
averaged over switching period Ts
Ideal rectifier (LFR)
⟨ ig(t)⟩Ts
s
⟨ p(t)⟩T
⟨ vg(t)⟩T
s
+
–
⟨ i2(t)⟩T
+
s
⟨ v(t)⟩T
C
Re (vcontrol )
s
Load
–
ac
input
dc
output
vcontrol
Ideal rectifier model, assuming that inner wide-bandwidth loop
operates ideally
High-frequency switching harmonics are removed via averaging
Ac line-frequency harmonics are included in model
Nonlinear and time-varying
Fundamentals of Power Electronics
66
Chapter 18: PWM Rectifiers
Predictions of large-signal model
Ideal rectifier (LFR)
⟨ ig(t)⟩Ts
If the input voltage is
vg(t) = 2 vg,rms sin ωt
⟨ p(t)⟩T
⟨ vg(t)⟩T
s
+
–
Re (vcontrol )
⟨ i2(t)⟩T
s
+
s
C
⟨ v(t)⟩T
s
–
Then the
instantaneous power
is:
vg(t)
ac
input
dc
output
vcontrol
2
v 2g,rms
p(t) T =
=
1 – cos 2ωt
s
Re(vcontrol(t)) Re(vcontrol(t))
Ts
which contains a constant term plus a secondharmonic term
Fundamentals of Power Electronics
67
Chapter 18: PWM Rectifiers
Load
Separation of power source into its constant and
time-varying components
⟨ i2(t)⟩T
s
+
V 2g,rms
–
cos 2 2ωt
Re
V 2g,rms
Re
C
⟨ v(t)⟩Ts
Load
–
Rectifier output port
The second-harmonic variation in power leads to second-harmonic
variations in the output voltage and current
Fundamentals of Power Electronics
68
Chapter 18: PWM Rectifiers
Removal of even harmonics via averaging
v(t)
⟨ v(t)⟩Ts
⟨ v(t)⟩T
2L
t
T2L =
Fundamentals of Power Electronics
1
2
69
2π = π
ω ω
Chapter 18: PWM Rectifiers
Resulting averaged model
⟨ i2(t)⟩T2L
+
V 2g,rms
Re
C
⟨ v(t)⟩T2L
Load
–
Rectifier output port
Time invariant model
Power source is nonlinear
Fundamentals of Power Electronics
70
Chapter 18: PWM Rectifiers
Perturbation and linearization
The averaged model predicts that the rectifier output current is
p(t)
i 2(t)
T 2L
=
v(t)
T 2L
T 2L
=
v 2g,rms(t)
Re(vcontrol(t)) v(t)
= f vg,rms(t), v(t)
Let
T 2L
, vcontrol(t))
T 2L
with
v(t)
i 2(t)
T 2L
T 2L
= V + v(t)
V >> v(t)
= I 2 + i 2(t)
I 2 >> i 2(t)
vg,rms = Vg,rms + vg,rms(t)
vcontrol(t) = Vcontrol + vcontrol(t)
Fundamentals of Power Electronics
Vg,rms >> vg,rms(t)
Vcontrol >> vcontrol(t)
71
Chapter 18: PWM Rectifiers
Linearized result
vcontrol(t)
I 2 + i 2(t) = g 2vg,rms(t) + j2v(t) –
r2
where
df vg,rms, V, Vcontrol)
g2 =
=
dvg,rms
– 1 =
r2
v g,rms = V g,rms
df Vg,rms, v
d v
, Vcontrol)
T 2L
=–
T 2L
dvcontrol
Fundamentals of Power Electronics
I2
V
v T =V
2L
df Vg,rms, V, vcontrol)
j2 =
Vg,rms
2
Re(Vcontrol) V
v control = V control
72
V 2g,rms
dRe(vcontrol)
=–
VR 2e (Vcontrol) dvcontrol
v control = V control
Chapter 18: PWM Rectifiers
Small-signal equivalent circuit
i2
+
r2
j2 vcontrol
g 2 vg,rms
C
v
R
–
Rectifier output port
Predicted transfer functions
Control-to-output
v(s)
1
= j2 R||r 2
vcontrol(s)
1 + sC R||r 2
Line-to-output
v(s)
1
= g 2 R||r 2
vg,rms(s)
1 + sC R||r 2
Fundamentals of Power Electronics
73
Chapter 18: PWM Rectifiers
Model parameters
Table 18.1 Small-signal model parameters for several types of rectifier control schemes
Controller type
g2
j2
r2
Average current control with
feedforward, Fig. 18.14
0
Pav
VVcontrol
V2
Pav
Current-programmed control,
Fig. 18.16
2Pav
VVg,rms
Pav
VVcontrol
V2
Pav
Nonlinear-carrier charge control
of boost rectifier, Fig. 18.21
2Pav
VVg,rms
Pav
VVcontrol
V2
2Pav
Boost with critical conduction mode
control, Fig. 18.20
2Pav
VVg,rms
Pav
VVcontrol
V2
Pav
DCM buck-boost, flyback, SEPIC,
or Cuk converters
2Pav
VVg,rms
2Pav
VD
V2
Pav
Fundamentals of Power Electronics
74
Chapter 18: PWM Rectifiers
Constant power load
ig(t)
⟨ pac(t)⟩Ts
+
iac(t)
vg(t)
vac(t)
pload(t) = VI = Pload
