EASTERN MEDITERRANEAN UNIVERSITY EE 529 Circuit and Systems Analysis Lecture 2 Mustafa Kemal Uyguroğlu

Mathematical Model of Multi- Terminal Components Measurement GraphA 5-terminal network element

Mathematical Model of Multi- Terminal Components Measurement Graph One of the terminal trees of the 5- terminal component

First Postulate of Network Theory All the properties of an n-terminal component can be described by a mathematical relation between a set of (n-1) voltage and a set of (n- 1) current variables.

Terminal Equation of Multi-terminal Components First Postulate of Network Theory shows that the mathematical model of an n-terminal component consists of a terminal graph (a tree) and the mathematical relations, (n-1) in numbers, between 2(n-1) terminal variables which describe the physical behaviour of the component. Hence the terminal equations of an n-terminal component may have the following general forms:

Terminal Equation of Multi-terminal Components If column matrices or vectors are used to denote the totality of the terminal voltage and current variables as

Terminal Equation of Multi-terminal Components Then the terminal equations can be written in a more compact form as follows:

Power and Energy The mathematical model of an n-terminal component contains two fundamental parts: A topological tree, called the terminal graph which gives information about how the terminal voltage and current measurement are made at the terminals. The terminal equations associated with this terminal graph which describe the relationships between the measured terminal voltages and terminal currents.

Power and Energy Since 2(n-1) terminal variables are related through (n-1) terminal equations, all of the terminal variables cannot be chosen arbitrarily. In general one can select only (n-1) of them arbitrarily. Once such a selection has been made, then rest of the (n-1) terminal variables are determined through these (n-1) equations.

Power and Energy Let the terminal variables be divided into two groups each containing (n-1) terminal variables such that if k=1,2,...,n-1 and the terminal voltage (current) at the k-port v k (t) (i k (t)) is one group, then the terminal current (voltage) at the same k-th port i k (t) (v k (t)) be in the other group. In general in each group there will be a mixture of the terminal variables. Let two vectors defined by the variables in each group be denoted by x(t) and y(t). Hence, if the k-th component of the vector x(t) is x k (t)=v k (t) (or x k (t)=i k (t) ), then the k-th component of y(t) will be y k (t)=i k (t) (or y k (t)=v k (t)). For this reason the vectors x(t) and y(t) are called complementary vectors.

Example The terminal equations of a 3-terminal component are given below. a1a1 a2a2

Example Write these equations in the following forms and indicate the conditions which the coefficents c i and b i (i =1,2,3,4) should be satisfied. (a) (b) (c) (d)

Hint The inverse of a 2×2 matrix. The matrix 2×2 matrix A will have an inverse so long as ad - bc  0.

Power and Energy Definition: For an n-terminal component, the instantaneous power p(t) is defined as the scalar product of the current and voltage vectors corresponding to a selected terminal tree.

Power and Energy The definition of p(t) can be expressed in a more general form. If the mixed terminal variables are used:

Power and Energy Definition: If the instantaneous power of an n- terminal component id p(t) and t indicating any instant of time in the interval [t 0,  ], then the scalar function where w(t 0 ) is the energy accumulated by the component at t 0

Notes of Graph Theory The discipline of mathematics that deals with the topology (the manner that the components are interconnected in the system) of a system is called graph theory.

Leonhard Euler Euler became the father of graph theory when he has presented his famous Konigsberg bridge problem

Konigsberg Bridge Problem In the town Konigsberg, there are two islands on Pregel river which are linked to each other and to the banks of the river by seven bridges as shown:

Konigsberg Bridge Problem Some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler ( )(pronounced "oiler"), a notable genius of the eighteenth-century.

The graph of Konigsberg Bridge In proving that this problem is impossible to solve Euler laid the foundations of graph theory.

Konigsberg Bridge Problem Euler recognized that in order to succeed, a traveler in the middle of the journey must enter a land mass via one bridge and leave by another, thus that land mass must have an even number of connecting bridges. Further, if the traveler at the start of the journey leaves one land mass, then a single bridge will suffice and upon completing the journey the traveler may again only require a single bridge to reach the ending point of the journey. The starting and ending points then, are allowed to have an odd number of bridges. But if the starting and ending point are to be the same land mass, then it and all other land masses must have an even number of connecting bridges.

Graphs In order to understand the idea of graphs let us, as an example, consider the railroads of a country. Let us assume that there are six stations in this country and the interconnection of these stations are as shown.

Graphs The vertices and the lines denote the railroad stations and railroads respectively. One interesting fact about this specific graph is that each line and vertex represent a physical concept, the railroad or the station. However this is not essential.

Graphs Assume that there are six men, and A is a friend of B; B is a friend of A and C; C is a friend of B, D, E, and F; D is a friend of C and E; E is a friend of F, C, and D; and F is a friend of C and E. Then the graph can be considered as the mathematical representation of these mutual friendships.

Basic Concepts of the Graph Theory Definition: A line segment together with its two distinct end points is called an edge. Definition: An end point of an edge is called a vertex (node). V2V2 V1V1 e e: edge v 1, v 2 : vertices

Basic Concepts of the Graph Theory Definition: A vertex v i and an edge e i are incident with each other if v i is one of the two end points of the edge e i. Definition: A linear topological graph G is a collection of edges and vertices, where two or more of the edges may be incident on the same vertex, and two edges have a point in common which is not a vertex.

