1. T RANSIENTS AND S TEP R ESPONSES ELCT222- Lecture Notes University of S. Carolina Spring 2012

2. O UTLINE RC transients charging RC transients discharge RC transients Thevenin P-SPICE RL transients charging RL transients discharge Step responses. P-SPICE simulations Applications Reading: Boylestad Sections 10.5, 10.6, 10.7, 10.9,10.10 24.1-24.7

3. 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.

4. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.27 v C during the charging phase.

5. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.28 Universal time constant chart.

6. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE TABLE 10.3 Selected values of e -x.

7. 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: The larger R is, the lower the charging current, longer time to charge The larger C is, the more charge required for a given V, longer time.

8. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.29 Plotting the equation y C = E(1 – e - t/ t ) versus time (t).

9. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

10. FIG. 10.32 Revealing the short-circuit equivalent for the capacitor that occurs when the switch is first closed.

11. 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.

12. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

13. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE U SING THE C ALCULATOR TO S OLVE E XPONENTIAL F UNCTIONS FIG. 10.34 Calculator key strokes to determine e -1.2.

14. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE U SING THE C ALCULATOR TO S OLVE E XPONENTIAL F UNCTIONS FIG. 10.35 Transient network for Example 10.6.

15. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE U SING THE C ALCULATOR TO S OLVE E XPONENTIAL F UNCTIONS FIG. 10.36 v C versus time for the charging network in Fig. 10.35.

16. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE U SING THE C ALCULATOR TO S OLVE E XPONENTIAL F UNCTIONS FIG. 10.37 Plotting the waveform in Fig. 10.36 versus time (t).

17. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE U SING THE C ALCULATOR TO S OLVE E XPONENTIAL F UNCTIONS FIG. 10.38 i C and y R for the charging network in Fig. 10.36.

18. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE We now investigate how to discharge a capacitor while exerting some control on 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—unless, of course, you are trained in the maneuver.

19. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG. 10.39 (a) Charging network; (b) discharging configuration.

20. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE For the voltage across the capacitor that is decreasing with time, the mathematical expression is:

21. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG. 10.40 y C, i C, and y R for 5 t switching between contacts in Fig. 10.39(a).

22. 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.

23. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE

24. FIG. 10.43 Effect of increasing values of C (with R constant) on the charging curve for v C.

25. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE FIG. 10.44 Network to be analyzed in Example 10.8.

26. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE FIG. 10.45 v C and i C for the network in Fig. 10.44.

27. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE FIG. 10.46 Network to be analyzed in Example 10.9. FIG. 10.47 The charging phase for the network in Fig. 10.46.

28. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE FIG. 10.48 Network in Fig. 10.47 when the switch is moved to position 2 at t = 1 t 1.

29. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE FIG. 10.49 v C for the network in Fig. 10.47.

30. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE T HE E FFECT OF ON THE R ESPONSE FIG. 10.50 i c for the network in Fig. 10.47.

31. 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.

32. INITIAL CONDITIONS FIG. 10.52 Example 10.10.

33. INITIAL CONDITIONS FIG. 10.53 v C and i C for the network in Fig. 10.52.

34. INITIAL CONDITIONS FIG. 10.54 Defining the parameters in Eq. (10.21) for the discharge phase.

35. INSTANTANEOUS VALUES Occasionally, you may need to determine the voltage or current at a particular instant of time that is not an integral multiple of t. FIG. 10.55 Key strokes to determine (2 ms)(log e 2) using the TI-89 calculator.

36. THÉVENIN EQUIVALENT: T = R T H 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.

37. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.56 Example 10.11.

38. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.57 Applying Thévenin’s theorem to the network in Fig. 10.56.

39. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.58 Substituting the Thévenin equivalent for the network in Fig. 10.56.

40. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.59 The resulting waveforms for the network in Fig. 10.56.

41. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.60 Example 10.12. FIG. 10.61 Network in Fig. 10.60 redrawn.

42. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.62 y C for the network in Fig. 10.60.

43. THÉVENIN EQUIVALENT: T = R T H C FIG. 10.63 Example 10.13.

44. 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:

45. THE CURRENT I C FIG. 10.64 v C for Example 10.14.

46. THE CURRENT I C FIG. 10.65 The resulting current i C for the applied voltage in Fig. 10.64.

47. I NDUCTORS

48. R-L TRANSIENTS: THE STORAGE PHASE The storage waveforms have the same shape, and time constants are defined for each configuration. Because these concepts are so similar (refer to Section 10.5 on the charging of a capacitor), you have an opportunity to reinforce concepts introduced earlier and still learn more about the behavior of inductive elements.

