Editing Total harmonic distortion

{{Short description|Measurement of the harmonic distortion present in a signal}}
The '''total harmonic distortion''' ('''THD''' or '''THDi''') is a measurement of the [[harmonic distortion]] present in a signal and is defined as the ratio of the sum of the powers of all harmonic components to the power of the [[fundamental frequency]]. '''Distortion factor''', a closely related term, is sometimes used as a synonym.

In audio systems, lower distortion means that the components in a loudspeaker, amplifier or microphone or other equipment produce a more accurate reproduction of an audio recording.

In radio communications, devices with lower THD tend to produce less unintentional interference with other electronic devices. Since harmonic distortion can potentially widen the frequency spectrum of the output emissions from a device by adding signals at multiples of the input frequency, devices with high THD are less suitable in applications such as [[spectrum sharing]] and [[spectrum sensing]].<ref name="iaroslav_04">{{cite journal |last1=Blagouchine |first1=Iaroslav V. |last2=Moreau |first2=Eric |title=Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues |journal=IEEE Transactions on Communications |date=September 2011 |volume=59 |issue=9 |pages=2478–2491 |doi=10.1109/TCOMM.2011.061511.100749 }}</ref>

In power systems, lower THD implies lower peak currents, less heating, lower electromagnetic emissions, and less core loss in motors.<ref name="aspowertechnologies.com">{{Cite web |url=https://www.aptsources.com/wp-content/uploads/pdfs/Total-Harmonic-Distortion-and-Effects-in-Electrical-Power-Systems.pdf |title=Total Harmonic Distortion and Effects in Electrical Power Systems – Associated Power Technologies}}</ref> It is a key metric in the stability and quality of the U.S. [[electrical grid]].<ref>{{Cite web|url=https://www.whiskerlabs.com/analysis-of-total-harmonic-distortion-on-the-us-electric-grid/|title=Analysis of Total Harmonic Distortion on the U.S. Electric Grid|website=www.whiskerlabs.com/}}</ref> IEEE Standard 519-2022 covers the recommended practice and requirements for harmonic control in electric power systems.<ref>{{cite journal
 | title = IEEE Standard for Harmonic Control in Electric Power Systems 
 | journal = IEEE STD 519-2022 (Revision of IEEE STD 519-2014)
 | year=2022
 | pages=1–31
 | doi=10.1109/IEEESTD.2022.9848440
| isbn = 978-1-5044-8727-6
 }}</ref>

==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>

==THD+N==
'''THD+N''' means total harmonic distortion plus noise. This measurement is much more common and more comparable between devices. It is usually measured by inputting a [[sine wave]], [[notch filter|notch-filtering]] the output, and comparing the ratio between the output signal with and without the sine wave:<ref>{{Cite web|url=http://www.rane.com/note145.html|title=Audio Specifications|website=www.rane.com}}</ref>
:<math>
 \text{THD+N} = \frac{\displaystyle\sum_{n=2}^\infty \text{harmonics} + \text{noise}}{\text{fundamental}}.
</math>

Like the THD measurement, this is a ratio of RMS amplitudes<ref name="MT-003"/><ref>[http://www.analog.com/static/imported-files/tutorials/MT-053.pdf Op Amp Distortion: HD, THD, THD + N, IMD, SFDR, MTPR].</ref> and can be measured as THD<sub>F</sub> (bandpassed or calculated fundamental as the denominator) or, more commonly, as THD<sub>R</sub> (total distorted signal as the denominator).<ref>[http://www.ap.com/solutions/introtoaudiotest/thd+n Introduction to the Basic Six Audio Tests]: "Since the sum of the distortion products will always be less than the total signal, the THD+N ratio will always be a negative decibel value, or a percent value less than 100%."</ref>

A meaningful measurement must include the [[Bandwidth (signal processing)|bandwidth]] of measurement. This measurement includes effects from [[Ground loop (electricity)|ground-loop]] power-line hum, high-frequency interference, [[intermodulation distortion]] between these tones and the fundamental, and so on, in addition to harmonic distortion. For psychoacoustic measurements, a weighting curve is applied such as [[A-weighting]] or [[ITU-R BS.468]], which is intended to accentuate what is most audible to the human ear, contributing to a more accurate measurement. A-weighting is a rough way to estimate the frequency sensitivity of every persons' ears, as it does not take into account the [[Equal-loudness contour|non-linear behavior of the ear]].<ref>{{cite book |doi=10.1533/9781782421955.208 |chapter=Low power transmission plastic gear trains: Which parameters affect the subjective acoustic quality? |title=International Gear Conference 2014: 26th–28th August 2014, Lyon |date=2014 |last1=Faventi |first1=R. |last2=Hopper |first2=H. |last3=Torrente Rodriguez |first3=M. |pages=208–218 |isbn=978-1-78242-194-8 }}</ref> The loudness model proposed by Zwicker includes these complexities. The model is described in the German standard DIN45631<ref>The loudness model proposed by Zwicker includes these complexities. The model is described in the German standard DIN45631.</ref>

For a given input frequency and amplitude, THD+N is reciprocal to [[SINAD]], provided that both measurements are made over the same bandwidth.

