Introduction to Electric Circuits

vt

This describes an exponential growth of voltage and is shown in Fig. 8.14.

0.368V

Vc

182 Transient analysis

0 T 5T "-t

Figure 8.14

The time constant of an RC circuit

The product RC is called the time constant (symbol z) of the RC circuit and it is

measured in seconds. When t = R C,

Vc = V[1 - exp (-1)] = 0.632 V

After a time equal to the time constant, therefore, the voltage across the

capacitor will have reached 63.2 per cent of its final steady state value V.

Compare this with the growth of current in an RL circuit when subjected to a

step input (Equation (8.3)).

After a period equal to 5RC, vc = V[1 - exp (-5)] = 0.993 V. To all intents

and purposes, therefore, the voltage across the capacitor will have reached its

steady state value after 5z seconds. The capacitor is then said to be fully

charged and there is no longer any movement of charge to its plates. The

current in the circuit is therefore zero and the voltage across the resistor is also

zero. Because the voltage across the capacitor is initially zero, the whole of the

applied voltage V appears across the resistor in accordance with KVL so V = iR

and the current immediately jumps to the value I = V/R. Thereafter it may be

found from

i= Cdvc//dt- C(d/dt){V[I- exp (-t/CR)]}

= CV[O - (-1/RC) exp (-t/RC)] = (V/R) exp (-t/RC)

Finally, since V/R - I, then

i - I exp (-t/RC) (8.15)

This shows that, as the charging process proceeds, the current decays exponen-

tially from its initial value of I(= V/R) to zero. The waveform of the current is

shown on the following page in Fig. 8.15.

The voltage across the resistor, vR is given by iR - IR exp (-t/CR) and, since

IR = V, then