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IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.6 JUNE 2005

1631

LETTER

Finding All DC Operating Points of Piecewise-Linear Circuits

Containing Neither Voltage nor Current Controlled Resistors

Kiyotaka YAMAMURA†a), Member and Daiki KAYA†∗, Nonmember

SUMMARY

finding all characteristic curves of one-port piecewise-linear resistive cir-

cuits. Using these algorithms, a middle scale one-port circuit can be repre-

sented by a piecewise-linear resistor that is neither voltage nor current con-

trolled. In this letter, an efficient algorithm is proposed for finding all dc

operating points of piecewise-linear circuits containing such neither volt-

age nor current controlled resistors.

key words: dc analysis, characteristic curve, piecewise-linear circuit, find-

ing all solutions

Recently, efficient algorithms have been proposed for

1.Introduction

This letter deals with the problem of finding all dc operat-

ing points of piecewise-linear (PWL) resistive circuits con-

taining neither voltage nor current controlled resistors, i.e.,

resistors whose v-i characteristics are represented by multi-

valued PWL functions.

In general, large scale circuits contain several often

used subcircuits that have already been analyzed, for ex-

ample, logic gates, amplifiers, analog-to-digital converters,

filters, and so on. By specifying these subcircuits by sim-

ple macromodels, large scale circuits can be analyzed ef-

ficiently [1]. The idea of finding characteristic curves of

a one-port circuit and then modeling it by a neither voltage

nor current controlledresistor is one of suchapproaches(see

Fig. 1).

Fortunately, studies on algorithms for finding all char-

acteristic curves of one-port PWL resistive circuits (as well

as algorithms for finding all dc operating points) have re-

markably developed; for example, see the references cited

in [2]. Especially, the algorithm proposed in Sect.5 of [2]

succeeded in finding all characteristic curves of one-port cir-

cuits containing 200 PWL resistors in practical computation

time. Thus, it is now possible to represent middle scale one-

port circuits by PWL resistors that are neither voltage nor

current controlled.

In this letter, we propose an efficient algorithm for find-

ing all dc operating points of PWL circuits containing such

neither voltage nor current controlled resistors by extending

the algorithm proposed in Sect.3 of [2], which is one of the

most efficient algorithms for finding all operating points of

PWL circuits. Effectiveness of the proposed algorithm is

Manuscript received November 30, 2004.

Final manuscript received February 21, 2005.

†The authors are with the Faculty of Science and Engineering,

Chuo University, Tokyo, 112-8551 Japan.

∗Presently, the author is with NEC Electronics Co., Ltd.

a)E-mail: yamamura@elect.chuo-u.ac.jp

DOI: 10.1093/ietfec/e88–a.6.1631

Fig.1

Macromodeling of a one-port circuit.

confirmed by several numerical examples.

2.Parametric Representation of Piecewise-Linear Re-

sistors

For multi-valued PWL characteristics, it is often effective

to express the voltage and the current using a new variable

called a parameter. In this section, we introduce the para-

metric representation of PWL resistors proposed in [3].

We first introduce the following notation for a real

number λ:

λ+= max{λ,0},

Assume that we are given a one-dimensional PWL

curve in Rlcharacterized by a set of m + 1 breakpoints

x0, x1,···, xmand two directions x−∞and x+∞(see Fig. 2).

Then, the parametric representation can proceed as fol-

lows [3].

λ−= max{−λ,0}.

(1)

1. Assign a parameter λ running from −∞ to +∞.

2. The part between λ = −∞ and λ = 1 is given by:

x = x0+ x−∞· λ−+ (x1− x0)λ+.

3. The direction of the curve between x0and x1is x1−x0.

From λ = 1 onwards, the direction of the curve has

to be corrected. This can be done by adding a term

(x2− 2x1+ x0) · (λ − 1)+:

x = x0+ x−∞· λ−+ (x1− x0) · λ+

+(x2− 2x1+ x0) · (λ − 1)+.

This formula is now valid between λ = −∞ and λ = 2.

4. This is continued to describe the complete curve:

(2)

(3)

x = x0+ x−∞· λ−+ (x1− x0) · λ+

m

?

+

k=2

(xk− 2xk−1+ xk−2) · (λ − k + 1)+

Copyright c ? 2005 The Institute of Electronics, Information and Communication Engineers