Introduction to Electric Circuits

8.4 The Laplace transform 195

Capacitance

If the current is given by i(t) then the voltage across the plates will be given by

Vc(t) = (1/C)fi dt. Taking Laplace transforms and using number 5 from the

table we have

vc(s) = (1/Cs)i(s) + vc(O)/s

If the capacitor is initially uncharged then Vc(0) = 0 and the second term on the

right-hand side disappears.

Example 8.10

Obtain the transform circuit for the circuit shown in Fig. 8.31

R L C

sv()

Figure 8.31

Solution

The voltage source is a step function of amplitude V. From Table 8.1 we see
that this transforms to V/s (pair number 3 with A = V).
The resistor R in the original circuit remains unchanged in the transform
circuit.
The inductor L transforms to an element Ls in series with a source Li(O).
This source is short circuited if the current is initially zero.
The capacitor C transforms to an element 1/Cs in series with a source
vc(O)/s. This source is short circuited if the capacitor is initially uncharged.
The transform circuit therefore takes the form shown in Fig. 8.32.

v

R Ls 1/Cs

~(s) l I! t ~ i!

u(o)

3

()VCs,~

Figure 8.32