Download Stability Analysis of Nonlinear Microwave Circuits (Artech by Almudena Suarez, Raymond Quere PDF

By Almudena Suarez, Raymond Quere

While you're operating with circuits in line with good country units, diodes and transistors, designing radio-frequency circuits, or on a regular basis thinking about any region requiring a useful wisdom of nonlinear oscillations and balance techniques, this quantity bargains an in-depth examine the very advanced and unexpected behaviour of nonlinear circuits. there's designated assurance of circuits utilized in strength amplifiers, voltage-controlled oscillators, frequency dividers, frequency multipliers, self-oscillating mixers, and phase-locked loops. This source should still supply the reader higher perception into the layout recommendations that nonlinear circuits supply, featuring factors and detection ideas for plenty of "anomalous" phenomena frequently encountered in perform. It additionally goals to supply a greater figuring out of nonlinear oscillation thought and balance matters as you study new innovations for the effective steady-state simulation of those circuits.

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Example text

When the equilibrium point is nonhyperbolic, higher-order terms must be considered in the Taylor-series development of the vector field for the determination of its stability [9]. 4). 28), the two eigenvalues associated with the equilibrium point will be real (for D > 0) or complex-conjugate (for D < 0). The different situations are analyzed in the following [1, 9]. 23), are real and different with the same sign (either positive or negative), the corresponding equilibrium point is called a node equilibrium point.

The concept of system flow is essential for the characterization of a given nonlinear system. The sets that remain invariant under the application of the flow are very special sets. Two examples have already been seen. One of them is the equilibrium point or dc solution because x o = ϕt ( x o ) for all t. Another invariant set is the limit cycle because once the solution is in the cycle, it remains there for all t. Points and closed orbits are invariant sets under the flow action. However, any set consisting of trajectories extending to t → − ∞ or to t → ∞ would also be invariant under the action of the flow.

Now, when x o varies, a function can be defined ϕt :R n → R n . This function is called system flow [9]. 9, where a subset Xo of x o values is transformed into the subset X after the precise time interval t through the action of the system flow. For other time value t ′, the flow would provide a different image subset X′. It is assumed that for any finite t, the inverse function ϕt−1 exists; also, both the ∂ϕ Jacobian matrix [Dϕt ] = t and the inverse [ Dϕt ]−1 exist and are continu∂x ous. In the case of a nonautonomous system, with periodic forcing (with a periodic input generator), the solution can be expressed x (t ) = ϕt ( x o , θ o ) with ( x , θ ) ∈ R n × S .

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