Radio Frequency Integrated Circuit Design - Webs

262 Radio Frequency Integrated Circuit Design
would like to make them as large as possible to minimize the loss (and the
noise). We get large R p by having large inductor Q, and we get large r e,tank by
using a large value of the capacitive transformer (by making C 2 bigger than
C 1). Note that, as before, the value of g m as specified in (8.38) or (8.39) is the
value that makes loop gain equal to 1, which is the condition for marginal
oscillation. To guarantee startup, loop gain is set greater than 1 or g m is set
greater than the value specified in the above equations.
Note in (8.39) that re seems to have disappeared; however, it was absorbed
by assuming that g m = 1/re .
8.8 Negative Resistance Generated by the Amplifier
In the next few sections, we will explicitly derive formulas for how much negative
resistance is generated by each type of oscillator.
8.8.1 Negative Resistance of Colpitts Oscillator
In this section, an expression for the negative resistance of the oscillators will
be derived. Consider first the common-base Colpitts configuration with the
negative resistance portion of the circuit replaced by its small-signal model
shown in Figure 8.17. Note that v�′ and the current source have both had their
polarity reversed for convenience.
An equation can be written for v�′ in terms of the current flowing through
this branch of the circuit.
ii + g mv�′ =j�C2v�′ + v�′
Figure 8.17 Small-signal model for the Colpitts common-base negative resistance cell.
re
(8.40)