5.2 The Source-Free Responses of RC and RL Circuits 5.4 Step Response of an RC Circuit 5.1 Capacitors and Inductors 5.3 Singularity Functions 5.5 Complete Response of an RL Circuit Chapter 5 First-Order Circuits 一阶电路

In this chapter,we shall examine two types of simple circuits: a circuit comprising a resistor and a capacitor and a circuit comprising a resistor and an inductor. These are called RC and RL circuits. We carry out the analysis of RC and RL circuits by applying Kirchhoff’s laws, and producing differential equations. The differential equations resulting from analyzing RC and RL circuits are of the first order. Hence, the circuits are collectively known as first-order circuits.

Ⅰ. Capacitors A capacitor consists of two conducting plates separated by an insulator. 绝缘体 Measured in farads (F) Capacitance A capacitor properties: 特性 5.1 Capacitors and Inductors Insulator 绝缘体 Conducting Plates 导电极板 vcvc Memory Storage element Open circuit to dc The voltage on a capacitor cannot change abruptly 电容和电感 电容 电容（值） 法拉 记忆 储能元件

Ⅱ. Inductors Measured in henrys (H) Inductance An inductor properties: An inductor consists of a coil of conducting wire. Memory Storage element Short circuit to dc The current through an inductor cannot change abruptly 电感 电感（值） 亨利 记忆 储能元件 iLiL N +–v Magnetic linkage 磁链

Resistor(R)Capacitor(C)Inductor(L) dissipate powerstorage element no memorymemory sameopen circuit to dcshort circuit to dc Summary

5.2 The Source-Free Responses of RC and RL Circuits t >0: Ⅰ.The Source-Free RC Circuit Time constant 一阶电路的零输入响应 时间常数

If there are many resistors in the circuit: R eq is the equivalent resistance of resistors. The key to working with a source-free RC circuit is finding: 1.The initial voltage v(0 + ) across the capacitor. 2. The time constant . Source-free response: The response is due to the initial energy stored and the physical characteristics of the circuit and not due to some external sources.

The time constant of a circuit is the time required for the response to decay to a factor of 1/e or 36.8 percent of its initial value.

Ⅱ. The Source-Free RL Circuit i L (0 + ) : The initial current through the inductor Time constant 时间常数

Example 5.1 The switch in the circuit has been closed for a long time. At t=0, the switch is opened. Calculate i(t) for t>0. Solution: hence Thus,

1. The step function u(t)u(t) 1 0t 5.3 Singularity Functions The delayed step function: 延迟阶跃函数 u(t-t0)u(t-t0) t0t0 1 0t The general step function: A 0 t t0t0 奇异函数 阶跃函数

Replace a switch by the step function:

2. The impulse function  (t) (1) 0t The delayed impulse function: tt0t0 0  (t-t 0 ) (1)(1) 冲激函数

5.4 Step Response of an RC Circuit The complete response 全响应 The temporary response V s ： The steady-state response (The forced response ) (The natural response ) 阶跃响应 稳态响应 强制响应 暂态响应 自由响应

0 vc(t)vc(t) V0V0 VSVS t V0<VSV0<VS vc(t)vc(t) VSVS 0 t V0V0 V0>VsV0>Vs

Source-free response 零输入响应 Zero-state response 零状态响应 Forced response 强制响应 Natural response 自由响应 The complete response 全响应

1. The initial capacitor voltage v(0 + ). 2. The final capacitor voltage v(  ). 3. The time constant . The complete response of an RC circuit requires three things: If the switch changes position at time t=t 0, the equation is:

Example 5.2 The circuit is in steady-state, switch S is closed at t=0. Calculate. when. Solution:

Example 5.3 The circuit is in steady-state, switch S moves from position 1 to 2 at t=0. Calculate when Solution:

5.5 Complete Response of an RL Circuit Three-factor method 三要素法 1. The initial value f(0 + ). 2. The final value f(  ). 3. The time constant .

Example 5.4 Find i(t) in the circuit for t>0. Assume that the switch has been closed for a long time. Solution: Thus,

Example 5.5 At t=0, switch 1 is closed, and switch 2 is closed 4s later. Find i(t) for t>0. Calculate i for t=2s and t=5s. Solution: Thus, For

Hence, For At node a :

At t=2s, At t=5s,

部分电路图和内容参考了： 电路基础（第 3 版），清华大学出版社 电路（第 5 版），高等教育出版社 特此感谢！