ABSTRACT
A sine-wave signal will have only a single-frequency component in its spectrum; that is, the frequency of the tone. However, if the sine wave is transmitted through a system (such as an amplifier) having some nonlinearity, then the signal emerging from the output of the system will no longer be a pure sine wave. That is, the output signal will be a distorted representation of the input signal. Since only a pure sine wave can have a single component in its frequency spectrum, this situation implies that the output must have other frequencies in its spectral composition. In the case of
harmonic distortion,
the frequency spectrum of the distorted signal will consist of the fundamental (which is the same frequency as the input sine wave) plus harmonic frequency components that are at integer multiples of the fundamental frequency. Taken together, these will form a Fourier representation of the distorted output signal. This phenomenon can be described mathematically. Refer to Figure 17.1, which depicts a sine-wave input signal
x
(
t
) at frequency
f
applied to the input of a system
A
(
x
), which has an output
y
(
t
). Assume that system
A
(
x
) has some nonlinearity. If the nonlinearity is severe enough, then the output
y
(
t
) might have excessive harmonic distortion such that its shape no longer resembles the input sine wave. Consider the example where the system
A
(
x
) is an audio amplifier and
x
(
t
) is a voice signal. Severe distortion can result in a situation where the output signal
y
(
t
) does not represent intelligible speech. The
total harmonic distortion
(THD) is a figure of merit that is indicative of the quality with which the system
A
(
x
) can reproduce an input signal
x
(
t
). The output signal
y
(
t
) can be expressed as:
(17.1)
where the
a
,
k
= 0, 1, …,
N
are the magnitudes of the Fourier coefficients, and
q
,
k
= 0, 1, …,
N
are the corresponding phases. The THD is defined as the percentage ratio of the rms voltage of all harmonics components above the fundamental frequency to the rms voltage of the fundamental. Mathematically, the definition is written:
(17.2)
y t a a kf t N
( ) = + +( ) =
kcos p q
THD
k= ¥= Âa
a
If the system has good linearity (which implies low distortion), then the THD will be a smaller number than that for a system having poorer linearity (higher distortion). To provide the reader with some feeling for the order of magnitude of a realistic THD, a reasonable audio amplifier for an intercom system might have a THD of about 2% or less, while a high-quality sound system might have a THD of 0.01% or less. For the THD to be meaningful, the bandwidth of the system must be such that the fundamental and the harmonics will lie within the passband. Therefore, the THD is usually used in relation to low-pass systems, or bandpass systems with a wide bandwidth. For example, an audio amplifier might have a 20 Hz to 20 kHz bandwidth, which means that a 1 kHz input sine wave could give rise to distortion up to the 20
harmonic (i.e., 20 kHz), which can lie within the passband of the amplifier. On the other hand, a sine wave applied to the input of a narrow-band system such as a radio frequency amplifier will give rise to harmonic frequencies that are outside the bandwidth of the amplifier. These kinds of narrow-band systems are best measured using
intermodulation distortion,
which is treated elsewhere in this Handbook. For the rest of the discussion at hand, consider the example system illustrated in Figure 17.1 which shows an amplifier system
A
(
x
) that is intended to be linear but has some undesired nonlinearities. Obviously, if a linear amplifier is the design objective, then the THD should be minimized.
