B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

228 ANGLE MODULATION AND DEMODULATION
One can construct a VCO using an operational amplifier and a hysteretic comparator, 9 (such
as a Schmitt trigger circuit). Another way of accomplishing the same goal is to vary one of the
reactive parameters ( C or L) of the resonant circuit of an oscillator. A reverse-biased semiconductor
diode acts as a capacitor whose capacitance varies with the bias voltage. The capacitance
of these diodes, known under several trade names (e.g., Varicap, Varactor, Voltacap), can be
approximated as a linear function of the bias voltage m(t) over a limited range. In Hartley or
Colpitt oscillators, for instance, the frequency of oscillation is given by
1
wo = --
./LC
If the capacitance C is varied by the modulating signal m(t), that is, if
then
C = Co - km(t)
wo = ---;:======
LCo [ 1 - k ; t) ]
1
[ km(t)
112
1- -- ]
Co
1 km(t) [ l + ]
2Co
km(t)
-- «1
Co
Here we have applied the Taylor series approximation
with n = I /2. Thus,
(1 +x? 1 +nx lxl « 1
[ km(t) 1
wo =we 1 +-- ] where Ct.) e - --
2Co
kwe
2Co
with kJ = -
Because C = Co - km(t), the maximum capacitance deviation is
Hence,
1'1C
Co
21'11
le
In practice, 1'11 /l e is usually small, and, hence, 1'1C is a small fraction of Co, which helps limit
the harmonic distortion that arises because of the approximation used in this derivation.