A Theory of Electrical Circuits with Resistively Coupled Distributed Structures: Delay Time Predicting

This paper deals with qualitative properties ( existence, uniqueness, and especially, stability) and numerical solution of a circuit consisting of a resistive multiport with r-c-g "exactly modeled" distrib­ uted elements connected to its terminals. This kind of results are useful, for instance, when we study the effect of interconnections on the speed of transient process from an integrated structure. A formula to evaluate the delay time as a global parameter of the circuit is given and verified by numerical calculus.


I. INTRODUCTION
T HE technology of integrated circuits is structurally involved in dealing with structures with distributed parameters.For instance, in MOS integrated circuits, a given inverter or logic node may drive several gates through polysilicon thin-film wires.This thin-film technology re sults in interconnection lines with parameters distributed over their entire length.Also, in monolithic structures a resistor or a capacitor appears unavoidably as a compo nent presenting as well a distributed resistance ( r ), a distributed capacitance ( c ), and a distributed conductance (g).With the advances in technology, the delay time associated with interconnections becomes significant com pared to the device delay time and in certain cases dominates the chip performance [1].To predict this extra devices delay for an integrated structure, we can replace the devices with their resistive models and thus we have to study a network composed by a resistive multiport with r-c-g distributed structures ("lines") connected to its termi nals (see Fig. 1).
The mathematical model of this class of circuits (for mulated in Section II) consists of a parabolic system of equations with nonstandard boundary conditions.The cor rectness of this model is given by existence, uniqueness and stability theorems enounced in Section III where the restrictions of hypotheses are also commented.
Our theorem regarding exponential asymptotic stability (the main result of this paper) suggests a definition and allows an evaluation of this delay time of the circuit under study.This parameter characterizes the rate of the evolu tion of the state vector to the stationary regime.As the upper bound of this characteristic depends upon all the circuit parameters and is easy to compute, such an ap proach seems to be quite useful in circuit design.
The proofs (presented in Section IV) use essentially the dissipative operator approach (see [3], (41) and its relations to abstract differential equation arising from our model.In Section V a numerical algorithm of the type finite elements in space, finite differences in time is formulated in order to obtain the solution of the problem under study.This algorithm is then used in Section VI in validating the evaluation of the response time computed for several ex amples.
The main mathematical notations and definitions are relegated to Appendix where a useful lemma is also proved.
If we try to establish the relations of this paper with mathematical and technical literature, let us mention at the beginning that the nonclassical kind of boundary condi tions appearing in our model were considered in [5]- (8] but for hyperbolic problems which cannot come to our one. The stability of the circuits with distributed structures was studied by using different methods.In the remarkable paper (9], for instance, the authors use a Lyapunov theory for a functional-differential equation of retarded type that describe a large class of such circuits.A small signal stability criterion (in terms of the roots of characteristic equation) is derived in [10] and input-output stability results are given in [11] for circuits with parasitic elements, by applying the theory of singular perturbations.
Analytical estimations of the transient response of cir cuits containing distributed structures (r-c lines) were ob tained especially for the case of a single such distributed element.A pioneering work in this field is (12].For a nonuniform open circuited r-c line the paper (13] shows the step response to be monotonously increasing.In [14] the bounds of this response were obtained. The engineering literature shows lately an intense preoc cupation regarding the estimation of signals delay in MOS integrated circuits (see [2] The delay estimation in lumped parameter RC circuits allows efficient using of CAD programs ([1], [17], [18] and references quoted therein).A fundamental paper in this respect is [2], where bounds of the transient response at a step-variation input are derived for each node of a lumped RC fanout circuit.The methods and results of this paper were then extended to mesh-type networks, [18], and to RC ladder networks with nonlinearities, [19].Another extension which takes into account the effect of initial condition is presented in [17].
There are two main directions of study that arise natu rally; first, to consider circuits whose distributed elements are "exact" models (r-c-g lines) and, second, to consider the dynamic behavior excited by step-variations simulta neously applied to more than one input.With such signifi cant extensions of the model, the present paper gives the estimation of the delay time defined as a global parameter of the circuit.
We shall report elsewhere the extension of below results to the circuits supplementary containing lumped capaci tors.A preliminary restricted version of this paper were [21], (30].

