Total harmonic distortion

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In radiocommunications, lower THD means the pure signal emission without causing interferences to other electronic devices. Moreover, the problem of distorted and not eco–friendly radio–emissions appear to be also very important in the context of spectrum sharing and spectrum sensing.[1]


Definitions and examples[edit]



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

where Vn is the RMS voltage of nth harmonic and n = 1 is the fundamental frequency. For many standard signals, the above criterion may be calculated analytically in a closed-form.[1] For example, a pure square wave has THDF equal to

\mathrm{THD_F} \,= \,\sqrt{\frac{\,\pi^2}{8}-1\,}\approx \, 0.483\,=\,48.3\%

The sawtooth signal possesses

\mathrm{THD_F} \,= \,\sqrt{\frac{\,\pi^2}{6}-1\,}\approx \, 0.803\,=\,80.3\%

The pure symmetrical triangle wave has THDF of

\mathrm{THD_F} \,= \,\sqrt{\frac{\,\pi^4}{96}-1\,}\approx\,0.121\,= \, 12.1\%


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


\mathrm{THD_F}\,= \,
\sqrt{\frac{\,\pi^2}{3} - \pi\coth\pi\,}\,\approx\,0.370\,= \, 37.0\%


-\cot\dfrac{\pi}{\sqrt{2\,}} - \coth\dfrac{\pi}{\sqrt{2\,}}\;}
\,+\,\frac{\,\pi^2}{3} \,-\, 1\;} 
\;\approx\;0.181\,= \, 18.1\%


\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} 


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

see[1] for more details.


Measurements for calculating the THD are made at the output[disambiguation needed] of a device under specified conditions. The THD is usually expressed in percent or in dB relative to the fundamental as distortion attenuation.


\mathrm{THD_R} \,=\,
\frac{ \sqrt{V_2^2 + V_3^2 + V_4^2 + \ldots + V_n^2+ \ldots} }{\sqrt{V_1^2 + V_2^2 + V_3^2 + \ldots + V_n^2+ \ldots}}\,
= \,\frac{\mathrm{THD_F}}{\sqrt{1 + \mathrm{THD}^2_\mathrm{F}}}





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



For a given input frequency and amplitude, THD+N is equal to SINAD, provided that both measurements are made over the same bandwidth.[6][dubious ]








See also[edit]


  12. ^ 39/41B Power Meter Glossary
  14. ^ Electric Machines By Charles A. Gross
  15. ^ Calculation of harmonic amplitude sum
  16. ^ Total Harmonic Distortion of a square wave
  17. ^ Distortion factor
  18. ^ IEEE 519
  19. ^ Harmonics and IEEE 519
  23. ^ Distortion - Valves vs. Transistors

External links[edit]