Radio Frequency Integrated Circuit Design - Webs

336 Radio Frequency Integrated Circuit Design
9.6 Linearity of the Negative Resistance Circuits
All receive-path circuits, including the filter resonator, have to process signals.
If a very large signal is present on the resonator, it will cease to work correctly.
Such large signals will tend to change the effective g m of the transistors in the
resonator, such as Q 3 and Q 4 of Figure 9.11, and therefore the negative resistance.
Thus, as signals get larger, we can expect degradation of image rejection.
To determine the maximum signal size the circuit can handle, we need
to do a large signal analysis. If a transistor is driven with a voltage source v in
and has no degeneration, then the output current can be expressed as an infinite
series:
ic = IC� 1 + v in
vT
+ 1
2�v in
vT� 2
+ 1
6�v in
vT� 3
+ ...�
(9.31)
We now find the g m of this circuit without making a small-signal assumption:
g m = dic
= IC� dv in
1
vT
+ v in
v 2 +
T
1
2
v 2
in
v 3 + ...�
T
= g mss�1 + v in
+
vT
1
2
v 2
in
v 2 + ...�
T
(9.32)
If v in remains relatively small, this takes on the small-signal value g mss of
IC /vT . However, as the signal grows, this value changes. Thus, in the case of
the Colpitts-style resonator as shown in Figure 9.9, the amount of negative
resistance generated R neg relative to the small-signal negative resistance R negss
is
R neg
R negss =
−g m
� 2 C 1C2
−g mss
� 2 C 1C2
=� 1 + v in
+
vT 1 v
2
2
in
v 2 + ...�
T
(9.33)
The current flowing into the resonator will cause a voltage of v be3 =
i in /sC 1 to be developed across the base emitter (assuming that the impedance
of C 1 is much lower than the transistor). As this voltage approaches vT , the
operating point of this transistor will start to shift and its effectiveness will
degrade. Therefore, an input current of
i in_max = v T sC 1
(9.34)