Loop and cut set Analysis - Department of Electrical ...eestaff.kku.ac.th/~jamebond/182304/Loop...

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 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.


Fig.1 Examples of Tree


Fig.2 Not a Tree


Properites 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 constitutesa unique loop called “fundamental loop”.

� Every Tree branch of together with some links defines a unique cut setof 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


Fig.3 Fundamental cut set

1b

Loop Analysis

Consider 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 of

tree branches defines a fundamental loop.

Fig.4 Fundamental loop

The graph of Fig. 4 illustrates fundamental loop for the chosen Tree

Loop Analysis

Assign 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= −l 1

tree branches

tn n= −

Loop Analysis

The 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 Fl

The KCL can be written in the form

T

T

= =

1j B i i

F

l

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 Analysis

In 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 networks

In a resistive circuit, the branch equations are of the form

s s= + −v Rj v Ri

Premultiply by B and apply KCL and KVL yields

Ts s= − +BRB Bv BRJ

ors

=Z i el

TZ BRBl @

where

Zl 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.

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


l� � � � � �

++++−

++++++−

+++++++−−

−−−−++

=

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

+ + − − − − − − − + + + + + + + − = − + + + + + + −

− + + + +


Properties of the loop impedance matrix

For a RLC networks in sinusoid steady state the loop impedance matrix

( ) ( )T

bj jω ω=Z BZ Bl 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 jωZ

)( ωjZ ii is the sum of impedance in the loop and i

( )ikZ jω 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 jωZ

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

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:

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

0

ikq

= −

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

[ ]n=Q 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 T=v Q eKVL

Cut-set analysis of linear time invariant networks

In Cut set analysis Kirchhoff’s laws are

=Qj 0KCL: KVL: T=v Q e

And the branch equations

s s= + −j Gv j Gv

Combine KCL KVL and branch equations to obtain

T

s s= −QGQ e QGv Qj

orq = sY e i

whereT

qY QGQ@ s s s−i QGv Qj@

qY is the cut set admittance matrix and is the current source vectorsi


Properties of cut set matrix

For RLC circuit with sinusoid sources in steady state the properties of theCut set admittance matrix areqY

( ) ( )T

q bj jω ω=Y QY Q

� If the network has no coupling element the branch admittance isdiagonal and the cut set admittance matrix is symmetric

� If there are no coupling can be written by inspection qY

( )iiY jω is the sum of admittance in the cut set and

( )ikY jω 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


Example2

Fig. 7

1sv

8sj

Write the cut set equation of Fig. 7 by inspection.


−=

+++−−

++++−−−

−−−−−++++−−

−−++

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 0T =QB 0and

Loop and cut set Analysis - Department of Electrical ...eestaff.kku.ac.th/~jamebond/182304/Loop...

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