Editing Total harmonic distortion (section)

==Definitions and examples==
To understand a system with an input and an output, such as an audio amplifier, we start with an ideal system where the [[Transfer Function|transfer function]] is [[LTI system theory|linear and time-invariant]]. When a sinusoidal signal of frequency ''ω'' passes through a non-ideal, non-linear device, additional content is added at multiples ''nω'' (harmonics) of the original frequency. THD is a measure of that additional signal content not present in the input signal.

When the main performance criterion is the "purity" of the original sine wave (in other words, the contribution of the original frequency with respect to its harmonics), the measurement is most commonly defined as the ratio of the [[RMS amplitude]] of a set of higher [[harmonic]] frequencies to the RMS amplitude of the first harmonic, or [[fundamental frequency|fundamental]] frequency<ref name="iaroslav_04" /><ref name="aspowertechnologies.com"/><ref name="eng.tau.ac.il">{{cite journal |title=On the definition of total harmonic distortion and its effect on measurement interpretation |journal=IEEE Transactions on Power Delivery |date=January 2005 |volume=20 |issue=1 |pages=526–528 |doi=10.1109/TPWRD.2004.839744 |quote=It has been shown that THD<sub>F</sub> is a much better measure of harmonics content. Employment of THD<sub>R</sub> in measurements may yield high errors in significant quantities such as power factor and distortion factor}}</ref><ref>{{cite book
  | last = Slone
  | first = G. Randy
  | title = The audiophile's project sourcebook
  | publisher = McGraw-Hill/TAB Electronics
  | year = 2001
  | isbn = 0-07-137929-0
  | page = 10
  | quote = This is the ratio, usually expressed in percent, of the summation of the root mean square (RMS) voltage values for all harmonics present in the output of an audio system, as compared to the RMS voltage at the output for a pure sinewave test signal that is applied to the input of the audio system.}}
</ref><ref>{{Cite web |last=Nachbaur |first=Fred |title=THD Measurement and Conversion |url=http://www.dogstar.dantimax.dk/tubestuf/thdconv.htm |access-date=2024-06-05 |website=Fred's Vacuum |quote=This number indicates the RMS voltage equivalent of total harmonic distortion power, as a percentage of the total output RMS voltage.}}</ref><ref name="MT-003">{{cite web
  | title = Tutorial MT-003: Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so You Don't Get Lost in the Noise Floor
  | first = Walt
  | last = Kester
  | url = http://www.analog.com/static/imported-files/tutorials/MT-003.pdf
  | publisher = [[Analog Devices]]
  | access-date = 1 April 2010}}
</ref><ref>IEEE 519 and other standards ([https://web.archive.org/web/20050106134937/http://grouper.ieee.org/groups/harmonic/single/docs/P1495D2.doc draft]): "distortion factor: The ratio of the root-mean-square of the harmonic content to the root-mean-square value of the fundamental quantity, often expressed as a percent of the fundamental. Also referred to as total harmonic distortion."</ref><ref>[http://static.schneider-electric.us/assets/consultingengineer/appguidedocs/section11_0307.pdf Section 11: Power Quality Considerations]. Bill Brown, P.E., Square D Engineering Services.</ref>
:<math>
 \mathrm{THD_F} = \frac{\sqrt{V_2^2 + V_3^2 + V_4^2 + \cdots}}{V_1},
</math>
where ''V<sub>n</sub>'' is the RMS value of the ''n''th harmonic voltage, and ''V''<sub>1</sub> is the RMS value of the fundamental component.

In practice, the THD<sub>F</sub> is commonly used in audio distortion specifications (percentage THD); however, THD is a non-standardized specification, and the results between manufacturers are not easily comparable. Since individual harmonic amplitudes are measured, it is required that the manufacturer disclose the test signal frequency range, level and gain conditions, and number of measurements taken. It is possible to measure the full 20&nbsp;Hz–20&nbsp;kHz range using a sweep (though distortion for a fundamental above 10&nbsp;kHz is inaudible).

Measurements for calculating the THD are made at the output of a device under specified conditions. The THD is usually expressed in [[percent]] or in [[Decibel|dB]] relative to the fundamental as distortion attenuation.

