Electric Circuit Analysis

13.18    Analysis of Dynamic Circuits by Laplace Transforms

Example: 13.6-2

The resistor value in Fig. 13.6-1 under Example 13.6-1 is changed to 2W. (i) Find the step response of

v

o

(t) (ii) Determine the zero-state response of v

o

(t) if v

S

(t) = e

-t

u(t) V.

Solution

The differential equation governing the output voltage with 2W resistor instead of 3W is

d v t

dt

dv t

dt

v t v t

o o

o s

2

2

2

( ) ( )

( ) ( )+ + = .

The System Function H s

V s

V s

s s s

o

s

( )

( )

( )

( )

= =

+ +

=

+

1

2 1

1

1

2 2

.

The roots of denominator polynomial are –1 and –1. The factors of the denominator polynomial are

(s + 1) and (s + 1). Therefore, the root at –1 has a multiplicity of 2.