EE 529 Circuits and Systems Analysis

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EE 529 Circuits and Systems Analysis. Mustafa Kemal Uyguroğlu. Physical System. interconnection of physical devices or components Electrical System: interconnection of electrical elements. Phsical System. Mechanical System: interconnection of mechanical components. Phsical System.

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### 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

Phsical System

Analogous Equations

Phsical System

Analogous Equations

E

C

A

D

F

B

Phsical System
• In general, A system is an interconnection of components.
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
• 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

Circuit Elements and Their Mathematical Models
• Mathematical Model of the Component consists of the terminal graph and the terminal equation.

constitute the mathematical model of the component

Mathematical Model of Multi-Terminal Components

A 5-terminal network element

Measurement Graph

Mathematical Model of Multi-Terminal Components

One of the terminal trees of the 5-terminal component

Measurement Graph

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: