Chapter 21, Problem 48P

To determine

(a) To determine:

To determine the time constant of the given circuit.

Answer to Problem 48P

Solution:

t = 1.85 ms

Explanation of Solution

Given:

Time $t=2.56ms=0.00256\text{ s }$

Inductance, $L=0.031H$

Formula:

The time constant may be determined from the relation

$I=I_0\left(1-e^{\left(-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.\right)}\right)$

Where, $\text{I}$ is the current in the circuit

${\text{I}}_0$ is the maximum current drawn by the circuit

$\text{E}$ is a constant and is equal to 2.718

$t$ is the time taken to reach the maximum current

$\tau$ is the time constant of the circuit

Calculation:

It is given that the current increases to maximum value of 0.75 from zero.

Therefore, $I=0.75I_0$.

To find the time constant of the circuit, we use the relation

$I=I_0\left(1-e^{\left(-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.\right)}\right)$

Substituting all the known values in the above relation, we get

$0.75I_0=I_0\left(1-{2.718}^{\left(-\raisebox{1ex}{$0.00256$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.\right)}\right)$

$e^{\left(-\raisebox{1ex}{$0.00256$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.\right)}=0.25$

$\frac{-0.00256}{\tau }=\ln \left(0.25\right)$

Therefore, the time constant, $\tau =1.85ms$.

To determine

(b) To determine:

To determine the resistance of the circuit.

Answer to Problem 48P

Solution:

The resistance of the circuit is $16.8\Omega$

Explanation of Solution

Given:

Time $t=2.56ms=0.00256\text{ s }$

Inductance, $L=0.031H$

Formula:

The time constant may be determined from the relation

$I=I_0\left(1-e^{\left(-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.\right)}\right)$

Where, $\text{I}$ is the current in the circuit

${\text{I}}_0$ is the maximum current drawn by the circuit

$\text{E}$ is a constant and is equal to 2.718

$t$ is the time taken to reach the maximum current

$\tau$ is the time constant of the circuit

Calculation:

The time constant of the LR circuit is given by

$\tau =\frac{L}{R}$

Substituting the values of L and t in the above relation, we get

$1.85\times {10}^{-3}=\frac{0.031}{R}$

The resistance (R)of the circuit is $16.8\Omega$