Fundamentals of Power Electronics
Second edition
Robert W. Erickson
Dragan Maksimovic
University of Colorado, Boulder
Fundamentals of Power Electronics
1
Chapter 1: Introduction
Chapter 1: Introduction
1.1.
Introduction to power processing
1.2.
Some applications of power electronics
1.3.
Elements of power electronics
Summary of the course
Fundamentals of Power Electronics
2
Chapter 1: Introduction
1.1 Introduction to Power Processing
Power
input
Switching
converter
Power
output
Control
input
Dc-dc conversion:
Ac-dc rectification:
Dc-ac inversion:
Change and control voltage magnitude
Possibly control dc voltage, ac current
Produce sinusoid of controllable
magnitude and frequency
Ac-ac cycloconversion: Change and control voltage magnitude
and frequency
Fundamentals of Power Electronics
3
Chapter 1: Introduction
Control is invariably required
Power
input
Switching
converter
Power
output
Control
input
feedforward
feedback
Controller
reference
4
Fundamentals of Power Electronics
Chapter 1: Introduction
High efficiency is essential
1
η=
Pout
Pin
η
0.8
1 –1
Ploss = Pin – Pout = Pout η
0.6
High efficiency leads to low
power loss within converter
Small size and reliable operation
is then feasible
Efficiency is a good measure of
converter performance
0.4
0.2
0
0.5
1
1.5
Ploss / Pout
Fundamentals of Power Electronics
5
Chapter 1: Introduction
A high-efficiency converter
Pin
Converter
Pout
A goal of current converter technology is to construct converters of small
size and weight, which process substantial power at high efficiency
Fundamentals of Power Electronics
6
Chapter 1: Introduction
+
–
Devices available to the circuit designer
DT
Resistors
Capacitors
Fundamentals of Power Electronics
Magnetics
T
s
s
LinearSwitched-mode
mode
Semiconductor devices
7
Chapter 1: Introduction
+
–
Devices available to the circuit designer
DT
Resistors
Capacitors
Magnetics
T
s
s
LinearSwitched-mode
mode
Semiconductor devices
Signal processing: avoid magnetics
Fundamentals of Power Electronics
8
Chapter 1: Introduction
+
–
Devices available to the circuit designer
DT
Resistors
Capacitors
Magnetics
T
s
s
LinearSwitched-mode
mode
Semiconductor devices
Power processing: avoid lossy elements
Fundamentals of Power Electronics
9
Chapter 1: Introduction
Power loss in an ideal switch
Switch closed:
v(t) = 0
Switch open:
i(t) = 0
+
i(t)
In either event: p(t) = v(t) i(t) = 0
Ideal switch consumes zero power
v(t)
–
10
Fundamentals of Power Electronics
Chapter 1: Introduction
A simple dc-dc converter example
I
10A
+
Vg
100V
Dc-dc
converter
+
–
R
5Ω
V
50V
–
Input source: 100V
Output load: 50V, 10A, 500W
How can this converter be realized?
11
Fundamentals of Power Electronics
Chapter 1: Introduction
Dissipative realization
Resistive voltage divider
I
10A
+
+
Vg
100V
+
–
50V –
Ploss = 500W
R
5Ω
V
50V
–
Pin = 1000W
Fundamentals of Power Electronics
Pout = 500W
12
Chapter 1: Introduction
Dissipative realization
Series pass regulator: transistor operates in
active region
+
I
10A
50V –
+
Vg
100V
linear amplifier
and base driver
+
–
–
+
Vref
R
5Ω
V
50V
Ploss ≈ 500W
–
Pin ≈ 1000W
Pout = 500W
13
Fundamentals of Power Electronics
Chapter 1: Introduction
Use of a SPDT switch
I
10 A
1
+
Vg
100 V
+
2
+
–
vs(t)
R
–
vs(t)
v(t)
50 V
–
Vg
Vs = DVg
0
DTs
(1 – D) Ts
t
1
2
1
switch
position:
14
Fundamentals of Power Electronics
Chapter 1: Introduction
The switch changes the dc voltage level
vs(t)
Vg
Vs = DVg
0
(1 – D) Ts
DTs
switch
position:
1
t
2
1
D = switch duty cycle
0≤D≤1
Ts = switching period
fs = switching frequency
= 1 / Ts
DC component of vs(t) = average value:
Vs = 1
Ts
Ts
vs(t) dt = DVg
0
Fundamentals of Power Electronics
15
Chapter 1: Introduction
Addition of low pass filter
Addition of (ideally lossless) L-C low-pass filter, for
removal of switching harmonics:
i(t)
1
+
Vg
2
+
–
100 V
+
L
vs(t)
R
C
v(t)
–
Pin ≈ 500 W
–
Pout = 500 W
Ploss small
•
Choose filter cutoff frequency f 0 much smaller than switching
frequency fs
•
This circuit is known as the “buck converter”
16
Fundamentals of Power Electronics
Chapter 1: Introduction
