University of Groningen
An Intrinsic Hamiltonian Formulation of the Dynamics of LC-Circuits
Maschke, B.M.; Schaft, A.J. van der; Breedveld, P.C.
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IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
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Maschke, B. M., Schaft, A. J. V. D., & Breedveld, P. C. (1995). An Intrinsic Hamiltonian Formulation of the
Dynamics of LC-Circuits.
IEEE Transactions on Circuits and Systems I: Fundamental Theory and
Applications
, 73-82.
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IEEE
TRANSACTIONS
ON
CIRCUITS
AND
SYSTEMS-I:
FUNDAMENTAL
THEORY
AND
APPLICATIONS, VOL.
42,
NO.
2,
FEBRUARY
1995
13
An Intrinsic Hamiltonian
Formulation
of
the Dynamics
of
LC-Circuits
B.
M.
Maschke,
Member. IEEE,
A.
J.
van der Schaft,
Member. IEEE,
and
P.
C.
Breedveld
Member. IEEE
Abstract-First, the dynamics
of
LC-circuits are formulated
as
a Hamiltonian system defined with respect to a Poisson bracket
which may
be degenerate, i.e., nonsymplectic. This Poisson
bracket is deduced from the network graph
of
the circuit and
captures the dynamic invariants due to KirchhoWs laws. Second,
the antisymmetric relations defining the Poisson bracket are
realized
as
a physical network using the gyrator element and
partially dualizing the network graph constraints. From the
network realization
of
the Poisson bracket, the reduced standard
Hamiltonian system
as
well as the realization
of
the embedding
standard Hamiltonian system are deduced.
I.
INTRODUCTION
HE
STUDY
of the dynamic equations of electrical cir-
T
cuits-in particular, weakly damped circuits-is often
carried out on their lossless part, i.e., by neglecting all re-
sistive effects. This had led to a surprisingly large variety
of Lagrangian or Hamiltonian formulations
[1]-[4],
which
differ in the choice
of
variables and generating functions
(i.e., the Lagrangian or Hamiltonian functions) and in the
underlying geometric structure (Riemannian or symplectic) of
the state-space. However these formulations are dissatisfying
in the sense that they depend on strong assumptions on the
constitutive relations of the multiports and that the variables
and structure are not easily interpretable in terms
of
the
original description of the electrical circuit: a set of charges and
flux linkages describing the electrical and magnetic energies
in the circuit and the interconnection constraints expressed in
Kirchhoffs laws. For instance
the
dynamics of LC-circuits
is described in terms of a constrained Lagrangian system
with generalized coordinates being fictitious flux linkages
associated with capacitors and fictitious charges with the
inductors
[2].
A
similar Lagrangian formulation was proposed
in terms of the more natural variables being capacitors’ charges
and inductors’ flux linkages, which however still relies on the
existence of some co-energy functions, that is on the assump-
tion on the invertibility
of
the constitutive relations of the
inductors and capacitors
[4].
In
the Hamiltonian formulation
proposed in
[l],
the variables are some linear combinations
of
the charges and flux linkages in the circuit.
In
the present
paper we propose an intrinsic Hamiltonian formulation of the
Manuscript received August 3, 1993; revised May
5,
1994 and October
20,
1994. This paper was recommended by Associate Editor Shinsaku Mori.
B.
M. Maschke is with the Laboratoire d’Automatisme Industriel, Conser-
vatoire National des
Arts
et MCtiers, F-75013
Paris,
France.
A.
J.
van der Schaft is with the Department
of
Applied Mathematics,
University
of
Twente, 7500
AE
Enschede, The Netherlands.
P.
C.
Breedveld is with the Department
of
Electrical Engineering, University
of
Twente, 7500
AE
Enscbede, The Netherlands.
IEEE
Log
Number 9408386.
dynamics of LC-circuits which is directly related with the
charges and flux linkages in the circuit and the interconnection
constraints expressed in Kirchhoff‘s laws.
The first main result
of
the present paper is the direct
formulation of the dynamics of LC-circuits as a Hamiltonian
system defined with respect to a Poisson bracket which may
be degenerate, i.e., nonsymplectic. This Poisson bracket is
uniquely determined by the network graph of the circuit and
takes account
of
the dynamic invariants defined by Kirchhoff
s
laws. The state variables are simply the capacitors’ charges
and the inductors’ fluxes.
