This article was downloaded by: [University of New Hampshire] On: 10 March 2013, At: 03:05 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/nmcm20 Modelling and simulation of transient noise in circuit simulation Georg Denk a & Renate Winkler b a Infineon Technologies, Memory Products, D-81726, München, Germany b Humboldt-Universität Berlin, Institut für Mathematik, D-10099, Berlin, Germany Version of record first published: 30 Aug 2007. To cite this article: Georg Denk & Renate Winkler (2007): Modelling and simulation of transient noise in circuit simulation, Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences, 13:4, 383-394 To link to this article: http://dx.doi.org/10.1080/13873950500064400 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and- conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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This article was downloaded by: [University of New Hampshire]On: 10 March 2013, At: 03:05Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Mathematical and Computer Modellingof Dynamical Systems: Methods, Toolsand Applications in Engineering andRelated SciencesPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/nmcm20
Modelling and simulation of transientnoise in circuit simulationGeorg Denk a & Renate Winkler ba Infineon Technologies, Memory Products, D-81726, München,Germanyb Humboldt-Universität Berlin, Institut für Mathematik, D-10099,Berlin, GermanyVersion of record first published: 30 Aug 2007.
To cite this article: Georg Denk & Renate Winkler (2007): Modelling and simulation of transientnoise in circuit simulation, Mathematical and Computer Modelling of Dynamical Systems: Methods,Tools and Applications in Engineering and Related Sciences, 13:4, 383-394
To link to this article: http://dx.doi.org/10.1080/13873950500064400
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
This paper presents a new approach to the transient noise analysis of integrated circuits. Thisapproach consists of two parts, the modelling of noise sources in the time domain and thedevelopment of numerical schemes for stochastic differential-algebraic equations. The noisesources include thermal noise, shot noise, and flicker noise and their modelling is based ongeneralized stochastic processes. Brownian motion is the starting point for the modelling ofwhite-noise sources (thermal and shot noise), while fractional Brownian motion is used forflicker noise sources. The numerical schemes employed for the computation of solution pathsadapt well-known methods for stochastic differential equations to the specific situation withincircuit simulation. Under the assumption of small noise the convergence properties of the drift-implicit Euler scheme and the drift-implicit Milstein scheme are proved. Finally numericalexperiments with real-world circuits are presented.
Simulation plays an important role in the development of integrated circuits; it helps toreduce the time necessary for the design as well as reducing production costs. A keystep in the design process is circuit simulation, that is, the simulation of the chip at thetransistor level. Circuit simulation helps to verify the physical behaviour of the circuitbefore producing test chips and allows for early corrections. Circuit analysis in the timedomain (transient analysis) is usually performed without taking noise effects intoaccount. However, due to both decreasing supply voltages and the increasingimportance of parasitics, this is no longer possible. The signal-to-noise ratio is gettingso small that the noise effects must also be simulated. In most simulators this is donewithin the small-signal analysis in the frequency domain, which means that only a
linearization of the circuit is handled. Another approach is the non-linear frequencyanalysis for oscillatory circuits, which restricts the application to a special class ofchips. To overcome this restriction, we will present a non-linear noise analysiscombined with the transient simulation of the circuit. This analysis computes the so-called path of the noisy signals and allows the computation of the statistical momentsof the signals in a post-processing step. A direct computation of the moments hasbeen presented in [1]. This approach, however, requires a linearization which is notacceptable for some circuits as noise may influence them in an essentially non-linearway. Another problem is the treatment of flicker noise which must be approximated bywhite-noise sources. The method presented here allows the direct modelling of allimportant noise sources.An important prerequisite for transient noise simulation is the proper modelling of
