Question 8.3: Design a 1-GHz quadrature oscillator using simplified models...
Design a 1-GHz quadrature oscillator using simplified models to demonstrate phase shift and amplitude theory. Change the capacitance by 1%, and observe and explain the resulting amplitude and phase mismatch.
A 1-GHz quadrature oscillator is built with a resonant tank made up of 5 nH and 5.06 pF. Feedback and cross-coupling transconductors have transfer functions i = 0.005v − 0.0005v^{3} . Initially, the circuit is run open loop with a voltage representing VCO_{2} , with that voltage adjusted to equal the oscillating amplitude of VCO_{1} , and magnitude and phase results are obtained as in Figure 8.28.
Thus, where amplitude is rolled off by 3 dB (down to about 2.1V), phase is at 90°, as is expected from the explanation around Figure 8.26. When connected as an injection-locked oscillator, the oscillating frequency is 1.05144 GHz, at the frequency where the phase shift is 90° with an amplitude of about 2.2V, also as expected. The phase shift between the two oscillators is exactly 90° within the simulation limits (better than a thousandth of a degree). When capacitance of one oscillator is increased by 1%, the frequency decreases to 1.0487 GHz, and the phase shift is now 93.93°. This can be explained by examining a zoom in of the phase versus frequency plot derived from (8.39),
\phi _{inj} = 2\phi _{osc}=- 2\tan^{-1}\left[\left(\omega C- \frac{1}{\omega L} \right)R \right]and shown in Figure 8.29. When both capacitors are at 5.06 pF, each phase shift is 90° at a predicted frequency of 1.0519 GHz, close to the simulated frequency of 1.05144 GHz. When one capacitor is high by a fraction \delta (in this case, 1%) at 5.1106 pF, its resonant frequency decreases by about \delta /2 (in this case, by 0.5%); consequently, there is more phase shift at the frequency of interest. The total phase still has to be 90°; hence, frequencies adjusts themselves until the sum of the two phase shifts is 180°. This new frequency can be found by noting where the average of the two phase shifts goes through approximately 90° at a frequency shift of \delta /4 (or 0.25%). From the starting frequency of 1.0519 GHz, a 0.25% shift will move it to 1.0493 GHz, while the equation and Figure 8.29 predict a new frequency of 1.0492 GHz. Both are close to the simulated frequency of 1.0487 GHz. Also of importance, the phase shift across the two oscillators is now seen to be about 87° and 93° for a total phase of 180°, again in agreement with the simulations. Phase shift can be shown to be related to bandwidth by
\left|\frac{d\phi }{d\omega } \right|= \frac{2Q }{\omega _{\omicron } } = \frac{2}{B} (8.45)
Hence, the phase shift is estimated at
\Delta \phi = \frac{\Delta \omega }{B/2} = \frac{\omega _{matched}. \delta /4}{B/2} (8.46)
where \delta is the capacitor mismatch (0.01), \omega _{matched} is the quadrature oscillator frequency with components matched (2\pi × 1.0519 GHz), and B/2 is the difference between the LC resonant frequency and the free-running frequency (2\pi × 51.6 MHz). This equation predicts a phase offset of 2.92°, which is quite close to the value predicted from Figure 8.29. The simulated phase change is slightly larger at 3.93° but still illustrates the usefulness of this estimate.
Similarly, the change of phase shift is also related to a change of amplitude, as seen in Figure 8.28. Simulated results show the amplitudes are now 2.08V and 2.35V, in agreement with the above explanation.
