Power Computations
2.11 Bibliography
M. E. Balci and M. H. Hocaoglu, “Comparison of Power Definitions for Reactive Power Compensation in Nonsinusoidal Circuits,” International Conference on Harmonics and Quality of Power, Lake Placid, New York, 2004.
L. S. Czarnecki, “Considerations on the Reactive Power in Nonsinusoidal Situations,”
International Conference on Harmonics in Power Systems, Worcester Polytechnic Institute, Worcester, Mass., 1984, pp. 231–237.
A. E. Emanuel, “Powers in Nonsinusoidal Situations, A Review of Definitions and Physical Meaning,” IEEE Transactions on Power Delivery, vol. 5, no. 3,
July 1990.
G. T. Heydt, Electric Power Quality, Stars in a Circle Publications, West Lafayette, Ind., 1991.
W. Sheperd and P. Zand, Energy Flow and Power Factor in Nonsinusoidal Circuits, Cambridge University Press, 1979.
Problems
Instantaneous and Average Power
2-1. Average power generally is not the product of average voltage and average current. Give an example of periodic waveforms for v(t) and i(t) that have zero average values and average power absorbed by the device is not zero. Sketch v(t), i(t), and p(t).
2-2. The voltage across a 10-resistor is v(t) 170 sin (377t) V. Determine (a) an expression for instantaneous power absorbed by the resistor, (b) the peak power, and (c) the average power.
2-3. The voltage across an element is v(t) 5 sin (2t) V. Use graphing software to graph instantaneous power absorbed by the element, and determine the average power if the current, using the passive sign convention, is (a) i(t) 4 sin (2t) A and (b) i(t) 3 sin (4t) A.
2-4. The voltage and current for a device (using the passive sign convention) are periodic functions with T100 ms described by
Determine (a) the instantaneous power, (b) the average power, and (c) the energy absorbed by the device in each period.
i(t)b04 A 50ms0tt10050msms v(t)b10 V0 70ms0t70ms
t100ms
60 C H A P T E R 2 Power Computations
2-5. The voltage and current for a device (using the passive sign convention) are periodic functions with T20 ms described by
Determine (a) the instantaneous power, (b) the average power, and (c) the energy absorbed by the device in each period.
2-6. Determine the average power absorbed by a 12-V dc source when the current into the positive terminal of the source is that given in (a) Prob. 2-4 and (b) Prob. 2-5.
2-7. A current of 5 sin (260t) A enters an element. Sketch the instantaneous power and determine the average power absorbed by the load element when the element is (a) a 5-resistor, (b) a 10-mH inductor, and (c) a 12-V source (current into the positive terminal).
2-8. A current source of i(t) 2 6 sin(260t) A is connected to a load that is a series combination of a resistor, an inductor, and a dc voltage source (current into the positive terminal). If R4 , L15 mH, and Vdc6 V, determine the average power absorbed by each element.
2-9. An electric resistance space heater rated at 1500 W for a voltage source of v(t) 120 sin (260t) V has a thermostatically controlled switch. The heater periodically switches on for 5 min and off for 7 min. Determine (a) the maximum in stantaneous power, (b) the average power over the 12-min cycle, and (c) the electric energy converted to heat in each 12-min cycle.
Energy Recovery
2-10. An inductor is energized as in the circuit of Fig. 2-4a. The circuit has L100 mH, R20 , VCC90 V, t14 ms, and T40 ms. Assuming the transistor and diode are ideal, determine (a) the peak energy stored in the inductor, (b) the energy absorbed by the resistor in each switching period, and (c) the average power supplied by the source. (d) If the resistor is changed to 40 , what is the average power supplied by the source?
2-11. An inductor is energized as in the circuit of Fig. 2-4a. The circuit has L10 mH and VCC14 V. (a) Determine the required on time of the switch such that the peak energy stored in the inductor is 1.2 J. (b) Select a value for Rsuch that the switching cycle can be repeated every 20 ms. Assume the switch and the diode are ideal.
2-12. An inductor is energized as in the circuit of Fig. 2-5a. The circuit has L50 mH, VCC90 V, t14 ms, and T50 ms. (a) Determine the peak energy stored in the inductor. (b) Graph the inductor current, source current, inductor
instantaneous power, and source instantaneous power versus time. Assume the transistors are ideal.
12
v(t)e10V 0t14 ms
0 14mst20 ms
i(t)c7A 0t6 ms
5A 6mst10 ms
4A
10 mst20 ms
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Problems 61
2-13. An alternative circuit for energizing an inductor and removing the stored energy without damaging a transistor is shown in Fig. P2-13. Here VCC12 V, L75 mH, and the zener breakdown voltage is VZ20 V. The transistor switch opens and closes periodically with ton20 ms and toff50 ms.
(a) Explain how the zener diode allows the switch to open. (b) Determine and sketch the inductor current iL(t) and the zener diode current iZ(t) for one switching period. (c) Sketch (p)(t) for the inductor and the zener diode.
(d) Determine the average power absorbed by the inductor and by the zener diode.
VCC
iZ iL
L
2-14. Repeat Prob. 2-13 with VCC20 V, L50 mH, VZ30 V, ton15 ms, and toff60 ms.
Effective Values: RMS
2-15. The rms value of a sinusoid is the peak value divided by . Give two examples to show that this is generally not the case for other periodic waveforms.
2-16. A three-phase distribution system is connected to a nonlinear load that has line and neutral currents like those of Fig. 2-8. The rms current in each phase is 12 A, and the resistance in each of the line and neutral conductors is 0.5 . Determine the total power absorbed by the conductors. What should the resistance of the neutral conductor be such that it absorbs the same power as one of the phase conductors?
2-17. Determine the rms values of the voltage and current waveforms in Prob. 2-4.
