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Chap. 7 Response of First-Order RL and RC Circuits

Chap. 7 Response of First-Order RL and RC Circuits. C ontents. 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

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Chap. 7 Response of First-Order RL and RC Circuits

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  1. Chap.7Response of First-Order RLand 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。

  2. First-Order Circuits Two forms of the circuits for natural response. 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。 Four possible first-order circuits. 2

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

  4. Power and Energy Delivered to the Resistor

  5. The Significance of the Time Constant Transient response 暫態響應 Steady-state response 穩態響應 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。 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。 5

  6. EX 7.1 Determining the Natural Response of an RL Circuit Hint: • 找出流經電感器的初始電流I0。 • 2. 找出電路的時間常數，τ= L/R。 • 3. 從I0和τ得到i (t) = I0 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: 6

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

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

  9. EX 7.3 Determining the Natural Response of an RC Circuit Hint: • 找出跨在電容器的初始電壓V0 • 2. 找出電路的時間常數，τ= RC • 3. 從V0和τ得到v (t) = V0 e - t /τ。 The switch has been in position x for a long time. At t = 0, the switch moves to positiony. & By voltage division, The power dissipated in the 60k-resistor: The total energy dissipated in the 60k-resistor: 9

  10. EX 7.4 Determining the Natural Response of an RC Circuit with Series Capacitors The initial energy stored in C1: The initial energy stored in C2: 10

  11. 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

  12. The Step Response of an RL Circuit If initial I0 = 0, At t = , 斜率 If initial I0 = 0, At t = , 斜率

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

  14. 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

  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 position2. Norton Equivalent OR 15

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

  17. 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。 • 決定變數的初始值，也就是t0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 = t0-和t = t0+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 = t0+。 • 計算出變數的最終值，亦即 t →∞的值。 • 計算出電路的時間常數。 17

  18. EX 7.7 Using the General Solution Method to Find an RCCircuit’s Step Response The switch has been in position afor a long time. At t = 0, the switch moves to positionb. 18

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

  20. EX 7.9Using the General Solution Method to Find an RLCircuit’s Step Response The switch has been openfor a long time. Also, 20

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

  22. 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(t0)。 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

  23. EX 7.11 RL-Sequential Switching (Contd.) What percentage of the initial energy stored in the inductor is dissipated in R3? For 0 < t < 35 ms For t  35 ms 23

  24. 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 positionc. For 0  t  15 ms For 15ms  t 24

  25. 7.6 Unbounded Response RTH = vT / iT 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 vofor t ≥ 0. For 0  t 25

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

  27. 7.7 The Integrating Amplifier (Contd.) 27

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