Total harmonic distortion

The total harmonic distortion (THD or THDi) is a measurement of the harmonic distortion present in a signal; THD 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 for THD.

In audio systems, equipment with lower distortion—such as loudspeakers, amplifiers, and microphones—reproduces an audio recording more accurately.

In radio communications, devices with lower THD tend to produce less unintentional interference with other electronic devices. Harmonic distortion can potentially widen the frequency spectrum of the output emissions from a device by adding signals at multiples of the input frequency. For this reason, devices with high THD are less suitable in applications such as spectrum sharing and spectrum sensing.

In power systems, lower THD implies lower peak currents, less heating, lower electromagnetic emissions, and less core loss in motors. THD is a key metric for the stability and quality of the U.S. electrical grid. The Institute of Electrical and Electronics Engineers (IEEE) Standard 519-2022 covers the recommended practice and requirements for harmonic control in electric power systems.

Definitions and examples

In an ideal system with an input and an output, such as an audio amplifier, the transfer function is linear and time-invariant. When a sinusoidal signal of frequency ω passes through a nonideal, nonlinear device, additional content is added at integer multiples nω (harmonics) of the original frequency. THD is a measure of this additional signal content that is absent from 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—THD is most commonly defined as the ratio of the root mean square (RMS) amplitude of a set of higher harmonic frequencies to the RMS amplitude of the first harmonic (the fundamental frequency)

T H D_{F} = \frac{\sqrt{V_{2}^{2} + V_{3}^{2} + V_{4}^{2} + \dots}}{V_{1}},

where V_n is the RMS value of the nth harmonic voltage, and V₁ is the RMS value of the fundamental component.

In practice, THD_F (where "F" stands for "fundamental") is commonly used in audio distortion specifications (as percentage THD); however, THD is not a standardized specification, and results across manufacturers are difficult to compare. Because individual harmonic amplitudes are measured, manufacturers need to disclose several values—the test signal frequency range, the level and gain conditions, and the number of measurements taken—for any results to be meaningful. It is possible to measure the full 20 Hz–20 kHz audio range using a sweep (though distortion for a fundamental frequency higher than 10 kHz is inaudible).

Measurements for calculating THD are taken at the output of a given device under specified conditions. THD is usually expressed as a percentage or in decibels (dB) relative to the fundamental as distortion attenuation.

A variant definition uses the fundamental plus harmonics as the reference:

T H D_{R} = \frac{\sqrt{V_{2}^{2} + V_{3}^{2} + V_{4}^{2} + \dots}}{\sqrt{V_{1}^{2} + V_{2}^{2} + V_{3}^{2} + \dots}} = \frac{T H D_{F}}{\sqrt{1 + T H D_{F}^{2}}} .

The two definitions can be distinguished as THD_F and THD_R (where "R" stands for "root mean square"). THD_R cannot exceed 100%. At low distortion levels, the difference between the two calculation methods is negligible. For example, a signal with a THD_F of 10% has a very similar THD_R of 9.95%. However, at higher distortion levels, the discrepancy becomes large. For example, a signal with a THD_F of 266% has a THD_R of 94%. As another example, a pure square wave with infinite harmonics has a THD_F of 48.3% and a THD_R of 43.5%.

Some use the term distortion factor as a synonym for THD_R, while others use this term as a synonym for THD_F.

In addition, the International Electrotechnical Commission (IEC) defines the term total harmonic factor as 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.

THD + N

THD + N refers to total harmonic distortion plus noise. This measurement is much more common and more comparable across devices. THD + N is usually measured by first inputting a sine wave, then notch-filtering the output, and finally comparing the ratio between the output signal with and without this sine wave:

THD+N = \frac{\sum_{n = 2}^{\infty} harmonics + noise}{fundamental} .

Like THD, this measurement is a ratio of RMS amplitudes; it can be measured as THD_F (with bandpassed or calculated fundamental as the denominator) or, more commonly, as THD_R (with total distorted signal as the denominator).

