Radio Frequency Integrated Circuit Design - Webs

268 Radio Frequency Integrated Circuit Design
Thus, we can see from this example that a −Gm oscillator can start with
half as much collector current in each transistor as a Colpitts oscillator under
the same loading conditions.
8.9 Comments on Oscillator Analysis
It has been shown that closed-loop analysis agrees exactly with the open-loop
analysis. It can also be shown that analysis by negative resistance produces
identical results. This analysis can be extended. For example, in a negative
resistance oscillator, it is possible to determine if oscillations will be stable as
shown by Kurokawa [1], with detailed analysis shown by [2]. However, what
does it mean to have an exact analysis? Does this allow one to predict the
frequency exactly? The answer is no. Even if one could take into account
RF model complexities including parasitics, temperature, process, and voltage
variations, the nonlinearities of an oscillator would still change the frequency.
These nonlinearities are required to limit the amplitude of oscillation, so they
are a built-in part of an oscillator. Fortunately, for a well-designed oscillator,
the predicted results will give a reasonable estimate of the performance. Then,
to refine the design, it is necessary to simulate the circuit.
Example 8.5 Oscillator Frequency Shifts and Open-Loop Gain
Explore the predicted frequency with the actual frequency of oscillation by
doing open-loop and closed-loop simulation of an oscillator. Compare the results
to the simple formula. This example can also be used to explore the amplitude
of oscillation and its relationship to the open-loop gain.
Solution
For this example, the previously found capacitor and inductor values are used
in the circuit shown in Figure 8.20.
Loop gain can be changed by adjusting g m or the tank resistance R p . Both
will also affect frequency somewhat. R p will affect �c through Q L and g m will
affect �c indirectly, since re = 1/g m . In this case, we varied both R p and g m .
Results are plotted in Figure 8.21.
It can be seen from Figure 8.21(a) that the open-loop simulations consistently
predict higher oscillating frequencies than the closed-loop simulations.
Thus, nonlinear behavior results in the frequency being decreased. We note
that the initial frequency estimate using the inductor and capacitor values and
adding an estimate for the parasitic capacitance results in a good estimate of
final closed-loop oscillating frequency. In fact, this estimate of frequency is
better than the open-loop small-signal prediction of frequency. It can also be
seen from Figure 8.21(b) that output signal amplitude is related to the openloop
gain, and as expected, as gain drops to 1 or less, the oscillations stop.