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Stationary principles and potential functions for nonlinear networks
Stationary Principles and Potential Functions for Nonlinear Networks* by L. 0.
CHUA
Department of Electrical Engineering and Computer Sciences University of California, Berkeley This
ABSTRACT:
stationary content,
paper
principles co-content
nonlinear
and Moser
on potential
networks
functions
for
functions
solvability
complete
topological
matrices. Simple
to a legitimate set of equations
current-controlled
complete
voltage-controlled
dynamic
nonlinear
equations
in terms of a single mixed potential
q-port.
of total
to that of total parametric
content
criterion
for
to non-complete
conditions
conditions resistance
is shown in terms
to be the
of standard
are given which require matrix
conductance also guarantee
by a system
paeudo-
which each of these
function
given explicitly
au.cient
can be represented
The results by Brayton
are generalized
potential
and the incremental These
various
namely, the pseudo-content,
the incremental
complete
network
p-port
of
deriving concepts
elements.
The precise
circuit theoretic
the positive-de$niiteneaa
for The
m-ports
functions;
approach networks.
multivalued
content.
reduces
of an implicit
theoretic
nonlinear
are generalized
containing
and the pseudo-hybrid
pseudo-potential
merely
content
graph type
in terms of three pseudopotential
co-content unique
a novel
single-element
and hybrid
for resistive m-ports
presents
for
matrix
of an
auxiliary
of an auxiliary
that a non-complete
of normal form
differential
function.
I. Introduction
It is well known that the solution of various classes of nonlinear networks is identical to the stationary points of a suitably defined scalar “potential function” (l-4) of a complete set of network variab1es.t A comprehensive theory of nonlinear networks based on potential functions was presented by Brayton and Moser in their 1964 paper (5). Some interesting related results were also given by Stern (6). More recently, the properties of network potential functions have been explored by several authors (7-11). Stationary principles may be derived by an application of the properties of line integrals of state functions (12) or by variational techniques (13). The main objective in this paper is to unify and generalize the existing results in this area using a surprisingly simple but novel approach. Only Kirchhoff’s laws and the vector form of the chain rule for taking the gradient and the Jacobian of composite functions are used in the proofs. * This work was supported in part by the National Science Foundation under Grant GK-32236 and the United States Naval Electronic Systems Command under Contract N00039-71-C-0255. t A set of variables i,, i,, . . . , i,, v,+,, v,+, . . . , vda is “complete” if they can be chosen independently without leading to a violation of Kirchhoff’s laws and if they determined in each branch at least one of the two variables, the current or the voltage (5).
91
L. 0. Chua Only networks containing two-terminal uncoupled elements are considered in this paper. Most of the results will be stated only for resistive networks even though they are generally applicable to single-element type networks as well. Section II contains a new formulation of several well-known stationary principles and a generalization in terms of the total parametric content of networks containing multivalued elements. Section III introduces the concept of the pseudo-content and pseudo-co-content for non-complete m-ports containing only current-controlled resistors, or voltage-controlled resistors, respectively. This concept is generalized in Sect. IV to that of a pseudo-hybrid content for noncomplete m-ports containing both currentcontrolled and voltage-controlled resistors. The precise criterion for which the pseudo-hybrid content can be reduced to a legitimate hybrid content is also presented in this section. Among other things, this criterion provides an answer to the heretofore unresolved question as to when does the mixed potential of Brayton and Moser (5) exist for non-complete dynamic networks. More convenient circuit theoretic sufficient conditions for satisfying this important criterion are derived in Sect. V. II.
Stationary
Principles
of Nonlinear
Networks
Let NR be a resistive network containing n nodes and b branches. Let Y be a tree of NR and 9 its associated co-tree.* Let us label the links from 1 to 1, and the tree branches from I+ 1 to b. Let i, i, and i, denote the branch current vector, the co-tree current vector and the tree current vector, respectively. Let v, v, and vF denote the branch voltage vector, the co-tree voltage vector and the tree voltage vector, respectively. The two Kirchhoff Laws can be written as Bv = 0 and Qi = 0, where B and Q are the fundamental loop matrix and the fundamental cut set matrix, respectively (14). (A) Principle
of stationary content (co-content)
Let iVR contain only current-controlled (voltage-controlled) resistors and independent voltage (current) sources so that the content G,($) = j$vi(ij) di, (co-content Gj(vj)~J~ij(vj) dvi) of each element is well defined. Let and define the total content $(i9) G(i) 3 XF==,Gj(j(if) (G(v) 3 &Gj(vj)) (total co-content ‘3(vY)) by g(i,)
= G(i) liCBtia=
GO (BtiY),
where “0” denotes the composition
~(v,)=G(v)Iv=ntvY
= Go(QtvY),
(1)
operation.
Theorem I (Millar’s theorem on stationary content (2, 4) ). If NR contains only current-controlled resistors and independent-voltage sources, then a vector i, = I, is a solution of NR if, and only if, it is a stationary point of the total content 3(iZ). * We assume throughout this paper that the network NE is connected. The term tree always means a spanning tree; i.e. it connects all nodes. Any subset of branches in a spanning tree is called a subtree in this paper.
92
Journal of The Franklin Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
Proof: If we take the gradient of $9(iu) and make use of the vector form of the chain rule (12), we obtain Z?‘(i,)/ai,
= ZY~ (Bti,)/ai,
= B 83(i)/ai lizsti9= Bvo (Btiy).
(2)
We now make the important observation that if the last expression in (2) is equated to zero, we would obtain the system of independent loop equations which completely determined the solution of NR. Consequently, any stationary point of 9(i9) is a solution of NR, and vice versa. Q.E.D. Using a dual proof, we obtain Theorem II. Theorem II (Millar’s theorem on stationary co-content (2,4)). If NR contains only voltage-controlled resistors and independent current sources, then a vector vF = E, is a solution of NR if, and only if, it is a stationary point of the total co-content 9(vg). (B) Principle
of stationary hybrid content
In order to derive a stationary principle that would allow both currentcontrolled and voltage-controlled elements, we follow the approach taken by Brayton and Moser (5). Let 5c.F be any subtree (which need not be connected) made up exclusively of voltage-controlled resistors and let 4, the remaining tree branches, be made up of current-controlled resistors, where q u& = F. Let P1 be all those voltage-controlled links which formed closed loops with elements of K only and let 9a be the remaining links so that 9ruZa = 9. Hence, all elements in .& are voltage-controlled and all elements in 9a are current-controlled. If we let vY1, vPa, vg, and vg, and iyJ denote the branch voltage (current) vectors associated (iF1, iZS, i,, with elements in dq, gZ, ~9 and %, respectively, then Bv = 0 and Qi = 0
=[I
r
0
0
(3)
’
L
where the symbols 1 and 0 denote the unit and null matrix respectively. Observe that OFIZ is a null matrix because each element in DLplforms a fundamental loop with only elements in <. Using the above notations and letting i&z -iya, we define the total hybrid content X(vS1, i&) associated with the network NR to be the scalar function:
Vol.296,
No.
2,
August 1973
93
sum of co-contents of elements in g1 1’
1
sum of conjugate
*
contents
( of elements in L$
I
2
sum of co-contents 1of elements in & I
%-,h-J = j~a%(ij) =
sum of contents of * elements in &
Theorem III (Principle of stationary hybrid content). A vector (EF1,IZZ) is a solution of the network NR described in the preceding paragraph if, and only if, (EF1, I&) is a stationary point of the total hybrid content X(vF1, i&), where 1%a = -1 -% Proof: Taking the gradient of X(vY1, i&) with respect to vF, and is2 respectively, we obtain
Now, an examination of (3), (5) and (6) sh ows that if we equate the righthand side of (5) and (6) to zero, we would obtain the fundamental cut-set equations corresponding to the fundamental cut-sets formed by the tree branches in q, and the fundamental loop equations corresponding to the fundamental loops formed by the links in LX2.Since these mixed systems of equations completely determined the solution of NR, it follows that (EY1, I& is a solution of NR if, and only if, (EF1, I$,) is a stationary point of the total hybrid content. Q.E.D. We remark that the total hybrid content defined in (4) is related but not identical to the mixed potential function of Brayton and Moser (5) and the dissipation function of Stern (6). We also remark that Theorem III remains valid if the total hybrid content is replaced by several related hybrid potential functions defined in (15, 16). (C) Princigde of stationary parametric content We now generalize the preceding stationary principles to allow all elements in NR to be parametrizable.? In particular, let each resistor be characterized * We define the conjugate content G,*(ij*) of a current-controlled resistor to be Observe t&t dGt(i,*)/dif = w,(iJ. G:(ij*) s j$’ wj( - ij) di,, where ij* 3 -i,. t A curve I7 representing the locus of some relationship f(~, y) = 0 is said to be perametrizsble if, and only if, I? can be described by two continuous functions of a third parameter p; namely, z = z(p) and y = y(p), p E [a, b] c (-co, CO). An element characterized by a parametrizable curve is said to be a parametrizable element.
94
Journal of The Franklin
Institute
Principles
Stationary
and Potential Functions for Nonlinear Networks
by vi = vi(pj) and ii = ii( The content Gi(pj) and co-content I are defined respectively by the Stieltjes integrals (17) Gi(pj) = J”@Vj(pi)dii(pi) and cj(pi) e J$ij(pi) dvj(pi). In terms of the parametric vectors Pi = (plP pz, . . . . P~++#
and
Pi
= (P~-~+~,
P~-~+~,
...) ~2
the equations of motion for NR are given by
1
b4P.9)
VdP.Pe) _.._____________ = 0, [-B$l,]
[l, /B,l
[ V94P.T)
_-_-_________-= 0.
[
i.54P.F) 1
Using the above notation, we define the total parametric associated with NR to be the scalar function: ~(PJZ, P.F) = I
content g(pp,
- ~(P.F) + i&(pJ B, vr(ps),
(7) pF)
(8)
where
Theorem IV (Principle of stationary parametric content). Every solution (pu, pY) of NR is a stationary point of the total parametric content Y(ppe, ps). Conversely, every stationary point (ir,, pF) of B(p,, ps) is a solution of NR provided (1) i$j)#O,j=
1,2 ,..., b-n+1
and (2) w(i(&)#O,j
= b-n+2,b-n+3
,..., b.
Proof: Let us take the gradient of B(p,, pF) with respect to pu and pY and make use of the relationships Gi(pi) = vj(pi) i;(pi) and am = ii v(i(pi) for Stieltjes integrals (12) to obtain @YP2?
Ps)/aPY
= wz4Pa)/~P~l
Wp9p)
P.&/~P~ = II-- ~vAP.GP.A
[V9(PY) L-(PA
+B.T
V,(P.T_)l~ -%-4dp.d.
