Introduction to Electric Circuits

8.2 Circuits containing resistance and inductance 173

v(

i R L

)

..q[..__.___

VR VL

i

0
Figure 8.1 Figure 8.2

,i...._ r

t

Applying KVL to the circuit we see that

V- VR- VL~-O

where

VL = L(di/dt)

and

VR = in

so that

V- iR - L(di/dt) = 0 (8.1)

Rearranging, we get

di/dt = (V- iR)/L -[(V/R)- i]/(L/R)

Separating the variables di and dt we have

di/[(V/R) - i] = R dt/L

Integrating, and remembering that f (dx/x)- ln x + a constant, and that

f dx = x + a constant, we have

-ln[(V/R- i] = (R/L)t + C

where C is a constant. Therefore

ln[(V/R)- i] = -(R/L)t- C

Taking antilogs we get

(V/R) - i - exp [(-R/L)t - C] = exp (-Rt/L) exp (-C)

Now at t = O, i = 0 so that exp (-C) = V/R. Therefore

(V/R) - i= exp (-Rt/L) x V/R

i- V/R - (V/R)exp (-Rt/L)

= (V/R)[1 - exp (-Rt/L)]

(V/R) is the value reached by the current when all transients have died away,