Addition of control system
for regulation of output voltage
Power
input
Switching converter
Load
+
vg
+
–
i
v
H(s)
–
Transistor
gate driver
Error
signal
ve
Pulse-width vc G (s)
c
modulator
Compensator
δ
–+
δ(t)
dTs Ts
Sensor
gain
Hv
Reference
vref
input
t
17
Fundamentals of Power Electronics
Chapter 1: Introduction
The boost converter
2
+
L
1
Vg
+
–
C
R
V
–
5Vg
4Vg
V
3Vg
2Vg
Vg
0
0
0.2
0.4
0.6
0.8
1
D
Fundamentals of Power Electronics
18
Chapter 1: Introduction
A single-phase inverter
vs(t)
1
+
+
–
Vg
2
–
+
v(t)
–
2
1
load
“H-bridge”
vs(t)
Modulate switch
duty cycles to
obtain sinusoidal
low-frequency
component
t
19
Fundamentals of Power Electronics
Chapter 1: Introduction
1.2 Several applications of power electronics
Power levels encountered in high-efficiency converters
• less than 1 W in battery-operated portable equipment
• tens, hundreds, or thousands of watts in power supplies for
computers or office equipment
• kW to MW in variable-speed motor drives
• 1000 MW in rectifiers and inverters for utility dc transmission
lines
20
Fundamentals of Power Electronics
Chapter 1: Introduction
A laptop computer power supply system
Inverter
iac(t)
vac(t)
Display
backlighting
Charger
Buck
converter
PWM
Rectifier
ac line input
85–265 Vrms
Fundamentals of Power Electronics
Boost
converter
Lithium
battery
21
Microprocessor
Power
management
Disk
drive
Chapter 1: Introduction
Power system of an earth-orbiting spacecraft
Dissipative
shunt regulator
+
Solar
array
vbus
–
Battery
charge/discharge
controllers
Dc-dc
converter
Dc-dc
converter
Payload
Payload
Batteries
22
Fundamentals of Power Electronics
Chapter 1: Introduction
An electric vehicle power and drive system
ac machine
Inverter
ac machine
Inverter
control bus
battery
µP
system
controller
+
3øac line
Battery
charger
DC-DC
converter
vb
50/60 Hz
Vehicle
Low-voltage electronics
dc bus
–
Inverter
Inverter
ac machine
ac machine
Variable-frequency
Variable-voltage ac
Fundamentals of Power Electronics
23
Chapter 1: Introduction
1.3 Elements of power electronics
Power electronics incorporates concepts from the fields of
analog circuits
electronic devices
control systems
power systems
magnetics
electric machines
numerical simulation
Fundamentals of Power Electronics
24
Chapter 1: Introduction
Part I. Converters in equilibrium
Inductor waveforms
vL(t)
Averaged equivalent circuit
RL
DTs
1
iL(t)
t
2
+
–
D' : 1
+
R
Predicted efficiency
100%
–V
L
0
V
I
–
∆iL
Vg – V
L
Vg
1
iL(DTs)
I
iL(0)
D' RD
D'Ts
–V
switch
position:
D' VD
D Ron
+
–
Vg – V
0.002
90%
0.01
Ts
DTs
80%
t
0.02
70%
0.05
60%
η
50%
RL/R = 0.1
40%
Discontinuous conduction mode
30%
20%
Transformer isolation
10%
0%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D
25
Fundamentals of Power Electronics
Chapter 1: Introduction
Switch realization: semiconductor devices
iA(t)
The IGBT
collector
Switching loss
transistor
waveforms
Qr
Vg
gate
iL
vA(t)
0
0
emitter
t
Emitter
diode
waveforms
iL
iB(t)
vB(t)
Gate
0
0
t
n
p
n
n
n-
p
area
–Qr
n
–Vg
minority carrier
injection
tr
p
pA(t)
= vA iA
area
~QrVg
Collector
area
~iLVgtr
t0
Fundamentals of Power Electronics
26
t1 t2
t
Chapter 1: Introduction
Part I. Converters in equilibrium
2. Principles of steady state converter analysis
3. Steady-state equivalent circuit modeling, losses, and efficiency
4. Switch realization
5. The discontinuous conduction mode
6. Converter circuits
Fundamentals of Power Electronics
27
Chapter 1: Introduction
Part II. Converter dynamics and control
Closed-loop converter system
Power
input
Averaging the waveforms
Switching converter
Load
gate
drive
+
vg(t) +
–
v(t)
R
feedback
connection
–
δ(t)
compensator
pulse-width vc G (s)
c
modulator
δ(t)
v
averaged waveform <v(t)>T
with ripple neglected s
voltage
reference vref
vc(t)
dTs Ts
t
actual waveform v(t)
including ripple
–+
transistor
gate driver
t
t
t
Controller
Vg – V d(t)
L
D' : 1
+
–
1:D
+
Small-signal
averaged
equivalent circuit
vg(t)
+
–
I d(t)
I d(t)
C
v(t)
R
–
28
Fundamentals of Power Electronics
Chapter 1: Introduction
Part II. Converter dynamics and control
7.