i2(t)
Re
C
–
+
+
vC(t)
Pload V +
–
v(t) load
–
–
Ideal rectifier (LFR)
Energy storage
capacitor
i(t)
Dc-dc
converter
Rectifier and dc-dc converter operate with same average power
Incremental resistance R of constant power load is negative, and is
2
V
R=–
Pav
which is equal in magnitude and opposite in polarity to rectifier
incremental output resistance r2 for all controllers except NLC
Fundamentals of Power Electronics
75
Chapter 18: PWM Rectifiers
Transfer functions with constant power load
When r2 = – R, the parallel combination r2 || R becomes equal to zero.
The small-signal transfer functions then reduce to
j
v(s)
= 2
vcontrol(s) sC
g
v(s)
= 2
vg,rms(s) sC
Fundamentals of Power Electronics
76
Chapter 18: PWM Rectifiers
18.5 RMS values of rectifier waveforms
Doubly-modulated transistor current waveform, boost rectifier:
iQ(t)
t
Computation of rms value of this waveform is complex and tedious
Approximate here using double integral
Generate tables of component rms and average currents for various
rectifier converter topologies, and compare
Fundamentals of Power Electronics
77
Chapter 18: PWM Rectifiers
RMS transistor current
RMS transistor current is
I Qrms =
1
Tac
T ac
iQ(t)
i 2Q(t)dt
0
Express as sum of integrals over
all switching periods contained
in one ac line period:
I Qrms =
1 T
Tac s
T ac/T s
∑
n=1
1
Ts
t
nT s
i 2Q(t)dt
(n-1)T s
Quantity in parentheses is the value of iQ2, averaged over the nth
switching period.
Fundamentals of Power Electronics
78
Chapter 18: PWM Rectifiers
Approximation of RMS expression
I Qrms =
T ac/T s
∑
1 T
Tac s
1
Ts
n=1
nT s
i 2Q(t)dt
(n-1)T s
When Ts << Tac, then the summation can be approximated by an
integral, which leads to the double-average:
I Qrms ≈
=
=
Fundamentals of Power Electronics
1 lim T
Tac T s→0 s
1
Tac
T ac
0
i 2Q(t)
79
Ts
1
Ts
T ac/T s
∑
n=1
t+T s
1
Ts
nT s
i 2Q(τ)dτ
(n-1)T s
i 2Q(τ)dτ dt
t
T ac
Chapter 18: PWM Rectifiers
18.5.1 Boost rectifier example
For the boost converter, the transistor current iQ(t) is equal to the input
current when the transistor conducts, and is zero when the transistor
is off. The average over one switching period of iQ2(t) is therefore
t+T
i
2
Q T
s
s
1
=
i 2Q(t)dt
Ts t
= d(t)i 2ac(t)
If the input voltage is
vac(t) = VM sin ωt
then the input current will be given by
VM
i ac(t) =
sin ωt
Re
and the duty cycle will ideally be
V =
1
vac(t) 1 – d(t)
Fundamentals of Power Electronics
80
(this neglects
converter dynamics)
Chapter 18: PWM Rectifiers
Boost rectifier example
Duty cycle is therefore
d(t) = 1 –
VM
sin ωt
V
Evaluate the first integral:
i
2
Q T
s
V 2M
V
= 2 1 – M sin ωt
V
Re
sin 2 ωt
Now plug this into the RMS formula:
I Qrms =
=
I Qrms =
Fundamentals of Power Electronics
1
Tac
T ac
i 2Q
0
T ac
1
Tac
0
2
M
2
e
2 V
Tac R
81
Ts
dt
V 2M
VM
1
–
sin ωt
V
R 2e
T ac/2
sin 2 ωt –
sin 2 ωt dt
VM
sin 3 ωt dt
V
0
Chapter 18: PWM Rectifiers
Integration of powers of sin θ over complete half-cycle
n
1
π
π
n
sin (θ)dθ =
0
2 2⋅4⋅6 (n – 1) if n is odd
π 1⋅3⋅5 n
1⋅3⋅5 (n – 1)
if n is even
2⋅4⋅6 n
Fundamentals of Power Electronics
82
1
π
π
sin n (θ)dθ
0
1
2
π
2
1
2
3
4
3π
4
3
8
5
16
15π
6
15
48
Chapter 18: PWM Rectifiers
Boost example: transistor RMS current
I Qrms =
VM
2 Re
V
1– 8 M
3π V
= I ac rms
V
1– 8 M
3π V
Transistor RMS current is minimized by choosing V as small as
possible: V = VM. This leads to
I Qrms = 0.39I ac rms
When the dc output voltage is not too much greater than the peak ac
input voltage, the boost rectifier exhibits very low transistor current.
