1. Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal 4.2 Phasors 4.3 Phasor Relationships for R, L and C 4.4 Impedance 4.5 Parallel and Series Resonance 4.6 Examples for Sinusoidal Circuits Analysis References References: Hayt-Ch7; Gao-Ch3; Engineering Circuit Analysis

2. Ch4 Sinusoidal Steady State Analysis Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid –All steady state voltages and currents have the same frequency as the source In order to find a steady state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency) We do not have to find this differential equation from the circuit, nor do we have to solve it Instead, we use the concepts of phasors and complex impedances Phasors and complex impedances convert problems involving differential equations into circuit analysis problems  Focus on steady state;  Focus on sinusoids.

3. 4.1 Characteristics of Sinusoidal Key Words Key Words: Period: T, Frequency: f, Radian frequency  Phase angle Amplitude: V m I m Ch4 Sinusoidal Steady State Analysis

4. 4.1 Characteristics of Sinusoidal v、iv、i t t1t1 t2t2 0 Both the polarity and magnitude of voltage are changing. Ch4 Sinusoidal Steady State Analysis

5. 4.1 Characteristics of Sinusoidal Radian frequency(Angular frequency):  = 2  f = 2  /T (rad/s ） Period: T — Time necessary to go through one cycle. (s) Frequency: f — Cycles per second. (Hz) f = 1/T Amplitude: V m I m i = I m sin  t ， v =V m sin  t v、iv、i tt  22 0 Vm、ImVm、Im Ch4 Sinusoidal Steady State Analysis

6. 4.1 Characteristics of Sinusoidal Effective Roof Mean Square (RMS) Value of a Periodic Waveform — is equal to the value of the direct current which is flowing through an R-ohm resistor. It delivers the same average power to the resistor as the periodic current does. Effective Value of a Periodic Waveform Ch4 Sinusoidal Steady State Analysis

7. 4.1 Characteristics of Sinusoidal Phase (angle) Phase angle  <0 00 Ch4 Sinusoidal Steady State Analysis

8. 4.1 Characteristics of Sinusoidal Phase difference — v(t) leads i(t) by (  1 -  2 ), or i(t) lags v(t) by (  1 -  2 ) v、iv、i tt v i Out of phase 。 tt v、iv、i v i v、iv、i tt v i In phase. — v(t) lags i(t) by (  2 -  1 ), or i(t) leads v(t) by (  2 -  1 ) Ch4 Sinusoidal Steady State Analysis

9. 4.1 Characteristics of Sinusoidal Review The sinusoidal waves whose phases are compared must: ① Be written as sine waves or cosine waves. ② With positive amplitudes. ③ Have the same frequency. 360°—— does not change anything. 90° —— change between sin & cos. 180°—— change between + & - Ch4 Sinusoidal Steady State Analysis

10. 4.1 Characteristics of Sinusoidal Phase difference P4.1, Find Ch4 Sinusoidal Steady State Analysis If

11. 4.1 Characteristics of Sinusoidal Phase difference P4.2, v、iv、i tt v i -  /3  /3  Ch4 Sinusoidal Steady State Analysis

12. 4.2 Phasors Key Words Key Words: Complex Numbers Rotating Vector Phasors A sinusoidal voltage/current at a given frequency, is characterized by only two parameters :amplitude an phase Ch4 Sinusoidal Steady State Analysis

13. 4.2 Phasors E.g. voltage response A sinusoidal v/i Complex transform Phasor transform By knowing angular frequency ω rads/s. Time domain Frequency domain Complex form: Phasor form: Angular frequency ω is known in the circuit. Ch4 Sinusoidal Steady State Analysis

14. 4.2 Phasors Rotating Vector i ImIm t1t1 i tt  ImIm  tt x y A complex coordinates number: Real value: Ch4 Sinusoidal Steady State Analysis i(t1)i(t1) Imag

15. 4.2 Phasors Rotating Vector VmVm x y 0   Ch4 Sinusoidal Steady State Analysis

16. 4.2 Phasors Complex Numbers — Rectangular Coordinates — Polar Coordinates conversion ：  |A| a b real axis imaginary axis Ch4 Sinusoidal Steady State Analysis

