# Harmonic Distortion

When a signal x(t) of a particular frequency f1 passes through a nonlinear system, the output of the system consists of f1 and its harmonics. The following expression describes the relationship between f1 and its harmonics.

f1, f2 = 2f1, f3 = 3f1, f4 = 4f1, …, fn = nf1

The degree of nonlinearity of the system determines the number of harmonics and their corresponding amplitudes the system generates. In general, as the nonlinearity of a system increases, the harmonics become higher. As the nonlinearity of a system decreases, the harmonics become lower.

The following figure illustrates an example of a nonlinear system where the output y(t) is the cube of the input signal x(t).

The following equation defines the input for the system shown in the figure above.

x(t) = cos(ωt) | (A)

Equation B defines the output of the system shown in the figure above.

x3(t) = 0.5cos(ωt) + 0.25[cos(ωt) + cos(3ωt) ] | (B)

In Equation B, the output contains not only the input fundamental frequency ω but also the third harmonic 3ω.

A common cause of harmonic distortion is clipping. Clipping occurs when a system is driven beyond its capabilities. Symmetrical clipping results in odd harmonics. Asymmetrical clipping creates both even and odd harmonics.

## THD

To determine the total amount of nonlinear distortion, also known as total harmonic distortion (THD), a system introduces, measure the amplitudes of the harmonics the system introduces relative to the amplitude of the fundamental frequency. The following equation yields THD.

| (C)

where A1 is the amplitude of the fundamental frequency, A2 is the amplitude of the second harmonic, A3 is the amplitude of the third harmonic, A4 is the amplitude of the fourth harmonic, and so on.

You usually report the results of a THD measurement in terms of the highest order harmonic present in the measurement, such as THD through the seventh harmonic.

The following equation yields the percentage total harmonic distortion (%THD).

| (D)

## THD + N

Real-world signals usually contain noise. A system can introduce additional noise into the signal. THD + N measures signal distortion while taking into account the amount of noise power present in the signal. Measuring THD + N requires measuring the amplitude of the fundamental frequency and the power present in the remaining signal after removing the fundamental frequency. The following equation yields THD + N.

| (E)

where N is the noise power.

A low THD + N measurement means that the system has a low amount of harmonic distortion and a low amount of noise from interfering signals, such as AC mains hum and wideband white noise.

As with THD, you usually report the results of a THD + N measurement in terms of the highest order harmonic present in the measurement, such as THD + N through the third harmonic.

The following equation yields percentage total harmonic distortion + noise (%THD + N).

| (F)