Efficiency of the boost rectifier is then quite high, and 95% is typical in
a 1kW application.
Fundamentals of Power Electronics
83
Chapter 18: PWM Rectifiers
Table of rectifier current stresses for various topologies
Tabl e 18. 3
Summary of rectifier current stresses for several converter topologies
rms
Average
Peak
CCM boost
Transistor
I ac rms
Diode
I dc
VM
1 – 8
3π V
V
I ac rms 2 π2 1 – π M
8 V
16 V
3π V M
I dc
I ac rms 2 π2
I ac rms
Inductor
I ac rms 2
2 I dc V
VM
I ac rms 2
CCM flyback, with n:1 isolation transformer and input filter
Transistor,
xfmr primary
I ac rms
L1
C1
Diode,
xfmr secondary
Fundamentals of Power Electronics
I ac rms
I dc
V
1+ 8 M
3π nV
I ac rms 2 π2
I ac rms
I ac rms 2 π2
8 VM
3π nV
0
3 + 16 nV
2 3π V M
I dc
84
I ac rms 2 1 +
V
n
I ac rms 2
I ac rms 2 max 1,
VM
nV
2I dc 1 + nV
VM
Chapter 18: PWM Rectifiers
Table of rectifier current stresses
continued
CCM SEPIC, nonisolated
Transistor
V
1+ 8 M
3π V
I ac rms
I ac rms
L1
C1
8 VM
3π V
I ac rms
L2
I dc
I ac rms 2 1 +
I ac rms 2 π2
I ac rms 2
0
VM 3
V 2
I ac rms V M
2 V
3 + 16 V
2 3π V M
I dc
I ac rms
Diode
I ac rms 2 π2
I ac rms max 1,
I ac rms
VM
V
VM
V
VM
2
V
2I dc 1 + V
VM
CCM SEPIC, with n:1 isolation transformer
transistor
I ac rms
V
1+ 8 M
3π nV
I ac rms
L1
C1,
xfmr primary
Diode,
xfmr secondary
I ac rms
I dc
8 VM
3π nV
3 + 16 nV
2 3π V M
I ac rms 2 π2
I ac rms 2 1 +
I ac rms 2 π2
I ac rms 2
0
I dc
VM
nV
I ac rms 2 max 1,
n
2I dc 1 + nV
VM
I ac rms
= 2 V , ac input voltage = V M sin(ω t)
VM
I dc
dc output voltage = V
with, in all cases,
Fundamentals of Power Electronics
85
Chapter 18: PWM Rectifiers
Comparison of rectifier topologies
Boost converter
•
Lowest transistor rms current, highest efficiency
•
Isolated topologies are possible, with higher transistor stress
•
No limiting of inrush current
•
Output voltage must be greater than peak input voltage
Buck-boost, SEPIC, and Cuk converters
•
Higher transistor rms current, lower efficiency
•
Isolated topologies are possible, without increased transistor
stress
•
Inrush current limiting is possible
•
Output voltage can be greater than or less than peak input
voltage
Fundamentals of Power Electronics
86
Chapter 18: PWM Rectifiers
Comparison of rectifier topologies
1kW, 240Vrms example. Output voltage: 380Vdc. Input current: 4.2Arms
Converter
Transistor rms
current
Transistor
voltage
Diode rms
current
Transistor rms
current, 120V
Diode rms
current, 120V
Boost
2A
380 V
3.6 A
6.6 A
5.1 A
Nonisolated
SEPIC
5.5 A
719 V
4.85 A
9.8 A
6.1 A
Isolated
SEPIC
5.5 A
719 V
36.4 A
11.4 A
42.5 A
Isolated SEPIC example has 4:1 turns ratio, with 42V 23.8A dc load
Fundamentals of Power Electronics
87
Chapter 18: PWM Rectifiers
18.6 Modeling losses and efficiency
in CCM high-quality rectifiers
Objective: extend procedure of Chapter 3, to predict the output
voltage, duty cycle variations, and efficiency, of PWM CCM low
harmonic rectifiers.
