linear passive networks: functional...

7 2 PROCEEDINGS O F THE IEEE, VOL. 64, NO. 1, JANUARY 1976

sional realization theory," &YAM J. Cone., vol. 13, pp. 221-241, Jan. 1965. H. J. Sussmann, "Existence and uniqueness of minimal realiza- tions of nonlinear systems-Part I: Initialized systems," J. Math. Syst. Theory, t o be published. - , "Orbits of families of vector fields and integrability of systems with singularities," Bull. Amer. Math. Soc., vol. 79, pp. 197-199, 1973. P. Stefan, "Two proofs of Chow's theorem," in Ge0mem.c Methods in System Theory, D. Q. Mayne and R. W. Brockett, Eds. Dordrecht, Holland: Reidel, 1973. H. J. Sussmann, "On quotients of manifolds: A generalization of the closed subgroup theorem," J . Differential Geo., vol. 10, Mar. 1975. R. W. Brockett, "On the algebraic structure of bilinear systems," in Theory and Applications of Variable Srructure Systems, R. Mohler and A. Ruberti, Eds. New York: Academic Press, 197:; M. Fleiss, "Sur la rialization des systimes dynamiques biiiniares, C. R. Acad. Sc. Paris, vol. A-277, pp. 923-926, 1973. W. A. Porter, "An overview of polynornic system theory," this issue. R. W. Brockett, Finite Dimensional Linear Systems. New York:

J. Wiley, 1970. (251 J. M. C. Clark, "An introduction to stochastic differential equa-

tions on manifolds," in Geometric Methods in System Theory, D. Q. Mayne and R. W. Brockett, Eds. Dordrecht, Holland: Reidel, 1973.

[26] D. Elliott, "Diffusions on manifolds arising from controllable systems," in Geometric Methods in System Theory, D. Q . Mayne and R. W. Brockett, Eds. Dordrecht, Holland: Reidel, 1973.

I271 -, "Controllable systems driven by vhi te noise," Ph.D. dissertation, Univ. of Calif., Los Angeles, 1969.

[28] R. W. Brockett, "Lie theory and control systems defined on spheres," SZAM J. Appl. Math.,vol. 25, pp. 213-225, Sept. 1973.

[29] L. Hijrmander, "Hypoelliptic second-order differential equations," Acta. Math., vol. 119, pp. 147-171, 1967.

[30] J. T. Lo and A. Willsky, "Estimation for rotational processes with one degree of freedom-Parts 1-111," ZEEE Trans. Automat. Conrr., vol. 20, pp. 10-33, Feb. 1975.

[31] I. M. Singer and J. A. Thorpe, Lecture Notes on Elementary Topology and Geometry. Glenview, Illinois: Scott, Foresman and Co., 1967.

[32] L. Auslander and R. E. MacKenzie,Zntroduction to Differentiable. Manifolds. New York: McGraw-Hill, 1963.

Linear Passive Networks: Functional Theory

Abstmct-Linear passive timevariable networks are imstigated pri- 16. Sec. 3.5 1 and sometimes in the time domain 171. Manv of - , - - &rough the use of dii iutiond kernels as to the the results based upon distribution theory to date are sum-

scattering matrix treated in the time domain. Necessary and sufficient for prssivity are obtained, md the scattering matrix is shown marized in the books of Doleial [8] and, more recently,

to be a measure mtisfying an energy form constraint. Lossless con- Zemanian [91. stnints pertinent to synthesis are deveioped while networks consisting For time-invariant networks, there are also available more or of a finite of circuit are in some less classical-type works [lo]-[ 151 based upon frequency do- Examples illustrating interesting behavior are presented. main concepts. These concepts have been taken over to the

I. INTRODUCTION HE MATHEMATICAL FIELD of functional analysis is now recognized as a rich one with a varied, though modem, history [ I ] . Within functional analysis, the

theory of distributions [2]-[4] plays a particularly interesting role, especially for physical systems. Indeed, motivation for the theory came, in part, from the circuit theory aspects of Heaviside's work [5] while more recent applications have led t o the development of the properties of passive networks in a distributional framework, sometimes in the frequency domain

Manuscript received January 3, 1975; revised March 21, 1975. This work was supported in part by the U.S. Air Force Office of Scientific Research under Grants AF-AFOSR-337-63 and AFOSR 70-1910, in part by the Australian Government Services Canteens Trust Fund, in part by the Australian Research Grants Committee, and in part by the U.S. Educational Foundation in Australia under a Fulbright Traveling Grant.

B. D. 0. Anderson is with the Department of Electrical Engineering, University of Newcastle, Newcastle, N.S.W., Australia.

R. W. Newcomb is with the Department of Electrical Engineering, College of Engineering, University of Maryland, College Park, MD 20742.

time domain characterizations, primarily through the state [16], such that extensions t o time-variable synthesis based upon the state [ 171, [ 181 become straightforward, albeit with strange results (as instabilities of passive structures [191). Likewise, there are recent applicable developments in operato!. theory [20], especially with regard to resolution space concepts [21]-[24] as well as some distributional treatments of time-variable networks [25]-[28 1. However, when one turns toward synthesis of time-variable networks, the functional analysis results are scarce [29]-[33]. Thus it seems that a functional analysis treatment of linear passive time-variable networks with an emphasis upon results important for synthesis is in order.

In this paper, some of the most important properties of linear passive networks are developed in terms of the time- varying scattering matrix [34] s ( t , r); this distributional kernel s appears to be one of the most generally useful descriptions available, especially for synthesis. The paper is structured such that Sections IV and V contain the general results. Section I1 essentially serves as a review section where the underlying concepts of interest are de f i ed ; among these are a network an4

ANDERSON AND NEWCOMB: LINEAR PASSIVE NETWORKS 7 3

its properties and distributional kernels. In Section 111, the scattering matrix is introduced. Because of the generality of the results of Section IV, where a complete characterization of linear, solvable, and passive networks is given in terms of s, one is naturally led to the scattering description of networks introduced in Section 111. An alternate characterization of the passive conditions on s is given in Section V in order t o allow the formulatian of a complete characterization of lossless networks, as given in Section VI. Throughout Section VI, atten- tion is focused on the external behavior of networks, while an investigation of the internal structure is begun in Section VII, where the general cascade load connection of networks is covered. Section VIII investigates some of the properties of arbitrary connections of a finite number of linear passive resistors, inductors, capacitors, transformers, and gyrators, where time-variable elements are allowed. There the quasi-lossless concept is introduced, and some important and useful lossless and quasi-lossless conditions for these finite networks are obtained, both on the scattering matrix and the impedance matrix. Examples are spread throughout while the final tech- nical section gives several more detailed examples in depth, to illuminate various portions of the theory, the one concerning time-variable delay being of especial physical significance while that of the opencircuit loaded transformer is of considerable theoretical interest.

The theory rests heavily on the theory of distributions and distributional kernels of Schwartz [2], [3], [351 and the theory of bounded transformations on Hilbert space [361, with which we assume some familiarity. Nevertheless, the deeper results of distribution theory are mainly used in proofs, and, consequently, we would hope that the results will be clear to those with an intuitive feel for the concepts.

11. PRELIMINARY DEFINITIONS, NOTATION, AND CONCEPTS We begin by introducing appropriate notation from which

we proceed to t h e definition of a network and the various properties of interest for the present study [ 121.

To set a precise framework for the study, we will at times be interested in excitations which are infinitely smooth and zero outside a bounded set, as

f( t) = exp (2) exp (2) ~ ( t - a)u (b - t ) t - a b - t

where u ( . ) is the unit step function anda and b are fixed real numbers. From these we may expect responses which are zero before the nonzero excitation occurs, such as

rhese latter may occur from a network described by a time- domain representation of its (transfer-function) characteristics, such as the distributional kernel t&'(t - T) + et-'u(t - T ) ,

where &I(.) denotes the second derivative of thk unit step u(.). Consequently, we are interested in the spaces 3, ID+, and 4)' which denote the spaces of real-valued n-vectors in one real variable whose entries are, respectively, infinitely differentiable functions zero outside a bounded set (compact support), infinitely differentiable functions zero until a finite value of the variable (support bounded on the left), and distributions [2]. The elements of the space 4) are called "testing functions" while the distributions, fD1, are the linear continuous functionals on 4); in other words, the testing functions and

distributions are topological duals of each other. Most often, the real variable will be taken as time t . Letting f and g denote - n-vectors and using a superscript tilde, , for matrix transpo- sition, we write (for later energy considerations)

which, for all finite t , is well defined if, for instance, f, g E 9,. We further write

I l f l l = l l f l l m (2.1 c)

and observe that I I - I l serves as a norm for the Hilbert space & of Lebesgue measurable n-vector functions f for which (f, f), is finite.

