1. Sinusoidal Steady-State Power Calculations
Chapter 10
Sinusoidal Steady-State Power Calculations

2. CHAPTER CONTENTS 10.1 Instantaneous Power
10.2 Average and Reactive Power
10.3 The rms Value and Power Calculations
10.4 Complex Power
10.5 Power Calculations
10.6 Maximum Power Transfer

3. CHAPTER OBJECTIVES
Understand the following ac power concepts, their relationships to one another, and how to calculate them in a circuit:
Instantaneous power;
Average (real) power;
Reactive power;
Complex power; and
Power factor.
Understand the condition for maximum real power delivered to a load in an ac circuit and be able to calculate the load impedance required to deliver maximum real power to the load.
Be able to calculate all forms of ac power in ac circuits with linear transformers and in ac circuits with ideal transformers.

4. 10.1 Instantaneous Power
Figure The black box representation of a circuit used for calculating power.

6. Figure Instantaneous power, voltage, and current versus vt for steady-state sinusoidal operation.

7. 10.2 Average and Reactive Power
Average (real) power
Reactive power
P is called the average power, and Q is called the reactive power. Average power is sometimes called real power.

8. Power for Purely Resistive Circuits
If the circuit between the terminals is purely resistive, he voltage and current are in phase, which means that qv = qi.
The instantaneous power expressed is referred to as the instantaneous real power.

9. Figure 10.3 Instantaneous real power and average power for a purely resistive circuit.

10. Power for Purely Inductive Circuits
If the circuit between the terminals is purely inductive, the voltage and current are out of phase by precisely 90°. In particular, the current lags the voltage by 90° (that is, qv = qi – 90°)
The instantaneous power
The average power P and reactive power Q carry the same dimension. To distinguish between average and reactive power, we use the units watt (W) for average power and var (volt-amp reactive, or VAR) for reactive power.

11. Figure Instantaneous real power, average power, and reactive power for a purely inductive circuit.

12. Power for Purely Capacitive Circuits
If the circuit between the terminals is purely capacitive, the voltage and current are precisely 90° out of phase. In this case, the current leads the voltage by 90° (that is, qv = qi + 90°)
The instantaneous power
Power engineers recognize this difference in the algebraic sign of Q by saying that inductors demand (or absorb) magnetizing vars, and capacitors furnish (or deliver) magnetizing vars.

13. Figure 10.5 Instantaneous real power and average power for a purely capacitive circuit.

14. The Power Factor The power factor angle: qv – qi
The cosine of this angle is called the power factor, pf.
The sine of this angle is called the reactive factor, rf.
Lagging power factor implies that current lags voltage—an inductive load. Leading power factor implies that current leads voltage—a capacitive load.
Both the power factor and the reactive factor are convenient quantities to use in describing electrical loads.

15. Example 10.1
a) Calculate the average power and the reactive power at the terminals of the network shown in Fig if b) State whether the network inside the box is absorbing or delivering average power. c) State whether the network inside the box is absorbing or supplying magnetizing vars.
Figure A pair of terminals used for calculating power.

16. Example 10.1

18. Appliance Ratings
Average power is used to quantify the power needs of household appliances.
The average power rating and estimated annual kilowatt-hour consumption of some common appliances are presented in Table 10.1.
The energy consumption values are obtained by estimating the number of hours annually that the appliances are in use.

19. Example 10.2
The branch circuit supplying the outlets in a typical home kitchen is wired with #12 conductor and is protected by either a 20 A fuse or a 20 A circuit breaker. Assume that the following 120 V appliances are in operation at the same time: a coffeemaker, egg cooker, frying pan, and toaster. Will the circuit be interrupted by the protective device?

20. Example 10.2

21. Example 10.2

22. 10.3 The rms Value and Power Calculations
Assume that a sinusoidal voltage is applied to the terminals of a resistor, as shown in Fig. 10.7
Figure A sinusoidal voltage applied to the terminals of a resistor.

