Powering Multiparameter Homotopy-Based Simulation with a Fast Path-Following Technique

The continuous scaling for fabrication technologies of electronic circuits demands the design of new and improved simulation techniques for integrated circuits. Therefore, this work shows how the hypersphere technique can be adapted and applied to trace a multiparameter homotopy. Besides, we present a path-following technique based on circles evolved from hypersphere , which is faster, and simpler to be implemented than hypersphere technique. Last, a comparative analysis between both techniques applied to simulation of circuits with bipolar transistors will be shown.


Introduction
The increment of the complexity of circuits influence the scientific progress in the simulation techniques area for integrated circuits. Also, homotopy techniques have been introduced as a useful tool in the area of operating point solution for circuits 1-5 , due to the Newton-Raphson NR method widely used which shows convergence problems 6 like oscillation and divergence.

Multiparameter Homotopy
The first step to formulate a homotopy is to establish the equilibrium equation to be solved; it is formulated from Kirchhoff laws, being defined as f x 0, where f : ∈ R n −→ R n , 2.1 2 ISRN Applied Mathematics where x represents the electrical variables of the circuit and n is the number of electrical variables. Multiparameter homotopies 7-9 are characterized by adding more than one extra homotopy parameter to the equilibrium equation. When homotopy parameters are adjusted to zero, the solution for H · becomes trivial, and when parameters reach value of one, then the operating point is located. The multiparameter homotopy function can be represented as where homotopy parameters are λ 1 , λ 2 , . . . , λ k ∈ 0, 1 and k is the number of homotopy parameters. Multiparameter homotopy 7 has been proposed in order to avoid fork bifurcations, singularities, among other problems that can be encountered with homotopy paths. Besides, as for the uniparametric 2 and multiparameter homotopies, the tracing technique 10, 11 is a fundamental tool capable of affecting the convergence, speed, and number of solutions located. Therefore, it is proposed to apply two tracing techniques for multiparameter homotopy, both will be described in the following sections.

Tracing Techniques
In order to apply tracing techniques described in this paper, a biparametric homotopy based in Newton's homotopy method will be used as an example: With the existence of two parameters λ 1 and λ 2 , two simultaneous deformations or transformations are produced: one in function f and another in function H. When x, λ 1 , λ 2 x i , 0, 0 , then Hence, homotopy function is satisfied. Besides, when λ 1 , λ 2 1, 1 becomes so the found solution of H is the solution of the equilibrium equation. Nonetheless, as function H has two extra variables, it is necessary to add two equations to the system H in order to be solved using more conventional techniques like NR.
Homotopy trajectory (1) Equation n 1 One equation is added to define path λ 1 − λ 2 , which will be named parametric function M λ 1 , λ 2 . This equation traverses three points λ 1 , λ 2 : where p 2 is defined by user, as shown in Figure 1 a . The range of values for A and B is 0, 1 .

(2) Equation n 2
Hypersphere equation is added 12 : where c is the center of the hypersphere which adjusts its value each iteration and r 1 is the hypersphere radius step size .
The summary of the procedures consists in the following steps 12 see Figure 1 b .
1 The first sphere is established S0 with center located at t0 x i , p 1 and the equation system is solved 3.1 , 3.4 , and 3.5 using the NR method setting t0 as an initial point , locating point t1.
2 A new hypersphere S1 is created with center at t1.
3 Using points t0 and t1, it is possible to create a prediction, which touches hypersphere S1 at point k 1 ; it is used as initial point for the NR method, until locating point t 2 on the homotopy path. 5 Points before and after p 3 are used to perform an interpolation 13 . The type of interpolation employed in this paper is linear multidimensional interpolation, which produces an approximation x a of solution x s for the equilibrium equation.
6 Finally, using as initial point x a in the NR method, the precision for the operating point x s is improved.
It is possible to replace 3.5 for the circle equation, in function of the homotopy parameters: where r 1. The rest of the steps to implement the numerical continuation are the same as the hypersphere technique already described.

Study Case: Circuit with Bipolar Transistors and a Diode
The following circuit 14 see Figure 2 contains nine solutions and has become the reference circuit for the homotopy applied to circuit analysis. Using the system reported by 14 , equilibrium equation is augmented    x i2 x i3 x s2 x s1 x s3 x s4 v 2 a Hypersphere technique x i1 x i2 x i3 x i4 x s2 x s1 x s3 x s4 Circles technique can be modified changing one of the homotopy parameters by an electrical variable of interest. For instance, the simulation was repeated from initial point x i1 , only changing the circle from 3.6 by Mathematics   7 where v 1 , r ∈ R. The result was that the homotopy path already known was traced see Figure 3 b with a total of 191 iterations locating the same solution at x s1 . Also, it is possible to use one of the two homotopy parameters with more than one electrical variable, to implement a reduced hypersphere. Therefore, in a forthcoming work the study of circles technique will be expanded and a possible application to simulate VLSI circuits will also be discussed.

Conclusion
This work showed that it is possible to use the hypersphere technique to trace multiparameter homotopies. Besides, a tracing technique derived from hypersphere circles was introduced, which is simpler to program and faster than the hypersphere technique. These results make the circles technique an attractive tool to trace multiparameter homotopies.