CHAPTER 7 Response of First-Order RL and RC Circuits
RL and RC Circuits RL (resistor inductor) and RC (resistor-capacitor) circuits. Figure 7.1 The two forms of the circuits for natural response. (a) RL circuit. (b) RC circuit.
Figure 7.2 Four possible first-order circuits. (a) An inductor connected to a Thévenin equivalent. (b) An inductor connected to a Norton equivalent. (c) A capacitor connected to a Thévenin equivalent. (d) A capacitor connected to a Norton equivalent.
7.1 The Natural Response of an RL Circuit Figure 7.3 An RL circuit. Figure 7.4 Te circuit shown in Fig. 7.3, for t 0.
Deriving the Expression for the Current The highest order derivative appearing in the equation is 1; hence the term first-order.
Initial inductor current
Natural response of an RL circuit Figure 7.5 The current response for the circuit shown in Fig. 7.4.
The Significance of the Time Constant Time constant for RL circuit
Transient response and steady-state response A momentary event and is referred to as the transient response of the circuit. The response that exists a long time after the switching has taken place is called the steady-state response. Figure 7.6 A graphic interpretation of the time constant of the RL circuit shown in Fig. 7.4.
Calculating the natural response of an RL circuit
Example 7.1 The switch in the circuit shown in Fig. 7.7 has been closed for a long time before it is opened at Find a) i L (t) for t 0, b) i o (t) for t 0 +, c) v o (t) for t 0 +, d) the percentage of the total energy stored in the 2 H inductor that is dissipated in the 10 Ω resistor. Figure 7.7 The circuit for Example 7.1.
Example 7.1
Example 7.1
Example 7.1
Example 7.2 In the circuit shown in Fig. 7.8, the initial currents in inductors L 1 and L2 have been established by sources not shown. The switch is opened at t = 0. a) Find i 1, i 2, and i 3 for t 0. b) Calculate the initial energy stored in the parallel inductors. c) Determine how much energy is stored in the inductors as t. d) Show that the total energy delivered to the resistive network equals the difference between the results obtained in (b) and (c).
Example 7.2 Figure 7.8 The circuit for Example 7.2.
Example 7.2
Example 7.2 Figure 7.9 A simplification of the circuit shown in Fig. 7.8.
Example 7.2
Example 7.2
7.2 The Natural Response of an RC Circuit The natural response of an RC circuit is developed from the circuit shown in Fig. 7.10. Figure 7.10 An RC circuit. Figure 7.11 The circuit shown in Fig. 7.10, after switching.
Deriving the Expression for the Voltage Initial capacitor voltage
Time constant for RC circuit Natural response of an RC circuit Figure 7.12 The natural response of an RC circuit.
Calculating the natural response of an RC circuit
Example 7.3 The switch in the circuit shown in Fig. 7.13 has been in position x for a long time. At t = 0 the switch moves instantaneously to position y. Find a) v C (t) for t 0, b) v o (t) for t 0 +, c) i o (t) for t 0 +, d) the total energy dissipated in the 60 kω resistor. Figure 7.13 The circuit for Example 7.3.
Example 7.3
Example 7.3
Example 7.4 The initial voltages on capacitors C 1 and C 2 in the circuit shown in Fig. 7.14 have been established by sources not shown. The switch is closed at t = 0. a) Find v 1 (t), v 2 (t), and v(t) for t 0 and i(t) for for t 0 +. b) Calculate the initial energy stored in the capacitors C 1 and C 2. c) Determine how much energy is stored in the capacitors as t. d) Show that the total energy delivered to the 250 kω resistor is the difference between the results obtained in (b) and (c).
Example 7.4 Figure 7.14 The circuit for Example 7.4.
Example 7.4
Example 7.4 Figure 7.15 A simplification of the circuit shown in Fig. 7.14.
Example 7.4
Example 7.4
Example 7.4
7.3 The Step Response of RL and RC Circuits The Step Response of an RL Circuit Figure 7.16 A circuit used to illustrate the step response of a first-order RL circuit.
Step response of RL circuit
When the initial energy in the inductor is zero One time constant after the switch has been closed, the current will have reached approximately 63% of its final value,
Figure 7.17 The step response of the RL circuit shown in Fig. 7.16 when I 0 = 0.