the noise sources in the time domain. There are three main sources of noise in electroniccircuits: thermal noise, shot noise, and flicker noise (also called 1/f-noise). While themodelling of thermal and shot noise in the time domain is well understood and leads towhite-noise stochastic processes, the numerical treatment of flicker noise in the timedomain requires some new approaches. Similar to Brownian motion whose (general-ized) derivative is a white Gaussian process, we search for a stochastic processwhose derivative has a spectrum of 1/fb, 0 5 b 5 1. This process exists and is calledfractional Brownian motion. For a numerical simulation of flicker noise in the timedomain we must construct random variables fulfilling the properties of fractionalBrownian motion. This task can be accomplished with the help of an appropriatecovariance matrix.The first step in the simulation of circuits is to transfer the physical and topological
description of the circuit into mathematics. Most circuit simulators use modifiednodal analysis (MNA) to set up the equations. This means that Kirchhoff’s currentlaw is applied to every node of the circuit, replacing all branch currents withelement relations in admittance form and expressing branch voltages by the nodevoltages via Kirchhoff’s voltage law. Element equations for voltage-defining elementscomplete the set of equations. The charge-oriented formulation for the noiseless circuityields
A � _qðxÞ þ fðxÞ þ sðtÞ ¼ 0; xð0Þ ¼ x0; ð1Þ
where x2Rn is the vector of unknown nodal voltages and branch currents throughvoltage-defining elements, and t2 [0, tend] is the time, the independent variable.A2Rn,m is an incidence matrix describing the topology of the dynamic parts of thecircuit, and q(x):Rn?Rm is the vector of terminal charges and branch fluxes.f(x):Rn?Rn describes the static elements, and s(t):R?Rn is the vector ofindependent sources. x0 are the consistent initial values for equation (1). For detailson the set-up of equations see [2]. As the rank of the matrix A is generally not full,equation (1) is a differential-algebraic equation (DAE). In the following it will beassumed that the equations have an index of at most 1. Although DAEs with index 2often occur in circuit simulation, in the noisy case theoretical results for higher indexDAEs are not yet available.For a transient noise simulation, equation (1) must be extended by the noise sources.
These are shunt in parallel to the ideal, noiseless elements giving the stochasticdifferential-algebraic equation (SDAE)
A � _qðxÞ þ fðxÞ þ sðtÞ þ gðt; xÞ � xt ¼ 0; ð2Þ
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with g(t,x):R6Rn?Rn,k holding the noise densities, and xt2Rk is the vector ofGaussian white-noise processes. From a mathematical point of view, equation (2) is notcorrect as the noise processes are almost nowhere differentiable. Rewriting equation (2)as an integral equation
A � qðXtÞjt0 þZ t
0
fðXtÞ dtþZ t
0
sðtÞ dtþZ t
0
gðt;XtÞ dBt ¼ 0; ð3Þ
gives a mathematically proper formulation. Here, Xt denotes the stochastic pro-cess describing the circuit behaviour and Bt is a k-dimensional Brownian motionprocess.
In the next section we will investigate the modelling of thermal noise, shot noise, andflicker noise. Numerical schemes for solving equation (3) will be presented in Section 3.Some numerical experiments conclude the paper.
2. Modelling of white noise and flicker noise
In circuit simulation there are three different types of noise, namely thermal noise, shotnoise, and flicker noise (also known as 1/f-noise) [3]. Thermal noise has its foundationin the thermal motion of electrons, and it is most important within resistors. The(single-sided) noise density g(t,Xt) is given by
with k denoting Boltzmann’s constant, T the temperature and R the resistance.As the noise density does not depend on Xt, this type of noise is called additivenoise.
Shot noise is present in semiconducting elements and current sources and is due tothe discrete character of elementary charge. The (single-sided) noise density dependson the current and is given by
where q is the elementary charge and Idet(Xt) is the deterministic current through thedevice. As g depends on the stochastic process Xt, shot noise is called multiplicativenoise.
