Radio Frequency Integrated Circuit Design - Webs

Voltage-Controlled Oscillators
Figure 8.31 Colpitts common-collector oscillator with large-signal transconductance applied.
Note that we are using the T-model here for the transistor, so the current source
is between collector and base.
We first note that the resonator voltage will be the bias current at the
fundamental times the equivalent resonator resistance.
V tank = 2I biasRtotal
279
(8.56)
This resistance will be made up of the equivalent loading of all losses in
the oscillator and the loading of the transconductor on the resonator.
The transconductor presents the impedance
1
Gm� C 1 + C 2
C 2
� 2
=
1
G m n 2
where n is the equivalent impedance transformation ratio.
This is in parallel with all other losses in the resonator R p :
and that
1
R total = R p //
Gm n
2 =
R p
1 + G m n 2 R p
We can plug this back into the original expression:
Now we also know that
R p
V tank = 2I bias
1 + Gm n 2 R p
Gm = 2I bias
V 1
(8.57)
(8.58)
(8.59)
(8.60)