EE 529 Circuit and Systems Analysis

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

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. 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

4. 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.

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 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

8. 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

9. 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

10. (n+1) edges connected in series (x1,y1) (x2,y2) (x0,y0) (xn,yn)

11. (x0,y0) (xn,yn) (x2,y2) (x1,y1) (n+1) edges connected in parallel

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

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

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

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

16. 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)

17. 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

18. 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.

19. 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.

20. 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.

21. 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

22. 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

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

24. 1 2 3 6 4 5 7 8 Circuit Analysis • Circuit graph and a proper tree

25. 1 2 3 6 4 5 7 8 Circuit Analysis • Fundamental cut-set equations

26. 1 2 3 6 4 5 7 8 Circuit Analysis • Fundamental cut-set equations

27. 1 2 3 6 4 5 7 8 Circuit Analysis • Fundamental circuit equations

28. Circuit Analysis v3= 9.5639V v2=-8.1203 V

29. Circuit Analysis