(9)
(10)
Observe that if we equate the right-hand expressions (within the bracket) of (9) and (10) to zero, we would obtain (7). Hence, any solution of NR must be a stationary point of the total parametric content. The converse is also true provided conditions (1) and (2) are satisfied. Q.E.D. We remark that the total parametric content defined in (8) can be transformed with the help of the integration-by-parts formula for Stieltjes integrals (12) into the following equivalent forms:
~(P_D er) = I + ‘%P.T) %P_B ~~1 = - C~(P~) + PI
+ Wp~) By - G-(p~)l VAP.FL +C.dvAp~;P) +B~v.d~~)l.
(11)
(12)
Observe that if we let the parametric vectors pg and pF equal i, and i, for current-controlled resistors, or v* and vF for voltage-controlled resistors, then it is easily shown that (11) and (12) reduce respectively to 5!?‘(ip) and g(vY) as defined in (l), as they should.
Vol. 296, No. 2, August
1973
95
L. 0. Chua ZZZ. Pseudo-content
and Pseudo-co-content
of Non-complete
m-ports
An m-port is said to be complete if its port variables constitute a complete set (5). It is well known that the equations of motion of a dynamic nonlinear network NRLc can always be written in the normal form if the energy storage elements of NRLc can be extracted across a complete m-port. In this case, the state equations can be represented in a compact form in terms of a suitably defined potential function (5). Our objective in the following sections is to generalize this concept to the case where the extracted m-port N is not complete. When an m-port is not complete, it may not be possible to characterize it by a single-valued function of the independent port variables representing the sources connected across the ports. In such cases, it is meaningless to talk about a potential function. For example, consider a one-port made up of two tunnel diodes in series. It is well known that for some combination of tunnel diodes, the driving-point v -i characteristic of this non-complete one-port is a disconnected and multivalued function of both v and i (17). Since the v-i curve is not parametrizable, this one-port does not possess even a parametric content. The main objective in this paper is to show that a non-complete m-port can always be characterized by a scalar function X(x, ; x,)-henceforth called a pseudo-potential-of a port vector x1 representing the driving internal sources, and an auxiliary vector x2 representing appropriate variables.* Moreover, the port vector y1 associated with xl is simply the pseudo-gradient of S( x1 ; x2) obtained by taking partial derivatives with respect to x1; namely, Yl =
8S(x,;
x2)
-y1@1;
x2).
ax, Hence, the necessary and sufficient condition that the m-port be characterizable by a single-valued function of x1 is that it be possible to solve for x2 as a function of x,; namely, x2 = f(x,). Hence, in this case, we can write Yl = Yl(Xli f(x1)). Moreover, under this condition, reduces to a legitimate potential Wx,) y1=,,1=
we will prove that the pseudo-potential function P(xl) = X(x1; f(x,)) such that
8S(x,: f(x,)) = aS(x,;
x2)
xz=f(x1) ax, This equation implies that taking the derivative of X(x, ; x2) after substituting the relationship x2 = f(x,) is the same as substituting this relationship after taking the derivative. This property is remarkable because, in general, the derivative operation and the composition operation do not commute. Let the m-port N shown in Fig. l(a) be driven by m current sources which do not form a cut-set. We assume that there are b -m elements inside N ax,
* Although 8(x, ; x2) is a function of both x1 and x2, we have chosen the unconventional notation of a semi-colon as the delimiter here, rather than a comma, to emphasize that the elements of x2 are not port variables.
96
Journal of The Franklin Irwtitute
i2t
(!I
.
.
.
-
v2
+
+ iit q +I
-
(a)
sources
voltage
?-
$
i2
.
.
.
+
-“2
i-fm
I--im0 t -"m
i2
I
iI
-“I
CI+
i, t
(b).
of N)
branches
tree
011
(containing
NL7
-3
‘m+2
ib-n+l
.. .
“m+l
+
i m+l
in
co-tree
A
1
a!2 forming the remaining port of the
of
set
link resistors
(the
m-port N in (a) is redrawn in (b) by extracting all co-tree resistors inside N.
irn
:.
i2
iI
-“m
: .
-“2
independent
constont
1
.I
co- tree d
-“I
1
set of
current sources in Al forming port of the
(the
and
resistors
controlled
current
(containing
N
FIG. 1. The non-complete
i2
il
L. 0.Chua consisting exclusively of current-controlled resistors and independent constant-voltage sources which do not form closed-loops. If n is the total number of external and internal nodes, then a tree T of this network will contain (n- 1) branches and the corresponding co-tree _Y will contain b -n + 1 branches. We put the set P1 of m current sources in the co-tree and let the remaining set 2Z2of (b-n+ 1) -m link resistors be extracted, as shown in Fig. l(b). Let -VP1 = (-q, V& = (%+I,
-v‘& . ..) -?l,p, %+2,
*. ., %,+Jt
and v.9- =
b4~-_n+2>
vb-,+3,
* * * 9 vb)t
denote the branch voltage vector of elements in 2Z1,2T2and T, respectively.* Similarly, let i,, = (i1, i,, . . . , fJt, . ly2 = @m+l,&+2, ***, &+$, and iT = &.,-_n+$,&-_n+3,. . . , &? denote the branch current vector of elements in gl, AC2and T, respectively. Finally, let v = ( - vY1, vT2, v~)~ and i = (i91,i92,i9)t denote the branch voltage vector and the branch current vector of N, respectively. Let us write the fundamental loop equations Bv = 0 in the following partitioned form :
DeJinitim 1. We define the pseudo-content ‘3z1(iFp1;i& complete m-port N in Fig. l(a) to be the scalar function
9ze,(i,,;
i&
of the non-
= G(i) li=Btig= Go
(14)
where G(i) is the sum of the contents of all elements inside N.+ * The reason for choosing the variable - vy, is due to the fact that each port current ii in Fig. 1 is defined leaving the positive terminal of the associated current source (i.e. entering port j). This is opposite to the standard reference current direction required by the topological relationships used in this paper. t Observe that G,(i,) = 0 for j = 1,2, . . . . m because the content of a current source is zero (16). Hence, G( .) is really only a function of i2, and i,. However, to avoid introducing additional symbols, it is more convenient to consider G( 0) as a function of i. A dual remark applies to the variable v in Definition 2.
98
Journal
of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
Theorem V. The port voltage vector v4p1of the non-complete m-port N in Fig. l(a) is equal to the pseudo-gradient of the pseudo-content $Y.&,1; i9J of N; namely, (15)
VP, = @JP1(i9,; i_&ai,l.
If the internal link-current vector i,Z in Fig. l(b) can be uniquely solved in terms of the port current vector iPl; namely, i9, = f(ipl), where f( *) is a single-valued function, then the m-port N in Fig. l(a) is current-controlled and is characterized explicitly by v9p1= a3P&1;
i9J/G,
liP~=z=f(iP ) = b&J.
In this case, the non-complete m-port N is completely characterized potential function 99(ilp1),called the content of N, where %,J
and
vZ, = @G9,)/&rl
(16) by a
= Gpl(iPpI; f(i&) = ag91(i91; f(i_&/&,.
(17)
Proof: The proof of this theorem is a special case of the proof of Theorem VII given in Sect. IV, and will not be given here due to lack of space. A complete proof can be found in Reference (16). Corollary 1. The non-complete m-port N in Fig. l(a) is characterized by a state function (representing the gradient of a potential function-the content of N), if and only if, iT2 is a single-valued function of iT1. In this case, the content S(i,l) of N corresponding to any given value of iP1 is equal to the sum of the contents of the elements inside N, evaluated at the corresponding operating point, and an additive constant which is independent of the operating points. Proof: The sufficiency part of this corollary follows immediately from Theorem 4 and the properties of state functions (12). To prove the necessity, we observe that if f( .) is a multivalued function of i91, then the pseudocontent defined in (14) will also be a multivalued function of iP1, and therefore, cannot reduce to a single-valued potential function. Q.E.D. We remark that since an m-port containing only two-terminal currentcontrolled elements is always reciprocal, the condition that i,, be a singlevalued function of iP1 is a necessary and sufficient condition only for the existence of a legitimate potential function, and is certainly not necessary for reciprocity (18). The following implicit relationship between iP1 and iPZ can be used to determine whether iZZ is a single-valued function of ipI: v~,(i~,) + Jb2.+w
(Bb2.&,
+ B&&J
= h,.
(18)
Observe that in the special case where N is complete, then the conclusion of Corollary 1 is always true. Observe also that Millar’s theorem on the superposition of contents for one-ports is a very special case of Corollary 1 (2). Example. A non-complete 2-port is shown in Fig. 2(a) and its associated 3-port is shown in Fig. 2(b) with the nonlinear link resistor extracted across
Vol. 296, No. 2, August 1973
99
L.
0.Chua
[1 “4
“5
[I i4 i5
L& = “3
v-i
curve of
R: v=i3-8i
J
v3-i3
curve:
&=
i3
v3=i:-8i3
lb).
(a L
FIG. 2. Example
i
of a non-complete S-port and some pertinent topological and relationships associated with its network graph.
matrices
port 3. The various topological matrices and relationships which are required in order to apply Theorem 1 and its corollaries are shown in Fig. 2(c). The total content is given by G(i) = iii - 4iZJ+ 2ii + 2iE. The pseudo-content defined by (14) is given by ?&,
*3 i2.2) = 2i,2+ 4i, i, + 4ig - 8(i2 + *ii) i, + *ii.
In order to determine whether this 2-port is characterizable by a content or not, we must solve for i, in terms of i, and i,. This is determined by writing down the implicit Eq. (18) : v9,(i& + B,, gvg o (B&, i,, + Bt,,
iyp,)= ii - Si, - 4i, = 0.
solving for i,, we obtain i, = f(&, iz) = 2(i, + &,)*. Hence, in view of Theorem 4, the content for this 2-port exists, and is given by (17), namely, 9(i,l)
= C?I(i,,is) = 991(il, i;; 2(i, + *i#)
= 2if + 4&i, + 4ig - lZ(i, + *i#.
