# Chap 7 Response of FirstOrder RL and RC

• Slides: 27

Chap. 7 Response of First-Order RL and RC Circuits Contents 7. 1 7. 2 7. 3 7. 4 The Natural Response of an RL Circuit The Natural Response of an RC Circuit The Step Response of RL and RC Circuits A General Solution for Step and Natural Responses 7. 5 Sequential Switching 7. 6 Unbounded Response 7. 7 The Integrating Amplifier Objectives 1. 能定出RL 和RC 電路的自然響應。 2. 能定出RL 和RC 電路的步階響應。 3. 知道如何分析具有順序切換的電路。 4. 能分析含有電阻和單一電容的運算放大器電路。 1

7. 1 The Natural Response of an RL Circuit t 0 By KVL: RL 電路的自然響應 3 3

Power and Energy Delivered to the Resistor 4

EX 7. 1 Determining the Natural Response of an RL Circuit Hint: 1. 找出流經電感器的初始電流 I 0。 The switch has been closed for a long time before it is opened at t = 0. 2. 找出電路的時間常數，τ= L/R 。 3. 從I 0 和τ得到i (t) = I 0 e - t /τ。 By current division, The initial energy stored in the inductor: 6

EX 7. 2 Determining the Natural Response of an RL Circuit with Parallel Inductors The initial energy stored in the inductors: 7

7. 2 The Natural Response of an RC Circuit By KCL: RC 電路的自然響應 Initial voltage Time constant 8 8

EX 7. 3 Determining the Natural Response of an RC Circuit Hint: 1. 找出跨在電容器的初始電壓 V 0 2. 找出電路的時間常數，τ= RC 3. 從V 0 和τ得到v (t) = V 0 e - t /τ。 The switch has been in position x for a long time. At t = 0, the switch moves to position y. & By voltage division, The power dissipated in the 60 k -resistor: The total energy dissipated in the 60 k -resistor: 9

EX 7. 4 Determining the Natural Response of an RC Circuit with Series Capacitors The initial energy stored in C 1: The initial energy stored in C 2: 10

7. 3 The Step Response of RL & RC Circuits The Step Response of an RL Circuit By KVL: RL 電路的步階響應 11

The Step Response of an RL Circuit If initial I 0 = 0, At t = , 斜率 If initial I 0 = 0, At t = , 斜率 12

EX 7. 5 Determining the Step Response of an RL Circuit t = ? when v = 24 V. 13

The Step Response of an RC Circuit By KCL: RC 電路的步階響應 OR 整式乘以C再微分 14

EX 7. 6 Determining the Step Response of an RL Circuit The switch has been in position 1 for a long time. At t = 0, the switch moves to position 2. Norton Equivalent OR 15

7. 4 A General Solution for Step and Natural Responses When x reach its final value xf (a constant), dx/dt =0. 自然和步階響應一般解 16

EX 7. 7 Using the General Solution Method to Find an RC Circuit’s Step Response The switch has been in position a for a long time. At t = 0, the switch moves to position b. 18

EX 7. 8 Using the General Solution Method with Zero Initial Conditions The switch has been open for a long time. Also, 19

EX 7. 9 Using the General Solution Method to Find an RL Circuit’s Step Response The switch has been open for a long time. Also, 20

EX 7. 10 Determining Step Response of a Circuit with Magnetically Coupled Coils The switch has been open for a long time. (Prob. 6. 46) Also, Since vo : Also, By KCL, 21

7. 5 Sequential Switching 順序切換(sequential switching): 指電路中的切換動作超過一次以上，重點在於求得初始值x(t 0)。 EX 7. 11 Analyzing an RL Circuit that has Sequential Switching The two switches in the circuit have been closed for a long time. At t = 0, switch 1 is opened. Then, 35 ms later, switch 2 is opened. t<0 22

EX 7. 11 RL-Sequential Switching (Contd. ) What percentage of the initial energy stored in the inductor is dissipated in R 3 ? For 0 < t < 35 ms For t 35 ms 23

EX 7. 12 Analyzing an RC Circuit that has Sequential Switching At t = 0, switch is moved to position b. Then, 15 ms later, switch is moved to position c. For 0 t 15 ms For 15 ms t ? ? 24

7. 6 Unbounded Response 電路的響應可以是指數增長型而非衰減型，這種響應稱為無限制響應 (unbounded response)，可能發生在含有相依電源的電路中。 EX 7. 13 Finding the Unbounded Response in an RC Circuit When the switch is closed, the voltage on the capacitor is 10 V. Find the expression for vo for t ≥ 0. RTH = v. T / i. T For 0 t 25

7. 7 The Integrating Amplifier (Contd. ) 27