Question

What is the time constant for charging an inductance L through a resistance R?

(a) LR

(b) L/R

(c) R/L

(d) RLC

Answer 1

For RL circuit

t=0

Applying KVL

and $V_L = L \frac{\mathrm{d} i}{\mathrm{d} t}$

$V = I*R+L * \frac{\mathrm{d} i}{\mathrm{d} t}$

$\frac{\mathrm{d} i}{\mathrm{d} t} =\frac{( V- i * R)}{L}$

$\frac{di}{V-iR}=\frac{dt}{L}$

Now integrating both side

$\int_{0}^{i} \frac{di}{V-iR}=\int_{0}^{t}\frac{dt}{L}$

$\ln (V-iR)_{0}^{i}\textrm{}=\frac{-Rt}{L}$

$\ln (V-iR) - \ln V =\frac{-Rt}{L}$

$\frac{-Rt}{L} = \ln(\frac{V-i R}{V})$

taking antilog

$e^{-\frac{Rt}{L}}=\frac{V-iR}{V}$

$V e^{-\frac{Rt}{L}}=V-iR$

$iR=V-V e^{-\frac{Rt}{L}}$

$i=\frac{V}{R} (1- e^{-\frac{Rt}{L}})$

or $i=\frac{V}{R} (1- e^{-\frac{t}{ \tau}})$

$\tau = L / R$

answer is B