Introduction to Electric Circuits

174 Transient analysis

i.e. it is the steady state value (I) of the current, so we have finally

i = 111 - exp (-Rt/L)] (8.2)

The time constant of an RL circuit

From Equation (8.2) we note that when t = L/R seconds,

i= 111 - exp (-1)] - 0.632 1 (8.3)

L/R is called the time constant of the circuit. Its symbol is ~-and it is measured

in seconds:

~"- L/R (8.4)

From Equation (8.3) we see that after a time equal to the time constant

following the sudden application of a step function, the current will have

reached 63.2 per cent of its steady state value. After a time equal to five time

constants (5r) the current will have reached [1 - exp (-5)] - 0.993 or 99.3 per

cent of its steady state value. Since mathematically the current can never reach

I we say that to all intents and purposes it has done so after 5~-.

The voltage across the resistor is VR = iR = I[1- exp (-Rt/L)]R and,

since IR = V,

V R = VII - exp (-Rt/L)] (8.5)

The voltage across the inductor is vL = V- VR = V- V[1 - exp (-Rt/L)], so

VL = V exp (-Rt/L) (8.6)

Equation (8.2) describes an exponential growth of current, Equation (8.5)

describes an exponential growth of voltage and Equation (8.6) describes an

exponential decay of voltage. The graphs of these are shown in Figs 8.3 and

8.4.

0.6321

f I

I

z 5z

Figure 8.3

VIr--

0.368V

VR

~t 0 ~ 5x ~--

Figure 8.4

The rate of change of current is obtained by differentiating Equation (8.2).
Thus, remembering that I = V/R

di/dt = (V/R)(R/L) exp (-Rt/L)
= (V/L) exp (-Rt/L) (8.7)