Chap. 7 Response of First-Order � RL and RC Circuits
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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。
First-Order Circuits
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Two forms of the circuits for
natural response.
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。
RL 和RC 電路稱為一階電路(first-order circuits)因為它的電壓和電流可由一階微分方程式來描述。
7.1 The Natural Response of an RL Circuit
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RL 電路的自然響應
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t ≥ 0
By KVL:
Power and Energy Delivered to the Resistor
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The Significance of the Time Constant
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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τ。
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
EX 7.1 Determining the Natural Response of an RL Circuit
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Hint:
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:
EX 7.2 Determining the Natural Response of an RL Circuit � with Parallel Inductors
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The initial energy stored in the inductors:
7.2 The Natural Response of an RC Circuit
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RC 電路的自然響應
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By KCL:
Initial voltage
Time constant
EX 7.3 Determining the Natural Response of an RC Circuit
Hint:
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。
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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 60kΩ-resistor:
&
The total energy dissipated in the 60kΩ-resistor:
EX 7.4 Determining the Natural Response of an RC Circuit� with Series Capacitors
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The initial energy stored in C1:
The initial energy stored in C2:
7.3 The Step Response of RL & RC Circuits
RL 電路的步階響應
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By KVL:
The Step Response of an RL Circuit
The Step Response of an RL Circuit
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If initial I0 = 0,
At t = τ,
斜率
If initial I0 = 0,
斜率
At t = τ,
EX 7.5 Determining the Step Response of an RL Circuit
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t = ? when v = 24 V.
The Step Response of an RC Circuit
RC 電路的步階響應
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By KCL:
OR
整式乘以C再微分
EX 7.6 Determining the Step Response of an RC Circuit
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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
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When x reach its final value
xf (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
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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+ 。
EX 7.7 Using the General Solution Method to Find an RC � Circuit’s Step Response
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The switch has been in position a for a long time.
At t = 0, the switch moves to position b.
EX 7.8 Using the General Solution Method with Zero Initial Conditions
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The switch has been open for a long time.
Also,
EX 7.9 Using the General Solution Method to Find an RL � Circuit’s Step Response
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The switch has been open for a long time.
Also,
EX 7.10 Determining Step Response of a Circuit� with Magnetically Coupled Coils
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The switch has been open for a long time.
Also,
(Prob. 6.46)
Since vo :
Also,
By KCL,
7.5 Sequential Switching
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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
EX 7.11 RL-Sequential Switching (Contd.)
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What percentage of the initial energy stored in the inductor is dissipated in R3Ω?
For t ≥ 35 ms
For 0 < t < 35 ms
EX 7.12 Analyzing an RC Circuit that has Sequential Switching
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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
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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 = vT / iT
For 0 ≤ t
7.7 The Integrating Amplifier
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。
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Ideal OP-Amp
7.7 The Integrating Amplifier (Contd.)
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EX 7.15 Analyzing an Integrating Amplifier that has Sequential � Switching
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At t = 9 ms, vo = -5 +9 = 4 V.
Saturation Time ts :
將輸入電阻Rs 和回授電容Cf 的位置對調，就可以把積分放大器轉換成微分放大器。