1 Chap. 7 Response of First-Order RL and RC Circuits Contents 7.1 The Natural Response of an RL Circuit 7.2 The Natural Response of an RC Circuit 7.3 The Step Response of RL and RC Circuits 7.4 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。

Two forms of the circuits for natural response. 2 Four possible first-order circuits. 自然響應 (natural response) ： 當電感器或電容器所 儲存的能量突然釋放 給電阻性網路時，該 電路電流和電壓的變 化。 步階響應 (step response) ： 當電感器或電容器 突然獲得來自直流 電壓源或電流源的 能量時，該電路電 流和電壓的變化。 First-Order Circuits

3 7.1 The Natural Response of an RL Circuit RL 電路的自然響應 3 t  0 By KVL:

Power and Energy Delivered to the Resistor 4

The Significance of the Time Constant 5 時間常數 (time constant) ： 在 e -(R/L)t 的項次中， t 的係數 R/L 將會決定電壓或電流趨近 於零的速率，此係數的倒數稱為電路的時間常數  。 對單一時間常數電路而言，經過了一個長時 間 (a long time) 或指經過了五倍以上的時間 常數，這時電流或電壓幾乎已經到達終值。 Transient response 暫態響應 Steady-state response 穩態響應

6 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 。 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 。 The switch has been closed for a long time before it is opened at t = 0. By current division, The initial energy stored in the inductor:

7 EX 7.2 Determining the Natural Response of an RL Circuit with Parallel Inductors The initial energy stored in the inductors:

8 7.2 The Natural Response of an RC Circuit RC 電路的自然響應 8 By KCL: Initial voltage Time constant

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 。 9 EX 7.3 Determining the Natural Response of an RC Circuit 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 60k  - resistor: & The total energy dissipated in the 60k  - resistor:

10 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 :

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

12 If initial I 0 = 0, At t = , 斜率 If initial I 0 = 0, 斜率 At t = ,

13 EX 7.5 Determining the Step Response of an RL Circuit t = ? when v = 24 V.

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

15 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

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

RL 和 RC 電路的自然和步階響應之計算步驟 確定電路中想要的變數，對 RC 電路而言，選 擇電容電壓最為方便；對 RL 電路而言，則最好 選擇電感電流。 2. 決定變數的初始值，也就是 t 0 時的值。 注意，如果選擇了電容電壓或電感電流作變數， 則不須區分 t = t 0 - 和 t = t 0 + 有何不同，因為它們 都是連續變數；如果選擇了其他變數，則必須 記住它的初始值定義於 t = t 0 + 。 3. 計算出變數的最終值，亦即 t →∞ 的值。 4. 計算出電路的時間常數。

18 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.

19 EX 7.8 Using the General Solution Method with Zero Initial Conditions The switch has been open for a long time. Also,

20 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,

21 EX 7.10 Determining Step Response of a Circuit with Magnetically Coupled Coils The switch has been open for a long time. Also, (Prob. 6.46) Since v o : Also, By KCL,

7.5 Sequential Switching 22 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

EX 7.11 RL-Sequential Switching (Contd.) 23 What percentage of the initial energy stored in the inductor is dissipated in R 3  ? For t  35 ms For 0 < t < 35 ms

EX 7.12 Analyzing an RC Circuit that has Sequential Switching 24 For 0  t  15 ms At t = 0, switch is moved to position b. Then, 15 ms later, switch is moved to position c. For 15ms  t ??

7.6 Unbounded Response 25 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 v o for t ≥ 0. R TH = v T / i T For 0  t

輸入電壓的積分乘上 ( 1 / R s C f ) 和負值後， 再加上電容器的初始電壓值。 7.7 The Integrating Amplifier 26 Ideal OP-Amp

7.7 The Integrating Amplifier (Contd.) 27