Radio Frequency Integrated Circuit Design - Webs

282 Radio Frequency Integrated Circuit Design
where R bias is the tail resistor that is serving as a current source, I BQ is the
quiescent bias current before oscillations begin, and VBEQ is the quiescent base
emitter voltage of Q 1 or Q 2. It is assumed that VBEQ is constant in this
expression, which is a good approximation for the half cycle when the transistor
is on. In the other half cycle, the denominator will force the expression to zero
for any reasonable value of V tank (V tank >> v T ), so the approximation will not
seriously affect the shape of the resulting waveform.
This expression can be used to find the average dc operating current in
the circuit (which will be higher than the quiescent current). This current also
depends on the final amplitude of the VCO.
2�
I AVE = 1
2� �
0
I BQ + |V tank cos (� )|
2R bias
V tank
v
cos (� )
1 + e T
d� ≈ I BQ + V tank
�R bias
(8.70)
The fundamental component of the current can also be extracted from
(8.69) as before.
i fund = 2
�
� �
�IBQ +
0 |Vtank cos (� )|
2R bias
V tank
1 + e
v T cos (� ) �cos (� ) d� ≈ 2
� I BQ + V tank
4R bias
(8.71)
This allows us to determine the ratio of current at the fundamental to
average current:
k = i fund
=
I AVE
2
� I BQ + V tank
4R bias
I BQ + V tank
�Rbias
(8.72)
In the limit of large and small V tank, it can be seen that k is bounded by
2 �
≤ k ≤
� 4
(8.73)
Therefore, the oscillation amplitude can once again be given in terms of
dc current as