Name: Energy Losses in a Transformer
Uploaded: 2019-07-19
Duration: 8 min 12 s
Description: There are different types of energy losses in a transformer. This video explains those losses in detail and also talks about the efficiency of ideal and prac...

Question

So that average power dissipated by a pure inductor and a pure capacitor is zero when they are connected to an a.c. voltage source.

Accepted Answer

Average power dissipated is the amount of energy lost due to unwanted means in an electronic device or circuit. This waste occurs because the energy gets converted to another form which is an unnecessary by-product, for example, heat, sound etc

In the case of a capacitor, the voltage applied is given by

$E = E_{0} \sin ω t$ and

$I = I_{0} \sin (ωt + ϕ)$

where $ϕ$ = phase difference between the voltage and the current. In case of a pure capacitor, the current leads the voltage by $\frac{π}{2}$ , hence we have,

$I = I_{0} \sin (wt + \frac{π}{2})$ .

The work done over a complete cycle is,

$W = \int_{0}^{T} EI = \int_{0}^{T} (E_{0} sinωt) (I_{0} \cos ωt) = \frac{E_{0} I_{0}}{2} \int_{0}^{T} 2 sinωtcosωt dt$ .

$= \frac{E_{0} I_{0}}{2} \int_{0}^{T} \sin 2 ωt dt$ .

$= \frac{E_{0} I_{0}}{2} {[\frac{- \cos 2 ωt}{2 ω}]}_{0}^{T} = zero$ ............................................................(A)

In the case of Pure inductor, the current lags behind the applied voltage, therefore we have,

$I = I_{0} \sin (ωt - ϕ)$ where $ϕ = \frac{π}{2}$

Power at an instant $t = \frac{dW}{dt} = EI = (E_{0} sinωt) \times I_{0} \sin (ωt - ϕ)$ .

On solving, we get for total work done,

$W = \int_{0}^{T} E_{0} I_{0} \sin^{2} ωtcosϕdt - \frac{E_{0} I_{0}}{} \int_{0}^{T} \frac{E_{0} I_{0}}{2} \sin 2 ωtsinϕdt$

= $E_{0} I_{0} cosϕ \int_{0}^{T} \sin^{2} ωtdt - \frac{E_{0} I_{0}}{2} sinϕ \int_{o}^{T} \sin 2 ωt dt$

= $E_{0} I_{0} cosϕ \times \frac{T}{2}$ .

Therefore the average power over the complete cycle

$P = \frac{W}{T} = \frac{E Icosϕ}{T} . \frac{T}{2} = \frac{E_{0}}{\sqrt{2}} . \frac{I_{0}}{\sqrt{2}} cosϕ = 0 as ϕ = \frac{π}{2}$

Therefore, the average power dissipation in a pure inductor is zero.

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So that average power dissipated by a pure inductor and a pure capacitor is zero when they are connected to an a.c. voltage source.

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