A voltage V = V₀sin(ωt) is applied to a series LCR circuit. Derive the expression for the average power dissipated over a cycle. Under what condition is (i) no power dissipated even though the current flows through the circuit, (ii) maximum power dissipated in the circuit?

Power factor of the circuit is given by: cosφ = R/Z = R/√(R² + (ωL - 1/ωC)²)
Current flowing in the circuit is given by: I = V/Z = V₀sin(ωt)/Z
Instantaneous real power dissipated in the circuit is: P = I²R = V₀²sin²(ωt)R/Z²
Average power dissipated in a cycle is given by:

= ∫₀²π/ω Pdt / ∫₀²π/ω dt = (V₀²R/Z²) x (2π/ω) ∫₀²π/ω (1 - cos(2ωt))dt

= VrmsIrmscos(φ)
(i) No power is dissipated when P = 0
This implies cosφ = 0
φ = π/2
That is the circuit is purely inductive or capacitive
(ii) Maximum power is dissipated when P is maximum.
This implies cosφ = 1
φ = 0
Circuit is purely resistive.