0195136047.pdf

148 TIME-DEPENDENT CIRCUIT ANALYSIS

With the aid of theorems concerning Laplace transforms and the table of transforms, lin-

ear differential equations can be solved by the Laplace transform method. Transformations of

the terms of the differential equation yields an algebraic equation in terms of the variables.

Thereafter, the solution of the differential equation is affected by simple algebraic manipulations

in thes-domain. By inverting the transform of the solution from thes-domain, one can get

back to the time domain. The response due to each term in the partial-fraction expansion is

determined directly from the transform table. There is no need to perform any kind of inte-

gration. Because initial conditions are automatically incorporated into the Laplace transforms

and the constants arising from the initial conditions are automatically evaluated, the resulting

response expression yields directly the total solution. The flow diagram is illustrated in Fig-

ure 3.3.1.

Eliminating the need to write the circuit differential equations explicitly,transformed net-

works,which are networks converted directly from the time domain to the frequency domain,

are used. For the three elementsR, L,andC,the transformed network equivalents, using Ta-

ble 3.3.1, are based on the Laplace transforms of their respective volt-ampere characteristics,

as follows:

L[v(t)=Ri(t)] → V(s)=RI(s) (3.3.28)

L

[

v(t)=L

di(t)

dt

]

→ V(s)=sLI(s)−Li

(

0 +

)

(3.3.29)

L

[

v(t)=

1

C

∫t

−∞

i(t)dt

]

→ V(s)=

1

Cs

I(s)+

v

(

0 +

)

s

(3.3.30)

Note that

∫ 0

−∞i(t)dt=q

(

0 −

)

andq

(

0 −

)

/C=v

(

0 −

)

=v( 0 +)because of continuity of

capacitor voltage. Figure 3.3.2 shows the time-domain networks and the transformed network

equivalents in the frequency domain for elementsR, L,andC.Notice the inclusion of the initial

inductor current by means of the voltage term [Li( 0 +)] and the initial capacitor voltage by the

voltage [v( 0 +)/s]. Source conversion can be applied to obtain alternate transformed equivalent

networks, as shown in Figure 3.3.2.

The following procedure is applied for the solution of network problems utilizing the

transformed networks in the frequency domain.

1. Replace the time-domain current and voltage sources by the Laplace transforms of their
   time functions; similarly, replace the dependent-source representations. Transform the
   circuit elements into the frequency domain with the use of Figure 3.3.2.


2. For the resultant transformed network, write appropriate KVL and KCL equations using
   the mesh and nodal methods of analysis.


3. Solve algebraically for the desired network response in the frequency domain.

Integro-differential

equation

Laplace

transformation

Agebraic

manipulation

Algebraic

equations

Initial

conditions

Inverse

transform

Standard

forms

Total

solution

Figure 3.3.1Flow diagram for

Laplace transformation method

of solving differential equations.