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Loop and cut set Analysis Fundamental theorem of graph theory Loop Analysis Two basic facts of loop analysis Loop analysis of linear time invariant networks Properties of the loop impedance matrix Cut set Analysis Two basic facts of cut-set analysis Cut-set analysis of linear time invariant networks Properties of the cut-set admittance matrix
Loop and cut set Analysis Loop and cut set are more flexible than node and mesh
analyses and are useful for writing the state equations of the circuit commonly used for circuit analysis with computers.
The loop matrix B and the cutset matrix Q will be introduced.
Fundamental Theorem of Graph Theory
A tree of a graph is a connected subgraph that contains all nodes of the graph and it has no loop. Tree is very important for loop and curset analyses. A Tree of a graph is generally not unqiue. Branches that are not in the tree are called links.
Loop and cut set Analysis
Fig.1 Examples of Tree
1 2
3
45
6
7
8
Graph G
1
3
4
7
8
1 2
4 6
2
4 68
1
3
45
Tree T1 Tree T2 Tree T3 Tree T4
Loop and cut set Analysis
1 2
3
45
6
7
8
Graph G
1
54
8
2
4
2
4 6
3
7
Fig.2 Not a Tree
Loop and cut set AnalysisProperites of loop and cut set
Give a connected graph G of nodes and branches and a tree of Gtn b T There is a unique path along the tree between any two nodes There are tree branches links. Every link of and the unique tree path between its nodes constitutes a unique loop called “fundamental loop”. Every Tree branch of together with some links defines a unique cut set of G . This cut set is called a fundamental cut set.
1tn 1tb n T
T
If G has nodes, branches and separate parts . Let tn b s sTTT ,....., 21be trees of each separate part. The set is called the forest of G sTTT ,...., 21
Loop and cut set Analysis
Fig.3 Fundamental cut set
T1
T21b
Cut set
Loop AnalysisConsider a connected graph with b branches and nt nodes. Pick a tree TThere are n = nt-1 tree branches and l = b-nt links. Number the links first to be1,2….l and number the tree from l+1 to b . Every link and a unique path oftree branches defines a fundamental loop.
Fig.4 Fundamental loop
The graph of Fig. 4 illustrates fundamental loop for the chosen Tree
6
21
3
4
5
7 8
Link 1,2,3,4
Tree 5,6,7,8
1
2
34
Loop AnalysisAssign the direction of loop current to the same as the direction of the linkthe KVL for each fundamental loop are.
0:4
0:3
0:2
0:1
764
8763
87652
651
vvvloop
vvvvloop
vvvvvloop
vvvloop
In matrix form
0
0
0
0
01101000
11100100
11110010
00110001
8
7
6
5
4
3
2
1
v
v
v
v
v
v
v
v
linksb n 1
tree branchestn n
Loop AnalysisThe l linear homogeneous algebraic equations in obtainedby applying KVL to each fundamental loop constitute a set of l linearly independent equation
bvvv ,..., 21
If the reference direction of the loop agrees with that of the link whichdefines it, the KVL is of the form.
Bv 0
B is l x b matrix called the fundamental loop matrix
0
1
1
ikb
If branch is in loop and reference direction agreek i
If branch is in loop and reference direction oppositek i
If branch is not in loopk i
Loop AnalysisThe fundamental loop matrix can be partitioned in to
B 1 F
The KCL can be written in the form
TT
1j B i i
F
The KCL for Fig.4 is
44
33
22
11
ij
ij
ij
ij
5 1 2
6 1 2 3 4
7 2 3 4
8 2 3
j i i
j i i i i
j i i i
j i i
Loop AnalysisIn the matrix form
4
3
2
1
8
7
6
5
4
3
2
1
0110
1110
1111
0011
1000
0100
0010
0001
i
i
i
i
j
j
j
j
j
j
j
j
Loop analysis of linear time invariant networksIn a resistive circuit, the branch equations are of the form
s s v Rj v RiPremultiply by B and apply KCL and KVL yields
Ts s BRB Bv BRJ
orsZ i eTZ BRB where
Z is called the loop impedance matrix and
se is the loop voltage source vector
s s s e Bv BRJ
Loop analysis of linear time invariant networksExample 1
Fig. 5
Write the fundamental loop equation for the circuit shown in Fig.5.
