0195136047.pdf

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.