1. Electric Circuits (EELE 2312)
Chapter 7 Sinusoidal Steady-State Analysis
Basil Hamed

2. Introduction
Thus far, we have focused on circuits with constant sources; in this chapter we are now ready to consider circuits energized by time-varying voltage or current sources. In particular, we are interested in sources in which the value of the voltage or current varies sinusoidally.
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3. Introduction
Sinusoidal sources and their effect on circuit behavior form an important area of study for several reasons
First, the generation, transmission, distribution, and consumption of electric energy occur under essentially sinusoidal steady-state conditions.
Second, an understanding of sinusoidal behavior makes it possible to predict the behavior of circuits with nonsinusoidal sources.
Third, steady-state sinusoidal behavior often simplifies the design of electrical systems.
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4. 7.1 The Sinusoidal Source
A sinusoidal voltage source (independent or dependent) produces a voltage that varies sinusoidally with time. A sinusoidal current source (independent or dependent) produces a current that varies sinusoidally with time.
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5. 7.1 The Sinusoidal Source
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6. Example 7.1
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7. Example 7.2
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8. Example 7.3
We can translate the sine function to the cosine function by subtracting 90o (π/2 rad) from the argument of the sine function.
Verify this translation by showing that sin(ωt+θ)=cos(ωt+θ-90o)
Use the result in (a) to express sin(ωt+30o) as a cosine function
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9. Example 7.4
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10. Example 7.4
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11. 7.2 The Sinusoidal Response
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12. 7.2 The Sinusoidal Response
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13. 7.3 The Phasor The Phasor Transform
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14. 7.3 The Phasor Inverse Phasor Transform
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15. Example 7.5
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16. Example 7.5
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17. Example 7.5
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18. 7.4 Passive Circuit Elements in f-Domain V-I Relationship for a Resistor
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19. 7.4 Passive Circuit Elements in f-Domain V-I Relationship for a Resistor
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20. Passive Circuit Elements in f-Domain V-I Relationship for an Inductor
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21. Passive Circuit Elements in f-Domain V-I Relationship for an Inductor
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22. 7.4 Passive Circuit Elements in f-Domain V-I Relationship for a Capacitor
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23. Basil Hamed

24. Impedance and Reactance
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25. 7.5 Kirchhoff’s Laws Kirchhoff’s Voltage Law in Frequency Domain

26. 7.5 Kirchhoff’s Laws Kirchhoff’s Current Law in Frequency Domain

27. 7.6 Circuit Simplifications Combining Impedances in Series

28. Example 7.6
A 90Ω resistor, a 32 mH inductor, and a 5 μF capacitor are connected in series across the terminals of a sinusoidal voltage source. The steady-state expression for the source voltage υs is 750 cos(5000t+30o) V.
Construct the frequency-domain equivalent circuit
Calculate the steady-state current i by the phasor method
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29. Example 7.6
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30. 7.6 Circuit Simplifications Combining Impedances in Parallel
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31. 7.6 Circuit Simplifications Combining Impedances in Parallel
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32. 7.6 Circuit Simplifications Combining Impedances in Parallel
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33. Example 7.7
The sinusoidal current source in the circuit produces the current is=8cos200000t A.
Construct the frequency-domain equivalent circuit
Find the steady-state expression for υ, i1, i2, and i3
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34. Example 7.7
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35. Example 7.7
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36. 7.6 Circuit Simplifications Source Transformation

37. 7.6 Circuit Simplifications Thevenin Equivalent Circuit

38. 7.6 Circuit Simplifications Norton Equivalent Circuit

39. Example 7.8
Use the concept of source transformation to find the phasor voltage Vo in the circuit.

40. Example 7.8
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41. Example 7.8
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42. Example 7.9
Find the Thevenin equivalent circuit with respect to terminals a,b for the circuit.

43. Example 7.9

44. Example 7.9
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45. Example 7.9
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46. 7.7 The Node-Voltage Method Example 7.10
Use the node-voltage method to find the branch currents Ia, Ib, and Ic in the circuit
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47. Example 7.10
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48. 7.8 The Mesh-Current Method Example 7.11
Use the mesh-current method to find the voltages V1, V2, and V3 in the circuit
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49. 7.8 The Mesh-Current Method Example 7.11
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50. 7.9 Instantaneous, Average, and reactive Power The Instantaneous Power
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51. 7.9 Instantaneous, Average, and reactive Power The Instantaneous Power
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52. Active and Reactive Power
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53. Power for Purely Resistive Circuit
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54. Power for Purely Resistive Circuit
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55. Power for Purely Inductive Circuit
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56. Power for Purely Inductive Circuit
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57. Power for Purely Capacitive Circuit
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58. Power for Purely Capacitive Circuit
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59. The Power Factor

60. Example 7.12
Calculate the average power and the reactive power at the terminals of the network shown given that: υ=100cos(ωt+15o) i=4sin(ωt-15o)
State whether the network is absorbing or delivering average power.
State whether the network is absorbing or delivering magnetizing vars.
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61. box delivers average power
Example 7.12
box delivers average power
box absorbs magnetizing vars
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62. Appliance Ratings

63. 7.10 The RMS Value & Power Calculation
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64. 7.10 The RMS Value & Power Calculation
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65. 7.10 The RMS Value & Power Calculation
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66. Example 7.13
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.
Repeat (a) by first finding the current in the resistor

67. 7.11 Complex Power & Power Calculations

68. 7.11 Complex Power & Power Calculations

69. 7.11 Complex Power & Power Calculations

70. Example 7.14
An electric load operates at 240 V rms. The load absorbs an average power of 8 kW at 0.8 lagging power factor.
Calculate the complex power of the load
Calculate the impedance of the load

71. Example 7.14

72. Power Calculations

73. Power Calculations

74. Alternate Forms for Complex Power

75. Alternate Forms for Complex Power

76. Example 7.15
In the circuit shown, 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.
Calculate the load current IL and voltage VL
Calculate the average and reactive power delivered to the load
Calculate the average and reactive power delivered to the line
Calculate the average and reactive power supplied by the source

77. Example 7.15

78. Example 7.16
The two loads in the circuit shown can be considered as follows: load 1 absorbs an average power of 8 kW at a leading power factor of 0.8 and load 2 absorbs 20 kVA at 0.6 lagging power factor.
Determine the power factor of the two load in parallel
Determine the apparent power required to supply the loads, the magnitude of the current, Is, and the average power loss in the transmission line
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

79. Example 7.17
Calculate the total average and reactive power delivered to each impedance
Calculate the average and reactive powers associated with each source
Verify that the average power delivered equals the average power absorbed, and that the magnetizing reactive power delivered equals the magnetizing reactive power absorbed

80. End Of
Chapter Seven