Ee 529 circuit and systems analysis
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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

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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)


N 1 edges connected in parallel

(x0,y0)

(xn,yn)

(x2,y2)

(x1,y1)

(n+1) edges connected in parallel


Mathematical model of a resistor

Mathematical Model of a Resistor

A

a

v(t)

i(t)

B

b


Mathematical model of an independent voltage source

a

v(t)

i(t)

b

Mathematical Model of an Independent Voltage Source

v(t)

Vs

i(t)


Mathematical model of an independent voltage source1

a

v(t)

i(t)

b

Mathematical Model of an Independent Voltage Source

v(t)

Is

i(t)


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


Circuit analysis14

Circuit Analysis


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