Form factor (electronics): Difference between revisions

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In [[electronics]] and [[electrical engineering]], the '''form factor''' of an [[alternating current]] waveform (signal) is the ratio of the RMS ([[Rootroot Meanmean Squaresquare]]) value to the [[Average rectified value|average value]] (mathematical mean of [[absolute value]]s of all points on the waveform).<ref>{{cite web|last=Stutz|first=Michael|title=Measurement of AC Magnitude|url=http://www.allaboutcircuits.com/vol_2/chpt_1/3.html|work=BASIC AC THEORY|accessdate=30 May 2012}}</ref> It identifies the ratio of the [[direct current]] of equal power relative to the given alternating current. The former can also be defined as the direct current that will produce equivalent heat.<ref name=Dusza>{{cite book|last=Dusza|first=Jacek|title=Podstawy Miernictwa (Foundations of Measurement)|year=2002|publisher=Wydawnictwo Politechniki Warszawskiej|___location=Warszawa|isbn=83-7207-344-9|pages=323|coauthorsauthor2=Grażyna Gortat, |author3=Antoni Leśniewski |pages=136-142136–142, 197-203197–203|language=Polish}}</ref>

== Calculating the form factor ==

For an ideal, continuous wave function over time T, the RMS can be calculated in [[integral]] form:<ref name="Jędrzejewski">{{cite book|last=Jędrzejewski|first=Kazimierz|title=Laboratorium Podstaw Pomiarow|year=2007|publisher=Wydawnictwo Politechniki Warszawskiej|___location=Warsaw|isbn=978-978-83-7207-43|pages=86–87|language=Polish}}</ref>:

<math>

X_\mathrm{rms} = \sqrt {{1 \over {T}} {\int_{t_0}^{t_0+T} {[fx(t)]}^2\, dt}}

</math>

The rectified average is then the mean of the integral of the function's absolute value:<ref name="Jędrzejewski" /> :

<math>

X_\mathrm{arv} = {1 \over {T}} {\int_{t_0}^{t_0+T} {|x(t)|\, dt}}

</math>

The [[quotient]] of these two values is the form factor, <math>k_fk_\mathrm{f}</math>, or in unambiguous situations, <math>k</math>.

<math>

k_fk_\mathrm{f} = \frac \mathrm{RMS} \mathrm{ARV} = \frac{\sqrt {{1 \over {T}} {\int_{t_0}^{t_0+T} {[fx(t)]}^2\, dt}}}{{1 \over {T}} {\int_{t_0}^{t_0+T} {|x(t)|\, dt}}} = \frac{\sqrt{T\int_{t_0}^{t_0+T}{{[fx(t)]}^2\, dt}}}{\int_{t_0}^{t_0+T} {|x(t)|\, dt}}

</math>

<math>X_\mathrm{rms}</math> reflects the variation in the function's distance from the average, and is disproportionately impacted by large deviations from the unrectified average value.<ref>{{cite web|title=Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE)|url=http://www.eumetcal.org/resources/ukmeteocal/verification/www/english/msg/ver_cont_var/uos3/uos3_ko1.htm|archive-url=https://web.archive.org/web/20070714143619/http://www.eumetcal.org/resources/ukmeteocal/verification/www/english/msg/ver_cont_var/uos3/uos3_ko1.htm|url-status=dead|archive-date=14 July 2007|publisher=The European Virtual Organisation for Meteorological Training|accessdate=30 May 2012}}</ref>

It will always be at least as large as <math>X_\mathrm{arv}</math>, which only measures the absolute distance from said average. The form factor thus cannot be smaller than 1 (a square wave where all momentary values are equally far above or below the average value; see below), and has no theoretical upper limit for functions with sufficient deviation.

<math>RMS_\mathrm{RMS}_\mathrm{total} = \sqrt{{RMS_1{\mathrm{RMS}_1}^2} + {RMS_2{\mathrm{RMS}_2}^2} + ... + {RMS_n{\mathrm{RMS}_n}^2}}</math>

can be used for combining signals of different frequencies<ref name=Two>{{cite web|title=Rms value of two sine waves|url=http://wiki.answers.com/Q/Rms_value_of_sum_of_two_sine_waves|publisher=Answers.com|accessdate=10 June 2012}}</ref> (for example, for harmonics<ref name=Dusza />), while for the same frequency, <math>RMS_\mathrm{RMS}_\mathrm{total} = RMS_1\mathrm{RMS}_1 + RMS_2\mathrm{RMS}_2 + ... + RMS_n\mathrm{RMS}_n</math>.

