Chapter 01

Fundamentals of Power Electronics Chapter 1: Introduction1
Fundamentals of Power Electronics
Second edition
Robert W. Erickson
Dragan Maksimovic
University of Colorado, Boulder

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

1.1 Introduction to Power Processing
Dc-dc conversion: Change and control voltage magnitude
Ac-dc rectification: Possibly control dc voltage, ac current
Dc-ac inversion: Produce sinusoid of controllable
magnitude and frequency
Ac-ac cycloconversion: Change and control voltage magnitude
and frequency
Switching
converter
Power
input
Power
output
Control
input

Control is invariably required
Switching
converter
Power
input
Power
output
Control
input
Controller
reference
feedbackfeedforward

High efficiency is essential
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 0.5 1 1.5
0.2
0.4
0.6
0.8
1
Ploss / Pout
ηη =
Pout
Pin
Ploss = Pin – Pout = Pout
1
η – 1

A high-efficiency converter
A goal of current converter technology is to construct converters of small
size and weight, which process substantial power at high efficiency
Converter
Pin Pout

Devices available to the circuit designer
DTs Ts
Resistors Capacitors Magnetics Semiconductor devices
Linear-
mode
+
–
Switched-mode

Signal processing: avoid magnetics
DTs Ts
Linear-
mode
+
–
Switched-mode

Power processing: avoid lossy elements
DTs Ts
Linear-
mode
+
–
Switched-mode

Power loss in an ideal switch
Switch closed: v(t) = 0
Switch open: i(t) = 0
In either event: p(t) = v(t) i(t) = 0
Ideal switch consumes zero power
+
v(t)
–
i(t)

A simple dc-dc converter example
Input source: 100V
Output load: 50V, 10A, 500W
How can this converter be realized?
+
–
R
5Ω
+
V
50V
–
Vg
100V
I
10A
Dc-dc
converter

Dissipative realization
Resistive voltage divider
+
–
R
5Ω
+
V
50V
–
Vg
100V
I
10A
+ 50V –
Ploss = 500W
Pout = 500WPin = 1000W

Dissipative realization
Series pass regulator: transistor operates in
active region
+
–
R
5Ω
+
V
50V
–
Vg
100V
I
10A+ 50V –
Ploss ≈ 500W
Pout = 500WPin ≈ 1000W
+–linear amplifier
and base driver
Vref

Use of a SPDT switch
+
–
R
+
v(t)
50 V
–
1
2
+
vs(t)
–
Vg
100 V
I
10 A
vs(t)
Vg
DTs
(1 – D) Ts
0
t
switch
position: 1 2 1
Vs = DVg

The switch changes the dc voltage level
D = switch duty cycle
0 ≤ D ≤ 1
Ts = switching period
fs = switching frequency
= 1 / Ts
Vs = 1
Ts
vs(t) dt
0
Ts
= DVg
DC component of vs(t) = average value:
vs(t)
Vg
DTs
(1 – D) Ts
0
t
switch
position: 1 2 1
Vs = DVg

Addition of low pass filter
Addition of (ideally lossless) L-C low-pass filter, for
removal of switching harmonics:
• Choose filter cutoff frequency f0 much smaller than switching
frequency fs
• This circuit is known as the “buck converter”
+
–
R
+
v(t)
–
1
2
+
vs(t)
–
Vg
100 V
i(t)
L
C
Ploss small
Pout = 500 WPin ≈ 500 W

Addition of control system
for regulation of output voltage
δ(t)
TsdTs t
+
–
+
v
–
vg
Switching converterPower
input
Load
–+
Compensator
vref
Reference
input
HvPulse-width
modulator
vc
Transistor
gate driver
δ Gc(s)
H(s)
ve
Error
signal
Sensor
gain
i

The boost converter
+
–
L
C R
+
V
–
1
2
Vg
D
0 0.2 0.4 0.6 0.8 1
V
5Vg
4Vg
3Vg
2Vg
Vg
0

A single-phase inverter
1
2
+
–
load
+ v(t) –
2
1
Vg
vs(t)
+ –
t
vs(t) “H-bridge”
Modulate switch
duty cycles to
obtain sinusoidal
low-frequency
component

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

A laptop computer power supply system
vac(t)
iac(t) Charger
PWM
Rectifier
Lithium
battery
ac line input
85–265 Vrms
Inverter
Buck
converter
Boost
converter
Display
backlighting
Microprocessor
Power
management
Disk
drive

Power system of an earth-orbiting spacecraft
Solar
array
+
vbus
–
Batteries
Battery
charge/discharge
controllers
Dc-dc
converter
Payload
Dc-dc
converter
Payload
Dissipative
shunt regulator

An electric vehicle power and drive system
3øac line
50/60 Hz
Battery
charger
battery
+
vb
–
Variable-frequency
Variable-voltage ac
Inverter
ac machine
Inverter
Inverter
ac machine
DC-DC
converter
µP
system
controller
Vehicle
electronicsLow-voltage
dc bus
control bus
ac machine ac machine
Inverter

