Form factor (electronics): Difference between revisions

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{{Refimprove|date=February 2013}}

In [[electronics]] orand [[electrical engineering]], the '''form factor''' of an [[alternating current]] waveform (signal) is the ratio of the RMS ([[root mean square]]) 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|author2=Grażyna Gortat |author3=Antoni Leśniewski |pages=136–142, 197–203|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>

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<math>a</math> represents the amplitude of the function, and any other coefficients applied in the vertical dimension. For example, <math>8 \sin(t)</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_\mathrm{f} = k_{\mathrm{f}_\mathrm{frs}}\sqrt{2}</math>.

{| class="wikitable"

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! [[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 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 1.571</math>

|-

| [[Rectifier#Full-wave 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>

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| [[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>

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| [[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>

|-

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

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| [[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{\frac{2}{\over\pi}}\sigma</math> || <math>\sqrt{\frac{\pi}{2\over2}}</math>

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