II. THE CIRCUIT MODEL
The circuit under study consists of a lumped parameter resistive multiport with any kind of sources as well as r-c-g distributed elements connected to its terminals, as shown in Fig. 1.
In Fig. 1, u k (t, x ) and i k (t, x ) is, respectively, the voltage and the current at the moment t E [O, oo) in the point x E [O, dd of the distributed structure, k E 1, Q n .This structure is mathematically described by the well known telegraph equations: (2.1) In the following we shall consider r k , c k , and g k constant parameters and we assume r k , c k > 0, g k ;;;;,: 0. From (2.1) we formally derive the system: 2) On the other hand, we shall suppose that the multiport (2n-port) is described by the following linear relation between the vector of the currents and the vector of voltages: [h]=...:..G[:']+B(t) hn 2n (2.3) where G is a 2n X 2n matrix of conductances and B is a 2n vector depending on sources.By observing in Fig. 1 that i k (t,O) = j 2k _ 1 (t) and i k (t, d k ) = -) 2k (t) and that u k (t,O) = u 2k _ 1 (t) and u k (t, d k ) = u 2k (t) we can derive from the first relation (2.1) and from (2.3) the following system of boundary conditions:

+B(t). (2.4)
Of course we have to add the initial condition: kEl,n and let u 0 ( x) the vector with u kO ( x) as components. (2.5) With these, our problem is the system (2.2) with boundary conditions (2.4) and with initial conditions (2.5).

III. STATEMENT OF THE RESULTS AND COMMENTS
In this section we refer to the solutions in the classical sense of our problem.The first result deals with the sufficient conditions for existence and uniqueness of such a solution.
Theo rem 3.1: Let us consider all independent sources having simultaneous step variation at t = 0. Then i) If G is a positive semidefinite matrix and u 0 is a C 2 function satisfying the boundary conditions, the dynamic problem (2.2), (2.4), (2.5) has a unique solution.ii) If G is positive semidefinite and g; > 0 for all i E 1, n, the steady-state solution of our problem exists and is unique.iii) If G is positive definite and there is i E 1, n for which g; = 0, there exists a unique steady-state solution.
Our second result establishes sufficient conditions for global exponential asymptotic stability of the network solutions.Before enouncing it, let us observe that if G is DRD we can find a unique y 1 E (0, 1r/2) such that for each j E 1, n.For our 2n X 2n matrix G with G; 1 elements we shall denote  In order to apply the previous results for circuit models validation, one has to check the assumptions used in obtaining those results.Consider the usual case of the MOS interconnections analysis [2], when: a) one of the multiport terminals is common for all external elements (this is the common "ground" of the r-c-g lines) b) all sources are independent.
A straightforward analysis of the resistive multiports hav ing the properties a) and b ), [22], shows that: 1) The existence of the matrix G is ensured if and only if the following hold: c) none of the O or drterminals is connected directly (i.e., through a zero-resistance branch) to the ground d) there is no direct connection between O and/or dr terminals.
2) If G does exist, then it is symmetric and WDRD, hence it is positively semidefinite.
All of above facts imply that if, besides the assumptions a)-d), is required that g; > 0 for all i then, according to Theorem 3.1, we obtain a broad class of circuits-let us say� -for which the dynamic and de solutions exist and are unique.
In order to ensure the realization of the assumptio:ns of Theorem 3.2 (thus allowing the delay time estimatio:n) one has to add the DRD condition for G to conditions a)-d).
Due to the fact that in this case G is positive definite, existence and uniqueness of the solution hold even if • g; = 0 for sotne i.
Consider, for instance, the simplest interconnection cir cuit of Fig. 2 where the three distributed elements are connected in A and the circuit loads are R", R III E (0, oo ]. If, as usually, we take R 1 = R 2 = R 3 = 0 and R' = oo, the condition c) is not fulfilled hence the matrix G does not exist.However if we assume-that R' = oo but R 1 , R 2 , R 3 =I= 0 (the contact resistances) and g 1 , g 2 , g 3 > 0, the circuit belongs to class �. hence its model is correct.Unfor tunately, one can still say nothing about the stability of the solutions.
Finally, if R 1 ,R 2 ,R 3 =1=0 and R',R",R 111 E(O,oo), we find We consider for the beginning the stationary (i.e., time independent) problem, where we take zero-dielectric con ductivity for the first k 0 distributed elements.So we have where G; = l/R;, i=l,2,3, G'=l/R', and This matrix is DRD and then all the qualitative proper ties are valid regardless of g; values.A finite R' can be "created" by concentrating the dielectric conductance g 1 to the point A, i.e., R' = ljg 1 d 1 .The distributed element is taken into account afterwards via the nonzero parameters r 1 , c 1 , d 1 , ar1d g 1 = 0.The same approach can be taken for getting finite values for R" and R III if the last two lines are originally open circuited. --- where we have denoted a k = g k /r k with a k = 0 for k E 1, k 0 and a k > 0 for k Ek 0 + 1, n.The following func tions where w is the '2n vector with componep.tsw k .The follow ing estimation is easy to be derived for the matrix P: where M 1 and M2 are strictly positive constants.Hence, the matrix P + G is positive definite under each of our h yp otheses ii) or iii), so that ( 4.4) has a solution which fixes the constants in (4.3).
The uniqueness of steady-state solution results from the dissipativity properties that follow.Let us define the oper ator A: .@(A)""'He° with .@(A)={ u = (u 1 , • •, u n ) EH; and satisfies (2.4)} and Au =  where u E .@(A)c H c 0 • If we take u, u E .@(A), then an elementary calculation shows, that In order to prove the m-dissipativity we have to find a function u E .@(A)such as for each component   The m-dissipativity implies that A generates a contraction semigroup of class C 0 , [3], and hence, (23] for any u 0 E �(A) there exists a unique solution of (4.9) belonging to class C([O,oo),Hc°)nC 1 ([0,oo),E&(A)).Finally, from the well-known a priori estimates for parabolic problems com bined with Sobolev's imbedding theorem, the desired dif ferentiability properties follow (see (24] and (25] for more classical approach).For recent study of Cauchy problem we refer to [29].

