Download - Network topology, cut-set and loop equation
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