Hence the obtained solution is asymptotically stable if all eigenvalues of the

Hence the obtained solution is asymptotically stable

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Hence, the obtained solution is asymptotically stable if all eigenvalues of the Jacobian matrix A satisfy i eal < 0 whereas the solution point is unstable if at least one eigenvalue i satisfies . Let us note that in the developed CAD tool, the eigenvalues are obtained using the MATLAB software function eig( ) . C. Simulation results In order to show the accuracy and the reliability of the proposed CAD tool, simulations using Agilent’s ADS software will be performed. In these conditions, let us consider the design of
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two identical differential oscillators coupled through a resistor R c /2 of 200 Ω, as shown in Fig. 4. The oscillators’ structure is based on a cross-coupled NMOS differential topology using a 0.35 μm BiCMOS SiGe process. The cross connected NMOS differential pair provides the negative resistance to compensate for the tank losses, and the tail current source is a simple NMOS current mirror which draws 28 mA. The frequency of oscillation is chosen to be close to 6 GHz and is determined by the LC tank at the drains. In these conditions, the inductance value, L 1,2 , is close to 0.8 nH and the capacitor value, C is close to 0.88 pF. The resistor value, R , is equal to 100 Ω so that the quality factor of the tank is equal to 3.3. A tail capacitor C T is used to attenuate both the high-frequency noise component of the tail current and the voltage variations on the tail node [21]. To ensure proper start-up of the oscillator, the transconductance of the NMOS transistor should be greater than R 1 . In these conditions, the sizes of NMOS transistors T 1 to T 4 are identical and chosen to be m m L W g 35 . 0 70 . Since the presented theory implemented in our CAD tool uses van der Pol oscillators to model microwave coupled oscillators, we performed the modeling of this structure as two differential van der Pol coupled oscillators as presented in [22], using ADS simulation results for one differential NMOS oscillator at the required synchronization frequency. As a consequence, the two coupled oscillators of Fig. 4 can be reduced into two differential van der Pol coupled oscillators as shown in Fig. 5. In this case, let us note that the value of the coupling resistor on each path is equal to Rc /2 to match well with the theory based on the use of two single-ended van der Pol oscillators. For the modelling of the active part, the I = f(Vd 1 - Vd 2 ) characteristic of one differential NMOS oscillator of Fig.4 at the required synchronization frequency was plotted
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leading to the typical cubic nonlinearity of a van der Pol oscillator. Hence, from this characteristic, the values of parameters a and b of the negative conductance presented by the active part of each oscillator were found to be respectively equal to 7.55 ּ 10 -3 and 4 ּ 10 -4 . Then, knowing the parameters 0 , a , a and b , the proposed CAD tool provides the cartography of the locked states of the two differential coupled oscillators. For instance, for a synchronization frequency of 5.97 GHz, with a = 5.68 ּ 10 9 rad/s and a coupling constant 0 = 0.5, the cartography of the oscillators’ locked states provided by the CAD tool is presented in Fig. 6.
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  • Spring '16
  • LC circuit, R. A. York

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