Nonlinearity causes harmonics and intermodulation (IMn)
We begin by considering a general electronic function. Signals x and y are the input and output powers, respectively, and A is the transfer function between them (i.e., the “gain” if the device is an amplifier). Referring to the discussion of the resistor in Figure 1, in all real-world devices the curve is not a nice straight line indicating that “y is proportional to x.” Instead, the curve is not perfect and becomes distorted when signals are large.
When x and y are small, the curve is close to a straight line, but not 100% straight. Whether or not the designer realizes it, there are nonlinearities. When x and y are large, however, the nonlinearities are highly visible. In general, the device saturates; the output cannot respond correctly to any further increase in the input signal. This phenomenon is better illustrated by the -1dB compression point which shows the upper limit of the applicable signals (i.e., the dynamic range) (Figure 2).
Figure 2. Figure shows nonlinearity versus ideal linearity behavior.
Generally speaking, one can write:
Unfortunately (as you now know), this is never entirely so; the terms in x2
etc. are present as well. Their magnitudes depend on the strength of A2, A3, A4 etc., and they are responsible for the deviation of the transfer function A away from the desired, perfect, proportional law.
Assume now that we are in sinusoidal world where x(t) is a sinewave signal. Here x(t) contains only one frequency, ω
. Therefore, by expressing it in a very general sinewave form:
We will focus only on the first term of the sum for the further discussions. (This simplifies the equation manipulations, since only the exponential effects will be used in our demonstrations.)
Let’s assume that the first term of the Euler form in x(t) is:
You see that y contains the same and unique frequency ω
. We can draw an important conclusion from this: a perfect linear function or device will never generate any other frequency by itself.
There are two important observations to be made now:
x contains two frequencies: ωa and ωb.
It is easy to show that if the device is linear, it does not matter; y will reproduce exactly the same two original frequencies, ω
a and ω
There are no other frequencies generated!
x contains multiple frequencies: ωa, ωb, ωc, ωd, ωn
Again, if the device is linear, the output remains a nice image with no distortion of x. The same original frequencies (no more and no less) are found in y.