SYMBOLIC ANALYSIS OF LARGE SIGNALS IN NONLINEAR SYSTEMS

Domine M. W. Leenaerts

Technical University Eindhoven, Department of EE, P.O. Box 513, 5600 MB, Eindhoven, the Netherlands

e-mail: dominel@eeb.ele.tue.nl

ABSTRACT

A method is proposed to obtain symboic expressions for the large signal behavior in nonlinear circuits. The expressions are described in terms of a nonlinear oper- ator. The method is valid when nonlinear component behavior is piecewise linear approximated. The method is a large extension of the technique as proposed by Fer- nandez [4].

1. INTRODUCTION

Symbolic analysis allows the circuit designer t o concen- trate on relations between design parameteirs and the speci ... cation parameters of the circuit [I]. This in con- trast t o numerical analysis which the designer provides with actual values of the performance for a chosen set of design parameters.

In the early days, symbolic analysis coulld only be applied t o linear networks. Because most circuits be- have nonlinear, the question arises how t o handle and solve nonlinear functions in a symbolic way.. Dinerent authors have focussed on this problem and started with weakly nonlinearities. This class of nonlinearities can be approximated by power series expansion up to a cer- tain order. One the possible approaches is to calculate the Volterra kernels that characterize the behavior in the frequency domain [2]. For the symbolic analysis of strong nonlinear circuits no good methods have been developed so far, although some approaches have been taken [3].

In [4] piecewise linear (PL) techniques are used t o ap- proximate the nonlinear behavior of the circuit. The re- sulting set of equations are then linear and can be han- dled by a symbolic simulator. However, the approach lacks the fact that the nonlinearity is not inicorporated into the resulting symbolic equations. I t is up to the designer t o understand when which linear description is valid. A much more convenient technique would be if the symbolic expressions include the nonlinearity, i.e. the expressions are valid over all PL segments.

In this paper a method will be presented which will give the symbolic expressions of PL circuits solutions. The solutions include the nonlinear behavior, i.e. they are valid over all PL segments. The method is based on a constructive proof to obtain explicit expressions for the solutions of piecewise linear networks if these are of class-P. The latter property is not limiting the applicability of the method because many transistor networks are of class-P.

We will start with some preliminaries on PL tech- niques in section 2. I t will be shown that any PL model is related t o the Linear Complementary Prob- lem (LCP). Then, in section 3 a short outline of the proof will be given to obtain the solutions of a class-P LCP in explicit formulas. To demonstrate the applica- bility of this result in symbolic analysis, two network examples will be provided and explained (section 4). We will end up with some conclusions.

2. PRELIMINARIES

The piecewise linear (PL) network or mapping f is as- sumed to be continuous. The function or mapping f can be written in a certain PL model description and is inherently confronted with the Linear Complementary Problem (LCP) [5]. In this paper the used format is

U o j o u j = o (2)

with which many network components can be modelled 161. In (I) , the ... rst equation de ... nes the linear relation between and 2. The second equation de ... nes in which region this linear relation holds. Changing from one re- gion into another will enect the linear relation via state vector U , resulting in a new linear relation valid for 'the new entered region. Close related t o PL modeling is the LCP.

The LCP is de ... ned as obtaining the solutions U and j of

j = D u + q (3)

0-7803-547 1 -0/99/$10.0001999 IEEE

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under restriction (2 ) , known as the complementary con- dition. As example consider the PL function of a non- linear resistor in Fig. 1, described as

+ ( $) + ( l ) u + l = O j = + U 1 (4)

The second equation of (4) together with (2) de ... nes which linear segment is valid. When is a t least 1 volt, U has to be equal t o zero. The current can then be obtained from the ... rst equation of (4), = 2 . When

. Using this result in the ... rst equation results in the new linear relation = 3 .

