Midterm Examination
Fall 2003
ECEN 5797
Power Electronics 1
Prof. Robert W. Erickson
Campus Box 425
Department of Electrical and Computer Engineering
University of Colorado, Boulder 80309-0425
(303) 492-7003, FAX: (303) 492-2758
rwe@colorado.edu
Instructions: This is a one week take-home exam. It is an open-book exam. You may refer to any materials that you wish, but you are on your honor not to consult with other people. Show all work: partial
credit will be given. Please write your name on all pages, and return this cover sheet with your exam.
On-campus students: The exam will be handed out on Friday, October 17, 2003, and will be due at the
beginning of class on Friday, October 24.
Off-campus students: This is a one-week take-home exam. It is not necessary to take the exam at the
same time as the on-campus students, but all students at a given location should take the exam at the
same time. You should arrange the starting and due dates with your company educational coordinator. If
you wish, you may request (via email) that Prof. Erickson transmit a pdf file containing the exam to you
via email; you will then be required to fax your completed exam to Prof. Erickson before 5 PM mountain
time, one week later.
This exam covers the material of Chapters 1 to 6 of the textbook. It contains three problems and four
pages.
1.
2.
3.
total
1
TAPPED-INDUCTOR BUCK CONVERTER FOR LOW-VOLTAGE APPLICATION
(35 POINTS)
Increasingly, it is required to supply a very low voltage at high current to a microprocessor. Often the
converter input voltage is relatively high, so that a nonisolated buck converter would operate with a very
small duty cycle in such an application. Poor switch utilization and low efficiency are the result. One
improvement that has been suggested by a number of people is the use of a tapped-inductor buck converter as illustrated in Fig. 1. By proper selection of turns ratio, the duty cycle, switch utilization, and
efficiency can be improved.
This converter is constructed and operated with the following element values and specifications:
Vg = 12 V
V=1V
Load power: P = 20 W
Turns ratio: n2 /n1 = 1/6
The transformer is constructed so that the magnetizing current ripple ∆iM is equal to 20% of the dc component of magnetizing current IM. You may neglect all losses.
(a)
Derive an expression for the conversion ratio V/Vg of this converter. For the operating conditions
listed above, compute the numerical value of the duty cycle D, and compare with the value that
would be required for the standard buck converter (i.e., with n1 = 0) (20 points)
(b)
Sketch the current i2(t) and label salient features (i.e., label the values of the current during each
interval, for the operating conditions listed above) (5 points)
(c)
Compute the numerical value of the peak transistor current. (5 points)
(d)
How large (numerically) is the off-state transistor voltage? (5 points)
{
{
Q1
n1
n2
turns turns
i2
+
Vg
+
–
D1
C
R
V
–
Fig. 1 Tapped-inductor buck converter for low-voltage application..
2
A DISCONTINUOUS CONDUCTION MODE (35 POINTS)
The converter illustrated in Fig. 2 is similar to the SEPIC, except that an additional diode is placed in
series with the input inductor L1. The objective of this problem is to analyze the discontinuous conduction mode associated with large ripple in the inductor current i1(t).
i1
Vg
L1
D1
+
–
+
D2
C1
C2
L2
R
v
Q1
–
Fig. 2 Circuit for problem 2.
i1(t)
0
Conducting
devices:
DTs
D1Ts
Q1, D1
Ts
D2Ts
D1, D2
t
D 3T s
D2
Fig. 3 Inductor current waveform i1(t).
For this problem, you may assume that the switching ripples in the current of inductor L2, the voltage of
capacitor C1, and the voltage of capacitor C2, are negligible. Figure 3 depicts the inductor current waveform i1(t) and the sequence of conducting devices for the discontinuous conduction mode that is the subject of this problem.
(a)
Derive boundary between continuous and discontinuous conduction mode, for the discontinuous conduction mode illustrated in Fig. 3. Express your result in terms of parameters K and
Kcrit(D), in the usual manner, and give expressions for K and Kcrit. (15 points)
(b)
Derive the system of equations that relate the dc components of the important waveforms of the
circuit in the discontinuous conduction mode of Fig. 3, and solve to find the conversion ratio
M(D,K). (20 points)
3
CONVERTER MODELING (30 POINTS)
Later this semester, we will study core loss in magnetic devices. An approximate equivalent circuit sometimes used to model core losses is a resistor Rc placed in parallel with the inductance of the device. The
objective of this problem is to model the effects of core loss on the efficiency of a buck-boost converter.
Figure 4 illustrates a conventional buck-boost converter, with core loss modeled by resistor Rc.
You may assume that the converter operates in continuous conduction mode, and you may neglect all
losses other than the core loss.
+
Vg
+
–
L
Rc
C
R
v
–
Fig. 4 Buck-boost converter, with inductor core loss modeled by resistor Rc.
(a)
Derive a complete steady-state equivalent circuit model for this converter. (15 points)
(b)
Solve your model to derive an analytical expression for the efficiency. Your result should be
expressed in terms of R, Rc , and the duty cycle, but no other quantities. (10 points)
(c)
For the values R = 10 Ω and Rc = 50 Ω, plot the efficiency over the range 0 < D < 1. What is the
efficiency at D = 0.5? (5 points)