As a side comment, it should be observed that II.I), serves as a norm for the Hilbert spaces, L:, which are subspaces of L2 having their functions f (A) zero for > t. Thus using 4 and 9' as identity and projection operators, we can write f: = (4 - T t ) & in which case there is placed a resolution on the identity of l2 making L2 into a resolution Hilbert space [ 2 1 ] .

The norm 11 TI1 of a bounded linear transformation T[ ] of f E L2 into T[f ] E Lz is defined in the customary manner as [36, p. 2001

I l Tll = sup I 1 T[f I ll. Ilf ll=1

The passive networks we will consider will define, through the scattering matrix, such a transformation on f z . However, these scattering matrices also give a distributional kernel representation in which case it is of interest and importance to tie together the transformation and kernel interpretations. To- ward this we proceed to a more precise meaning of the network concept.

A physical n-port [12, p. 71 places constraints CN on the components of the n-vector voltages v and currents i at its n- ports; mathematically, we can consider the constraints as a binary relation [37, p. 101 and represent the relationship by vCNi Given a binary relation CN, the mathematical represerr tation of an n-port network N, or, for our purposes, simply N itself, is defined as the set of voltage and current couples [v, i ] satisfying vCNi; that is, N is the set of [v, i l allowed at the ports by the network constraints. Precisely

Among the constraints always assumed will be v, i E 9, and a choice of variables such that 5i has the physical interpretation of the total instantaneous power entering N. An example is the time-variable 2-port transformer NT of turns ratio T(t):

This 2-port transformer is illustrated in its uses of Figs. 3(a), 4, 5 below.

Using this definition of a network in terms of allowed or admissable pairs of current and voltage, the important properties of linearity, solvability, passivity, and losslessness can be defined in their most general context.

PROCEEDINGS OF THE IEEE, JANUARY 1976

Fig. 1. Augmented network.

(a) N is linear if for every [v, , il I, [v2, i 2 ] E N and all real constants a and p

[avl + flu2, a i l , +Pi2 I E N .

the augmenting voltage sources e vanish suitably fast at infinity. As an example, the transformer 2-port introduced above is passive and lossless, as &(t) 0. We comment that other definitions of losslessness exist, some of which [39 I contain that of quasilosslessness to be introduced in Section VIII.

In the sequel, we will have heavy use for distributional kernels. These are n X n matrices k(t, T) of real-valued distributions in two real variables [35], where we use the word "distribution" in its now established mathematical meaning [4] synonymous with "generalized function" [40, p. 221. The space of distributions includes all normal functions as well as impulses, doublets, other singularity functions, etc. Any

(b) N is solvable if for every e E 3 + there is a unique [v, i ] E N such that

(2.4) distributional kernel defines a h e a r continuous mapping

It is this latter property which allows the introduction of the scattering operator which in turn will be linear, giving the scattering matrix, if N is linear. The last equation (2.5), allows e t o be considered an excitation for an augmented network N, defined by [e, il E N, if [e - i, i l E N for each e E 0,. Physically N, represents N with a unit resistor con- nected in series with each port. The connection is illustrated in Fig. 1 where 1, is the n X n identity matrix representing the impedance of the unit resistors. If N is linear then so is N,, as a simple application of (2.4) and (2.5) shows. In pass- ing, we remark that similar results to what will be obtained below occur for a shunt augmentation where the unit resistors are placed across the ports. rather than in series with them as - . in Fig. 1 [38].

The physical meaning of

of x E 3, the input space (considered in a rigorous way in its strong topology), into y E B', the output space (considered in its weak topology). We have, in (2.9a), given the intuitive meaning of this composition operation as an integral, with which it coincides when all quantities are normal integrable functions. However, to make (2.9a) completely precise we let ( y , q ) denote the scalar product between a distribution, y E 3 ' , and a testing function, cpE 9 , analogous to ( , ), of (2.la), and ((k(t, T), $(t, 7))) the same for two variables. Then (2.9a) comes exactly from

t

&(t) = lw $r)i(r) d r = (v, i ) t (2.6a) The converse is also true; that is, any linear continuous m a p ping of 0 into 0' can be described by a distributional kernel through (2.9a) [35, p. 2231, this result being known as the

is the total energy input into N Up t o time t, assuming the net- "Schwartz kernel theorem," work is relaxed at t = --, as is assumed for u, i E 3+ for which By applying another kernel h to y of (2.9a) we obtain the also the energy always exists. This leads into the definition of (Volterra) composition h o k (35, p. 2291. In particular passivity.

(c) N is passive if for every [v, i ] E N and every finite t

It is customary to call N active if it is not passive. If N is passive then so is N, since

(e, i ) t = llill: + (v, i)t. (2.7a)

Of more weight is the relation [ 1 I , p. 1 11 I

IIeII: = IIvII: +IIiII: + 2 ( v , i h (2.7b)

which will allow various conclusions in the sequel, one of which is that for a passive N, e E & implies v, i E &, since the left is finite while each term on the right is positive. In such a case, &(m) is well defined for the following lossless definition.

m

h o k = l _ h ( t , h ) k ( h , r ) d h (2. lob)

defines h o k as the unique kernel mapping x into z, when such a mapping can be performed. Although h o k cannot always be formed, we note that if k and h both map 3, into 4)+ then h o k exists and also maps 3, into 3,. As a caution, this composition need not always be associative (as an example after Theorem 5 below shows), but it will be when all kernels map fb+ into fb+ [35, p. 2291. Since 6 I,, with 6 the unit impulse, acts as the identity map, 61, can be composed with any kernel and merely reproduces the kernel acted upon. Again, the integral notation of (2.10b) is merely symbolic

(d) N is lossless if it is passive and solvable and if for every serving to supply intuitive meaning to the composition 0, this inhitive meaning being exact, though, when the integration

e = v + i € f ) + n X 2 (2.8a) can actually be performed.

a(-) = 0. In much of the following we will use these integral nota- (2'8b) tions and often will display the variables, as in y ( t ) =

Physically, (2.8) states that the energy put into a lossless N is k(t, 7) x (7). We will also customarily drop the bold-face returned and dissipated in the resistors used t o obtain N,, if type when considering the 1-port, n = 1, case.

ANDERSON A N D NEWCOMB: LINEAR PASSIVE NETWORKS 7 5

It is appropriate to define a left inverse k;' and a right inverse k;' under composition by

If k;' = k;' , we call these the inverse k-' . Depending upon the domain of definition considered, one kernel may have several inverses. Consequently, we will assume, unless otherwise mentioned, that if k is a mapping of d+ into 9, then k-' (if it exists) is also a mapping of 9, into P)+ (there is, at most, one inverse with this range and domain). For such a mapping (2.1 la) has the meaning that for any x E P),, k-' (k.x)=(k- ' o k) x = x .

As simple examples of these compositions, consider

0 6 ( t - T I ] [6 ' ( t -T) 0 ] h (t, T) = k(t, 7) =

tu(t - 7) 0 e7u(t - 7) 0

Then

e7u(t - 7) 0 0 6'(t - 7)

t6(t - 7)

To close this section, we mention that every distribution, as k x in one variable or k(t, 7) in two variables, is, on any open set (of compact closure), a derivative of some order of a continuous function [2, p. 521. Such a representation for the scattering matrix is developed in Section IV.

From this point on, unless otherwise mentioned, we assume that N is linear, solvable, and passive. Under these assump- tions, we show in this section that N possesses a scattering matrix s(t, 7).