A meaningful measurement must include its bandwidth. This measurement includes effects from ground-loop power-line hum, high-frequency interference, and intermodulation distortion between these tones and the fundamental, in addition to harmonic distortion. For psychoacoustic measurements, a weighting curve is applied—such as A-weighting or ITU-R BS.468 noise weighting—which is intended to accentuate frequencies that are most audible to the human ear, giving a more accurate measurement. A-weighting is only a rough estimate of the frequency sensitivity of human hearing, because it does not take into account the nonlinear behavior of the ear. The loudness model proposed by the scientist Eberhard Zwicker, described in the German standard DIN 45631 and the international standard ISO 532-1, accounts for these effects.

For a given input frequency and amplitude, THD + N is the reciprocal of the signal-to-noise and distortion ratio (SINAD), provided that both measurements are taken over the same bandwidth.

Measurement

The distortion of a waveform relative to a pure sine wave can be measured in one of two ways: using a THD analyzer to separate the output wave into its constituent harmonics, and then noting the amplitude of each harmonic relative to the fundamental; or canceling out the fundamental using a notch filter and then measuring the remaining signal, which is the total aggregate harmonic distortion plus noise.

A sine wave generator with very low inherent distortion can be used as the input to amplification equipment, whose distortion at different frequencies and signal levels can then be measured by examining the output waveform.

Dedicated electronic equipment exists both to generate sine waves and to measure distortion; however, a general-purpose digital computer with a sound card and suitable software can also perform harmonic analysis. In addition, software can generate sine waves, although their inherent distortion may be too high for measuring 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 distortion at the same THD, because the harmonics produced by crossover distortion are nearly as strong at higher-frequency harmonics—such as 10 to 20 times the fundamental—as they are at lower-frequency harmonics, such as 3 or 5 times the fundamental. Those harmonics appearing far in frequency from a fundamental (a desired signal) are less easily masked by that fundamental. In contrast, at the onset of clipping, harmonics first appear at low-order frequencies and then gradually start to occupy higher-frequency harmonics. Therefore, a single THD number is inadequate to specify audibility and must be interpreted with care. In a listening study in 2003, Geddes and Lee found no significant relationship between listeners' ratings of distorted audio and measurements of this audio's THD or intermodulation distortion. Measuring THD at different output levels can reveal the source of the distortion: clipping, which decreases with a decreasing output level; or crossover, which stays constant with varying output levels and is thus a greater percentage of the sound produced at low volumes.

THD weights all harmonics equally, even though research has found that lower-order harmonics are harder to hear than higher-order ones at the same level. In addition, even-order harmonics are said to be generally harder to hear than odd-order ones.^{[better source needed]} A number of methods have been developed to estimate the actual audibility of THD, and they are used to quantify crossover distortion or loudspeaker rub and buzz. These methods include high-order harmonic distortion (HOHD) or higher harmonic distortion (HHD), which measures only the 10th and higher harmonics, as well as metrics that apply psychoacoustic loudness curves to the residual signal.

Examples

For many standard signals, the THD criterion may be calculated analytically in closed form. For example, a pure square wave has a THD_F equal to

T H D_{F} = \sqrt{\frac{π^{2}}{8} - 1} \approx 0.483 = 48.3 % .

The sawtooth signal possesses a

T H D_{F} = \sqrt{\frac{π^{2}}{6} - 1} \approx 0.803 = 80.3 % .

The pure symmetrical triangle wave has a

T H D_{F} = \sqrt{\frac{π^{4}}{96} - 1} \approx 0.121 = 12.1 % .

