Time Constant

Let us take, a simple RC circuit as shown below.
RC circuit
Let us assume, the capacitor is initially unchanged and the switch S is closed at time t = 0. After closing the switch, electric current i(t) starts flowing through the circuit. Applying Kirchhoff Voltage Law in that single mesh circuit, we get,

Differentiating both sides with respect to time t, we get,

Integrating both side we get,

Now, at t = 0, the capacitor behaves as short circuit, so, just after closing the switch, the current through the circuit will be,

Now, putting this value in equation (i) we get,

Putting the value of k at equation (i) we get,

Now, if we put t = RC in the final expression of circuit current i(t), we get,

From the above mathematical expression it is clear that RC is the time in second during which the current in a charging capacitor diminishes to 36.7 percent from its initial value. Initial value means, current at the time of switching on the unchanged capacitor.
This term is quite significant in analyzing behavior of capacitive as well as inductive circuit. This term is known as time constant. So time constant is the duration in seconds during which the current through a capacities circuit becomes 36.7 percent of its initial value. This is numerically equal to the product of resistance and capacitance value of the circuit. Time constant is normally denoted by τ (tau). So,

In complex RC circuit, time constant will be product of equivalent resistance and capacitance of circuit.
Let us discuss significance of time constant in more details for that, we plot current i(t).
charging current plot
At t = 0, the current through the capacitor circuit is

At t = RC, the current through the capacitor is

Let us consider, another RC circuit.
RC circuit
Circuit equations using KVL of the above the circuits are,


From (iii) and (v)

Differenting both sides with respect to time t, we get,

Integrating both sides we get,

At t = 0,

Time constant of this circuit would be 2RC/3 sec. Now, the equivalent resistance of the circuit is,

Time constant of the circuit has become

Let us consider an example of series RL circuit
RL circuit
Applying Kirchhoff Voltage Law in the above circuit. we get,

The equation can also be solved Laplace Transformation technique. For that we have to take Laplace Transformation of the equation at both sides,

Hence, in this equation.

Since, current just after switch is on, current through the inductor will be zero.

Taking inverse Laplace of the above equation, we get,

Now, if we put,

We get,

So, at RL circuit, at time = L/R sec the current becomes 63.3% of its final steady state value. The L/R is known as time constant of an LR circuit. Let us plot the current of inductor circuit.
current of inductor circuit
The time constant of an LR circuit is the ratio of inductance to resistance of the circuit. Let us take another
This circuit can be redrawn as,

So, time constant of the circuit would be

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