We first show in the following arguments that a distributional kernel y, is defined for any linear solvable passive network. By the solvability of the network N, the augmented network N, defines a transformation 5,[ ] which maps each source voltage e E P), into a unique port current i E 0,

By the linearity of N we have seen at (2.5) that N, is linear too, in which case %,[ ] is a linear transformation. By the passivity of N, this transformation is continuous from the set of testing functions e E 4) (strong topology) into the set of distributions i E b ' (weak topology). This continuity is seen by fust noting that d C 9, n L2. Then consider a sequence {ei), ei E 9 with all ei of fixed support and converging with all derivatives uniformly to 0 (this is what is meant by strong convergence in 9). Now (2.7b), Ilell: = Ilvll: + 1 1 ill: + 2(v, i),, shows, on using passivity, which is (v, i), > 0, that ii = 9, [ei] converges to 0, this convergence in fact implying

for all p E 9 . This is what is meant by weak convergence to 0 in 9'.

The mapping 3,[ ] is then a linear continuous mapping of e E 0 C d+ into i E 9 + C d' and, therefore, has a kernel representation [35, p. 2231, [41, p. 1431

From e = v + i we then obtain

We call y,(t, 7) the augmented admittance mam'x and note, since any kernel can be composed with the unit imprhe, that the physical meaning of the (i, j ) entry of y , is the current at port i at time t due to a unit impulse of voltage applied to port j of N, at time 7, all other such port voltages being zero. Al- though y, has been properly defined only as a map of d into d+, the properties of ! fa [ ] allow y, to be immediately considered as a map of 9, into d+ [35, p. 2241. This extension of the domain of definition of y, will always be assumed and allows y, to be applied to v of ( 3 . 1 ~ ) and 61, - y, to i of (3. l b) to obtain

If y,' exists, then an impedance matrix z can be defined through v = z i, and, similarly, i = y v defines anadmittance matrix y if (61, - yo)-' exists:

Because neither z nor y need exist, as will be shown below by the ideal transformer, and because we can obtain the most general and complete results in simple form, we make a change to scattering variables, that is to incident and reflected voltages v' and vr, respectively, defined by

or on solving

Inserting these latter into (3.2) serves to define the scattering mamx s(t , r ) in terms of y, through

As we have seen earlier, every linear solvable passive N possesses an augmented admittance matrix, which, by the comment after (2.7b), maps d + n & into 9 + fl d;. Conse- quently, we conclude from (3.5b) that every linear solvable passive N possesses a time-variable scattenng matrix s which is a distributional kemel that defines a linear continuous map of incident voltages d E I)+ into reflected voltages v r E 9,. By the properties of y,, s also defines a map of P)+ n L2 into 9, n 1,.

In terms of the new (scattering) variables, the energy of (2.6a), $ = (u, i),, takes the useful form


TABLE 1 SUMMARY OF DESCRIPTION INTERRELATIONS

If we precompose (3.2) with any kernel c having an inverse under composition, we can obtain a slightly more general darcrip tion

For example, every linear solvable passive network has, by (3.2), this general description witha = y a , b = 61, - y a . How- ever, not all linear passive networks possess an augmented admittance, as shown by the 1-port norator [12, p. 131, [42] which has the general description 0 v = 0 i. Equation (3.7a) yield another convenient formula for the scattering matrix. On substitution of port voltage and current variables in terms of scattering variables (3.4c, d), we directly get

which on using (3.2) checks (3.5b). As an example, consider the (n +m)port transformer de-

scribed through a time-variable m X n turns ratio matrix T(t). This is precisely defined through the general description of (3.7a) as [12, p. 491

where the port variables v and i are partitioned as the transformer ports (into n primary and m seconda~y ports) and 0, denotes the n X n zero matrix. Equation (3.7b) then yields [12 , p. 511

where T = T(t), and the inverses are those for matrices. In this case, no y or z exists because both a and b are identically singular.

In Table I, we summarize, for easy reference, the various interrelations between the descriptions introduced. The various matrices composed internally in the table commute, for instance, r = (z - S 1,) o (z + 6 In)-'.

By definition, Nd is the dual of N if every [vd, i d ] E Nd is of the fo,rm [id, vd] EN. Equations (3.4a, b) then show that v i = v', U: = -vr and hence

Similarly

which, by Table I, checks sd = -s. It is also clear, say from (3.6), that the energy input t o a network is a selfdual concept.

To conclude this section, we point out that the foregoing definitions and formulae all include the well-known time- invariant results when the two arguments (t, ?) become the one argument (t - 7).

IV. PROPERTIES O F s

In this section, the necessary and sufficient conditions are developed, as given in Theorem 1 , on the scattering matrix to guarantee N passive. From these, some useful properties are obtained, such as the fact that s(t , 7) is a measure in both variables and s2 o s l is passive with s l and 52 passive.

We begin with some preliminaries, the first of which con- cerns the support of s for a passive network. If we let 0, denote the n X n zero matrix then

s(t, 7) = On, when t < T. (4.1)

The same result holds for y a by (3.5b). In essence, (4.1) states that N is in some sense casual, at least when viewed through the augmenting resistors. More precpely, s is antecedal [43]; that is, if the incident voltage, v', is zero before time to, then so will be the reflected voltage, vr.

The validity of (4.1) follows from (3.6). Thus with to fixed, if vi(t) = 0 for t < to (3.6) shows vr(r) = 0 for t < to , since &(t) 2 0 by assumption. Therefore, setting x = d,

whenever cp(t) = 0 for t > to and x (7) = 0 for to > 7 with cp, x E 9. As this is true irrespective of the values of q(t) and x(7) for t < to < 7, (4.1) follows [2, p. 261. We remark that it is sufficient t o test sij with separable \kij(t, 7) = qi(t)xj(r) by the denseness of finite sums of such in nonseparable qij( t , 7) [2, p. 1081.

Because s is a linear continuous transformation on 9 into b', it has an adjoint % defined, in the standard manner, by

for allx, cpE 9. In fact

In other words, the adjoint of the scattering matrix is found by reversing the excitation and response times in the transpose

sd =-s. (3.9a) of the original scattering matrix.

ANDERSON AND NEWCOMB: LINEAR PASSIVE NETWORKS 77

This expression for the adjoint follows from ( 2 . 9 ~ ) on making an interchange in dummy variables and indices:

Because of (4.11, it is now clear that sa(t, T) = 0, for t > T.

The boundedness of the scattering operator as a linear continuous transformation on E2 can also be obtained. From L3.6) and the passivity of N

1 1 vill: > Ilvrll: = 11s . vill:. (4.4)

Choosing e = 2vi E 0, n E2 and letting t = show that s defines a bounded linear continuous transformation on a subset of e2. We can, therefore, make another extension [36, p. 2981 defining s in a bounded manner for all u' €1,. Observing (4.3a) for x, q E d; shows that sa is also a bounded linear continuous transformation on E2. Then, noting that s and sa have the same norm [36, p. 2011 and comparing (4.4) when t = - with the definition of the norm of a transformation shows that, when s is considered as an E2 transformation,

llsll= llsall 1. (4.5)

Equation (4.5) is a necessary condition for the passivity of N; essentially it is also sufficient if N is linear and solvable.

Theorem 1

A linear solvable network N is passive if and only if the following is true.

1 ) A scattering matrix s exists mapping vi E 0, into vr E 9, 2) s maps E2 into E2. 3) s(t, T) = 0, for t < T. 4) llsll < 1 regarded as an L2 map.

Proof: The only if portion has been proven by the reasoning leading to (4.5). To show the if portion, we first observe that if conditions 1)-3) are not satisfied then N must fail to be either linear or solvable or passive.

As we are assuming N linear and solvable, we are led to consider the existence of an s satisfying llsll < 1 but which is 9ot passive; we search for a contradiction. Under the violation of ,passivity, there is some pair [vl , il I EN, obtainable from 2v: = v l + i l E 0 + , and some finite constant time T such that ( v l , i l ) T = $ l ( T ) < O . ,

Let a second excitation u: E 4) be defined as

for arbitrarily small E > 0 (v; is defined in an infinitely differentiable manner in T < t < T + E). Then 2vi = v2 + i2 ; by the uniqueness in solvability, v2 = vl and i2 = il for t < T. Then

$2(T)= $l (T) (4.7a)

82(-) < $2(T) + ~ ( € 1 (4.7b)

where Y(E) can be made arbitrarily small by properly choosing e since $(t) is evaluated as an integral. But v: E L2 and thus also v; E L2 by Ilv; I 1 G llsllllv~ I I . Equation (3.6) then shows, by Ilsll G 1,

Choosing 0 < Y(E) < - d 1 (T) shows that the assumption of (T) < 0 is violated, and hence N must be passive. Q.E.D.

