RMS and AC Power

Overview

RMS and AC Power explains how alternating current is assigned an effective value for heating and power calculations.

For a sinusoidal alternating current, the instantaneous current changes continuously and may be positive or negative. Since the mean current over one complete cycle is zero, a better quantity is needed to compare AC with DC.

That quantity is the root mean square value.

This page supports Alternating Current and links naturally to DC Circuits.

Definition

The rms value of an alternating current is the value of steady direct current that would produce the same average heating effect in a resistor.

Mathematically:

$$I_{\text{rms}}=\sqrt{⟨I^2⟩}$$

Similarly:

$$V_{\text{rms}}=\sqrt{⟨V^2⟩}$$

where $⟨⟩$ means average over one complete cycle.

Why It Matters

A resistor heats according to:

$$P=I^{2}R$$

Heating depends on the square of current, so negative current still produces positive power dissipation.

Hence:

• mean current may be zero
• average heating is not zero

Rms values therefore measure the effective current or voltage for real power transfer.

Key Representations

Derivation for Sinusoidal AC

For sinusoidal current:

$$I=I_0\sin (\omega t)$$

Then:

$$I^2=I_0^2{\sin }^2(\omega t)$$

Average over one cycle:

$$⟨{\sin }^2(\omega t)⟩=\frac{1}{2}$$

So:

$$⟨I^2⟩=\frac{I_0^2}{2}$$

Taking the square root:

$$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$$

RMS Voltage for Sinusoidal AC

If:

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

then similarly:

$$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}$$

Key Relationships

Current:

$$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$$ $$I_0=\sqrt{2}{I}_{\text{rms}}$$

Voltage:

$$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}$$ $$V_0=\sqrt{2}{V}_{\text{rms}}$$

Average Power in a Resistor

For a resistor:

$$P=IV$$

For pure resistance, current and voltage are in phase, so average power is:

$$P_{\text{avg}}=I_{\text{rms}}^{2}R$$

Also:

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

These are the AC versions of the familiar DC power equations.

Mean Power for Sinusoidal AC

For a resistor:

$$I=I_0\sin (\omega t)$$

Instantaneous power:

$$P=I^{2}R=I_0^{2}R{\sin }^2(\omega t)$$

Maximum power:

$$P_{\max }=I_0^{2}R$$

Since the average of ${\sin }^2$ over a cycle is $\frac{1}{2}$:

$$P_{\text{avg}}=\frac{1}{2}{P}_{\max }$$

Peak and RMS Comparison

Quantity	Peak Value	RMS Value
Current	$I_0$	$I_0/\sqrt{2}$
Voltage	$V_0$	$V_0/\sqrt{2}$
Used in waveform equation	yes	no
Used in power rating	rarely	yes

Common Mistakes

Using peak voltage in a power formula

Use $V_{\text{rms}}$ unless the question explicitly works from instantaneous expressions.

Using mean current equals zero

This does not mean no heating.

Forgetting rms is smaller than peak

For sinusoidal AC:

$$\text{rms}=0.707\times \text{peak}$$

Mixing peak current with rms voltage

Use one consistent set of quantities.

Worked Example

A heater has resistance $48\Omega$ connected to a $240\text{V}$ rms supply.

Power:

$$P=\frac{V_{\text{rms}}^2}{R}$$ $$P=\frac{240^2}{48}=1200\text{W}$$

Use rms voltage directly.

Links

• Alternating Current
• DC Circuits
• Alternating Current Common Exam Traps

Summary

Core results:

$$I_{\text{rms}}=\sqrt{⟨I^2⟩}$$ $$V_{\text{rms}}=\sqrt{⟨V^2⟩}$$

For sinusoidal AC:

$$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$$ $$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}$$

Power in a resistor:

$$P=I_{\text{rms}}^{2}R$$ $$P=V_{\text{rms}}{I}_{\text{rms}}$$

Rms values are the effective AC values used in almost all practical calculations.