Radio Frequency Integrated Circuit Design - Webs

Voltage-Controlled Oscillators
Figure 8.6 Linear model of an oscillator as a feedback control system.
The gain of the system in Figure 8.6 is given by
Vout(s)
Vin(s) =
H1(s)
1 − H1(s)H2(s)
249
(8.5)
We can see from the equation that if the denominator approaches zero,
with finite H1(s), then the gain approaches infinity and we can get a large
output voltage for an infinitesimally small input voltage. This is the condition
for oscillation. By solving for this condition, we can determine the frequency
of oscillation and the required gain to result in oscillation.
More formally, the system poles are defined by the denominator of (8.5).
To find the poles of the closed-loop system, one can equate this expression to
zero, as in
1 − H1(s)H2(s) = 0 (8.6)
For sustained oscillation at constant amplitude, the poles must be on the
j� axis. To achieve this, we replace s with j� and set the equation equal to
zero.
For the open-loop analysis, rewrite the above expression as
and that
H1( j�)H2( j�) = 1 (8.7)
Since in general H1( j�) and H 2( j�) are complex, this means that
|H1( j�)||H2( j�)| = 1 (8.8)
∠H1( j�)H 2( j�) = 2n� (8.9)
where n is a positive integer.
These conditions for oscillation are known as the Barkhausen criterion,
which states that for sustained oscillation at constant amplitude, the gain around