Alternating Current

Lesson

Part 1 — how an alternating current is described and measured, and how resistors, coils and capacitors share an AC supply.

In this lesson you will

Describe a sinusoidal AC by its peak, period, frequency and phase, and derive its half-cycle mean and rms values
Find the phase and the reactance for a pure resistor, inductor and capacitor, and read reactance–frequency graphs
Use phasors to find the impedance, current and phase angle of RL, RC and series LCR circuits

Diagram

Sinusoidal AC: period, frequency and phase

A sinusoidal current is written $i = I_0\sin(\omega t + \phi_0), \qquad \omega = 2\pi f = \dfrac{2\pi}{T}$ $I_0$ is the peak value, $T$ the period, $f$ the frequency and $(\omega t + \phi_0)$ the phase.

A phasor is a vector of length $I_0$ turning anticlockwise at $\omega$ ; its projection on a fixed line gives the instantaneous value.
Two quantities of the same frequency differ in phase by a fixed angle $\phi$ ; the one whose phasor is ahead leads.
Indian mains: 230 V (rms), $f = 50$ Hz, so $T = 20$ ms and $\omega \approx 314$ rad s $^{-1}$ .

Concept

Mean value over a half cycle

Over a full cycle the positive and negative halves cancel, so the mean current is zero. The useful mean is over a half cycle: $\bar I = \dfrac{1}{T/2}\int_0^{T/2} I_0\sin\omega t\,dt = \dfrac{2I_0}{\omega T}\Big[-\cos\omega t\Big]_0^{T/2} = \dfrac{2I_0}{\pi}$

$\bar I = 2I_0/\pi \approx 0.637\,I_0$ ; likewise $\bar V = 2V_0/\pi$ .
It gives the charge carried in one half cycle, $\bar I \times T/2$ : the shaded area under the curve.

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