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 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 电容

A capacitor consists of two conducting plates separated by an insulator. 绝缘体

dt

dvCi c

Insulator 绝缘体Conducting Plates 导电极板

Measured in Farads(F) 法拉

Capacitance 电容（值）

A capacitor properties: 特性

5.1 Capacitors and Inductors 电容和电感

)0()0( cc vv

vc Memory 记忆 Storage element 储能元件 Open circuit to dc The voltage on a capacitor cannot change abruptly

Ⅱ. Inductors 电感

Measured in henrys(H) 亨利

Inductance 电感（值）

An inductor properties:

An inductor consists of a coil of conducting wire.

dt

diLv L

)0()0( LL ii

length, l

Core material μ

Cross-sectional area, A

Number of turns, N

Memory 记忆 Storage element 储能元件 Short circuit to dc

The current through an inductor cannot change abruptly.

5.2 The Source-Free Response of RC and RL Circuits 一阶电路的零输入响应

0)0( Vvc

t >0: RC vv

dt

dvRCv c

c

RCtc eVtv /

0)( /e)0( t

cc vv

0)0()0( Vvv cc

RC

Ⅰ.The Source-free RC Circuit

Time constant 时间常数

)(sCReq

If there are many resistors in the circuit:

•Req 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 .

)(tvt

0

0

0

0

0

V00674.05

V01832.04

V04979.03

V13534.02

V36788.0

)0()( tsCReq

/0)( t

c eVtv

Ⅱ. The Source-Free RL Circuit /)0()( t

LL eiti

)(/ sRL eqiL(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:

:0t

:0t

3124

124

A832

401

i

A6412

12)( 1

iti

A6)0()0( ii

816//)412(eqR

sR

L

eq 4

1

8

2

A6)0()( 4/ tt eeiti

Hence

Thus,

1. The step function 阶跃函数

0,1

0,0)(

t

ttu

u(t)1

0 t

5.3 Singularity Functions 奇异函数

The delayed step function:延迟阶跃函数

u(t-t0)

t0

1

0 t00

0

0,( )

1,

t tu t t

t t

The general step function:

0

00 ,

,0)(

ttA

ttttAu

A

0 tt0

00

0

,V

,0)(

tt

tttv

Replace a switch by the step function:

)(V)( 00 ttutv

2. The impulse function 冲激函数

(t) (1)

0 t

0)( t

1)()(0

0

dttdtt

0t

The delayed impulse function:

1)()(0

0

00

t

t

dtttdttt

0)( t 0tt

t t00

(t-t0)(1)

dt

tdut

)()(

t

dtttu )()(

0)0()0( Vvv CC

5.4 Step Response of an RC Circuit 阶跃响应

sCR Vvv :0t

sC VvRi

sCC Vv

dt

dvRC

)0()()( /0 teVVVtv t

SSC )(sRC

The complete response 全响应

The temporary response 暂态响应

Vs ： The steady-state response 稳态响应

:)( /0

tS eVV

(The forced response ) 强制响应

(The natural response ) 自由响应

0)(

0)(

/0

0

teVVV

tVtv

tSS

c

0

vc(t)

V0

VS

t

V0<VS

vc(t)

VS

0t

V0

V0>Vs

)0()()( /0 teVVVtv t

SSC

)0()1()( //0 teVeVtv t

St

C

Source-free response零输入响应

Zero-state response零状态响应

nfc vvtv )(

Forced response 强制响应

Natural response 自由响应

The complete response 全响应

/)]()0([)()( tevvvtv

1. The initial capacitor voltage v(0+). 2. The final capacitor voltage v(). 3. The time constant .

/)(0

0)]()([)()( ttevtvvtv

The complete response of an RC circuit requires three things:

If the switch changes position at time t=t0,so the equation is

5.5 Complete Response of an RL Circuit

)0()0()0( 0 tIii

/)]()0([)()( teiiiti

eqRL /

Three-factor method 三要素法/)]()0([)()( teffftf

1. The initial value f(0+). 2. The final value f(). 3. The time constant .

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

.

Solution:

V51020101201010

10)( 33

cv

s1.0101010)1010//(20 63 CReq/)]()0([)()( t

cccc evvvtv

V155155 101.0/ tt ee

V201011020)0()0( 33 cc vv

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

when

Solution:

V8)0()0( cc vv

V12624)( 111 iiivc

sCReq 11.010 /)]()0([)()( t

cccc evvvtv

V2012)128(12 tt ee

110iv

101i

vReq

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！