# 总谐波失真

## 定義及例子

${\displaystyle \mathrm {THD_{F}} \,=\,{\frac {\sqrt {V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots }}{V_{1}}}}$

${\displaystyle \mathrm {THD_{R}} \,=\,{\frac {\sqrt {V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots }}{\sqrt {V_{1}^{2}+V_{2}^{2}+V_{3}^{2}+\cdots }}}\,=\,{\frac {\mathrm {THD_{F}} }{\sqrt {1+\mathrm {THD} _{\mathrm {F} }^{2}}}}}$

## THD+N

THD+N代表总谐波失真再加上雜訊。相較於THD，此量測比較容易在不同的設備之間比較。一般是輸入正弦曲線，將輸出經過带阻滤波器，再比較輸出信號本身和沒有弦波成份輸出信號之間的比例[20]

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

THD+N類似THD，都是均方根值振幅的比值[6][21]，也可以用THDF（分母是計算後的基頻振幅）或THDR（以總信號為分母）計算，後者比較常用。例如，音響精密量測會用THDR[22]

## 例子

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{2}}{8}}-1\,}}\approx \,0.483\,=\,48.3\%}$

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{2}}{6}}-1\,}}\approx \,0.803\,=\,80.3\%}$

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{4}}{96}}-1\,}}\approx \,0.121\,=\,12.1\%}$

${\displaystyle \mathrm {THD_{F}} \,(\mu )={\sqrt {{\frac {\mu (1-\mu )\pi ^{2}\,}{2\sin ^{2}\pi \mu }}-1\;}}\,,\qquad 0<\mu <1}$

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{2}}{3}}-\pi \coth \pi \,}}\,\approx \,0.370\,=\,37.0\%}$

${\displaystyle \mathrm {THD_{F}} \,={\sqrt {\pi \,{\frac {\;\cot {\dfrac {\pi }{\sqrt {2\,}}}\cdot \coth ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}-\cot ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}\cdot \coth {\dfrac {\pi }{\sqrt {2\,}}}-\cot {\dfrac {\pi }{\sqrt {2\,}}}-\coth {\dfrac {\pi }{\sqrt {2\,}}}\;}{{\sqrt {2\,}}\left(\!\cot ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}+\coth ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}\!\right)}}\,+\,{\frac {\,\pi ^{2}}{3}}\,-\,1\;}}\;\approx \;0.181\,=\,18.1\%}$

${\displaystyle \mathrm {THD_{F}} \,(\mu ,p)=\csc \pi \mu \,\cdot \!{\sqrt {\mu (1-\mu )\pi ^{2}-\,\sin ^{2}\!\pi \mu \,-\,{\frac {\,\pi }{2}}\sum _{s=1}^{2p}{\frac {\cot \pi z_{s}}{z_{s}^{2}}}\prod \limits _{\scriptstyle l=1 \atop \scriptstyle l\neq s}^{2p}\!{\frac {1}{\,z_{s}-z_{l}\,}}\,+\,{\frac {\,\pi }{2}}\,\mathrm {Re} \sum _{s=1}^{2p}{\frac {e^{i\pi z_{s}(2\mu -1)}}{z_{s}^{2}\sin \pi z_{s}}}\prod \limits _{\scriptstyle l=1 \atop \scriptstyle l\neq s}^{2p}\!{\frac {1}{\,z_{s}-z_{l}\,}}\,}}}$

${\displaystyle z_{l}\equiv \exp {\frac {i\pi (2l-1)}{2p}}\,,\qquad l=1,2,\ldots ,2p}$

[1]中有更多的細節說明。

## 參考資料

1. Iaroslav Blagouchine and Eric Moreau. Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Transactions on Communications, vol. 59, no. 9, pp. 2478—2491, September 2011.
2. Total Harmonic Distortion and Effects in Electrical Power Systems - Associated Power Technologies
3. ^ Slone, G. Randy. The audiophile's project sourcebook. McGraw-Hill/TAB Electronics. 2001: 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 页面存档备份，存于互联网档案馆 "This number indicates the RMS voltage equivalent of total harmonic distortion power, as a percentage of the total output RMS voltage."
5. 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). 亚德诺半导体. [1 April 2010].
6. ^ 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."
7. ^ Section 11: Power Quality Considerations Bill Brown, P.E., Square D Engineering Services
8. ^ VOLTAGE WAVE QUALITY IN LOW VOLTAGE POWER SYSTEMS José M. R. Baptista, Manuel R. Cordeiro, and A. Machado e Moura
9. ^ The Power Electronics Handbook edited by Timothy L. Skvarenina "This definition is used by the Canadian Standards Association and the IEC"
10. ^ 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."
11. ^ 39/41B Power Meter Glossary
12. ^ Total Harmonic Distortion Calculation by Filtering for Power Quality Monitoring
13. ^ Electric Machines By Charles A. Gross
14. ^ Calculation of harmonic amplitude sum
15. ^ Total Harmonic Distortion of a square wave
16. ^ Distortion factor
17. ^ IEEE 519
18. ^ Harmonics and IEEE 519
19. ^ Rane audio's definition of both THD and THD+N
20. ^ Op Amp Distortion: HD, THD, THD + N, IMD, SFDR, MTPR
21. ^ Introduction to the Basic Six Audio Tests 页面存档备份，存于互联网档案馆 "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%."
22. ^ Distortion - Valves vs. Transistors