**Rise of Current in an Inductive Circuit:** In figure 1 is shown a resistance of R in series with a coil of self-inductance L henry, the two being put across a battery of V volt. The R-L combination becomes connected to battery when switch SW is connected to terminal ‘a’ and is short-circuited when SW is connected to ‘b’. the inductive coil is assumed to be resistance-less, its actual small resistance being included in R.

When SW1 is connected to ‘a’ the R-L combination is suddenly put across the voltage of V volt.

### Derivation of Rise of Current in Inductive Circuit

Let us take the instant of closing switch SW1 as the starting zero time. It is found that current does not reach its maximum value instantaneously but take some finite time. It is easily explained by recalling that the coil possesses electrical inertia i.e. self-inductance and hence, due to the production of the counter e.m.f. of self-inductance, delays the instantaneous full establishment of current through it.

We will now investigate the growth of current I through such an inductive circuit.

The applied voltage *V* must, at any instant, supply not only the ohmic drop iR over the resistance R but must also overcome the e.m.f. of self-inductance i.e. .

Therefore,

Or,

Therefore, …(i)

Multiplying both sides by (-R). we gate

Integrating both sides, we get

Therefore, …(ii)

Where *e* is the Napierian logarithmic base = 2.718 and *K* is constant of integration whose value can be found from the initial known conditions.

To begin with, when t=0, i=0, hence putting these values in (ii) above, we get

Substituting this value of *K* in the above equation, we have

.

Or, ’time constant’

Therefore, \dfrac{V – iR}{V} = e^{-t\lambda} or i = \dfrac{V}{R}(1 – e^{dfrac{-t}{\lambda}})[/latex]

Now, represent the maximum steady value of current I_{m} that would eventually be established through the R – L circuit.

Therefore, …(iii)

This is an exponential equation whose graph is shown in figure 2. It is seen from it that current rise is rapid at first and then decreases until at , it becomes zero. Theoretically, current does not reach its maximum steady value *I _{m} *until infinite time. However, in practice, it reaches this value in a relatively short time of about .

The rate of rise of current at any stage can be found by differentiating Eq. (iii) above w.r.t. time. However, the initial rate of rise of current can be obtained by putting t = 0 and i = 0 in (i) above.

The constant is known as the time-constant of the circuit. It can be variously defined as:

- It is the time during which current would have reached its maximum value of had it maintained its initial rate or rise

But actually the current takes makes more time because its rate of rise decrease gradually. In actual practice, in a time equal to the time constant, it merely reaches 0.632 of its maximum value as shown below:

Putting in equation (iii) above, we get

- Hence, the time constant of an R-L circuit may also be defined as the time during which the current actually rises to 0.632 of its maximum steady value (Figure 1).

This delay rise of current in an inductive circuit is utilized in providing time lag in the operation of electric relays and trip coils etc.

### Decay of current in an inductive circuit

When the switch SW is connected to point ‘b’, the *R-L* circuit is short-circuited. It’s found that the current does not cease immediately, as it would do in a non-inductive circuit, but continues to flow and is reduced to zero only after an appreciable time has elapsed since the instant of short-circuit.

### Derivation of Decay of Current in Inductive Circuit

The equation for decay of current with time is found by putting V = 0

Integrating both sided, we have

Therefore, …..(i)

Now, at the instant of switching off the circuit, i = I_{m }and if time is counted from this instant, then t = 0

Therefore,

Putting the value of K in Eq (i) above, we get,

It is a decaying exponential function and is plotted in figure 3. It can be shown again that theoretically, current should take infinity time to reach zero value although, in actual practice, it does so in a relatively short time of about

Again, putting in equation (ii) above, we get

Hence, time constant of an *R-L* circuit may also be defined as the tie during which current falls to 0.37 or 37% of its maximum steady value while decaying.

### Electronics Pani Note:

Initial value of can also be found by differentiating equation (iii) and putting t = 0 in it. In fact, the three quantities *V, L, R* gives the following various combinations:

(the maximum final steady current.)

= initial rate of rise of current

= time constant of the circuit

The first rule of switching is that the current flowing through an inductance cannot change instantaneously. The second rule of switching is that the voltage across a capacitor cannot change instantaneously.