We can recover the state function by using either (16) and CYP1(iYl;iyE) or (17) and 3(iPp,). In either case, we obtain w1= 4i, + 4i, - 8(i, + &i,)* and w2= 4i,+ 8i,- 16(i, + iiJ+. Observe that (16) and (17) imply that taking the derivative of ‘YPI(i,,, . iyz) after substituting the relationship i9, = f(i&
100
Journal of The Franklin
Institute
Xtationary Principles
and Potential Functions for Nonlinear Networks
is the same as substituting this relationship after taking the derivative. In other words, Eqs. (16) and (17) reveal the remarkable commutative property alluded earlier between the differentiation operation and the composition operation. We close this section by considering briefly the dual case of an m-port driven by m voltage sources which do not form a closed loop, as shown in Fig. 3(a). We assume that there are b-m elements inside N consisting exclusively of voltage-controlled resistors and independent constant current sources which do not form cut sets. Let us put the set ~9 of m voltage sources in a tree Y of N and let the remaining set s of (n - 1 - m) tree resistors be extracted as shown in Fig. 3(b), where n is the total number of nodes. We number the tree branches first and adopt the obvious dual notations to write Qi = 0 in the following partitioned form:
DeJinition 2. We define the pseudo-co-content gY,(vg,; complete m-port N in Fig. 3(a) to be the scalar function
vY.J of the non-
(20) where o(v) is the sum of the co-contents
of all elements inside N.
Theorem VI. The port current vector irl of the non-complete m-port N in Fig. 3(a) is equal to the pseudo-gradient of the pseudo-co-content gY1(vY1; vg,) of N; namely, . rrl = a%l(vfl$ v.9@vsl. (21) If the internal tree-branch voltage vector vg, in Fig. 3(b) can be uniquely solved in terms of the port voltage vector vg-,; then the m-port N in Fig. 3(a) is voltage-controlled and is characterized explicitly by 19-1= @&,l;
vg_,)Pv& l”&=f(“~-ll =i9+&
In this case, the non-complete m-port N is completely characterized potential function $?(v,,), called the co-content of N, where %.%-J = Y&,l;
(22) by a
C-S,))
and vy, = @(v.&/&1
= @.&sl;
f(vs,))Pv,,*
(23)
Corollary 2. The non-complete m-port N in Fig. 3(a) is characterized by a state function (representing the gradient of a potential function-the
Vol.
296, No.
2, August
1973
101
4n
i! -
Vm-+
(a).
sources
current
constant
1
independent
and
resistors
controlled
voltage -
(containing
N
FIG. 3. The non-complete
a
.
.
.
l-----l
m-port
V
N in (a) is redrawn
(the set of voltage sources in 3, forming part of the tree .T)
(b). in (b) by extracting
-im
of N)
I inks
011
all tree
resistors
in-1
inside
N.
tree resistors in g2 forming the remaining part of the tree l7 1
(the set of
Stationary Principles
and Potential Functions for Nonlinear
Networks
co-content of N), if, and only if, vY2 is a single-valued function of vY1. In this case, the co-content B(v,,) of N corresponding to any given value of vY1 is equal to the sum of the co-contents of the elements inside N, evaluated at the corresponding operating point, and an additive constant which is independent of the operating points. The dual implicit relationship between vYz and vY1 is now given by
IV.
Pseudo-hybrid
Content
of Non-complete
m-Ports
Now consider the more general hybrid m-port N shown in Fig. 4(a) driven by m, voltage sources which do not form closed loops and m2 current sources which do not form cut-sets, where m = m, +m,. Let us choose a particular tree Y by first including in it the set & of all m, port voltage sources, and let the set 2Z2of all m2 port current sources be included as part of the co-tree .2? associated with 7. Next we add as many voltage-controlled resistors as tree branches, forming another sub-tree &. The remaining subset -IL”1of voltage-controlled resistors which cannot be included in the tree must form closed loops with branches in K u &, and must, therefore, be placed in the co-tree 9. Let us next fill up the tree with a sub-tree .99 chosen from the remaining elements, which are current-controlled, such that 9 = q u J< u ~9. Similarly, the co-tree 9 = g1 u_EPZU~~,where the subset ,Ep3would include the remaining unlocated elements, which are all current-controlled. Let us label the elements as shown in Fig. 4(b) and define the branch voltage vector v = (v&p - v&*3v& v&, v&, vtY,)t and the branch current vector
i = (i$,, i&, if,, - i$,, itr2,i$Jt. Observe that the vectors vY1, vY2, irl and iz2 in Fig. 4(b) are identical to the port voltage and current vectors defined in Fig. 4(a). Let Bv = 0 and Qi = 0 be written in the following partitioned form:
Vol. 296, No.
2,
August 1973
103
L. 0. Chua
104
Journal of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
and where Gj(vj), G:(ij*) and Gj($) d enote the co-content, and content, respectively.
conjugate
content
Theorem VII. The port current vector iyl and the port voltage vector vP2 are equal to the pseudo-gradients of the pseudo-hybrid content with respect to v9-, and i&, respectively; namely, . vp2 = a~l~~;pz(~~l, i&; vrsz, i&)lai&. lfl = a %192(vyl, i& 3 * vr2, %J~v~-,j (27) subvector vY2 and co-tree subvector i$, in Fig. 4(b) can be uniquely solved in terms of the port voltage vector vyl and the port current vector i$,, namely, vr2 = f9JvFl, i&) and it&= fp3(v,l, i&), then the m-port N in Fig. 4(a) is characterized explicitly by
If the internal tree-branch
.
vpee
=
a~~I~2evyI~
i&i
vyez,
i&)/ai$
_ I Ivy2-fys(vr,,
i>J,,, iips=fipsh.9-1, =
v&y,,
&I i&I.
(29)
this case, the non-complete hybrid m-port N is completely characterized by a potential function 2’?(vYI, i&), called the hybrid content of N, where
In
Vol. 296, No.
6
2, August
1973
105
L. 0. Chua proof: Equation
(27) can be proved by taking the gradient of
~&92(v.&z;
v9-,&)
with respect to vYI and is, and making use of (25)
: . 1.9-l - ~@&&_T-,,iA ; vr8, i&p,)/~vrI= i,,- Pv/av~l]~ P~d~vl = iFI -t B&IF1 bI + BkZr1b2+ B!Y~.F~~Lc~
=v 22- B,yl VT,- Jbez.~z v~ipa - %a~-~VT, = 0 92’ (34) To derive (28), (29), (31) and (32), we observe that a~~l~~(v~l,i&;
f~,(v~,,i&),f,,(v,,,i~,)) av,1
(35) and
(36) But a~&&%%
= [av/av,,it
[a%l,,/avi
= - Bklr2 ipl + ir2 - B$2r2 ipz - B&r2 ip3 = O,,
(37)
and asY,,,/ai&
= Wai<
[a*,z/aii
= VY, + B _Y3r1v3-1+B 9k.9-2vr.z+ B 93Y3vYz = 0 93’
(38)
Substituting (37) and (38) into (35) and (36), we obtain (28), (29), (31) and (32). Q.E.D. Corollary 3. The non-complete m-port N in Fig. 4(a) is characterized by a state function representing the gradient of a potential function-the
106
Journal of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
hybrid content of N, if, and only if, vY2 and iys associated with the extracted elements in J< and ,Ep3of Fig. 4(b) are single-valued functions of vJ1 and i,s. i*SS) of N corresponding to any given In this case, the hybrid content X(v,,, value of vY1 and it& is equal to the sum of the co-contents of all voltagecontrolled resistors in Z1 and 3, plus the sum of conjugate contents of all current-controlled resistors in .&, plus the sum of the contents of all currentcontrolled resistors in J<, plus a cross coupling term equal to
all evaluated at the corresponding operating point, plus an additive constant which is independent of the operating points. The implicit relationships among the four vectors v~,, vr2, ix and i& which must be used to solve for f,,( *) and fp3(- ), if they exist, can be obtained from (25) ; namely,
The necessary and sufficient condition that the hybrid m-port in Fig. 4(b) possesses a hybrid content is clearly that (39) and (40) be uniquely solvable for vY2 and i$, as a function of vY1 and is,. Here, the global implicit function theorem (19) may be invoked. If we replace the voltage and current sources in Fig. 4(a) by capacitors and inductors, respectively, we would obtain the same formulation used by Brayton and Moser (5). The only difference is in the reference current direction chosen in Fig. 4(a)-which is opposite to that adopted by Brayton and Moser. Consequently, the hybrid content can also be used to derive the normal form equations of the dynamic network and the following explicit criterion for determining when the mixed potential formulation is valid follows trivially. Theorem VIII. If we replace the voltage sources by voltage-controlled capacitors and current sources by current-controlled inductors in Fig. 4(b), then the necessary and sufficient condition that the normal form equations of a non-complete dynamic nonlinear network exist and are representable in terms of a mixed potential is precisely that (39) and (40) be uniquely solvable for vY2 and i$, in terms of vY1 and i&.
V. Simple n-ports
Circuit
Theoretic
Su.cient
Conditions
in Terms
Let us use the relationships i& = -i,, and i& = -iz3 (40) into the following more convenient form:
Vol. 296, No. 2, August
1973
of Complete
to rewrite (39) and
107
L. 0. Chua
J(vr2,iyaP,) =
J3-,kT,~ M
VLTJ
-Mt
J,$,s, &,I I ’
-I-JbQ-ps3 0w&T&~ + %,, bp,,lR&9-~~ M = K&9-2’
(44) (45)
Observe that both J,& vr2, vrI) and J9,(ipa, i22) are symmetric even though the Jacobian matrix J(v,,, ize,) is not. To avoid working with the relatively large and non-symmetric Jacobian matrix, we now derive a sufficient condition in terms of only the two smaller submatrices J~2(v~2, vFI) and J.dba~ i~2). But first we need the following definition: DeJinition 4. A 2-terminal resistor is said to be ultimately strongly passive if, and only if, its v-i curve eventually lies in the first and the third quadrants, and jil-tco as IvI-tco. Theorem IX. If the non-complete hybrid m-port N in Fig. 4(a) contains only ultimately strongly passive resistors and a consistent set of independent voltage and current sources, and if the two matrices J,&vr2, vg,) and ) are positive definite for all values of (Vet, vyl) and (ip8,i9J, then J&v i_cz;pz h(v,,, i ~3) is a one-to-one function of vrz and i,,, for any assumed value of vYI and i,,. Moreover, (39) and (40) can be solved uniquely for vrS and i$, in terms of vg, and is,. Proof:
We first show that J(vg2, i_& is positive definite by observing that x
[ Y I
t
Jtak3-2z’v.4 [
M
-Me
J&b,, i.& I
X
[ Y I
= xt Jr2(vr2, vrI) x - xt Mt y + yt Mx + yt J9Q(i-Lp3, i& y > 0,
(46)
for all x #O and y #0 since both JY2(vF2,vrI) and J9,(iz3, i& are positive definite and since ytMx = (Y~Mx)~ = xtMt y. TO show that h(vrZ, i& =h(z) is one-to-one, where zs (v,,,i&, suppose the contrary; i.e. suppose there exist two vectors z1 and z2 such that z1 # z2 and h(z,) = h(z,). Let
108
Journal of The Franklin
Institute
Stationary
and applying q(l) -q(O)
Principles
the mean-value
and Potential Functions for Nonlinear Networks theorem,
we obtain
=
[IddVW P -01 = ([aho(z+X(z,-z,))/azl k,-z,l>” (zz-zd
=
(zZ-~l)t[Jo(~+J;(~2-~l))]t(~2-~1),
k(O,l).
