1. The Basic Elements and Phasors

2. OBJECTIVES
Become familiar with the response of a resistor, an inductor, and a capacitor to the application of a sinusoidal voltage or current.
Learn how to apply the phasor format to add and subtract sinusoidal waveforms.
Understand how to calculate the real power to resistive elements and the reactive power to inductive and capacitive elements.
Become aware of the differences between the frequency response of ideal and practical elements.
Become proficient in the use of a calculator to work with complex numbers.

3. DERIVATIVE
To understand the response of the basic R, L, and C elements to a sinusoidal signal, you need to examine the concept of the derivative in some detail.
Recall from Section that the derivative dx/dt is defined as the rate of change of x with respect to time.
If x fails to change at a particular instant, dx = 0, and the derivative is zero.

4. DERIVATIVE
FIG Defining those points in a sinusoidal waveform that have maximum and minimum derivatives.

5. DERIVATIVE
FIG Derivative of the sine wave of Fig

6. DERIVATIVE
FIG Effect of frequency on the peak value of the derivative.

7. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
Resistor
Inductor
Capacitor
FIG Determining the sinusoidal response for a resistive element.

8. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG The voltage and current of a resistive element are in phase.

9. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG Defining the opposition of an element to the flow of charge through the element.
FIG Defining the parameters that determine the opposition of an inductive element to the flow of charge.

10. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG Investigating the sinusoidal response of an inductive element.
FIG For a pure inductor, the voltage across the coil leads the current through the coil by 90°.

11. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG Defining the parameters that determine the opposition of a capacitive element to the flow of charge.

12. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG The current of a purely capacitive element leads the voltage across the element by 90°.
FIG Investigating the sinusoidal response of a capacitive element.

13. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG Example 14.1(a).
FIG Example 14.1(b).

14. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG Example 14.3(b).
FIG Example 14.3(a).

15. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
FIG Example 14.5.
FIG Example 14.7.

16. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Ideal Response
Resistor R
Inductor L
Capacitor C

17. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Ideal Response
FIG XL versus frequency.
FIG R versus f for the range of interest.

18. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Ideal Response
FIG Effect of low and high frequencies on the circuit model of an inductor.

19. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Ideal Response
FIG XC versus frequency.

20. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Ideal Response
FIG Effect of low and high frequencies on the circuit model of a capacitor.

21. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Practical Response
Resistor R
Inductor L
Capacitor C
ESR

22. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Practical Response
FIG Typical resistance-versus-frequency curves for carbon composition resistors.

23. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Practical Response
FIG Practical equivalent for an inductor.
FIG ZL versus frequency for the practical inductor equivalent of Fig

24. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Practical Response
FIG Practical equivalent for a capacitor; (a) network; (b) response.

25. FREQUENCY RESPONSE OF THE BASIC ELEMENTS Practical Response
FIG ESR. (a) Impact on equivalent model; (b) Measuring instrument.

26. AVERAGE POWER AND POWER FACTOR
FIG Demonstrating that power is delivered at every instant of a sinusoidal voltage waveform (except vR = 0V).

27. AVERAGE POWER AND POWER FACTOR
FIG Power versus time for a purely resistive load.

28. AVERAGE POWER AND POWER FACTOR
FIG Determining the power delivered in a sinusoidal ac network.

29. AVERAGE POWER AND POWER FACTOR
FIG Defining the average power for a sinusoidal ac network.

30. AVERAGE POWER AND POWER FACTOR
Resistor
Inductor
Capacitor
Power Factor

31. AVERAGE POWER AND POWER FACTOR
FIG Purely inductive load with Fp = 0.
FIG Purely resistive load with Fp = 1.

32. AVERAGE POWER AND POWER FACTOR
FIG Example 14.12(a).
FIG Example 14.12(b).

33. AVERAGE POWER AND POWER FACTOR
FIG Example 14.12(c).

34. COMPLEX NUMBERS
A complex number represents a point in a two-dimensional plane located with reference to two distinct axes.
This point can also determine a radius vector drawn from the origin to the point.
The horizontal axis is called the real axis, while the vertical axis is called the imaginary axis.

35. COMPLEX NUMBERS
FIG Defining the real and imaginary axes of a complex plane.

36. RECTANGULAR FORM
The format for the rectangular form is:

37. RECTANGULAR FORM FIG. 14.39 Defining the rectangular form.
FIG Example 14.13(a).

38. RECTANGULAR FORM FIG. 14.41 Example 14.13(b).
FIG Example 14.13(c).
FIG Example 14.13(b).

39. POLAR FORM
The format for the polar form is:

40. POLAR FORM FIG. 14.43 Defining the polar form.
FIG Demonstrating the effect of a negative sign on the polar form.
FIG Defining the polar form.

41. POLAR FORM
FIG Example 14.14(b).
FIG Example 14.14(a).

42. POLAR FORM
FIG Example 14.14(c).

43. CONVERSION BETWEEN FORMS
Rectangular to Polar
Polar to Rectangular
FIG Conversion between forms.

44. CONVERSION BETWEEN FORMS
FIG Example
FIG Example

45. CONVERSION BETWEEN FORMS
FIG Example
FIG Example

46. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS
Complex Conjugate
Reciprocal
Addition
Subtraction
Multiplication
Division

47. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS
FIG Defining the complex conjugate of a complex number in rectangular form.
FIG Defining the complex conjugate of a complex number in polar form.

48. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS
FIG Example 14.19(b).
FIG Example 14.19(a).

49. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS
FIG Example 14.20(b).
FIG Example 14.20(a).

50. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS
FIG Example 14.21(a).
FIG Example 14.21(b).

51. CALCULATOR METHODS WITH COMPLEX NUMBERS Calculators
FIG TI-89 scientific calculator.

52. CALCULATOR METHODS WITH COMPLEX NUMBERS Calculators
FIG Setting the DEGREE mode on the TI-89 calculator.

53. CALCULATOR METHODS WITH COMPLEX NUMBERS Calculators
Rectangular to Polar Conversion
FIG Converting 3 + j 5 to the polar form using the TI-89 calculator.

54. CALCULATOR METHODS WITH COMPLEX NUMBERS Calculators
Polar to Rectangular Conversion
FIG Converting 5<3.1° to the rectangular form using the TI 89 calculator.

55. CALCULATOR METHODS WITH COMPLEX NUMBERS Calculators
Mathematical Operations
FIG Performing the operation (10< 50°)(2< 20°).

56. CALCULATOR METHODS WITH COMPLEX NUMBERS Calculators
FIG Performing the operation (5<3.1°)(2 + j 2).
FIG Verifying the results of Example 14.26(c).

57. PHASORS
FIG Adding two sinusoidal waveforms on a point-by-point basis.

58. PHASORS
FIG (a) The phasor representation of the sinusoidal waveforms of part (b); (b) finding the sum of two sinusoidal waveforms of v1 and v2.

59. PHASORS
FIG Adding two sinusoidal currents with phase angles other than 90°.

60. PHASORS
FIG Example

61. PHASORS
FIG Solution to Example

62. PHASORS
FIG Example

63. PHASORS
FIG Solution to Example

64. COMPUTER ANALYSIS PSpice
FIG Using PSpice to analyze the response of a capacitor to a sinusoidal ac signal.

65. COMPUTER ANALYSIS PSpice
FIG A plot of the voltage, current, and power for the capacitor in Fig

66. COMPUTER ANALYSIS Multisim
FIG Using Multisim to review the response of an inductive element to a sinusoidal ac signal.