EASTERN MEDITERRANEAN UNIVERSITY

EE 529 Circuit and Systems Analysis

Lecture 4

Matrices of Oriented Graphs

THEOREM: In a graph G let the fundamental circuit and cut-set matrices with respect to a tree to be written as

( ) ( )

( )

and

If the column orderings are identical then

r

r

r

f

f

T

B = B U

A = U A

A = B

Matrices of Oriented Graphs

Consider the following graph

e3

v1

v2

v3

v0

e1

e2

e4e5

e3

v1

v2

v3

v0

e1

e2

e4e5

e6

5 3 62 4 1

2

4

5

1 0 0 1 1 0

0 1 0 1 0 1

0 0 1 0 1 1

e e ee e e

e

e

e

fA

5 3 62 4 1

1

3

6

1 1 0 1 0 0

1 0 1 0 1 0

0 1 1 0 0 1

e e ee e e

e

e

e

fB

FUNDAMENTAL POSTULATES

Now, Let G be a connected graph having e edges and let

be two vectors where xi and yi, i=1,...,e, correspond to the across and through variables associated with the edge i respectively.

1 2

1 2

, ,

and

, ,

Te

Te

x t x t x t

y t y t y t

x

y

FUNDAMENTAL POSTULATES

2. POSTULATE Let B be the circuit matrix of the graph G having e edges then we can write the following algebraic equation for the across variables of G

3. POSTULATE Let A be the cut-set matrix of the graph G having e edges then we can write the following algebraic equation for the through variables of G

Bx = 0

Ay = 0

FUNDAMENTAL POSTULATES

2. POSTULATE is called the circuit equations of electrical system. (is also referred to as Kirchoff’s Voltage Law)

3. POSTULATE is called the cut-set equations of electrical system. (is also referred to as Kirchoff’s Current Law)

Fundamental Circuit & Cut-set Equations

Consider a graph G and a tree T in G. Let the vectors x and y partitioned as

where xb (yb) and xc (yc) correspond to the across (through) variables associated with the branches and chords of the tree T, respectively.

Then

T T Tb c

T T Tb c

x x x

y y y

0

b

c

c b

xB U

x

x Bx

b

c

b c

yU A = 0

y

y = -Ay

and

fundamental circuit equation

fundamental cut-set equation

Series & Parallel Edges Definition: Two edges ei and ek are said to be

connected in series if they have exactly one common vertex of degree two.

ei

ek

v0

Series & Parallel Edges Definition: Two edges ei and ek are said to be

connected in parallel if they are incident at the same pair of vertices vi and vk.

ei

ek

vi

vk

(n+1) edges connected in series

(x1,y1)

(x2,y2)

(xn,yn)

(x0,y0)

0

1

2

3

1 1 1 1 1 0

n

x

x

x

x

x

0

1

2

3

1 1 0 0 0

1 0 1 0

1 0 0 1 0

1 0 0 0 0

1 0 0 0 1 n

y

y

y

y

y

0

01

n

ii

x x

0 1 2 ny y y y

(n+1) edges connected in parallel

0

1

2

3

1 1 1 1 1 0

n

y

y

y

y

y

0

1

2

3

1 1 0 0 0

1 0 1 0

1 0 0 1 0

1 0 0 0 0

1 0 0 0 1 n

x

x

x

x

x

0

01

n

ii

y y

0 1 2 nx x x x

(x0,y0) (x1,y1) (x2,y2) (xn,yn)

Mathematical Model of a Resistor

A

B

a

b

v(t)

i(t)

( ) ( )v t Ri t

Mathematical Model of an Independent Voltage Source

a

b

v(t)

i(t)

B

A

Vs

v(t)

i(t)

Vs

Mathematical Model of an Independent Voltage Source

a

b

v(t)

i(t)

B

A

Is

v(t)

i(t)

Is

Circuit Analysis

A-Branch Voltages Method:

Consider the following circuit.

