Find the value of t/τ for which the current in an LR circuit builds
up to
(a) 90%
(b) 99%
(c) 99.9%
of the steady-state value.
For a series LR circuit, the current across the inductor varies as a
function of time. The current across the inductor at time t will
be
…(i)
where i0 is the current at time t=0(also called the steady state value), R is the resistance of the resistor and L is the inductance of the inductor.
We can define a quantity called the time constant for a series LR circuit. It is given as
So equation(i) becomes
…(ii)
We have to find the values of 
for three different values of current
(a)- when i is 90% of i0
90% of i0 is 
i0, so 
Putting these values in eq(ii)
Taking natural logarithm on both sides
(as ln (1) is equal to 0)
The value of 
for which the current is 90% of steady state
value is 2.3.
(b)- when i is 99% of i0
99% of i0 is 
i0, so 
Putting these values in eq(ii)
Taking natural logarithm on both sides
(as ln (1) is equal to 0)
The value of 
for which the current is 99% of steady state
value is 4.6.
(c)- when i is 90% of i0
99.9% of i0 is 
i0, so 
Putting these values in eq(ii)
Taking natural logarithm on both sides
(as ln (1) is equal to 0)
The value of 
for which the current is 99.9% of steady state
value is 6.9.