Hence, the obtained solution is asymptotically stable if all eigenvalues of the Jacobian matrixAsatisfy ieal< 0 whereas the solution point is unstable if at least one eigenvalue isatisfies .Let us note that in the developed CAD tool, the eigenvalues are obtained using the MATLABsoftware function eig().C.Simulation resultsIn order to show the accuracy and the reliability of the proposed CAD tool, simulations usingAgilent’s ADS software will be performed. In these conditions, let us consider the design of
two identical differential oscillators coupled through a resistor Rc/2 of 200 Ω, as shown in Fig.4. The oscillators’ structure is based on a cross-coupled NMOS differential topology using a 0.35μm BiCMOS SiGe process. The cross connected NMOS differential pair provides the negativeresistance to compensate for the tank losses, and the tail current source is a simple NMOScurrent mirror which draws 28 mA. The frequency of oscillation is chosen to be close to 6 GHzand is determined by the LCtank at the drains. In these conditions, the inductance value, L1,2, isclose to 0.8 nH and the capacitor value, Cis close to 0.88 pF. The resistor value, R, is equal to100 Ω so that the quality factor of the tank is equal to 3.3. A tail capacitor CTis used to attenuateboth the high-frequency noise component of the tail current and the voltage variations on the tailnode . To ensure proper start-up of the oscillator, the transconductance of the NMOStransistor should be greater than R1. In these conditions, the sizes of NMOS transistors T1to T4are identical and chosen to be mmLWg35.070.Since the presented theory implemented in our CAD tool uses van der Pol oscillators to modelmicrowave coupled oscillators, we performed the modeling of this structure as two differentialvan der Pol coupled oscillators as presented in , using ADS simulation results for onedifferential NMOS oscillator at the required synchronization frequency. As a consequence, thetwo coupled oscillators of Fig. 4 can be reduced into two differential van der Pol coupledoscillators as shown in Fig. 5. In this case, let us note that the value of the coupling resistor oneach path is equal to Rc/2 to match well with the theory based on the use of two single-ended vander Pol oscillators. For the modelling of the active part, the I = f(Vd1- Vd2)characteristic of onedifferential NMOS oscillator of Fig.4 at the required synchronization frequency was plotted
leading to the typical cubic nonlinearity of a van der Pol oscillator. Hence, from thischaracteristic, the values of parameters aand bof the negative conductance presented by theactive part of each oscillator were found to be respectively equal to 7.55ּ10-3and 4ּ10-4.Then, knowing the parameters 0, a, aand b, the proposed CAD tool provides the cartographyof the locked states of the two differential coupled oscillators. For instance, for a synchronizationfrequency of 5.97 GHz, with a= 5.68ּ109rad/s and a coupling constant 0= 0.5, thecartography of the oscillators’ locked states provided by the CAD tool is presented in Fig. 6.
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