Mathematical background

According to Filippov theory ^[1] Saltation Matrix is a peculiar concept to extend to non smooth dynamical systems the fundamental matrix solution associated to the linearizazion of a smooth dynamical system.

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Saltation Matrix can be easily introduced starting from the simplest example of non smooth system that is obtained when an hypersurface partitions the phase space in two regions.

\left\{{\begin{array}{l}\displaystyle {{\dot {x}}(t)=f(x(t))=\left\{{\begin{array}{l}f_{1}(x(t)),\quad x(t)\in R_{1}(t)\\f_{2}(x(t)),\quad x(t)\in R_{2}(t)\\\end{array}}\right.}\\\displaystyle {x(t_{0})=x_{0}}\\\end{array}}\right.

where $x\in \mathbb {R} ^{N}$ , $f_{1}$ and $f_{2}$ are assumed to be smooth in $\mathbb {R} ^{N}$ .

Simulation of mixed signal analog/digital circuits

The possibility of simulating mixed signal analog/digital circuits determining, if it exists, their periodic steady state working condition by resorting to shooting methods ^[2]. and the evaluation of noise and its phase and amplitude components by resorting to variational equations has been proposed in ^[3]. This has been possible thanks to saltation matrix introduction and its extension to Differential Algebraic Equations (DAE) that naturally occur when a circuit is described by Modified Nodal Analysis (MNA). The starting point of this important advance in Analog Mixed Signal (AMS) simulation is the choice of hybrid systems as a natural modeling solution of analog/digital circuits. In this class of circuits the analog portion is modeled in a continuous time domain while the digital one in a discrete time domain. Proper interfaces are inserted between the two different circuit descriptions: pseudo analog-to-digital (pa2d) and digital-to-analog (pd2a) converters. On the analog side, step variations of the pd2a outputs often lead to jumps in the vector field, so that capacitor branch currents and/or inductor branch voltages show discontinuities (switching phenomena). On the digital side, if the registers are modeled by discrete state variables, impacts can be observed whenever variations of the binary code at the output of the pa2d converters (re)set those registers. These particular state variables, belonging to the digital part of the circuit, can assume only a discrete set of values and exhibit an always null time derivative, but for a set of null support, corresponding to impact events, where an instantaneous change of value takes place (and the derivative is not defined).

V=x^{5}+ax^{3}+bx^{2}+cx\,

Shooting method

$\left\{{\begin{array}{l}x(1)-x(0)=0\\s(x(0),T)=0\\\end{array}}\right.$

Noise

PAC and PNOISE

References

^ A.F. Filippov, Differential Equations with Discontinuous Right-hand Sides, Kluwer, 1988, ISBN 978-90-277-2699-5.
^ T.J. Aprille e T.N. Trick, Steady-state analysis of nonlinear circuits with periodic inputs, in Proceedings of the IEEE, vol. 60, n. 1, 1972, pp. 108–114, DOI:10.1109/PROC.1972.8563.
^ [Bizzarri], A. Brambilla e G. Storti Gajani, Steady State Computation and Noise Analysis of Analog Mixed Signal Circuits, in Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 59, n. 3, 2012, pp. 541–554, DOI:10.1109/TCSI.2011.2167273.

External links

[Simulator PAN is available at the URL: http://brambilla.ws.dei.polimi.it]

[Filippov88-1] A.F. Filippov, Differential Equations with Discontinuous Right-hand Sides, Kluwer, 1988, ISBN 978-90-277-2699-5.

[Aprille1972-2] T.J. Aprille e T.N. Trick, Steady-state analysis of nonlinear circuits with periodic inputs, in Proceedings of the IEEE, vol. 60, n. 1, 1972, pp. 108–114, DOI:10.1109/PROC.1972.8563.

[Bizzarri2011-3] [Bizzarri], A. Brambilla e G. Storti Gajani, Steady State Computation and Noise Analysis of Analog Mixed Signal Circuits, in Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 59, n. 3, 2012, pp. 541–554, DOI:10.1109/TCSI.2011.2167273.

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[2]

[3]

Utente:Gian/Urgh

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