Radio Frequency Integrated Circuit Design - Webs

332 Radio Frequency Integrated Circuit Design
circuit coupled into the resonator with the aid of an on-chip transformer. The
−Gm cell generates a negative resistance of −2/g m , which is then transformed
through the transformer to the emitter of the transistor Q 1. If the transformer
is assumed to have an inductance ratio of 1 for simplicity (note that nonunity
inductance ratios can result in linearity, stability, and noise advantages, but a
unity ratio will keep the math simpler), then (9.9) can be modified by first
noting that the total resistance associated with the resonator will now be
R Total =− 2
g m
// R E =
2R E
2 − g m R E
(9.18)
Note that R E is assumed to be the total positive resistance associated with
the resonator. Now (9.9) becomes
T(s) =
−g m1RL
2R E
where D = 2 − g m R E
2C E R E
1
and F =
C�1 2 − g m R E� s 2 +
C E
s 3 + Ds 2 + Es + F
s 1
+
2R E L E C E�
2 − g m R E
1
+
R S� 1
2 − g m R E
1 2R E
+ , E =
C�1 C E�
C E C�1RS
(9.19)
+ g m1 1
+
L E C E ,
.
L E C E C�1R S
The stability analysis of this circuit has already been considered in Section
9.3.3. Though reassuring, a plot of the poles fails to provide much design
insight into the problem of making the filter stable. In this section, a simpler
interpretation of stability will be presented. The mechanism for damping the
oscillation must come from either the source or load impedance. In the previous
circuit, the cascode transistor provided damping for the resonator. In this circuit,
the source impedance must damp the filter, as shown in Figure 9.12; however,
this analysis proceeds much like the one in the previous section.
For perfect notching, the negative resistance must equal the tank losses
R E , so,
2
g m = R E
Thus, the optimal current must be
(9.20)