Radio Frequency Integrated Circuit Design - Webs

High-Frequency Filter Circuits
where � is the process tolerance of Isharp, which can have value in this case of
anywhere between 1.01 > � > 0.99. At a bias current of 3 mA, g m2 will be
120 mA/V; thus, the image rejection is given by
| R(1
IR = 20 log
− �) + 1
| g m2
R(1 − �)
Plugging in numbers at both extremes gives an image rejection of 44.5
dB. Thus, the circuit will give good image rejection provided that tolerances
can be small. This is a big improvement over the previous design that had
stability problems as well as tolerance problems, but more is still needed to
make this practical.
Formulas for image rejection for the circuit of Figure 9.11 can also be
developed. Noting that well below resonance an LC tank will have an impedance
given roughly by the reactance of the inductor, and well above resonance it will
have an impedance roughly that of its capacitor, we can develop the following
equations. First, the gain in the passband G PB of the filter is given by
G PB = R L
Z E
=
R L
� PBLE
since the resonator in the emitter is below resonance there.
The gain in the stop band G SB is given by
G SB = Z L
Z E
= R L
R Total
335
(9.28)
(9.29)
where the resonator in the emitter is now resonating with total resistance R Total
(which is made up of resonator losses R E and negative resistance generated by
active circuitry).
Thus, making use of (9.18), (9.28), and (9.29), the image rejection IR
can be approximated as
IR = 20 log | R L
� PBLE
� R Total
R L | = 20 log |
As before, this will be limited by process tolerance.
2R E
(2 − g m R E )�PBLE | (9.30)