1. EASTERN MEDITERRANEAN UNIVERSITY EE 529 Circuits and Systems Analysis Mustafa Kemal Uyguroğlu

2. Physical System interconnection of physical devices or components Electrical System: interconnection of electrical elements

3. Phsical System Mechanical System: interconnection of mechanical components

4. 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.

5. Phsical System Analogous Quantities Electrical Quantity Mechanical Analog Voltage, eVelocity, v Current, iForce, f Resistance, R Lubricity, 1/B (Inverse friction) Capacitance, CMass, M Inductance, L Compliance, 1/K (Inverse spring constant) Transformer, N1:N2Lever, L1:L2

6. Phsical System Analogous Equations

7. Phsical System Analogous Equations

9. Phsical System In general, A system is an interconnection of components. A B C D E F

10. 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.

11. 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.

12. 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.

13. 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.

14. Solution By solving the system equations, the responses (outputs) can be expressed uniquely in terms of the excitations (inputs).

15. 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.

16. Circuit Elements and Their Mathematical Models

17. 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

18. 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

19. Example: Mathematical Model of a Diode mathematical model

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

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

22. 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.

23. 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:

24. 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

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