Radio Frequency Integrated Circuit Design - Webs

as
High-Frequency Filter Circuits
325
Z in = Z�1 + Z E (1 + g m1Z�1) (9.6)
If Z�1 is assumed to be capacitive, then this expression can be rewritten
Z in =
� 1
+
C�1
1
C E� s 2 1
+
+ g
R m1
E �
s�s 2 s
+ +
C E R E
s + � C�1CE
1
L E C E�
1
L E C E C�1
(9.7)
This formula will simplify to (6.61) (without L b present) if C E = 0 and
R E =∞. Thus, as shown in Chapter 6, it can be seen that inductive degeneration
can generate positive resistance looking into the base. Similarly, capacitive degeneration
generates negative resistance looking into the base. One of the major
reasons why capacitive degeneration of amplifiers is not commonly used is
because this negative resistance can lead to instability. Fortunately, for this filter
circuit, negative resistance is only generated above the notch frequency. Above
the resonant frequency of the resonator in the emitter, the overall resistance
will be negative over a narrow frequency band if the inductance and capacitance
ratio is chosen properly in this circuit. Since the input will resonate in the
passband frequencies below where the input resistance goes negative, the circuit
can be designed to be stable. Nevertheless, a more rigorous analysis follows.
The transfer function T(s) for this circuit can be derived using Figure
9.5. With only minimal effort, it can be shown that
T(s) = vout(s)
v in(s) = −g m Z� R L
Z in + R S
Substituting (9.7) into (9.8) and after much manipulation,
1
where A =
C E R E
1
C =
.
L E C E C�1R S
T(s) =
−g m1RL
C� R S � s 2 +
s
C E R E
As 3 + Bs + C
+
1
L E C E�
+ 1
R S� 1
+
C�1
1
1
R E
, B =
C E� + g m1
C E C�1R S
(9.8)
(9.9)
1
+ , and
L E C E