EE 529 Circuit and Systems Analysis Lecture 4

EE 529 Circuit and Systems Analysis Lecture 4 EASTERN MEDITERRANEAN UNIVERSITY

Matrices of Oriented Graphs n 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 Graphs n. Consider the following graph v 1 e 2 e 1 v 0 e 5 e 4 v 2 e 3 e 6 v 3

FUNDAMENTAL POSTULATES n. 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 n 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 n 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 n 2. POSTULATE is called the circuit equations of electrical system. (is also referred to as Kirchoff’s Voltage Law) n 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 n Consider a graph G and a tree T in G. Let the vectors x and y partitioned as n where xb (yb) and xc (yc) correspond to the across (through) variables associated with the branches and chords of the tree T, respectively. n Then and fundamental circuit equation fundamental cut-set equation

Series & Parallel Edges n Definition: Two edges ei and ek are said to be connected in series if they have exactly one common vertex of degree two. v 0 ei ek

Series & Parallel Edges n 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 (x 1, y 1) (x 2, y 2) (x 0, y 0) (xn, yn)

(n+1) edges connected in parallel (x 0, y 0) (x 1, y 1) (x 2, y 2) (xn, yn)

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

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

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

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

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

Circuit Analysis A-Branch Voltages Method: 1. Select a proper tree: (n-1=4 branches) Ø Place voltage sources in tree Ø Place current sources in co-tree Ø Complete the tree from the resistors 2 a 3 b 4 c 1 5 d 8 6 7 e

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 a 3 b 4 c 1 5 d 8 6 7 e

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

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

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

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

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

Circuit Analysis n Circuit graph and a proper tree 1 2 3 6 4 7 8 5

Circuit Analysis n Fundamental cut-set equations 1 2 3 6 4 7 8 5

Circuit Analysis n Fundamental cut-set equations 1 2 3 6 4 7 8 5

Circuit Analysis n Fundamental circuit equations 1 2 3 6 4 7 8 5

Circuit Analysis v 3= 9. 5639 V v 2=-8. 1203 V

Circuit Analysis