What have you learned in this lecture until now? Circuit theory Undefined quantities ( , ) Axioms ( , ) Two kinds of information: Nature of elements and connection structure We use graph theory for understanding the connection structure. How to obtain equations from the connection structure: Fundamental loops Fundamental cut-sets

Equivalent axiom: Kirchhoff’s current law for cut-sets Theorem: KCL for Gaussian surfaces KCL for nodes KCL for cut-sets Proof: (1) (2) Assume that KCL for Gaussian surfaces is true. Choose Gaussian surfaces as those that contain only one node. Then, we obtain KCL for nodes. (2) (3) Assume that KCL for nodes is true. Let C be a cut-set. Consider the KCL equations for the nodes on one side of the cut-set. The sum of all these equations gives KCL for cut-set C. (3) (1) Assume that KCL for cut-sets are true. Since a Gaussian surface that covers nodes can also be seen as a cut-set, KCL for Gaussian surfaces is also true. On the other hand, to a Gaussian surface that does not cover a node, no current enters, hence the sum is zero. For all lumped circuits, for all cut-sets, for all times t, the algebraic sum of all the currents flowing through a cut-set at time t is equal to zero. Choose one of the KCL’s as an axiom then the others are theorems! In other words:

Linearly Independent Equations We say that the equations are linearly independent if implies that If m equations are not linearly independent then Example: Are these equations linearly independent? How many equations do we need?

Are the KCL equations for nodes linearly independent? A ext What are the dimensions? i 0 k’th element (k=5) k’th edge Ending node of the edge Starting node of the edge

How can we check whether these equations are linearly independent or not? If r<n then In order to find r do elementary row operations until you get a lower triangular form. Then r is node 3.node 4.node 5.node A n = number of rows r = rank of the matrix

1.node 2.node 3.node 5.node A What is the dimension of A? What is the rank? Ai=0 Reference node is 4. In order to write the reduced node matrix A choose a reference node!

KVL in terms of node voltages M

Consider KCL equations: CS 1 CS 2 CS 3 CS 4 CS 5 CS 6 CS 7 How many linearly independent equations are there? QaQa KCL and KVL equations for cut-sets and loops

Choose a tree and specify the fundemental cut-sets Tree: {1,3,4,5} Fundemental cut-sets: FC 1 : {1,2,8,9} FC 2 : {3,7,8} FC 3 : {4,6,7,9} FC 4 : {5,6} IQ* Q

Write KVL equations: Specify some loops L 1 : {1,2} L 2 : {2,3,8} L 3 : {3,4,7} L 4 : {4,5,6} L 5 : {7,8,9} L 6 : {1,3,8} L 7 : {1,4,7,8} L 8 : {1,5,6,7,8} L 9 : {2,4,9} L 10 : {2,5,6,9} L 11 : {2,5,6,9} How many linearly independent equations are there? 5

Again, specify the fundemental loops: Tree: {1,3,4,5} Fundemantal loops: FL 1 : {1,2} FL 4 : {1,3,8} FL 3 : {3,4,7} FL 2 : {4,5,6} FL 5 : {1,4,9} chords: {2,6,7,8,9} IB* B Note that

1- a) Draw the circuit graph for the circuit above. b) Determine a tree. c) Write KVL for fundamental loops in matrix form. d) Write KCL for fundamental cut-sets in matrix form.

Proof: Choose a reference node and construct the node matrix A. Then Tellegen’s Theorem

Note that in Tellegen’s Theorem v and i seem unrelated: v is a possible solution for voltages and i is a possible solution for currents. They don’t have to exist together in the circuit. In other words, if v’ and v’’ satisfy KVL, i’ and i’’ satisfy KCL, then If v and i are solutions of the same circuit then Tellegen Theorem implies Let’s prove that using Tellegen’s Theorem.