1. EASTERN MEDITERRANEAN UNIVERSITY EE 529 Circuit and Systems Analysis Lecture 4

2. 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

3. Matrices of Oriented Graphs Consider the following graph e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e3e3 v1v1 v2v2 v3v3 v0v0 e1e1 e2e2 e4e4 e5e5 e6e6

4. FUNDAMENTAL POSTULATES Now, Let G be a connected graph having e edges and let be two vectors where x i and y i, i=1,...,e, correspond to the across and through variables associated with the edge i respectively.

5. 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

6. 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)

7. Fundamental Circuit & Cut-set Equations Consider a graph G and a tree T in G. Let the vectors x and y partitioned as where x b (y b ) and x c (y c ) correspond to the across (through) variables associated with the branches and chords of the tree T, respectively. Then and fundamental circuit equation fundamental cut-set equation

8. Series & Parallel Edges Definition: Two edges e i and e k are said to be connected in series if they have exactly one common vertex of degree two. eiei ekek v0v0

9. Series & Parallel Edges Definition: Two edges e i and e k are said to be connected in parallel if they are incident at the same pair of vertices v i and v k. eiei ekek vivi vkvk

10. (n+1) edges connected in series (x 1,y 1 ) (x 2,y 2 ) (x n,y n ) (x 0,y 0 )

11. (n+1) edges connected in parallel (x 0,y 0 ) (x 1,y 1 ) (x 2,y 2 ) (x n,y n )

12. Mathematical Model of a Resistor A B a b v(t) i(t)

13. Mathematical Model of an Independent Voltage Source a b v(t) i(t) v(t) i(t) VsVs

14. Mathematical Model of an Independent Voltage Source a b v(t) i(t) v(t) i(t) IsIs

15. Circuit Analysis A-Branch Voltages Method: Consider the following circuit.

16. Circuit Analysis A-Branch Voltages Method: 1. Draw the circuit graph 1 2 3 4 5 6 7 8 a b c d e There are: 5 nodes (n) 8 edges (e) 3 voltage sources (n v ) 1 current source (n i )

17. Circuit Analysis A-Branch Voltages Method: 1.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 1 2 3 4 5 6 7 8 a b c d e

18. Circuit Analysis A-Branch Voltages Method: 2. Write the fundamental cut-set equations for the tree branches which do not correspond to voltage sources. 1 2 3 4 5 6 7 8 a b c d e

19. Circuit Analysis A-Branch Voltages Method: 2. Write the currents in terms of voltages using terminal equations. 1 2 3 4 5 6 7 8 a b c d e

20. Circuit Analysis A-Branch Voltages Method: 2. Substitute the currents into fundamental cut-set equation. 1 2 3 4 5 6 7 8 a b c d e 3. v 3, v 5, and v 6 must be expressed in terms of branch voltages using fundamental circuit equations.

21. Circuit Analysis A-Branch Voltages Method: 1 2 3 4 5 6 7 8 a b c d e Find how much power the 10 mA current source delivers to the circuit

22. Circuit Analysis A-Branch Voltages Method: 1 2 3 4 5 6 7 8 a b c d e Find how much power the 10 mA current source delivers to the circuit

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

24. Circuit Analysis Circuit graph and a proper tree 1 2 3 4 5 6 78

25. Circuit Analysis Fundamental cut-set equations 1 2 3 4 5 6 78

26. Circuit Analysis Fundamental cut-set equations 1 2 3 4 5 6 78

27. Circuit Analysis Fundamental circuit equations 1 2 3 4 5 6 78

28. Circuit Analysis v 3 = 9.5639V v 2 = -8.1203 V

29. Circuit Analysis