Figure 7.18 Inductor voltage versus time.
Example 7.5 The switch in the circuit shown in Fig. 7.19 has been in position a for a long time. At t = 0, the switch moves from position a to position b. The switch is a make-before-break type; that is, the connection at position b is established before the connection at position a is broken, so there is no interruption of current through the inductor. a) Find the expression for i(t) for t 0. b) What is the initial voltage across the inductor just after the switch has been moved to position b?
Example 7.5 c) How many milliseconds after the switch has been moved does the inductor voltage equal 24 V? d) Does this initial voltage make sense in terms of circuit behavior? e) Plot both i(t) and v(t) versus t. Figure 7.19 The circuit for Example 7.5.
Example 7.5
Example 7.5
Example 7.5
Example 7.5 Figure 7.20 The current and voltage waveforms for Example 7.5.
The Step Response of an RC Circuit Figure 7.21 A circuit used to illustrate the step response of a first-order RC circuit.
Example 7.6 The switch in the circuit shown in Fig. 7.22 has been in position 1 for a long time. At t = 0 the switch moves to position 2. Find a) v o (t) for t 0 and b) i o (t) for t 0 +. Figure 7.22 The circuit for Example 7.6.
Example 7.6
Example 7.6
Example 7.6
Example 7.6 Figure 7.23 The equivalent circuit for t > 0 for the circuit shown in Fig. 7.22.
7.4 A General Solution for Step and Natural Responses Figure 7.24 Four possible first-order circuits. Figure (a) 7.24 An inductor Four possible connected first-order to a Thévenin circuits. equivalent. (a) An inductor connected to a Thévenin (b) An inductor equivalent. connected (b) to An a Norton inductor equivalent. connected to a Norton equivalent. (c) A (c) capacitor A capacitor connected connected to a Thévenin to equivalent. a Thévenin equivalent. (d) A capacitor (d) connected A capacitor connected to a Norton to a Norton equivalent. equivalent
General solution for natural and step responses of RL and RC circuits
Calculating the natural or step response of RL or RC circuits
Example 7.7 The switch in the circuit shown in Fig. 7.25 has been in position a for a long time. At t = 0 the switch is moved to position b. a) What is the initial value of v C? b) What is the final value of v C? c) What is the time constant of the circuit when the switch is in position b? d) What is the expression for v C (t) when t 0? e) What is the expression for i(t) when t 0 +?
Example 7.7 f) How long after the switch is in position b does the capacitor voltage equal zero? g) Plot v C (t) and i(t) versus t. Figure 7.25 The circuit for Example 7.7.
Example 7.7
Example 7.7
Example 7.7
Example 7.7
Example 7.7 Figure 7.26 The current and voltage waveforms for Example 7.7.
Example 7.8 The switch in the circuit shown in Fig. 7.27 has been open for a long time. The initial charge on the capacitor is zero. At t = 0 the switch is closed. Find the expression for a) i(t) for t 0 + and b) v(t) when t 0 +. Figure 7.27 The circuit for Example 7.8.
Example 7.8
Example 7.8
Example 7.8
Example 7.9 The switch in the circuit shown in Fig. 7.28 has been open for a long time. At t = 0 the switch is closed. Find the expression for a) v(t) for t 0 + and b) i(t) when t 0. Figure 7.28 The circuit for Example 7.9.
Example 7.9
Example 7.9
Example 7.10 There is no energy stored in the circuit in Fig. 7.29 at the time the switch is closed. a) Find the solutions for i o, v o, i 1 and i 2. b) Show that the solutions obtained in (a) make sense in terms of known circuit behavior. Figure 7.29 The circuit for Example 7.10.
Example 7.10 Figure 7.30 The circuit in Fig. 7.29 with the magnetically coupled coils replaced by an equivalent coil.
Example 7.10
Example 7.10
Example 7.10
Example 7.10
Example 7.10
7.5 Sequential Switching Whenever switching occurs more than once in a circuit, we have sequential switching. For example, a single, two-position switch may be switched back and forth, or multiple switches may be opened or closed in sequence. Recall that anything but inductive currents and capacitive voltages can change instantaneously at the time of switching. Thus solving first for inductive currents and capacitive voltages is even more pertinent in sequential switching problems.
