Download EE 529 Circuits and Systems Analysis

EE 529 Circuits and
Systems Analysis
Mustafa Kemal Uyguroğlu
EASTERN MEDITERRANEAN UNIVERSITY
Physical System
 interconnection of physical devices or
components
 Electrical System: interconnection of electrical
elements
Phsical System
 Mechanical System: interconnection of
mechanical components
Phsical System
 It is possible to make electrical and
mechanical systems using analogs. An
analogous electrical and mechanical system
will have differential equations of the same
form. The analogous quantities are given
below.
Phsical System
Analogous Quantities
Electrical
Quantity
Mechanical
Analog
Voltage, e
Velocity, v
Current, i
Force, f
Resistance, R
Lubricity, 1/B
(Inverse friction)
Capacitance, C
Mass, M
Inductance, L
Compliance, 1/K
(Inverse spring constant)
Transformer, N1:N2
Lever, L1:L2
Phsical System
Analogous Equations
Electrical
Equation
Mechanical
Analog I
(Force-Current)
Phsical System
Analogous Equations
voltage of ground=0
velocity of ground=0
(you can apply any current to
ground and voltage remains 0)
(you can apply any force to
ground and velocity remains 0)
Conversion from Electrical to Mechanical 1 -- Visual Method
Start with an electrical
circuit. Label all node voltages.
a
Draw over circuit, replacing
electrical elements with their
analogs; current sources replaced by
force generators, voltage sources by
input velocities, resistors with
friction elements, inductors with
springs, and capacitors (which must
be grounded) by capacitors. Each
node becomes a position.
Label currents, positions, and
mechanical elements as they were in
the original electrical circuits.
Phsical System
 In general, A system is an interconnection of
components.
E
C
A
D
B
F
System Description and Analysis
Procedure
 In order to analyze a system, System will
have the following properties:



It will be composed of connected assembly of
finite number of components
The pattern of component interconnection is
recongnizable
Each component can be characterized in a
manner entirely independent of any other
component connected to it.
System Description and Analysis
Procedure
 The analysis procedure



Modeling: The characterization of components
by mathemical models
Formulation: The development of sets of
equations describing the overall system
Solution: The mathematical procedures of
solving the equations formulated.
Modeling
 Each component in a system can be studied
in isolation and a mathematical model can be
develop for it. The procedure to obtain the
mathematical model is either experimental
i.e., determined after performing certain tests
on the component, or, based upon the
knowledge of the physics of the components.
Formulation
 When the mathematical models of all the
components in the system are established, a
set of equations called system equations is
derived by combining the mathematical
model of the components with the equations
describing the interconnection pattern of
these components.
Solution
 By solving the system equations, the
responses (outputs) can be expressed
uniquely in terms of the excitations (inputs).
Circuit Elements and Their
Mathematical Models
 Circuit elements or components are the
building blocks of a network.
 As explained, their properties can be put into
a mathematical representation by making a
number of observations (electrical
measurements) at the terminals of the
components.
Circuit Elements and Their
Mathematical Models
A1
A
Exciting
Circuit
V
A2
+
a1
A1
A
+
V
i(t)
v(t)
a2
A2
connection of Ammeter and Voltmeter
terminal graph
to the two-terminal component
Circuit Elements and Their
Mathematical Models
 i(t) and v(t) are terminal variables
 The ralation between the terminal variables is
called terminal equation.
 The terminal equation of a two-terminal
component is
f(v,i)=0
or
dv di 

f  v, i , ,   0
dt dt 

Circuit Elements and Their
Mathematical Models
 Mathematical Model of the Component
consists of the terminal graph and the
terminal equation.
terminal graph; terminal equation
constitute the mathematical model of the
component
Example: Mathematical Model of a
Diode
a
A
i(t)
B
v(t)
b
mathematical model
Mathematical Model of MultiTerminal Components
A1
1
a1
A2
5
a2
7
6
5 TC
2
9
8
10
A5
A3
a5
a3
4
A4
A 5-terminal network element
3
a4
Measurement Graph
Mathematical Model of MultiTerminal Components
1
a1
5
7
6
a1
a2
1
2
9
8
8
10
10
a5
a3
4
3
a4
Measurement Graph
a2
a5
a3
4
a4
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

f1  i1 , i2 ,

, in 1 , v1 , v2 ,
, vn 1 ,
di1 di2
,
,
dt dt
,
din 1 dv1 dv2
,
,
,
dt dt dt
,
dvn 1 
,t   0
dt


f 2  i1 , i2 ,

, in 1 , v1 , v2 ,
, vn 1 ,
di1 di2
,
,
dt dt
,
din 1 dv1 dv2
,
,
,
dt dt dt
,
dvn 1 
,t   0
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

