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140 TIME-DEPENDENT CIRCUIT ANALYSIS


FromiL( 0 +)=0 andvC( 0 +)=8 V, it follows thatA 1 +A 2 =0 andB 1 +B 2 + 4 =8.
Simultaneous solution yieldsA 1 =− 2 ,A 2 = 2 ,B 1 =6, andB 2 =−2. Thus we have, fort>0,
iL(t)=

(
− 2 e−t+ 2 e−^3 t

)
A; vC(t)=

(
4 + 6 e−t− 2 e−^3 t

)
V

In order to represent the abrupt changes in excitation, encountered when a switch is opened
or closed and in individual sequences of pulses,singularity functionsare introduced. This type of
representation leads to thestepandimpulsefunctions. The methods for evaluating the transient
(natural) and steady-state (forced) components of the response can also be applied to these
excitations.
Theunit-stepfunction, represented byu(t), and defined by

u(t)=

{
0 ,t< 0
1 ,t> 0
(3.2.39)

is shown in Figure 3.2.4. The physical significance of the unit step can be associated with the
turning on of something (which was previously zero) att=0. In general, a source that is applied
att=0 is represented by
v(t) or i(t)=f (t)u(t) (3.2.40)
Adelayed unit step u(t−T), shown in Figure 3.2.5, is defined by

u(t−T)=

{
1 ,t>T
0 ,t<T

(3.2.41)

A widely used excitation in communication, control, and computer systems is therectangular
pulse,whose waveform is shown in Figure 3.2.6 and whose mathematical expression is given by

f(t)=

{ 0 , −∞<t< 0
A, 0 <t<T
0 ,T<t<∞

(3.2.42)

Equation (3.2.42) can be expressed as the sum of two step functions,
f(t)=Au(t)−Au(t−T) (3.2.43)

f(t)

t

1

0

Figure 3.2.4Unit-step function.

f(t)

t

1

T

Figure 3.2.5Delayed unit-step function.
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