B. Proof of Theorem 3.2
Let u = ( u 1 , • •, u 11 ) be the classical solution of our problem (2.2), (2.4), (2.5).For every k E G we take where a k will be conveniently chosen.We obtain a new problem  The following result is essential for our aims: Lemma 4.1: We suppose that G is DRD and for every j EI� we denote a 1 = y 1 -c where y 1 E (0, '7T /2) is the solution of (3.1) and c > 0 is such that a 1 E (0, 'TT/2).If we take then, the operator A -wJ constructed with these a 1 is dissipative on Z.
Proof' If we take v, iJ E £&(A) and suppose 'the non trivial case w = v -iJ if:.0, according with Lemma A (see the Appendix) we have to prove From here, with the inequality a P < tana P we derive that a P .

V. APPROXIMATION OF THE SOLUTION
We shall approximate the solution u of the problem (2.2), (2.4), (2.5) by the finite difference method in time and the finite element method in space.If cf, = ( cf, 1, If we formally integrate by parts, then according with (2.5) we get that for all cf, E H 1 , where P( u ( t) ,cJ, ) and k=l c k To approximate (5.2) we suppose that a family of finite dimensional subspaces {S h } h > 0 of H 1 is given such that for a small positive h there is a constant c , independent of h and v for which inf { !Iv -cJ,II H o + h l lv -'P II H'} � c h s llvll w </> E Sh where v En s for s = 2.For examples of such subspaces, which include the usual piecewise-linear finite element spaces see [28).
We may now pose the semi-discrete problem corre sponding to (5.2).Find u h (t) E S h for t> 0, such that { !_ (u h (t) , cp ) H o + P( u h ( t),cp) = R (B ( t ) , cp)  The matrix G and the vector B have the following terms: On the other hand, we have computed numerically our problem by the method presented in Section V.In Fig. 4 we give the space variation of the solution for the different values t; = idt of the time in the case of Example I.
The discretization parameters were h = 0.0202 • d and /it= 8.52.For the same example, in Fig. 5 the numerically comptued D(t), T 0 _ 1 , T 0 _ 5 together with their a priori upper bounds are given.
The results for the delay time in Examples I-III are summarized in Table II.
As expected by intuition, the delay Tx is (ap proximately) the same in Examples II and III, where the global parameters rd, cd, and gd have been chosen to be the same.A high mark of our bound f x is the fact that it is also the same for the two cases.
Let .us consider now the simplest amplifier with one bipolar transistor and with two distributed elements (Fig. 6).
In Fig. 7 the same circuit is plotted with the incremental equivalent model for the transistor.
The physical parameters are: • for the resistances: R With these parameters G is a nonsymmetric but positive definite and DRD matrix: G 11 = 2x10-2 n-1, G 22 = l.6XOn the other hand, the application of the numerical method of Section V provides the results shown in Fig. 8 for u 1 ,h(t, x) and in Fig. 9 for u 2 ,h(t, x).The discretization parameters are h 1 = 0.0202•d 1 , h 2 = 0.0202•d 2 , dt = 1.8X 10-6 .Numerical calculus gives T 0 _ 1 = 14.4 µs and T 0 _ 5 = 7.2 µs.
In Fig. 10 we can see the computed D(t) and its a priori bound.
All above examples show that our bounds are suffi ciently tight for a first evaluation, if we take into account how simple they are to calculate.Certainly, a similar lower bound of the delay time is desirable.