17. 4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d) Real Axis Imaginary Axis AB A + B Ch4 Sinusoidal Steady State Analysis

18. 4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d) Real Axis Imaginary Axis AB A - B Ch4 Sinusoidal Steady State Analysis

19. 4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Multiplication : A = A m     B = B m    A  B = (A m  B m )  (     ) Division: A = A m   , B = B m    A / B = (A m / B m )  (     ) Ch4 Sinusoidal Steady State Analysis

20. 4.2 Phasors Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid: Phasor Diagrams A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). A phasor diagram helps to visualize the relationships between currents and voltages. Ch4 Sinusoidal Steady State Analysis

21. 4.2 Phasors Complex Exponentials A real-valued sinusoid is the real part of a complex exponential. Complex exponentials make solving for AC steady state an algebraic problem. Ch4 Sinusoidal Steady State Analysis

22. 4.3 Phasor Relationships for R, L and C Key Words Key Words: I-V Relationship for R, L and C, Power conversion Ch4 Sinusoidal Steady State Analysis

23. 4.3 Phasor Relationships for R, L and C v~i relationship for a resistor Relationship between RMS : Wave and Phasor diagrams ： v、iv、i tt v i Resistor Suppose Ch4 Sinusoidal Steady State Analysis

24. 4.3 Phasor Relationships for R, L and C Time domain frequency domain Resistor With a resistor θ ﹦ φ, v(t) and i(t) are in phase. Ch4 Sinusoidal Steady State Analysis

25. 4.3 Phasor Relationships for R, L and C Power Resistor p0p0 v、iv、i tt v i P=IV Average Power Transient Power Ch4 Sinusoidal Steady State Analysis Note: I and V are RMS values.

26. 4.3 Phasor Relationships for R, L and C Resistor P4.4 ，, R=10  ， Find i and P 。 Ch4 Sinusoidal Steady State Analysis

27. 4.3 Phasor Relationships for R, L and C v~i relationship Inductor Suppose Ch4 Sinusoidal Steady State Analysis

28. 4.3 Phasor Relationships for R, L and C v~i relationship Inductor Relationship between RMS: For DC ， f = 0 ，  X L = 0. v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º Ch4 Sinusoidal Steady State Analysis

29. 4.3 Phasor Relationships for R, L and C v ~ i relationship Inductor i(t) = I m e j  t Represent v(t) and i(t) as phasors: The derivative in the relationship between v(t) and i(t) becomes a multiplication by j  in the relationship between and. The time-domain differential equation has become the algebraic equation in the frequency-domain. Phasors allow us to express current-voltage relationships for inductors and capacitors in a way such as we express the current-voltage relationship for a resistor. Ch4 Sinusoidal Steady State Analysis

30. 4.3 Phasor Relationships for R, L and C v ~ i relationship Inductor v、iv、i tt v i eLeL Wave and Phasor diagrams ： Ch4 Sinusoidal Steady State Analysis

31. 4.3 Phasor Relationships for R, L and C Power Inductor P tt v、iv、i tt v i ++ -- Energy stored: Average Power Reactive Power （ Var ） Ch4 Sinusoidal Steady State Analysis

32. 4.3 Phasor Relationships for R, L and C Inductor P4.5 ， L = 10mH ， v = 100sin  t ， Find i L when f = 50Hz and 50kHz. Ch4 Sinusoidal Steady State Analysis

33. 4.3 Phasor Relationships for R, L and C v ~ i relationship Capacitor Suppose: i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º Relationship between RMS: For DC ， f = 0 ，  X C   Ch4 Sinusoidal Steady State Analysis

34. v(t) = V m e j  t Represent v(t) and i(t) as phasors: The derivative in the relationship between v(t) and i(t) becomes a multiplication by j  in the relationship between and. The time-domain differential equation has become the algebraic equation in the frequency-domain. Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. 4.3 Phasor Relationships for R, L and C v ~ i relationship Capacitor Ch4 Sinusoidal Steady State Analysis

35. 4.3 Phasor Relationships for R, L and C v ~ i relationship Capacitor v、iv、i tt v i Wave and Phasor diagrams ： Ch4 Sinusoidal Steady State Analysis

