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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

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

2

b

a

4

3

c

1

7

6

5

e

d

8

Circuit AnalysisA-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.

Circuit Analysis

v3= 9.5639V v2=-8.1203 V

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