VII. CONCLUSIONS
The complexity of circuits models whose consistency is assured by our theorems is limited by the existence of the description matrix of the resistive multiport and by its properties needed as hypotheses.As we saw, the area of applicability of the results is large enough.
As regards the delay time, in many cases this quantity is useful as a comparison criterium.For instance, in circuit design, in view to obtain from a circuit C 1 "a two times faster" circuit 2 , we must change the parameters such that T;/T{ = 2.If the equality T;jT{ = T;/T{ is (at least approximately) satisfied, then T >-.itself can be considered as " the delay time" of the network.Examples I, II, III (regarding the same circuit) show that this is really the case.X-+ X is dissipative if a( xy, f ( x ) -f (y)) � 0 for any x, y E P).If in addition Range (J -/) = X, then/ is said to be m-dissipative.If a is a real number, it is easily to show that / -al is dissipative iff for every x , y E P), a (x-y ,f(x)-f( y )) �allx-yll x • For a Hilbert space X with the inner product ( •, •) x, we find l l a ll xa(a,b)=(a,b)x, so that / is dissipative iff ( x -y ,f ( x ) -f( y) )x� 0 for any x , y E P).
Finally, if Y is a compact space, let us consider the Banach space C = C(Y, R n )={/: X--+ R n with continu ous /; components} with the usual supremum norm.In order " to calculate" the a functional in this space, we shall give a result which is an extension of a theorem of Sato (26], with a simpler proof.

2nS; 4 )
= L I G i j l J-1,J*i and we shall say that G is diagonally row sum dominant (DRD) if for each i we have G;; > S ;.If G;; � S ;, G is said to be weakly diagonally row sum dominant (WDRD).Also, if we denote(3.2)wehave:Theorem 3.2:Let G be a DRD matrix and u, u be two solutions of the problem (2.2), (2.4), (2.5) for two different C 2 initial conditions u 0 and u 0 , respectively, and for the same continuous sources.Then, max max ju;(t,x;)-u; ( t,x;)I ew01 1 E; i ,E; n O � X; � d; Moreover, the time function from the left-hand side of above inequality is strictly decreasing on [O, oo ).Now let us suppose that u; are the components of the solution for step sources, U;o are their initial values and u; =f:.U;o are the components of the solution of the time independent problem with the same step sources.The above theorem gives lim,-00 u;(t,x)=u;(x) for.everyi E 1, n and x E [O, d;].The evolution of all u; functions (and, therefore, the evolution of the state of our network) from the initial state U;o towards the steady state u; can be globally described by the function D: [O, oo) --+ (0, 1] with max max ju;(t,x;) -u;(x;) I D( t ) = _1_..; _ ; _..; _n _o _..; _x_;_..; _d _, -�• ----max max ju; 0 (x;)-u;(x;) I l�1�n O�x;�d; In th. e assumptions of Theorem 3.2 it holds e W o l D(t) �D(t) = -.--If we fix A E (0, 1) let us define the "A delay time" of the circuit as the last instant when D( t) passes through A value, i.e., T h = sup{t;D(t) = A }.
Theorem 3.2 allows us to write T h = n-1 (A) and, more important, provides us an upper bound T h of this parame ter, namely, ln A min cos Y; T ,,,T _ 1..;,..;n h"" h -(3.6)It is clear from its definition that T h is a measure for the speed of signals propagation in our network.As its upper bound T h can be easily computed from circuit parameters (see (3.1), (3.2), (3.6)) and is sufficiently tight (see examples in Section VI), it can be useful for initial design stages.
fii x , verify (4.1).k Er:Tc; kEk0 +1,n (4.3)By checking the boundiµy conditions ( 4.2) for the func tions (4.3) and using the notations w fk-l = B k and w 2k =A k d k + B k for k E 1, k 0 and also w 2k -l = A k + B k and w 2k = A k e/ii; dk + B k e -jik dk for k Ek 0 + 1, n, we obtain a linear equation in R 2 " of the form Pw = -Gw+B(4.4)

(4. 5 )
where u k are the components of vector u = u -u.When G is positive semidefinite (4.5) implies that A is dissipa tive.The notation Ht k (O, d k ), I ;;i, 0, is used for the usual Sobolev space (with a weight cd of real functions defined on (0, d k ).Particularly L� k (O, d k ) = H c �(O, d k ).Consider also Ht = nz_ 1 Ht k (O, d k ) and H 1 = f1Z� 1 Hf(O, d k ).The inner product in H c 0 will be (u, u) H o = [Z = hft k u(x) u(x) dx.