For the rectangular pulse train with the duty cycle μ (sometimes called the cyclic ratio), THD_F has the form

T H D_{F} (μ) = \sqrt{\frac{μ (1 - μ) π^{2}}{2 \sin^{2} π μ} - 1}, 0 < μ < 1,

and it reaches its minimum (≈0.483) when the signal is symmetrical (μ = 0.5)—that is, a pure square wave. Appropriate filtering of these signals may drastically reduce the resulting THD. For example, the pure square wave filtered by the Butterworth low-pass filter of the second order (with the cutoff frequency set equal to the fundamental frequency) has a THD_F of 5.3%; by contrast, the same signal filtered by the fourth-order filter has a THD_F of 0.6%. However, analytic computation of THD_F for complicated waveforms and filters is often difficult, and the resulting expressions can be lengthy. For example, the closed-form expression for the THD_F of the sawtooth wave filtered by the first-order Butterworth low-pass filter is simply

T H D_{F} = \sqrt{\frac{π^{2}}{3} - π \coth π} \approx 0.370 = 37.0 %;

in comparison, the closed-form expression for the same signal filtered by the second-order Butterworth filter is given by the rather cumbersome formula

T H D_{F} = \sqrt{π \frac{\cot \frac{π}{\sqrt{2}} \cdot \coth^{2} \frac{π}{\sqrt{2}} - \cot^{2} \frac{π}{\sqrt{2}} \cdot \coth \frac{π}{\sqrt{2}} - \cot \frac{π}{\sqrt{2}} - \coth \frac{π}{\sqrt{2}}}{\sqrt{2} (\cot^{2} \frac{π}{\sqrt{2}} + \coth^{2} \frac{π}{\sqrt{2}})} + \frac{π^{2}}{3} - 1} \approx 0.181 = 18.1 % .

The closed-form expression for the THD_F of the pulse train filtered by the pth-order Butterworth low-pass filter is even more complicated:

T H D_{F} (μ, p) = \csc π μ \cdot \sqrt{μ (1 - μ) π^{2} - \sin^{2} π μ - \frac{π}{2} \sum_{s = 1}^{2 p} \frac{\cot π z_{s}}{z_{s}^{2}} \prod_{\binom{l = 1}{l \neq s}}^{2 p} \frac{1}{z_{s} - z_{l}} + \frac{π}{2} Re \sum_{s = 1}^{2 p} \frac{e^{i π z_{s} (2 μ - 1)}}{z_{s}^{2} \sin π z_{s}} \prod_{\binom{l = 1}{l \neq s}}^{2 p} \frac{1}{z_{s} - z_{l}}},

where μ is the duty cycle, 0 < μ < 1, and

z_{l} \equiv \exp \frac{i π (2 l - 1)}{2 p}, l = 1, 2, \dots, 2 p .