(4’)
Since J(z) is positive definite, (47) implies that g(l)-g(O)> 0. But since h(z,) = h(z,), we have g( 1) -g(O) = [h(z& - h(zl)lt (z2- zl) = 0, which is and the vector-valued function absurd. Hence, h(vYa, ia,) is one-to-one representing the left-hand sides of (39) and (40) is a one-to-one function of vrZ and i&, for any assumed value of vY1 and i&. This means that (39) and (40) can have at most one solution for each assumed value of vg, and i&. Now, since N contains only ultimately strongly passive resistors, we know (39) and (40) have at least one solution (20). Hence, given any value of vY1 and i&, (39) and (40) have one, and only one solution. Q.E.D. We remark that the first part of the hypotheses for Theorem IX can be relaxed somewhat by replacing it with any other less stringent sufficient condition which guarantees the existence of at least one solution (21). The second part of the hypotheses can be given a very simple circuit theoretic interpretation by constructing an auxiliary p-port N, as shown in Fig. 5(a) and an auxiliary q-port Np.as shown in Fig. 5(b). The p-port N, is obtained from the network in Fig. 4(b) by short-circuiting all elements in -Epr,< and Y2 and open-circuiting all elements in ,Eaz. The p = (y-p) ports are created by pliers-type entry through each resistor in .&. The q-port Nq is obtained by open-circuiting all elements in YZ, Pa and z and short-circuiting all elements in ~5. The q = (E-S) ports are created by soldering-iron-type entry across each resistor in &. If we define the port voltage vector
for NP and the port current vector iYZ = (i,,,, &, . . . . s,)t for N,, then the terminal equations for Np and and N, are easily seen to be given by
(49) We now make the important observation that if we set vY1 = Or1 in (43) and iYs = O,, in (44), then J~a(v~2, 0,) and J,&,8, O,,) are identical to the Jacobian matrix of (48) and (49), respectively.* Moreover, JrZ(vg2, Oyl) is simply the incremental conductance matrix of N, and JIps(i~~,O_rp) is simply the incremental resistance matrix of NP. Observe next that both NP and Ngare complete and hence NP is characterized by a content potential function 99(i) = 9(!_2 and N, in Fig. 5(b) is characterized by a co-content potential function CC?(V) = g(vY,). Now since the Jacobian matrices Jy3(i93,0,,) and * Observe that since vrI and iz2 are independent of vs2 and i2a in Eq. (41), the positive definiteness of the Jacobian matrix J(vr2, i.& does not depend on vrI and i,,. Consequently, we can assume vrI = i,, = 0 for testing purposes.
Vol.
296, No.
2, August
1973
109
current-controlled
p-po!t
Np(p=
7-
8)
[b).
“Y Auxiliary
1:
voltage-COntrOlW
.
. . . +
q-port
:
+ “c+2
+
Nq tq=a-8)
in
resistors
FIG. 5. Two auxiliary complete p- and p-ports N, end N, obtained from the non-complete m-ports in Fig. 4. For typographical reasons, all symbols with a hat in this drawing have been snbstitut,ed by symbols with & bar in the text.
((11. Auxiliary
i.
+ “r4
“ai2
“#++I
“a
l
. :iB=O
“ai2
i.+2=0 +
Stationary Principles
and Potential Functions for Nonlinear Networks
J9-,@.92O,,)
are the Hessian of ‘2&J and $3(v,,), respectively, the positivedefinite hypothesis in Theorem IX implies that both N, and Nz are strictly convex (22).
N
P+qT
+
1’1-
i.. “2
“2
. . “a
P=Y-P a=r-S
. i, 1
+
il
“a
i2 . . .
i
.+I
ia
:-.~
I
=o
voltage-controlled tree resistors in CJ2
+
“.+I
open 011
I4 .+2=0 L
‘B+l
1
1
currentcontrolled co-tr~e
t-5 +
Vr _I
(curr;;nt
c+I
1
controlled tree
FIG. 6. The auxiliary complete hybrid (p + p)-port IV,,, obtained from the non-complete m-port in Fig. 4. For typographical reasons, all symbols with a hat in this drawing have been substituted by symbols with a bar in the text.
Observe next that instead of testing two auxiliary p- and q-ports, we can construct one auxiliary hybrid (p + q)-port A$,+, as shown in Fig. 6. This (p + q)-port is obtained from Fig. 4(b) by open-circuiting all current sources in 3?‘, and short-circuiting all voltage-sources in q. The current-driven ports and the voltage-driven ports are created as in Figs. 5(a) and (b). Observe that this (p + q)-port is complete and hence N,,, is completely characterized
Vol.
296,
No.
2, August
1973
111
L. 0. Chua by a hybrid content potential function &(va, iz) = Z(vy2;, is3). The terminal equations for N,,, are easily seen to be given by I,, = l.&Ya)
- R&9-* i& - R&Y* i,I o ( - B,,,
vYa)’
Y.v~= v_~,(i_& f I&Y, van + I&37.3vr-,o (B$&&,
(50) (51)
where I, = (i,,,, fsf2, . . . . i,)t and YPs = (Cl+,, G~+~,. . . . $)t. As before, if we set vY1 = O,, and ip2 = O,, in (43) and (44), we found the resulting Jacobian matrix J(vYZ, ig8) in (42) is exactly the Jacobian matrix associated with (50) and (51). We now summarize the above observations as follows: Theorem X. The incremental hybrid matrix of the auxiliary complete hybrid (1, + q)-port N,,, is positive definite if both the incremental resistance matrix of the complete current-controlled p-port NP and the incremental conductance matrix of the complete voltage-controlled q-port NQare positive definite. Under this condition, the content of NP, and the co-content of Np are strictly convex. If the non-complete hybrid m-port N in Fig. 4 contains only ultimately strongly passive resistors and a consistent set of independent voltage and current sources, and if the two Jacobian matrices Jyl(vgB, vg,) and J,3(i,3, i_& as defined in (43) and (44) are positive definite for all values of (vgB,vY1) and (i93,i,,), then (39) and (40) can be solved uniquely for vYs and iw in terms of vg, and i&. Moreover, the normal form equations of the non-complete dynamic network constructed by replacing the voltage sources in Fig. 4(a) by voltage-controlled capacitors and current sources in Fig. 4(a) by current-controlled inductors exist and are representable in terms of a mixed potential function as presented in Brayton and Moser. To illustrate the application of the preceding circuit theoretic interpretations, consider again the non-complete S-port shown in Fig. 2(b). Recasting this circuit into the framework of Fig. 4(b), we immediately identified the two current sources as &, the current-controlled nonlinear resistor as s3 and the two linear resistors as ~5. Observe that for this simple network, ~5, 4 and Z1 are empty sets. To test whether this 2-port admits a content potential function, we identified the auxiliary p-port (p = 1 in this case) as consisting of the nonlinear resistor in series with the two S-ohm linear resistors. The second auxiliary q-port is irrelevant in this case since q = 0. The resulting l-port N, is characterized by V = (i3-%)+4i+4i
= ia.
Since this characteristic represents a one-to-one function, we arrive at the same conclusion as in the example of Sect. III that the 2-port shown in Fig. 2(b) has a well defined content potential function. Notice, however, that in contrast to the topological formulation shown in Fig. 2(c), only straightforward circuit theoretic approach is invoked here. Observe also that the incremental resistance of N, is positive semi-definite but not positive definite in this case. Hence, the positive definite hypothesis of Theorem X is not a necessary condition for the existence of a hybrid content potential function.
112
Journal
of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
It can be shown, however, that this hypothesis is not too conservative fact is about as best as could be stated for a general m-port.
VI.
Concluding
and in
Remarks
The content, co-content and hybrid content for complete nonlinear m-ports have been generalized to pseudo-content, pseudo co-content and pseudo hybrid content of non-complete m-ports. The precise criterion for which these pseudo-potential functions can be reduced to a legitimate potential function is shown to be the unique solvability of a set of implicit equations given explicitly in (18), (24), (39) and (40). Simple circuit theoretic sufficient conditions for unique solvability of the non-complete hybrid m-port are given in terms of the positive definiteness of the incremental resistance matrix of a complete auxiliary p-port and the incremental conductance matrix of a complete auxiliary q-port. Moreover, since a dynamic network containing no “capacitor-voltage source loops” or “inductor-current source cut sets” can always be viewed as an m-port-generally non-complete-with the capacitors and inductors connected across its ports, it follows that the same conditions which guarantee the existence of a hybrid content potential function also apply to the existence of the normal form equations of the associated dynamic network.
References (1) J. C. Maxwell, “A Treatise on Electricity and Magnetism”, Vol. 1, 3rd ed., pp. 407-408, Clarendon Press, Oxford, 1891. (2) W. Millar, “Some general theorems for nonlinear systems possessing resistance”, Phil. Mug., Ser. 7, Vol. 42, pp. 1150-1160, Oct. 1951. (3) C. Cherry, “Some general theorems for nonlinear systems possessing reactance”, Phil. Mug., Ser. 7, Vol. 42, pp. 1161-1177, Oct. 1951. (4) A. R. Oliver, “Energy methods in electrical networks”, Proc. IEE, Part B, Vol. 109, pp. 537-538, Nov. 1962. (5) R. K. Brayton and J. K. Moser, “A theory of nonlinear networks-I and II”, Quart. A&. Math. Vol. 22, pp. l-33, April 1964, and pp. 81-104, July 1964. (6) T. E. Stern, “Theory of Nonlinear Networks and Systems”, Addison-Wesley, Reading, Mass., 1965. (7) J. 0. Flower and F. J. Evans, “Derivation of certain nonlinear circuit equations”, Electronics Letts, Vol. 3, p. 265, June 1967. (8) J. 0. Flower, “The topology of the mixed potential function”, Proc. IEEE (Letts), Vol. 56, pp. 1721-1722, Oct. 1968. (9) J. 0. Flower, “The existence of the mixed potential function”, Proc. IEEE (Letts), Vol. 56, pp. 1735-1736, Oct. 1968. (10) A. G. J. MacFarlane, “Dual-system methods in dynamical analysis, Part IVariational principles and their applications to nonlinear network theory”, Proc. IEE, Vol. 116, pp. 1453-1457, Aug. 1969. (11) D. L. Jones and F. J. Evans, “Variational principle for nonconservative networks”, Electronic Let&, Vol. 6, pp. 163-164, March 1970. (12) T. M. Apostol, “Mathematical Analysis”, Addison-Wesley, Reading, Mass., 1957.