23. The rms value is also referred to as the effective value of the sinusoidal voltage (or current).

24. Figure 10. 8 demonstrates this equivalence
Figure 10.8 demonstrates this equivalence. Energywise, the effect of the two sources is identical. This has led to the term effective value being used interchangeably with rms value.
Figure The effective value of υs (100 V rms) delivers the same power to R as the dc voltage Vs (100 V dc).

25. The effective value of the sinusoidal signal in power calculations is so widely used that voltage and current ratings of circuits and equipment involved in power utilization are given in terms of rms values.

26. Example 10.3
a) A sinusoidal voltage having a maximum amplitude of 625 V is applied to the terminals of a 50 Ω resistor. Find the average power delivered to the resistor. b) Repeat (a) by first finding the current in the resistor.

27. Example 10.3

29. 10.4 Complex Power
Complex power is the complex sum of real power and reactive power

30. For the right triangle shown in Fig. 10
For the right triangle shown in Fig. 10.9, the magnitude of complex power is referred to as apparent power.
Apparent power
Figure A power triangle.

31. Example 10.4
An electrical load operates at 240 V rms. The load absorbs an average power of 8 kW at a lagging power factor of 0.8.
a) Calculate the complex power of the load.
b) Calculate the impedance of the load.

32. Example 10.4
Figure A power triangle.a

33. Example 10.4

34. Example 10.4

35. 10.5 Power Calculations

36. Figure 10.11 The phasor voltage and current associated with a pair of terminals.
Complex power

38. Alternate Forms for Complex Power
Figure The general circuit of Fig replaced with an equivalent impedance.

40. Example 10.5
In the circuit shown in Fig , a load having an impedance of 39 + j26 Ω is fed from a voltage source through a line having an impedance of 1 + j4 Ω. The effective, or rms, value of the source voltage is 250 V.
a) Calculate the load current IL and voltage VL.
b) Calculate the average and reactive power delivered to the load.
c) Calculate the average and reactive power delivered to the line.
d) Calculate the average and reactive power supplied by the source.

41. Example 10.5
Figure The circuit for Example 10.5.

42. Example 10.5

43. Example 10.5

44. Example 10.5

45. Example 10.6
The two loads in the circuit shown in Fig can be described as follows: Load 1 absorbs an average power of 8 kW at a leading power factor of 0.8. Load 2 absorbs 20 kVA at a lagging power factor of 0.6.
Figure The circuit for Example 10.6.

46. Example 10.6
a) Determine the power factor of the two loads in parallel. b) Determine the apparent power required to supply the loads, the magnitude of the current, Is, and the average power loss in the transmission line. c) Given that the frequency of the source is 60 Hz, compute the value of the capacitor that would correct the power factor to 1 if placed in parallel with the two loads. Recompute the values in (b) for the load with the corrected power factor.

47. Example 10.6

48. Example 10.6
Figure (a) The power triangle for load 1. (b) The power triangle for load 2. (c) The sum of the power triangles.

49. Example 10.6

50. Example 10.6

51. Example 10.6

52. Example 10.6
Figure (a) The sum of the power triangles for loads 1 and 2. (b) The power triangle for a µF capacitor at 60 Hz. (c) The sum of the power triangles in (a) and (b).

53. Example 10.6

54. Example 10.7
a) Calculate the total average and reactive power delivered to each impedance in the circuit shown in Fig
Figure The circuit, with solution, for Example 10.7.

55. Example 10.7
b) Calculate the average and reactive powers associated with each source in the circuit. c) Verify that the average power delivered equals the average power absorbed, and that the magnetizing reactive power delivered equals the magnetizing reactive power absorbed.

56. Example 10.7

57. Example 10.7

58. Example 10.7

59. Example 10.7

61. 10.6 Maximum Power Transfer
Figure A circuit describing maximum power transfer.
Condition for maximum average power transfer

62. Figure 10. 19 The circuit shown in Fig. 10
Figure The circuit shown in Fig , with the network replaced by its Thévenin equivalent.

