Derivation of Average Power Dissipated in a Series LCR Circuit:

In a series LCR circuit, a voltage V = V0 sin ωt is applied. The circuit consists of an inductor (L), a capacitor (C), and a resistor (R). We need to derive the expression for the average power dissipated over a cycle.

1. Instantaneous Power Dissipation:
The instantaneous power dissipated in the series LCR circuit can be calculated using the formula:
P(t) = V(t) * I(t)
where P(t) is the instantaneous power, V(t) is the voltage across the circuit, and I(t) is the current flowing through the circuit.

Since the voltage across the LCR circuit is given as V(t) = V0 sin ωt, and the current can be represented as I(t) = I0 sin (ωt + φ), where I0 is the maximum current and φ is the phase difference between the voltage and current, we can rewrite the equation for instantaneous power as:
P(t) = V0 sin ωt * I0 sin (ωt + φ)

2. Average Power Dissipation:
The average power dissipated over a cycle can be determined by integrating the instantaneous power over one complete cycle and then dividing it by the period of the waveform.

3. Integration of the Instantaneous Power:
Integrating the product of sin functions can be simplified by using the trigonometric identity:
sin α sin β = 1/2 [cos(α - β) - cos(α + β)]

Applying this identity to the equation for instantaneous power, we get:
P(t) = (V0 * I0 / 2) [cos φ - cos (2ωt + φ)]

4. Average Power Dissipation:
To find the average power dissipated, we need to integrate the instantaneous power over one complete cycle and divide by the period T.

Integrating over one complete cycle (0 to T), we get:
∫[0 to T] P(t) dt = (V0 * I0 / 2) [∫[0 to T] cos φ dt - ∫[0 to T] cos (2ωt + φ) dt]

The first integral on the right-hand side gives us:
∫[0 to T] cos φ dt = φT

The second integral on the right-hand side gives us:
∫[0 to T] cos (2ωt + φ) dt = 0

Since the integral of cos (2ωt + φ) over one complete cycle is zero, the average power dissipated over a cycle becomes:
P_avg = (V0 * I0 / 2) * φT / T = (V0 * I0 / 2) φ

Conditions for Power Dissipation:

(i) No Power Dissipated:
No power is dissipated when the phase difference between voltage and current is such that cos φ = 1. This occurs when the circuit is purely resistive (R) and there is no reactance from the inductor (L) or capacitor (C). In this case, the current is in phase with the voltage, and power is not dissipated.

(ii) Maximum Power Dissipated:
Maximum power