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Sec. 4–9 Nonlinear Distortion 261
then the second-order output term is
K 2 (A 1 sin v 1 t + A 2 sin v 2 t) 2 = K 2 (A 1 2 sin 2 v 1 t + 2A 1 A 2 sin v 1 t sin v 2 t + A 2 2 sin 2 v 2 t)
The first and last terms on the right side of this equation produce harmonic distortion at
frequencies 2f 1 and 2f 2 . The cross-product term produces IMD. This term is present only
when both input terms are present—thus, the name “intermodulation distortion.” Then the
second-order IMD is
2K 2 A 1 A 2 sin v 1 t sin v 2 t = K 2 A 1 A 2 {cos[(v 1 - v 2 )t] - cos [1v 1 + v 2 2t]}
It is clear that IMD generates sum and difference frequencies.
The third-order term is
K 3 v i 3 = K 3 (A 1 sin v 1 t + A 2 sin v 2 t) 3
= K 3 (A 1 3 sin 3 v 1 t + 3A 1 2 A 2 sin 2 v 1 t sin v 2 t
+ 3A 1 A 2 2 sin v 1 t sin 2 v 2 t + A 2 3 sin 3 v 2 t)
(4–49)
The first and last terms on the right side of this equation will produce harmonic distortion, and
the second term, a cross product, becomes
3K 3 A 1 2 A 2 sin 2 v 1 t sin v 2 t = 3 2 K 3A 1 2 A 2 sin v 2 t(1 - cos 2v 1 t)
= 3 2 K 3A 1 2 A 2 {sin v 2 t - 1 2 [sin(2v 1 + v 2 )t
- sin(2v 1 - v 2 )t]}
(4–50)
Similarly, the third term of Eq. (4–49) is
3K 3 A 1 A 2 2 sin v 1 t sin 2 v 2 t
= 3 2 K 3A 1 A 2 2 5sin v 1 t - 1 2 [ sin12v 2 + v 1 2t - sin(2v 2 - v 1 )t]6
(4–51)
The last two terms in Eqs. (4–50) and (4–51) are intermodulation terms at nonharmonic frequencies.
For the case of bandpass amplifiers where f 1 and f 2 are within the bandpass with f 1
close to f 2 (i.e., f 1 L f 2 0), the distortion products at 2f 1 + f 2 and 2f 2 + f 1 will usually fall
outside the passband and, consequently, may not be a problem. However, the terms at 2f 1 - f 2
and 2f 2 - f 1 will fall inside the passband and will be close to the desired frequencies f 1 and f 2 .
These will be the main distortion products for bandpass amplifiers, such as those used for RF
amplification in transmitters and receivers.
As Eqs. (4–50) and (4–51) show, if either A 1 or A 2 is increased sufficiently, the IMD
will become significant, since the desired output varies linearly with A 1 or A 2 and the IMD
output varies as A 2 or A 1 A 2 1 A 2 2 . Of course, the exact input level required for the intermodulation
products to be a problem depends on the relative values of K 3 and K 1 . The level may be
specified by the amplifier third-order intercept point, which is evaluated by applying two
equal amplitude test tones (i.e., A 1 = A 2 = A). The desired linearly amplified outputs will have