1. ELECTRICAL TECHNOLOGY EET 103/4
Define and explain sine wave, frequency, amplitude, phase angle, complex number
Define, analyze and calculate impedance, inductance, phase shifting
Explain and calculate active power, reactive power, power factor
Define, explain, and analyze Ohm’s law, KCL, KVL, Source Transformation, Thevenin theorem.

2. SERIES AND PARALLEL AC CIRCUITS
(CHAPTER 15)

3. Introduction
In this chapter, phasor algebra will be used to develop a quick, direct method for solving both the series and the parallel ac circuits.

4. 15.2 Impedance and the Phasor diagram
Resistive Elements
v and i are in phase
Or;
In phasor form;
Where;

5. 15.2 Impedance and the Phasor diagram
Resistive Elements
Applying Ohm’s law;
Since v and i are in phase;
Hence;

6. 15.2 Impedance and the Phasor diagram
The boldface Roman quantity ZR, having both magnitude and an associate angle, is referred to as the impedance of a resistive element.
ZR is not a phasor since it does not vary with time.
Even though the format R0° is very similar to the phasor notation for sinusoidal current and voltage, R and its associated angle of 0° are fixed, non-varying quantities.

7. 15.2 Impedance and the Phasor diagram
Example 15.1
Find i in the figure
Solution
Applying Ohm’s law;

8. 15.2 Impedance and the Phasor diagram
Example 15.1 – solution (cont’d)
Inverse-transform;

9. 15.2 Impedance and the Phasor diagram
Example 15.2
Find v in the figure
Solution

10. 15.2 Impedance and the Phasor diagram
Example 15.2 – solution (cont’d)
Applying Ohm’s law;

11. 15.2 Impedance and the Phasor diagram
Example 15.2 – solution (cont’d)
Inverse-transform;

12. 15.2 Impedance and the Phasor diagram
Inductive Reactance
By phasor transformation;
Where;

13. 15.2 Impedance and the Phasor diagram
Inductive Reactance
By Ohm’s law;

14. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Since v leads i by 90;
Hence;
Therefore;

15. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Example 15.3
Find i in the figure
Solution

16. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Example 15.3 – solution (cont’d)
Applying Ohm’s law;
Inverse-transform;

17. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Example 15.3 – solution (cont’d)

18. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Example 15.4
Find v in the figure
Solution

19. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Example 15.4 – solution (cont’d)
Applying Ohm’s law;
Inverse-transform;

20. 15.2 Impedance and the Phasor diagram
Inductive Reactance
Example 15.4 – solution (cont’d)
Inverse-transform;

21. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
By phasor transformation;
Where;

22. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
By Ohm’s law;

23. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Since i leads v by 90;
Hence;
Therefore;

24. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Example 15.5
Find i in the figure
Solution

25. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Example 15.5 – solution
Use phasor transformation;
By Ohm’s law;

26. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Example 15.5 – Solution (cont’d)
Inverse-transform;

27. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Example 15.6
Find v in the figure

28. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Example 15.6 – solution
Use phasor transformation;
By Ohm’s law;

29. 15.2 Impedance and the Phasor diagram
Capacitive Reactance
Example 15.6 – solution
Inverse- transform;

30. 15.2 Impedance and the Phasor diagram
Impedance Diagram
For any configuration (series, parallel, series-parallel, etc.), the angle associated with the total impedance is the angle by which the applied voltage leads the source current. For inductive networks, T will be positive, whereas for capacitive networks, T will be negative.

31. 15.2 Impedance and the Phasor diagram

32. 15.3 Series Configuration
The overall properties of series ac circuits are the same as those for dc circuits.
For instance, the total impedance of a system is the sum of the individual impedances:

33. 15.3 Series Configuration
In a series ac configuration having two impedances, the current is the same through each element (as it was for the series dc circuit) and is determined by Ohm’s Law:

34. 15.3 Series Configuration Example 15.7
Draw the impedance diagram and find the total impedance

35. 15.3 Series Configuration
Example 15.7 – solution
Impedance diagram

36. 15.3 Series Configuration Example 15.8
Draw the impedance diagram and find the total impedance

37. 15.3 Series Configuration
Example 15.8 – solution
Impedance diagram

38. 15.3 Series Configuration
Kirchhoff’s voltage law can be applied in the same manner as it is employed for a dc circuit
The power to the circuit can be determined by:
Where T is the phase angle between E and I.