The physical background of flicker noise is not as clear as that of thermal or shotnoise; it is mostly due to crystal defects. The noise density g(t,Xt) is given by
where K denotes a model parameter (typical range 10– 25 – 10– 19), a is a physicalconstant (0.5 – 2.0), b is in most cases &1, Idet(t) is the deterministic current through
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the device, and f is the frequency. As the density depends on the frequency f, a quantitynormally not present in transient analysis, the modelling is much more complicatedcompared to the other noise types.In order to model the noise sources as stochastic processes we introduce the concept
of generalized stochastic processes (for details see [4]).
Definition. Let K be the space of test functions w:R?R that are smooth andidentically 0 outside of a compact interval, and let (O, S, P) be a probability space. LetF be a function on K that assigns to every w2K a random variable,
w 7! FðwÞ random variable on ðO;S;PÞ:
F is called a generalized stochastic process, if it is linear and continuous.For every stochastic process Xt with continuous paths a generalized stochastic
process X can be defined as
XðwÞ :¼Z 1�1
wðtÞXt dt; w 2 K:
Starting with the standard Brownian motion (Bt)t50 we get the generalized Brownianmotion process B as
BðwÞ :¼Z 1�1
wðtÞBt dt:
The derivative of a generalized stochastic process always exists, and it can be shownthat the generalized white-noise process W :¼ _B is a generalized Gaussian process. Ithas the following properties:
. the expectation functional mW(w)=0, w2K;
. the covariance functional CWðw;cÞ ¼R10
R10 dðt� sÞwðtÞcðsÞ dt ds, w,c2K,
where d is the Dirac delta function;. _B ¼W, where _B is generalized Brownian motion; and. the spectral density SW(f)=1.
Comparing these properties with the noise densities for thermal and shot noise showsthat the white-noise process W is adequate for modelling these two noise sources.Although this modelling is well accepted, it is somehow physically incorrect, see e.g.[5,6]. Using a corpuscular model for thermal noise, which reflects the dynamicbehaviour of the electrons more precisely, we get a different stochastic processdescribed by the so-called Ornstein –Uhlenbeck equation
Xtjtend0 ¼ �Z tend
0
aXt dtþZ tend
0
sdBt
with some physical parameters a, s2R, a4 0. For s/a= const., a??, theOrnstein –Uhlenbeck process converges to the generalized process W describedabove. For typical devices a&1013, so the difference between both approaches is quitesmall.
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For flicker noise we repeat the strategy used for thermal and shot noise. The startingpoint is the fractional Brownian motion ðBH
t Þt�0. Fractional Brownian motion was firstdefined by Mandelbrot and van Ness [7] as a Gaussian process with expectation
EðBHt Þ ¼ 0; for all t � 0;
and covariance function
Covðs; tÞ ¼ 1
2ðs2H þ t2H � js� tj2HÞ; s; t � 0: ð4Þ
The Hurst parameter H is a fixed real number with 05H5 1. We have to distinguishthe three cases H=1/2, 05H5 1/2, and 1/25H5 1. The first case reduces tostandard Brownian motion; for 05H5 1/2 the increments of the fractional Brownianmotion are negatively correlated; while for 1/25H5 1 they are positively correlated.The generalized fractional Brownian motion process BH is then given as
BHðwÞ :¼Z 1�1
wðtÞBHt dt; w 2 K:
It can be shown [8] that the derived process WH :¼ _BH
with 1/25H5 1 is ageneralized Gaussian process and has the following properties:
. the expectation functional mWHðwÞ ¼ 0, w2K;
. the covariance functional
CWHðw;cÞ ¼Z 10
Z 10
Hð2H� 1Þjt� sj2H�1wðtÞcðsÞ dt ds; w;c 2 K;
. _BH ¼WH, where _B
His the generalized fractional Brownian motion; and
. the spectral density SWHðfÞ ¼ const � ð 1jfjÞ2H�1
.
With 2H – 1= b this generalized processWH can be used for modelling the flicker noisein electric devices, and this completes the modelling of the noise sources for transientnoise analysis.