==Measurement==
The distortion of a [[waveform]] relative to a pure sinewave can be measured either by using a [[THD analyzer]] to [[Fourier analysis|analyse the output wave into its constituent harmonics]] and noting the amplitude of each relative to the fundamental; or by cancelling out the fundamental with a [[notch filter]] and measuring the remaining signal, which will be total aggregate harmonic distortion plus noise.

Given a sinewave generator of very low inherent distortion, it can be used as input to amplification equipment, whose distortion at different frequencies and signal levels can be measured by examining the output waveform.

There is electronic equipment both to generate sinewaves and to measure distortion; but a general-purpose [[digital computer]] equipped with a [[sound card]] can carry out harmonic analysis with suitable software. Different software can be used to generate sinewaves, but the inherent distortion may be too high for measurement of very low-distortion amplifiers.

=== Interpretation ===
For many purposes, different types of harmonics are not equivalent. For instance, [[crossover distortion]] at a given THD is much more audible than [[Clipping (audio)|clipping]] distortion at the same THD, since the harmonics produced by crossover distortion are nearly as strong at higher-frequency harmonics, such as 10× to 20× the fundamental, as they are at lower-frequency harmonics like 3× or 5× the fundamental. Those harmonics appearing far away in frequency from a fundamental (desired signal) are not as easily [[Auditory masking|masked]] by that fundamental.<ref>{{Cite web |last=Elliott |first=Rod |date=2009 |title=Valves vs. Transistors (Part 1) |url=https://sound-au.com/valves/valve-trans.html |access-date=2024-12-12 |website=Elliott Sound Products |quote=[Crossover distortion] may barely register on a distortion meter, so the figures looked excellent.  Unfortunately … listeners could hear the distortion - it was plainly audible, and sounded dreadful.}}</ref> In contrast, at the onset of clipping, harmonics first appear at low-order frequencies and gradually start to occupy higher-frequency harmonics. A single THD number is therefore inadequate to specify audibility and must be interpreted with care. Taking THD measurements at different output levels would expose whether the distortion is clipping (which decreases with an decreasing level) or crossover (which stays constant with varying output level, and thus is a ''greater percentage'' of the sound produced at low volumes).

THD is a summation of a number of harmonics equally weighted, even though research performed decades ago identifies that lower-order harmonics are harder to hear at the same level, compared with higher-order ones. In addition, even-order harmonics are said to be generally harder to hear than odd-order.<ref>{{Cite web |url=https://gearspace.com/board/so-much-gear-so-little-time/382595-odd-vs-even-harmonic-distortion.html |title=Odd vs Even harmonic distortion |website=Gearspace.com}}</ref> A number of methods have been developed to estimate the actual audibility of THD, used to quantify crossover distortion or loudspeaker rub and buzz, such as "high-order harmonic distortion" (HOHD) or "higher harmonic distortion" (HHD) which measures only the 10th and higher harmonics, or metrics that apply psychoacoustic loudness curves to the residual.<ref>{{Cite web |title=Raising the Bar for Rub & Buzz Defect Detection |url=https://www.ap.com/news/raising-the-bar-for-rub-buzz-defect-detection |access-date=2024-12-12 |website=Audio Precision |language=en-us |quote=High-Order Harmonic Distortion (HOHD…) – A classic method for rub & buzz detection, HOHD uses the THD (Total Harmonic Distortion) ratio but only of harmonics above the 10th, 10-35, 20-200, etc., which is a simple way to account for frequency masking effects. … Rub & Buzz Loudness … Applies a psycho-acoustic loudness model to the residual signal to calculate the perceived level of the rub and buzz}}</ref><ref>{{Cite web |title=RTA Window |url=https://www.roomeqwizard.com/help/help_en-GB/html/spectrum.html |access-date=2024-12-12 |website=www.roomeqwizard.com |quote=HHD (higher harmonic distortion for harmonics from the 10th up to at most the 50th)}}</ref>

