Alternating Current — Part 2: Power, resonance and the AC generator: visual lesson, ISC Class 12 Physics

Lesson

AC Power, Resonance and the Generator

Part 2 — how much power an AC circuit really takes, how a circuit can pick out one frequency, and where alternating current comes from.

In this lesson you will

Derive and use the average power V_rms I_rms cos φ, the power factor R/Z and the idea of wattless current, and explain the choke coil
Find the resonant frequency, bandwidth and Q factor of a series LCR circuit, and describe LC oscillations qualitatively
Explain the construction and working of the AC generator, derive its sinusoidal emf, and compare AC with DC

Diagram

Average power in an AC circuit

v / V₀
i / I₀ (lags by 60°)
p / V₀I₀ = vi / V₀I₀

p = vi; the average is V_rms I_rms cos φ.

With $v = V_0\sin\omega t$ and $i = I_0\sin(\omega t - \phi)$ , the instantaneous power is $p = vi = V_0I_0\sin\omega t\sin(\omega t - \phi) = \dfrac{V_0I_0}{2}\left[\cos\phi - \cos(2\omega t - \phi)\right]$ The second term averages to zero over a cycle, so $P = \dfrac{V_0I_0}{2}\cos\phi = V_\text{rms}I_\text{rms}\cos\phi$

$V_\text{rms}I_\text{rms}$ is the apparent power (volt-ampere, VA); $P$ is the true (average) power, in watts.

Concept

Power factor

The factor $\cos\phi$ is the power factor. For a series LCR circuit $\cos\phi = \dfrac{R}{Z} = \dfrac{P}{V_\text{rms}I_\text{rms}}, \qquad P = I_\text{rms}^2R$

Pure resistor: $\cos\phi = 1$ . Pure inductor or capacitor: $\cos\phi = 0$ , so no average power: energy taken in one quarter cycle is returned in the next.
Only the resistance turns electrical energy into heat or work.

Table of the phase angle, power factor and average power for a pure resistor, a pure inductor or capacitor, a general series LCR circuit and an LCR circuit at resonance
Load	Phase angle φ	Power factor cos φ	Average power
Pure resistor	0	1	$V_\text{rms}I_\text{rms}$
Pure inductor or pure capacitor	90°	0	zero (wattless)
Series LCR	$\tan^{-1}\dfrac{X_L - X_C}{R}$	$R/Z$	$I_\text{rms}^2R$
Series LCR at resonance	0	1	$V_\text{rms}^2/R$

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