查看“︁总谐波失真”︁的源代码

'''总谐波失真'''（{{lang|en|total harmonic distortion, THD}}）是電氣信号[[失真#諧波失真|谐波失真]]的一项指标，常見的定義方式表达为所有[[諧波 (電力)|諧波]]成分功率之和与[[基本频率]]信号功率的比值。有時也會用'''失真因素'''（Distortion factor）來表示。总谐波失真越大，表示谐波成份的比例越大。

较低的总谐波失真使得音响、[[电子放大器]]或麦克风等设备产生更加精确、较少谐波、与原始采样信号接近的输出信号。

在無線通訊系統中，較低的总谐波失真表示訊號傳送時，比較不會干擾其他的電子設備。而且在頻譜共享（spectrum sharing）及频谱感知（spectrum sensing）的應用中，所發射無線電訊號的失真是嚴重的問題<ref name="iaroslav_04">{{Cite web |url=https://www.researchgate.net/publication/260672713_Analytic_Method_for_the_Computation_of_the_Total_Harmonic_Distortion_by_the_Cauchy_Method_of_Residues |title=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. |access-date=2018-12-04 |archive-date=2019-06-09 |archive-url=https://web.archive.org/web/20190609194934/https://www.researchgate.net/publication/260672713_Analytic_Method_for_the_Computation_of_the_Total_Harmonic_Distortion_by_the_Cauchy_Method_of_Residues |dead-url=no }}</ref>。

在電力系統中，較低的总谐波失真表示其峰值電流、發熱、能耗以及馬達[[鐵損]]比較少<ref name="aspowertechnologies.com">{{Cite web |url=http://www.aspowertechnologies.com/resources/pdf/Total%20Harmonic%20Distortion.pdf |title=Total Harmonic Distortion and Effects in Electrical Power Systems - Associated Power Technologies |access-date=2018-12-04 |archive-date=2012-01-05 |archive-url=https://web.archive.org/web/20120105183946/http://www.aspowertechnologies.com/resources/pdf/Total%20Harmonic%20Distortion.pdf |dead-url=no }}</ref>。
==定義及例子==
針對輸入及輸出的系統（例如音響放大器），最單純的假設是[[传递函数]]為[[线性时不变系统理论|线性时不变]]的理想系統，此時輸出信號的大小及相位可能和輸入信號不同，但其頻率不變。

若訊號通過的是非線性、非理想的系統，輸出除了原有的頻率外，會出現其他的諧波頻率，而总谐波失真就是描述這些谐波成份比例的工具。

若原始弦波信號的「乾淨程度」（也就是原始頻率能量相對諧波頻率能量的比例），其量測一般會定義為[[谐波]]頻率的均方根振幅，除以[[基本頻率]]（第一諧波）[[振幅]]的比例<ref name="iaroslav_04" /><ref name="aspowertechnologies.com"/><ref name="eng.tau.ac.il">[http://www.eng.tau.ac.il/~shmilo/10.pdf On the Definition of Total Harmonic Distortion and Its Effect on Measurement Interpretation] {{Wayback|url=http://www.eng.tau.ac.il/~shmilo/10.pdf |date=20160418193957 }}, Doron Shmilovitz</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>[http://www.dogstar.dantimax.dk/tubestuf/thdconv.htm THD Measurement and Conversion] {{Wayback|url=http://www.dogstar.dantimax.dk/tubestuf/thdconv.htm |date=20080418095502 }} "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
  | format = PDF
  | publisher = [[亚德诺半导体]]
  | accessdate = 1 April 2010
  | archive-date = 2013-02-25
  | archive-url = https://web.archive.org/web/20130225165934/http://www.analog.com/static/imported-files/tutorials/MT-003.pdf
  | dead-url = no
  }}</ref><ref>IEEE 519 and other standards ([http://grouper.ieee.org/groups/harmonic/single/docs/P1495D2.doc draft] {{Wayback|url=http://grouper.ieee.org/groups/harmonic/single/docs/P1495D2.doc |date=20160304115009 }}): "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>{{Cite web |url=http://static.schneider-electric.us/assets/consultingengineer/appguidedocs/section11_0307.pdf |title=Section 11: Power Quality Considerations Bill Brown, P.E., Square D Engineering Services |access-date=2018-12-04 |archive-url=https://web.archive.org/web/20131202222140/http://static.schneider-electric.us/assets/consultingengineer/appguidedocs/section11_0307.pdf |archive-date=2013-12-02 |dead-url=yes }}</ref>
:<math>
\mathrm{THD_F} \,= \,\frac{ \sqrt{V_2^2 + V_3^2 + V_4^2 + \cdots} }{V_1}
</math>
其中''V<sub>n</sub>''是n次諧波的RMS電壓，而''n''&nbsp;=&nbsp;1 即為基本頻率。