49. R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.31 Basic R-L transient network. Remember, for an inductor

50. R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.32 i L, y L, and y R for the circuit in Fig. 11.31 following the closing of the switch.

51. R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.33 Effect of L on the shape of the i L storage waveform. τ=L/R If L is large, more flux needed for transient If R is large, i L is small, Δi L small, fast.

52. R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.34 Circuit in Figure 11.31 the instant the switch is closed. Fast times, open circuit High Frequency, open circuit Long times, short circuit Low Frequency, short circuit Because it’s a wound wire. Current cannot change instantly. Why?

53. R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.35 Circuit in Fig. 11.31 under steady-state conditions. FIG. 11.36 Series R-L circuit for Example 11.3.

54. R-L TRANSIENTS: THE STORAGE PHASE FIG. 11.37 i L and v L for the network in Fig. 11.36.

55. INITIAL CONDITIONS Since the current through a coil cannot change instantaneously, the current through a coil begins the transient phase at the initial value established by the network (note Fig. 11.38) before the switch was closed. It then passes through the transient phase until it reaches the steady-state (or final) level after about five time constants. The steadystate level of the inductor current can be found by substituting its shortcircuit equivalent (or R l for the practical equivalent) and finding the resulting current through the element.

56. INITIAL CONDITIONS FIG. 11.38 Defining the three phases of a transient waveform.

57. INITIAL CONDITIONS FIG. 11.39 Example 11.4.

58. INITIAL CONDITIONS FIG. 11.40 i L and v L for the network in Fig. 11.39.

59. R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.41 Demonstrating the effect of opening a switch in series with an inductor with a steady-state current.

60. Current wants to go to 0, but cannot, produces a spark due to large di L /dt Introduce intentional “discharge” path. R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.43 Network in Fig. 11.42 the instant the switch is opened.

61. R-L TRANSIENTS: THE RELEASE PHASE FIG. 11.42 Initiating the storage phase for an inductor by closing the switch.

62. FIG. 11.43 Network in Fig. 11.42 the instant the switch is opened. R-L TRANSIENTS: THE RELEASE PHASE Apply KVL, and remember that i L cannot change instantly. Why? τ’=L/(R 1 +R 2 ) You can write this down. How?

63. R-L TRANSIENTS: THE RELEASE PHASE

64. FIG. 11.45 The various voltages and the current for the network in Fig. 11.44.

65. S TEP R ESPONSES

66. OBJECTIVES Become familiar with the specific terms that define a pulse waveform and how to calculate various parameters such as the pulse width, rise and fall times, and tilt. Be able to calculate the pulse repetition rate and the duty cycle of any pulse waveform. Become aware of the parameters that define the response of an R-C network to a square-wave input. Understand how a compensator probe of an oscilloscope is used to improve the appearance of an output pulse waveform.

67. IDEAL VERSUS ACTUAL The ideal pulse in Fig. 24.1 has vertical sides, sharp corners, and a flat peak characteristic; it starts instantaneously at t 1 and ends just as abruptly at t 2. FIG. 24.1 Ideal pulse waveform.

68. IDEAL VERSUS ACTUAL FIG. 24.2 Actual pulse waveform.

69. IDEAL VERSUS ACTUAL Amplitude Pulse Width Base-Line Voltage Positive-Going and Negative-Going Pulses Rise Time (t r ) and Fall Time (t f ) Tilt

70. IDEAL VERSUS ACTUAL FIG. 24.3 Defining the base- line voltage.

71. IDEAL VERSUS ACTUAL FIG. 24.4 Positive- going pulse.

72. IDEAL VERSUS ACTUAL FIG. 24.5 Defining t r and t f.

73. IDEAL VERSUS ACTUAL FIG. 24.6 Defining tilt.

74. IDEAL VERSUS ACTUAL FIG. 24.7 Defining preshoot, overshoot, and ringing.

75. IDEAL VERSUS ACTUAL FIG. 24.8 Example 24.1.

76. IDEAL VERSUS ACTUAL FIG. 24.9 Example 24.2.

77. PULSE REPETITION RATE AND DUTY CYCLE A series of pulses such as those appearing in Fig. 24.10 is called a pulse train. The varying widths and heights may contain information that can be decoded at the receiving end. If the pattern repeats itself in a periodic manner as shown in Fig. 24.11(a) and (b), the result is called a periodic pulse train.

