Transcript EE 529 Circuit and Systems Analysis Lecture 2

EE 529
Circuit and Systems Analysis
Lecture 2
Mustafa Kemal Uyguroğlu
EASTERN MEDITERRANEAN UNIVERSITY
Mathematical Model of MultiTerminal Components
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5 TC
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A 5-terminal network element
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Measurement Graph
Mathematical Model of MultiTerminal Components
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Measurement Graph
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One of the terminal trees of the 5terminal 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 (n1) 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
di
dv
di di
dv dv


f1  i1 , i2 , , in 1 , v1 , v2 , , vn 1 , 1 , 2 , , n 1 , 1 , 2 , , n 1 , t   0
dt dt
dt dt dt
dt


di
dv
di di
dv dv


f 2  i1 , i2 , , in 1 , v1 , v2 , , vn 1 , 1 , 2 , , n 1 , 1 , 2 , , n 1 , t   0
dt dt
dt dt dt
dt



f n 1  i1 , i2 ,

, in 1 , v1 , v2 ,
di1 di2
, vn 1 , ,
,
dt dt
din 1 dv1 dv2
,
,
,
,
dt dt dt
dvn 1 
,
,t   0
dt

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
 i1 (t ) 
 i (t ) 
i (t )   2  ,




i
(
t
)
 n 1 
 v1 (t ) 
 v (t ) 
v (t )   2  ,




v
(
t
)
 n 1 
 f1 (.) 
 f (.) 
f (t )   2 




f
(.)
 n 1 
Then the terminal equations can be written in a more compact form as
follows:
d d


f  i , v, i , v, t   0
dt dt


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 vk(t) (ik(t)) is one group, then the terminal
current (voltage) at the same k-th port ik(t) (vk(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
xk(t)=vk(t) (or xk(t)=ik(t) ), then the k-th component of
y(t) will be yk(t)=ik(t) (or yk(t)=vk(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.
c1v1 (t )  c2v2 (t )  c3i1 (t )  c4i2 (t )  0
b1v1 (t )  b2v2 (t )  b3i1 (t )  b4i2 (t )  0
a1
A1
3 TC
A3
A2
a2
1
2
a3
Example
 Write these equations in
(a)
the following forms and
indicate the conditions
which the coefficents ci
and bi (i =1,2,3,4) should
be satisfied.
 v1 (t )  
v (t )   
 2  
  i1 (t ) 
 i (t ) 
 2 
(b)
 i1 (t )  
i (t )   
2  
  v1 (t ) 
 v (t ) 
 2 
 i1 (t )  
v (t )   
 2  
 v1 (t ) 
  i (t ) 
 2 
v1 (t )  
 i (t )   
2  
  i1 (t ) 
 v (t ) 
 2 
(c)
(d)
Hint
 The inverse of a 2×2 matrix.
a b 
A

c d 
A1 
1  d b 
ad  bc  c a 
 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:
p(t )  x (t )y(t )  y (t )x(t )
T
T
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 [t0,], then
the scalar function
t
w   p(t )dt  w  t0 
t0
where w(t0) is the energy accumulated
by the component at t0
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
1707 - 1783
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 (17071783)(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.
 http://www.contracosta.cc.ca.us/math/konig.htm
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.
C
B
D
A
F
E
 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.
C
B
D
A
F
E
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).
V2
V1
e
e: edge
v1, v2: vertices
Basic Concepts of the Graph Theory
 Definition: A vertex vi and an edge ei are
incident with each other if vi is one of the two
end points of the edge ei.
 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
V1
V3
V4
V3
V4
e1
e1
e2
e2
V2
V1
V2
e
V1
e
V2
V1
V2
Basic Concepts of the Graph Theory
 Degenerate cases

Isolate vertex : There are no edges incident on
it.
Vi

Self-loop: which may occur when the two
verices of an edge coincide.
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.
v2
v4
e4
e3
e1
(v=5, e=7)
v3
e6
e5
e2
v1
e7
v5
v: number of
vertices
e: number of
edges
Basic Concepts of the Graph Theory
 Remove the edge e1 from the graph
 The removal of an edge ei implies that, the
vertices vj and vk which are incident with the
edge are first splitted into two vertices as
vj=vj’ =vj’’and vk=vk’ =vk’’and then ei is
removed.
Basic Concepts of the Graph Theory
v2’
v2’’
v4
e4
e3
e1
v3
e6
e5
e2
v1’
v1’’
e7
v5
 Definition: A subgraph Gs of a graph G, denoted by Gs G
contains a subset of the edges of G. Conversely, G is a
supergraph of Gs and is denoted by G  Gs.
Basic Concepts of the Graph Theory
 Let G be a graph composed of the following
set of edges e1,e2,...,em; then it is convenient
to denote this particular graph G by
G={e1,e2,...,em} or G={ei}, i=1,2,...,m.
 Definition: Consider the graphs
G1  e1i , e2j  and G2  e1i , e3k 
 where i=1,2,...,m; j= 1,2,...,n; k=1,2,...,l.
Basic Concepts of the Graph Theory
 The graph G defined by
G  G1  G2  G2  G1  e1i , e2j , e3k 
is called the union of G1 and G2.
 The graph G defined by
G  G1  G2  G2  G1  e1i 
is called the intersection of G1 and G2.
 Note that if
G  G1  G2  0
then G is called a null graph
Basic Concepts of the Graph Theory
 Definition: Let G1 and G2 be two subgraphs of
G. If
G1  G2  G, and G1  G2  0,
then, G1 and G2 are said to be the complement
of each other.
 In the degenerate case let G1 = G and G2 be
a null graph. Note that G1 and G2 still satisfy
the Definition above. In such a case G1 is
said to be an improper subgraph of G. On the
other hand if G2 is not a null graph, then G1 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(v1)=3, d(v2)=3,
d(v3)=2.
v2
v4
e4
e3
e1
v3
e6
e5
e2
v1
e7
v5
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.
e6
e3
e2
e1
m(ei): multiplicty of the
edge ei.
m(e1)=1
e5
e4
m(e2)=1
e7
m(e3)=2
m(e4)=2
m(e5)=2
m(e6)=1
m(e7)=1
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.
e6
e3
e2
In the graph, the edge
sequence {e1,e2,e3,e4,e7,e5}
is an edge train. Note that
an edge train of a graph G is
a subgraph of G.
e5
e4
final
vertex
e7
e1
e1
initial vertex
e4
e2
e3
Basic Concepts of the Graph Theory
e1
e4
e2
final
vertex
e3
initial 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.
Set of edges:
v2
v4
e4
{e7,e5,e4,e3}
e3
e1
{e1,e2}
are PATHS.
v3
e6
e5
e2
The sets:
{e1,e5}
{e1,e2,e7}
v1
e7
v5
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
If G=G1G2 G3 G4
e2
e1
then p=4
e6
e4
e3
e7
G2
e8
G1
e5
e9
e14
e12
G3
e10
e19
e11
e13
e15
e20
e18
e16
e17
G4
e21
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.