39. 15.3 Series Configuration
R-L
Phasor Notation

40. 15.3 Series Configuration
R-L ZT
Impedance diagram:

41. 15.3 Series Configuration
R-L I

42. 15.3 Series Configuration
R-L
VR and VL

43. 15.3 Series Configuration
R-L
Kirchhoff’s voltage law
Or;

44. 15.3 Series Configuration R-L Kirchhoff’s voltage law
From the above calculation;

45. 15.3 Series Configuration
R-L
Phasor diagram

46. 15.3 Series Configuration
R-L
Power
Or;

47. 15.3 Series Configuration
R-L
Power factor

48. 15.3 Series Configuration
R-C
Phasor Notation

49. 15.3 Series Configuration
R-C ZT
Impedance diagram:

50. 15.3 Series Configuration
R-C E

51. 15.3 Series Configuration
R-C
VR and VC

52. 15.3 Series Configuration
R-C
Phasor diagram

53. 15.3 Series Configuration
R-C
Time domain

54. 15.3 Series Configuration
R-C
Time domain plot

55. 15.3 Series Configuration
R-C
Kirchhoff’s voltage law
Or;

56. 15.3 Series Configuration
R-C
Power
Or;

57. 15.3 Series Configuration
R-C
Power factor

58. 15.3 Series Configuration
R-L-C
TIME DOMAIN
PHASOR DOMAIN

59. 15.3 Series Configuration
R-L-C

60. 15.3 Series Configuration
R-L-C
Impedance diagram:

61. 15.3 Series Configuration
R-L-C

62. 15.3 Series Configuration
R-L-C

63. 15.3 Series Configuration
R-L-C

64. 15.3 Series Configuration
R-L-C

65. 15.3 Series Configuration
R-L-C
Phasor diagram

66. 15.3 Series Configuration
R-L-C
Time-domain plot

67. 15.3 Series Configuration
R-L-C
Total Power
Or;
Power factor

68. 15.3 Series Configuration R-L-C Example 15.11(a) I, VR, VL and VC;
Calculate
I, VR, VL and VC;

69. 15.3 Series Configuration R-L-C Example 15.11(a) – solution
Combined the R’s, L’s and C’s;

70. 15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d)
Transform the circuit into phasor domain;

71. 15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d)
Find the total impedance;

72. 15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d)
Apply Ohm’s law;

73. 15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d)
Apply Ohm’s law;

74. 15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d)
Apply Ohm’s law;

75. 15.3 Series Configuration R-L-C Example 15.11(a) – solution (cont’d)
Apply Ohm’s law;

76. 15.3 Series Configuration R-L-C Example 15.11(b) Calculate
the total power factor.

77. 15.3 Series Configuration R-L-C Example 15.11(b) – solution
The total power factor;

78. 15.3 Series Configuration R-L-C Example 15.11(c) Calculate
the total average power delivered to the circuit.

79. 15.3 Series Configuration R-L-C Example 15.11(c) – solution
The total average power delivered to the circuit.

80. 15.3 Series Configuration R-L-C Example 15.11(d)
Draw the phasor diagram;

81. 15.3 Series Configuration R-L-C Example 15.11(d) – solution
The phasor diagram;

82. 15.3 Series Configuration R-L-C Example 15.11(e)
Obtain the phasor sum of VR, VL and VC and show that it equals the input voltage E.

83. 15.3 Series Configuration
R-L-C
Example 15.11(e) – solution

84. 15.3 Series Configuration R-L-C Example 15.11(f)
Find VR and VC using voltage divider rule.

85. 15.3 Series Configuration
R-L-C
Example 15.11(f) – solution

86. 15.3 Series Configuration
R-L-C
Example 15.11(f) – solution

87. 15.6 Summaries of Series ac Circuits
For a series ac circuit with reactive elements:
The total impedance will be frequency dependent.
The impedance of any one element can be greater than the total impedance of the network.
The inductive and capacitive reactances are always in direct opposition on an impedance diagram.
Depending on the frequency applied, the same circuit can be either predominantly inductive or predominantly capacitive.

88. 15.6 Summaries of Series ac Circuits
For a series ac circuit with reactive elements:
At lower frequencies the capacitive elements will usually have the most impact on the total impedance, while at high frequencies the inductive elements will usually have the most impact.
The magnitude of the voltage across any one element can be greater than the applied voltage.

89. 15.6 Summaries of Series ac Circuits
The magnitude of the voltage across an element as compared to the other elements of the circuit is directly related to the magnitude of its impedance; that is, the larger the impedance of an element , the larger the magnitude of the voltage across the element.
The voltages across a coil or conductor are always in the direct opposition on a phasor diagram.

90. 15.6 Summaries of Series AC Circuits
The current is always in phase with the voltage across the resistive elements, lags the voltage across all the inductive elements by 90°, and leads the voltage across the capacitive elements by 90°.
The larger the resistive element of a circuit compared to the net reactive impedance, the closer the power factor is to unity.