8 Transient analysis
8.1 INTRODUCTION
We saw in Chapter 2 that the potential difference across a capacitor cannot
change instantaneously (Equation 2.26) and that the current through an
inductor cannot change suddenly (Equation 2.35). If, therefore, when a circuit
containing capacitance or/and inductance is operating in the steady state and
conditions change for some reason, requiring the current and voltage values to
change, there will be a finite period of time during which these changes take
place. This period is called a period of transient operation.
Two obvious examples of transient operation are (1) when a circuit contain-
ing capacitance or inductance is initially switched on and (2) when such a
circuit, having been operating in the steady state for some time, is suddenly
switched off.
These transient conditions are associated with the changes in the energy
stored in the capacitor or the inductor, and circuits containing either of these
elements are called single energy circuits. Circuits containing both of these
elements are called double energy circuits. Because there is no energy stored in
purely resistive circuits, currents and voltages in such circuits are able to change
without periods of transient operation (i.e. instantaneously).
8.2 CIRCUITS CONTAINING RESISTANCE AND
INDUCTANCE
The sudden application of a step voltage
The series RL circuit shown overleaf in Fig. 8.1 is connected to a d.c. voltage
source V such that when the switch S is closed (at an instant t = 0 say) a voltage
V is suddenly applied to the circuit. This is called a step function and is shown in
Fig. 8 2. For this step function,
V(t) = O fort<0
V(t) = V fort->0