network topology, cut-set and loop equation network topology, cut-set and loop equation 20050300...
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Network topology, cut-Network topology, cut-set and set and
loop equation loop equation
20050300 HYUN KYU SHIM
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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.
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(A) Connected Graph (B) Disconnected Graph
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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General CaseGeneral Case
t)all(for 0)()()( 321 tvtvtv
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t)all(for 0)()()( 321 tititi
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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
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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
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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
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Hence,
We obtain cutset equations
btbb QtvQQytQi )(`)(0
btb QtvQQy )(`
)(`)( tvQtvib
bib QtvQQy
)(`
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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
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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
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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