1. Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] Introductory Circuit Analysis, 12/e Boylestad Chapter 10 Capacitors

2. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] OBJECTIVES Become familiar with the basic construction of a capacitor and the factors that affect its ability to store charge on its plates. Be able to determine the transient (time-varying) response of a capacitive network and plot the resulting voltages and currents. Understand the impact of combining capacitors in series or parallel and how to read the nameplate data. Develop some familiarity with the use of computer methods to analyze networks with capacitive elements.

3. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INTRODUCTION The capacitor has a significant impact on the types of networks that you will be able to design and analyze. Like the resistor, it is a two-terminal device, but its characteristics are totally different from those of a resistor. In fact, the capacitor displays its true characteristics only when a change in the voltage or current is made in the network.

4. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THE ELECTRIC FIELD FIG. 10.1 Flux distribution from an isolated positive charge. Electric field (E) ⇨ electric flux lines ⇨ to indicate the strength of E at any point around the charged body. Denser flux lines ⇨ stronger E.

5. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THE ELECTRIC FIELD FIG. 10.2 Determining the force on a unit charge r meters from a charge Q of similar polarity.

6. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THE ELECTRIC FIELD FIG. 10.3 Electric flux distributions: (a) opposite charges; (b) like charges. Electric flux lines always extend from a +ve charged body to a -ve charged body, ⊥ to the charged surfaces, and never intersect.

7. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITANCE FIG. 10.4 Fundamental charging circuit. ⇨ V=IR

8. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITANCE FIG. 10.7 Effect of a dielectric on the field distribution between the plates of a capacitor: (a) alignment of dipoles in the dielectric; (b) electric field components between the plates of a capacitor with a dielectric present.

9. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITANCE TABLE 10.1 Relative permittivity (dielectric constant) Σr of various dielectrics.

10. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITOR Construction FIG. 10.9 Example 10.2. ⇨ R =ρL/A

11. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS Types of Capacitors Capacitors, like resistors, can be listed under two general headings: fixed and variable. FIG. 10.11 Symbols for the capacitor: (a) fixed; (b) variable.

12. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS Types of Capacitors FIG. 10.12 Demonstrating that, in general, for each type of construction, the size of a capacitor increases with the capacitance value: (a) electrolytic; (b) polyester- film; (c) tantalum.

13. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS Types of Capacitors FIG. 10.20 Variable capacitors: (a) air; (b) air trimmer; (c) ceramic dielectric compression trimmer. [(a) courtesy of James Millen Manufacturing Co.] Variable Capacitors –All the parameters can be changed to create a variable capacitor. –For example; the capacitance of the variable air capacitor is changed by turning the shaft at the end of the unit.

14. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS Leakage Current and ESR FIG. 10.21 Leakage current: (a) including the leakage resistance in the equivalent model for a capacitor; (b) internal discharge of a capacitor due to the leakage current.

15. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS Capacitor Labeling FIG. 10.23 Various marking schemes for small capacitors.

16. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS Measurement and Testing of Capacitors The capacitance of a capacitor can be read directly using a meter such as the Universal LCR Meter. FIG. 10.24 Digital reading capacitance meter. (Courtesy of B+K Precision.)

17. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE The placement of charge on the plates of a capacitor does not occur instantaneously. Instead, it occurs over a period of time determined by the components of the network. FIG. 10.26 Basic R-C charging network.

18. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.27 v C during the charging phase. The current ( i c ) through a capacitive network is essentially zero after five time constants of the capacitor charging phase.

19. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.28 Universal time constant chart.

20. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE TABLE 10.3 Selected values of e -x.

21. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE The factor t, called the time constant of the network, has the units of time, as shown below using some of the basic equations introduced earlier in this text:

22. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.29 Plotting the equation y C = E(1 – e -t/t ) versus time (t).

23. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

24. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.31 Demonstrating that a capacitor has the characteristics of an open circuit after the charging phase has passed.

25. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.32 Revealing the short-circuit equivalent for the capacitor that occurs when the switch is first closed.

26. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

27. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG. 10.35 Transient network for Example 10.6.

28. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG. 10.36 v C versus time for the charging network in Fig. 10.35.

29. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG. 10.37 Plotting the waveform in Fig. 10.36 versus time (t).

30. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG. 10.38 i C and y R for the charging network in Fig. 10.36.

31. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE How to discharge a capacitor and how long the discharge time will be. You can, of course, place a lead directly across a capacitor to discharge it very quickly—and possibly cause a visible spark. For larger capacitors such those in TV sets, this procedure should not be attempted because of the high voltages involved.

32. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG. 10.39 (a) Charging network; (b) discharging configuration. For the voltage across the capacitor that is decreasing with time, the mathematical expression is:

33. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG. 10.40 y C, i C, and y R for 5t switching between contacts in Fig. 10.39(a).

34. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG. 10.41 v C and i C for the network in Fig. 10.39(a) with the values in Example 10.6.

35. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response

36. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.43 Effect of increasing values of C (with R constant) on the charging curve for v C.

37. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.44 Network to be analyzed in Example 10.8.

38. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.45 v C and i C for the network in Fig. 10.44.

39. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.46 Network to be analyzed in Example 10.9. FIG. 10.47 The charging phase for the network in Fig. 10.46.

40. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.48 Network in Fig. 10.47 when the switch is moved to position 2 at t = 1t 1.

41. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.49 v C for the network in Fig. 10.47.

42. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.50 i c for the network in Fig. 10.47.

43. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS The voltage across the capacitor at this instant is called the initial value, as shown for the general waveform in Fig. 10.51. FIG. 10.51 Defining the regions associated with a transient response.

44. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS FIG. 10.52 Example 10.10.

45. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS FIG. 10.53 v C and i C for the network in Fig. 10.52.

46. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] INITIAL CONDITIONS FIG. 10.54 Defining the parameters in Eq. (10.21) for the discharge phase.

47. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C You may encounter instances in which the network does not have the simple series form in Fig. 10.26. You then need to find the Thévenin equivalent circuit for the network external to the capacitive element.

48. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.56 Example 10.11.

49. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.57 Applying Thévenin’s theorem to the network in Fig. 10.56.

50. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.58 Substituting the Thévenin equivalent for the network in Fig. 10.56.

51. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.59 The resulting waveforms for the network in Fig. 10.56.

52. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.60 Example 10.12. FIG. 10.61 Network in Fig. 10.60 redrawn.

53. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.62 yC for the network in Fig. 10.60.

54. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN EQUIVALENT: t =R Th C FIG. 10.63 Example 10.13.

55. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THE CURRENT i C There is a very special relationship between the current of a capacitor and the voltage across it. For the resistor, it is defined by Ohm’s law: i R = v R /R. The current through and the voltage across the resistor are related by a constant R—a very simple direct linear relationship. For the capacitor, it is the more complex relationship defined by:

56. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THE CURRENT i C FIG. 10.64 v C for Example 10.14.

57. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THE CURRENT i C FIG. 10.65 The resulting current i C for the applied voltage in Fig. 10.64.

58. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL Capacitors, like resistors, can be placed in series and in parallel. Increasing levels of capacitance can be obtained by placing capacitors in parallel, while decreasing levels can be obtained by placing capacitors in series.

59. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.66 Series capacitors.

60. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.67 Parallel capacitors.

61. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.68 Example 10.15. FIG. 10.69 Example 10.16.

62. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.70 Example 10.17. FIG. 10.71 Reduced equivalent for the network in Fig. 10.70.

63. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.72 Example 10.18. FIG. 10.73 Determining the final (steady-state) value for yC.

64. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.74 Example 10.19.

65. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] ENERGY STORED BY A CAPACITOR An ideal capacitor does not dissipate any of the energy supplied to it. It stores the energy in the form of an electric field between the conducting surfaces. A plot of the voltage, current, and power to a capacitor during the charging phase is shown in Fig. 10.75. The power curve can be obtained by finding the product of the voltage and current at selected instants of time and connecting the points obtained. The energy stored is represented by the shaded area under the power curve.

66. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] ENERGY STORED BY A CAPACITOR FIG. 10.75 Plotting the power to a capacitive element during the transient phase.

67. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS Touch Pad FIG. 10.77 Laptop touch pad.

68. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS Touch Pad FIG. 10.78 Matrix approach to capacitive sensing in a touch pad.

69. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS Flash Lamp FIG. 10.81 Flash camera: general appearance.

70. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS Flash Lamp FIG. 10.82 Flash camera: basic circuitry.

71. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] APPLICATIONS Flash Lamp FIG. 10.83 Flash camera: internal construction.