20.4.1 A.C. Through a Resistor

When an alternating voltage is applied across a pure resistor, the behaviour of the circuit is the simplest of all AC circuit elements.

Applied Voltage and Resulting Current

Let the instantaneous voltage applied across a resistor $R$ be:

$V=V_0\sin (\omega t)$

By Ohm's Law, the instantaneous current is:

$I=\frac{V}{R}=\frac{V_0}{R}\sin (\omega t)=I_0\sin (\omega t)$

where $I_0=\frac{V_0}{R}$ is the peak current.

Phase Relationship

Both $V$ and $I$ vary as $\sin (\omega t)$, so they are in phase with each other. The phase difference $\varphi =0°$. This means:

• Voltage and current reach their maximum values at the same instant.
• They pass through zero at the same instant.
• They reach their minimum (negative peak) at the same instant.

Phasor Diagram

In a phasor diagram, the voltage phasor $V_0$ and the current phasor $I_0$ are drawn parallel to each other (in the same direction), confirming zero phase difference.

Ohm's Law in RMS Form

Since $V_{rms}=\frac{V_0}{\sqrt{2}}$ and $I_{rms}=\frac{I_0}{\sqrt{2}}$, Ohm's Law holds in RMS form:

$V_{rms}=I_{rms}R$

This is identical in form to the DC case.

Frequency Independence of Resistance

The resistance $R$ of a pure resistor is determined by the material, length, and cross-sectional area of the conductor. It does not depend on the frequency of the AC supply. This distinguishes a resistor from inductors and capacitors, whose opposition to AC (reactance) is frequency-dependent.

Power Dissipation

The instantaneous power is:

$P_{inst}=VI=V_0{I}_0{\sin }^2(\omega t)$

The average power over a complete cycle is:

$P_{avg}=V_{rms}{I}_{rms}=I_{rms}^{2}R=\frac{V_{rms}^2}{R}$

Equivalently, since $V_{rms}=V_0/\sqrt{2}$ and $I_{rms}=I_0/\sqrt{2}$:

$P_{avg}=\frac{V_0{I}_0}{2}=\frac{1}{2}{V}_0{I}_0$

Because $\varphi =0°$, the power factor $\cos (0°)=1$, meaning all the energy supplied by the source is dissipated as heat in the resistor. Power is never negative in a purely resistive AC circuit.

Summary Table