Vol.
296, No.
2, August
1973
113
CHUA
Department of Electrical Engineering and Computer Sciences University of California, Berkeley This
ABSTRACT:
stationary content,
paper
principles co-content
nonlinear
and Moser
on potential
networks
functions
for
functions
solvability
complete
topological
matrices. Simple
to a legitimate set of equations
current-controlled
complete
voltage-controlled
dynamic
nonlinear
equations
in terms of a single mixed potential
q-port.
of total
to that of total parametric
content
criterion
for
to non-complete
conditions
conditions resistance
is shown in terms
to be the
of standard
are given which require matrix
conductance also guarantee
by a system
paeudo-
which each of these
function
given explicitly
au.cient
can be represented
The results by Brayton
are generalized
potential
and the incremental These
various
namely, the pseudo-content,
the incremental
complete
network
p-port
of
deriving concepts
elements.
The precise
circuit theoretic
the positive-de$niiteneaa
for The
m-ports
functions;
approach networks.
multivalued
content.
reduces
of an implicit
theoretic
nonlinear
are generalized
containing
and the pseudo-hybrid
pseudo-potential
merely
content
graph type
in terms of three pseudopotential
co-content unique
a novel
single-element
and hybrid
for resistive m-ports
presents
for
matrix
of an
auxiliary
of an auxiliary
that a non-complete
of normal form
differential
function.
I. Introduction
It is well known that the solution of various classes of nonlinear networks is identical to the stationary points of a suitably defined scalar “potential function” (l-4) of a complete set of network variab1es.t A comprehensive theory of nonlinear networks based on potential functions was presented by Brayton and Moser in their 1964 paper (5). Some interesting related results were also given by Stern (6). More recently, the properties of network potential functions have been explored by several authors (7-11). Stationary principles may be derived by an application of the properties of line integrals of state functions (12) or by variational techniques (13). The main objective in this paper is to unify and generalize the existing results in this area using a surprisingly simple but novel approach. Only Kirchhoff’s laws and the vector form of the chain rule for taking the gradient and the Jacobian of composite functions are used in the proofs. * This work was supported in part by the National Science Foundation under Grant GK-32236 and the United States Naval Electronic Systems Command under Contract N00039-71-C-0255. t A set of variables i,, i,, . . . , i,, v,+,, v,+, . . . , vda is “complete” if they can be chosen independently without leading to a violation of Kirchhoff’s laws and if they determined in each branch at least one of the two variables, the current or the voltage (5).
91
L. 0. Chua Only networks containing two-terminal uncoupled elements are considered in this paper. Most of the results will be stated only for resistive networks even though they are generally applicable to single-element type networks as well. Section II contains a new formulation of several well-known stationary principles and a generalization in terms of the total parametric content of networks containing multivalued elements. Section III introduces the concept of the pseudo-content and pseudo-co-content for non-complete m-ports containing only current-controlled resistors, or voltage-controlled resistors, respectively. This concept is generalized in Sect. IV to that of a pseudo-hybrid content for noncomplete m-ports containing both currentcontrolled and voltage-controlled resistors. The precise criterion for which the pseudo-hybrid content can be reduced to a legitimate hybrid content is also presented in this section. Among other things, this criterion provides an answer to the heretofore unresolved question as to when does the mixed potential of Brayton and Moser (5) exist for non-complete dynamic networks. More convenient circuit theoretic sufficient conditions for satisfying this important criterion are derived in Sect. V. II.
Stationary
Principles
of Nonlinear
Networks
Let NR be a resistive network containing n nodes and b branches. Let Y be a tree of NR and 9 its associated co-tree.* Let us label the links from 1 to 1, and the tree branches from I+ 1 to b. Let i, i, and i, denote the branch current vector, the co-tree current vector and the tree current vector, respectively. Let v, v, and vF denote the branch voltage vector, the co-tree voltage vector and the tree voltage vector, respectively. The two Kirchhoff Laws can be written as Bv = 0 and Qi = 0, where B and Q are the fundamental loop matrix and the fundamental cut set matrix, respectively (14). (A) Principle
of stationary content (co-content)
Let iVR contain only current-controlled (voltage-controlled) resistors and independent voltage (current) sources so that the content G,($) = j$vi(ij) di, (co-content Gj(vj)~J~ij(vj) dvi) of each element is well defined. Let and define the total content $(i9) G(i) 3 XF==,Gj(j(if) (G(v) 3 &Gj(vj)) (total co-content ‘3(vY)) by g(i,)
= G(i) liCBtia=
GO (BtiY),
where “0” denotes the composition
~(v,)=G(v)Iv=ntvY
= Go(QtvY),
(1)
operation.
Theorem I (Millar’s theorem on stationary content (2, 4) ). If NR contains only current-controlled resistors and independent-voltage sources, then a vector i, = I, is a solution of NR if, and only if, it is a stationary point of the total content 3(iZ). * We assume throughout this paper that the network NE is connected. The term tree always means a spanning tree; i.e. it connects all nodes. Any subset of branches in a spanning tree is called a subtree in this paper.
92
Journal of The Franklin Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
Proof: If we take the gradient of $9(iu) and make use of the vector form of the chain rule (12), we obtain Z?‘(i,)/ai,
= ZY~ (Bti,)/ai,
= B 83(i)/ai lizsti9= Bvo (Btiy).
(2)
We now make the important observation that if the last expression in (2) is equated to zero, we would obtain the system of independent loop equations which completely determined the solution of NR. Consequently, any stationary point of 9(i9) is a solution of NR, and vice versa. Q.E.D. Using a dual proof, we obtain Theorem II. Theorem II (Millar’s theorem on stationary co-content (2,4)). If NR contains only voltage-controlled resistors and independent current sources, then a vector vF = E, is a solution of NR if, and only if, it is a stationary point of the total co-content 9(vg). (B) Principle
of stationary hybrid content
In order to derive a stationary principle that would allow both currentcontrolled and voltage-controlled elements, we follow the approach taken by Brayton and Moser (5). Let 5c.F be any subtree (which need not be connected) made up exclusively of voltage-controlled resistors and let 4, the remaining tree branches, be made up of current-controlled resistors, where q u& = F. Let P1 be all those voltage-controlled links which formed closed loops with elements of K only and let 9a be the remaining links so that 9ruZa = 9. Hence, all elements in .& are voltage-controlled and all elements in 9a are current-controlled. If we let vY1, vPa, vg, and vg, and iyJ denote the branch voltage (current) vectors associated (iF1, iZS, i,, with elements in dq, gZ, ~9 and %, respectively, then Bv = 0 and Qi = 0
=[I
r
0
0
(3)
’
L
where the symbols 1 and 0 denote the unit and null matrix respectively. Observe that OFIZ is a null matrix because each element in DLplforms a fundamental loop with only elements in <. Using the above notations and letting i&z -iya, we define the total hybrid content X(vS1, i&) associated with the network NR to be the scalar function:
Vol.296,
No.
2,
August 1973
93
sum of co-contents of elements in g1 1’
1
sum of conjugate
*
contents
( of elements in L$
I
2
sum of co-contents 1of elements in & I
%-,h-J = j~a%(ij) =
sum of contents of * elements in &
Theorem III (Principle of stationary hybrid content). A vector (EF1,IZZ) is a solution of the network NR described in the preceding paragraph if, and only if, (EF1, I&) is a stationary point of the total hybrid content X(vF1, i&), where 1%a = -1 -% Proof: Taking the gradient of X(vY1, i&) with respect to vF, and is2 respectively, we obtain
Now, an examination of (3), (5) and (6) sh ows that if we equate the righthand side of (5) and (6) to zero, we would obtain the fundamental cut-set equations corresponding to the fundamental cut-sets formed by the tree branches in q, and the fundamental loop equations corresponding to the fundamental loops formed by the links in LX2.Since these mixed systems of equations completely determined the solution of NR, it follows that (EY1, I& is a solution of NR if, and only if, (EF1, I$,) is a stationary point of the total hybrid content. Q.E.D. We remark that the total hybrid content defined in (4) is related but not identical to the mixed potential function of Brayton and Moser (5) and the dissipation function of Stern (6). We also remark that Theorem III remains valid if the total hybrid content is replaced by several related hybrid potential functions defined in (15, 16). (C) Princigde of stationary parametric content We now generalize the preceding stationary principles to allow all elements in NR to be parametrizable.? In particular, let each resistor be characterized * We define the conjugate content G,*(ij*) of a current-controlled resistor to be Observe t&t dGt(i,*)/dif = w,(iJ. G:(ij*) s j$’ wj( - ij) di,, where ij* 3 -i,. t A curve I7 representing the locus of some relationship f(~, y) = 0 is said to be perametrizsble if, and only if, I? can be described by two continuous functions of a third parameter p; namely, z = z(p) and y = y(p), p E [a, b] c (-co, CO). An element characterized by a parametrizable curve is said to be a parametrizable element.
94
Journal of The Franklin
Institute
Principles
Stationary
and Potential Functions for Nonlinear Networks
by vi = vi(pj) and ii = ii( The content Gi(pj) and co-content I are defined respectively by the Stieltjes integrals (17) Gi(pj) = J”@Vj(pi)dii(pi) and cj(pi) e J$ij(pi) dvj(pi). In terms of the parametric vectors Pi = (plP pz, . . . . P~++#
and
Pi
= (P~-~+~,
P~-~+~,
...) ~2
the equations of motion for NR are given by
1
b4P.9)
VdP.Pe) _.._____________ = 0, [-B$l,]
[l, /B,l
[ V94P.T)
_-_-_________-= 0.
[
i.54P.F) 1
Using the above notation, we define the total parametric associated with NR to be the scalar function: ~(PJZ, P.F) = I
content g(pp,
- ~(P.F) + i&(pJ B, vr(ps),
(7) pF)
(8)
where
Theorem IV (Principle of stationary parametric content). Every solution (pu, pY) of NR is a stationary point of the total parametric content Y(ppe, ps). Conversely, every stationary point (ir,, pF) of B(p,, ps) is a solution of NR provided (1) i$j)#O,j=
1,2 ,..., b-n+1
and (2) w(i(&)#O,j
= b-n+2,b-n+3
,..., b.
Proof: Let us take the gradient of B(p,, pF) with respect to pu and pY and make use of the relationships Gi(pi) = vj(pi) i;(pi) and am = ii v(i(pi) for Stieltjes integrals (12) to obtain @YP2?
Ps)/aPY
= wz4Pa)/~P~l
Wp9p)
P.&/~P~ = II-- ~vAP.GP.A
[V9(PY) L-(PA
+B.T
V,(P.T_)l~ -%-4dp.d.
