A battery with emf is connected in series with an inductance L and a resistance

Question

A battery with emf is connected in series with an inductance L and a resistance R. (a) Assuming the current has reached steady state when it is at 99% of its maximum value, how long does it take to reach steady state, assuming the initial current is zero? (Use any variable or symbol stated above as necessary. To represent, use E.) t9996 = (b) If an emergency power circuit needs to reach steady state within 2.4 ms of turning on and the circuit has a total resistance of 68 , what values of the total inductance of the circuit are needed to satisfy the requirement? (Give the maximum value.)

Explanation / Answer

In L,R circuit  

we have the condition for current i(t ) = I0 (1-e^(-t/T))

initially at t=0 , i = 0

to reach steady state that is 99% of I0 , it took time t

and T = L/R which is time constant of LR circuit , it t = T , so that the currnt in the circuit reaches its 63% of maximum value (I0)

a)

(99/100)I0 = I0(1-e^(-t/(L/R)))

0.99 = 1-e^(-t/(L/R))

(-t/(L/R)) = ln(0.01 )

t = 4.60*(L/R)

given R= 68 ohm , time to reach the steady state is t = 28 ms , so

b)

0.99 = 1-e^(-t/(L/R))

0.99 = 1-e^(-0.028/(L/68))

L = 413 mH

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