Thus this figure presents the variations of the phase shift the oscillators

Thus this figure presents the variations of the phase

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Thus, this figure presents the variations of the phase shift, , the oscillators’ amplitudes A 1 and A 2 , and the synchronization frequency f s in function of f 01 and f 02 . In order to validate the results provided by our CAD tool, we compared them to the simulation results of the two coupled differential oscillators of Fig.4, obtained with Agilent’s ADS software. Let us note that with ADS, only a transient analysis of one point at a time of synchronization region allows to verify the synchronization results obtained with the CAD tool. The amplitudes of the oscillators’ output voltages as well as the phase shift between them and the synchronization frequency are provided for each combination of f 01 and f 02 . For instance, let us now consider Fig. 6 where the point marked with an arrow in the four subplots represents a free- running frequency f 01 = 5.73 GHz for oscillator 1 and f 02 = 6.21 GHz for oscillator 2. Let us note that these two free-running frequencies correspond to the equally spaced tunings line as presented in [11]. The marked points lead to a phase shift of 32.39°, a synchronization frequency of the coupled oscillators of 5.97 GHz and an amplitude of 2.68 V at the output of each of the coupled oscillators.
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In the same conditions, the two differential NMOS coupled oscillators of Fig.4 simulated with ADS have lead to two sinusoidal waves at a synchronization frequency of 5.98 GHz, a phase shift of 31.23° and an amplitude of 2.72 V at the output of each oscillator, as presented in Fig. 7. Thus, the errors regarding the phase shift and the amplitude predictions are only 3.71 % and 1.49 %, respectively. Furthermore, Table I shows a comparison between the synchronization frequency, the phase shift and the amplitudes obtained in simulations with ADS for the two coupled NMOS differential oscillators, and with our CAD tool. Let us note that the free-running frequencies chosen for this table correspond to the equally spaced tunings line for which the synchronization frequency is equal to 5.97 GHz. Table II shows the same comparison as Table I, except that the free-running frequencies are not referred to the equally spaced tunings line. Hence, a good agreement was found between the results presented in both tables showing the reliability and the accuracy of the presented CAD tool. Nevertheless, as we approach the locking-region boundary, one can observe that the difference between the theoretical and simulated results is increasing. This is mainly due to the fact that the modeling of each NMOS differential oscillator as a differential van der Pol oscillator has been performed only at one frequency, i.e. the desired synchronization frequency. Anyway, these results show that it is possible to adjust, with a relatively high accuracy, the free- running frequencies of the two differential NMOS oscillators required to achieve the desired phase shift, amplitudes and synchronization frequency. The main advantage of this CAD tool is that, in an extremely short simulation time, one can obtain all the phase shifts, synchronization frequencies and amplitudes of the differential coupled system, in function of the free-running
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  • Spring '16
  • LC circuit, R. A. York

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