We point out that, in this proof, the assumption of a solvable N has allowed the truncated v' of (4.6) in the domain of definition of Na. Consequently, for the purposes of this paper, b( t ) > 0 in the definition of passive N, (2.6b) could be replaced by $(-) > 0 [ 11, p. 1 101. However, such a modified definition would restrict us from the beginning to L2 functions, in contrast to 0, functions, would not generalize to nonsolvable N and seems physically unappealing. It is worth observing that the - 2 4 resistor has s(t , T) = +36(t - 7) which satisfies conditions 11-31 of the theorem but has llsll = 3; this checks the active nature of a negative resistor.

For conciseness, it is convenient to call s passive if it satisfies the conditions of Theorem 1. Note that (3.9a) shows, with Theorem 1, that sd is passive with s , a fact which is also clear from the definition of a dual network.

A result of some use in synthesis, as well as analysis, is the following [44], [45, ch. 1 1 1, [46], [47].

Theorem 2

If sl and s 2 are passive scattering matrices then

is passive. Proof: We must show that s satisfies Theorem 1. Clearly s

maps 9, into 0, as the composition of two such scattering matrices; similarly for L2 into 8 2 . In obvious notation U: = v', v: = v;, U; = vr, or from (3.6)

&(t) = d 1 (t) + $2 (t) 2 0. (4.9)

The argument for (4.1) then shows that s(t, T) = 0, for t < 7, and the argument for (4.5) shows Ilsll< 1. This latter fact also follows from

We observe that N is defined through (3.4) and is linear and solvable with s as its scattering matrix. Q.E.D.

Theorem 2 admits of a nice physical interpretation, which we, however, postpone for Section IX, (see Example 4 and Fig. 7). The next result is of assistance in exhibiting the form of s (specifically for the finite N treated in Section VIII), as well as for ruling out nonpassive s. For Theorem 3, by a measure essentially means a distribution which is no more "singular" than the impulse, that is, which is no higher derivative of step discontinuities than the first (thus 6 ' is not a measure).

Theorem 3

A passive s(t, T) is a measure in r for each fixed T, and a measure in T for each fixed t.

Proof: The meaning of the theorem is that each component sij of s has the stated properties [ 2 , p. 151. By a theorem of Schwartz [2 , p. 251 s (t, T ) will be a measure in T if for each t the sequence of 9 + functions (s x,) converges to 0 when-

7 8 PROCEEDINGS OF THE IEEE, JANUARY 1976

ever the sequence {x,), x, E d , converges to 0 uniformly irrespective of convergence of the derivatives of x,. However, for each cp E 0 , considered also as a subspace of 12, we have by Schwarz's inequality [36, p. 1981 and by the passivity constraint of (4.5)

Vj(t)) = I(s x,, lp)l (4.1 la)

Thus if x, tends uniformly to 0, (by the passivity) (s x,, cp) tends to 0 for almost all t as a function. Since this result holds for all cp, s x, converges to zero for almost all t. We wish to see that this holds for all t. For this, consider, by the distributional rule for calculating the derivative (denoted as above by a prime) [2, P. 351

I I = I x,, I x a l l l l l . (4.1 1c)

Therefore, (s x,)' converges to 0 for almost all t when x, converge to 0 uniformly; integrating then shows that s x, converges to 0 for each t. Considering sa x, we obtain the same result on p (7 , t) = ~ ( t , 7); that is, 9 (7 , t) is a measure in t for each fixed 7. Q.E.D.

We can further see that s(t , 7) is also a measure in both variables simultaneously over any compact set K of the (t, 7)- plane. Toward this consider the restriction of s to K defined by

where a and p are nonnegative infinitely differentiable functions bounded by unity such that a6 = 1 over K and a0 = 0 outside a square containing K. The scattering matrix SK is passive since Theorem 2 applies to

where

Both s, and sp come from passive networks (resistors, in fact) since, for instance,

= llui 11; - IIvL 11: 2 11 vk 11: > 0. (4.13)

Now SK, having compact support, can be convoluted (denoted by *) with unit step (Heaviside) functions u to give

(rather than a sequence of two convolutions, one for each u, one can consider (4.14) as a two-dimensional convolution of sK(t, 7) with the tensor product [2, p. 1141 u(t) @u(7) = u(t, 7); that is, sK (t, 7) * u(t, 7) ) . By Theorem 3, (4.14) defines a function of bounded variation in both t and 7, showing that sK(t, 7)is ameasure jointly in t and 7 [3, p. 451.

We can go further and obtain a more explicit representation for SK. By passivity, (4.1) applies to SK and we can now write

where g~ consists, at most, of functions of bounded variation in t and 7. We have immediately

which, on performing the indicated differentiations, shows gK(t, t ) = On since 6'(t - 7) is not a measure in t for each 7. As shown by the example of s(t, 7) = 6 ( t - d(t) - r), to be dis- cussed in Section IX, not much more can be generally con- cluded concerning g ~ , except that it is infinitely differentiable for finite N, as will be seen in Section VIII. Nevertheless, a double integration of g ~ u , over t and 7, shows that SK is, at most, the fourth derivative of a continuous function over K , verifying the result mentioned at the end of Section 11.

Because of the insight it yields and as a preparation for the lossless constraints, we here express the passivity requirement, formerly given by the unity bound on the norm of the scattering matrix, llsll< 1, in an alternative manner in terms of a nonnegative kernel Qt obtained by expressing the energy integral as a scalar product.

By definiticn, a real distributional kernel k(a, 0) which is self-adjoint, k(P, a ) = k(a,P) is called nonnegative, written k 0, if for every x E 0

Here, as above, ( . ,; ) denotes the scalar product between distributions y €0 ' , with now y = k x, and testing functions x E 0 . The nonnegativity definition agrees with one of Schwartz put forward in a relatively unavailable work [48, p. 451 while agreeing with a similarly known one in the case that the composition coincides with convolution *, as ih does for time-invariant systems [3, p. 13 1 1.

Considering the energy integral expression of (3.6) we can write the energy as

where the integral over (-a, t ) has been replaced by one over (-w,m) by inserting the unit step function u. We are then able to use the adjoint definition (4.3a) with terms reversed, allowing s(A, a ) to be shifted off d ( a ) in the final portion of (5.2b). This shows that the composition (in A) of s" (a, A) =?(A, a) , and u(t - h)s(A, P) exists. Collecting all terms composed of ui(o), we arrive at

Equation ( 5 . 2 ~ ) holds for all dEP)+, and one easily checks that for each finite t , Qt is self-adjoint. Consequently, for each finite t, Qt is nonnegative for a passive N. In such a case, d E L2 implies that Qt(a, p) u'u) E g2 since f, and hence each term on the right of (5.2d) maps l2 into 12, and we can again apply the reasoning of Theorem 3 to show that Qr(a, P) is a measure in a and p because

ANDERSON A N D NEWCOMB: LINEAR PASSIVE NETWORKS

Since 8 ( t ) is simply related to Qt and since 4 E 3, can be considered as the limit of a sequence of c/ E 9, we can re- phrase Theorem 1.

Theorem 4

A linear solvable network N is passive if and only if conditions 1)-3) of Theorem 1 are satisfied and for all finite t

One advantage of the formulation of this section is that it shows the nature of results when z or y of (3.3) exist. Thus using 8 ( t ) = ( v , i)t = ( i , v ) ~ , the manipulations of (5.2), and a result similar to (4.3b), we see that an equivalent statement of condition 4') is the following. For each t, the form

From (5.4), one sees that the necessary and sufficient condition for a linear solvable N completely described by an impedance matrix z, (assumed causal, i.e., with z(t, ~ ) = 0 , for t < T), to be passive is the satisfaction of (5.4). This situation is in contrast to that of Zemanian [7, footnote 17, p. 2691 where the time-domain description does not rule out z(t, r ) = 6(3)(t - 7) which is excluded by (5.4), as proven in Appendix 1.

It is worth noting that

From this, some further insight into the Qt > 0 constraint can be gained by relating to familiar frequency domain results for the time-invariant situation. In the time-invariant situation, when the Fourier transform S(jw) = 3[s(t, O)] describes the network, the resistivity matrix R ( j o ) = 1, - s ( - j o ) ~ ( j w ) must be positive semidefinite for almost all w[49, p. 571. But R(jw) =3 [Q,(t, O)] follows from (5.5) since S(- jw) = 3[sU(t , 011.