A variant definition uses the fundamental plus harmonics as the reference:<ref name="eng.tau.ac.il"/><ref>{{cite journal |last1=Baptista |first1=José MR |first2=Manuel R. |last2=Cordeiro |first3=A. |last3=Machado e Moura |title=Voltage Wave Quality in Low Voltage Power Systems |journal=Renewable Energy and Power Quality Journal |volume=1 |issue=1 |date=2003 |pages=117–122 |doi=10.24084/repqj01.317 |url=https://repqj.com/index.php/repqj/article/view/25/BAPTISTA317.pdf |quote=Two equations exist to calculate the THD… }}</ref><ref>{{Cite book |last=Skvarenina |first=Timothy L. |url=https://books.google.com/books?id=AXjLBQAAQBAJ&dq=%22This+definition+is+used+by+the+Canadian+Standards+Association+and+the+IEC%22&pg=SA17-PA6 |title=The Power Electronics Handbook |date=2018-10-03 |publisher=CRC Press |isbn=978-1-4200-3706-7 |language=en |quote=In the opinion of some, [THD<sub>F</sub>] exaggerates the harmonic problem. … [THD<sub>R</sub>] is used by the Canadian Standards Association and the IEC.}}</ref>
:<math>
 \mathrm{THD_R} = \frac{\sqrt{V_2^2 + V_3^2 + V_4^2 + \cdots}}{\sqrt{V_1^2 + V_2^2 + V_3^2 + \cdots}}
 = \frac{\mathrm{THD_F}}{\sqrt{1 + \mathrm{THD_F^2}}}.
</math>
These can be distinguished as '''THD<sub>F</sub>''' (for "fundamental"), and '''THD<sub>R</sub>''' (for "root mean square").<ref>[http://panelmeters.weschler.com/Asset/AEMC-605-UserManual.pdf AEMC 605 User Manual]. "THDf: Total harmonic distortion with respect to the fundamental. THDr: Total harmonic distortion with respect to the true RMS value of the signal."</ref><ref>{{Cite web |title=39/41B Power Meter Glossary |url=http://www.atecorp.com/ATECorp/media/pdfs/data-sheets/Fluke-39-41B_Datasheet.pdf |quote=%THD-F … ratio of the harmonic components … to the voltage … of the fundamental alone. … %THD-R … ratio of the harmonic components … to the total voltage … including the fundamental and all harmonics.}}</ref> THD<sub>R</sub> cannot exceed 100%. At low distortion levels, the difference between the two calculation methods is negligible.  For instance, a signal with THD<sub>F</sub> of 10% has a very similar THD<sub>R</sub> of 9.95%. However, at higher distortion levels the discrepancy becomes large. For instance, a signal with THD<sub>F</sub> 266% has a THD<sub>R</sub> of 94%.<ref name="eng.tau.ac.il"/> A pure [[Square wave (waveform)|square wave]] with infinite harmonics has THD<sub>F</sub> of 48.3%<ref name="iaroslav_04" /><ref>{{Cite web|url=http://www.eletrica.ufpr.br/edu/artigos/TeD2004_artigo282.pdf|title=Total Harmonic Distortion Calculation by Filtering for Power Quality Monitoring}}</ref><ref>{{Cite book |url=https://books.google.com/books?id=_LhFxN7sUXEC&dq=43.5+%22square+wave%22+THD&pg=PA178 |title=Electric Machines |first=Charles A. |last=Gross |date=October 20, 2006 |publisher=CRC Press |isbn=9780849385810 |via=Google Books}}</ref> and THD<sub>R</sub> of 43.5%.<ref>{{Cite web |url=https://www.wolframalpha.com/ |title=sqrt((1/3)^2 (1/5)^2 (1/7)^2 (1/9)^2 ...)/sqrt(1^2 (1/3)^2 (1/5)^2 (1/7)^2 (1/9)^2 ...) in percent |website=Wolfram&#124;Alpha}}</ref><ref>{{Cite web |url=http://vk1od.net/measurement/SquareWave/THD.htm |title=Total Harmonic Distortion of a square wave |date=September 11, 2012 |archive-url=https://web.archive.org/web/20120911204258/http://vk1od.net/measurement/SquareWave/THD.htm |archive-date=2012-09-11 }}</ref>

Some use the term "distortion factor" as a synonym for THD<sub>R</sub>,<ref>{{Cite web |url=https://www.amplifier.cd/Tutorial/Klirrfaktor/distortion_factor.htm |title=Distortion factor |website=www.amplifier.cd}}</ref> while others use it as a synonym for THD<sub>F</sub>.<ref>{{Cite web|url=http://energylogix.ca/harmonics_and_ieee.pdf|title=Harmonics and IEEE 519}}</ref>

The [[International Electrotechnical Commission]] (IEC) also defines another term ''total harmonic factor'' for the "ratio of the RMS value of the harmonic content of an alternating quantity to the RMS value of the quantity" using a different equation.<ref>{{Cite web |url=http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=103-07-32 |title = IEC 60050 – International Electrotechnical Vocabulary. Details for IEV number 103-07-32: "total harmonic factor"}}</ref>