EE 529 Circuit and Systems Analysis

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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

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
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

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.

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
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

and

fundamental cut-set equation

fundamental circuit 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.

v0

ek

ei

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.

vi

ek

ei

vk

(n+1) edges connected in series

(x1,y1)

(x2,y2)

(x0,y0)

(xn,yn)

Circuit Analysis

A-Branch Voltages Method:

Consider the following circuit.

2

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)
Circuit Analysis
• 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

2

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.
2

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.
2

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.

2

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

2

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

Circuit Analysis
• Example: Consider the following circuit. Find ix in the circuit.
1

2

3

6

4

5

7

8

Circuit Analysis
• Circuit graph and a proper tree
1

2

3

6

4

5

7

8

Circuit Analysis
• Fundamental cut-set equations
1

2

3

6

4

5

7

8

Circuit Analysis
• Fundamental cut-set equations
1

2

3

6

4

5

7

8

Circuit Analysis
• Fundamental circuit equations
Circuit Analysis

v3= 9.5639V v2=-8.1203 V