KAHIBARO
Table of Contents
Why RMS is Needed
In alternating current, voltage and current change with time and usually reverse direction. Because of this, a simple average over a full cycle is often not very useful. For example, a sinusoidal current spends as much time positive as negative, so its average value over one complete cycle is zero.
But a zero average does not mean zero effect. An AC current can still heat a resistor, light a bulb, or deliver power. To describe the effective size of an AC voltage or current, we use the RMS value, which stands for root mean square.
The RMS value tells us which DC value would produce the same heating effect in a resistor.
Definition of RMS
For a time dependent quantity $x(t)$, the RMS value over one period $T$ is
$$
x_{\mathrm{rms}} = \sqrt{\frac{1}{T}\int_0^T x^2(t)\,dt}
$$
The name comes from the three steps in the definition. First square the quantity, then take the mean, then take the square root.
This procedure is important because squaring removes the sign. Both positive and negative parts contribute to the result.
The RMS value of an alternating quantity is not the same as its ordinary average over a full cycle.
For many AC waveforms, especially sine waves,
$$
x_{\text{average over full cycle}} = 0
$$
but
$$
x_{\mathrm{rms}} \neq 0
$$
RMS of Sinusoidal Voltage and Current
Consider a sinusoidal voltage
$$
v(t) = V_0 \sin(\omega t)
$$
where $V_0$ is the peak voltage. Its RMS value is
$$
V_{\mathrm{rms}} = \sqrt{\frac{1}{T}\int_0^T V_0^2 \sin^2(\omega t)\,dt}
$$
Since the average value of $\sin^2(\omega t)$ over one full cycle is $\frac{1}{2}$,
$$
V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}}
$$
Similarly, for a sinusoidal current
$$
i(t) = I_0 \sin(\omega t)
$$
the RMS current is
$$
I_{\mathrm{rms}} = \frac{I_0}{\sqrt{2}}
$$
Numerically,
$$
\frac{1}{\sqrt{2}} \approx 0.707
$$
So the RMS value is about $70.7\%$ of the peak value for a sine wave.
For a sinusoidal waveform,
$$
V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}}, \qquad I_{\mathrm{rms}} = \frac{I_0}{\sqrt{2}}
$$
where $V_0$ and $I_0$ are the peak values.
RMS and Heating Effect
The main physical meaning of RMS comes from power in a resistor. Instantaneous power is
$$
p(t) = i^2(t)R = \frac{v^2(t)}{R}
$$
Because power depends on the square of current or voltage, negative current does not cancel positive current in heating.
For a resistor, the average power over one cycle is
$$
P_{\mathrm{avg}} = I_{\mathrm{rms}}^2 R = \frac{V_{\mathrm{rms}}^2}{R}
$$
This is exactly the same form as for DC circuits. That is why RMS values are so useful.
If a resistor connected to AC has $I_{\mathrm{rms}} = 2\,\text{A}$, it heats the same way as it would with a steady DC current of $2\,\text{A}$.
RMS values are effective values for power calculations in resistive circuits:
$$
P_{\mathrm{avg}} = I_{\mathrm{rms}}^2 R = \frac{V_{\mathrm{rms}}^2}{R}
$$
Relation Between Peak, Peak to Peak, and RMS
Several different sizes are used for AC signals. It is important not to confuse them.
| Quantity | Symbol | Meaning for a sine wave |
|---|---|---|
| Peak value | $V_0$, $I_0$ | Maximum magnitude |
| Peak to peak value | $V_{pp}$, $I_{pp}$ | Difference from highest to lowest value |
| RMS value | $V_{\mathrm{rms}}$, $I_{\mathrm{rms}}$ | Effective DC equivalent |
For a sine wave,
$$
V_{pp} = 2V_0, \qquad I_{pp} = 2I_0
$$
and therefore
$$
V_{\mathrm{rms}} = \frac{V_{pp}}{2\sqrt{2}}, \qquad
I_{\mathrm{rms}} = \frac{I_{pp}}{2\sqrt{2}}
$$
Common Example, Household AC
When a wall outlet is labeled as $120\,\text{V}$ or $230\,\text{V}$, this usually means the RMS voltage, not the peak voltage.
For example, if the RMS voltage is $230\,\text{V}$, then the peak voltage is
$$
V_0 = \sqrt{2}\,V_{\mathrm{rms}} \approx 1.414 \times 230 \approx 325\,\text{V}
$$
So the instantaneous voltage actually rises above $230\,\text{V}$ and falls below it during each cycle.
Visual Meaning
A sine wave reaches its peak only for an instant, but the RMS value represents a steady equivalent effect.
RMS for Different Waveforms
The factor $\frac{1}{\sqrt{2}}$ is true only for sine waves. Different waveforms have different RMS values.
| Waveform | Peak value | RMS value |
|---|---|---|
| Sine wave | $A$ | $\dfrac{A}{\sqrt{2}}$ |
| Square wave | $A$ | $A$ |
| Triangular wave | $A$ | $\dfrac{A}{\sqrt{3}}$ |
A square wave has constant magnitude $A$, changing only sign, so its RMS value equals its peak value.
Do not use
$$
x_{\mathrm{rms}} = \frac{x_0}{\sqrt{2}}
$$
for every AC waveform. This is valid only for a sinusoidal waveform.
A Short Calculation Example
Suppose an AC source provides a sinusoidal voltage with peak value $V_0 = 10\,\text{V}$ across a resistor $R = 5\,\Omega$.
First find the RMS voltage:
$$
V_{\mathrm{rms}} = \frac{10}{\sqrt{2}} \approx 7.07\,\text{V}
$$
Then the RMS current is
$$
I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}}{R} = \frac{7.07}{5} \approx 1.41\,\text{A}
$$
The average power is
$$
P_{\mathrm{avg}} = \frac{V_{\mathrm{rms}}^2}{R} = \frac{(7.07)^2}{5} \approx 10\,\text{W}
$$
So this AC source produces the same average heating in the resistor as a $7.07\,\text{V}$ DC source.
Key Idea to Remember
RMS quantities are the effective values of alternating voltage and current. They allow AC circuits, especially resistive ones, to be treated in power formulas just like DC circuits.
For sinusoidal AC,
$$
V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}}, \qquad
I_{\mathrm{rms}} = \frac{I_0}{\sqrt{2}}
$$
and for a resistor,
$$
P_{\mathrm{avg}} = V_{\mathrm{rms}} I_{\mathrm{rms}} = I_{\mathrm{rms}}^2 R = \frac{V_{\mathrm{rms}}^2}{R}
$$