若諧波RMS電壓大於基頻電壓時，THD有可能超過100%。

實務上，THD<sub>F</sub>常用在音響失真量的規格中（THD百分比）。不過THD不是標準化的規格，不同製造商的結果也不容易互相比較。因為量測的是個別的諧波振幅，因此需要製造商揭露其測試信號的頻率範圍、準位及增益條件，以及量測的信號數量。有可能是用掃頻的方式量測20–20&nbsp;kHz的頻段<!--（though distortion for a fundamental above 10&nbsp;kHz is inaudible）-->。

总谐波失真的計算是在特定條件下，量測設備的輸出。总谐波失真一般會用[[百分比]]或是[[分貝]]，以基頻為準，描述谐波所佔的比例。

另外一種算法是在分母考慮基頻以及諧波的成分，不過較不鼓勵使用此定義<ref name="eng.tau.ac.il"/><ref>{{Cite web |url=http://www.icrepq.com/pdfs/BAPTISTA317.pdf |title=VOLTAGE WAVE QUALITY IN LOW VOLTAGE POWER SYSTEMS José M. R. Baptista, Manuel R. Cordeiro, and A. Machado e Moura |access-date=2018-12-04 |archive-date=2018-06-04 |archive-url=https://web.archive.org/web/20180604001317/http://www.icrepq.com/pdfs/BAPTISTA317.pdf |dead-url=no }}</ref><ref>[https://books.google.com/books?id=xxbvM40Wwa8C&dq=thdf+thdr&source=gbs_navlinks_s The Power Electronics Handbook] {{Wayback|url=https://books.google.com/books?id=xxbvM40Wwa8C&dq=thdf+thdr&source=gbs_navlinks_s |date=20190612013427 }} edited by Timothy L. Skvarenina "This definition 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}^2_\mathrm{F}}}
</math>
這二種算法可以用'''THD<sub>F</sub>'''（分母為基頻）及'''THD<sub>R</sub>'''（分母為均方根值）來識別<ref>[http://panelmeters.weschler.com/Asset/AEMC-605-UserManual.pdf AEMC 605 User Manual] {{Wayback|url=http://panelmeters.weschler.com/Asset/AEMC-605-UserManual.pdf |date=20131203033121 }} "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 |url=http://www.atecorp.com/ATECorp/media/pdfs/data-sheets/Fluke-39-41B_Datasheet.pdf |title=39/41B Power Meter Glossary |access-date=2018-12-04 |archive-date=2020-11-28 |archive-url=https://web.archive.org/web/20201128222235/http://www.atecorp.com/ATECorp/media/pdfs/data-sheets/Fluke-39-41B_Datasheet.pdf |dead-url=no }}</ref>。THD<sub>R</sub>不會超過100%。若是諧波成份不高，這二種算法的差異很小，可以省略，例如THD<sub>F</sub>為10%的信號，其THD<sub>R</sub>也很接近，為9.95%。不過若是諧波成份很高，兩者差異就很大，例如THD<sub>F</sub>為266%的信號，其THD<sub>R</sub>為94%<ref name="eng.tau.ac.il"/>。純方波有無限次的諧波，其THD<sub>F</sub>為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 |access-date=2018-12-04 |archive-date=2015-09-23 |archive-url=https://web.archive.org/web/20150923235343/http://www.eletrica.ufpr.br/edu/artigos/TeD2004_artigo282.pdf |dead-url=no }}</ref><ref>{{Cite web |url=https://books.google.com/books?id=_LhFxN7sUXEC&lpg=PA178&ots=ovMKpXD1QA&dq=43.5%20%22square%20wave%22%20THD&pg=PA178#v=onepage&q=43.5%20%22square%20wave%22%20THD&f=false |title=Electric Machines By Charles A. Gross |access-date=2018-12-04 |archive-date=2019-06-05 |archive-url=https://web.archive.org/web/20190605224210/https://books.google.com/books?id=_LhFxN7sUXEC&lpg=PA178&ots=ovMKpXD1QA&dq=43.5%20%22square%20wave%22%20THD&pg=PA178#v=onepage&q=43.5%20%22square%20wave%22%20THD&f=false |dead-url=no }}</ref>，而THD<sub>R</sub>為43.5%<ref>{{Cite web |url=http://www.wolframalpha.com/input/?i=sqrt%28%281%2F3%29%5E2%2B%281%2F5%29%5E2%2B%281%2F7%29%5E2%2B%281%2F9%29%5E2%2B...%29%2Fsqrt%281%5E2+%2B+%281%2F3%29%5E2%2B%281%2F5%29%5E2%2B%281%2F7%29%5E2%2B%281%2F9%29%5E2%2B...%29+in+percent |title=Calculation of harmonic amplitude sum |access-date=2021-10-02 |archive-date=2020-12-01 |archive-url=https://web.archive.org/web/20201201064540/https://www.wolframalpha.com/input/?i=sqrt%28%281%2F3%29%5E2%2B%281%2F5%29%5E2%2B%281%2F7%29%5E2%2B%281%2F9%29%5E2%2B...%29%2Fsqrt%281%5E2+%2B+%281%2F3%29%5E2%2B%281%2F5%29%5E2%2B%281%2F7%29%5E2%2B%281%2F9%29%5E2%2B...%29+in+percent |dead-url=no }}</ref><ref>[https://web.archive.org/web/20120911204258/http://vk1od.net/measurement/SquareWave/THD.htm Total Harmonic Distortion of a square wave]</ref>。

