1. Losses in a transformer
A transformer is connected to a voltage source and a load as illustrated in Fig. 2. The primary
winding is excited by the voltage v1(t) whose waveform is illustrated in Fig. 3. The switching
frequency is fs = 1/Ts = 200 kHz, and the duty cycle is D = 1/3. The load cu
ent is a 200 kHz
sinusoid having amplitude 5 A rms.
v1(t)
Figure 3 Primary voltage waveform v1(t) for
Problem 1.
The transformer consists of a fe
ite PQ 26/25 core, with flat copper (ri
on) windings. The
primary winding consists of two turns of flat copper of rectangular cross-section, with a
copper width of 1.25 cm and a copper thickness of 0.07 cm. The secondary winding consists
of eight turns of flat copper also of rectangular cross-section, with a copper width of 1.25 cm
and a copper thickness of 0.017 cm. Each turn comprises one layer in the winding. You may
assume that the transformer operates at a temperature of 100◦C. The PQ 26/25 core
dimensions are:
• Core cross-sectional area Ac = 1.18cm2
• Core window area WA = 0.503cm2
• Core mean-length per turn MLT = 5.62cm
• Core magnetic path length m = 5.55cm
The core loss data for this core operating at 200 kHz is plotted in Fig. 4.
The primary and secondary windings are interleaved as follows:
• Three layers of secondary
• One layer of primary
• Two layers of secondary
• One layer of primary
Transformer circuit of Problem 1
+
–
n 1 : n 2
v 1 ( t ) load
t
+24 V
–12 V
T s DT s
Page 2
• Three layers of secondary
(a) Find the peak ac flux density ∆B and the
core loss Pfe for this transformer.
Figure 4 Core loss vs. peak ac flux density for Problem 1.
∆B, Tesla
(b) Find the dc resistance Rdc and ϕ for each layer.
(c) Sketch the MMF diagram for this transformer and find the effective m for each layer.
(d) Compute the total power loss in each layer, and the total transformer loss, in Watts.
XXXXXXXXXX
0.01
0.1
1
Page 3
2. Analysis of a converter feedback system
The buck-boost converter of Fig. 5 operates in the continuous conduction mode, with the
element values shown. The nominal input voltage is Vg = 48V, and it is desired to regulate
the output voltage at V = −15V.
The compensator gain Gc(s) is:
The compensator parameters are:
• Gcm = 2.5
• fL = ωL/2π = 500Hz
• fcz = 3kHz
• fp1 = 100kHz
Computer simulations will not be accepted for this problem. You must construct the Bode
diagram and its asymptotes manually, using the methods taught in this class.
Figure 5 Closed-loop buck-boost converter system of Problem 2.
+
–
R C L
+
v
–
v g
f s = 200 kHz
220 µ F 5 Ω 50 µ H
Compensator
v ref
Hv Pulse-width
modulator
v c
Transistor
gate driver
δ G c ( s )
H ( s )
v e
V M = 3 V
5 V
Page 4
(a) The feedback gain H(s) is realized with a simple resistive voltage divider circuit having
no dynamics. Specify the value of H required to cause the regulated output voltage V to
e the value specified above.
(b) Carefully construct a Bode plot of the magnitude and phase of the loop gain T(s) of this
system, using the attached semilog axes. Carefully label all asymptote slopes and
eak
frequencies, for both the magnitude asymptotes and the phase asymptotes.
(c) Use your Bode plot to determine the crossover frequency fc.
(d) Use your Bode plot to determine the phase margin ϕm.
Page 5
3. Boost converter with diode forward voltage drop
In the boost converter illustrated in Fig.1, the diode has a forward voltage drop that can be
modeled as a constant voltage VD in series with a resistive element RD. All other elements
should be modeled as ideal. The converter operates in the continuous conduction mode. In
this problem, you will show how this diode drop affects the ac model of the converter.
Figure 1 Boost converter of Problem 1.
(a) Derive the small-signal ac equations that describe this converter.
(b) Derive the small-signal ac model of this converter.
(c) Manipulate your model into canonical form. Give expressions for the source coefficients
e(s) and j(s) in your model.
