# 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.

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]

## Explanation

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 the ratio of the sum of the powers of all higher harmonic frequencies to the power at the first harmonic, or fundamental, frequency:[2]

$\mbox{THD} = \frac{P_2 + P_3 + P_4 + \cdots + P_\infty}{P_1} = \frac{\displaystyle\sum_{i=2}^\infty P_i}{P_1}$

which can equivalently be written as

$\mbox{THD} = \frac{P_\mathrm{total} - P_1}{P_1}$

if there is no source of power other than the signal and its harmonics.

Measurements based on amplitudes (e.g. voltage or current) must be converted to powers to make addition of harmonics distortion meaningful. For a voltage signal, for example, the ratio of the squares of the RMS voltages is equivalent to the power ratio:

$\mbox{THD} = \frac{V_2^2 + V_3^2 + V_4^2 + \cdots + V_\infty^2}{V_1^2}$

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

THD is also commonly defined as an amplitude ratio rather than a power ratio,[3] resulting in a definition of THD which is the square root of that given above:

$\mbox{THD} = \frac{ \sqrt{V_2^2 + V_3^2 + V_4^2 + \cdots + V_\infty^2} }{V_1}$

This latter definition is commonly used in audio distortion (percentage THD) specifications. It is unfortunate that these two conflicting definitions of THD (one as a power ratio and the other as an amplitude ratio) are both in common usage.

As a result, 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. For all signal processing equipment, except microphone preamplifiers, the preferred gain setting is unity. For microphone preamplifiers, standard practice is to use maximum gain.

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

## 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:[4]

$\mathrm{THD+N} = \frac{\displaystyle\sum_{n=2}^\infty{\text{harmonic powers}} + \text{noise power}}{\text{fundamental power}}$

A meaningful measurement must include the bandwidth of measurement. This measurement includes effects from intermodulation distortion, and so on, in addition to harmonic distortion. In Europe, it is preferable to apply a ITU-R BS.468 weighed curve,[citation needed] which is intended to accentuate what is most audible to the human ear, contributing to a more accurate measurement. However, as the weight of the curve adds 12 dB of gain to the critical midband,[citation needed] making THD+N measurements bigger, manufacturers object to its use and have widely prevented its adoption in American and Asian markets.

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.[6] 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]