Ac modeling
8.
Converter transfer functions
9.
Controller design
10.
Input filter design
11.
Ac and dc equivalent circuit modeling of the discontinuous
conduction mode
12.
Current-programmed control
29
Fundamentals of Power Electronics
Chapter 1: Introduction
Part III. Magnetics
n1 : n2
transformer
design
iM(t)
i1(t)
the
proximity
effect
i2(t)
LM
R1
R2
3i
layer
3
–2i
2Φ
layer
2
2i
–i
ik(t)
Φ
layer
1
Rk
d
current
density
J
: nk
i
4226
Pot core size
3622
0.1
2616
2616
2213
2213
1811
0.08
0.06
1811
0.04
Bmax (T)
transformer
size vs.
switching
frequency
0.02
0
25kHz
50kHz
100kHz
200kHz
250kHz
400kHz
500kHz
1000kHz
Switching frequency
Fundamentals of Power Electronics
30
Chapter 1: Introduction
Part III. Magnetics
13.
Basic magnetics theory
14.
Inductor design
15.
Transformer design
31
Fundamentals of Power Electronics
Chapter 1: Introduction
Part IV. Modern rectifiers,
and power system harmonics
Pollution of power system by
rectifier current harmonics
A low-harmonic rectifier system
boost converter
i(t)
ig(t)
+
iac(t)
vac(t)
L
vg(t)
+
D1
Q1
vg(t)
multiplier
v(t)
R
–
–
vcontrol(t)
C
ig(t)
Rs
X
PWM
va(t)
v (t)
+– err
Gc(s)
vref(t)
= kx vg(t) vcontrol(t)
compensator
controller
Harmonic amplitude,
percent of fundamental
100%
100%
91%
80%
THD = 136%
Distortion factor = 59%
73%
60%
52%
40%
32%
19% 15% 15%
13% 9%
20%
0%
1
3
5
7
Ideal rectifier (LFR)
iac(t)
+
9
11
13
15
17
19
Model of
the ideal
rectifier
vac(t)
p(t) = vac2 / Re
+
v(t)
Re(vcontrol)
–
–
ac
input
Harmonic number
i(t)
dc
output
vcontrol
Fundamentals of Power Electronics
32
Chapter 1: Introduction
Part IV. Modern rectifiers,
and power system harmonics
16.
Power and harmonics in nonsinusoidal systems
17.
Line-commutated rectifiers
18.
Pulse-width modulated rectifiers
Fundamentals of Power Electronics
33
Chapter 1: Introduction
Part V. Resonant converters
The series resonant converter
Q1
L
Q3
D1
C
1:n
D3
+
Vg
+
–
R
Q2
V
–
Q4
D2
Zero voltage
switching
D4
1
vds1(t)
Q = 0.2
Vg
0.9
Q = 0.2
0.8
0.35
M = V / Vg
0.7
0.2
0.1
0
1
0.5
0.4
0.75
1
1.5
2
3.5
5
10
Q = 20
0
Q1
Q4
conducting
devices:
0.75
0.5
0.3
Dc
characteristics
0.5
0.35
0.6
t
X D2
D3
turn off
Q 1, Q 4
1.5
commutation
interval
2
3.5
5
10
Q = 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F = fs / f0
34
Fundamentals of Power Electronics
Chapter 1: Introduction
Part V. Resonant converters
19.
20.
Resonant conversion
Soft switching
35
Fundamentals of Power Electronics
Chapter 1: Introduction
Appendices
RMS values of commonly-observed converter waveforms
Simulation of converters
Middlebrook’s extra element theorem
L
1
2
Magnetics design tables
50 µH
2 CCM-DCM1
+
–
5
28 V
20 dB
|| Gvg ||
1
Vg
Open loop, d(t) = constant
–20 dB
–60 dB
–80 dB
5 Hz
8
vx
5 kHz
85 kΩ
R3
C3
120 kΩ
7
VM = 4 V
500 Hz
6
vz
–vy
LM324
1.1 nF
5
vref
+
–
value = {LIMIT(0.25 vx, 0.1, 0.9)}
f
C2
2.7 nF
+12 V
Epwm
50 kHz
v
–
R2
L = 50 µΗ
fs = 100 kΗz
R = 25 Ω
Closed loop
50 Hz
R
11 kΩ
4
Xswitch
R=3Ω
–40 dB
+
R1
3
4
0 dB
iLOAD
3
C
500 µF
+
–
A.
B.
C.
D.
R4
47 kΩ
5V
.nodeset v(3)=15 v(5)=5 v(6)=4.144 v(8)=0.536
Fundamentals of Power Electronics
36
Chapter 1: Introduction