==Examples==
For many standard signals, the above criterion may be calculated analytically in a closed form.<ref name="iaroslav_04" /> For example, a pure [[Square wave (waveform)|square wave]] has THD<sub>F</sub> equal to
:<math>
 \mathrm{THD_F} = \sqrt{\frac{\pi^2}{8} - 1} \approx 0.483 = 48.3\%.
</math>
The [[Sawtooth wave|sawtooth signal]] possesses
:<math>
 \mathrm{THD_F} = \sqrt{\frac{\pi^2}{6} - 1} \approx 0.803 = 80.3\%.
</math>
The pure symmetrical [[triangle wave]] has
:<math>
 \mathrm{THD_F} = \sqrt{\frac{\pi^4}{96} - 1} \approx 0.121 = 12.1\%.
</math>
For the rectangular [[pulse train]] with the ''[[duty cycle]]'' ''μ'' (called sometimes the ''cyclic ratio''), the THD<sub>F</sub> has the form
:<math>
 \operatorname{THD_F}(\mu) = \sqrt{\frac{\mu(1 - \mu)\pi^2}{2\sin^2\pi\mu} -1}, \quad 0 < \mu < 1,
</math>
and logically, reaches the minimum (≈0.483) when the signal becomes symmetrical ''μ''&nbsp;=&nbsp;0.5, i.e. the pure [[Square wave (waveform)|square wave]].<ref name="iaroslav_04" /> Appropriate filtering of these signals may drastically reduce the resulting THD. For instance, the pure [[Square wave (waveform)|square wave]] filtered by the [[Butterworth filter|Butterworth low-pass filter]] of the second order (with the [[cutoff frequency]] set equal to the fundamental frequency) has THD<sub>F</sub> of 5.3%, while the same signal filtered by the fourth-order filter has THD<sub>F</sub> of 0.6%.<ref name="iaroslav_04" /> However, analytic computation of the THD<sub>F</sub> for complicated waveforms and filters often represents a difficult task, and the resulting expressions may be quite laborious to obtain. For example, the closed-form expression for the THD<sub>F</sub> of the [[sawtooth wave]] filtered by the first-order [[Butterworth filter|Butterworth low-pass filter]] is simply
:<math>
 \mathrm{THD_F} = \sqrt{\frac{\pi^2}{3} - \pi\coth\pi} \approx 0.370 = 37.0\%,
</math>
while that for the same signal filtered by the second-order [[Butterworth filter]] is given by a rather cumbersome formula<ref name="iaroslav_04" />
: <math>
 \mathrm{THD_F} =
 \sqrt{\pi \frac{\cot\dfrac{\pi}{\sqrt{2}} \cdot \coth^2 \dfrac{\pi}{\sqrt{2}} -
                 \cot^2 \dfrac{\pi}{\sqrt{2}} \cdot \coth\dfrac{\pi}{\sqrt{2}} -
                 \cot\dfrac{\pi}{\sqrt{2}} - \coth\dfrac{\pi}{\sqrt{2}}}
                {\sqrt{2} \left(\cot^2 \dfrac{\pi}{\sqrt{2}} +
                                \coth^2 \dfrac{\pi}{\sqrt{2}}\right)} +
       \frac{\pi^2}{3} - 1}
 \approx 0.181 =  18.1\%.
</math>
Yet, the closed-form expression for the THD<sub>F</sub> of the [[pulse train]] filtered by the ''p''th-order [[Butterworth filter|Butterworth low-pass filter]] is even more complicated and has the following form:<ref name="iaroslav_04" />
: <math>
 \operatorname{THD_F}(\mu, p) =
 \csc\pi\mu \cdot
 \sqrt{\mu(1 - \mu)\pi^2 - \sin^2 \pi\mu -
       \frac{\pi}{2} \sum_{s=1}^{2p} \frac{\cot \pi z_s}{z_s^2}
        \prod\limits_{\scriptstyle l=1\atop\scriptstyle l\neq s}^{2p} \frac{1}{z_s - z_l} +
       \frac{\pi}{2} \operatorname{Re} \sum_{s=1}^{2p} \frac{e^{i\pi z_s(2\mu - 1)}}{z_s^2 \sin \pi z_s}
        \prod\limits_{\scriptstyle l=1\atop\scriptstyle l\neq s}^{2p} \frac{1}{z_s - z_l}},
</math>
where ''μ'' is the [[duty cycle]], 0 < ''μ'' < 1, and
: <math>
 z_l \equiv \exp{\frac{i\pi(2l - 1)}{2p}}, \quad l = 1, 2, \ldots, 2p.
</math>

==See also==
* [[Audio system measurements]]
* [[Signal-to-noise ratio]]
* [[Timbre]]

==References==
{{Reflist|30em}}

==External links==
* [http://www.sengpielaudio.com/calculator-thd.htm Conversion: Distortion attenuation in dB to distortion factor THD in %]
* [http://www.rohde-schwarz.com/appnote/1GA56 Swept Harmonic Distortion Measurements]
* [http://www.rohde-schwarz.com/appnote/1GA55 Harmonic Distortion Measurements in the Presence of Noise]

[[Category:Electrical parameters]]
[[Category:Audio amplifier specifications]]