4.2 Pure inductor circuit

  Alternating Current

The circuit which contains only inductor in the AC circuit is known as pure inductor circuit.

Consider a pure inductor of self induction $i$ and zero resistance is connected to an alternating current as shown in the figure.

Let the instantaneous current $i$ flows in the circuit when the AC power supply is $V = {V_0}\sin \omega t$ at any time $t$.

So, from Kirchoff's loop law we can write,

$$V - L\frac{{di}}{{dt}} = 0$$

Note: $\left( { - L\frac{{di}}{{dt}}} \right)$: Potential drop across the inductor of inductance $L$ when current $i$ is flowing through it.

So, $${V_0}\sin t - L\frac{{di}}{{dt}} = 0$$$$\frac{{{V_0}}}{L}\sin \omega tdt = di$$ or $$\int {di} = \frac{{{V_0}}}{L}\int {\sin \omega tdt} $$$$i = \frac{{{V_0}}}{L}\left( { - \frac{{\cos \omega t}}{\omega }} \right) + c$$$$i = - \frac{{{V_0}}}{{\omega L}}\cos \omega t + c$$

where, $c=0$ as the average of $i$ must be zero over one time period.

$$i = - \frac{{{V_0}}}{{\omega L}}\cos \left( {\frac{\pi }{2} - \omega t} \right)$$$$i = \frac{{{V_0}}}{{\omega L}}\cos \left( {\omega t - \frac{\pi }{2}} \right)\quad ...(ii)$$ Also, $$i = {i_0}\cos \left( {\omega t - \frac{\pi }{2}} \right)$$

Thus, the alternating current lags behind the alternating voltage by a phase angle $\left( {\frac{\pi }{2}} \right)$ when an alternating current flows through a pure inductor circuit.

We can also say that the alternating voltage leads the current by a phase angle of $\left( {\frac{\pi }{2}} \right)$ when an alternating current flows through a pure inductor circuit.

Inductive reactance: It is the opposition offered by the inductor to the flow of alternating current through it.

$${X_L} = \omega L = 2\pi fL$$

The inductive reactance is zero for D.C. as $\left( {f = 0} \right)$ and has a finite value for A.C.

So, we can draw the phasor diagram as,

If we are only intrested in phase relationship, the phasor diagram may also be represented as,