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Analysis of First- and Second-Order

Transient Circuits

< Chapter-7 >

Jongsik Kim

High-Speed Integrated Circuits and Systems Lab.

Department of Wireless Communications Engineering

Kwangwoon University

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Outline

• First- and Second-Order Circuits

• Charge and Discharge Time

• General Form of the Response Equations

• Examples

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1st- and 2nd- Order Circuits

• First-order circuit : single storage element

First-order differential equation

• Circuit’s time constant – Transient analysis

How fast the circuit will respond to changes

• Second-order circuit : an inductor and a capacitor are present

simultaneously

• Laplace transform

• PSPICE

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Charge and Discharge Time

free-body diagram

ex : flash circuit in a camera

( ) ( )0

( ) 1( ) 0

C C

CC

d t tC

dt R

d tt

dt RC

KCL

0( ) t RC

C t V e

Decaying exponentially

with function of RC

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General Form of the Response Equations (1/4)

• Solution of the first-order circuitsFirst-order differential equation

( )( ) ( )

dx tax t f t

dt

If x(t)=xp(t) and x(t)=xc(t)

( )( ) 0,

dx tax t

dt ( ) ( ) ( )p cx t x t x t

( ) ( )( ) , ( ) 0

p cp c

dx t dx tax t A ax t

dt dt

xp(t) : particular integral solution

xc(t) : complementary solution, natural response

f(t)=A, constant

Since A is constant, xp(t) must also be a constant

1 1( ) , p

Ax t K K

a

2

( ), ln ( )

( )

ln ( )

( )

cc

c

c

at

c

dx t dt da x t a

x t dt

x t at c

x t K e

Hence,

2

( ) ( ) ( )

p c

at

x t x t x t

AK e

a

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General Form of the Response Equations (2/4)

1 2( ) tx t K K e

• General form of the solution of first-order transient circuits

K1 : steady-state solution for t infinite

: time constant

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General Form of the Response Equations (3/4)

( )( )0

( ) ( )

S

S

t Vd tC

dt R

Vd t t

dt RC RC

For time t>0 2

( )( )

( )

S

Rt

LS

di tL Ri t V

dt

Vi t K e

R

For time t>0

Using same manner

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General Form of the Response Equations (4/4)

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Example 7.1 (1/2)

• Calculate i(t) for t>0

At t = 0-

*Capacitor is open for DC

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Example 7.1 (2/2)

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Example 7.2 (1/2)

• Find the output voltage v0(t) for t>0*Circuit modification to Thevenin equvalent

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Example 7.2 (2/2)