6
R2
R3
R4
R5
R7 R8
Link 1,2,3,4
Tree 5,6,7,8
1
2
34
R1
+-1sv
8sj
Loop analysis of linear time invariant networksThe branch equations are
88
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
s
s
jR
v
j
j
j
j
j
j
j
j
R
R
R
R
R
R
R
R
v
v
v
v
v
v
v
v
Loop analysis of linear time invariant networksZ =BRBT
76476766
7687638766
768768765265
6665651
RRRRRRRR
RRRRRRRRRR
RRRRRRRRRRRR
RRRRRRR
And the loop equations are
1 5 6 5 6 6 6 1 1
5 6 2 5 6 7 8 6 7 8 6 7 2 8 8
6 6 7 8 3 6 7 8 6 7 3 8 8
6 6 7 6 7 4 6 7 4 0
s
s
s
R R R R R R R i v
R R R R R R R R R R R R i R j
R R R R R R R R R R i R j
R R R R R R R R i
Loop analysis of linear time invariant networksProperties of the loop impedance matrix
For a RLC networks in sinusoid steady state the loop impedance matrix
( ) ( ) Tbj j Z BZ B and has the following properties
If there is no coupling element the matrix is diagonal and the loop impedance matrix is symmetric. If there is no coupling element the matrix can be written by inspection
( )b jZ
)( jZ ii is the sum of impedance in the loop and i
( )ikZ j is the sum or negative sum of impedance of branch impedance common to loop the plus sign applied if the branch direction agree with the loop direction.
ik
k If all current sources are converted to Thevenin voltage sources, then is the sum of voltage sources forcing the current flow in the loop .
ske
( )b jZ
Cut set Analysis Cut set analysis is a dual of loop analysis Every tree branch defines a unique cut set
Fig. 6
The fundamental cut set of the circuit of Fig.4 is shown in Fig.6
6
2
3
4
5
7
8
Link 1,2,3,4Tree 5,6,7,8
1
Cut set 3Cut set 4
Cut set AnalysisKCL can be written for each cut set as shown
Cut set 1:
Cut set 2:
Cut set 3:
Cut set 4:
0521 jjj
1 2 3 4 5 0j j j j j
2 3 4 7 0j j j j 2 3 8 0j j j
In matrix form
0
0
0
0
10000110
01001110
00101111
00010011
8
7
6
5
4
3
2
1
j
j
j
j
j
j
j
j
Or
Qj 0
Cut set Analysis The n linear homogeneous algebraic equations in
obtained by applying KCL to each fundamental cut set constitute a set of n linearly independent equations.
bjjj ,...., 21
The fundamental cut set matrix Q is defined by
1
1
0ikq
If branch belongs to cut set and reference direction agreek i
If branch belongs to cut set but reference direction oppositek i
If branch does not belong to cut setk i
The cut set matrix can be partitioned by
nQ E 1 l link n cut set
Since the voltage of each branch is a linear combination of tree branchvoltages and if tree branch voltages are then for Fig.61 2, ,.... ne e e
Cut set Analysis
48
37
26
15
32764
4328763
432187652
21651
ev
ev
ev
ev
eevvv
eeevvvv
eeeevvvvv
eevvv
or
1
2
3
4
1 1 0 0
1 1 1 1
0 1 1 1
0 1 1 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
e
e
e
e
v
KVL
or Tv Q eKVL
Cut-set analysis of linear time invariant networksIn Cut set analysis Kirchhoff’s laws are
Qj 0KCL: KVL: Tv Q e
And the branch equations
s s j Gv j GvCombine KCL KVL and branch equations to obtain
Ts s QGQ e QGv Qj
orq sY e i
whereT
qY QGQ s s si QGv Qj
qY is the cut set admittance matrix and is the current source vectorsi
Cut-set analysis of linear time invariant networksProperties of cut set matrix
For RLC circuit with sinusoid sources in steady state the properties of theCut set admittance matrix areqY
( ) ( ) Tq bj j Y QY Q
If the network has no coupling element the branch admittance is diagonal and the cut set admittance matrix is symmetric If there are no coupling can be written by inspection qY
( )iiY j is the sum of admittance in the cut set and
( )ikY j is the sum or negative sum of branch admittance commonto cut set and cut set the plus sign applied if the branch and branch has the same direction.
kk
If all voltage sources are converted to Norton sources, then is the algebraic sum of those currents in opposite to the direction of the cut set.
i
ii
ski
qY
Cut-set analysis of linear time invariant networksExample2
Fig. 7
G6
G2
G3
G4
G5
G7G8
Link 1,2,3,4Tree 5,6,7,8
G1
Cut set 3Cut set 4
+-1sv
8sj
Write the cut set equation of Fig. 7 by inspection.
Cut-set analysis of linear time invariant networks
8
11
11
4
3
2
1
83232322
3274324322
324326432121
2221521
0
s
s
s
j
vG
vG
e
e
e
e
GGGGGGGG
GGGGGGGGGG
GGGGGGGGGGGG
GGGGGGG
Comments on loop and cut set analysis
Loop and cut set analysis are more general than node and mesh analysissince Tree can be selected in many ways . With certain Tree the loop analysisBecomes the mesh analysis and Cut set analysis becomes the node analysis.
Relation between B and Q
T BQ 0 T QB 0and