The second general result is that the antisymmetric relations
defining the Poisson bracket may be realized by a physical
network using the gyrator element
[5], [6].
But this physical
network is obtained by partial dualization of the network graph
of the circuit and requires a more general notation: the bond
graph
[7],
[SI.
In this way the dynamics of LC-circuits may be
related to the dynamics of more general conservative systems
[91-[111.
In
Section
11,
we recall the definition of Hamiltonian systems
defined with respect to general Poisson brackets
[12], [13],
possibly degenerate.
It
is recalled that the degeneracy of
the Poisson bracket is related to the existence of invariant
functions, different from the energy function, and to the
reduction of standard Hamiltonian systems.
In Section
111,
the dynamic equations of the capacitors’
charges and the inductors’ flux linkages are formulated as
a Hamiltonian system, generated by the total energy of the
circuit with respect to a Poisson bracket, uniquely defined by
the network graph.
As
a consequence the state-variables are
directly interpretable and the order of the Hamiltonian systems
is equal to the (possibly odd) order of the circuit.
Section
IV
gives a network realization of the antisymmetric
relations associated with the Poisson bracket using the gyrator
element
[5], [6].
The bond graph representation is used in order
to ensure a graphical representation
of
the dual of any network
graph. Finally, the bond graph plays also a key role for the
network realization of the embedding standard Hamiltonian
system, by enabling to represent the set of symmetry variables
conjugated to the invariants of the system as port variables
of
a network.
11.
NONSTANDARD
HAMILTONIAN
SYSTEMS
This section recalls very briefly the definition of general
Poisson manifolds and the Hamiltonian dynamic systems
defined on it. For a detailed treatment of this subject the reader
may consult
[
121-[ 141.
1057-7122/95$04.00
0
1995 IEEE
14
IEEE TRANSACTIONS ON CIRCUITS
AND
SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS,
VOL.
42,
NO.
2.
FEBRUARY
1995
Dejnition
1:
Let
M
be a smooth (i.e.,
C")
manifold
and let
C"(M)
denote the set of the smooth real functions
on
M.
A
Poisson bracket
on
M
is a bilinear map from
C"(M)
x
C"(M) into
C"(M),
denoted as:
constant skew-symmetric (m
x
m) matrix defines a Poisson
bracket on
R".
From
(4) it follows that in the local coordinates:
21,
-
1
.
,
x,,
the Hamiltonian vector field
XH(~)
is given by the following
(F,
G)
{F,
G}
E
C"(M),
F,
G
E
C"(M)
which verifies
the
following properties:
-skew-symmetry:
{
F,
G}
=
-
{
G,
F}
(1)
(2)
-Leibniz rule:
{F,G.H}
=
{F,G}.H+G*{F,H}.
(3)
-.lacobi identity:
{
F,
{
G,
H}}
+
{
G,
{
H,
F}}
+
{H,
{F,
G)}
=
0
Definition
2:
A
Poisson manifold
is a smooth manifold M
endowed with a Poisson bracket.
Proposition-Definition
3:
Let
M
be
a Poisson manifold
with Poisson bracket
[,
1.
Then any function
H
E
C"(M)
defines a mapping:
X+): C"(M)
+
w
SUC~
that:
XH(F)(Z)
=
{F,
H}(z),
VF
E
C"(M).
(4)
Because of
(3)
XH
is a vector field on
M,
called the
Hamiltonian vector-field
with respect to
H
and the Poisson
bracket.
It follows that the Hamiltonian function
H
is a conserved
quantity for the Hamiltonian vector field
XH.
Indeed one
obtains from
(1):
XH(H)(z)
=
{H,H}(z)
-{H,H}(Z)
=
0.
(5)
vector:
and thus the dynamical equations of motion determined by the
Hamiltonian vector field
XH
are given in local coordinates by:
These equations may also be interpreted being a local matrix
representation of a bundle map from the cotangent bundle
T*M
to the tangent bundle
TM
mapping the differential
of a Hamiltonian function
H
to its Hamiltonian vector field
XH(Z).
The structure matrix
J(z)
of the Poisson bracket is
the representation of this map in local coodinates.