Approach: Use the models developed in Chapter 3. Integrate over
one ac line cycle to determine steady-state waveforms and average
power.
Boost example
RL
L
D1
ig(t)
i(t)
RL
DRon
D' : 1
VF
i(t)
+
–
ig(t)
+
+
vg(t)
+
–
Q1
C
R
v(t) vg(t)
+
–
R
–
–
Dc-dc boost converter circuit
Fundamentals of Power Electronics
Averaged dc model
88
v(t)
Chapter 18: PWM Rectifiers
Modeling the ac-dc boost rectifier
Boost
rectifier
circuit
ig(t)
+
iac(t)
id(t)
RL
+
L
D1
vg(t)
vac(t)
i(t)
Q1
C
v(t)
R
–
–
controller
RL
d(t) Ron
Averaged
model
vg(t)
+
–
d'(t) : 1
VF
+
–
ig(t)
id(t)
i(t) = I
+
C
(large)
R
v(t) = V
–
Fundamentals of Power Electronics
89
Chapter 18: PWM Rectifiers
Boost rectifier waveforms
vg(t)
ig(t)
300
10
vg(t)
Typical waveforms
8
200
ig(t)
(low frequency components)
6
4
100
ig(t) =
2
0
0
0°
d(t)
vg(t)
Re
30°
60°
90°
120°
150°
180°
1
6
0.8
5
id(t)
4
0.6
i(t) = I
3
0.4
2
0.2
1
0
0
0°
30°
60°
90°
Fundamentals of Power Electronics
120°
150°
180°
90
0°
30°
60°
90°
120°
150°
180°
ωt
Chapter 18: PWM Rectifiers
Example: boost rectifier
with MOSFET on-resistance
ig(t)
+
id(t)
d(t) Ron
vg(t)
i(t) = I
d'(t) : 1
+
–
C
(large)
R
v(t) = V
–
Averaged model
Inductor dynamics are neglected, a good approximation when the ac
line variations are slow compared to the converter natural frequencies
Fundamentals of Power Electronics
91
Chapter 18: PWM Rectifiers
18.6.1 Expression for controller duty cycle d(t)
ig(t)
Solve input side of
model:
d(t) Ron
i g(t)d(t)Ron = vg(t) – d'(t)v
with
ig(t) =
vg(t)
Re
vg(t)
i(t) = I
d'(t) : 1
+
–
+
id(t)
C
(large)
R
v(t) = V
–
vg(t) = VM sin ωt
eliminate ig(t):
solve for d(t):
v – vg(t)
d(t) =
R
v – vg(t) on
Re
vg(t)
d(t)Ron = vg(t) – d'(t)v
Re
Again, these expressions neglect converter dynamics, and assume
that the converter always operates in CCM.
Fundamentals of Power Electronics
92
Chapter 18: PWM Rectifiers
18.6.2 Expression for the dc load current
Solve output side of
model, using charge
balance on capacitor C:
I = id T
ac
ig(t)
d(t) Ron
vg(t)
+
–
i(t) = I
d'(t) : 1
+
id(t)
C
(large)
R
vg(t)
i d (t) = d'(t)i g(t) = d'(t)
Re
v(t) = V
–
Butd’(t) is:
hence id(t) can be expressed as
R
vg(t) 1 – on
Re
d'(t) =
R
v – vg(t) on
Re
Ron
Re
v 2g(t)
i d (t) =
Re
Ron
v – vg(t)
Re
1–
Next, average id(t) over an ac line period, to find the dc load current I.