As ARV's on the same ___domain can be summed as

<math>ARV_\mathrm{ARV}_\mathrm{total} = ARV_1\mathrm{ARV}_1 + ARV_2\mathrm{ARV}_2 + ... + ARV_n\mathrm{ARV}_n</math>,

the form factor of a complex wave composed of multiple waves of the same frequency can sometimes be calculated as

<math>k_{f_\mathrm{f}_\mathrm{tot}} = \frac{RMS_\mathrm{RMS}_\mathrm{tot}}{ARV_\mathrm{ARV}_\mathrm{tot}} = \frac{RMS_1\mathrm{RMS}_1 + ... + RMS_n\mathrm{RMS}_n}{ARV_1\mathrm{ARV}_1 + ... + ARV_n\mathrm{ARV}_n}</math>.

== UsageApplication ==

Digital AC measuring instruments are often built with specific waveforms in mind. For example, many digitalmultimeters on their AC multimetersranges are specifically scaled to display the RMS value of a sine wave. Since the RMS calculation can be difficult to achieve digitally, the absolute average is calculated instead and the result multiplied by the form factor of a sinusoid. This method will give less accurate readings for waveforms other than a sinewave, and the instruction plate on the rear of an [[Avometer]] states this explicitly. <ref>{{cite web|last=Tanuwijaya|first=Franky|title=True RMS vs AC Average Rectified Multimeter Readings when a Phase Cutting Speed Control is Used|url=http://www.escoglobal.com/resources/pdf/white-papers/True_G2.pdf|publisher=Esco Micro Pte Ltd|accessdate=2012-12-13/12/12}}</ref>

== Properties ==

As discussed above, the form factor is the quotient of the RMS and the ARV. The independent properties and similarities of these two values define the properties of the form factor.

For example, both RMS and ARV are directly proportional to the [[Amplitude]] <math>a</math>. However, their division removes the amplitude from the equation, meaning that form factor of a given waveform is the same regardless of how large or small the alternating current or voltage may be.

The squaring in RMS and the absolute value in ARV mean that both the values and the form factor are independent of the wave function's sign (and thus, the electrical signal's direction) at any point. For this reason, the form factor is the same for a direction-changing wave with a regular average of 0 and its fully rectified version.

The form factor, <math>k_fk_\mathrm{f}</math>, is the smallest of the three wave factors, the other two being [[crest factor]] <math>k_ak_\mathrm{a} = \frac{X_\mathrm{max}}{X_\mathrm{rms}}</math> and the lesser-known averaging factor <math>k_\mathrm{av} = \frac{X_\mathrm{max}}{X_\mathrm{arv}}</math>.

<math>k_\mathrm{av} \ge k_ak_\mathrm{a} \ge k_fk_\mathrm{f}</math><ref name=Dusza />

Due to their definitions (all relying on the [[Root Mean Square]], [[Average rectified value]] and maximum [[amplitude]] of the waveform), the three factors are related by <math>k_\mathrm{av} = k_ak_fk_\mathrm{a} k_\mathrm{f}</math>,<ref name=Dusza /> so the form factor can be calculated with <math>k_fk_\mathrm{f} = \frac{k_ak_\mathrm{av}}{k_\mathrm{ava}}</math>.

== Specific form factors ==

<math>a</math> represents the amplitude of the function, and any other coefficients applied in the vertical dimension. For example, <math>8sin8 \sin(at)</math> can be analyzed as <math>f(t) = a* \sin(t),\ a = 8</math>. As both RMS and ARV are directly proportional to it, it has no effect on the form factor, and can be replaced with a normalized 1 for calculating that value.

<math>D = \frac{\tau}{\over T}</math> is the [[duty cycle]], the ratio of the "pulse" time <math>\tau</math> (when the function's value is not zero) to the full wave [[Periodic function|period]] <math>T</math>. Most basic wave functions only achieve 0 for infinitely short instants, and can thus be considered as having <math>\tau = T, D = 1</math>. However, any of the non-pulsing functions below can be appended with <math>\frac{\sqrt{D}}{\over D} = \frac{1}{\over\sqrt{D}} = \sqrt{\frac{T}{\over\tau}}</math>

to allow pulsing. This is illustrated with the half-rectified sine wave, which can be considered a pulsed full-rectified sine wave with <math>D = \frac{1}{2\over2}</math>, and has <math>k_fk_\mathrm{f} = k_{f_\mathrm{f}_\mathrm{frs}}\sqrt{2}</math>.