3. Numerical schemes for transient noise analysis
First, we discuss numerical schemes for the SDAE (3) driven only by Gaussian white-noise processes such as thermal noise and shot noise. We present adaptations of knownschemes for stochastic differential equations (SDEs) that are implicit in thedeterministic and explicit in the stochastic part of the SDAE (3). Designing themethods such that the iterates have to fulfil the constraints of the SDAE at the currenttime-point is the key idea of adapting known methods for SDEs to (3). This isimplemented by an implicit Euler discretization of the deterministic part.
Numerical schemes for flicker noise are not yet available, as the necessary calculusrequires further investigation (see e.g. [9]). In order to obtain at least an approximationfor flicker noise, this type of noise is treated similarly to white noise.
For the convergence results given below we assume that the SDAE (3) possessesnoise-free constraints. This guarantees a solution process that is not directly affected by
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the white-noise process, which is true for the MNA system if there are alwayscapacitances in parallel with a noise source. This is quite restrictive in the actual noisemodelling. Nevertheless, it is also possible to handle many situations where thiscondition is violated. Often noisy constraints are only needed for the determination ofalgebraic solution components that do not interact with the dynamical ones. Futurework should be dedicated to the classification of such situations.The noise densities given in Section 2 contain small parameters, and noise should not
be dominant in well-designed real-world circuits. Efficient integration schemes musttake advantage of this characteristic. To exploit the smallness of the noise in theanalysis of discretization errors we express noise densities in the form
gðt; xÞ :¼ E~gðt; xÞ; E� 1: ð5Þ
3.1 Drift-implicit Euler scheme
On the deterministic grid 0= t05 t15 � � � 5 tN= tend the drift-implicit Euler schemefor (3) is given by
AqðX‘Þ � qðX‘�1Þ
h‘þ fðX‘Þ þ sðt‘Þ þ gðt‘�1;X‘�1Þ
DB‘h‘¼ 0; ‘ ¼ 1; . . . ;N; ð6Þ
where h‘= t‘ – t‘ – 1, DB‘=B(t‘) –B(t‘ – 1), and X‘ denotes the approximation to Xt‘ .Realizations of DB‘ are simulated as N(0,h‘,I)-distributed random variables, seeSection 3.3. The Jacobian of (6) is the same as in the deterministic setting. In general,the Jacobian is solution-dependent and differs from path to path.The scheme (6) for the SDAE (3) possesses the same convergence properties as the
drift-implicit Euler scheme for SDEs [10,11]. In general, its order of strong convergenceis 1/2, i.e.
jjXt‘ � X‘jjL2ðOÞ :¼ ðEjXt‘ � X‘j2Þ1=2 � c � h1=2; h :¼ max‘¼1;...;N
h‘;
holds for the mean square norm of global errors. For additive noise, i.e. g(t,x)= g(t),the order of strong convergence is 1, for small noise (5) the error is bounded byO(h+ e2h1/2) (see [12] or [13] for related results).The smallness of the noise also allows special estimates of local error terms, which
can be used to control the step size. By means of Ito – Taylor expansions the local errorfor the Euler scheme applied to SDEs with small noise is analysed in [12]. For SDAEsthe local error in the so-called inherent SDE that describes the inherent dynamics of theSDAE in a lower-dimensional subspace must be considered [11]. As long as step sizeswith
h‘ � E2
are used, the dominating local error (per unit step) term of (6) is
where A7 denotes a suitable pseudo-inverse of A. For e? 0 it approaches the knownerror estimate in the deterministic setting. If an ensemble of solution paths is computedsimultaneously, the estimate Z‘ can be approximated and may be used to control thelocal error corresponding to a given tolerance. This results in an adaptive step sizesequence which is uniform for all solution paths.