References

1 2 3 4 5 6 7 8 Blagouchine, Iaroslav V.; Moreau, Eric (September 2011). "Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues". IEEE Transactions on Communications. 59 (9): 2478–2491. doi:10.1109/TCOMM.2011.061511.100749.
1 2 "Total Harmonic Distortion and Effects in Electrical Power Systems – Associated Power Technologies" (PDF).
↑ "Analysis of Total Harmonic Distortion on the U.S. Electric Grid". www.whiskerlabs.com/.
↑ "IEEE Standard for Harmonic Control in Electric Power Systems". IEEE STD 519-2022 (Revision of IEEE STD 519-2014): 1–31. 2022. doi:10.1109/IEEESTD.2022.9848440. ISBN 978-1-5044-8727-6.{{cite journal}}: CS1 maint: periodical has ISBN (link)
1 2 3 "On the definition of total harmonic distortion and its effect on measurement interpretation". IEEE Transactions on Power Delivery. 20 (1): 526–528. January 2005. doi:10.1109/TPWRD.2004.839744. It has been shown that THD_F is a much better measure of harmonics content. Employment of THD_R in measurements may yield high errors in significant quantities such as power factor and distortion factor
↑ Slone, G. Randy (2001). The audiophile's project sourcebook. McGraw-Hill/TAB Electronics. p. 10. ISBN 0-07-137929-0. 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.
↑ Nachbaur, Fred. "THD Measurement and Conversion". Fred's Vacuum. Retrieved 2024-06-05. This number indicates the RMS voltage equivalent of total harmonic distortion power, as a percentage of the total output RMS voltage.
1 2 Kester, Walt. "Tutorial MT-003: Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so You Don't Get Lost in the Noise Floor" (PDF). Analog Devices. Retrieved 2010-04-01.
↑ IEEE 519 and other standards (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."
↑ Section 11: Power Quality Considerations. Bill Brown, P.E., Square D Engineering Services.
↑ Baptista, José MR; Cordeiro, Manuel R.; Machado e Moura, A. (2003). "Voltage Wave Quality in Low Voltage Power Systems" (PDF). Renewable Energy and Power Quality Journal. 1 (1): 117–122. doi:10.24084/repqj01.317. Two equations exist to calculate the THD…
↑ Skvarenina, Timothy L. (2018-10-03). The Power Electronics Handbook. CRC Press. ISBN 978-1-4200-3706-7. In the opinion of some, [THD_F] exaggerates the harmonic problem. … [THD_R] is used by the Canadian Standards Association and the IEC.
↑ 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."
↑ "39/41B Power Meter Glossary" (PDF). %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.
↑ "Total Harmonic Distortion Calculation by Filtering for Power Quality Monitoring" (PDF).
↑ Gross, Charles A. (2006-10-20). Electric Machines. CRC Press. ISBN 9780849385810 – via Google Books.
↑ "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". Wolfram|Alpha.
↑ "Total Harmonic Distortion of a square wave". 2012-09-11. Archived from the original on 2012-09-11.
↑ "Distortion factor". www.amplifier.cd.
↑ "Harmonics and IEEE 519" (PDF).
↑ "IEC 60050 – International Electrotechnical Vocabulary. Details for IEV number 103-07-32: "total harmonic factor"".
↑ "Audio Specifications". www.rane.com.
↑ Op Amp Distortion: HD, THD, THD + N, IMD, SFDR, MTPR.
↑ 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%."
↑ Faventi, R.; Hopper, H.; Torrente Rodriguez, M. (2014). "Low power transmission plastic gear trains: Which parameters affect the subjective acoustic quality?". International Gear Conference 2014: 26th–28th August 2014, Lyon. pp. 208–218. doi:10.1533/9781782421955.208. ISBN 978-1-78242-194-8.
↑ "ISO 532-1:2017 Acoustics – Methods for calculating loudness – Part 1: Zwicker method". International Organization for Standardization. June 2017. Retrieved 2026-10-12.
↑ Elliott, Rod (2009). "Valves vs. Transistors (Part 1)". Elliott Sound Products. Retrieved 2024-12-12. [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.
↑ Geddes, Earl R.; Lee, Lidia W. (October 2003). Auditory Perception of Nonlinear Distortion. 115th Audio Engineering Society Convention. Audio Engineering Society. Paper 5891.
↑ "Odd vs Even harmonic distortion". Gearspace.com.
↑ "Raising the Bar for Rub & Buzz Defect Detection". Audio Precision. Retrieved 2024-12-12. 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 ...
↑ "RTA Window". www.roomeqwizard.com. Retrieved 2024-12-12. HHD (higher harmonic distortion for harmonics from the 10th up to at most the 50th)

External links

[iaroslav_04-1] 1 2 3 4 5 6 7 8 Blagouchine, Iaroslav V.; Moreau, Eric (September 2011). "Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues". IEEE Transactions on Communications. 59 (9): 2478–2491. doi:10.1109/TCOMM.2011.061511.100749.

[aspowertechnologies.com-2] 1 2 "Total Harmonic Distortion and Effects in Electrical Power Systems – Associated Power Technologies" (PDF).

[3] "Analysis of Total Harmonic Distortion on the U.S. Electric Grid". www.whiskerlabs.com/.

