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
Loop and cut set Analysis
Fig.2 Not a Tree
Loop and cut set Analysis
Properites of loop and cut set
Give a connected graph G of
nt nodes and b
branches and a tree T of G
There is a unique path along the tree between any two nodes
There are n − 1 tree branches b − n + 1 links.
t
t
Every link of T and the unique tree path between its nodes constitutes
a unique loop called “fundamental loop”.
Every Tree branch of T together with some links defines a unique cut set
of G . This cut set is called a fundamental cut set.
If G has nt nodes, b branches and s separate parts . Let T1 , T2 ,.....Ts
be trees of each separate part. The set {T1 , T2 ,....Ts } is called the forest of G
Loop and cut set Analysis
b1
Fig.3 Fundamental cut set
Loop Analysis
Consider a connected graph with b branches and nt nodes. Pick a tree T
There are n = nt-1 tree branches and l = b-nt links. Number the links first to be
1,2….l and number the tree from l+1 to b . Every link and a unique path of
tree branches defines a fundamental loop.
The graph of Fig. 4 illustrates fundamental loop for the chosen Tree
Fig.4 Fundamental loop
Loop Analysis
Assign the direction of loop current to the same as the direction of the link
the KVL for each fundamental loop are.
loop 1 : v1 − v5 + v6 = 0
loop 2 : v2 + v5 − v6 + v7 + v8 = 0
loop 3 : v3 − v6 + v7 + v8 = 0
loop 4 : v4 − v6 + v7 = 0
In matrix form
1
0

0

0
0 0 0 −1
1
1 0 0
1
−1 1
0 1 0
0
−1 1
0 0 1
0
−1 1
l = b − n links
0
n = nt − 1
 v1 
v 
 2
0  v3  0
   

1   v 4  0 
=


0 
1  v5
   
0   v 6  0 
v 
 7
 v8 
tree branches
Loop Analysis
The l linear homogeneous algebraic equations in v1 , v 2 ,...vb obtained
by applying KVL to each fundamental loop constitute a set of l linearly
independent equation
If the reference direction of the loop agrees with that of the link which
defines it, the KVL is of the form.
Bv = 0
B is l x b matrix called the fundamental loop matrix
 1

bik =  − 1
 0

If branch k is in loop i and reference direction agree
If branch k is in loop i and reference direction opposite
If branch k is not in loop
i
Loop Analysis
The fundamental loop matrix can be partitioned in to
B = [1l F]
The KCL can be written in the form
 1l 
j = B i =  T i
F 
T
The KCL for Fig.4 is
j1 = i1
j5 = −i1 + i2
j 2 = i2
j6 = i1 − i2 − i3 − i4
j3 = i3
j7 = i2 + i3 + i4
j 4 = i4
j8 = i2 + i3
Loop Analysis
In the matrix form
 j1   1 0 0 0 
j  

0
1
0
0
 2 

 j3   0 0 1 0   i1 
  
 
 j4  =  0 0 0 1  i2 
 j5  − 1 1 0 0  i3 
  
 
 j6   1 − 1 − 1 − 1 i4 
j   0 1 1 1 
 7 

 j8   0 1 1 0 
Loop analysis of linear time invariant networks
In a resistive circuit, the branch equations are of the form
v = Rj + v s − Ri s
Premultiply by B and apply KCL and KVL yields
BRBT = −Bv s + BRJ s
or
where
Zl i = e s
Z l @ BRBT
e s = −Bv s + BRJ s
Z l is called the loop impedance matrix and
e s is the loop voltage source vector
Loop analysis of linear time invariant networks
Example 1
Write the fundamental loop equation for the circuit shown in Fig.5.
vs1
js 8
Fig. 5
Loop analysis of linear time invariant networks
The branch equations are
 v1   R1
v  
 2 
 v3  
  
v 4  = 
v5  
  
 v6  
v  
 7 
 v8  
0
R2
R3
R4
R5
0
R6
R7
  j1  v s1   0 
 j     0 
 2   0  

  j3   0   0 
    

j
0
 4  +  0  + 

  j5   0   0 
    

  j6   0   0 
 j   0   0 
 7    

R8   j8   0   R8 j s8 
Loop analysis of linear time invariant networks
l
 R1 + R5 + R6
 −R −R
5
6
=

− R6

− R6

− R5 − R6
− R6
R2 + R5 + R6 + R7 + R8
R6 + R7 + R8
R6 + R7 + R8
R3 + R6 + R7 + R8
R6 + R7
R6 + R7
− R6

R6 + R7 
R6 + R7 

R4 + R6 + R7 
And the loop equations are
 R1 + R5 + R6
 −R − R
5
6


− R6

− R6

− R5 − R6
R2 + R5 + R6 + R7 + R8
− R6
R6 + R7 + R8
R6 + R7 + R8
R3 + R6 + R7 + R8
R6 + R7
R6 + R7
− R6
R6 + R7
  i1   −vs1 
 i  