{| class="wikitable"

Line 62 ⟶ 57:

! [[Waveform]] !! Image !! [[Root Mean Square|RMS]] !! [[Average rectified value|ARV]] !! Form Factor

|-

| [[Sine wave]] || [[File:Simple sine wave.svg|100px]] || <math>\frac{a}{\over\sqrt{2}}</math><ref name=Dusza /> || <math>a{2a\frac{2}{over\pi}</math><ref name=Dusza /> || <math>\frac{\pi}{2\over2\sqrt{2}} \approx 1.11072073</math><ref name="Jędrzejewski" />

|-

| [[Rectifier#Half-wave_rectificationwave rectification|Half-wave rectified sine]] || [[File:Simple half-wave rectified sine.svg|100px]] || <math>\frac{a}{2\over2}</math> || <math>\frac{a}{\over\pi}</math> || <math>\frac{\pi\over2}{2} \approx 1.5707963</math>

|-

| [[Rectifier#Full-wave_rectificationwave rectification|Full-wave rectified sine]] || [[File:Simple full-wave rectified sine.svg|100px]] || <math>\frac{a}{\over\sqrt{2}}</math> || <math>a{2a\frac{2}{over\pi}</math>|| <math>\frac{\pi}{2\over2\sqrt{2}}</math>

|-

| [[Square wave (waveform)|Square wave]], constant value || [[File:Square wave.svg|100px]] || <math>a</math> || <math>a</math> || <math>\frac{a}{a} = 1</math>

|-

| [[Pulse wave]] || [[File:Pulse wide wave.svg|100px]] || <math>a\sqrt{D}</math><ref>{{cite web|last=Nastase|first=Adrian|title=How to Derive the RMS Value of Pulse and Square Waveforms|url=http://masteringelectronicsdesign.com/how-to-derive-the-rms-value-of-pulse-and-square-waveforms/|accessdate=9 June 2012}}</ref> || <math>aD</math> || <math>\frac{1}{\over\sqrt{D}} = \sqrt{\frac{T}{\over\tau}}</math>

|-

| [[Triangle wave]] || [[File:Triangle wave.svg|100px]] || <math>\frac{a}{\over\sqrt{3}}</math><ref>{{cite web|last=Nastase|first=Adrian|title=How to Derive the RMS Value of a Triangle Waveform|url=http://masteringelectronicsdesign.com/how-to-derive-the-rms-value-of-a-triangle-waveform/|accessdate=9 June 2012}}</ref> || <math>\frac{a}{2\over2}</math> || <math>\frac{2}{\over\sqrt{3}} \approx 1.15470054</math>

|-

| [[Sawtooth wave]] || [[File:Sawtooth wave.svg|100px]] || <math>\frac{a}{\over\sqrt{3}}</math> || <math>\frac{a}{2\over2}</math> || <math>\frac{2}{\over\sqrt{3}}</math>

|-

| [[WhiteUniform random noise]] ''U''(-1a,1a) || || <math>{a\frac{1}{over\sqrt{3}}</math> || <math>{a\frac{1}{2over2}</math> || <math>\frac{2}{\over\sqrt{3}}</math>

|-

| [[Gaussian white noise]] ''G''(σ) || || <math>\sigma</math><ref>{{cite web|author1-link=Ronald Aarts|last=Aarts|first=Ronald|title=Tracking and estimation of frequency, amplitude, and form factor of a harmonic time series. IEEE SPS Magazine, 38(5), pp. 86-91, Sept. 2021, DOI 10.1109/MSP.2021.3090681 |url=https://www.sps.tue.nl/rmaarts/RMA_papers/aar21pu6.pdf}}</ref>|| <math>\sqrt{{2\over\pi}}\sigma</math> || <math>\sqrt{{\pi\over2}}</math>

|}

== See also References==

* [[Crest factor]]

== External links ==

* [http://www.daycounter.com/Calculators/RMS-Calculator.phtml RMS Calculator]

== References ==

{{Reflist}}

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