[4] "IEEE Standard for Harmonic Control in Electric Power Systems". IEEE STD 519-2022 (Revision of IEEE STD 519-2014): 1–31. 2022. doi:10.1109/IEEESTD.2022.9848440. ISBN 978-1-5044-8727-6.{{cite journal}}: CS1 maint: periodical has ISBN (link)

[eng.tau.ac.il-5] 1 2 3 "On the definition of total harmonic distortion and its effect on measurement interpretation". IEEE Transactions on Power Delivery. 20 (1): 526–528. January 2005. doi:10.1109/TPWRD.2004.839744. It has been shown that THD_F is a much better measure of harmonics content. Employment of THD_R in measurements may yield high errors in significant quantities such as power factor and distortion factor

[6] Slone, G. Randy (2001). The audiophile's project sourcebook. McGraw-Hill/TAB Electronics. p. 10. ISBN 0-07-137929-0. 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.

[7] Nachbaur, Fred. "THD Measurement and Conversion". Fred's Vacuum. Retrieved 2024-06-05. This number indicates the RMS voltage equivalent of total harmonic distortion power, as a percentage of the total output RMS voltage.

[MT-003-8] 1 2 Kester, Walt. "Tutorial MT-003: Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so You Don't Get Lost in the Noise Floor" (PDF). Analog Devices. Retrieved 2010-04-01.

[9] IEEE 519 and other standards (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."

[10] Section 11: Power Quality Considerations. Bill Brown, P.E., Square D Engineering Services.

[11] Baptista, José MR; Cordeiro, Manuel R.; Machado e Moura, A. (2003). "Voltage Wave Quality in Low Voltage Power Systems" (PDF). Renewable Energy and Power Quality Journal. 1 (1): 117–122. doi:10.24084/repqj01.317. Two equations exist to calculate the THD…

[12] Skvarenina, Timothy L. (2018-10-03). The Power Electronics Handbook. CRC Press. ISBN 978-1-4200-3706-7. In the opinion of some, [THD_F] exaggerates the harmonic problem. … [THD_R] is used by the Canadian Standards Association and the IEC.

[13] 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."

[14] "39/41B Power Meter Glossary" (PDF). %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.

[15] "Total Harmonic Distortion Calculation by Filtering for Power Quality Monitoring" (PDF).

[16] Gross, Charles A. (2006-10-20). Electric Machines. CRC Press. ISBN 9780849385810 – via Google Books.

[17] "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". Wolfram|Alpha.

[18] "Total Harmonic Distortion of a square wave". 2012-09-11. Archived from the original on 2012-09-11.

[19] "Distortion factor". www.amplifier.cd.

[20] "Harmonics and IEEE 519" (PDF).

[21] "IEC 60050 – International Electrotechnical Vocabulary. Details for IEV number 103-07-32: "total harmonic factor"".

[22] "Audio Specifications". www.rane.com.

[23] Op Amp Distortion: HD, THD, THD + N, IMD, SFDR, MTPR.

[24] 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%."

[25] Faventi, R.; Hopper, H.; Torrente Rodriguez, M. (2014). "Low power transmission plastic gear trains: Which parameters affect the subjective acoustic quality?". International Gear Conference 2014: 26th–28th August 2014, Lyon. pp. 208–218. doi:10.1533/9781782421955.208. ISBN 978-1-78242-194-8.

[26] "ISO 532-1:2017 Acoustics – Methods for calculating loudness – Part 1: Zwicker method". International Organization for Standardization. June 2017. Retrieved 2026-10-12.

[27] Elliott, Rod (2009). "Valves vs. Transistors (Part 1)". Elliott Sound Products. Retrieved 2024-12-12. [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.

[28] Geddes, Earl R.; Lee, Lidia W. (October 2003). Auditory Perception of Nonlinear Distortion. 115th Audio Engineering Society Convention. Audio Engineering Society. Paper 5891.

[29] "Odd vs Even harmonic distortion". Gearspace.com.

[30] "Raising the Bar for Rub & Buzz Defect Detection". Audio Precision. Retrieved 2024-12-12. 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 ...

[31] "RTA Window". www.roomeqwizard.com. Retrieved 2024-12-12. HHD (higher harmonic distortion for harmonics from the 10th up to at most the 50th)