Basic Concepts of the Graph Theory V3V3 V4V4 e2e2 V1V1 V2V2 e1e1 V4V4 V3V3 e2e2 V1V1 V2V2 e1e1 V1V1 V2V2 e V1V1 V2V2 e

Degenerate cases Isolate vertex : There are no edges incident on it. Self-loop: which may occur when the two verices of an edge coincide. ViVi v e

Basic Concepts of the Graph Theory For simplicity the word graph will be used instead of linear topological graph. Consider the following graph G. v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 (v=5, e=7) v: number of vertices e: number of edges

Basic Concepts of the Graph Theory Remove the edge e 1 from the graph The removal of an edge e i implies that, the vertices v j and v k which are incident with the edge are first splitted into two vertices as v j =v j ’ =v j ’’and v k =v k ’ =v k ’’and then e i is removed.

Basic Concepts of the Graph Theory Definition: A subgraph G s of a graph G, denoted by G s  G contains a subset of the edges of G. Conversely, G is a supergraph of G s and is denoted by G  G s. v 1 ’’ v 2 ’’ v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 v2’v2’ v1’v1’

Basic Concepts of the Graph Theory Let G be a graph composed of the following set of edges e 1,e 2,...,e m ; then it is convenient to denote this particular graph G by G={e 1,e 2,...,e m } or G={e i }, i=1,2,...,m. Definition: Consider the graphs where i=1,2,...,m; j= 1,2,...,n; k=1,2,...,l.

Basic Concepts of the Graph Theory The graph G defined by is called the union of G 1 and G 2. The graph G defined by is called the intersection of G 1 and G 2. Note that if then G is called a null graph

Basic Concepts of the Graph Theory Definition: Let G 1 and G 2 be two subgraphs of G. If then, G 1 and G 2 are said to be the complement of each other. In the degenerate case let G 1 = G and G 2 be a null graph. Note that G 1 and G 2 still satisfy the Definition above. In such a case G 1 is said to be an improper subgraph of G. On the other hand if G 2 is not a null graph, then G 1 is said to be proper subgraph of G.

Basic Concepts of the Graph Theory Definition: The degree of a vertex is defined as the number of edges incident on it and is denoted by d(v). As an example in the graph, d(v 1 )=3, d(v 2 )=3, d(v 3 )=2. v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7

Basic Concepts of the Graph Theory Definition: If the edges of a graph or subgraph are ordered such that each edge has a vertex in common with the preceeding edge (in the ordered sequence) and the other vertex in common with the succeeding edge, then this set of edges is called an edge sequence. Note that in an edge sequence any edge may appear a number of times.

Basic Concepts of the Graph Theory Definition: The number of times an edge appears in an edge sequence is called the multiplicity of the edge. e1e1 e2e2 e3e3 e5e5 e6e6 e7e7 m(e i ): multiplicty of the edge e i. m(e1)=1 m(e2)=1 m(e3)=2 m(e4)=2 m(e5)=2 m(e6)=1 m(e7)=1 e4e4

Basic Concepts of the Graph Theory Definition: If each edge of an edge sequence is of multiplicity one, then the sequence is called an edge train. In the graph, the edge sequence {e 1,e 2,e 3,e 4,e 7,e 5 } is an edge train. Note that an edge train of a graph G is a subgraph of G. e1e1 e2e2 e3e3 e5e5 e6e6 e7e7 e4e4 e1e1 e2e2 e3e3 e4e4 initial vertex final vertex

Basic Concepts of the Graph Theory e1e1 e2e2 e3e3 e4e4 initial vertex final vertex terminal vertices If terminal vertices are distinct, then the edge train is called an open edge train. If the terminal vertices are coincident, then the edge train is called a closed edge train.

Basic Concepts of the Graph Theory Definition: If in an open edge train, the degree of each nonterminal vertex is exactly two, then it is called a path. v1v1 v2v2 v3v3 v4v4 v5v5 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 Set of edges: {e 1,e 2 } {e 7,e 5,e 4,e 3 } are PATHS. The sets: {e 1,e 5 } {e 1,e 2,e 7 } are NOT paths.

Basic Concepts of the Graph Theory Definition: If in a graph G there exists at least one path between any two vertices then G is called a connected graph. Conversely, if G is not connected then it is called ab unconnected or separated graph. The connected subgraphs of G are called the connected parts. Let the number of connected parts of a graph G be denoted by p.

Basic Concepts of the Graph Theory e1e1 e2e2 e3e3 e5e5 e4e4 e6e6 e7e7 e8e8 e9e9 e 10 e 11 e 12 e 13 e 14 e 15 e 16 e 17 e 18 e 19 e 20 e 21 G1G1 G2G2 G3G3 G4G4 If G=G 1  G 2  G 3  G 4 then p=4

Basic Concepts of the Graph Theory Definition: A path which its two terminal vertices coincident is called a circuit, closed path or loop. Definition: In a connected graph G of v vertices the subgraph T that satisfies the following properties is called a tree.  T is connected  T contains all the vertices of G  T contains no circuit,  T contains exactly v-1 number of edges.