1. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Chapter 5 Steady-State Sinusoidal Analysis 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state ac circuits using phasors and complex impedances.

2. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis 4. Find Thévenin and Norton equivalent circuits. Lightly. 5. Determine load impedances for maximum power transfer. 3. Compute power for steady-state ac circuits.

3. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

4. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis SINUSOIDAL CURRENTS AND VOLTAGES Vt = Vm cos(ωt +θ) V m is the peak value ω is the angular frequency in radians per second θ is the phase angle

5. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Frequency Angular frequency

6. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Root-Mean-Square Values

7. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis RMS Value of a Sinusoid The rms value for a sinusoid is the peak value divided by the square root of two. This is not true for other periodic waveforms such as square waves or triangular waves.

8. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

9. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Phasor Definition

10. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Adding Sinusoids Using Phasors Step 1: Determine the phasor for each term. Step 2: Add the phasors using complex arithmetic. Step 3: Convert the sum to polar form. Step 4: Write the result as a time function.

11. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Using Phasors to Add Sinusoids

12. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

13. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Sinusoids can be visualized as the real- axis projection of vectors rotating in the complex plane. The phasor for a sinusoid is a snapshot of the corresponding rotating vector at t = 0.

14. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

15. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Phase Relationships To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise. Then when standing at a fixed point, if V 1 arrives first followed by V 2 after a rotation of θ, we say that V 1 leads V 2 by θ. Alternatively, we could say that V 2 lags V 1 by θ. (Usually, we take θ as the smaller angle between the two phasors.)

16. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

17. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis To determine phase relationships between sinusoids from their plots versus time, find the shortest time interval t p between positive peaks of the two waveforms. Then, the phase angle is θ = (t p /T ) × 360°. If the peak of v 1 (t) occurs first, we say that v 1 (t) leads v 2 (t) or that v 2 (t) lags v 1 (t).

18. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

19. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis COMPLEX IMPEDANCES

20. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

21. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

22. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

23. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

24. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

25. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Kirchhoff’s Laws in Phasor Form We can apply KVL directly to phasors. The sum of the phasor voltages equals zero for any closed path. The sum of the phasor currents entering a node must equal the sum of the phasor currents leaving.

26. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Circuit Analysis Using Phasors and Impedances 1. Replace the time descriptions of the voltage and current sources with the corresponding phasors. (All of the sources must have the same frequency.)

27. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis 2. Replace inductances by their complex impedances Z L = jωL. Replace capacitances by their complex impedances Z C = 1/(jωC). Resistances have impedances equal to their resistances. 3. Analyze the circuit using any of the techniques studied earlier in Chapter 2, performing the calculations with complex arithmetic.

28. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

29. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

30. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

31. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

32. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

33. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

34. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

35. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

36. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis “SPOOKY” And “SPECTRE”

37. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis AC-130 GUNSHIP AC Power

38. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

39. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

40. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

41. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis AC Power Calculations

42. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

43. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

44. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

45. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

46. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

47. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

48. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

49. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

50. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

51. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis THÉVENIN EQUIVALENT CIRCUITS

52. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis The Thévenin voltage is equal to the open-circuit phasor voltage of the original circuit. We can find the Thévenin impedance by zeroing the independent sources and determining the impedance looking into the circuit terminals.

53. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis The Thévenin impedance equals the open-circuit voltage divided by the short-circuit current.

54. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

55. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

56. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

57. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

58. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Maximum Power Transfer If the load can take on any complex value, maximum power transfer is attained for a load impedance equal to the complex conjugate of the Thévenin impedance. If the load is required to be a pure resistance, maximum power transfer is attained for a load resistance equal to the magnitude of the Thévenin impedance.

59. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

60. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

61. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis BALANCED THREE-PHASE CIRCUITS Much of the power used by business and industry is supplied by three-phase distribution systems. Plant engineers need to be familiar with three-phase power.

62. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

63. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Phase Sequence Three-phase sources can have either a positive or negative phase sequence. The direction of rotation of certain three-phase motors can be reversed by changing the phase sequence.

64. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

65. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis Wye–Wye Connection Three-phase sources and loads can be connected either in a wye configuration or in a delta configuration. The key to understanding the various three- phase configurations is a careful examination of the wye–wye circuit.

66. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

67. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

68. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

69. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

70. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

71. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

72. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

73. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

74. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis

75. ELECTRICA L ENGINEERING Principles and Applications SECOND EDITION ALLAN R. HAMBLEY ©2002 Prentice-Hall, Inc. Chapter 5 Steady-State Sinusoidal Analysis