Radio Frequency Integrated Circuit Design - Webs

330 Radio Frequency Integrated Circuit Design
In this case, the gain at the resonance frequency will go to zero when the
zeros are on the j� axis. This will happen when all the circuit losses are perfectly
canceled and the LC resonator makes a perfect short circuit to ground. In this
case,
R =
g m3
� 2 C 1C2
= Isharp_opt
vT � 2 C 1C2
Therefore, the current to give the deepest notch is
Isharp_opt = Rv T � 2 C 1C 2
(9.13)
(9.14)
Note that even in the case with the zeros on the j� axis, the poles of the
system are still safely in the left half plane. For the system to be unstable, the
negative resistance generated by the Colpitts stage needs to be equal to
g m3
� 2 C 1C 2
= R + 1
g m2
or the current to create an unstable notch filter would be
1
Isharp_osc =� R +
g m2� � 2 C 1C2vT
(9.15)
(9.16)
By taking the ratio of these two currents, it is possible to see that the
current Isharp_osc, which generates oscillations, is always bigger than the current
Isharp_opt, which produces the perfect notch. The ratio of these two currents is
given by
Isharp_osc
Isharp_opt
=
R + 1
g m2 1
= 1 +
R g m2 R
(9.17)
Thus, there is a safety margin, as the ratio is always bigger than 1. However,
g m2 should not be allowed to become too large, as this results in less separation
between the notch current and the oscillation current and in higher possibility
for instability.
Conceptually, what happens is that even though the path looking into
the series resonator is a perfect short circuit (or even has a small negative
resistance), the circuit is still loaded with the emitter resistance of Q 2 that
serves to damp the circuit which therefore cannot oscillate, as shown in
Figure 9.10.