R
j
−
  2  =  8 s8 
R6 + R7   i3   − R8 js8 
  

R4 + R6 + R7  i4   0 
Loop analysis of linear time invariant networks
Properties of the loop impedance matrix
For a RLC networks in sinusoid steady state the loop impedance matrix
Z l ( jω ) = BZb ( jω )BT and has the following properties
If there is no coupling element the matrix Zb ( jω ) is diagonal and the loop
impedance matrix is symmetric.
If there is no coupling element the matrix Zb ( jω ) can be written by inspection
Z ii ( jω )
Zik ( jω )
is the sum of impedance in the loop
i and
is the sum or negative sum of impedance of branch k
impedance common to loop i the plus sign applied
if the branch k direction agree with the loop direction.
If all current sources are converted to Thevenin voltage sources, then esk
is the sum of voltage sources forcing the current flow in the loop .
Cut set Analysis
Cut set analysis is a dual of loop analysis
Every tree branch defines a unique cut set
The fundamental cut set of the circuit of Fig.4 is shown in Fig.6
Fig. 6
Cut set Analysis
KCL can be written for each cut set as shown
Cut set 1:
Cut set 2:
Cut set 3:
Cut set 4:
j1 − j2 + j5 = 0
− j1 + j2 + j3 + j4 + j5 = 0
− j2 − j3 − j4 + j7 = 0
− j2 − j3 + j8 = 0
In matrix form
Or
Qj = 0
 1 −1 0 0
− 1 1 1 1

 0 −1 −1 −1

 0 −1 −1 0
1 0 0
0 1 0
0 0
1
0 0 0
 j1 
j 
 2
0   j3   0 
   

0   j 4  0 
=


0 
0  j5



 
1  j6  0
j 
 7
 j8 
Cut set Analysis
The n linear homogeneous algebraic equations in j1 , j 2 ,.... jb
obtained by applying KCL to each fundamental cut set constitute a
set of n linearly independent equations.
The fundamental cut set matrix Q is defined by
 1

qik =  −1
 0

If branch k belongs to cut set i and reference direction agree
i but reference direction opposite
If branch k does not belong to cut set i
If branch k belongs to cut set
The cut set matrix can be partitioned by
Q = [ E 1n ]
l link n cut set
Since the voltage of each branch is a linear combination of tree branch
voltages and if tree branch voltages are e1 , e2 ,....en then for Fig.6
Cut set Analysis
KVL
v1 = v5 − v6 = e1 − e2
v2 = −v5 + v6 − v7 − v8 = −e1 + e2 − e3 − e4
v3 = v6 − v7 − v8 = e2 − e3 − e4
v4 = v6 − v7 = e2 − e3
v5 = e1
v 6 = e2
v7 = e3
v8 = e4
or
KVL
or
v = QT e
 1 −1 0 0 
 −1 1 −1 −1


 0 1 −1 −1  e1 

 
e
−
0
1
1
0
 2
v=
 1 0 0 0  e3 

 
 0 1 0 0  e4 
 0 0 1 0


 0 0 0 1
Cut-set analysis of linear time invariant networks
In Cut set analysis Kirchhoff’s laws are
Qj = 0
KCL:
KVL:
v = QT e
And the branch equations
j = Gv + js − Gv s
Combine KCL KVL and branch equations to obtain
QGQT e = QGv s − Qjs
or
Yq e = i s
where
Yq @ QGQT
i s @ QGv s − Qjs
Yq is the cut set admittance matrix and i s is the current source vector
Cut-set analysis of linear time invariant networks
Properties of cut set matrix
For RLC circuit with sinusoid sources in steady state the properties of the
Cut set admittance matrix Yq are
Yq ( jω ) = QYb ( jω )QT
If the network has no coupling element the branch admittance is
diagonal and the cut set admittance matrix Yq is symmetric
If there are no coupling Yq can be written by inspection
Yii ( jω )
Yik ( jω )
is the sum of admittance in the cut set
i and
is the sum or negative sum of branch admittance common
to cut set i and cut set k the plus sign applied
if the branch i and branch k has the same direction.
If all voltage sources are converted to Norton sources, then isk is the
algebraic sum of those currents in opposite to the direction of the cut set.
Cut-set analysis of linear time invariant networks
Example2
Write the cut set equation of Fig. 7 by inspection.
vs1
js 8
Fig. 7
Cut-set analysis of linear time invariant networks
G1 + G2 + G5
 −G −G
1
2


G2

G2

− G1 − G2
G1 + G2 + G3 + G4 + G6
− G2 − G3 − G4
− G2 − G3
G2
− G2 − G3 − G4
G2 + G3 + G4 + G7
G2 + G3
G2
  e1   G1v s1 
− G2 − G3  e2  − G1v s1 
=
 0 



G2 + G3
e3

  
G2 + G3 + G8  e4   j s8 
Comments on loop and cut set analysis
Loop and cut set analysis are more general than node and mesh analysis
since Tree can be selected in many ways . With certain Tree the loop analysis
Becomes the mesh analysis and Cut set analysis becomes the node analysis.
Relation between B and Q
BQ = 0
T
and
QBT = 0