Radio Frequency Integrated Circuit Design - Webs

278 Radio Frequency Integrated Circuit Design
Note that in a real oscillator, a sinusoid driving the base is a good approximation,
provided the resonator has a reasonable Q. This results in all other frequency
components being filtered out and the voltage (although not the current) is
sinusoidal even in the presence of strong nonlinearity.
It is assumed that the transistor is being driven by a large voltage, so it
will only be on for a very small part of the cycle, during which time it produces
a large pulse of current. However, regardless of what the current waveform
looks like, its average value over a cycle must still equal the bias current.
Therefore,
ic = 1
T � T
0
ic (t) dt = Ibias
(8.52)
The part of the current at the fundamental frequency of interest can be
extracted by multiplying by a cosine at the fundamental and integrating.
i fund = 2
T � T
0
i c (t) cos (�t) dt (8.53)
This can be solved by assuming a waveform for ic (t). However, solving
this equation can be avoided by noting that the current is only nonzero when
the voltage is almost at its peak value. Therefore, the cosine can be approximated
as unity and integration simplifies:
i fund ≈ 2
T � T
0
ic (t) dt = 2I bias
(8.54)
With this information, it is possible to define a large signal transconductance
for the transistor given by
Gm = i fund
V1 = 2I bias
V1
(8.55)
Since g m = I c /v T and since G m can never be larger than g m , it becomes
clear that this approximation is not valid if V1 is less then 2vT .
We can now apply this to the case of the Colpitts common-collector
oscillator as shown in Figure 8.10. We draw the simplified schematic replacing
the transistor with the large-signal transconductance, as shown in Figure 8.31.