Network topology, cut-Network topology, cut-set and set and

loop equation loop equation

20050300 HYUN KYU SHIM

DefinitionsDefinitionsConnected Graph : A lumped network

graph is said to be connected if there exists at least one path among the branches (disregarding their orientation ) between any pair of nodes.

Sub Graph : A sub graph is a subset of the original set of graph branches along with their corresponding nodes.

(A) Connected Graph (B) Disconnected Graph

Cut – SetCut – Set

Given a connected lumped network graph, a set of its branches is said to constitute a cut-set if its removal separates the remaining portion of the network into two parts.

Tree Tree

Given a lumped network graph, an associated tree is any connected subgraph which is comprised of all of the nodes of the original connected graph, but has no loops.

LoopLoop

Given a lumped network graph, a loop is any closed connected path among the graph branches for which each branch included is traversed only once and each node encountered connects exactly two included branches.

TheoremsTheorems(a) A graph is a tree if and only if

there exists exactly one path between an pair of its nodes.

(b) Every connected graph contains a tree.

(c) If a tree has n nodes, it must have n-1 branches.

Fundamental cut-setsFundamental cut-sets

Given an n - node connected network graph and an associated tree, each of the n -1 fundamental cut-sets with respect to that tree is formed of one tree branch together with the minimal set of links such that the removal of this entire cut-set of branches would separate the remaining portion of the graph into two parts.

Fundamental cutset Fundamental cutset matrixmatrix

.cutset

withassociatedbranch tree theas cutset

defining surface closed the toregardh wit

onoriientati opposite thehas and cutset in is branch if : 1

.cutset in not is branch if : 0

.set -cut with associatedbranch

tree theas cutset defining surface closed the toregard

n with orientatio same thehas and cutset in is branch if : 1

i

i

ij

ij

i

i

ij

ijq

Nodal incidence matrixNodal incidence matrix

The fundamental cutset equations may be obtained as the appropriately signed sum of the Kirchhoff `s current law node equations for the nodes in the tree on either side of the corresponding tree branch, we may always write

(A is nodal incidence matrix)

aWAQ

Loop incidence matrixLoop incidence matrix

Loop incidence matrix defined by

loop. theasdirection opposite in the

oriented is and loopin is branch if : 1-

. loopin not is branch if : 0

loop. theasdirection same in the

oriented is and loopin is branch if : 1

ij

ij

ij

bij

Loop incidence matrix & Loop incidence matrix & KVLKVL

We define branch voltage vector

We may write the KVL loop equations conveniently in vector – matrix form as

)]`(),...,(),([)( 21 tvtvtvtv bb

tallfor 0)( tvB ba

General CaseGeneral Case

t)all(for 0)()()( 321 tvtvtv

t)all(for 0)()()( 321 tititi

To obtain the cut set equations for an n-node , b-branch connected lumped network, we first write Kirchhoff `s law

The close relation of these expressions with

0)( tQib )(`)( tvQtv tb

0)( tAib )(`)( tvAtv nb

bbbb tvyti )()(

)( kb ydiagy

sourcecurrent t independenan containsbranch th if : 0

L valueof inductancean containsbranch th if : L

1

R valueof resistance a containsbranch th if : R1

C valueof ecapacitanc a containsbranch th if : C

source. voltageindepedentan containsbranch th if : 0

kk

kk

kk

k

kD

k

kD

k

yk

And current vector is specified as

follows

b

function timeby the specified source

currentt independenan containsbranch th if : )(

)(tcondition initial

with theinductancean containsbranch th if :

resistance a containsbranch th if : 0

ecapacitanc a containsbranch th if : 0

source t voltageindependenan containsbranch th if : )(

00

0

k

k

kk

k

k

k

i

kti

ii

ki

k

k

kti

Hence,

We obtain cutset equations

btbb QtvQQytQi )(`)(0

btb QtvQQy )(`

)(`)( tvQtvib

bib QtvQQy

)(`

ExampleExample

0

)(

)(

0

)(

)(

0 0 0 0

0 1

0 0 0

0 0 0 0 0

0 0 0 1

0

0 0 0 0 0

)(

04

1

ti

ti

ti

tv

CDLD

Rti bb

hence the fundamental cutset matrix

yields the cutset equations

1- 1- 1- 1 0

1- 1- 1- 0 1Q

)()(

)()()(

)(

)(11

1

1

1

04

041

2 titi

tititi

tv

tv

CDLDR

CDLD

CDLD

CDLD

In this case we need only solve

for the voltage function to obtain

every branch variable.

tt

tt titi

dttvd

CdvLdt

tdvCdv

Ltv

R 0 0

)()()(

)(1)(

)(1

)(1

042

22

2v