VI. LOSSLESS N

Here we show in Theorem 5 that a necessary and sufficient

v' €9, fl J z , (2.8) and ( 5 . 2 ~ ) yield, for losslessness,

Therefore, [2, p. 261, Q,((u, P) = 0 for all such vi and thus Q,(a, p) is independent of a. Being self-adjoint, it is independent of 8, and we conclude that Q,(t, r ) = 0, is a necessary and sufficient condition for a passive s to be lossless. The conclusion Q, = 0, also follows from a similar result on Hilbert spaces [SO, p. 2671. Evaluating Q, = 6 1, - sa o s = 0, from (5.5) gives the following result which is of considerable importance for synthesis [5 1 ] 4 5 3 1 .

Theorem 5

A passive s is lossless if and only if

In essence, a lossless s can be viewed as an isometry [52, p. 151 in the standard Hilbert space terminology. We comment too that f o s is well defined since, for all x, 9 E 9, (s x , s 9 ) = (x, (% 0 S) 9) by use of the adjoint definition (4.3a) and the definition of composition. Although % is a left inverse for s, since % is generally not antecedal (i.e., does not satisfy (4.1)), s" is clearly not generally the inverse mapping 9 + into 9 +.

Consequently, associativity does not generally hold in s 0 [P 0 s ] = s, and one cannot generally conclude that sa is a right inverse of s under the conditions of Theorem V. As examples of the types of behavior possible, we observe that the lossless 1-port delay s(t, 7) = 6(t - d - T), with constant d > 0, shows that f can map 9 + into + and that sa can be a right inverse. The unit inductor s(t, 7) = 6(t - 7) - 2 (exp [T - t ] ) u(t - 7) has sa(t, 7) = 6(t - 7) - 2 (exp [ t - 71 ) - U(T - b) which is a left and a right inverse, but which does not map 9, into 9,. The example

condition for a passive scattering matrix s to be lossless is that with 9 any smooth square-integrable function, for instance

the adjoint % acts as a left inverse of s under composition. d t ) = 1 / [ 1 + tZ I , is lossless, but % is not a right inverse. For simplicity, we will call s lossless if it represents a lossless We observe, from = that a lossless must have an

N. First we investigate the behavior of Ilsll. BY the meaning of impulsive term present ins. Furthermore, Q, = can not be losslessness, 8(=) = 0 of (2.8), and the energy constraints of used directly to obtain results such as z + za = 0, (which is in- (3.6) and (4.5) we have for all vi E .& correct) on impedances since z, in contrast to s, does not map

we recall that the domain of s was extended to g2 below (4.4)). Therefore, a passive s is lossless only if

llsll = 1. (6.2)

Unfortunately, llsll = 1 is not a sufficient condition for losslessness, as is seen by networks which over some interval of time behave in a "nondissipative" manner, irrespective of their behavior over all time. For example, the 1-port resistor of resistance

r(t) ~ ( t ) u( 1 - t) exp [- 1 /tZ I exp [- 1 /(t - 1)' ]

has

s(t , 7) = 6(t - ~ ) [ r ( t ) - 1 l /[r( t) + 1 I with Ilsll= 1.

Consequently, we turn to the more useful result obtained from the energy kernel Qt, as defined in Section V. For all

ez into e z , as seen by the series tuned circuit described by z(t, 7) = 6'(t - 7) + uft - 7). For finite networks, a condition for z to be lossless will, however, be obtained in Section VIII.

For further insight into the meaning of the lossless constraint of c6.4), we note that in the time-invariant case % 0 s = 61, corresponds to the Laplace transform para-unitary relationship .!?(-p)~@) = I,, when S(p) is meromophic [ 1 1, p. 1231, [12, p. 1011.

VII. CASCADE LOADING

In order to obtain somewhat more specific results for synthesis and for specific networks of considerable importance, we turn to a useful and general method of combining networks, that of cascade loading. In particular, we calculate the input scattering matrix s , when a certain inverse exists, of a loaded coupling network, the result being given in (7.4). It is shown that, under such a condition, s is passive when the subnetworks are.


:+p~q I r s I

I I X I

N I L ,---------- A

Fig. 2. Cascade loading.

Referring to Fig. 2, consider the coupling (n + m)-port Nz whose variables are partitioned as the ports, that is, i;k = [<\, 3 ] ,% = [<: , z; 1 with the subscripts 1 and 2, respectively, denoting n- and m-vectors. Then, an m-port Nl is said to cascade load Nz if

4 = v: and v; = vi (7. la)

where 4 and I( are incident and reflected voltages for the load networks NI. This connection defines a new n-port N, as illustrated in Fig. 2, whose incident and reflected variables are

d = d and v r = 4 (7.1 b)

subject to the constraints placed on the coupling network NL: by loading it with Nl. Partitioning the scattering matrix 2 of Nz according to its port partition and defining 81, and s as the scattering matrices of NI and N lead to

where (7.1) have been used in L$ = E v b . In order to gain some insight into various of the following manipulations, we write these out fully as

We wish to solve for I? in terms of d , to equate to (7.2c), by eliminating 4. Doing this when the indicated inverse exists yields

As an example of the use of this cascade-load formula, consider the loading of a 2n-port transformer with an orthogonal turns ratio matrix F = T-' . Equation (3.8b), describing the transformer through its scattering matrix, becomes

Loading at the final n ports by a network of scattering matrix sr(t, T), the input scattering matrix is by (7.4)

In essence, the loading of an orthogonal transformer results in a congruency transformation on the load scattering matrix, this also being a similarity transformation by virtue of the orthogonality of the turns ratio matrix.

Unfortunately, the inverse needed for (7.4) may not exist even though Nz and N1 are passive. Although an interconnec-

tion of the type under consideration need not possess a scattering matrix (see Example 3 of Section IX), still the cascade load connection will often be described by s, even though the inverse in (7.4) may not exist. If we let di[ 1 denote the range space, then a sufficient condition for the cascade load connection of passive networks to possess a scattering matrix is

To see this, we comment that always (see Appendix 2)

where T( [ ] denotes the null space. Thus if vi is fixqd but arbitrary in 0+, then, by (7.5a), there is at least one 4 E 9, satisfying (7.3b). If there are two or more such 4, their differ- ence is in n [61, - ZZ2 0 s ~ ] , and (7.5b) then shows tha (2'2 ? sr) v{ is uniquely determined. Consequently, given any v' E 0, there exists a unique v' E 0, determined by (7.5); this shows that s for ( 7 . 2 ~ ) is well defined as a map of 9, into 9, whenever (7.5a) holds. Interpreting the exponent - 1 of (7.4) as a type of pseudo-inverse, we will understand (7.4) to mean the process just described, whenever (7.5a) holds. Be- cause (7.5a) always holds in the time-invariant case [55 ] , or in the case where the range spaces are closed, the above considerations are often of importance. A very complete and general mathematical treatment of cascade loading is given in [26].

We conclude this section with the physically obvious fact that N is passive if it is constructed by cascade loading a passive Nz by a passive Nl.

Theorem 6

If Z and SI are passive, then the cascade loaded N is passive; s of (7.4) is passive when it exists.

Proof: Even though s need not exist, N is always passive since, in obvious notation, we have by (3.6) and (7.1)

By definition, s is then passive when it exists. Q.E.D. The generality of the cascade-load connection is worth

observing. Thus by suitably manipulating the subnetworks, virtually every interconnection of importance can be represented in cascade form, for example, the series and paralle connections of n-ports [ 12, p. 641. Example 4 of Section IX will further illustrate the use of cascade-loading for the realization of the scattering matrix product. Consequently, the cascade loading concept is useful for actual designs [561, the most general synthesis [571, and practical computer-aided analysis [58].

VIII. FINITE NETWORKS --QUASI-LOSSLESSNESS

In this section, we consider the interconnection of a finite number of common circuit elements while introducing the concept of quasi-lossless networks. For the latter, which are interconnections of lossless circuit elements as well as for lossless networks, a useful set of necessary and sufficient conditions are developed in terms of the scattering matrix. From such a development, a similar constraint on the impedance of a finite network is also obtained. To proceed without undue

ANDERSON AND NEWCOMB: LINEAR PASSIVE NETWORKS

delay, we bypass considerable detail and assume the meaning of various physical concepts to be familiar from classical network theory [12], [59].

