17.6 AC Through Inductor

17.6 AC Through Inductor

An inductor, also called a coil, or choke is a passive two terminal electrical component having a large value of self-inductance and negligible resistance that stores energy in a magnetic field when electric current flows through it. The inductor is used to slow down current surges or spikes by temporarily storing energy in an electromagnetic field and then releasing it back into the circuit. Suppose an inductor of inductance $L$ is connected with an AC source of frequency $f$ with a negligible resistance as shown in Fig. 17.7 (a). Suppose the current is:

If $L$ is the inductance of the coil, then changing current sets up a back emf in the coil of magnitude: $V = L \dfrac{\Delta I}{\Delta t}$ To maintain a constant current, the applied voltage must be equal to the back emf . The magnitude of applied voltage across the coil must have value given by As $I = I_{o} \sin\omega t$; so $V = L \dfrac{\Delta}{\Delta t} (I_{o} \sin\omega t)$

As $\dfrac{\Delta}{\Delta t} \sin(\omega t) = \omega \cos(\omega t)$ , therefore, $V = \omega L I_{o} \cos(\omega t)$ As $\omega L I_{o} = V_{o}$ , so $V = V_{o} \cos(\omega t)$ As $\cos\omega t = \sin(\omega t + \dfrac{\pi}{2})$ , so

(Fig. 17.7 (a): Inductor $L$ is connected with an AC source.)

(Fig. 17.7(b): Graphically showing voltage is leading current in an inductor)

It is seen from Eq. 17.15, the voltage across the inductor $L$ is leading the current, or in other words, current is lagging behind the voltage in AC circuit containing an inductor as shown graphically in Figs. 17.7(b) and 17.7 (c). The current in an inductor always lags behind the voltage by $90^{\circ}$ or $\pi/2$ rad. The resistance offered by an inductor is called inductive reactance denoted by $X_{L}$ and is given as

where $V_{\mathrm{rms}}$ is the rms value of alternating voltage in the inductor and $I_{\mathrm{rms}}$ is the rms current passing through it. As $V_{\mathrm{rms}} = 0.707 V_{o}$ and $I_{\mathrm{rms}} = 0.707 I_{o}$, so $X_{L} = \dfrac{0.707 V_{o}}{0.707 I_{o}} = \dfrac{V_{o}}{I_{o}}$ As $V_{o} = \omega L I_{o}$, so $X_{L} = \dfrac{\omega L I_{o}}{I_{o}}$ Hence

(Fig. 17.7(c): vector representation of $V$ and $I$ in an inductor.)

It shows that inductive reactance $X_{L}$ is directly proportional to both, frequency of current and the inductance $L$ of the inductor. In case of DC, $f = 0$, so $X_{L} = 0$, while in case of large AC frequency, $X_{L}$ is also large. Thus, we conclude that an inductor allows DC but blocks the AC. The unit of $X_{L}$ is ohm.