Proposition-Dejnition
5:
The
rank
of
a
Poisson bracket
[,
]
at a point
x
E
M
is defined as the rank of the structure
matrix at this point (which may be shown to be independent
of the choice of local coordinates).
By the skew-symmetry of the Poisson bracket
(8),
the rank
of a Poisson bracket at any point is even.
A
Poisson bracket
on
M
whose rank is equal everywhere to the dimension of
M
is called nondegenerate, and in this case the Poisson manifold
M
is called a
iymplectic manifold.
Thus the dimension of a
symplectic manifold is necessarily even.
The definition of canonical coordinates
as
used for
standard
Hamiltonian systems (i.e., Hamiltonian vector fields on a
Definition
4:
The
structure matrix
J(z)
=
(Jkl(z))k,l=l,
...,-
associated with the Poisson bracket
[,
]
and coordinate functions
XI,
. . .
,
x,
is defined by:
In local coordinates
21,
. .
.
,
x,
the Poisson bracket of two
smooth real functions
F
and G may now be expressed in terms
of the structure matrix
as
follows:
which makes clear that
{F,
G}(z)
only depends on the differ-
entials of
F
and G in
z.
It follows from (l), respectively (2), that the structure matrix
satisfies the
two
following conditions:
skew-symmetry:
&(2)
=
-JIk(Z)
k,
1
=
1,.
. .
,
m
(8)
symplectic maifold) may be generalized to Poisson brackets
with the aid of the following proposition (see e.g.,
[
121-[ 141).
Proposition
1:
Let
M
be an m-dimensional Poisson
manifold. Suppose the Poisson bracket has constant rank
2n
in a neighborhood of a point
xo
E
M.
Then lo-
cally around
x~
one can find coordinates
(q,p,r)
=
(qlr...,Qn,pl,".,pn,rl,...,rz)
where:
(2n
+
1)
=
m,
which satisfy:
{qi,Pj}
=
Sij
{4i,
4j)
=
{Pi,Pj}
=
0,
{(&,?-if}
=
{Pi,Ti!}
=
{?-j/,?-i/}
=
0
i,j
=
1,.
.
.
,n,
i',j'=
1,..-,1, (12)
m
Jacobi identities:
(Jlj(z)~(z)
aJik
+
Jri(z)=(z)
8Jkj
or equivalently, the
m
x
m structure matrix
J(q,p,
r)
is given
as follows:
1=1
i,j,
k
=
1,.
,m.
(9)
Conversely if a matrix
J
with coefficients in C"(M)
satisfies
(8)
and
(9),
then it defines locally a Poisson bracket
according to
(7).
It may
be
noted that, in consequence, any
Dejnition
6:
The coordinates
(q,p,
r)
satisfying
(12)
are
called
canonical coordinates
of the Poisson manifold with
Poisson bracket
[,
1.
MASCHKE
et
al.:
INTRINSIC
HAMILTONLUG
FORMULATION
OF
THE
DYNAMICS
OF
LC-CIRCUITS
Remark:
As
noted before, any skew-symmetric constant
(m
x
m)
matrix defines a Poisson bracket on
R".
In this
particular case, Proposition
1
reduces to the well-known fact
from linear algebra that there exist linear (and thus
global!)
coordinates for
R"
in which
J
takes the form (13).
Let
M
be
an m-dimensional Poisson manifold with the
Poisson bracket of constant rank
2n
in a neighborhood of
a point
zcg
E
M,
and canonical coordinates
(q,p,r),
then it
follows from (1
1)
and (13) that every Hamiltonian vector field
XH
has the following expression in the coordinates
(4,
p,
T):
If
M
is a symplectic manifold then
2n
=
m
=
dim
M
and
1
=
0, and
(14)
reduces to the standard Hamiltonian equations.
DeJnition
7:
Let
M
be a Poisson manifold. The
distin-
guished
or
Casimirfunctions
are those smooth functions
F
E
C"(M)
which satisfy:
{F,
G}
=
0,
VG
E
C"(M)
(15)
(or equivalently, since
{F,
G}
=
-XF(G),
all those functions
F such that
XF
=
0).
Thus the Casimir functions correspond to the kernel of
the map:
F
H
XF
from
Cm(M)
modulo
R
to the vector
fields on
M
given by (4). In local terms
F
is a Casimir
function if dF(z) is in the kernel of the map
J(x):T,M
H
T,M
for every
z
in
M.