3.2 Drift-implicit Milstein scheme
The Milstein scheme for SDEs possesses strong convergence of order 1. It differs fromthe Euler scheme by an additional correction term for the stochastic part, whichincludes double stochastic integrals. For additive noise the additional term vanishesand both schemes coincide.
where g ¼ ðg1; . . . ; gkÞ, I‘j ¼ ðI‘j;iÞki¼1, and I‘j;i ¼
R t‘t‘�1
R st‘�1
dBiðtÞdBjðsÞ.We point out the use of the derivatives ðgjÞy, j=1, . . . , k, and of the double
stochastic integrals in the scheme. Unless the commutativity condition
ðgiÞy gj�ðgjÞy gi ¼ 0; 8i 6¼ j;
is fulfilled, k(k – 1)/2 double stochastic integrals must be approximated in general. Thismakes the implementation of this scheme for systems involving a large number of noisesources rather costly.
To adapt the drift-implicit Milstein scheme to the SDAE (3) we intend to design thismethod in such a way that it implicitly realizes the drift-implicit Milstein scheme for theinherent SDE. Except for higher order terms this is realized by (cf. [10,11])
AqðX‘Þ � qðX‘�1Þ
h‘þ fðX‘Þ þ sðt‘Þ þ gðt‘�1;X‘�1Þ
DB‘h‘
�Xkj¼1ðgjÞxðAqx þ hfxÞ�1gðx‘�1; t‘�1Þ
I‘jh‘¼ 0;
where g=(g1, . . . , gk). In the last term the Jacobian Aqx þ hfx of the previous iteratecan be reused. The approach in [14] results in a similar approximation to the Milsteinscheme.
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The higher order 1 of strong convergence of this scheme must be paid for with theuse of a large number of double stochastic integrals and the use of the derivatives of thenoise densities. In an application with a large number of small noise sources we mustpay much for a mostly theoretical gain in accuracy.
3.3 Generation of random numbers
For a numerical simulation of thermal, shot, and flicker noise we need two types ofrandom numbers, namely N(0,1)-distributed random numbers for thermal and shotnoise, and random numbers with a distribution according to fractional Brownianmotion. While there are many efficient implementations available for the computationof N(0,1)-distributed pseudo-random numbers (e.g. [15]), the direct implementation ofrandom numbers for fractional Brownian motion leads to a very inefficient transientnoise analysis. The reason is that the random numbers of successive time steps are nolonger independent since the covariance matrix (cf. equation (4)) must be respected.The dimension of that matrix is the number of integration steps already computed,which can be very large for real applications. The consequence is that the simulatorspends most of the CPU time in calculating the random numbers instead of evaluatingthe circuit equations. Although it is possible to reduce the effort by using Schur-complement techniques, the effort is proportional to N2 with N denoting the number ofalready computed time steps.As the influence of time steps on the 1/f-distributed random numbers decreases with
the number of intermediate time steps, it makes sense to restrict the number of timesteps which enter into the computation of the random numbers. This defines a mask onthe covariance matrix and thus restricts the computational effort necessary for thecomputation of the random numbers. After an initialization phase the mask is shiftedby 1 after each time step. As the submatrices can be efficiently updated from step tostep with the Sherman –Morrison –Woodbury formula, the CPU time necessary forthe computation of the random numbers is quite small compared to the evaluation ofthe device equations and it is independent of the number of time steps. The errorintroduced by the mask can be controlled by the user through the mask size. Althougha rigorous error estimation is not yet available, a size of 100, for example, gives a goodcompromise between accuracy and runtime. As numerical tests have shown, the resultsof a computation with and without such a mask lie within the error tolerances used forsimulation.
4. Numerical experiments
We start with a textbook example of a ring-oscillator consisting of three coupledinverter steps with simple MOSFET models [14,16]. Only thermal noise in theMOSFETs and the resistors is considered. The corresponding circuit diagram is givenin figure 1.The system only partly fulfils the assumptions made in Section 3. The three thermal
resistance noise sources directly affect the current through the voltage source. However,the direct noise occurring in this variable is harmless in the sense that this variable doesnot influence others. Omitting this critical variable together with the nodal equation fornode 4 would lead to a system without direct noise. To make the differences betweenthe solutions of the noisy and the noise-free model for this simple example more visible,
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we dealt with a system where the diffusion coefficients had been scaled by a factor of1000.