(9)
(10)
Observe that if we equate the right-hand expressions (within the bracket) of (9) and (10) to zero, we would obtain (7). Hence, any solution of NR must be a stationary point of the total parametric content. The converse is also true provided conditions (1) and (2) are satisfied. Q.E.D. We remark that the total parametric content defined in (8) can be transformed with the help of the integration-by-parts formula for Stieltjes integrals (12) into the following equivalent forms:
~(P_D er) = I + ‘%P.T) %P_B ~~1 = - C~(P~) + PI
+ Wp~) By - G-(p~)l VAP.FL +C.dvAp~;P) +B~v.d~~)l.
(11)
(12)
Observe that if we let the parametric vectors pg and pF equal i, and i, for current-controlled resistors, or v* and vF for voltage-controlled resistors, then it is easily shown that (11) and (12) reduce respectively to 5!?‘(ip) and g(vY) as defined in (l), as they should.
Vol. 296, No. 2, August
1973
95
L. 0. Chua ZZZ. Pseudo-content
and Pseudo-co-content
of Non-complete
m-ports
An m-port is said to be complete if its port variables constitute a complete set (5). It is well known that the equations of motion of a dynamic nonlinear network NRLc can always be written in the normal form if the energy storage elements of NRLc can be extracted across a complete m-port. In this case, the state equations can be represented in a compact form in terms of a suitably defined potential function (5). Our objective in the following sections is to generalize this concept to the case where the extracted m-port N is not complete. When an m-port is not complete, it may not be possible to characterize it by a single-valued function of the independent port variables representing the sources connected across the ports. In such cases, it is meaningless to talk about a potential function. For example, consider a one-port made up of two tunnel diodes in series. It is well known that for some combination of tunnel diodes, the driving-point v -i characteristic of this non-complete one-port is a disconnected and multivalued function of both v and i (17). Since the v-i curve is not parametrizable, this one-port does not possess even a parametric content. The main objective in this paper is to show that a non-complete m-port can always be characterized by a scalar function X(x, ; x,)-henceforth called a pseudo-potential-of a port vector x1 representing the driving internal sources, and an auxiliary vector x2 representing appropriate variables.* Moreover, the port vector y1 associated with xl is simply the pseudo-gradient of S( x1 ; x2) obtained by taking partial derivatives with respect to x1; namely, Yl =
8S(x,;
x2)
-y1@1;
x2).
ax, Hence, the necessary and sufficient condition that the m-port be characterizable by a single-valued function of x1 is that it be possible to solve for x2 as a function of x,; namely, x2 = f(x,). Hence, in this case, we can write Yl = Yl(Xli f(x1)). Moreover, under this condition, reduces to a legitimate potential Wx,) y1=,,1=
we will prove that the pseudo-potential function P(xl) = X(x1; f(x,)) such that
8S(x,: f(x,)) = aS(x,;
x2)
xz=f(x1) ax, This equation implies that taking the derivative of X(x, ; x2) after substituting the relationship x2 = f(x,) is the same as substituting this relationship after taking the derivative. This property is remarkable because, in general, the derivative operation and the composition operation do not commute. Let the m-port N shown in Fig. l(a) be driven by m current sources which do not form a cut-set. We assume that there are b -m elements inside N ax,
* Although 8(x, ; x2) is a function of both x1 and x2, we have chosen the unconventional notation of a semi-colon as the delimiter here, rather than a comma, to emphasize that the elements of x2 are not port variables.
96
Journal of The Franklin Irwtitute
i2t
(!I
.
.
.
-
v2
+
+ iit q +I
-
(a)
sources
voltage
?-
$
i2
.
.
.
+
-“2
i-fm
I--im0 t -"m
i2
I
iI
-“I
CI+
i, t
(b).
of N)
branches
tree
011
(containing
NL7
-3
‘m+2
ib-n+l
.. .
“m+l
+
i m+l
in
co-tree
A
1
a!2 forming the remaining port of the
of
set
link resistors
(the
m-port N in (a) is redrawn in (b) by extracting all co-tree resistors inside N.
irn
:.
i2
iI
-“m
: .
-“2
independent
constont
1
.I
co- tree d
-“I
1
set of
current sources in Al forming port of the
(the
and
resistors
controlled
current
(containing
N
FIG. 1. The non-complete
i2
il
L. 0.Chua consisting exclusively of current-controlled resistors and independent constant-voltage sources which do not form closed-loops. If n is the total number of external and internal nodes, then a tree T of this network will contain (n- 1) branches and the corresponding co-tree _Y will contain b -n + 1 branches. We put the set P1 of m current sources in the co-tree and let the remaining set 2Z2of (b-n+ 1) -m link resistors be extracted, as shown in Fig. l(b). Let -VP1 = (-q, V& = (%+I,
-v‘& . ..) -?l,p, %+2,
*. ., %,+Jt
and v.9- =
b4~-_n+2>
vb-,+3,
* * * 9 vb)t
denote the branch voltage vector of elements in 2Z1,2T2and T, respectively.* Similarly, let i,, = (i1, i,, . . . , fJt, . ly2 = @m+l,&+2, ***, &+$, and iT = &.,-_n+$,&-_n+3,. . . , &? denote the branch current vector of elements in gl, AC2and T, respectively. Finally, let v = ( - vY1, vT2, v~)~ and i = (i91,i92,i9)t denote the branch voltage vector and the branch current vector of N, respectively. Let us write the fundamental loop equations Bv = 0 in the following partitioned form :
DeJinitim 1. We define the pseudo-content ‘3z1(iFp1;i& complete m-port N in Fig. l(a) to be the scalar function
9ze,(i,,;
i&
of the non-
= G(i) li=Btig= Go
(14)
where G(i) is the sum of the contents of all elements inside N.+ * The reason for choosing the variable - vy, is due to the fact that each port current ii in Fig. 1 is defined leaving the positive terminal of the associated current source (i.e. entering port j). This is opposite to the standard reference current direction required by the topological relationships used in this paper. t Observe that G,(i,) = 0 for j = 1,2, . . . . m because the content of a current source is zero (16). Hence, G( .) is really only a function of i2, and i,. However, to avoid introducing additional symbols, it is more convenient to consider G( 0) as a function of i. A dual remark applies to the variable v in Definition 2.
98
Journal
of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
Theorem V. The port voltage vector v4p1of the non-complete m-port N in Fig. l(a) is equal to the pseudo-gradient of the pseudo-content $Y.&,1; i9J of N; namely, (15)
VP, = @JP1(i9,; i_&ai,l.
If the internal link-current vector i,Z in Fig. l(b) can be uniquely solved in terms of the port current vector iPl; namely, i9, = f(ipl), where f( *) is a single-valued function, then the m-port N in Fig. l(a) is current-controlled and is characterized explicitly by v9p1= a3P&1;
i9J/G,
liP~=z=f(iP ) = b&J.
In this case, the non-complete m-port N is completely characterized potential function 99(ilp1),called the content of N, where %,J
and
vZ, = @G9,)/&rl
(16) by a
= Gpl(iPpI; f(i&) = ag91(i91; f(i_&/&,.
(17)
Proof: The proof of this theorem is a special case of the proof of Theorem VII given in Sect. IV, and will not be given here due to lack of space. A complete proof can be found in Reference (16). Corollary 1. The non-complete m-port N in Fig. l(a) is characterized by a state function (representing the gradient of a potential function-the content of N), if and only if, iT2 is a single-valued function of iT1. In this case, the content S(i,l) of N corresponding to any given value of iP1 is equal to the sum of the contents of the elements inside N, evaluated at the corresponding operating point, and an additive constant which is independent of the operating points. Proof: The sufficiency part of this corollary follows immediately from Theorem 4 and the properties of state functions (12). To prove the necessity, we observe that if f( .) is a multivalued function of i91, then the pseudocontent defined in (14) will also be a multivalued function of iP1, and therefore, cannot reduce to a single-valued potential function. Q.E.D. We remark that since an m-port containing only two-terminal currentcontrolled elements is always reciprocal, the condition that i,, be a singlevalued function of iP1 is a necessary and sufficient condition only for the existence of a legitimate potential function, and is certainly not necessary for reciprocity (18). The following implicit relationship between iP1 and iPZ can be used to determine whether iZZ is a single-valued function of ipI: v~,(i~,) + Jb2.+w
(Bb2.&,
+ B&&J
= h,.
(18)
Observe that in the special case where N is complete, then the conclusion of Corollary 1 is always true. Observe also that Millar’s theorem on the superposition of contents for one-ports is a very special case of Corollary 1 (2). Example. A non-complete 2-port is shown in Fig. 2(a) and its associated 3-port is shown in Fig. 2(b) with the nonlinear link resistor extracted across
Vol. 296, No. 2, August 1973
99
L.
0.Chua
[1 “4
“5
[I i4 i5
L& = “3
v-i
curve of
R: v=i3-8i
J
v3-i3
curve:
&=
i3
v3=i:-8i3
lb).
(a L
FIG. 2. Example
i
of a non-complete S-port and some pertinent topological and relationships associated with its network graph.
matrices
port 3. The various topological matrices and relationships which are required in order to apply Theorem 1 and its corollaries are shown in Fig. 2(c). The total content is given by G(i) = iii - 4iZJ+ 2ii + 2iE. The pseudo-content defined by (14) is given by ?&,
*3 i2.2) = 2i,2+ 4i, i, + 4ig - 8(i2 + *ii) i, + *ii.
In order to determine whether this 2-port is characterizable by a content or not, we must solve for i, in terms of i, and i,. This is determined by writing down the implicit Eq. (18) : v9,(i& + B,, gvg o (B&, i,, + Bt,,
iyp,)= ii - Si, - 4i, = 0.
solving for i,, we obtain i, = f(&, iz) = 2(i, + &,)*. Hence, in view of Theorem 4, the content for this 2-port exists, and is given by (17), namely, 9(i,l)
= C?I(i,,is) = 991(il, i;; 2(i, + *i#)
= 2if + 4&i, + 4ig - lZ(i, + *i#.
We can recover the state function by using either (16) and CYP1(iYl;iyE) or (17) and 3(iPp,). In either case, we obtain w1= 4i, + 4i, - 8(i, + &i,)* and w2= 4i,+ 8i,- 16(i, + iiJ+. Observe that (16) and (17) imply that taking the derivative of ‘YPI(i,,, . iyz) after substituting the relationship i9, = f(i&
100
Journal of The Franklin
Institute
Xtationary Principles
and Potential Functions for Nonlinear Networks
is the same as substituting this relationship after taking the derivative. In other words, Eqs. (16) and (17) reveal the remarkable commutative property alluded earlier between the differentiation operation and the composition operation. We close this section by considering briefly the dual case of an m-port driven by m voltage sources which do not form a closed loop, as shown in Fig. 3(a). We assume that there are b-m elements inside N consisting exclusively of voltage-controlled resistors and independent constant current sources which do not form cut sets. Let us put the set ~9 of m voltage sources in a tree Y of N and let the remaining set s of (n - 1 - m) tree resistors be extracted as shown in Fig. 3(b), where n is the total number of nodes. We number the tree branches first and adopt the obvious dual notations to write Qi = 0 in the following partitioned form:
DeJinition 2. We define the pseudo-co-content gY,(vg,; complete m-port N in Fig. 3(a) to be the scalar function
vY.J of the non-
(20) where o(v) is the sum of the co-contents
of all elements inside N.