Furthermore, under the assumptions
of Proposition 1, the Casimir functions are all functions
depending only on
TI,
.
,
T'
where
(q,
p,
T)
are canonical
coordinates.
Proposition
1
has some interesting consequences concerning
the local
reduction
of the generalized Hamiltonian (14) to
lower-dimensional
standard
Hamiltonian equations, as well
as
concerning the local
embedding
of (14) into
higher-
dimensional
standard
Hamiltonian equations, see also
[
1
13.
Indeed, by the local projection
T:
W2"+'
+
R2"
defined as:
R2"
inherits
the Poisson bracket defined by (12) in
R2"+'
by
leaving out
TI,
. .
,
rl.
In fact this Poisson bracket on
R2"
is
nondegenerate
and the dynamics (14) projects locally to the
standard Hamiltonian equations on
R2":
with the Hamiltonian function:
Hr(q,p)
=
H(q,p,r)
param-
eterized
by
T.
On the other hand, consider locally the
embed-
-
ding
space
R2n+2Z
with linear coordinates
(q1,
.
. .
,
qn,
pl
,
.
.
,
p,,
TI,
. . .
,
rl,
sl,
. .
.
,
SI)
and nondegenerate Poisson bracket
defined by (12) and the additional relations:
Then (14) can be locally embedded into the standard Hamil-
tonian equations on
R'"+~'
given as:
with Hamiltonian:
He(q,p,
T,
s)
=
H(q,p,
T),
i.e., not depend-
ing on
s.
Furthermore, we note that, conversely, the transition from
the standard Hamiltonian system (1 8)
R2"+"
to the standard
Hamiltonian system (16),
via
the nonstandard Hamiltonian
system (14) can be interpreted as the
canonical reduction
of
order caused by the
symmetry
of the Hamiltonian
He
with
respect to translations in the s-coordinates, see e.g.,
[13]
for
an introduction into this subject. From this point of view the
Casimir functions can be seen as the
conserved quantities
corresponding to the infinitesimal symmetries:
6,
. .
,
%.
a
111. LC-CIRCUIT DYNAMICS
This section recalls first the definition and assumptions on
the LC-circuits considered in this paper. Then the dynamic
equations are formulated in "natural" coordinates, i.e., in
terms of the energy variables of the elements (charges of the
capacitors and flux linkages of the inductors), as a Hamiltonian
system defined on a Poisson manifold. The Poisson bracket
is shown to be determined by the constraint relations, i.e.,
Kirchhoff
s
laws induced by the network graph, among the
inductors' voltages and among the capacitors' currents. Finally
this formulation is compared with the standard Hamiltonian
equations proposed in
[l].
An LC-circuit is composed of a set of multiport inductors
and capacitors interconnected through their ports by a graph
I?
called
nerwork graph
[15]
or port
connection graph
[16].
The capacitor and inductor elements are defined by their
constitutive relation, whereas the network graph defines the
relations among their port variables arising from Kirchhoff
s
laws.
DeJnition
8:
An
n-port
capacitor
(respectively
inductor)
is defined by a set of energy variables, the charge:
q
E
R"
(resp. the flux linkage:
4
E
R"),
an energy function:
&(q)
E
Cm(Rn)
(resp.
EL(^)
E
C"(R"))
and two sets of port
variables: the current
ic
E
R"
(resp.
i~
E
R")
and the voltage
16
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY
AND
APPLICATIONS,
VOL.
42,
NO.
2,
FEBRUARY
1995
vc
E
R"
(resp.
VL
E
R"),
related by the constitutive relations:
respectively for the inductor:
It may be noted that, as the capacitors' and inductors' con-
stitutive relations are derived from the stored energy function,
they satisfy Maxwell's reciprocity equations-resulting from
energy conservation, hence they define reciprocal multiports.
For the sake of simplicity, we shall group in the sequel all
the capacitors
of
the circuit into one nc-port with the energy-
function
Ec(q)
equal to the sum of the energy-functions
of
the capacitors and all the inductors into one n~-port with
the energy-function
EL(^)
equal to the sum of the energy-
functions of the inductors.