Fundamentals of Power Electronics
93
Chapter 18: PWM Rectifiers
Dc load current I
Now substitute vg (t) = VM sin ωt, and integrate to find ⟨id(t)⟩Tac:
T ac/2
V 2M
Re
I = id T = 2
ac
Tac
1–
v–
0
Ron
sin 2 ωt
Re
VM Ron
sin ωt
Re
dt
This can be written in the normalized form
T ac/2
2
M
R
V
I= 2
1 – on
Tac VRe
Re
with
a=
Fundamentals of Power Electronics
VM
V
sin 2 ωt
1 – a sin ωt
0
dt
Ron
Re
94
Chapter 18: PWM Rectifiers
Integration
By waveform symmetry, we need only integrate from 0 to Tac/4. Also,
make the substitution θ = ωt:
π/2
2
M
V
R 2
I=
1 – on π
VRe
Re
0
sin 2 θ
1 – a sin θ
dθ
This integral is obtained not only in the boost rectifier, but also in the
buck-boost and other rectifier topologies. The solution is
π/2
4
π
0
sin 2 θ
dθ = F(a) = 22
aπ
1 – a sin θ
1 – a2
• a is typically much smaller than
unity
• Result is in closed form
• a is a measure of the loss
resistance relative to Re
Fundamentals of Power Electronics
– 2a – π +
4 sin – 1 a + 2 cos – 1 a
95
Chapter 18: PWM Rectifiers
The integral F(a)
π/2
4
π
0
sin 2 θ
dθ = F(a) = 22
aπ
1 – a sin θ
Approximation via
polynomial:
Fundamentals of Power Electronics
1 – a2
1.15
1.1
F(a) ≈ 1 + 0.862a + 0.78a 2
For | a | ≤ 0.15, this
approximate expression is
within 0.1% of the exact
value. If the a2 term is
omitted, then the accuracy
drops to ±2% for | a | ≤ 0.15.
The accuracy of F(a)
coincides with the accuracy
of the rectifier efficiency η.
– 2a – π +
4 sin – 1 a + 2 cos – 1 a
1.05
F(a)
1
0.95
0.9
0.85
–0.15
–0.10
–0.05
0.00
0.05
0.10
0.15
a
96
Chapter 18: PWM Rectifiers
18.6.3 Solution for converter efficiency η
Converter average input power is
V 2M
Pin = pin(t) T =
ac
2Re
Average load power is
Pout = VI = V
V 2M
R F(a)
1 – on
VRe
Re
2
VM
a=
V
with
Ron
Re
So the efficiency is
η=
Pout
R
= 1 – on F(a)
Pin
Re
Polynomial approximation:
R
η ≈ 1 – on
Re
Fundamentals of Power Electronics
V R
V R
1 + 0.862 M on + 0.78 M on
V Re
V Re
97
2
Chapter 18: PWM Rectifiers
Boost rectifier efficiency
η
1
η=
.05
R on /R e = 0
0.95
R on/R e
0.9
= 0.1
• To obtain high
efficiency, choose V
slightly larger than VM
15
0.
/R e =
R on
0.85
=
/R e
0.2
• Efficiencies in the range
90% to 95% can then be
obtained, even with Ron
as high as 0.2Re
R on
0.8
0.75
0.0
0.2
0.4
0.6
0.8
VM /V
Fundamentals of Power Electronics
Pout
R
= 1 – on F(a)
Pin
Re
98
1.0
• Losses other than
MOSFET on-resistance
are not included here
Chapter 18: PWM Rectifiers
18.6.4 Design example
Let us design for a given efficiency. Consider the following
specifications:
Output voltage
390 V
Output power
500 W
rms input voltage
120 V
Efficiency
95%
Assume that losses other than the MOSFET conduction loss are
negligible.
Average input power is
P
Pin = ηout = 500 W = 526 W
0.95
Then the emulated resistance is
V 2g, rms (120 V) 2
Re =
=
= 27.4 Ω
Pin
526 W
Fundamentals of Power Electronics
99
Chapter 18: PWM Rectifiers
Design example
Also,
η
VM 120 2 V
=
= 0.435
V
390 V
95% efficiency with
VM/V = 0.435 occurs
with Ron/Re ≈ 0.075.
1
.05
R on /R e = 0
0.95
R on/R e
0.9
So we require a
MOSFET with on
resistance of
= 0.1
Ron ≤ (0.075) Re
15
0.
/R e =
R on
0.85
=
/R e
= (0.075) (27.4 Ω) = 2 Ω
0.2
R on
0.8
0.75
0.0
0.2
0.4
0.6
0.8
1.0
VM /V
Fundamentals of Power Electronics
100
Chapter 18: PWM Rectifiers