36. 4.3 Phasor Relationships for R, L and C Power Capacitor Average Power: P=0 Reactive Power （ Var ） P tt v、iv、i tt v i ++ -- Energy stored: Ch4 Sinusoidal Steady State Analysis

37. 4.3 Phasor Relationships for R, L and C Capacitor P4.7 ， Suppose C=20  F ， AC source v=100sin  t ， Find X C and I for f = 50Hz, 50kHz 。 Ch4 Sinusoidal Steady State Analysis

38. 4.3 Phasor Relationships for R, L and C Review (v-I relationship) Time domainFrequency domain,,, v and i are in phase., v leads i by 90°., v lags i by 90°. R C L Ch4 Sinusoidal Steady State Analysis

39. 4.3 Phasor Relationships for R, L and C Summary R：R： L：L： C：C： Frequency characteristics of an Ideal Inductor and Capacitor: A capacitor is an open circuit to DC currents; A Inducter is a short circuit to DC currents. Ch4 Sinusoidal Steady State Analysis

40. 4.4 Impedance Key Words Key Words: complex currents and voltages. Impedance Phasor Diagrams Ch4 Sinusoidal Steady State Analysis

41. AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law: 4.4 Impedance Complex voltage ， Complex current ， Complex Impedance Z is called impedance. measured in ohms (  ) Ch4 Sinusoidal Steady State Analysis

42. 4.4 Impedance Complex Impedance Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor) Impedance is a complex number and is not a phasor (why?). Impedance depends on frequency Ch4 Sinusoidal Steady State Analysis

43. 4.4 Impedance Complex Impedance Z R = R  ＝ 0; or Z R = R  0 Resistor——The impedance is R or Capacitor——The impedance is 1/j  C or Inductor——The impedance is j  L Ch4 Sinusoidal Steady State Analysis

44. 4.4 Impedance Complex Impedance Impedance in series/parallel can be combined as resistors. Current divider: Voltage divider: Ch4 Sinusoidal Steady State Analysis

45. 4.4 Impedance Complex Impedance P4.8, Ch4 Sinusoidal Steady State Analysis

46. 4.4 Impedance Complex Impedance Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage and voltage from current 20k  + - 1F1F10V  0  VCVC + -  = 377 Find V C P4.9 How do we find V C ? First compute impedances for resistor and capacitor: Z R = 20k  = 20k  0  Z C = 1/j (377 *1  F) = 2.65k  -90  Ch4 Sinusoidal Steady State Analysis

47. 4.4 Impedance Complex Impedance 20k  + - 1F1F10V  0  VCVC + -  = 377 Find V C P4.9 20k  0  + - 2.65k  -90  10V  0  VCVC + - Now use the voltage divider to find V C : Ch4 Sinusoidal Steady State Analysis

48. 4.4 Impedance Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state. All the analysis techniques we have learned for the linear circuits are applicable to compute phasors –KCL & KVL –node analysis / loop analysis –superposition –Thevenin equivalents / Norton equivalents –source exchange The only difference is that now complex numbers are used. Complex Impedance Ch4 Sinusoidal Steady State Analysis

49. 4.4 Impedance Kirchhoff’s Laws KCL and KVL hold as well in phasor domain. KVL ： v k - Transient voltage of the # k branch KCL: i k - Transient current of the # k branch Ch4 Sinusoidal Steady State Analysis

50. 4.4 Impedance Admittance I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S) Resistor: –The admittance is 1/R Inductor: –The admittance is 1/j  L Capacitor: –The admittance is j  C Ch4 Sinusoidal Steady State Analysis

51. 4.4 Impedance Phasor Diagrams A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). A phasor diagram helps to visualize the relationships between currents and voltages. 2mA  40  – 1F1F VCVC + – 1k  VRVR + + – V I = 2mA  40 , V R = 2V  40  V C = 5.31V  -50 , V = 5.67V  -29.37  Real Axis Imaginary Axis VRVR VCVC V Ch4 Sinusoidal Steady State Analysis

52. 4.5 Parallel and Series Resonance Key Words Key Words: RLC Circuit, Series Resonance Parallel Resonance Ch4 Sinusoidal Steady State Analysis

53. 4.5 Parallel and Series Resonance v vRvR vLvL vCvC Phasor  (2nd Order RLC Circuit ) Series RLC Circuit Ch4 Sinusoidal Steady State Analysis