有些文獻會用「失真因素」來作為THD<sub>R</sub>的同義詞<ref>{{Cite web |url=http://www.amplifier.cd/Tutorial/Klirrfaktor/distortion_factor.htm |title=Distortion factor |access-date=2018-12-04 |archive-date=2020-12-02 |archive-url=https://web.archive.org/web/20201202181807/https://www.amplifier.cd/Tutorial/Klirrfaktor/distortion_factor.htm |dead-url=no }}</ref>，不過也有些會用來表示THD<sub>F</sub><ref>IEEE 519</ref><ref>{{Cite web |url=http://energylogix.ca/harmonics_and_ieee.pdf |title=Harmonics and IEEE 519 |access-date=2018-12-04 |archive-url=https://web.archive.org/web/20131202234418/http://energylogix.ca/harmonics_and_ieee.pdf |archive-date=2013-12-02 |dead-url=yes }}</ref>。

==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<ref>{{Cite web |url=http://www.rane.com/note145.html |title=Rane audio's definition of both THD and THD+N |access-date=2012-03-03 |archive-url=https://web.archive.org/web/20140223174547/http://www.rane.com/note145.html |archive-date=2014-02-23 |dead-url=yes }}</ref>：
:<math>
\mathrm{THD\!\!+\!\!N} = \frac{\displaystyle\sum_{n=2}^\infty{\text{harmonics}} + \text{noise}}{\text{fundamental}}
</math>

THD+N類似THD，都是均方根值振幅的比值<ref name="MT-003"/><ref>{{Cite web |url=http://www.analog.com/static/imported-files/tutorials/MT-053.pdf |title=Op Amp Distortion: HD, THD, THD + N, IMD, SFDR, MTPR |access-date=2018-12-26 |archive-date=2014-06-11 |archive-url=https://web.archive.org/web/20140611022213/http://www.analog.com/static/imported-files/tutorials/MT-053.pdf |dead-url=no }}</ref>，也可以用THD<sub>F</sub>（分母是計算後的基頻振幅）或THD<sub>R</sub>（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THD<sub>R</sub><ref>[http://www.ap.com/solutions/introtoaudiotest/thd+n Introduction to the Basic Six Audio Tests] {{Wayback|url=http://www.ap.com/solutions/introtoaudiotest/thd+n |date=20160506164140 }} "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>。

有意義的量測資訊需要包括量測的[[带宽]]。量測除了谐波失真外，也會包括{{link-en|接地迴路|Ground loop (electricity)}}的電源線噪音、高頻干擾、高頻和基頻之間{{link-en|交调失真|intermodulation distortion}}等雜訊來源。若是針對心理聲學的量測，會配合像{{link-en|A加權|A-weighting}}或{{link-en|ITU-R BS.468|ITU-R BS.468}}的加權曲線，會強調人耳可以聽到的聲音，讓相關的分析更加準確。

針對相同的輸入頻率及振幅，THD+N是[[SINAD]]的倒數，前提是二個量測都是在相同的帶寛下進行。

==量測==
波形相對弦波的扭曲程度可以用{{le|THD分析儀|THD analyzer}}，將信號用[[傅里叶分析]]分解成基頻及諧波成份。並且計算各諧波相對於基頻的比例。或是用[[带阻滤波器]]濾掉基頻，再量測過濾後的信號，即為各諧波成份的加總。