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

Part I. Converters in equilibrium
iL(t)
t0 DTs
Ts
I
iL(0) Vg – V
L
iL(DTs)
∆iL
– V
L
vL(t) Vg – V
t
– V
D'TsDTs
switch
position: 1 2 1
RL
+
–Vg
D' RD
+
–
D' VDD Ron
R
+
V
–
I
D' : 1
Inductor waveforms Averaged equivalent circuit
D
η RL/R = 0.1
0.02
0.01
0.05
0.002
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Predicted efficiency
Discontinuous conduction mode
Transformer isolation

Switch realization: semiconductor devices
Collector
p
n-
n n
p p
Emitter
Gate
nn
minority carrier
injection
collector
emitter
gate
The IGBT
t
iL
–Vg
0
iB(t)
vB(t)
area
–Qr
0
tr
t
iL
Vg
0
iA(t)
vA(t)
Qr
0
t
area
~QrVg
area
~iLVgtr
t0 t1 t2
transistor
waveforms
diode
waveforms
pA(t)
= vA iA
Switching loss

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

Part II. Converter dynamics and control
+
–
+
v(t)
–
vg(t)
Switching converterPower
input
Load
–+
R
compensator
Gc(s)
vref
voltage
reference
v
feedback
connection
pulse-width
modulator
vc
transistor
gate driver
δ(t)
δ(t)
TsdTs t t
vc(t)
Controller
t
t
gate
drive
actual waveform v(t)
including ripple
averaged waveform <v(t)>Ts
with ripple neglected
+
– I d(t)vg(t)
+
–
L
Vg – V d(t)
+
v(t)
–
RCI d(t)
1 : D D' : 1
Closed-loop converter system Averaging the waveforms
Small-signal
averaged
equivalent circuit

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

Part III. Magnetics
Φ
i
–i
3i
–2i
2i
2Φ
current
density
J
dlayer
1
layer
2
layer
3
0
0.02
0.04
0.06
0.08
0.1
Switching frequency
Bmax(T)
25kHz 50kHz 100kHz 200kHz 250kHz 400kHz 500kHz 1000kHz
Potcoresize
4226
3622
2616
2213
1811 1811
2213
2616
n1 : n2
: nk
R1 R2
Rk
i1(t) i2(t)
ik(t)
LM
iM(t)transformer
design
transformer
size vs.
switching
frequency
the
proximity
effect

Part III. Magnetics
13. Basic magnetics theory
14. Inductor design
15. Transformer design

Part IV. Modern rectifiers,
and power system harmonics
100%
91%
73%
52%
32%
19% 15% 15% 13% 9%
0%
20%
40%
60%
80%
100%
1 3 5 7 9 11 13 15 17 19
Harmonic number
Harmonicamplitude,
percentoffundamental
THD = 136%
Distortion factor = 59%
Pollution of power system by
rectifier current harmonics
Re(vcontrol)
+
–
vac(t)
iac(t)
vcontrol
v(t)
i(t)
+
–
p(t) = vac
2
/ Re
Ideal rectifier (LFR)
ac
input
dc
output
boost converter
controller
Rvac(t)
iac(t) +
vg(t)
–
ig(t)
ig(t)vg(t)
+
v(t)
–
i(t)
Q1
L
C
D1
vcontrol(t)
multiplier X
+–
vref(t)
= kx vg(t) vcontrol(t)
Rs
va(t)
Gc(s)
PWM
compensator
verr(t)
A low-harmonic rectifier system
Model of
the ideal
rectifier

Part IV. Modern rectifiers,
and power system harmonics
16. Power and harmonics in nonsinusoidal systems
17. Line-commutated rectifiers
18. Pulse-width modulated rectifiers

Part V. Resonant converters
L
+
–
Vg
CQ1
Q2
Q3
Q4
D1
D2
D3
D4
1 : n
R
+
V
–
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
M=V/Vg
F = fs / f0
Q = 20
10
5
3.5
2
1.5
1
0.75
0.5
0.35
Q = 0.2
Q = 20
10
5
3.5
2
1.5
1
0.75
0.5
0.35
Q = 0.2
The series resonant converter
Dc
characteristics
conducting
devices:
t
Vg
vds1(t)
Q1
Q4
D2
D3
turn off
Q1
, Q4
commutation
interval
X
Zero voltage
switching

Part V. Resonant converters
19. Resonant conversion
20. Soft switching

Appendices
A. RMS values of commonly-observed converter waveforms
B. Simulation of converters
C. Middlebrook’s extra element theorem
D. Magnetics design tables
2
1
3
4
5
CCM-DCM1
+
–
+
–
+
–
C2
50 µH
11 kΩ
500 µF
Vg
28 V
L
C
R
vref
5 V
+12 V
LM324
R1
R2
R3 C3
R4
85 kΩ
1.1 nF2.7 nF
47 kΩ
120 kΩ
vz
–vyvx
Epwm
VM = 4 V
value = {LIMIT(0.25 vx, 0.1, 0.9)}
+
v
–
iLOAD1 2 3
4
5678
.nodeset v(3)=15 v(5)=5 v(6)=4.144 v(8)=0.536
Xswitch
L = 50 µΗ
fs = 100 kΗz
f
|| Gvg ||
–60 dB
–80 dB
0 dB
20 dB
5 Hz 50 Hz 5 kHz 50 kHz500 Hz
R = 3 Ω
R = 25 Ω
–40 dB
–20 dB
Open loop, d(t) = constant
Closed loop

Chapter 01

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