54. 4.5 Parallel and Series Resonance (2nd Order RLC Circuit ) Z X = X L -X C R   Phase difference: X L >X C   >0 ， v leads i by  ——Inductance Circuit X L <X C   <0 ， v lags i by  ——Capacitance Circuit X L =X C   =0 ， v and i in phase——Resistors Circuit Series RLC Circuit Ch4 Sinusoidal Steady State Analysis

55. 4.5 Parallel and Series Resonance (2nd Order RLC Circuit ) v vRvR vLvL vCvC Series RLC Circuit Ch4 Sinusoidal Steady State Analysis

56. 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) P4.9, R. L. C Series Circuit ， R = 30  ， L = 127mH ， C = 40  F ， Source. Find 1) X L 、 X C 、 Z ； 2) and i; 3) and v R ; and v L ; and v C ; 4) Phasor diagrams; P4.10 ， Computing by (complex numbers) Phasors v vRvR vLvL vCvC Ch4 Sinusoidal Steady State Analysis

57. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) and —— Series Resonance Resonance condition f0f0 f X Resonance frequency Ch4 Sinusoidal Steady State Analysis

58. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Z min ； when V=constant, I=I max =I 0 。 When,  Quality factor Q, Resonance condition: Ch4 Sinusoidal Steady State Analysis

59. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

60. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

61. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

62. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

63. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

64. 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

65. 4.5 Parallel and Series Resonance Parallel RLC Circuit   Parallel Resonance Parallel Resonance frequency In generally When, In phase with Z max I min : Ch4 Sinusoidal Steady State Analysis

66. 4.5 Parallel and Series Resonance Parallel RLC Circuit   Z  .Z  . Quality factor Q, Ch4 Sinusoidal Steady State Analysis

67. 4.5 Parallel and Series Resonance Parallel RLC Circuit P4.10, v  i i i 1   i2 i2 Find i 1 、 i 2 、 i Ch4 Sinusoidal Steady State Analysis

68. 4.5 Parallel and Series Resonance Parallel RLC Circuit Review For sinusoidal circuit ， Series ： ？ Two Simple Methods: Phasor Diagrams and Complex Numbers Parallel : Ch4 Sinusoidal Steady State Analysis

69. 4.6 Examples for Sinusoidal Circuits Analysis Key Words Key Words: Bypass Capacitor RC Phase Difference Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

70. P4.11, Let 4.6 Examples for Sinusoidal Circuits Analysis Bypass Capacitor f ＝ 500Hz ， Determine V AB before the C is connected. And V AB after parallel C = 30  F v ii Before C is connected After C is connected Ch4 Sinusoidal Steady State Analysis

71. P4.12, 4.6 Examples for Sinusoidal Circuits Analysis RC Phase Difference f = 300Hz, R = 100  。 If  vo -  vi =  /4 ， C = ？ Ch4 Sinusoidal Steady State Analysis

72. P4.13, The voltage sources are v i =240+100sin2  100t(V), R ＝ 200  ， C ＝ 50  F ， Determine V AC and V DC in output voltage v o. 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter V DC = 240V R  C---- High-Pass Filter Ch4 Sinusoidal Steady State Analysis

73. 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

74. 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

75. 4.6 Examples for Sinusoidal Circuits Analysis Ch4 Sinusoidal Steady State Analysis

76. 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

77. 4.6 Examples for Sinusoidal Circuits Analysis Ch4 Sinusoidal Steady State Analysis

78. 4.6 Examples for Sinusoidal Circuits Analysis Ch4 Sinusoidal Steady State Analysis

79. 4.6 Examples for Sinusoidal Circuits Analysis v 1 =120sin  t v2v2  i3i3  i1 i1  i2 i2 P4.14, Find in the circuit of the fo ll owing fig. Complex Numbers Analysis Ch4 Sinusoidal Steady State Analysis

80. 4.6 Examples for Sinusoidal Circuits Analysis P4.15, Let V m ＝ 100V. Use Thevenin’s theorem to find v(t)= 100sinwt V v Complex Numbers Analysis Ch4 Sinusoidal Steady State Analysis