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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

EE 529 Circuit and Systems Analysis

Lecture 4

EASTERN MEDITERRANEAN UNIVERSITY

matrices of oriented graphs
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
matrices of oriented graphs1
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
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 postulates1
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 postulates2
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
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
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 edges1
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
(n+1) edges connected in series

(x1,y1)

(x2,y2)

(x0,y0)

(xn,yn)

circuit analysis
Circuit Analysis

A-Branch Voltages Method:

Consider the following circuit.

circuit analysis1
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 analysis2
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

circuit analysis3
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.
circuit analysis4
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.
circuit analysis5
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.

circuit analysis6
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 analysis7
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 analysis8
Circuit Analysis
  • Example: Consider the following circuit. Find ix in the circuit.
circuit analysis9
1

2

3

6

4

5

7

8

Circuit Analysis
  • Circuit graph and a proper tree
circuit analysis10
1

2

3

6

4

5

7

8

Circuit Analysis
  • Fundamental cut-set equations
circuit analysis11
1

2

3

6

4

5

7

8

Circuit Analysis
  • Fundamental cut-set equations
circuit analysis12
1

2

3

6

4

5

7

8

Circuit Analysis
  • Fundamental circuit equations
circuit analysis13
Circuit Analysis

v3= 9.5639V v2=-8.1203 V

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