Question

Obtain an expression for average power of AC over a cycle. Discuss its special cases.

Answer 1

It is given by the product of the voltage and current. In an AC circuit, the voltage and current vary continuously with time. Let us first calculate the power at an instant and then it is averaged over a complete cycle.

The alternating voltage and alternating current in the series RLC circuit at an instant are given by

υ = V_m sin ωt and i = I_m sin (ωt + 4>)

where φ is the phase angle between υ and i. The instantaneous power is then written as

P = υi = V_m I_m sin ωt sin(ωt + φ)

= V_m I_m sin ωt (sin ωt cos φ – cos ωt sin φ)

P = V_m I_m (cos φ sin² ωt – sin ωt cos ωt sin φ) …… (1)

Here the average of sin² ωt over a cycle is `1/2` and that of sin ωt cos ωt is zero. Substituting these values, we obtain average power over a cycle.

P_av = V_m I_m cos φ `xx 1/2 = "V"_"m"/sqrt2 "I"_"m"/sqrt2` cos φ

P_av = V_RMS I_RMS cos φ …… (2)

where V_RMS I_RMS is called apparent power and cos φ is power factor. The average power of an AC circuit is also known as the true power of the circuit.

Special Cases:

1. For a purely resistive circuit, the phase angle between voltage and current is zero and cos
   φ = 1.
   ∴ P_av = V_RMS I_RMS
   
2. For a purely inductive or capacitive circuit, the phase angle is 
   `+- pi/2 and cos (+- pi/2) = 0`
   ∴ P_av = 0
3. For series RLC circuit, the phase angle φ = tan^-1 `(("x"_"L" - "x"_"c")/"R")`
   

   ∴ P_av = V_RMS I_RMS cos φ

4. For series RLC circuit at resonance, the phase angle is zero and cos φ = 1.
   ∴ P_av = V_RMS I_RMS