Theorem VI. The port current vector irl of the non-complete m-port N in Fig. 3(a) is equal to the pseudo-gradient of the pseudo-co-content gY1(vY1; vg,) of N; namely, . rrl = a%l(vfl$ v.9@vsl. (21) If the internal tree-branch voltage vector vg, in Fig. 3(b) can be uniquely solved in terms of the port voltage vector vg-,; then the m-port N in Fig. 3(a) is voltage-controlled and is characterized explicitly by 19-1= @&,l;
vg_,)Pv& l”&=f(“~-ll =i9+&
In this case, the non-complete m-port N is completely characterized potential function $?(v,,), called the co-content of N, where %.%-J = Y&,l;
(22) by a
C-S,))
and vy, = @(v.&/&1
= @.&sl;
f(vs,))Pv,,*
(23)
Corollary 2. The non-complete m-port N in Fig. 3(a) is characterized by a state function (representing the gradient of a potential function-the
Vol.
296, No.
2, August
1973
101
4n
i! -
Vm-+
(a).
sources
current
constant
1
independent
and
resistors
controlled
voltage -
(containing
N
FIG. 3. The non-complete
a
.
.
.
l-----l
m-port
V
N in (a) is redrawn
(the set of voltage sources in 3, forming part of the tree .T)
(b). in (b) by extracting
-im
of N)
I inks
011
all tree
resistors
in-1
inside
N.
tree resistors in g2 forming the remaining part of the tree l7 1
(the set of
Stationary Principles
and Potential Functions for Nonlinear
Networks
co-content of N), if, and only if, vY2 is a single-valued function of vY1. In this case, the co-content B(v,,) of N corresponding to any given value of vY1 is equal to the sum of the co-contents of the elements inside N, evaluated at the corresponding operating point, and an additive constant which is independent of the operating points. The dual implicit relationship between vYz and vY1 is now given by
IV.
Pseudo-hybrid
Content
of Non-complete
m-Ports
Now consider the more general hybrid m-port N shown in Fig. 4(a) driven by m, voltage sources which do not form closed loops and m2 current sources which do not form cut-sets, where m = m, +m,. Let us choose a particular tree Y by first including in it the set & of all m, port voltage sources, and let the set 2Z2of all m2 port current sources be included as part of the co-tree .2? associated with 7. Next we add as many voltage-controlled resistors as tree branches, forming another sub-tree &. The remaining subset -IL”1of voltage-controlled resistors which cannot be included in the tree must form closed loops with branches in K u &, and must, therefore, be placed in the co-tree 9. Let us next fill up the tree with a sub-tree .99 chosen from the remaining elements, which are current-controlled, such that 9 = q u J< u ~9. Similarly, the co-tree 9 = g1 u_EPZU~~,where the subset ,Ep3would include the remaining unlocated elements, which are all current-controlled. Let us label the elements as shown in Fig. 4(b) and define the branch voltage vector v = (v&p - v&*3v& v&, v&, vtY,)t and the branch current vector
i = (i$,, i&, if,, - i$,, itr2,i$Jt. Observe that the vectors vY1, vY2, irl and iz2 in Fig. 4(b) are identical to the port voltage and current vectors defined in Fig. 4(a). Let Bv = 0 and Qi = 0 be written in the following partitioned form:
Vol. 296, No.
2,
August 1973
103
L. 0. Chua
104
Journal of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
and where Gj(vj), G:(ij*) and Gj($) d enote the co-content, and content, respectively.
conjugate
content
Theorem VII. The port current vector iyl and the port voltage vector vP2 are equal to the pseudo-gradients of the pseudo-hybrid content with respect to v9-, and i&, respectively; namely, . vp2 = a~l~~;pz(~~l, i&; vrsz, i&)lai&. lfl = a %192(vyl, i& 3 * vr2, %J~v~-,j (27) subvector vY2 and co-tree subvector i$, in Fig. 4(b) can be uniquely solved in terms of the port voltage vector vyl and the port current vector i$,, namely, vr2 = f9JvFl, i&) and it&= fp3(v,l, i&), then the m-port N in Fig. 4(a) is characterized explicitly by
If the internal tree-branch
.
vpee
=
a~~I~2evyI~
i&i
vyez,
i&)/ai$
_ I Ivy2-fys(vr,,
i>J,,, iips=fipsh.9-1, =
v&y,,
&I i&I.
(29)
this case, the non-complete hybrid m-port N is completely characterized by a potential function 2’?(vYI, i&), called the hybrid content of N, where
In
Vol. 296, No.
6
2, August
1973
105
L. 0. Chua proof: Equation
(27) can be proved by taking the gradient of
~&92(v.&z;
v9-,&)
with respect to vYI and is, and making use of (25)
: . 1.9-l - ~@&&_T-,,iA ; vr8, i&p,)/~vrI= i,,- Pv/av~l]~ P~d~vl = iFI -t B&IF1 bI + BkZr1b2+ B!Y~.F~~Lc~
=v 22- B,yl VT,- Jbez.~z v~ipa - %a~-~VT, = 0 92’ (34) To derive (28), (29), (31) and (32), we observe that a~~l~~(v~l,i&;
f~,(v~,,i&),f,,(v,,,i~,)) av,1
(35) and
(36) But a~&&%%
= [av/av,,it
[a%l,,/avi
= - Bklr2 ipl + ir2 - B$2r2 ipz - B&r2 ip3 = O,,
(37)
and asY,,,/ai&
= Wai<
[a*,z/aii
= VY, + B _Y3r1v3-1+B 9k.9-2vr.z+ B 93Y3vYz = 0 93’
(38)
Substituting (37) and (38) into (35) and (36), we obtain (28), (29), (31) and (32). Q.E.D. Corollary 3. The non-complete m-port N in Fig. 4(a) is characterized by a state function representing the gradient of a potential function-the
106
Journal of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
hybrid content of N, if, and only if, vY2 and iys associated with the extracted elements in J< and ,Ep3of Fig. 4(b) are single-valued functions of vJ1 and i,s. i*SS) of N corresponding to any given In this case, the hybrid content X(v,,, value of vY1 and it& is equal to the sum of the co-contents of all voltagecontrolled resistors in Z1 and 3, plus the sum of conjugate contents of all current-controlled resistors in .&, plus the sum of the contents of all currentcontrolled resistors in J<, plus a cross coupling term equal to
all evaluated at the corresponding operating point, plus an additive constant which is independent of the operating points. The implicit relationships among the four vectors v~,, vr2, ix and i& which must be used to solve for f,,( *) and fp3(- ), if they exist, can be obtained from (25) ; namely,
The necessary and sufficient condition that the hybrid m-port in Fig. 4(b) possesses a hybrid content is clearly that (39) and (40) be uniquely solvable for vY2 and i$, as a function of vY1 and is,. Here, the global implicit function theorem (19) may be invoked. If we replace the voltage and current sources in Fig. 4(a) by capacitors and inductors, respectively, we would obtain the same formulation used by Brayton and Moser (5). The only difference is in the reference current direction chosen in Fig. 4(a)-which is opposite to that adopted by Brayton and Moser. Consequently, the hybrid content can also be used to derive the normal form equations of the dynamic network and the following explicit criterion for determining when the mixed potential formulation is valid follows trivially. Theorem VIII. If we replace the voltage sources by voltage-controlled capacitors and current sources by current-controlled inductors in Fig. 4(b), then the necessary and sufficient condition that the normal form equations of a non-complete dynamic nonlinear network exist and are representable in terms of a mixed potential is precisely that (39) and (40) be uniquely solvable for vY2 and i$, in terms of vY1 and i&.
V. Simple n-ports
Circuit
Theoretic
Su.cient
Conditions
in Terms
Let us use the relationships i& = -i,, and i& = -iz3 (40) into the following more convenient form:
Vol. 296, No. 2, August
1973
of Complete
to rewrite (39) and
107
L. 0. Chua
J(vr2,iyaP,) =
J3-,kT,~ M
VLTJ
-Mt
J,$,s, &,I I ’
-I-JbQ-ps3 0w&T&~ + %,, bp,,lR&9-~~ M = K&9-2’
(44) (45)
Observe that both J,& vr2, vrI) and J9,(ipa, i22) are symmetric even though the Jacobian matrix J(v,,, ize,) is not. To avoid working with the relatively large and non-symmetric Jacobian matrix, we now derive a sufficient condition in terms of only the two smaller submatrices J~2(v~2, vFI) and J.dba~ i~2). But first we need the following definition: DeJinition 4. A 2-terminal resistor is said to be ultimately strongly passive if, and only if, its v-i curve eventually lies in the first and the third quadrants, and jil-tco as IvI-tco. Theorem IX. If the non-complete hybrid m-port N in Fig. 4(a) contains only ultimately strongly passive resistors and a consistent set of independent voltage and current sources, and if the two matrices J,&vr2, vg,) and ) are positive definite for all values of (Vet, vyl) and (ip8,i9J, then J&v i_cz;pz h(v,,, i ~3) is a one-to-one function of vrz and i,,, for any assumed value of vYI and i,,. Moreover, (39) and (40) can be solved uniquely for vrS and i$, in terms of vg, and is,. Proof:
We first show that J(vg2, i_& is positive definite by observing that x
[ Y I
t
Jtak3-2z’v.4 [
M
-Me
J&b,, i.& I
X
[ Y I
= xt Jr2(vr2, vrI) x - xt Mt y + yt Mx + yt J9Q(i-Lp3, i& y > 0,
(46)
for all x #O and y #0 since both JY2(vF2,vrI) and J9,(iz3, i& are positive definite and since ytMx = (Y~Mx)~ = xtMt y. TO show that h(vrZ, i& =h(z) is one-to-one, where zs (v,,,i&, suppose the contrary; i.e. suppose there exist two vectors z1 and z2 such that z1 # z2 and h(z,) = h(z,). Let
108
Journal of The Franklin
Institute
Stationary
and applying q(l) -q(O)
Principles
the mean-value
and Potential Functions for Nonlinear Networks theorem,
we obtain
=
[IddVW P -01 = ([aho(z+X(z,-z,))/azl k,-z,l>” (zz-zd
=
(zZ-~l)t[Jo(~+J;(~2-~l))]t(~2-~1),
k(O,l).