We consider the basic elements of interest to be the linear 1-port resistor, inductor, and capacitor, the linear 2-port gyrator, and the linear (n +m)-port transformer (previously treated in (3.8)). We call these circuit elements and define them respectively by

v = ri [resistor] (8.1 a)

v = d[li] ldt [inductor] (8.1 b)

i = d [cvl ldt [capacitor] (8.1 c)

u = [' i [gyrator] -7 0

Here the scalar parameters r(t) [resistance], l(t) [inductance], c(t) [capacitance], ~ ( t ) [gyration resistance], and the m X n matrix T(t) = [tg(t)] [turns ratios] are infinitely differentiable real-valued functions of time. We assume the transformer ports partitioned as for Nz of Fig. 2. The gyrator and transformer are both lossless, while the passivity conditions on the remaining elements are [29, pp. 10-2 1 1, [30] (see also Exam- ple 1 of Section IX for comments on the derivation of (8.2b))

(4 Fig. 3. Transformer replacements.

r 2 O (8.2a) where C and D are n X m matrices, polynomial in the deriva-

1 2 0 and 1 ' 2 0 tive operator p = dldt and with time-varying coefficients. By (8'2b) the measure and casual properties of (4.1 5b) and the separabil-

c > O and c l>O. ( 8 . 2 ~ ) ity of the impulse responses of differential equations [29, p. 76-86], [61, ch. 61 we can write

Under these conditions, all circuit elements have scattering matrices which are relatively easy to calculate, that for the s ( t , ~ ) = A ( t ) 6 ( t - ~ ) + @ ( t ) G ( r ) u ( t - T ) (8.4) transformer being previously given at (3.8b) while that for the inductor is obtained in Example 1 of Section IX.

Interconnecting a finite number of circuit elements subject to Kirchhoff's laws yields a finite circuit; attaching ports yields a "network." Therefore, we define a finite network as a network which has a finite circuit representation. Of special interest is the fact that every finite network has a representation in the cascade loaded form of Fig. 2, where Nz consists of transformers and NI consists of constant parameter uncoupled resistors, inductors, capacitors, and gyrators. To obtain this representation when all elements are passive, one replaces each element by its equivalent circuit [30] as given in Fig. 3, plat- ing all transformers in the coupling network, N z . Often this leads to a simple method of finding the scattering matrix of a finite network, when it exists, since (7.4) applies. In particular, this shows that s exists for a finite network constructed from time-invariant circuit elements, that is, with all r, I, c, y, and T constant [55]. Nevertheless, whether s exists or not, this shows that a finite network having a circuit representation completely in terms of passive circuit elements is passive, since Theorem 6 applies. This passivity also follows from the addi- tivity of energy in each circuit element through a direct application of Tellegen's theorem [60, p. 3961.

Equation (8.1) shows that a finite passive N possessing a scattering matrix has d and vr related by an ordinary differential equation, expressed in general description form, similar to (3.7a), as

C03, t ) J ( t ) = D(P, t) d ( t ) (8.3)

where in fact A , @, and \k are infinitely differentiable. We comment that the cascade loaded equivalent circuit of a finite network mentioned above virtually guarantees the existence of the general description of (3.7) which converts to (8.3) under the change of voltage and current variables to incident and reflected voltages, since distributional kernel! have 6(i), the jth derivative of the impulse, substituted for f l of the differential polynomials comprising C and D. With this replacement s = C-' o D. Further s of (8.4) is zero for t < T and hence represents an antecedal impulse response for (8.3); a non- antecedal impulse response results from changing u(t - T) to -U(T - t) in s of (8.4). In other words, there are several impulse responses to the scattering system differential equations (8.3) with that of (8.4) representing the antecedal impulse response.

Using this idea of unit step function replacement, we may associate in a natural way with sa(t, T)=?(T, t) an antecedal adjoint impulse response s:, associated with the adjoint system, by

(If terms B(t ) 6(j1(t - T) were present in s they would be replaced by &T) 6(j)(r - f) for sg.)

Turning to the lossless property, we note that, by direct integration in forming the composition and using (8.4),

sa o s = 6 1, = a t ) A (t) 6 (t - T)

82 PROCEEDINGS OF THE IEEE, JANUARY 1976

where

Since a linear passive N is solvable if and only if s exists, and a lossless network must be passive and solvabie, we conclude the following, as we know f o r = 6 1,.

Theorem 7

A finite passive N is lossless if and only if

and

From the last two equalities, the association used to obtain the antecedal adjoint gives an interesting interpretation which we can now describe. By direct calculation

4 o s = X(t) A(t) 6 (t - r ) + [F(t, r ) - F(T, t)] u( t - r )

Consequently, in the case of finite lossless networks 4, given by (8.5), is necessarily an antecedal left inverse of s mapping 9, into 9 +. If an impedance matrix exists, we can write, from Table I and (8.7d)

61, = [z: +61,]-I o [z: - 61,] o [z +61,] o [z +61,]-'

(8.8) where we have used (h 0 k): = k: o h: which is valid in the finite case under consideration, as simple calculations show. Since all terms in this last expansion of 61, are 9, into 0 , mappings, the product is associative. Precomposing in (8.8) by 2% +61, and post composingbyz+61, gives 1 2 9 , ~ . 571

2: + z = 0, (8.9)

as a necessary condition to be satisfied by finite lossless z. Note that we cannot use the same argument on f 0 s to obtain za + z = 0, (which is false) [62, p. 851 in view of the equation corresponding to (8.8) having nonassociative prod- ucts. It is also to be noted that 4 need not exist for nonf i i te lossless N, as shown by s(t, r ) = 6(t - d - r ) with constant d > 0. Nevertheless, whenever & and z exist, (8.9) is valid and shows that even s o 4 = 6 1, for such N.

We further remark that (8.7), or equivalently (8.9), is not a sufficient condition for s to be lossless, as, besides the non- associativity just mentioned, nonzero F which is skew under the adjoint operation satisfy (8.7d). For example,

is not the impedance of a lossless network [as will be seen by (9.5b) with T(t) = 1/(1 + but does satisfy (8.9).

Consequently, let us term a passive scattering matrix s satisfying 4 0 s = 6 1, or an impedance (or admittance) matrix satisfying z: + z = 0, quasi-lossless, and apply the same term to the corresponding network. We have seen above that the class of f i t e quasi-lossless networks includes the class of finite lossless networks as a strict subclass, and the question arises as

to what is the physical significance of the quasi-lossless property [63]. This is taken up in the next theorem [32] [44].

Theorem 8

Let N be a finite network composed only of lossless circuit elements and possessing a scattering matrix s. Then s is quasi- lossless. Conversely, let s be a quasi-lossless scattering matrix having the separable form of (8.5). Then there exists a network N, with s as scattering matrix, constructed using lossless circuit elements only.

Before outlining a proof of the direct part of the theorem, we make several comments.

1) Because the class of lossless finite networks is strictly in- cluded in the class of quasi-lossless finite networks, we see that there are interconnections of lossless circuit elements which are no longer lossless. This is paradoxical but nevertheless true; a basic example appears as Example 2 of the next section.

2) An analogous theorem can be stated for impedance, admittance, or hybrid matrices.

3) We shall not prove here the converse part of the theorem, which amounts to a synthesis. But explicit synthesis proce- dures for quasi-lossless scattering matrices and quasi-lossless impedance matrices can be found in [32], [44], [64], [65, p. 1421.

We turn now to an outline of the direct statement of the theorem. Think of N as a cascade load connection with the load comprising the inductors, capacitors, gyrators, and transformers of N, all uncoupled from one another, and the coupling network comprising only the wires which interconnect these circuit elements, all assumed lossless. With the notation of Section VII both x and sl are lossless and, therefore, quasi- lossless. Next, one can calculates via (7.4). Formal evaluation of s: o s and use of the quasi4ossless property of x and sl yields 4 o s = 6 1,. Note that this line of argument will not work to show that s is lossless, for one would have to assume associativity of the resulting composition operations to carry through similar manipulations in a context in which this assumption is unwarranted.

