Editing Root mean square (section)

==In common waveforms==
[[File:Waveforms.svg|thumb|right|400px|[[Sine wave|Sine]], [[Square wave (waveform)|square]], [[triangle wave|triangle]], and [[sawtooth wave|sawtooth]] waveforms.  In each, the centerline is at 0, the positive peak is at <math>y = A_1</math> and the negative peak is at <math>y = -A_1</math> ]]
[[File:Dutycycle.svg|thumb|right|400px|A rectangular pulse wave of duty cycle D, the ratio between the pulse duration (<math>\tau</math>) and the period (T); illustrated here with ''a'' = 1.]]
[[File:Sine wave voltages.svg|thumb|right|400px|Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.]]
If the [[waveform]] is a pure [[sine wave]], the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous [[period (physics)|periodic]] wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak [[amplitude]] is:
:''Peak-to-peak'' <math> = 2 \sqrt{2} \times \text{RMS} \approx 2.8 \times \text{RMS}.</math>

For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave:
:''Peak-to-peak'' <math> = 2 \sqrt{3} \times \text{RMS} \approx 3.5 \times \text{RMS}.</math>

{| class="wikitable"
|-
! Waveform !! Variables and operators !! RMS
|-
| [[Direct current|DC]] || <math>y = A_0\,</math> || <math>A_0\,</math>
|-
| [[Sine wave]]
| <math>y = A_1 \sin(2\pi ft)\,</math>
| <math>\frac{A_1}{\sqrt{2}}</math>
|-
| [[Square wave (waveform)|Square wave]]
| <math>y = \begin{cases}
 A_1 & \operatorname{frac}(ft) < 0.5 \\
-A_1 & \operatorname{frac}(ft) > 0.5
\end{cases}</math>
| <math>A_1\,</math>
|-
| DC-shifted square wave
| <math>y = A_0 + \begin{cases}
 A_1 & \operatorname{frac}(ft) < 0.5 \\
-A_1 & \operatorname{frac}(ft) > 0.5
\end{cases}</math>
| <math>{\sqrt{A_0^2 + A_1^2}}\,</math>
|-
| [[Inverter (electrical)#Modified sine wave|Modified sine wave]]
| <math>y = \begin{cases}
 0   & \operatorname{frac}(ft) < 0.25 \\
 A_1 & 0.25 < \operatorname{frac}(ft) < 0.5 \\
 0   & 0.5 < \operatorname{frac}(ft) < 0.75 \\
-A_1 & \operatorname{frac}(ft) > 0.75
\end{cases}</math>
| <math>\frac{A_1}{\sqrt{2}}</math>
|-
| [[Triangle wave]]
| <math>y = \left|2 A_1 \operatorname{frac}(ft) - A_1\right|</math>
| <math>A_1 \over \sqrt 3</math>
|-
| [[Sawtooth wave]]
| <math>y = 2 A_1 \operatorname{frac}(ft) - A_1\,</math>
| <math>A_1 \over \sqrt 3</math>
|-
| [[Pulse wave]]
| <math>y = \begin{cases}
 A_1 & \operatorname{frac}(ft) < D \\
 0   & \operatorname{frac}(ft) > D
\end{cases}</math>
| <math>A_1 \sqrt D</math>
|-
| [[Three-phase electric power|Phase-to-phase sine wave]]
| <math>y = A_1 \sin(t) - A_1 \sin\left(t - \frac{2\pi}{3}\right)\,</math>
| <math>A_1 \sqrt{\frac{3}{2}}</math>
|-
| colspan=3 | where: {{unbulleted list
|''y'' is displacement,
|''t'' is time,
|''f'' is frequency,
|''A{{sub|i}}'' is amplitude (peak value),
|''D'' is the [[duty cycle]] or the proportion of the time period (1/''f'') spent high,
|frac(''r'') is the [[fractional part]] of ''r''.
}}
|}
<!--fixme: add more waveforms-->

===In waveform combinations===
Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of the component RMS values, if the component waveforms are [[orthogonal functions|orthogonal]] (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).<ref>
{{cite web
| last1=Nastase |first1=Adrian S.
| title=How to Derive the RMS Value of Pulse and Square Waveforms
| url=https://masteringelectronicsdesign.com/how-to-derive-the-rms-value-of-pulse-and-square-waveforms/
| website=MasteringElectronicsDesign.com
| access-date=21 January 2015
}}</ref>
:<math>\text{RMS}_\text{Total} =\sqrt{\text{RMS}_1^2 + \text{RMS}_2^2 + \cdots + \text{RMS}_n^2}</math>

Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly.