CHAPTER 9. Sinusoidal Steady-State Analysis

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1 CHAPTER 9 Sinusoidal Steady-State Analysis

2 9.1 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.

3 Figure 9.1 A sinusoidal voltage.

4 The sinusoidal function repeats at regular intervals is called periodic. The length of time is referred to as the period of the function and is denoted T. The angular frequency of the sinusoidal function

5 The angle f is know as the phase angle of the sinusoidal voltage. Figure 9.2 The sinusoidal voltage from Fig. 9.1 shifted to the right when ϕ = 0.

6 Root mean square (rms) The rms value of a periodic function is defined as the square root of the mean value of the squared function. rms value of a sinusoidal voltage source

7 Example 9.1 A sinusoidal current has a maximum amplitude of 20 A.The current passes through one complete cycle in 1 ms. The magnitude of the current at zero time is 10 A. a) What is the frequency of the current in hertz? b) What is the frequency in radians per second? c) Write the expression for i(t) using the cosine function. Express f in degrees. d) What is the rms value of the current?

8 Example 9.1

9 Example 9.2 A sinusoidal voltage is given by the expression v = 300 cos (120pt + 30 ). a) What is the period of the voltage in milliseconds? b) What is the frequency in hertz? c) What is the magnitude of at t = ms? d) What is the rms value of v?

10 Example 9.2

11 Example 9.3 We can translate the sine function to the cosine function by subtracting 90 (p/2 rad) from the argument of the sine function. a) Verify this translation by showing that sin (ωt + θ) = cos (ωt + θ 90 ). b) Use the result in (a) to express sin (ωt + 30 ) as a cosine function.

12 Example 9.3

13 Example 9.4 Calculate the rms value of the periodic triangular current shown in Fig Express your answer in terms of the peak current I p. Figure 9.3 Periodic triangular current.

14 Example 9.4

15 Example 9.4 Figure 9.4 i 2 versus t.

16 Example 9.4

17 9.2 The Sinusoidal Response Figure 9.5 An RL circuit excited by a sinusoidal voltage source. Kirchhoff s voltage law transient component steady-state component

18 Transient & Steady-state component 1. The steady-state solution is a sinusoidal function. 2. The frequency of the response signal is identical to the frequency of the source signal. If R, L, and C, are constant. 3. The maximum amplitude of the response signal is Vm / R L and the maximum amplitude of the signal source is V m. 4. The phase angle of the current is f q and that of the voltage source is f.

19 9.3 The Phasor The phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function.1 The phasor concept is rooted in Euler s identity, which relates the exponential function to the trigonometric function:

20 The phasor transform transfers the sinusoidal function from the time domain to the complex-number domain, which is also called the frequency domain.

21 Phasor transform the phasor transform of V m cos (ωt + θ)

22 Inverse Phasor Transform finding the inverse phasor transform the inverse phasor transform of j Ve f. m Equation 9.17 indicates that to find the inverse phasor j t transform, we multiply the phasor by e and then extract the real part of the product.

23 The transient component vanishes as time elapses, so the steady state component of the solution must also satisfy the differential equation. In a linear circuit driven by sinusoidal sources, the steadystate response also is sinusoidal, and the frequency of the sinusoidal response is the same as the frequency of the sinusoidal source. Using the notation introduced in Eq. 9.11, we can postulate that the steady-state solution is of the form where A is the maximum amplitude of the response and is the phase angle of the response. When we substitute the postulated steady-state solution j t into the differential equation, the exponential term e cancels out, leaving the solution for A and b in the domain of complex numbers.

24

25 Example 9.5 If y 1 = 20 cos (ωt 30 ) and y 2 = 40 cos (ωt + 60 ) express y = y 1 + y 2 as a single sinusoidal function. a) Solve by using trigonometric identities. b) Solve by using the phasor concept.

26 Example 9.5 Figure 9.6 A right triangle used in the solution for y.

27 Example 9.5

28 9.4 The Passive Circuit Elements in the Frequency Domain The V-I Relationship for a Resistor If i = I m cos (ωt + q i ), Figure 9.7 A resistive element carrying a sinusoidal current. The phasor transform of this voltage is Relationship between phasor voltage and phasor current for a resistor

29 The signals are said to be in phase because they both reach corresponding values on their respective curves at the same time. Figure 9.8 The frequencydomain equivalent circuit of a resistor. Figure 9.9 A plot showing that the voltage and current at the terminals of a resistor are in phase.

30 The V-I Relationship for an Inductor Relationship between phasor voltage and phasor current for an inductor.

31 voltage leading current or current lagging voltage Figure 9.10 The frequencydomain equivalent circuit for an inductor. Figure 9.11 A plot showing the phase relationship between the current and voltage at the terminals of an inductor (θ i = 60 ).

