Total harmonic distortion

The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. THD is used to characterize the linearity of audio systems and the power quality of electric power systems. Distortion factor is a closely related term, sometimes used as a synonym.

In audio systems, lower THD means the components in a loudspeaker, amplifier or microphone or other equipment produce a more accurate reproduction by reducing harmonics added by electronics and audio media.

In power systems, lower THD means reduction in peak currents, heating, emissions, and core loss in motors.[1]

Definitions

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 is linear and time-invariant. When a signal passes through a non-ideal, non-linear device, additional content is added at the harmonics of the original frequencies. THD is a measurement of the extent of that distortion.

When the input is a pure sine wave, 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:[2][3][4][5][6][7][8]

$\mathrm{THD_F} = \frac{ \sqrt{V_2^2 + V_3^2 + V_4^2 + \cdots + V_n^2} }{V_1}$

where Vi is the RMS voltage of ith harmonic and i = 1 is the fundamental frequency.

This measurement is commonly used in audio distortion (percentage THD) specifications, 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 20–20 kHz range using a sweep (though distortion for a fundamental above 10 kHz is inaudible). For all signal processing equipment, except microphone preamplifiers, the preferred gain setting is unity.[citation needed] For microphone preamplifiers, standard practice is to use maximum gain.[citation needed]

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 dB relative to the fundamental as distortion attenuation.

A variant definition uses the fundamental plus harmonics as the reference, though usage is discouraged:[9][10][11]

$\mathrm{THD_R} = \frac{ \sqrt{V_2^2 + V_3^2 + V_4^2 + \cdots + V_n^2} }{\sqrt{V_1^2 + V_2^2 + V_3^2 + \cdots + V_n^2}}$

These can be distinguished as THDF (for "fundamental"), and THDR (for "root mean square").[12][13] THDR cannot exceed 100%. At low distortion levels, the difference between the two calculation methods is negligible. For instance, a signal with THDF of 10% has a very similar THDR of 9.95%. However, at higher distortion levels the discrepancy becomes large. For instance, a signal with THDF 266% has a THDR of 94%. A pure square wave has THDF of 48.34%,[14][15][16] or THDR of 43.52%.[17][18]

Some use the term "distortion factor" as a synonym for THDR,[19] while others use it as a synonym for THDF.[20][21]

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 filtering the output, and comparing the ratio between the output signal with and without the sine wave:[22]

$\mathrm{THD\!\!+\!\!N} = \frac{\displaystyle\sum_{n=2}^\infty{\text{harmonics}} + \text{noise}}{\text{fundamental}}$

Like the THD measurement, this is a ratio of RMS amplitudes.[5][23]

A meaningful measurement must include the bandwidth of measurement. This measurement includes effects from intermodulation distortion, 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.

For a given input frequency and amplitude, THD+N is equal to SINAD, provided that both measurements are made over the same bandwidth.[5]

Measurement

The distortion of a waveform relative to a pure sinewave can be measured either by using a THD analyzer to 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 distortion at the same THD, since the harmonics produced are at higher frequencies, which are not as easily masked by the fundamental.[24] A single THD number is 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 increases with level) or crossover (which decreases with level).

THD is an average 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.[citation needed] A number of formulas that attempt to correlate THD with actual audibility have been published, however none have gained mainstream use.[citation needed]

References

1. ^ Total Harmonic Distortion and Effects in Electrical Power Systems - Associated Power Technologies
2. ^
3. ^ 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."
4. ^ THD Measurement and Conversion "This number indicates the RMS voltage equivalent of total harmonic distortion power, as a percentage of the total output RMS voltage."
5. ^ a b c 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 1 April 2010.
6. ^ Total Harmonic Distortion and Effects in Electrical Power Systems - Associated Power Technologies
7. ^ 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."
8. ^ Section 11: Power Quality Considerations Bill Brown, P.E., Square D Engineering Services
9. ^
10. ^ VOLTAGE WAVE QUALITY IN LOW VOLTAGE POWER SYSTEMS José M. R. Baptista, Manuel R. Cordeiro, and A. Machado e Moura
11. ^ The Power Electronics Handbook edited by Timothy L. Skvarenina "This definition is used by the Canadian Standards Association and the IEC"
12. ^ 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."
13. ^ 39/41B Power Meter Glossary
14. ^ http://www.wolframalpha.com/input/?i=sqrt%28%281%2F3%29^2%2B%281%2F5%29^2%2B%281%2F7%29^2%2B%281%2F9%29^2%2B...%29%2F1+in+percent
15. ^ http://www.eletrica.ufpr.br/edu/artigos/TeD2004_artigo282.pdf