IX. EXAMPLES To illustrate the previous results in more depth we present,

in some detail, the five examples of this section. We begin by considering the inductor, determining its descriptions and lossless constraints. Some of the calculations for the inductor carry over to the capacitor loaded transformer of Example 2, which illustrates many points of concern for lossless networks. The next example, that of a transformer of turns ratio falling to zero cascade loaded in an opencircuit, demonstrates that interconnections of networks with scattering matrices need not have scattering matrices. Following this, the 3n-port circulator is covered, allowing a physical interpretation of Theorem 2. We close the section with a consideration of the time-variable delay which illustrates many of the differences between finite and nonfinite networks.

Example I: Time- Variable Inductor

The time-variable inductor is described by

which gives


the latter of which follows on integrating the describing equation, (9. l a). Choosing i large but almost constant near a given t and then i small but of large derivative gives, using (9. la), the passive constraints I > 0, 1' > 0 of (8.2b). These passivity conditions can also be derived by physically reasoning upon an equivalent circuit for (9.1 b) comprising a cascaded loaded transformer loaded in time-invariant resistors and inductors [30], as in Fig. 3(b) but with similar series active components added (with R = L = - 1) to take care of times when I or 1' are negative.

Assuming I > 0, we can solve e = dxldt + x/l, x = li, by standard means to get

.'?j,;%, Fig. 4. Capacitor loaded transformer.

Fig. 5 . Open-circuit loaded transformer.

ratio T(t), which may even fall to zero (Appendix III), and, ~ ( t ) = I- {exp [-r $11 ~ ( t - 7) e(r) d r consequently, the combination is linear, solvable, and passive.

-- Note also that (9.4) agrees with the general passive form of

from which s results by (3.5b) (8.5) and that the coefficient of 6 is orthogonal, as in (8.7a). However, the network need not be lossless, as is seen physically if T(t) = 0 for t > to # -, in which case charge can be

s(t, 7) = 6(t - 7) - - exp - ] e x [ ] t - 7). trapped on the capacitor causing P(m) * 0 with square- I(t) integrable incident voltages.

(9.2) For (8.7b), one calculates (see Appendix 111)

Here a is any finite real number. Clearly s exists if I > 0, as Theorem 1 requires. To find the condition under which a - 2T(t) exp [it T2 (A) d A] T(r) exp [lT T2 (A) dh] time-variable inductor is lossless, we apply the constraint of F ( ~ , = (8.7b), F = On, for which

~ X P [Zlm T'(A) dh]

(9.5a)

Since F(t, 7) = 0 is required for losslessness, as in (8.7b), one concludes from (9.5a) that a necessary and sufficient condi- [I.' I(x) "1 + exp [it &] [(XI [lr *] I(x tion for the network of Fig. 4 to be lossless is [ 6 2 ]

Setting this to zero yields, on differentiating, 1' = 0 as the lossless constraint; that is, I is a positive constant for a (nonzero) lossless inductor.

Since the capacitor with c = I is the dual of the inductor, the above results apply to the capacitor with the duality relationship sd = - r , (3.9a), showing that s of (9.2), is replaced by its negative for the capacitor.

for all real finite a. Irrespective of satisfaction of the lossless condition of (9.5b),

the network is quasi-lossless, being an interconnection of two lossless circuit elements. One can also see this by noting, from (8.7c), that quasi-losdessness holds if and only if F(t, 7 ) = s(7 , t); this condition is easily checked using (9.5a).

Example 3: Open-Circuit Loaded Transformer

Example 2: Unit Capacitor Loaded Transformer If the turns ratio T(t) falls to zero, as shown in Fig. 5(b), then the open-circuit loaded transformer 1-port of Fig. 3(a)

Consider the 2-port transformer cascade loaded by a capaci- has rather degenerate behavior. We first observe that the tor with c = 1, as shown in Fig. 4. We calculate, as for Exam- 2-port transformer, which is cascade loaded by an open-circuit ple 1 (see Appendix 3) to form N, is a solvable network described by (applying (3.8b)

to a scalar T)

exp [l' T' (A) u(t - 7) (9.4) Since

for any real finite constant a. Note that s exists for any turns an inverse of S - E22 o SI = 2T26/(T2 + 1) does not exist for


t > to, when T(t) = 0. Since any function in the range of 6 - Cz2 0 sl = 2T26/(T2 + 1) must fall to zero faster than any corresponding function in the range of CZ1 = 2 ~ 6 / ( ~ ~ + l ) , (7.5a) does not hold and the given methods of Section VII fail to yield an s. In actual fact, s does not exist because the 9. constraint requires e(to) = 0 in e = v + i; the given network is not solvable in spite of the fact that the transformer and opencircuit are since only this zero value of e is allowed for t > to.

Some comments on this behavior are in order. We first observe that one would intuitively say N behaves as a short- circuit (v = 0) for t > to , which indeed is the case in the formulation being considered. However, one can extend the network to square-integrable vr and vi, in which case one essentially has to postulate the behavior of N for t > to. For instance, one can logically postulate that N behaves as an open circuit for t > to, which, in fact, is consistent with assuming T = E > 0 for t > to, using (7.4) to get

and then letting E converge to 0. Other methods yield other results, and we can only conclude that no unique scattering matrix exists for the given network in any reasonable context.

Physically, this example points up the ideal nature of the transformer. In actual fact, such a device is the limit of mutu- ally coupled coils with the mutual inductance infinite [59, p. 1741. To get a turns ratio behavior as shown in Fig. 3(b), the mutual inductance must flip from being infinite before t = to to being zero after t = to. This type of behavior seems to be unknown in the physical world, where every network appears to be solvable. Nevertheless, this example shows the type of behavior which must necessarily be considered when using a workable mathematical theory to model reality.

Example 4: Realization of Scattering Matrix Product

We define the 3n-port circulator by

Since sa o s = 6 13,, this is clearly lossless, as it is passive by Theorem 1. The 3n-port circulator, represented in Fig. 6, can . -

be considered as the juxtaposition of n 3-port circulators, which can be physically realized [66, p. 5201.

Toward a realization of the composition of two scattering matrices, consider Fig. 7 where we first consider N2 as a cascade load on the circulator. Equation (7.4) gives

which is actually more easily obtained by physically reasoning on the structure. Again, applying (7.4) to N1 loading Nz gives

Consequently, Fig. 7 realizes the product considered in Theorem 2. Note that the passivity of each subnetwork in Fig. 7 makes the passivity of s physically obvious. This connection has been considered in the time-invariant case by Belevitch [45 1, [67, p. 2801 whose corresponding results indi- cate that a factorization theory of passive s may be fruitful for general time-variable synthesis. Indeed, a factorization theory has been successfully used as the basis for a synthesis of time-

Fig. 6. 3n-port circulator symbol.

Fig. 7. Interpretation of Theorem 2 .

varying quasi-lossless scattering matrices [44] , [68, p. 26 1 I and quasi-lossless transfer scattering matrices [69 1.

Using the connection of Fig. 7, one can obtain a physical proof of the associativity of the composition of passive scattering matrices. This is shown in Fig. 8 where a 4n-port circulator has been introduced in Fig. 8(b).

Example 5: Time- Variable Delay

Consider a 1-port described by

where d(t) is a real-valued infinitely differentiable function of time with (for convenience) d(fm) Z f m. Since

vr(t) = vi(t - d(t)) (9.121

the name time-variable delay is descriptive when d > 0, which must be the case if s is passive by the antecedence constraint of (4.1 ). By a change of variable x = h - d(A) we have.

where a(h - d(h)) = A is the inverse of the change of variable. We observe from (9.13) that this is a lossless network if and only if the delay d is constant, since it is passive by the following results.

To find the passivity constraint on s, it is easiest to directly' calculate the energy in terms of scattering variables as expressed in (3.6).

t - d ( t ) -dt(a(t)) . = 1- 1 - dt(a(t))

[u'(T)] ' d r + lt [vi(r)] dh. t - d ( t )

Since d 0, the final term in (9.14b) i s nonnegative. Conse- quently, we choose the support of vz in [-00, t - d(t)l and let the square of vz approach an impulse to see that -dl/[ 1 - d'] > 0 (for all t) is necessary for & ( t ) > 0 (clearly i t


Fig. 8. Associativity of composition for passive N.

also a sufficient condition). Then either d' 2 1 or d' < 0. In the former case

and letting t approach - with a fixed contradicts d > 0. We conclude that s is passive if and only if

and

for all t. Physically, relaxing the d' G 0 constraint would mean that input (incident) signals become stretched in time when being changed into output (reflected) signals, allowing more nergy to come out of the network than would have gone in. Because s is not of the form of (8.5), we know that the time

variable delay, with d > 0, is not a finite network. Neverthe- less, it is of value in modeling the reflection characteristics of time-varying media, such as the ionosphere, or of moving targets in radar studies [70].

