BASIC ELECTRIC CIRCUITS
SUMMARY/REVIEW QUESTIONS 1 What is the function of PPUs?
2. What are the sub-blocks of PPUs?
3. What are the roles of the rectifier and the filter-capacitor sub-blocks?
4. Qualitatively, how does a switch-mode amplifier differ from a linear amplifier?
5. Why does operating transistors as switches result in much smaller losses compared to operating them in their linear region?
6. How is a bi-positional switch realized in a converter pole? 7. What is the gain of each converter pole?
8. How does a switch-mode converter pole approach the output of a linear amplifier?
9. What is the meaning of vAN( )t ?
10. How is the pole output voltage made linearly proportional to the input control signal?
11. What is the physical significance of the duty-ratio, for example d t ? A( ) 12. How is pulse-width-modulation (PWM) achieved and what is its function? 13. Instantaneous quantities on the two sides of the converter pole, for example
pole-A, are related by the switching signal q t . What relates the average A( ) quantities on the two sides?
14. What is the equivalent model of a switch-mode pole in terms of its average quantities?
15. How is a switch-mode dc-dc converter which can achieve an output voltage of either polarity and an output current flowing in either direction realized? 16. What is the frequency content of the output voltage waveform in dc-dc
17. In a dc-drive converter, how is it possible to keep the ripple in the output current small, despite the output voltage pulsating between 0 and V , or 0 d
and − , during each switching cycle? Vd
18. What is the frequency content of the input dc current? Where does the pulsating ripple component of the dc-side current flow through?
19. How is bi-directional power flow achieved through a converter pole?
20. What makes the average of the dc-side current in a converter pole related to the average of the output current by its duty-ratio?
21. How are three-phase, sinusoidal ac output voltages synthesized from a dc voltage input?
22. What are the voltage and current ratings and the switching speeds of various power semiconductor devices?
REFERENCES
1. N. Mohan, T. Undeland, and W. P. Robbins, Power Electronics: Converters,
Applications and Design, 2nd edition, 1995, John Wiley & Sons, New York, NY.
2. N. Mohan, Power Electronics: Computer Simulation, Analysis and Education
using PSpice, January 1998, distributed through www.mnpere.com.
3. Proceedings of the NSF-Faculty Workshop, Teaching of First Courses on
Power Electronics, University of Minnesota, June 25-28, 1998.
4. N. Mohan, A 4-1/2 hour video course on Power Electronics (parts 1 and 2,
approximately 2 hours long), distributed through www.mnpere.com.
PROBLEMS
4-1 In a switch-mode converter pole-A, 150Vd = V, ˆVtri =5V, and fs =20kHz. Calculate the values of the control signal vc A, and the pole duty-ratio d A
during which the switch is in its top position, for the following values of the average output voltage: vAN =125V and vAN =50V.
4-2 In a converter pole, the ( )i t waveform is as shown in Fig. 4-4a. Including A
DC-DC Converters (Four-Quadrant Capability)
4-3 A switch-mode dc-dc converter uses a PWM-controller IC which has a triangular waveform signal at 25 kHz with ˆV = 3 V. If the input dc source tri
voltage 150Vd = V , calculate the gain kPWM of this switch-mode amplifier.
4-4 In a switch-mode dc-dc converter, 0.8 ˆ
c
tri
v
V = with a switching frequency
20
s
f = kHz and Vd =150V. Calculate and plot the ripple in the output voltage ( )v t . o
4-5 A switch-mode dc-dc converter is operating at a switching frequency of 20
kHz, and V = 150 V. The average current being drawn by the dc motor is d
8.0 A. In the equivalent circuit of the dc motor, Ea = 100 V, Ra = 0.25 Ω , and La = 4 mH. (a) Plot the output current and calculate the peak-to-peak ripple and (b) plot the dc-side current.
4-6 In Problem 4-5, the motor goes into regenerative braking mode. The average current being supplied by the motor to the converter during braking is 8.0 A. Plot the voltage and current waveforms on both sides of this converter. Calculate the average power flow into the converter.
4-7 In Problem 4-5, calculate i , dA i , and (dB id =Id).
4-8 Repeat Problem 4-5 if the motor is rotating in the reverse direction, with the same current draw and the same induced emf E value of the opposite a
polarity.
4-9 Repeat Problem 4-8 if the motor is braking while it has been rotating in the reverse direction. It supplies the same current and produces the same induced emf E value of the opposite polarity. a
4-10 Repeat problem 4-5 if a bi-polar voltage switching is used in the dc-dc converter. In such a switching scheme, the two bi-positional switches are operated in such a manner that when switch-A is in the top position, switch- B is in its bottom position, and vice versa. The switching signal for pole-A is derived by comparing the control voltage (as in Problem 4-5) with the triangular waveform.