32 The V-I Relationship for a Capacitor Relationship between phasor voltage and phasor current for a capacitor. Figure 9.12 The frequency domain equivalent circuit of a capacitor.

33 The current leads the voltage by 90. Figure 9.13 A plot showing the phase relationship between the current and voltage at the terminals of a capacitor (θ i = 60 ).

34 Impedance and Reactance Definition of impedance where Z represents the impedance of the circuit element.

35 9.5 Kirchhoff s Laws in the Frequency Domain Kirchhoff s Voltage Law in the Frequency Domain Euler s identity

36 KVL in the frequency domain

37 Kirchhoff s Current Law in the Frequency Domain KCL in the frequency domain

38 9.6 Series, Parallel, and Delta-to-Wye Simplifications Combining Impedances in Series and Parallel Figure 9.14 Impedances in series.

39 Example 9.6 A resistor, a 32 mh inductor, and a 5 mf capacitor are connected in series across the terminals of a sinusoidal voltage source, as shown in Fig The steady-state expression for the source voltage v s is 750 cos (5000t + 30 ) V. a) Construct the frequency-domain equivalent circuit. b) Calculate the steady-state current i by the phasor method. Figure 9.15 The circuit for Example 9.6.

40 Example 9.6

41 Example 9.6 Figure 9.16 The frequency-domain equivalent circuit of the circuit shown in Fig

42 Example 9.6

43 Impedances connected in parallel Figure 9.17 Impedances in parallel.

44 Just two impedances in parallel Admittance, defined as the reciprocal of impedance and denoted Y G, is called conductance and whose imaginary part, B, is called susceptance.

45

46 Example 9.7 The sinusoidal current source in the circuit shown in Fig produces the current i s = 8 cos 200,000t A. a) Construct the frequency-domain equivalent circuit. b) Find the steady-state expressions for v, i 1, i 2, and i 3. Figure 9.18 The circuit for Example 9.7.

47 Example 9.7 Figure 9.19 The frequencydomain equivalent circuit.

48 Example 9.7

49 Example 9.7

50 Example 9.7

51 Example 9.7

52 9.7 Source Transformations and Thévenin-Norton Equivalent Circuits The source transformations and the Thévenin- Norton equivalent circuits can be applied to frequency-domain circuits Figure 9.24 A source transformation in the frequency domain.

53 Figure 9.25 The frequency-domain version of a Thévenin equivalent circuit. Figure 9.26 The frequency-domain version of a Norton equivalent circuit.

54 Example 9.9 Use the concept of source transformation to find the phasor voltage V 0 in the circuit shown in Fig Figure 9.27 The circuit for Example 9.9.

55 Example 9.9 Figure 9.28 The first step in reducing the circuit shown in Fig

56 Example 9.9

57 Example 9.9 Figure 9.29 The second step in reducing the circuit shown in Fig

58 Example 9.10 Find the Thévenin equivalent circuit with respect to terminals a,b for the circuit shown in Fig Figure 9.30 The circuit for Example 9.10.

59 Example 9.10

60 Example 9.10 Figure 9.31 A simplified version of the circuit shown in Fig

61 Example 9.10

62 Example 9.10

63 Example 9.10 Figure 9.32 A circuit for calculating the Thévenin equivalent impedance.

64 Example 9.10

65 Example 9.10 Figure 9.33 The Thévenin equivalent for the circuit shown in Fig

66 9.8 The Node-Voltage Method Example 9.11 Use the node-voltage method to find the branch currents I a, I b, and I c in the circuit shown in Fig Figure 9.34 The circuit for Example 9.11.

67 Example 9.11 Figure 9.35 The circuit shown in Fig. 9.34, with the node voltages defined.

68 Example 9.11

69 Example 9.11

70 9.9 The Mesh-Current Method Example 9.12 Use the mesh-current method to find the voltages V 1, V 2, and V 3 in the circuit shown in Fig Figure 9.36 The circuit for Example 9.12.

71 Example 9.12

72 Example 9.12 Figure 9.37 Mesh currents used to solve the circuit shown in Fig

73 Example 9.12

74 Example 9.12

75 9.10 The Transformer A transformer is a device that is based on magnetic coupling. In communication circuits, the transformer is used to match impedances and eliminate dc signals from portions of the system. In power circuits, transformers are used to establish ac voltage levels that facilitate the transmission, distribution, and consumption of electrical power.

76 The Analysis of a Linear Transformer Circuit Linear transformer is found primarily in communication circuits. Primary winding: the winding connected to the source Secondary winding: the winding connected to the load

77 Figure 9.38 The frequency domain circuit model for a transformer used to connect a load to a source.

78

79

80 Reflected Impedance The third term in Eq is called the reflected impedance (Z r ), because it is the equivalent impedance of the secondary coil and load impedance transmitted, or reflected, to the primary side of the transformer. Note that the reflected impedance is due solely to the existence of mutual inductance; that is, if the two coils are decoupled, M becomes zero, Z r becomes zero, and Z ab reduces to the self-impedance of the primary coil.