Along the same lines one can replace 6(t - T) in (9.8) by 6(t - d(t) - T) to obtain a time-variable circulator, the passive conditions again being as in (9.15) with d(f w) f f w. Choos- ing n = 1 and terminating port 3 in a unit resistor yields Fig. 9 which is described by

axch a 2-port device is matched at both ports and has one-way transmission which is a variable delay. The presence of the resistor as well as the time-variable circulator explains why s of (9.16) is not lossless, or quasi-lossless, although it is passive under (9.15).

Because of the generality of the scattering matrix s, as shown by the conditions for its existence of Section 111, we have concentrated on the properties of passive networks in terms of s. The most fundamental results are those of Theorem 1, which gives a complete characterization of (linear, solvable, and) passive networks in terms of s, and Theorem 5, which completely characterizes lossless networks through s. Since Theorem 1 relies on the difficult calculation of Ilsll, an dternate characterization in terms of an energy form Qt was

Fig. 9. Loaded time-variable circulator for time-variable delay.

given in Theorem 4. Although the properties of Q, may often be equally hard to find, it yields conditions on the impedance z and shows how the time-domain approach of this paper re- duces to the classical frequency-domain approach used for time-invariant networks.

Since finite networks are of considerable practical importance, general results for this class of time-variable networks were considered, and, in particular, lossless and quasi-lossless conditions were obtained both on s and z in Section VIII. Of most interest are Theorems 7 and 8 which give a complete characterization of finite lossless and quasi-lossless networks, the latter being interconnections of the former. For nonquasi- lossless finite networks results are not as compleJe, the most recent ones being for sl (t, T ) = 6(t - ~ ) 1 , + @(t) 9 ( r ) u(t - 7), that is s of the form of (8.4) but with A(t) = 1,; in this case, the necessary and sufficient passivity conditions are stated as (71 I

t

*t) G(t) > max [@(t)([_ $(a) *(a) da) &t),

As these appear rather complicated, though a synthesis results from them [71], the problems in obtaining general syntheses can be appreciated; we comment that if A is nonsingular in (8.4) then the situation reverts to that of (10.1) by writing s = ( A s ) 0 sl and using the realization for the composition of two scattering matrices of (9. lo), since 11A111 < 1 by passivity. Nevertheless, to date, syntheses have occurred for all quasi- lossless scattering matrices [44], [68] and a large class of general time-variable scattering matrices [29] , [33], [71].

Because the cascade load is of importance for such synthesis methods and because almost all interconnections of net-


works can be considered as of the cascade loaded form, the scattering matrix for such a connection was investigated in Section VII, where a method of finding the over all scattering matrix was given. Although this latter will usually exist for physically meaningful connections, Example 3 of Section IX shows that care must be used, as somewhat degenerate networks can result from the interconnection of reasonable networks. In particular, the connection of two solvable networks need not be solvable, this in spite of the fact that physically constructable networks seem to be solvable.

The remaining examples of Section IX show the ease with which the developed lossless criteria can be used and various tricks which are useful in avoiding some of the tedious calculations needed in applying passivity conditions. The essential dif- ference between finite and nonfinite networks is illustrated by the time-variable delay which also points out the generality of the theory. Example 2 , besides illustrating various calculations, points out that "obvious" results of time-invariant theory cannot be carried over to time-variable theory, since an interconnection of lossless networks need not be lossless, though it is always quasi-lossless.

In final summary, the material of this paper prepares a foundation for the synthesis of passive time-variable networks based upon distributional kernel theory and rigorously estab- lishes properties already used in time-varying quasi-lossless synthesis. As such, it generalizes known results of time- invariant networks while also opening up for investigation other known time-invariant results [ 131, [ 1.5 1, [72] -[74] for extension to time-variable situations. The results, although developed in a network context are, of course, applicable to any passive scattering system.

APPENDIX I ACTIVITY OF =6(3)

We show here that the impedance z ( t , T) = 8(3)(t - 7) fails t o satisfy the passivity criterion on Rt of (5.4). The method used is distinct from an essentially ad hoc procedure of Spauld- ing [ 29 , p. 481. We shall evaluate (Rt(a, 0) cp(P), d a ) ) , with Rt(a, p) = u(t - ~ ~ ) 8 ( ~ ) ( a - p) + u(t - ~ ) 8 ( ~ ) ( 0 - a), ex- plicitly for arbitrary p, and observe that the resulting expression is not necessarily positive.

We have

since p and all its derivatives are zero at the lower limit. Clearly this expression has arbitrary sign depending on the selection of Q.

APPENDIX I1 NULL SPACES FOR CASCADE LOADING

Without preamble, we carry over the notation of Section VII. First observe that since Z and sl are passive scattering

matrices, the same is true of

and thus also (by Theorem 2)

Consequently, to demonstrate

it is sufficient to show that

for any passive E. To prove this, we first establish a pre liminary result.

Lemma A2: If E is passive, then, in the sense defined at (5.11,

61, - Z42 0 Zp - Z!2 o E12 2 0 . (A2.4)

Proof: From Theorem 4 and (5.5) when applied to vectors of the form y"= [Cn, 21 , x E 9, the result follows immediately. Q.E.D.

A simple extension using condition 4') of Theorem 4 is, for any x E 9+,

Now to establish (A2.3), assume there is a 9, function x E ? I [ S l , - E n ] . T h e n x = Z Z 2 * x and t h u ~ ( x , x ) ~ - ( Z ~ ~ . x, EZ2 x)* = 0. From (A2.5) it follows that x Z12 x ) ~ = 0, thus Z12 x = 0, that is, x E ?I [Z12 I .

APPENDIX 111 UNIT CAPACITOR-LOADED TRANSFORMER

We consider the network shown in Fig. 4, under the assumption that the turns ratio T(t) is never zero. Then the voltage and current at the input port are related by

Replacing the variables v and i by v i and v' with the aid of (3.4), we obtain the following relation between v' and v':

Since s maps v i into v', s(r, 7) is the antecedal impulse response of the above equation. Methods of finding this impulse response are by now well known, being detailed in, for example, [ 6 1, p. 355 ] , and one may easily determine

e x J ~ ' ( h ) dh]

Here, the real finite constant a is arbitrary. One can apply the cascade load results to also obtain this, which, a l t h o u


the calculations are somewhat more tedious, shows that (9.4) still holds when T ( t ) is permitted to equal zero, since 6 - CZ2 o s l = 2 [ 6 ( t - 7 ) - ( 1 - T2(t)}e-( ' - ' ) u(t - T ) ] /

(1 + T 2 ( t ) ) is never singular. The adjoint is found in the usual fashion:

t T ) = - 6 ( t - T ) + { 2 T ( t ) exp [it T 2 ( h ) dh]

T ( 7 ) U ( T - t ) . (A3 .3 )

The composition product .P 0 s can now be formed; it consists ~f the sum of a S( t - 7 ) term, a term multiplied by u ( r - t ) , i nd a term multiplied by u( t - 7 ) . The last term is given, by inspection of (A3 .3 ) and (9.4), as

+ 4 T ( t ) exp [ [ I t T 2 0 ) d h ]

I t exp [ 2 lo T 2 ( A ) d h ]

- T ( r ) exp [[I T 2 ( h ) d h ] .

Now obsesve that

" exp [ 2 lo T 2 (A) d A ]

2 exp [ 2 it T 2 ( h ) d h ] 2 exp [ 2 lw T 2 ( h ) d h ]

and thus

- 2T( t ) exp [ I f 7 ' 0 ) d h ] T ( r ) exp [[' 7' ( A ) d h ] - -

exp [ 2 1- T 2 ( h ) dh]

The authors wish to thank B. Serrano-McKee for her excellent assistance in the initial preparation of the manuscript and B. Aycock and T. Casey for their excellent assistance in the final preparation of the manuscript.

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