Introduction to Electric Circuits

192 Transient analysis

8.4 THE LAPLACE TRANSFORM

It is a common technique to transform problems to a different form in order to

make their solution easier, even if the resulting processes are longer. One

example is the use of logarithms to transform the process of multiplication or

division into the simpler one of addition or subtraction. The method is to:

1 look up the logarithms of the numbers to be multiplied (or divided);

2 add (or subtract) the logarithms;

3 look up the antilogarithm of the result in order to obtain the answer to the

original problem.

The equations associated with the transient operation of electric circuits are

differential equations in the time domain. A stimulus which is a function of time

is applied to the circuit whose behaviour is then described by one or more

differential equations. These equations then have to be solved in order to

determine the response of the circuit to the stimulus. By means of the Laplace

transform it is possible to convert these differential equations into algebraic

equations involving a complex variable, s. After manipulation in order to solve

these algebraic equations (which are easier to solve than differential equations)

the inverse transform is found, which gives the time response to the original

stimulus. The method, then, is to:

1 set up the differential equations which describe the operation of the circuit;

2 look up the table of Laplace transforms in order to convert these to

algebraic equations;

3 solve the algebraic equations to find the response to the circuit in terms of

the complex variable, s;

4 look up the table of inverse transforms to find the time response of the

circuit to the original stimulus.

The Laplace transform is named after Pierre-Simon Laplace, a French

mathematician. It is written as

L[f(t)] : F(s) (8.22)

which is read 'the Laplace transform of the function of time f(t) is equal to a

function of s'. It is defined as

L[f(t)] : ff(t) exp (-st) dt (8.23)
0
Thus the original function f(t) is first multiplied by the exponential decay
exp (-st) and the result is integrated from zero to infinity. The value of s must
be such that Equation (8.23) converges to zero as t ---, ~ (it almost always does
in problems associated with electrical circuits).