Variability analysis of interconnect structures including general nonlinear elements in SPICE-type framework A. Biondi, P. Manfredi, D. Vande Ginste, D. De Zutter and F.G. Canavero A stochastic modelling method is developed and implemented in a SPICE framework to analyse variability effects on interconnect struc- tures including general nonlinear elements. Introduction: Over the past few decades, computer-aided simulation tools have become an important asset for the design, simulation and optimisation of complex electronic networks. Nonetheless, since large-scale integration and miniaturisation lead to an important impact of the manufacturing process on the system performance, appropriate instruments are needed to evaluate uncertainties of the circuit par- ameters. Typical tools to gather quantitative statistical information of the circuit response are based on the well-known Monte Carlo (MC) method. This is computationally demanding, especially when realistic, complex structures are analysed. Effective solutions to overcome these previous limitations have been proposed, leveraging the so-called poly- nomial chaos (PC) methods. In [1], a PC-based technique, called the stochastic Galerkin method (SGM), was developed in MATLAB for transmission lines terminated by linear loads. This technique for ‘distributed’ circuits including linear elements was modified in [2] to allow for implementation in a SPICE environment. A PC-method for lumped circuits consisting of dis- crete, linear and nonlinear elements was first reported in [3, 4], allowing the modelling of uncertainties in a small-signal regime or by approxi- mating the nonlinearities by means of Taylor expansions. Recently, an improved PC-technique for ‘distributed’ circuit elements terminated by ‘general’ nonlinear loads was conceived by the authors of the present contribution [5]. Unfortunately, this technique could only be implemented in MATLAB, as it relies on a finite-difference time- domain solver for transmission lines, making it also cumbersome to deal with lossy, dispersive lines and arbitrary circuit topologies. Therefore, in the present Letter, a SPICE-compatible method is devel- oped and implemented in a traditional environment allowing us for the first time to perform PC-based variability analyses of lossy, disper- sive multiconductor transmission lines (MTLs) terminated by general nonlinear loads. Stochastic modelling formalism and implementation: Consider a uniform MTL with the propagation direction along the z-axis. The MTL consists of N signal conductors and a reference conductor. (An example of such a line is given in Fig. 1, where N= 2.) Owing to man- ufacturing, one or more geometrical and/or material parameters are not known in a deterministic way, thus they have to be treated as stochastic random variables (RVs), characterised by a probability density function (PDF), rendering the Telegrapher’s equations non-deterministic. For ease of notation, in this Section, we consider a single lossless dispersion- free line (N= 1), affected by a single stochastic parameter β. We can then write the pertinent stochastic Telegrapher’s equations as follows: ∂ ∂z v(z, t, b) i(z, t, b)   =− 0 L(b) C(b) 0   ∂ ∂t v(z, t, b) i(z, t, b)   (1) where v and i are the voltage and current along the line, and with L and C the per-unit-of-length (PUL) transmission line parameters. Next to the position z along the line and the time t, we have also explicitly written down the dependence on the stochastic parameter β, of which only the PDF is known, hence prohibiting a straightforward solution of (1). h G t w tan d n in n NX n FX C L n out n s R g 1 R g 2 A¢ A L e r w Fig. 1 Cross-section AA′ (left) of source-line-load configuration (right) of pair of coupled microstrip lines To solve (1), we rely on the SGM which is detailed in [1]. As a result of applying this method, a novel set of deterministic Telegrapher’s equations arises: ∂ ∂z ˜ v(z, t ) ˜ i(z, t)   =− 0 ˜  L ˜  C 0   ∂ ∂t ˜ v(z, t ) ˜ i(z, t)   (2) The new unknowns ˜ v and ˜ i are (K + 1)-vectors, containing voltage coefficients ˜ v k (z, t ) and current coefficients ˜ i k (z, t)(k = 0, …, K ). The parameter K determines the number of terms in the so-called PC-expansions, as explained in [1]. ˜  L and ˜  C are the known (K + 1) × (K + 1) PUL matrices. The ‘augmented’ system (2) is now fully determi- nistic, no longer showing dependence on β, and it can be implemented in a SPICE framework. On solving for ˜ v and ˜ i, statistical information about v and i is readily obtained. To solve the set of the 2(K + 1) equations in (2), a set of proper 2(K + 1) boundary conditions (BCs) is required. These are obtained by adding terminations to the lines. Assume that a nonlinear load is attached to the far-end terminal, i.e. at z =L, with the following charac- teristic: i(L, t, b) = F (v(L, t , b)) (3) where F(·) represents a general nonlinear function. Then, it was shown in [5] that by means of the SGM a new set of K + 1 deterministic BCs is obtained, which can be cast in the form ∀m = 0, ... , K: ˜ i m (L, t) ≃  Q q=1 w (1) mq F  K k=0 w (2) kq ˜ v k (L, t)   (4) where w (1) mq and w (2) kq are the weights (k, m = 0, …, K; q = 1, …, Q) and Q is a parameter that determines the accuracy. A similar set of BCs can be obtained at the near-end z = 0, for any kind of (non)linear load and gen- erator. The novel BCs (4) connect all voltage and current expansion coefficients contained in ˜ v and ˜ i through the known nonlinear function F(·) and proper linear combinations. Hence, these deterministic BCs (4) are readily implemented in a SPICE framework by using dependent sources. This results in a somewhat more complex network in terms of number of nodes, but guarantees a very efficient simulation yielding comprehensive statistical information, and rendering this technique very useful for variability analysis during circuit design. 0 0.2 0.4 0.6 0.8 1 x 10 –9 –0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time, s averages of voltage waveforms, V v in v out v NX v FX Fig. 2 Averages of voltage waveforms v in (t), v out (t), v NX (t) and v FX (t) at four terminals of coupled microstrip lines of Fig.1 Circles ( ): average computed using SGM technique; full black lines: average computed using MC technique Numerical results: In this Section, the technique is validated by apply- ing it to the variability analysis of the pair of coupled copper microstrip lines illustrated in Fig. 1. The length L is 5 cm and the gap G between the lines and the relative permittivity e r of the substrate are considered to be two RVs uniformly distributed in the range [70, 90]μm and [3.7, 4.3], respectively. The first line, i.e. the active line, is excited by means of a voltage source v s (t ) that produces a ramped step, going from 0 to 1 V in a risetime of 100 ps, in series with impedance R g1 = 50 Ω. This active line is terminated by means of a forward biased diode described by the well-known Shockley model i = I s e (v/hVt ) − 1 ( ) , TechsetCompositionLtd,Salisbury Doc:{EL}ISSUE/50-4/Pagination/EL20133191.3d Circuits andsystems