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

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.

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 as𝑓(𝑥)=0,where𝑓∶∈ℜ^𝑛⟶ℜ^𝑛,(2.1) 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 as𝐻𝑓(𝑥),𝜆₁,𝜆₂,…,𝜆_𝑘=0,(2.2) where homotopy parameters are 𝜆₁,𝜆₂,…,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.

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:𝐻𝑓(𝑥),𝜆₁,𝜆₂=𝑓𝑥,𝜆₂−1−𝜆₁𝑓𝑥_𝑖,0,where𝐻∶∈ℜ^𝑛+1×ℜ⟶ℜ^𝑛.(3.1)

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]=[𝑥_𝑖,0,0], then𝐻𝑓(𝑥),𝜆₁,𝜆₂=𝑓𝑥_𝑖,0−𝑓𝑥_𝑖,0=0.(3.2) Hence, homotopy function is satisfied. Besides, when [𝜆₁,𝜆₂]=[1,1] becomes𝐻𝑓(𝑥),𝜆₁,𝜆₂=𝑓(𝑥),(3.3) 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) Equation 𝑛+1
One equation is added to define path 𝜆₁−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 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)),(3.4) where 𝑝₂ is defined by user, as shown in Figure 1(a). The range of values for 𝐴 and 𝐵 is [0,1].

(a)

(b)

Figure 1 

(a) Parametric function. (b) Hypersphere technique.

(2) Equation 𝑛+2
Hypersphere equation is added [12]: 𝑆(⋅)=𝑥₁−𝑐₁²+𝑥₂−𝑐₂²+⋯+𝜆₁−𝑐_𝑛+1²+𝜆₂−𝑐_𝑛+2²−𝑟²,(3.5) 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,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 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 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 point 𝑝₃. (5)Points before and after 𝑝₃ 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:𝐶(⋅)=𝜆₁−𝑐_𝑛+1²+𝜆₂−𝑐_𝑛+2²−𝑟²,(3.6) 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 augmented𝐟𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝜆₂=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑓¹𝑓=6.103168𝐼_𝑠𝑒^40𝑣1−1𝜆₂+4.36634𝑣₂+2.863168𝐼_𝑠𝑒^40𝑣2−1−12,₂=5.4𝑣₁+3.58𝐼_𝑠𝑒^40𝑣1−1𝜆₂+6.62𝐼_𝑠𝑒^40𝑣2−1+𝑣₃𝑓+0.7𝐼_𝑠𝑒^40𝑣3−1+0.5𝐼_𝑠𝑒^40𝑣4−1−22,₃𝑓=6.103168𝐼_𝑠𝑒^40𝑣3−1+2.863168𝐼_𝑠𝑒^40𝑣4−1𝜆₂+4.36634𝑣₄−12,₄=𝑣₁+0.7𝐼_𝑠𝑒^40𝑣1−1𝜆₂+0.5𝐼_𝑠𝑒^40𝑣2−1+5.4𝑣₃+3.58𝐼_𝑠𝑒^40𝑣3−1+6.62𝐼_𝑠𝑒^40𝑣4−1𝜆₂−20,(4.1) where 𝐼_𝑠=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.

Table 1 presents in summary the results of performing the tracing of four paths with different initial points (𝑥_𝑖1, 𝑥_𝑖2, 𝑥_𝑖3, and 𝑥_𝑖4), 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 𝑥_𝑖1), 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.

(a) Hypersphere technique

(b) Circle technique

Figure 3 

Homotopy trajectories 𝑣2−𝜆1.

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) by𝐶𝑣₁,𝜆₁=𝑣²¹+𝜆²¹−𝑟²,(4.2) 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 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.

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.