Derive an expression for power dissipated in LCR series AC circuit and explain power factor.

**OR**

Explain principle, construction and working of A.C. generator.

Question

Derive an expression for power dissipated in LCR series AC circuit and explain power factor.

**OR**

Explain principle, construction and working of A.C. generator.

Accepted Answer

>

> [!TLDR]
> The average power dissipated in a series LCR circuit depends on the phase angle between current and voltage through the power factor cosφ = R/Z, and is zero for a purely reactive (L or C only) circuit.
>
> **Setup:** In a series LCR circuit driven by $v = v_m\sin\omega t$, the current lags/leads the voltage by phase angle φ:
> $$i = i_m\sin(\omega t - \phi)$$
> where $i_m = v_m/Z$, $Z = \sqrt{R^2 + (X_L - X_C)^2}$ is the impedance, and $	an\phi = \dfrac{X_L - X_C}{R}$.
>
> **Instantaneous power:**
> $$p = vi = v_m\sin\omega t \cdot i_m\sin(\omega t - \phi)$$
> Expanding $\sin(\omega t - \phi) = \sin\omega t\cos\phi - \cos\omega t\sin\phi$:
> $$p = v_mi_m\left[\sin^2\omega t\cos\phi - \sin\omega t\cos\omega t\sin\phiight]$$
>
> **Averaging over a full cycle:** $\langle\sin^2\omega tangle = 1/2$ and $\langle\sin\omega t\cos\omega tangle = 0$, so
> $$\bar{P} = \frac{v_mi_m}{2}\cos\phi = \frac{v_m}{\sqrt2}\cdot\frac{i_m}{\sqrt2}\cos\phi = V_{rms}I_{rms}\cos\phi$$
>
> **Power factor:** The term $\cos\phi$ is called the **power factor**. Using the impedance triangle, $\cos\phi = R/Z$, so equivalently
> $$\bar{P} = I_{rms}^2 R$$ …

Q.Derive an expression for power dissipated in LCR series AC circuit and explain power factor. OR Explain principle, construction and working of A.C. generator.

Concept understanding — Average Power Absorption

Average Power Absorption – From Intuition to Precision

The Intuitive Picture

The Precise Definition

Why This Matters for Exams

The General Case (Phase Differences)

Why this formula?

Average Power Absorption: Why the Formula Holds

1. Instantaneous Power — The Starting Point

2. Why We Need an Average

3. The Key Derivation (Step-by-Step)

Step 1: Write the sinusoidal forms

Step 2: Instantaneous power

Step 3: Average over one cycle

4. The Standard Form Using RMS Values

5. What $\cos \phi$ Really Means

Q.Derive an expression for power dissipated in LCR series AC circuit and explain power factor. OR Explain principle, construction and working of A.C. generator.

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