Dejinition
9: The
network graph
is defined as an oriented
graph whose edges correspond one-to-one to the ports of the
capacitors and inductors and the orientation corresponds to the
sign convention of the voltage variables (and the opposite sign
convention of the current variables).
The network graph describes the connection constraints
among the port variables of the elements due to Kirchhoff
s
laws which may be formulated in the following generalized
form [15].
The sum
of
volt-
ages along any cycle (or loop) in the network graph vanishes.
The sum
of
cur-
rents along any cocycle (or cutset) in the network graph
vanishes.
Moreover, for the sake of simplicity, we shall make the
following assumptions on the electrical circuit:
(Hl) The network graph is connected.
(H2) The
nc
ports
of
the capacitors correspond to a tree,
denoted by
C,
in the network graph.
(H2) The
n~
ports of the inductors correspond to a cotree,
denoted by
L,
in the network graph, complementary
to
c.
The assumption (Hl) says that the circuit consists of one
part. The assumptions
(H2)
and (H3) say that there is no
capacitor loop or inductor cutset, i.e., that the topological
constraints induced by the circuit do not constrain the space
of
the energy variables
(q,(p)
to a proper subset of
RnC
x
RnL
which may be chosen as the state-space of the system. If this
is not the case, the results presented here remain valid by
replacing the space of energy-variables
RnC
x
RnL
by some
proper subspace
A4
of it [9].
Following the notation in [l], the fundamental loop matrix
corresponding to the capacitor's tree
C
may be written as:
B
=
(InL XRL BLC), and the fundamental cutset matrix cor-
responding to the complementary inductor's cotree
L
may be
written as:
Q
=
(-B~cIn,,,,), where
BLC
is an
n~
x
nc
matrix with coefficients in
(-1,
0,
l}. Then Kirchhoffs laws
imply the two following relations on the port variables
of
the
Proposition
2:
Kirchhoffs voltage law.
Proposition
3:
Kirchhoffs current law.
elements:
VL
=
-
BLCVC
(21)
ic
=
BECZh. (22)
Using the state-variables:
x
=
(q,4)
E
RnC
x
RnL,
i.e.,
the energy variables of the inductors and capacitors, and
assembling
(19)-(22), one obtains the following dynamical
equation of the LC-circuit:
k
=
JdH(x)
(23)
where
=
[
Oncxnc
-BLC
OnLxn,
BEc
1
and
Proposition
4: The dynamical equations (23) of the energy-
variables (charges in the capacitors and flux linkages in the
inductors) of an LC-circuit are a Hamiltonian system with
Hamiltonian function defined as the sum of the electrical and
magnetic energy functions of the circuit, with respect to the
Poisson bracket defined on the manifold
RnC
x
RnL
by the
structure matrix
J
defined by the fundamental loop matrix
according to
(24).
Indeed the matrix
J
is skew-symmetric and constant and
hence it verifies the conditions
(8),
(9) of a structure matrix.
Thus it defines a Poisson bracket on the state-space of the
energy-variables
RnC
x
RnL.
And according to Section 11,
(23) is the local representation of a Hamiltonian vector field.
Thus the "natural" dynamics associated with an electrical
circuit may be directly expressed as a Hamiltonian vector field
defined on a Poisson manifold. It is remarkable that using this
formalism, the two concepts of energy-storage elements (i.e.,
the inductors and capacitors) and network graph which may be
defined separately, correspond exactly to the distinct objects
of the Hamiltonian function, respectively Poisson bracket.
Indeed the constitutive relations of the inductors and capacitors
are defined by the energy functions defining the Hamiltonian
function, while the topological constraints on the port-variables
represented by the network graph are fully captured into the
structure matrix
of
the Poisson bracket.
The degeneracy of the Poisson bracket or its structure
matrix may be related to the topological constraints induced
by the network graph. For this purpose, consider the following
partition of the tree
C
and the cotree
L
[17]:
41
a maximal subset of
C
which forms a cotree
-,Cl
a maximal subset of
L
which forms a tree
42
the complement of
C1
in
C
-,Cz
the complement of
L1
in
L.
It may be shown [17] that:
3T
E
N;
IC11
=
/Cl(
=
T,
(27)
where
1x1
denotes the number of elements of any set
X
and
thus:
(28)
IC21
=
nc
-
T
=
s
and
=
n~
-
T
=
1.