Time Constant of the RL Circuit

Author: E.E. Kimberly

An examination of equation (8-2) will reveal that the time required for the current in an RL circuit to rise to a certain percentage of its final value is proportional to L and inversely proportional to R. The larger the ratio of the inductance to the resistance, the more time will be taken for the current to rise and the longer will be the time of the effective transient.

Fig, 8-3. Transient Rise of Current in an LR Circuit

The ratio L/R is called the time constant of the circuit. When t equals L/R equation (8-2) reduces to

or

(8-3)

Therefore, after a time t=L/R (beginning when t = 0), the current will have risen to 63.2 per cent of its final value.

Fig. 8-4. Time Constant in LR Circuit

The initial rate of rise of current after the circuit is closed may be found by differentiating equation (8-2) with respect to t. Thus,

A current rising at the rate of V/L amperes per second will require t₁ seconds to reach the steady-state value of V/R. This statement may be expressed by the relation

from which

It is true, therefore, that the time constant L/R of a circuit is equal to the time that would be required for the current i to reach its final value of V/R if it continued to rise at its initial rate when t = 0. Fig. 8-4 shows that relationship.

In the circuit of Fig. 8-3 the voltage Ri across the resistance, being proportional to i, will have a rising characteristic similar to that of i in

Fig. 8-4; this is shown in Fig. 8-5. The difference between the constant voltage V and the rising voltage drop Ri is the voltage

Fig. 8-5. Rate of Change of Voltage Across L and R With Transient Rise of Current