81 The load impedance in rectangular form: The reflected impedance in rectangular form:

82 Example 9.13 The parameters of a certain linear transformer are R 1 = 200 Ω, R 2 = 100 Ω, L 1 = 9 H, L 2 = 4 H and k = 0.5. The transformer couples an impedance consisting of an 800 Ω resistor in series with a 1 mf capacitor to a sinusoidal voltage source. The 300 V (rms) source has an internal impedance of j100 Ω and a frequency of 400 rad/s. a) Construct a frequency-domain equivalent circuit of the system. b) Calculate the self-impedance of the primary circuit. c) Calculate the self-impedance of the secondary circuit. d) Calculate the impedance reflected into the primary winding. e) Calculate the scaling factor for the reflected impedance. f) Calculate the impedance seen looking into the primary terminals of the transformer. g) Calculate the Thévenin equivalent with respect to the terminals c,d.

83 Example 9.13 Figure 9.39 The frequency-domain equivalent circuit for Example 9.13.

84 Example 9.13

85 Example 9.13

86 Example 9.13 Figure 9.40 The Thévenin equivalent circuit for Example 9.13.

87 9.11 The Ideal Transformer An ideal transformer consists of two magnetically coupled coils having N 1 and N 2 turns, respectively, and exhibiting these three properties: 1. The coefficient of coupling is unity (k = 1). 2. The self-inductance of each coil is infinite (L 1 = L 2 = ). 3. The coil losses, due to parasitic resistance, are negligible.

88 Exploring Limiting Values

89

90

91 Figure 9.41 The circuits used to verify the volts-per-turn and ampere-turn relationships for an ideal transformer.

92 Determining the Voltage and Current Ratios For unity coupling, the mutual inductance equals LL 1 2

93 Voltage relationship for an ideal transformer for k = 1, Current relationship for an ideal transformer

94 Figure 9.42 shows the graphic symbol for an ideal transformer. Figure 9.42 The graphic symbol for an ideal transformer.

95 Determining the Polarity of the Voltage and Current Ratios Dot convention for ideal transformers

96 Figure 9.43 Circuits that show the proper algebraic signs for relating the terminal voltages and currents of an ideal transformer.

97 Ratio of the turns Figure 9.44 Three ways to show that the turns ratio of an ideal transformer is 5.

98 Example 9.14 The load impedance connected to the secondary winding of the ideal transformer in Fig consists of a mω resistor in series with a 125 mh inductor. If the sinusoidal voltage source (v g ) is generating the voltage 2500 cos 400t V, find the steady-state expressions for: (a) i 1 ; (b) v 1 ; (c) i 2 ; and (d) v 2. Figure 9.45 The circuit for Example 9.14.

99 Example 9.14

100 Example 9.14 Figure 9.46 Phasor domain circuit for Example 9.14.

101 Example 9.14

102 The Use of an Ideal Transformer for Impedance Matching Figure 9.47 Using an ideal transformer to couple a load to a source.

103 9.12 Phasor Diagrams Constructing phasor diagrams of circuit quantities generally involves both currents and voltages. Figure 9.48 A graphic representation of phasors. Figure 9.49 The complex number 7 j 3 = 7.62 /

104 Example 9.15 For the circuit in Fig. 9.50, use a phasor diagram to find the value of R that will cause the current through that resistor, i R, to lag the source current. is, by 45 when = 5 krad/s. Figure 9.50 The circuit for Example 9.15.

105 Example 9.15

106 Example 9.15 Figure 9.51 The phasor diagram for the currents in Fig

107 Example 9.16 The circuit in Fig has a load consisting of the parallel combination of the resistor and inductor. Use phasor diagrams to explore the effect of adding a capacitor across the terminals of the load on the amplitude of V s if we adjust V s so that the amplitude of V L remains constant. Utility companies use this technique to control the voltage drop on their lines. Figure 9.52 The circuit for Example 9.16.

108 Example 9.16 Figure 9.53 The frequency-domain equivalent of the circuit in Fig

109 Example 9.16

110 Example 9.16

111 Example 9.16 Figure 9.54 The stepby-step evolution of the phasor diagram for the circuit in Fig

112 Example 9.16

113 Example 9.16 Figure 9.55 The addition of a capacitor to the circuit shown in Fig Figure 9.56 The effect of the capacitor current I c on the line current I.

114 Example 9.16 Figure 9.57 The effect of adding a load-shunting capacitor to the circuit shown in Fig if V L is held constant.

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