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Average Power Dissipated In A Pure Inductor

Average Power Dissipated In A Pure Inductor :- The instantaneous power of a purely inductive circuit,

$\displaystyle P=E\times I=\left( {{E}_{0}}sin\text{ }\omega t \right)\times \left[ {{I}_{0}}sin\left( \omega t-\frac{\pi }{2} \right) \right]$

$\displaystyle \Rightarrow P={{E}_{0}}{{I}_{0}}\left( sin\text{ }\omega t \right)\times \left[ sin\left( \omega t-\frac{\pi }{2} \right) \right]={{E}_{0}}{{I}_{0}}\left( sin\text{ }\omega t \right)\left( -cos\text{ }\omega t \right)$

$\displaystyle \Rightarrow P=-{{E}_{0}}{{I}_{0}}\left( sin\text{ }\omega t \right)\left( cos\text{ }\omega t \right)$

$\displaystyle \Rightarrow P=\frac{-{{E}_{0}}{{I}_{0}}}{2}\left( 2\text{ }sin\omega t\text{ }cos\omega t \right)$

$\displaystyle \Rightarrow P=\frac{-{{E}_{0}}{{I}_{0}}}{2}sin2\omega t$

Now, over one complete cycle, the value of (sin 2ωt) becomes zero; therefore, over one complete cycle, the average power of the circuit is :-

$\displaystyle {{P}_{avg.}}=\langle \frac{-{{E}_{0}}{{I}_{0}}}{2}sin2\omega t\rangle =\frac{-{{E}_{0}}{{I}_{0}}}{2}\langle sin2\omega t\rangle =\frac{-{{E}_{0}}{{I}_{0}}}{2}\times 0=0$

Thus, in one complete cycle, the average power dissipated in a pure inductor is zero. In fact, the energy that an inductor takes from the source during magnetization is returned to the source during demagnetization.

The magnetization and demagnetization of an inductor can be understood as follows :-

Similarly, in the following graph, one complete cycle of alternating voltage E, alternating current I, magnetic flux Φ, and electric power P is shown :-

In figure (a), the electric current enters the inductor at point A and leaves at point B. From time t = 0 to t = T/4, the values of current and magnetic flux increase from zero to their maximum. During this time, both the alternating voltage E and the alternating current I are positive, and therefore the electric power P is also positive. Hence, the inductor absorbs energy from the AC source, and the core of the inductor becomes magnetized.
In figure (b), during the interval from time t = T/4 to t = T/2, the direction of electric current remains the same, but both the current and the magnetic flux decreases from their maximum values to zero. During this time, the alternating voltage E is negative while the alternating current I is positive; therefore, the electric power P is negative. Hence, the inductor supplies energy to the AC source, and the core of the inductor gets demagnetized.
In figure (c), after the half cycle, the electric current now enters the inductor at point B and leaves at point A. From time t = T/2 to t = 3T/4, the current and magnetic flux increase from zero to their maximum values in the opposite direction. Since the direction of current has reversed, the polarity across the inductor has also reversed. During this time, both the alternating voltage E and the alternating current I are negative, and therefore the electric power P is positive. Hence, the inductor absorbs energy from the AC source, and the core of the inductor becomes magnetized.
In figure (d), during the interval from time t = 3T/4 to t = T, the current and magnetic flux decrease from their maximum value to zero. At this time, the alternating voltage E is positive and the alternating current I is negative; therefore, the electric power P is negative. Hence, the energy taken from the AC source during the interval t = T/2 to t = 3T/4 is returned to the source by the inductor, and the core of the inductor becomes demagnetized.

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Average Power Dissipated In A Pure Inductor

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