Efficient Transient Noise Analysis in Circuit Simulation
Transient noise analysis means time domain simulation of noisy electronic circuits. We consider mathematical models where the noise is taken into account by means of sources of Gaussian white noise that are added to the deterministic network equations, le
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imonda AG, Products, M¨ unchen, Germany, [email protected] Institute of Mathematics, Humboldt-Universit¨ at zu Berlin, Germany {romisch,sickenberger,winkler}@math.hu-berlin.de
Transient noise analysis means time domain simulation of noisy electronic circuits. We consider mathematical models where the noise is taken into account by means of sources of Gaussian white noise that are added to the deterministic network equations, leading to systems of stochastic differential algebraic equations (SDAEs). A crucial property of the arising SDAEs is the large number of small noise sources that are included. As efficient means of their integration we discuss adaptive linear multi-step methods, in particular stochastic analogues of the trapezoidal rule and the two-step backward differentiation formula, together with a new step-size control strategy. Test results including real-life problems illustrate the performance of the presented methods.
1 Transient Noise Analysis in Circuit Simulation The increasing scale of integration, high clock frequencies and low supply voltages cause smaller signal-to-noise ratios. Reduced signal-to-noise ratio means that the difference between the wanted signal and noise is getting smaller. A consequence of this is that the circuit simulation has to take noise into account. In several applications the noise influences the system behaviour in an essentially nonlinear way such that linear noise analysis is no longer satisfactory and transient noise analysis, i.e., the simulation of noisy systems in the time domain, becomes necessary (see [4, 16]). For an implementation of an efficient transient noise analysis in an analog simulator, both an appropriate modelling and integration scheme is necessary (see [3]). Here we deal with the thermal noise of resistors as well as the shot noise of semiconductors that are modelled by additional sources of additive or multiplicative Gaussian white noise currents that are shunt in parallel to the noise-free elements. Thermal noise ith of resistors is caused by the thermal motion of electrons and is described by Nyquist’s theorem. Shot noise ishot of
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pn-junctions, caused by the discrete nature of currents due to the elementary charge, is modelled by Schottky’s formula and inherits noise intensities that depend on the deterministic currents: 2kT ξ(t) , ishot = qe |idet (u)|ξ(t) . ith = (1) R Here ξ(t) is a standard Gaussian white noise process, R denotes the resistance, T is the temperature, k = 1.38 · 10−23 is Boltzmann’s constant, idet (u) is the characteristic of the noise-free current through the pn-junction and qe = 1.60 · 10−19 is the elementary charge. Combining Kirchhoff’s current law with the element characteristics and using the charge-oriented formulation yields a stochastic differential-algebraic equation (SDAE) of the form (see e.g. [15], or for the deterministic case [6]) A
m d q(x(t)) + f (x(t), t) + gr (x(t), t)ξr (t) = 0 , dt r=1
(2)
where A is a constant singular incidence matrix determined by the topology of the dynamic c
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