Alternating Current

Lesson

Part 1 — voltages and currents that reverse direction many times a second, and how resistors, coils and capacitors respond to them.

In this lesson you will

Relate the peak and rms values of an alternating current and voltage
Describe the phase relations for AC through a resistor, an inductor and a capacitor using phasors
Calculate inductive and capacitive reactance

Diagram

Alternating voltage and current

v = V₀ sin ωt

The rms value is 0.707 of the peak for a sinusoidal supply.

A sinusoidal supply gives $v = V_0\sin\omega t$ with $\omega = 2\pi f$ . In India $f = 50$ Hz, so $\omega \approx 314$ rad s $^{-1}$ . The current in a resistor $R$ is $i = \dfrac{V_0}{R}\sin\omega t = I_0\sin\omega t$ .

The average of $\sin\omega t$ over a cycle is zero, so the average current is zero.
The average of $\sin^2\omega t$ is 1/2, so the average power is not zero.

Concept

Peak and rms values

The rms (root-mean-square) value is the steady (DC) value that would give the same average heating in a resistor: $\bar P = I_\text{rms}^2 R = \dfrac{I_0^2 R}{2}\ \Rightarrow\ I_\text{rms} = \dfrac{I_0}{\sqrt2} \approx 0.707I_0,\qquad V_\text{rms} = \dfrac{V_0}{\sqrt2}$

Mains voltage “220 V” is an rms value; the peak is about 311 V.
Ammeters and voltmeters for AC read rms values.

v = V₀ sin ωt

The rms value is 0.707 of the peak for a sinusoidal supply.

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