Example 7.11 The two switches in the circuit shown in Fig. 7.31 have been closed for a long time. At t = 0 switch 1 is opened.then, 35 ms later, switch 2 is opened. a) Find il(t) for 0 t 35 ms. b) Find il for t 35 ms. c) What percentage of the initial energy stored in the 150 mh inductor is dissipated in the 18 Ω resistor? d) Repeat (c) for the 3 Ω resistor. e) Repeat (c) for the 6 Ω resistor.
Example 7.11 Figure 7.31 The circuit for Example 7.11. Figure 7.32 The circuit shown in Fig. 7.31, for t > 0.
Example 7.11
Example 7.11 Figure 7.33 The circuit shown in Fig. 7.31, for 0 t 35 ms. Figure 7.34 The circuit shown in Fig. 7.31, for t 35 ms.
Example 7.11
Example 7.11
Example 7.11
Example 7.11
Example 7.11
Example 7.11
Example 7.11
Example 7.12 The uncharged capacitor in the circuit shown in Fig. 7.35 is initially switched to terminal a of the threeposition switch. At t = 0, the switch is moved to position b, where it remains for 15 ms. After the 15 ms delay, the switch is moved to position c, where it remains indefinitely. a) Derive the numerical expression for the voltage across the capacitor. b) Plot the capacitor voltage versus time. c) When will the voltage on the capacitor equal 200 V?
Example 7.12 Figure 7.35 The circuit for Example 7.12.
Example 7.12
Example 7.12
Example 7.12 Figure 7.36 The capacitor voltage for Example 7.12.
Example 7.12
7.6 Unbounded Response A circuit response may grow, rather than decay, exponentially with time. This type of response, called an unbounded response, is possible if the circuit contains dependent sources. Thévenin equivalent resistance with respect to the terminals of either an inductor or a capacitor may be negative. This negative resistance generates a negative time constant, and the resulting currents and voltages increase without limit. In an actual circuit, the response eventually reaches a limiting value when a component breaks down or goes into a saturation state, prohibiting further increases in voltage or current.
Rather than using the step response solution given in Eq. 7.59, we derive the differential equation that describes the circuit containing the negative resistance and then solve it using the separation of variables technique.
Example 7.13 a) When the switch is closed in the circuit shown in Fig. 7.37, the voltage on the capacitor is 10 V. Find the expression for v o for t 0. b) Assume that the capacitor short-circuits when its terminal voltage reaches 150 V. How many milliseconds elapse before the capacitor short-circuits? Figure 7.37 The circuit for Example 7.13.
Example 7.13
Example 7.13 Figure 7.38 The test-source method used to find R TH. Figure 7.39 A simplification of the circuit shown in Fig. 7.37.
Example 7.13
7.7 The Integrating Amplifier Figure 7.40 An integrating amplifier.
We assume that the operational amplifier is ideal. Thus we take advantage of the constraints Because vp = 0 Hence, from Eqs. 7.61, 7.63, and 7.64,
Multiplying both sides of Eq. 7.65 by a differential time dt and then integrating from to t generates the equation
A step change Figure 7.41 An input voltage signal. When t lies between t 1 and 2t 2,
Figure 7.42 The output voltage of an integrating amplifier.
Example 7.14 Assume that the numerical values for the signal voltage shown in Fig. 7.41 are V m = 50 and t1 = 1 s. This signal voltage is applied to the integratingamplifier circuit shown in Fig. 7.40. The circuit parameters of the amplifier are R s = 100 k Ω, C f = 0.1 mf, and V CC = 6 V. The initial voltage on the capacitor is zero. a) Calculate v o (t). b) Plot v o (t) versus t.
Example 7.14
Example 7.14 Figure 7.43 The output voltage for Example 7.14.
Example 7.15 At the instant the switch makes contact with terminal a in the circuit shown in Fig. 7.44, the voltage on the 0.1 mf capacitor is 5 V. The switch remains at terminal a for 9 ms and then moves instantaneously to terminal b. How many milliseconds after making contact with terminal b does the operational amplifier saturate?
Example 7.15 Figure 7.44 The circuit for Example 7.15.
Example 7.15
Example 7.15