Question

Find the value of t/τ for which the current in an LR circuit builds up to (a) 90%, (b) 99% and (c) 99.9% of the steady-state value.

Answer 1

Value of $\frac {t}{\tau}$ is 6.9

Explanation:

l(t) = $\frac {V}{R}(1 - e^{\frac {-t}{L/R}})$  

$\frac {V}{R}(1 - e^{\frac {-t}{t_c}})$,

, where $t_c$ is time constant and $\frac {V}{R}$ is steady state current,

V is applied voltage and R is series resistance

 

if current reaches 99 % of steady state,  then

$(1 - e^{\frac {-t}{t_c}})$ = 0.99

or  

$e^{\frac {-t}{t_c}}$

= 0.01 = $10^-^2$  .

Hence $\frac {t}{t_c} = 2 \times ln10$ = 4.6

if current reaches 99.9 % of steady state,  then

$(1 - e^{\frac {-t}{t_c}})$ = 0.999 or

$e^{\frac {-t}{t_c}}$

= 0.001 = $10^-^3$  . Hence $\frac {t}{t_c} = 3 \times ln10$ = 6.9