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

Authors: Héctor Vázquez-Leal; Roberto Castañeda-Sheissa; Felipe Rabago-Bernal; Luis Hernández-Martínez; Arturo Sarmiento-Reyes; Uriel Filobello-Niño
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

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.


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## Body

## 1. 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.


## 2. 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(2.1)f(x)=0,wheref:∈Rn⟶Rn,
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(2.2)H(f(x),λ1,λ2,…,λk)=0,
where homotopy parameters are λ1,λ2,…,λk∈[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.


## 3. 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:(3.1)H(f(x),λ1,λ2)=f(x,λ2)-(1-λ1)f(xi,0),whereH:∈Rn+1×R⟶Rn.With the existence of two parameters (λ1 and λ2), two simultaneous deformations or transformations are produced: one in function f and another in function H. When [x,λ1,λ2]=[xi,0,0], then(3.2)H(f(x),λ1,λ2)=f(xi,0)-f(xi,0)=0.
Hence, homotopy function is satisfied. Besides, when [λ1,λ2]=[1,1] becomes(3.3)H(f(x),λ1,λ2)=f(x),
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.(1) Equationn+1
One equation is added to define pathλ1-λ2, which will be named parametric function M(λ1,λ2). This equation traverses three points [λ1,λ2]: p1=[0,0], p2=[A,B], and p3=[1,1]. The proposed equation is
(3.4)M(λ1,λ2)=-λ1+(λ2+(B(-1+A)/(AB+1-2A)))(-((-1+2A-B)λ2/(AB+1-2A))+2(B(-1+A)/(AB+1-2A))),
where p2 is defined by user, as shown in Figure 1(a). The range of values for A and B is [0,1].(a) Parametric function. (b) Hypersphere technique.
(a)(b)(2) Equationn+2
Hypersphere equation is added [12]:
(3.5)S(⋅)=(x1-c1)2+(x2-c2)2+⋯+(λ1-cn+1)2+(λ2-cn+2)2-r2,
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 establishedS0 with center located at t0=[xi,p1] 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 hypersphereS1 is created with center at t1.(3)
Using pointst0 and t1, it is possible to create a prediction, which touches hypersphere S1 at point k1; it is used as initial point for the NR method, until locating point t2 on the homotopy path.(4)
Steps 2 and 3 are successively repeated (updating hypersphere’s center after each iteration) until crossing pointp3.(5)
Points before and afterp3 are used to perform an interpolation [13]. The type of interpolation employed in this paper is linear multidimensional interpolation, which produces an approximation xa of solution xs for the equilibrium equation.(6)
Finally, using as initial pointxa in the NR method, the precision for the operating point xs is improved.It is possible to replace (3.5) for the circle equation, in function of the homotopy parameters:(3.6)C(⋅)=(λ1-cn+1)2+(λ2-cn+2)2-r2,
where r≪1. The rest of the steps to implement the numerical continuation are the same as the hypersphere technique already described.


## 4. 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(4.1)f(v1,v2,v3,v4,λ2)={f1=6.103168Is(e40v1-1)λ2+4.36634v2+2.863168Is(e40v2-1)-12,f2=5.4v1+3.58Is(e40v1-1)λ2+6.62Is(e40v2-1)+v3+0.7Is(e40v3-1)+0.5Is(e40v4-1)-22,f3=6.103168Is(e40v3-1)+2.863168Is(e40v4-1)λ2+4.36634v4-12,f4=v1+0.7Is(e40v1-1)λ2+0.5Is(e40v2-1)+5.4v3+3.58Is(e40v3-1)+6.62Is(e40v4-1)λ2-20,
where Is=10-6, [v1,v2,v3,v4] are the voltage drop between base-emitter terminals for each transistor in the circuit (see Figure 2) and λ2 is the second homotopy parameter. The complete homotopy formulation can be established by using the following equations:(i)
the augmented equilibrium equation (4.1),(ii)
the homotopy function equation (3.1),(iii)
the parametric function (3.4),(iv)
the hypersphere function (3.5) or circle function (3.6), which depends on the selected technique.Figure 2
Chua’s circuit.Table1 presents in summary the results of performing the tracing of four paths with different initial points (xi1, xi2, xi3, and xi4), each one. This process was repeated for both tracing techniques, showing the results in Figures 3(a) and 3(b). Comparing both techniques, there are two interesting conclusions to be highlighted: first, homotopy paths traced from the same initial point lead to the same solution; in fact, comparing paths point by point, it can be observed that it is the same path; second, despite that paths are identical, the circle technique required a fixed number of iterations (48); those are much less required than using the hypersphere method. In fact, from Table 1, it can be concluded that, at best (choosing initial point at xi1), the tracing technique for circles required 8.95 times less CPU time (by using MAPLE 15 software in an Intel Quad Core i7 processor at 2.6 GHz) than the hypersphere technique. Both tracing techniques employed radius of r=0.03 and a parametric function M with p2=[0.2,0.3].Table 1
Relevant points for homotopy simulations.
Hypersphere—Init. point where [λ1,λ2]=[0,0]No. IterTime  (Sec)Operating point[v1,v2,v3,v4]where [λ1,λ2]=[1,1]xi1=[-5,-5,-5,-5]5197.70xs1=[0.3830,-3.5446,0.3851,-4.0990]xi2=[-1,-2,-1,0]2023.59xs2=[0.3869,-4.6321,-0.8002,0.3775]xi3=[-5,-0.5,-5,0]2164.06xs3=[-0.5136,0.3775,-0.9682,0.3775]xi4=[-1,0,0,0]1683.13xs4=[-1.0510,0.3775,0.3845,-3.9542]Circle—Init. pointwhere [λ1,λ2]=[0,0]No. IterTime  (Sec)Operating point[v1,v2,v3,v4]where [λ1,λ2]=[1,1]xi1=[-5,-5,-5,-5]480.86xs1=[0.3830,-3.5446,0.3851,-4.0990]xi2=[-1,-2,-1,0]480.89xs2=[0.3869,-4.6321,-0.8002,0.3775]xi3=[-5,-0.5,-5,0]480.74xs4=[-0.5136,0.3775,-0.9682,0.3775]xi4=[-1,0,0,0]480.82xs4=[-1.0510,0.3775,0.3845,-3.9542]Homotopy trajectoriesv2-λ1.
(a)
Hypersphere technique(b)
Circle techniqueCircles technique can be modified changing one of the homotopy parameters by an electrical variable of interest. For instance, the simulation was repeated from initial pointxi1, only changing the circle from (3.6) by(4.2)C(v1,λ1)=v12+λ12-r2,
where v1,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 xs1). 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.


## 5. 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.



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*Source: 610637-2011-10-31.xml*