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Chapter 7. First and second order transient circuits

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2Contents

1. Introduction2. First order circuits3. Second order circuits4. Application examples

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31. Introduction

Camera flash discharge :

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4Analysis

0)()(

Rt

dttdC CC

0)(1)( t

RCdttd

CC

RCt

C eVt

0)(

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52. First order transient circuits

Solution to 1st order differential equation :

f (t) = 0 → homogeneous equationf (t) ≠ 0 → inhomogeneous equation

xh(t) or xc(t) → homogeneous or complementary solutionxp(t) → inhomogeneous or particular solution

)()()( tftaxdt

tdx

0)()( tax

dttdx

cc

)()()( txtxtx cp

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aAKtxp 1)(

Particular solution :

For polynomial functions f (t) , x(t) should also be a polynomial func-tion of the order of equal or lower degree.

cbtattxtttfex p 22 )(232)()

1)()() KtxAtfex p

AaKtaxdt

tdxAtftax

dttdx

pp 1)(

)()()()(

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Homogeneous solution :

CatCatc eKeKetx

22)(

00)()( c

cc

c axdtdxtax

dttdx

dtaxddtax

dxaxdtdx

cc

cc

c ][ln

Cattxc )(ln

atatcp eK

aAeKKtxtxtx 221)()()(

The constant K2 can be found, if the value of x(t) is known at one instant of time.

Integrating both sides with respect to the arguments,

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8General solution : 1st order differential equation

K1 : steady-state solution: x(t) → K1 as t→∞ when the second term becomes negligible.

τ : time constant

aeKKtx t 1)( /

21

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9Simple RC circuit

)0( t

Sp VKKt )(

01)( ttth e

RCeet

RC1

RCt

Shp eKVttt

2)()()(

0)()(

RVt

dttdC S

RCVt

RCdttd S )(1)(

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S

S

VKKVt

2

20)0(

RCt

SS eVVt

)(

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11Simple RL circuit

LVti

LR

dttdiVtRi

dttdiL S

S )()()()(

tLR

S eKR

Vti

2)(

0)0()0( 2 KR

Viti SR

VK S 2

tLR

St

LR

SS eR

VeR

VR

Vti 1)(

tLR

SR eVtiRt 1)()(

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12Example 7.1

Consider the circuit shown in Fig. 7.4a. Assuming that the switch has been in posi-tion 1 for a long time, at time t=0 the switch is moved to position 2. We wish to cal-culate the current i(t) for t > 0.

Initial condition :

][436

312)0( Vkk

kC

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0)()()(

21

R

tdt

tdCR

t

FCkRkR 100,3,6 21

0)(5)( t

dttd

tAet )(Suppose

0)5(0)(5)( tt eeAt

dttd

5

tt eAet 55 4)( ][4)0()0( VCC

][34)()( 5

2

mAeR

tti t

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14Example 7.2

The switch in the network in Fig. 7.5a opens at t=0. Let us find the output volt-age vo(t) for t > 0.

][34)0()0( Aii LL

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0)()()( 311

dttdiLtiRRVS

HLRR 2,2,2 31

6)(2)( ti

dttdi

teKti 223)(

][34)0()0( Aii LL 3

5334

343)0( 22 KKi

teti 2

353)(

6)(2)( ti

dttdi

to etit 2

3106)(2)(

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