1, the second equation in (4) be- comes j = U + q and is therefore a special case of the LCP (3). LCP (3) yields an unique solution for any q under the restriction that D P , where class-P is strictly de ... ned as in [5]. A de ... nition for class-P is that the principal minors of D must be positive.

is less then 1 volt, j is equal to zero and U = 1

De ... ning q =

3. EXPLICIT SOLUTIONS

Consider a network, consisting of PL components, re- sistors and (controlled) sources. Applying nodal analy- sis and taken the complementary variables U j of the PL components as the network variables, the complete behavior of the network can be given by (3) together with (2). Note that (2) retects a nonlinear operation and therefore this set of equations describes the com- plete large signal behavior of the nonlinear network.

For class-P networks, a technique does exist to ob- tain explicit expressions for the solutions of (3) [5],[7]. An outline of this proof is as follows. De ... ne the - operator as 2 = ( 2 + x) and consider the scalar problem j = U + q valid under condition (2). For q 0, the solution yields j = q U = 0 and for 4 < 0 we obtain j = 0 U = -. Consequently, the solution may be written in an explicit formula, namely

Consider now the two-dimensional problem (again valid under restriction (2))

then we may also write

jl = l l U l + j 2 = 21U1 + 22U2 + 4 2 (7)

with = q1 + 1 2 ~ 2 . De ... ne an other scalar LCP, to be derived from (7) by taking u1 = 0,

where the tilde reSects that the solution of (8) is only valid within the frame of the global solution of (6). Fur- thermore, le t = q1 + 1 2 ~ Starting from the unique solution U and j of ( 6 ) , determine the value of . Sup- pose u1 = 0. This result implies, comparing (7) with (8), that j 2 = j u2 = U .

Consequently, this equivalence yields that = 0 in which can be derived solely from the solutions of the reduced LCP (8). In a similar way the situation in which < 0 can be treated. Finally, combining the results yields the global solution

0 then from (7) j , =

Therefore, the solution of the 2-dimensional LCP (7) is expressed in terms of the solutions of the reduced order l-dimensional problem (8) that is solved by (5). In this way the solution can explicitly be written down in a recursive fashion. For (6) the solution yields

j l = 4 1 + 12 - j 2 = 4 2 + 21 - (10)

In [5] the proof is given for any n-dimensional LCP of class-P.

The discussed steps can completely be performed in a symbolic way. Obviously, in symbolic expressions, it is not always clear if the LCP is of class-P, but this can be checked afterwards by Jling in numerical values. The steps to analyze a PL network in a symbolic matter are the following:

Reformulate the network equations in terms of the state vectors U , and j of the PL models.

Solve the LCP j = DU + q, U

in an explicit way. 0, j 0, U j = 0

Use the explicit formulas to express other network voltages and currents in terms of the network pa- rameters.

4. EXAMPLES

Consider the PL network of Fig. 2 with two ideal diodes (U 0 j 0 ~j = 0). Applying Kirchhon's voltage law yields

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which can be reformulated as the LCP (together with (2)) U = j + with

- 1 - 1( 2 + 3 ) + 4 1 3 - 1 3 3 ( 1 + 2 )

(1 2) and = + + 3 . Note that this LCP is of class- P for any positive resistors values. According to the previous section (9). the solution for u1 can be given as

1 U1 = - ( 3 + 2)El + E2 1 3 j 2 (13)

where the solution for j 2 in (13) can be obtained from the reduced system (like (8))

1 1 u 2 = - [ 3 ( I + 2 ) ] j 2 + - ( 3Ei E2) (14)

This yields the intermediate result

E2 3El j 2 =

3 ( 1 + 2 )

Combining (13) with (15) ... nally yields the explicit symbolic solution for u1 as

1 U1 = - ( 3 + 2)Ei -t E2 1 3Pi (16)

E2 3El PI = 3 ( l f 2 )

and in the same way the explicit solution for u2 can be obtained as

( 3 + 2 ) E l E2 1( 2 + 3 ) + 4

P2 =

The node voltages and follow directly from (16)- (18) and (11). Note that the -operator de ... nes the nonlinear behavior of the network and that (16)-(18) are valid for any parameter value, i.e. large s8ignal be- havior. As example, choose l = 2 = == = l and sweep El E2. Node voltage has a 3-segment PL behavior, as can be seen from Fig. 3.