78. PULSE REPETITION RATE AND DUTY CYCLE FIG. 24.11 Periodic pulse trains. % of time voltage is high

79. PULSE REPETITION RATE AND DUTY CYCLE FIG. 24.12 Example 24.3.

80. PULSE REPETITION RATE AND DUTY CYCLE FIG. 24.13 Example 24.4.

81. PULSE REPETITION RATE AND DUTY CYCLE FIG. 24.14 Example 24.5.

82. AVERAGE VALUE The average value of a pulse waveform can be determined using one of two methods. The first is the procedure outlined in Section 13.7, which can be applied to any alternating waveform. The second can be applied only to pulse waveforms since it utilizes terms specifically related to pulse waveforms; that is,

83. AVERAGE VALUE FIG. 24.15 Example 24.6.

84. AVERAGE VALUE FIG. 24.16 Solution to part (b) of Example 24.7.

85. AVERAGE VALUE I NSTRUMENTATION The average value (dc value) of any waveform can be easily determined using the oscilloscope. If the mode switch of the scope is set in the ac position, the average or dc component of the applied waveform is blocked by an internal capacitor from reaching the screen. FIG. 24.17 Determining the average value of a pulse waveform using an oscilloscope.

86. TRANSIENT R-C NETWORKS In Chapter 10, the general solution for the transient behavior of an R-C network with or without initial values was developed. The resulting equation for the voltage across a capacitor is repeated here for convenience:

87. TRANSIENT R-C NETWORKS FIG. 24.18 Defining the parameters of Eq. (24.6).

88. TRANSIENT R-C NETWORKS FIG. 24.19 Example of the use of Eq. (24.6).

89. TRANSIENT R-C NETWORKS FIG. 24.20 Example 24.8.

90. TRANSIENT R-C NETWORKS FIG. 24.21 y C and i C for the network in Fig. 24.20.

91. TRANSIENT R-C NETWORKS FIG. 24.22 Example 24.9.

92. TRANSIENT R-C NETWORKS FIG. 24.23 v C for the network in Fig. 24.22.

93. R-C RESPONSE TO SQUARE-WAVE INPUTS The square wave in Fig. 24.24 is a particular form of pulse waveform. It has a duty cycle of 50% and an average value of zero volts, as calculated as follows: FIG. 24.24 Periodic square wave.

94. R-C RESPONSE TO SQUARE-WAVE INPUTS FIG. 24.25 Raising the base-line voltage of a square wave to zero volts.

95. R-C RESPONSE TO SQUARE-WAVE INPUTS FIG. 24.26 Applying a periodic square-wave pulse train to an R-C network.

96. T/2 > 5 T

97. T/2 = 5 T

98. T / 2 < 5 T

99. FIG. 24.30 v C for T/2 << 5t or T << 10t.

100. R-C RESPONSE TO SQUARE-WAVE INPUTS FIG. 24.31 Example 24.10.

101. R-C RESPONSE TO SQUARE-WAVE INPUTS FIG. 24.32 v C for the R-C network in Fig. 24.31.

102. R-C RESPONSE TO SQUARE-WAVE INPUTS FIG. 24.33 i C for the R-C network in Fig. 24.31.

103. R-C RESPONSE TO SQUARE-WAVE INPUTS

105. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE The X10 attenuator probe used with oscilloscopes is designed to reduce the magnitude of the input voltage by a factor of 10. If the input impedance to a scope is 1 MΩ, the X10 attenuator probe will have an internal resistance of 9 MΩ, as shown in Fig. 24.36.

106. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.36 X10 attenuator probe.

107. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.37 Capacitive elements present in an attenuator probe arrangement.

108. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.38 Equivalent network in Fig. 24.37.

109. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.39 Thévenin equivalent for C i in Fig. 24.38.

110. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.40 The scope pattern for the conditions in Fig. 24.38 with v t = 200 V peak.

111. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.41 Commercial compensated 10 : 1 attenuator probe. (Courtesy of Tektronix, Inc.)

112. OSCILLOSCOPE ATTENUATOR AND COMPENSATING PROBE FIG. 24.42 Compensated attenuator and input impedance to a scope, including the cable capacitance. Compensating delay pre-delays the input signal Allows scope electronics time to “catch up” And remove the RC charging distortion we saw previously.