+
–
+
v ( t )
–
v g ( t )
Controller
L
C R
d ( t )
Q 1
D 1
Fundamentals
of Powe
Electronics
Third Edition
Robert W. Erickson
Dragan Maksimović
Fundamentals of Power Electronics
Robert W. Erickson • Dragan Maksimović
Fundamentals of Powe
Electronics
Third Edition
Robert W. Erickson
Department of Electrical, Computer,
and Energy Engineering
University of Colorado Boulde
Boulder, CO, USA
Dragan Maksimović
Department of Electrical, Computer,
and Energy Engineering
University of Colorado Boulde
Boulder, CO, USA
ISBN XXXXXXXXXXISBN XXXXXXXXXXeBook)
https:
doi.org/10.1007/ XXXXXXXXXX
2nd edition:© Kluwer Academic Publishers 2001
© Springer Nature Switzerland AG 2020
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is
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1st edition:© Springer Science+Business Media Dordrecht 1997
https:
doi.org/10.1007/ XXXXXXXXXX
Dedicated to
Linda, William, and Richard
Lidija, Filip, Nikola, and Stevan
Preface
The objective of the First and Second Editions was to serve as a textbook for introductory powe
electronics courses where the fundamentals of power electronics are defined, rigorously pre-
sented, and treated in sufficient depth so that students acquire the knowledge and skills needed
to design practical power electronic systems. An additional goal was to contribute as a reference
ook for engineers who practice power electronics design, and for students who want to develop
their knowledge of the area beyond the level of introductory courses. In this Third Edition, the
asic objectives and philosophy of the earlier editions have not been changed.
Since we wrote the Second Edition, the field of power electronics has grown tremen-
dously, including new significant commercial applications such as electric vehicles, wireless
power transfer, and utility microgrids. Technical growth includes the commercialization of wide
andgap power semiconductors, widespread digital control of switching converters, and matu-
ation of converter modeling. Our university power electronics cu
iculum has evolved as well,
in content as well as in organization. This edition is a response to these changes, and represents
a significant revision relative to the previous edition.
As of 2020, at the University of Colorado we offer a sequence of three core graduate courses
in power electronics. The first course, Introduction to Power Electronics, covers basic converte
analysis, converter controllers, and magnetics. In the Third Edition, this material is presented in
Chaps. 1–12, at the level and in the order covered in this class. Our second course, Modeling and
Control of Power Electronics Systems, covers more advanced topics of power converter applica-
tions, control, and design-oriented analysis. This material is covered in detail in Chaps. 13–21
in the Third Edition; this portion of the text represents a major revision of technical material
and coverage. Our third course, Resonant and Soft Switching Phenomena in Power Electronics,
elies primarily on supplementary notes rather than this textbook. Chapters 22 and 23 of the
Third Edition cover a summary of a portion of this third course.
The coverage of power semiconductor devices in Chap. 4 has been bolstered and updated.
The discussion of power diode switching has been significantly expanded, leading into aver-
aged modeling of diode-induced switching loss. New material on wide bandgap devices and on
MOSFET gate drivers has been added. The discussion of switching loss mechanisms has been
updated and reorganized, and the MCT section is removed.
The Third Edition adopts a more mature viewpoint of averaging, based on the trapezoidal
moving average defined in Eq XXXXXXXXXXThe waveforms of the averaged model become true con-
tinuous quantities, with the approximations and logical steps clearly defined. New material in
vii
viii Preface
Chap. 7 includes a section on the averaging operator, and a new treatment of how the small-
ipple approximation works with the trapezoidal moving average. Additionally, the logical flow
of Chap. 7 has been significantly revised to conform to how we now teach this material in ou
on-campus courses, and new material on state-space averaging has been added. This new view-
point of averaging then is followed throughout the remainder of the book. Of most note, this
viewpoint leads to the cu
ent-programmed control model of Tan and Middle
ook. The cu
ent-
programmed control Chap. 18 has been significantly revised and updated accordingly. The high-
frequency effects of sampling are discussed as well, in connection with cu
ent-programmed
control and also with ac modeling of the discontinuous conduction mode.
The previous treatment of stability and phase margin would leave some students with mis-
conceptions; to alleviate this, we have introduced a new section on Nyquist stability. Instructors
may choose whether there is time to cover this material in a power