Power in an AC Circuit | Physics for JEE Main & Advanced PDF Download

7. Power in an AC circuit

In case of a steady current the rate of doing work is given by,

P = Vi

In an alternating circuit, current and voltage both vary with time, so the work done by the source in time interval dt is given by

dW= Vidt

Suppose in an ac, the current is leading the voltage by an angle φ. Then we can write,

V = V₀ sinωt

and i = i₀ sin(ωt + φ)

dW = V₀i₀ sin ωt sin (ωt + φ) dt

= V₀ i₀ (sin² ωt cos f + sinωt cos ωt sin φ) dt

The total work done in a complete cycle is

W = V₀i₀ cos + V₀i₀sin

+  = 

The average power delivered by the source is, therefore,

P =  =  = 

= V_rms i_rms cos φ

or <P>_{one cycle} = V_rms i_rms cos φ

Here, the term cos φ is known as power factor.

It is said to be leading if current leads voltage, lagging if current lags voltage. Thus, a power factor of 0.5 lagging means current lags the voltage by 60° (as cos^-10.5 = 60°). The product of V_rms and i_rms gives the apparent power. While the true power is obtained by multiplying the apparent power by the power factor cos φ. Thus,

and apparent power = V_rms × i_rms

True power = apparent power × power factor

For φ= 0°, the current and voltage are in phase. The power is thus, maximum (V_rms× i_rms). For

φ = 90°, the power is zero. The current is then stated as wattless. Such a case will arise when resistance in the circuit is zero. The circuit is purely inductive or capacitive.