Radio Frequency Integrated Circuit Design - Webs

Voltage-Controlled Oscillators
tions will be made by making use of the results shown in Section 8.7.2. As in
the previous section, two expressions are written: one for the feedforward gain
and one for the feedback gain.
If we assume we are operating above the capacitive feedback highpass
corner frequency, then the feedback gain is given by
ve
vc
=
C 1
C 1 + C 2
261
(8.35)
Under these conditions, it can be seen that the capacitive voltage divider
is a straight voltage divider with no phase shift involved. The loop gain can be
seen to be
C 1
C 1 + C 2
=� C 1
C 1 + C 2� �
H1H2 = g m
Y tank
1
R p
+
1
re,tank
g m
+ j�CT − j
�L
(8.36)
This can be set equal to 1 and solved for oscillating conditions. The
imaginary terms cancel, resulting in the expected expression for resonant
frequency:
� o = √ 1
C T L
(8.37)
The remaining real terms can be used to obtain an expression for the
required g m :
g m =� 1
R p
+
1
re,tank� �� C 1 + C 2
C � (8.38)
1
where C T , as before, is the series combination of C 1 and C 2. This final
expression can be manipulated to show that it is equal to (8.19) and (8.33)
and is here repeated:
g m = �(C 1 + C 2)
Q L
(8.39)
Here we can see that the transistor transconductance makes up for losses
in the resistors R p and r e,tank. Since they are in parallel with the resonator, we