30 V

2 k

4 k

1 k

15 V

20 V

3 k 10 mA

Circuit Analysis

A-Branch Voltages Method:

1. Draw the circuit graph

30 V

2 k

4 k

1 k

15 V

20 V

3 k 10 mA

1

2

3 4

56

7

8

a b

c

d e

There are:

•5 nodes (n)

•8 edges (e)

•3 voltage sources (nv)

•1 current source (ni)

Circuit Analysis

A-Branch Voltages Method:

1. Select a proper tree: (n-1=4 branches)

Place voltage sources in tree

Place current sources in co-tree

Complete the tree from the resistors

1

2

3 4

56

7

8

a b

c

d e

Circuit Analysis

A-Branch Voltages Method:

2. Write the fundamental cut-set equations for the tree branches which do not correspond to voltage sources.

1

2

3 4

56

7

8

a b

c

d e

2

3

5

6

7

1 1 1 1 1 0

i

i

i

i

i

Circuit Analysis

A-Branch Voltages Method:

2. Write the currents in terms of voltages using terminal equations.

1

2

3 4

56

7

8

a b

c

d e

22

33

55

66

7

2

4

1

310

vi

kv

ikv

ikv

ik

i mA

Circuit Analysis

A-Branch Voltages Method:

2. Substitute the currents into fundamental cut-set equation.

1

2

3 4

56

7

8

a b

c

d e

3 5 62 102 4 1 3

v v vvm

k k k k

3. v3, v5, and v6 must be expressed in terms of branch voltages using fundamental circuit equations.

Circuit Analysis

A-Branch Voltages Method:

1

2

3 4

56

7

8

a b

c

d e

3 2 1 2

5 4 2 2 2

6 2 1 8 2 2

30

1 15 30 15

30 20 50

v v v v

v v v v v

v v v v v v

3 5 62

2 3 5 6

2 2 2 2

2

2

2

12 102 4 1 3

6 3 12 4 120

6 3( 30) 12( 15) 4( 50) 120

6 3 12 4 90 180 200 120

25 350

35014

25

v v vvk m

k k k k

v v v v

v v v v

v

v

v V

Find how much power the 10 mA current source delivers to the circuit

Circuit Analysis

A-Branch Voltages Method:

1

2

3 4

56

7

8

a b

c

d e

7 8 1 2 20 30 14 36v v v v V

Find how much power the 10 mA current source delivers to the circuit

10 7 7 36 10 360mAP v i m mW

Circuit Analysis

Example: Consider the following circuit. Find ix in the circuit.

4 4

ix

2 20 V

3

15 V

5 10 V

i1

i2 i3

Circuit Analysis

Circuit graph and a proper tree

4 4

ix

2 20 V

3

15 V

5 10 V

i1

i2 i3

5xi i

12

3

4

5

6

7 8

Circuit Analysis

Fundamental cut-set equations

2 6 8 7

3 6 8

i i i i

i i i

322 3

6 76 7

88

2 4

4 5

3

vvi i

v vi i

vi

12

3

4

5

6

7 8

Circuit Analysis

Fundamental cut-set equations

6 8 72

3 6 8

2 4 3 5

4 4 3

v v vv

v v v

12

3

4

5

6

7 8

Circuit Analysis

Fundamental circuit equations1

2

3

4

5

6

7 8

6 3 1 2 3 2

8 4 3 1 2 5 3 2 3 2

7 1 2 5 2 2

20

15 20 10 15

20 10 30

v v v v v v

v v v v v v v v v v

v v v v v v

Circuit Analysis

3 2 3 22 2

3 3 2 3 2

2 3 2 3 2 2

3 2

3 3 2 3 2

3 2

20 15 3060

2 4 3 5

20 1512

4 4 3

30 15 15 300 20 20 300 12 360

35 77 960...................(1)

3 3 3 60 4 4 60

10 7 120..........

v v v vv v

v v v v v

v v v v v v

v v

v v v v v

v v

............(2)

v3= 9.5639V v2=-8.1203 V

Circuit Analysis

7 8 3 225 7 8

1530

5 3 5 3

v v v vvi i i

5 3.48

3.48x

i A

i A