DC-to-Three-Phase AC Inverters
4-11 Plot d t if the output voltage of the converter pole-A is A( )
1 ( ) 0.85 sin( ) 2 2 d d AN V V v t = + ωt , where ω1=2π×60rad s/ .
4-12 In the three-phase dc-ac inverter of Fig. 4-12a, Vd =300V, ˆVtri =1V , ˆ 0.75
c
V = V , and f1=45Hz. Calculate and plotd t d t d t , A( ), B( ), C( ) ( ), ( ), ( )
AN BN CN
v t v t v t , and vAn( )t , vBn( )t , and vCn( )t .
4-13 In the balanced three-phase dc-ac inverter shown in Fig. 4-12b, the phase-A average output voltage is ( ) 0.75sin( 1 )
2 d An V v t = ωt , where Vd =300V and 1 2 45rad s/
ω = π× . The inductance L in each phase is 5 mH. The ac- motor internal voltage in phase A can be represented as
0 1
( ) 106.14 sin( 6.6 )
A
e t = ωt− V , assuming this internal voltage to be purely sinusoidal. (a) Calculate and plot d t , ( )A( ) d t , and B d t , (b) sketch ( )C( ) i t , A
and (c) sketch idA( )t .
4-14 In Problem 4-13, calculate and plot ( )i t , which is the average dc current d
drawn from the dc capacitor bus in Fig. 4-12b. SIMULATION PROBLEMS
4-15 In a switch-mode converter pole-A, 300Vd = V and ˆVtri =5V. The values of s
f are specified below. The control voltage v t is a sine wave, with an c( ) amplitude ˆVc =3.75V and f1=60Hz. With respect to a hypothetical mid- point “o” of the dc input bus voltage, the idealized pole output is given as
N / 2 ( ) ( ) ˆ pole d Ao c tri k V v t v t V
= . Now, assume that the control voltage has a discrete and constant value during each switching-frequency time-period T . This s
value equals the value of the control voltage at the beginning of each T . s
value vAo k, over every k-th switching time-period. Plot vAo( )t and vAo k, . In
a separate figure, plot the difference of these two output voltage values. Repeat this procedure for two values of switching frequency, where both values are chosen to be extremely low (for modern power semiconductor devices) for ease of plotting: (a) fs =900Hz and (b) fs =1.8kHz. Which switching-frequency value leads to a better approximation of the idealized output? What would you expect if the switching frequency is increased to 18 kHz?
4-16 Simulate a two-quadrant pole of Fig. 4-6a in dc steady state. The nominal values are as follows: V = 200 V, 0.37d Ra = Ω , L = 1.5 mH, a E = 136 V, a
ˆ 1
tri
V = V , and vc A, =0.416 V. The switching frequency fs =20kHz. In dc steady state, the average output current is IA =10A. (a) Obtain the plot of
( )
AN
v t , ( )i t , and A idA( )t , (b) obtain the peak-peak ripple in ( )i t and A
compare it with its value obtained analytically, and (c) obtain the average values of ( )i t and A idA( )t , and show that these two averages are related by
the duty-ratio d . A
4-17 Repeat Problem 4-16 by calculating the value of the control voltage such that the converter pole is operating in the boost mode, with IA = −10A. DC-DC Converters
4-18 Simulate the dc-dc converter of Fig. 4-11a in dc steady state. The nominal values are as follows: V = 200 V, 0.37d Ra = Ω , L = 1.5 mH, ˆa Vtri =1V , and
, 0.416 c A
v = V . The switching frequency fs =20kHz. In the dc steady state, the average output current is IA =10A. (a) Obtain the plot of v t , o( )
( )
o
i t , and ( )i t , (b) obtain the peak-peak ripple in ( )d i t and compare it with o
its value obtained analytically, and (c) obtain the average values of ( )i t and o
( )
d
i t , and show that these two averages are related by the duty-ratio d in
Eq. 4-21.
4-19 In Problem 4-18, apply a step-increase in the control voltage to 0.6 V at 0.5
4-20 Repeat Problem 4-19 with each converter pole represented on its average basis.
DC-to-Three-Phase AC Inverters
4-21 Simulate the three-phase ac inverter on an average basis for the system described in Problem 4-13. Obtain the various waveforms.
4-22 Repeat Problem 4-21 for a corresponding switching circuit and compare the switching waveforms with the average waveforms in Problem 4-21.