As second example consider the single MOST ampli- ... er of Fig. 4. The behavior of the MOST is in i ts most simple way modelled as [6]

+ (0 0) + ( ) U + (0) = 0

0 1 t

(20) 1 0 j = + U +

with the threshold voltage, = 2L and where restriction (2) st i l l applies. The network equations are described by

E l = + 2 E 2 = + ( I + 2 ) (21)

and together with the second equation in (20), the LCP yields

j = 1+ 2 2 t E1 1 I + 1 t El -t E2

(22) Note that ,the LCP is of class-P for any solutions of (22) are given as

+. The

~ 1 = E -+1+ t E l + - + E2

~ 2 = E __ E2 + 1+ t E l + - -

E = E l t (23)

with = 1 + ( + 2 ) These intermediate so- lutions can be combined with the ... rst equation of (20) to express the current through the MOST: :=

u2). These type of symbolic formulas (allow thle designer to make several design large signal trade-on!;. In this case the formulas are rather coarse due to the simplicity of the used MOST model. However, adding more PL segments will result in a better approxima- tion, and yielding more accurate solutions.

For large networks, a symbolic analysis in all its components is most likely not worthwhile. The manly segments will result in high order nestings of the - operator and expressions become t o complex to undeir- stand them. Most designers will use symbolic: analysis up to a certain level by choosing already several pa- rameter values and explore the design space for the remaining parameters. For instance, suppose that for the MOST network El = 3 , E2 = 5 , =: 1 and

2 . The relation between the resistor val- ues and the current is depicted in Fig. 5.

(uI

= 10

5. CONCLUSION

A method is presented t o analyze in a symbolic manner large signal behavior of nonlinear networks. 'The non- linear behavior is piecewise linear approximated and this allows to ... nd the explicit formulas of the soh- tions in terms of the network voltages and currents. The symbolic expressions are described in terms of the nonlinear -operator, making the solutions valid over all PL segments. This is in great contrast t o the method proposed in [4], where the solutions are only valid within a certain region, i.e. the solutions do not contain a nonlinear operator t o make the solutions global valid.

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6. REFERENCES

[ I ] F.V. Fernandez, A. Rodriquez-Vdzguez, J.L. Huer- tas and G.Gielen, eds. Symbolic Analysis Tech- niques. Applications to Analog Design Automation IEEE Press, 1998

[2] P. Wambacq, Symbolic Analysis of Large and Weakly Nonlinear Analog Integrated Circuits, Ph.D. Dissertation Katholieke Universiteit Leuven, Belgium, 1996

[3] C. Borchers, L. Hedrich, E. Barke,'Equation-based model generation for nonlinear analog circuits,' Proc. 33rd Design Automation Conf.,pp.236-239, 1996

[4] F.V. Fernandez, B. Perez-Verdu, A. Rodriquez- Vazguez,'Behavioral Modeling of PWL Analog Cir- cuits using Symbolic Analysis,' Proc. ISCAS Mon- terey, 1998

[5] D.M.W. Leenaerts, W.M.G. van Bokhoven, Piece- wise Linear Modeling and Analysis, Kluwer Acad- emic Pub., New York, 1998

[6] W. Kruiskamp, D. Leenaerts, "Behavioral and macro modeling using piecewise linear techniques", Int. Journal on Analog Integrated Circuits and Sig- nal Processing, I O , no. 1/2, pp.67-76, 1996

[7] D.M.W. Leenaerts, "Further Extensions to Chua's Explicit Piecewise Linear Function Descriptions", Int. Journal of Circuit Theory and Appl., 24, pp.621-633,1996

u > o u = o /

Fig.1. A piecewise linear function and its electrical network equivalent based on the ideal diode

U1 v3- + R4

R E l v2 1/1

+ u2,

E2 -

I I,

F ig2 Network with two ideal diodes, representing the nonlinear behavior.

! I T------- .,........ , >-- ...... / - - - .... ........ ,___ , ,-- ......

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4

Fig.3. Behavior of 3, having a 3-segment PL behavior.

m 1 I I

Fig.4. Single MOST network example. $:?t$S

FigS. Relation between , and .

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