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# EE 529 Circuit and Systems Analysis - PowerPoint PPT Presentation

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. v 1. e 2. e 3. e 1. v 0. e 5. e 4. v 3. v 2. Matrices of Oriented Graphs.

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### EE 529 Circuit and Systems Analysis

Lecture 4

EASTERN MEDITERRANEAN UNIVERSITY

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

v1

e2

e3

e1

v0

e5

e4

v3

v2

Matrices of Oriented Graphs

• Consider the following graph

v1

e2

e3

e1

v0

e5

e4

v3

e6

v2

• 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.

• 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

• 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)

• 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

and

fundamental cut-set equation

fundamental circuit equation

• Definition: Two edges ei and ek are said to be connected in series if they have exactly one common vertex of degree two.

v0

ek

ei

• 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.

vi

ek

ei

vk

(x1,y1)

(x2,y2)

(x0,y0)

(xn,yn)

(x0,y0)

(xn,yn)

(x2,y2)

(x1,y1)

(n+1) edges connected in parallel

A

a

v(t)

i(t)

B

b

v(t)

i(t)

b

Mathematical Model of an Independent Voltage Source

v(t)

Vs

i(t)

v(t)

i(t)

b

Mathematical Model of an Independent Voltage Source

v(t)

Is

i(t)

A-Branch Voltages Method:

Consider the following circuit.

b

a

4

3

c

1

7

6

5

e

d

8

Circuit Analysis

A-Branch Voltages Method:

1. Draw the circuit graph

• There are:

• 5 nodes (n)

• 8 edges (e)

• 3 voltage sources (nv)

• 1 current source (ni)

• A-Branch Voltages Method:

• 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

2

b

a

4

3

c

1

7

6

5

e

d

8

b

a

4

3

c

1

7

6

5

e

d

8

Circuit Analysis

• A-Branch Voltages Method:

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

b

a

4

3

c

1

7

6

5

e

d

8

Circuit Analysis

• A-Branch Voltages Method:

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

b

a

4

3

c

1

7

6

5

e

d

8

Circuit Analysis

• A-Branch Voltages Method:

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

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

b

a

4

3

c

1

7

6

5

e

d

8

Circuit Analysis

• A-Branch Voltages Method:

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

b

a

4

3

c

1

7

6

5

e

d

8

Circuit Analysis

• A-Branch Voltages Method:

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

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

2

3

6

4

5

7

8

Circuit Analysis

• Circuit graph and a proper tree

2

3

6

4

5

7

8

Circuit Analysis

• Fundamental cut-set equations

2

3

6

4

5

7

8

Circuit Analysis

• Fundamental cut-set equations

2

3

6

4

5

7

8

Circuit Analysis

• Fundamental circuit equations

v3= 9.5639V v2=-8.1203 V