(4’)
Since J(z) is positive definite, (47) implies that g(l)-g(O)> 0. But since h(z,) = h(z,), we have g( 1) -g(O) = [h(z& - h(zl)lt (z2- zl) = 0, which is and the vector-valued function absurd. Hence, h(vYa, ia,) is one-to-one representing the left-hand sides of (39) and (40) is a one-to-one function of vrZ and i&, for any assumed value of vY1 and i&. This means that (39) and (40) can have at most one solution for each assumed value of vg, and i&. Now, since N contains only ultimately strongly passive resistors, we know (39) and (40) have at least one solution (20). Hence, given any value of vY1 and i&, (39) and (40) have one, and only one solution. Q.E.D. We remark that the first part of the hypotheses for Theorem IX can be relaxed somewhat by replacing it with any other less stringent sufficient condition which guarantees the existence of at least one solution (21). The second part of the hypotheses can be given a very simple circuit theoretic interpretation by constructing an auxiliary p-port N, as shown in Fig. 5(a) and an auxiliary q-port Np.as shown in Fig. 5(b). The p-port N, is obtained from the network in Fig. 4(b) by short-circuiting all elements in -Epr,< and Y2 and open-circuiting all elements in ,Eaz. The p = (y-p) ports are created by pliers-type entry through each resistor in .&. The q-port Nq is obtained by open-circuiting all elements in YZ, Pa and z and short-circuiting all elements in ~5. The q = (E-S) ports are created by soldering-iron-type entry across each resistor in &. If we define the port voltage vector
for NP and the port current vector iYZ = (i,,,, &, . . . . s,)t for N,, then the terminal equations for Np and and N, are easily seen to be given by
(49) We now make the important observation that if we set vY1 = Or1 in (43) and iYs = O,, in (44), then J~a(v~2, 0,) and J,&,8, O,,) are identical to the Jacobian matrix of (48) and (49), respectively.* Moreover, JrZ(vg2, Oyl) is simply the incremental conductance matrix of N, and JIps(i~~,O_rp) is simply the incremental resistance matrix of NP. Observe next that both NP and Ngare complete and hence NP is characterized by a content potential function 99(i) = 9(!_2 and N, in Fig. 5(b) is characterized by a co-content potential function CC?(V) = g(vY,). Now since the Jacobian matrices Jy3(i93,0,,) and * Observe that since vrI and iz2 are independent of vs2 and i2a in Eq. (41), the positive definiteness of the Jacobian matrix J(vr2, i.& does not depend on vrI and i,,. Consequently, we can assume vrI = i,, = 0 for testing purposes.
Vol.
296, No.
2, August
1973
109
current-controlled
p-po!t
Np(p=
7-
8)
[b).
“Y Auxiliary
1:
voltage-COntrOlW
.
. . . +
q-port
:
+ “c+2
+
Nq tq=a-8)
in
resistors
FIG. 5. Two auxiliary complete p- and p-ports N, end N, obtained from the non-complete m-ports in Fig. 4. For typographical reasons, all symbols with a hat in this drawing have been snbstitut,ed by symbols with & bar in the text.
((11. Auxiliary
i.
+ “r4
“ai2
“#++I
“a
l
. :iB=O
“ai2
i.+2=0 +
Stationary Principles
and Potential Functions for Nonlinear Networks
J9-,@.92O,,)
are the Hessian of ‘2&J and $3(v,,), respectively, the positivedefinite hypothesis in Theorem IX implies that both N, and Nz are strictly convex (22).
N
P+qT
+
1’1-
i.. “2
“2
. . “a
P=Y-P a=r-S
. i, 1
+
il
“a
i2 . . .
i
.+I
ia
:-.~
I
=o
voltage-controlled tree resistors in CJ2
+
“.+I
open 011
I4 .+2=0 L
‘B+l
1
1
currentcontrolled co-tr~e
t-5 +
Vr _I
(curr;;nt
c+I
1
controlled tree
FIG. 6. The auxiliary complete hybrid (p + p)-port IV,,, obtained from the non-complete m-port in Fig. 4. For typographical reasons, all symbols with a hat in this drawing have been substituted by symbols with a bar in the text.
Observe next that instead of testing two auxiliary p- and q-ports, we can construct one auxiliary hybrid (p + q)-port A$,+, as shown in Fig. 6. This (p + q)-port is obtained from Fig. 4(b) by open-circuiting all current sources in 3?‘, and short-circuiting all voltage-sources in q. The current-driven ports and the voltage-driven ports are created as in Figs. 5(a) and (b). Observe that this (p + q)-port is complete and hence N,,, is completely characterized
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L. 0. Chua by a hybrid content potential function &(va, iz) = Z(vy2;, is3). The terminal equations for N,,, are easily seen to be given by I,, = l.&Ya)
- R&9-* i& - R&Y* i,I o ( - B,,,
vYa)’
Y.v~= v_~,(i_& f I&Y, van + I&37.3vr-,o (B$&&,
(50) (51)
where I, = (i,,,, fsf2, . . . . i,)t and YPs = (Cl+,, G~+~,. . . . $)t. As before, if we set vY1 = O,, and ip2 = O,, in (43) and (44), we found the resulting Jacobian matrix J(vYZ, ig8) in (42) is exactly the Jacobian matrix associated with (50) and (51). We now summarize the above observations as follows: Theorem X. The incremental hybrid matrix of the auxiliary complete hybrid (1, + q)-port N,,, is positive definite if both the incremental resistance matrix of the complete current-controlled p-port NP and the incremental conductance matrix of the complete voltage-controlled q-port NQare positive definite. Under this condition, the content of NP, and the co-content of Np are strictly convex. If the non-complete hybrid m-port N in Fig. 4 contains only ultimately strongly passive resistors and a consistent set of independent voltage and current sources, and if the two Jacobian matrices Jyl(vgB, vg,) and J,3(i,3, i_& as defined in (43) and (44) are positive definite for all values of (vgB,vY1) and (i93,i,,), then (39) and (40) can be solved uniquely for vYs and iw in terms of vg, and i&. Moreover, the normal form equations of the non-complete dynamic network constructed by replacing the voltage sources in Fig. 4(a) by voltage-controlled capacitors and current sources in Fig. 4(a) by current-controlled inductors exist and are representable in terms of a mixed potential function as presented in Brayton and Moser. To illustrate the application of the preceding circuit theoretic interpretations, consider again the non-complete S-port shown in Fig. 2(b). Recasting this circuit into the framework of Fig. 4(b), we immediately identified the two current sources as &, the current-controlled nonlinear resistor as s3 and the two linear resistors as ~5. Observe that for this simple network, ~5, 4 and Z1 are empty sets. To test whether this 2-port admits a content potential function, we identified the auxiliary p-port (p = 1 in this case) as consisting of the nonlinear resistor in series with the two S-ohm linear resistors. The second auxiliary q-port is irrelevant in this case since q = 0. The resulting l-port N, is characterized by V = (i3-%)+4i+4i
= ia.
Since this characteristic represents a one-to-one function, we arrive at the same conclusion as in the example of Sect. III that the 2-port shown in Fig. 2(b) has a well defined content potential function. Notice, however, that in contrast to the topological formulation shown in Fig. 2(c), only straightforward circuit theoretic approach is invoked here. Observe also that the incremental resistance of N, is positive semi-definite but not positive definite in this case. Hence, the positive definite hypothesis of Theorem X is not a necessary condition for the existence of a hybrid content potential function.
112
Journal
of The Franklin
Institute
Stationary Principles
and Potential Functions for Nonlinear Networks
It can be shown, however, that this hypothesis is not too conservative fact is about as best as could be stated for a general m-port.
VI.
Concluding
and in
Remarks
The content, co-content and hybrid content for complete nonlinear m-ports have been generalized to pseudo-content, pseudo co-content and pseudo hybrid content of non-complete m-ports. The precise criterion for which these pseudo-potential functions can be reduced to a legitimate potential function is shown to be the unique solvability of a set of implicit equations given explicitly in (18), (24), (39) and (40). Simple circuit theoretic sufficient conditions for unique solvability of the non-complete hybrid m-port are given in terms of the positive definiteness of the incremental resistance matrix of a complete auxiliary p-port and the incremental conductance matrix of a complete auxiliary q-port. Moreover, since a dynamic network containing no “capacitor-voltage source loops” or “inductor-current source cut sets” can always be viewed as an m-port-generally non-complete-with the capacitors and inductors connected across its ports, it follows that the same conditions which guarantee the existence of a hybrid content potential function also apply to the existence of the normal form equations of the associated dynamic network.
References (1) J. C. Maxwell, “A Treatise on Electricity and Magnetism”, Vol. 1, 3rd ed., pp. 407-408, Clarendon Press, Oxford, 1891. (2) W. Millar, “Some general theorems for nonlinear systems possessing resistance”, Phil. Mug., Ser. 7, Vol. 42, pp. 1150-1160, Oct. 1951. (3) C. Cherry, “Some general theorems for nonlinear systems possessing reactance”, Phil. Mug., Ser. 7, Vol. 42, pp. 1161-1177, Oct. 1951. (4) A. R. Oliver, “Energy methods in electrical networks”, Proc. IEE, Part B, Vol. 109, pp. 537-538, Nov. 1962. (5) R. K. Brayton and J. K. Moser, “A theory of nonlinear networks-I and II”, Quart. A&. Math. Vol. 22, pp. l-33, April 1964, and pp. 81-104, July 1964. (6) T. E. Stern, “Theory of Nonlinear Networks and Systems”, Addison-Wesley, Reading, Mass., 1965. (7) J. 0. Flower and F. J. Evans, “Derivation of certain nonlinear circuit equations”, Electronics Letts, Vol. 3, p. 265, June 1967. (8) J. 0. Flower, “The topology of the mixed potential function”, Proc. IEEE (Letts), Vol. 56, pp. 1721-1722, Oct. 1968. (9) J. 0. Flower, “The existence of the mixed potential function”, Proc. IEEE (Letts), Vol. 56, pp. 1735-1736, Oct. 1968. (10) A. G. J. MacFarlane, “Dual-system methods in dynamical analysis, Part IVariational principles and their applications to nonlinear network theory”, Proc. IEE, Vol. 116, pp. 1453-1457, Aug. 1969. (11) D. L. Jones and F. J. Evans, “Variational principle for nonconservative networks”, Electronic Let&, Vol. 6, pp. 163-164, March 1970. (12) T. M. Apostol, “Mathematical Analysis”, Addison-Wesley, Reading, Mass., 1957.
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2, August
1973
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