In figures 2 and 3 we present numerical results obtained with the drift-implicit Eulerscheme. A mean-square estimate of the dominating local error term (7) was used tocontrol the step size according to the relative tolerance 10– 4. We plotted values of thenodal potential V(1) at node 1 versus the time. The solution of the noise-free system isgiven by the dashed line. In figure 2 we present the mean m and the boundaries of theinterval [m – 3s, m+3s], where s is the estimate for the standard deviation. Thesequantities are in fact moments of the solution and could be obtained more easily byweakly convergent numerical schemes. The mean appears damped and differs consi-derably from the noiseless, deterministic solution.
In figure 3 we present two sample paths (the solid lines). They cannot be consideredas small perturbations of the deterministic solution; phase-noise is highly visible.
Figure 1. Thermal noise sources in a MOSFET ring-oscillator model
Figure 2. The mean over 100 sample paths (E V(1)), the 3s range (+ 3s), and the noiseless voltage (det V(1))at node 1
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Although the new numerical schemes for SDAEs are developed independently ofcircuit simulators, only an implementation within such a simulator enables a test to becarried out on real-life examples. A slightly modified version of scheme (6) has beenimplemented in Infineon’s in-house simulator TITANTM. Assuming small noise – whichshould be true for well-designed chips – the step size control of the deterministictransient analysis gives quite reasonable results, cf. Section 3.1. The iterative methodused to solve the non-linear equation system (6) for X‘ needs some modifications in theconvergence tests as the noise currents which are added to the node equations play aspecial role. Currently, only the computation of a single path is available in TITAN; thesimultaneous computation of an ensemble of solution paths is under development.The cross-section of a 256 Mb DRAM chip has been used as a test application. This
circuit contains approximately 37 000 devices (mostly resistors and MOSFETs), whichgives k&150 000 thermal, shot and flicker noise sources. The dimension n of the systemis nearly 17 000. The mask size used for the computation of the 1/f – distributedrandom numbers was set to 100. The transient noise analysis requires roughly 30%more CPU time than the simulation without noise sources. Figure 4 gives the path of anodal voltage. The simulation without noise sources is plotted with a solid line; thedotted line represents the simulation with noise sources. The waveform of the noiseanalysis shows only small deviations from the deterministic result. Since the signalshown belongs to the digital part of the chip we expect this result.
4. Summary
We have presented a new approach to the pathwise transient noise analysis ofelectronic circuits. The modelling of the noise sources in the time domain is based ongeneralized stochastic processes which enables a direct simulation of all relevant noisesources. The numerical schemes presented here are based on well-known methods forstochastic differential equations. They are modified in such a way that they take into
Figure 3. Two sample paths of the voltage in node 1 (V(1)), the mean over 100 sample paths (E V(1)), and thenoiseless voltage (det V(1))
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account the specific situation in circuit simulation, namely the charge-orientedformulation as a stochastic differential-algebraic equation. Local error terms usablefor step-size control have been shown under the assumption of small noise. Furtherresearch should be conducted for a better understanding of the generalized stochasticprocess based on fractional Brownian motion as well as the higher index case.Improvements of the numerical methods should also be investigated in order to obtainefficient schemes with a uniform step-size control for ensembles of solution paths.
Acknowledgements
The authors acknowledge the valuable discussions and fruitful contributions given byU. Feldmann, Infineon Technologies. The first author is indebted to St. Schaffler, D.Meintrup, and C. Hillermeier, UniBw Munchen, for many helpful discussions. Thework of the second author was funded by the German Federal Ministry for Educationand Research (BMBF) which is appreciated.
References
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Figure 4. Simulation result for a cross-section of a 256 Mb DRAM: without noise (solid line); with noise(dotted line)
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