2-18. Determine the rms values of the voltage and current waveforms in Prob. 2-5.
Nonsinusoidal Waveforms
2-19. The voltage and current for a circuit element are v(t) 2 5 cos (260t) 3cos(460t45) V and i(t) 1.5 2cos(260t 20) 1.1cos(460t 20) A.
(a) Determine the rms values of voltage and current. (b) Determine the power absorbed by the element.
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62 C H A P T E R 2 Power Computations
2-20. A current source i(t) 3 4 cos(260t) 6 cos (460t) A is connected to a parallel RCload with R100 and C50 F. Determine the average power absorbed by the load.
2-21. In Fig. P2-21, R4 , L10 mH, Vdc12 V, and vs(t) 50 30 cos (460t) 10 cos(860t) V. Determine the power absorbed by each component.
vs
Vdc L R
+
−
+
− Figure P2-21
2-22. A nonsinusoidal periodic voltage has a Fourier series of v(t) 6 5 cos(260t) 3cos(660t). This voltage is connected to a load that is a 16-resistor in series with a 25-mH inductor as in Fig. 2-11. Determine the power absorbed by the load.
2-23. Voltage and current for a device (using the passive sign convention) are
Determine the average power based on the terms through n4.
2-24. Voltage and current for a device (using the passive sign convention) are
Determine the average power based on the terms through n4.
2-25. In Fig. P2-21, R20 , L25 mH, and Vdc36 V. The source is a periodic voltage that has the Fourier series
vs(t)50a
q
n1a400
nbsinA200ntB
v(t)50a
q
n1a50
n bcos(nt) V
i(t)10a
q
n1a10
n2bcosAnttan1n>2B
v(t)20a
q
n1a20
n bcos(nt) V
i(t)5a
q
n1a5
n2bcos(nt) A
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Problems 63
Using the Fourier series method, determine the average power absorbed by R, L, and Vdcwhen the circuit is operating in the steady state. Use as many terms in the Fourier series as necessary to obtain a reasonable estimate of power.
2-26. A sinusoidal current of 10 A rms at a 60-Hz fundamental frequency is contaminated with a ninth harmonic current. The current is expressed as
Determine the value of the ninth harmonic rms current I9if the THD is (a) 5 percent, (b) 10 percent, (c) 20 percent, and (d) 40 percent. Use graphing software or PSpice to show i(t) for each case.
2-27. A sinusoidal voltage source of v(t) 170 cos (260t) V is applied to a nonlinear load, resulting in a nonsinusoidal current that is expressed in Fourier series form as i(t) 10 cos (260t30) 6 cos (460t45) 3 cos (860t20) A.
Determine (a) the power absorbed by the load, (b) the power factor of the load, (c) the distortion factor, and (d) the total harmonic distortion of the load current.
2-28. Repeat Prob. 2-27 with i(t) 12 cos (260t40) 5 sin (460t) 4 cos (860t) A.
2-29. A sinusoidal voltage source of v(t) 240 sin (260t) V is applied to a nonlinear load, resulting in a current i(t) 8 sin (260t) 4 sin (460t) A.
Determine (a) the power absorbed by the load, (b) the power factor of the load, (c) the THD of the load current, (d) the distortion factor of the load current, and (e) the crest factor of the load current.
2-30. Repeat Prob. 2-29 with i(t) 12 sin (260t) 9 sin (460t) A.
2-31. A voltage source of v(t) 5 25 cos (1000t) 10 cos (2000t) V is connected to a series combination of a 2-resistor, a 1-mH inductor, and a 1000-F capacitor.
Determine the rms current in the circuit, and determine the power absorbed by each component.
PSpice
2-32. Use PSpice to simulate the circuit of Example 2-1. Define voltage and current with PULSE sources. Determine instantaneous power, energy absorbed in one period, and average power.
2-33. Use PSpice to determine the instantaneous and average power in the circuit elements of Prob. 2-7.
2-34. Use PSpice to determine the rms values of the voltage and current waveforms in (a) Prob. 2-5 and (b) Prob. 2-6.
2-35. Use PSpice to simulate the circuit of Prob. 2-10. (a) Idealize the circuit by using a voltage-controlled switch that has Ron0.001 and a diode with n0.001.
(b) Use Ron0.5 and use the default diode.
2-36. Use PSpice to simulate the circuit of Fig. 2-5a. The circuit has VCC75 V, t040 ms, and T100 ms. The inductance is 100 mH and has an internal resistance of 20 . Use a voltage-controlled switch with Ron1 for the transistors, and use the PSpice default diode model. Determine the average power absorbed by each circuit element. Discuss the differences between the behavior of this circuit and that of the ideal circuit.
i(t)1022sin(260t)I922sin(1860t) A
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64 C H A P T E R 2 Power Computations
2-37. Use PSpice to simulate the circuit of Prob. 2-13. Use Ron0.001 for the switch model and use n 0.001, BV 20 V for the breakdown voltage and IBV 10 A for the current at breakdown for the zener diode model. (a) Display iL(t) and iZ(t). Determine the average power in the inductor and in the zener diode. (b) Repeat part (a) but include a 1.5-series resistance with the inductor and use Ron0.5 for the switch.
2-38. Repeat Prob. 2-37, using the circuit of Prob. 2-14.
2-39. Use PSpice to determine the power absorbed by the load in Example 2-10.
Model the system as a voltage source and four current sources in parallel.
2-40. Modify the switch model so Ron1 in the PSpice circuit file in Example 2-13.
Determine the effect on each of the quantities obtained from Probe in the example.
2-41. Demonstrate with PSpice that a triangular waveform like that of Fig. 2-9ahas an rms value of Vm/ . Choose an arbitrary period T,and use at least three values of t1. Use a VPULSE source with the rise and fall times representing the triangular wave.
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