64. The Maximum Average Power Absorbed

65. Maximum Power Transfer When Z is Restricted
A second type of restriction occurs when the magnitude of ZL can be varied but its phase angle cannot. Under this restriction, the greatest amount of power is transferred to the load when the magnitude of ZL is set equal to the magnitude of ZL that is, when

66. Example 10.8
a) For the circuit shown in Fig , determine the impedance ZL that results in maximum average power transferred to ZL. b) What is the maximum average power transferred to the load impedance determined in (a)?
Figure The circuit for Example 10.8.

67. Example 10.8
Figure A simplification of Fig by source transformations.

68. Example 10.8

69. Example 10.8
Figure The circuit shown in Fig , with the original network replaced by its Thévenin equivalent.

70. Example 10.9
a) For the circuit shown in Fig , what value of ZL results in maximum average power transfer to ZL? What is the maximum power in milliwatts? b) Assume that the load resistance can be varied between 0 and 4000 Ω and that the capacitive reactance of the load can be varied between 0 and –2000 Ω What settings of RL and XL transfer the most average power to the load? What is the maximum average power that can be transferred under these restrictions?

71. Example 10.9
Figure The circuit for Examples 10.9 and

72. Example 10.9

73. Example 10.9

74. Example 10.10
A load impedance having a constant phase angle of –36.87° is connected across the load terminals a and b in the circuit shown in Fig The magnitude of ZL is varied until the average power delivered is the most possible under the given restriction.
a) Specify ZL in rectangular form.
b) Calculate the average power delivered to ZL.

75. Example 10.10

77. Example 10.11
The variable resistor in the circuit in Fig is adjusted until maximum average power is delivered to RL.
a) What is the value of RL in ohms?
b) What is the maximum average power (in watts) delivered to RL?
Figure The circuit for Example

78. Example 10.11
Figure The circuit used to find the Thévenin voltage.

79. Example 10.11

80. Example 10.11
Figure The circuit used to calculate the short circuit current.

81. Example 10.11

82. Example 10.11
Figure The Thévenin equivalent loaded for maximum power transfer.

84. Summary
Instantaneous power is the product of the instantaneous terminal voltage and current, or p = ±vi. The positive sign is used when the reference direction for the current is from the positive to the negative reference polarity of the voltage. The frequency of the instantaneous power is twice the frequency of the voltage (or current).

85. Summary
Average power is the average value of the instantaneous power over one period. It is the power converted from electric to nonelectric form and vice versa. This conversion is the reason that average power is also referred to as real power. Average power, with the passive sign convention, is expressed as

86. Summary
Reactive power is the electric power exchanged between the magnetic field of an inductor and the source that drives it or between the electric field of a capacitor and the source that drives it. Reactive power is never converted to nonelectric power. Reactive power, with the passive sign convention, is expressed as

87. Summary
Both average power and reactive power can be expressed in terms of either peak (Vm, Im) or effective (Veff, Ieff) current and voltage. Effective values are widely used in both household and industrial applications. Effective value and rms value are interchangeable terms for the same value.

88. Summary
The power factor is the cosine of the phase angle between the voltage and the current: The terms lagging and leading added to the description of the power factor indicate whether the current is lagging or leading the voltage and thus whether the load is inductive or capacitive.
The reactive factor is the sine of the phase angle between the voltage and the current:

89. Summary
Complex power is the complex sum of the real and reactive powers, or
Apparent power is the magnitude of the complex power:

90. Summary
The watt is used as the unit for both instantaneous and real power. The var (volt amp reactive, or VAR) is used as the unit for reactive power. The volt-amp (VA) is used as the unit for complex and apparent power.
Maximum power transfer occurs in circuits operating in the sinusoidal steady state when the load impedance is the conjugate of the Thévenin mpedance as viewed from the terminals of the load impedance.