若有弦波產生器，可以產生固有失真低的弦波，可以以此為輸入送到放大設備中，再量測輸出信號各諧波的分量，也可以計算总谐波失真。

有電子設備可以同時產生弦波並且量測失真，不過通用的[[電腦]]配合[[声卡]]，就可以用特定的軟體進行諧波分析。可以用不同的軟體來產生弦波，不過其固有失真太高，不適合量測低失真的放大機。

=== 詮釋 ===
在許多的應用中，各谐波成份不是等效的。例如在总谐波失真中，相同THD的{{link-en|交调失真|intermodulation distortion}}要比削波失真（clipping distortion）更容易聽到，因為其谐波的頻率較高，基頻的[[掩蔽效应]]無法蓋過該諧波<ref>{{Cite web |url=http://sound.whsites.net/valves/valve-trans.html#s33 |title=Distortion - Valves vs. Transistors |access-date=2018-12-26 |archive-date=2019-02-12 |archive-url=https://web.archive.org/web/20190212182846/http://sound.whsites.net/valves/valve-trans.html#s33 |dead-url=no }}</ref>。單一的THD數字無法代表特定聲音的可聽性，需要更多資料加以分析。量測不同輸出下THD的可以分辨失真屬於削波失真（隨音量而增加）或是交调失真（隨音量而減少）。

==例子==
對於許多常見的信號，可以找到其总谐波失真的解析解<ref name="iaroslav_04" />，例如[[方波]]的THD<sub>F</sub> 為
:<math>
\mathrm{THD_F} \,= \,\sqrt{\frac{\,\pi^2}{8}-1\,}\approx \, 0.483\,=\,48.3\%
</math>
[[锯齿波]]的THD<sub>F</sub>則是
:<math>
\mathrm{THD_F} \,= \,\sqrt{\frac{\,\pi^2}{6}-1\,}\approx \, 0.803\,=\,80.3\%
</math>
對稱的[[三角波]]THD<sub>F</sub>為
:<math>
\mathrm{THD_F} \,= \,\sqrt{\frac{\,\pi^4}{96}-1\,}\approx\,0.121\,= \, 12.1\%
</math>
[[占空比]]''μ''的方波[[脈波]]，其THD<sub>F</sub>為
: <math> 
\mathrm{THD_F}\,(\mu)=\sqrt{\frac{\mu(1-\mu)\pi^2\,}{2\sin^2\pi\mu}-1\;}\,,\qquad 0<\mu<1
</math>
若方波脈波對稱（''μ''=0.5），THD<sub>F</sub>有最小值（≈0.483），也就是純[[方波]]的THD<sub>F</sub><ref name="iaroslav_04" />。將訊號經過過適當的濾波可以使其总谐波失真大幅下降。例如[[方波]]若用二階[[巴特沃斯滤波器]]濾波（[[截止頻率]]等於基頻），其THD<sub>F</sub>可降到5.3%，若用四階巴特沃斯滤波器濾波，THD<sub>F</sub>為0.6%<ref name="iaroslav_04" />。不過若是複雜的信號或是複雜的濾波器，要找解析解並不容易，要計算其結果也很不容易。例如[[锯齿波]]用一階巴特沃斯滤波器濾波後，其THD<sub>F</sub>為
:<math>
\mathrm{THD_F}\,= \,
\sqrt{\frac{\,\pi^2}{3} - \pi\coth\pi\,}\,\approx\,0.370\,= \, 37.0\%
</math>
若用二階巴特沃斯滤波器濾波後，會得到更複雜的式子<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>
而脈波用p階[[巴特沃斯滤波器]]濾波後的解析解更加複雜，式子如下
: <math> 
\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\,}\,}
</math>
其中''μ''為[[占空比]], 0<''μ''<1，而且 
: <math> 
z_l\equiv \exp{\frac{i\pi(2l-1)}{2p}}\,, \qquad l=1, 2,\ldots, 2p 
</math>
<ref name="iaroslav_04" />中有更多的細節說明。

== 參考資料 ==
{{reflist}}
== 相關條目==
*{{le|音響系統測量|Audio system measurements}}
*[[信噪比]]
*[[音色]]
== 外部链接 ==
* [https://web.archive.org/web/20080418095502/http://www.dogstar.dantimax.dk/tubestuf/thdconv.htm Explanation of THD measurements]
* [https://web.archive.org/web/20140223174547/http://www.rane.com/note145.html Rane audio's definition of both THD and THD+N]
* [http://www.sengpielaudio.com/calculator-thd.htm Conversion: Distortion attenuation in dB to distortion factor THD in %] {{Wayback|url=http://www.sengpielaudio.com/calculator-thd.htm |date=20100115081729 }}
[[Category:电量参数|T]]