( 4 . 6 )
where u k E H c �(O, d k ) and a k > 0. This problem splits into two simpler problems: and Uk - problem (4,7) is trivial; upon introducing its solu tion u in the boµndary condition from (4.8), a problem of the type (4.1), (4.2) (with k 0 = 0) is obtained.With similar arguments we conclude that, for semi-positive G, the oper ator A is m-dissipative in He°, On the other hand, our dynamic problem (2.2), (2.4), (2.5) is equivalent with a Cauchy problem on H c 0 , namely, d .=Au(t,) { du(t,) u(�,) = u o ( , ).

P 2 - 1 -) cos 2 aP + S 2 P
sup [Av-AiJL(x)sgn wix)<w,llwllz • (4.15) (j; x) E M(w) Let us take the pair (p; y) E M(w), i.e., max max lw; (x;)l=lw/y) l = ll w ll2 (4.16) 1 � I � 11 0 ::s;;; X; � d;and for which clearly we have w/y) if:.0. If we suppose already proved that y if:.0, d P , two cases may appear.For the first one, i.e., w/y) > 0, y is a maximum point for w P in (0, d p ) and then, from (4.11) we have [ Av -A v] P ( y) sgn w P ( y) 1 a 2 w P ( 2 a P a P x P ) aw P =---2 (y)---tg --(y) P ( y ) I d P r P c P c P ::::; -( d:� + g P ) I w p ( Y ) I < w,ll w llz• The same inequality can be obtained in the second case w/y) < 0, when y is a minimum point for w P .Thus we have shown (4.15).It remains to prove that y if:.0, d P .Let us suppose y = 0. From odd rows of (4.12), following (4.16)we easily derive 1 aw -; a; ( O ) w p ( O) < ( -G2 p -1,2 p -1 + S2 p -1)l w / O) I p and from DRD property of G it results 1 aw P ---(O)w (0) < 0. r a x p p (4.17)But if w/0) > 0, then w/0);;, w; (x) for any i and x and [-2._ lim w p ( x P )-w p ( O ) ] w (0) ;;, 0 r P xP ..... o + x P Pthat contradicts (4.17).Also if w/0) < 0, then w P (O)::::;; w; ( x) for any i and x and we obtain again a contradiction.Hence y if:.0. Let us suppose now y = d P -Also, from boundary condi tions (4.12) (namely from even rows) and from (4.16) we may derive the following inequality: 2._ a a w p (d p )w p (d p )cosa p r p X < ( -G2 p ,Z p cos a P + S2 P + ..!.L sina P ) I w P ( d P ) ( (4.18) r P d P On the other hand, we have cos a P > cos Y p = the right-hand side member of (3.1) (with j = p) and this implies -(G2P,2P +cosa P + � < 0. r P d P r P P d P )w P ( d P ) cos a P <O.rp X But, as above, is easy to see that we have obtained a contradiction with (p; d P ) E M( w ).Then y * d P and the proof of Lemma 4.1 is finished.Now we are in position to obtain the result of Theorem 3.2.If we denote by v and v the correspondent functions of u and u, respectively, via (4.10) with a k = Y k -£ and Y k given by (3.1), these functions satisfy (4.14) resulting in d dt [ v ( t, •) -v ( t,)] = Av ( t, ) -Av ( t, ) .( 4 .19 ) On the other hand, the norm llv (t, ) -v (t, ) llz is differen tiable on A = [O, oo ) -JV where JV is an at most count able set.A known result of Coppel [27] combined with (4.18) gives us d dt II v ( t, .)-• v( t, .) 11 2 dt (u h ( O), cp ) H o = (u 0 , cp ) H o , ( 5.3 ) for all cf, E S h .Once a basis for the finite-dimensional space { ,i.l • • • ,i.l • • • ,1,n • • • ,1is given (Iv;= N;(h) is the dimension of Sk), we can write n N, u h ( , ) = I: I: Q5 ( t)cp;.i=lj=l In this way, the problem (5.3) comes to a system of ordinary differential equations of the form: { MQ' ( t)+KQ ( t) =F ( t) ( 5.4 ) Q ( O) = Q o where the mass and the stiffness matrices are, respectively, M= { ( cp/ , cp 1 k ) H o } and K= {P(cp/ , cf,' k)} .The system (5.4) can be solved numerically, for example, by implicit Euler difference scheme.If At denotes the time mesh size, and Q k = Q(klit), p k = F(klit), then { ( : +K)Q k +I =p k +I + : Q k , Q o =Q o k=O,l,00• (5.5)

Fig. 3 .
Fig. 3.The network for Examples I, II, and III.

APPENDIX
If X is a real Banach space with the norm II II x, let us consider the functional a: XX X--+ R with a( a , b) = lim h -+ o+ ( Il a + h b ll xll a ll x )/h.We shall say that /: P) c