1. EE 529 Circuit and Systems Analysis Lecture 6
Mustafa Kemal Uyguroğlu
EASTERN MEDITERRANEAN UNIVERSITY

2. Mathematical Models of Electrical Components
A. Nullator a b

3. Mathematical Models of Electrical Components
B. Norator a b
Nullator and Norator are conceptual elements. They are used to represent some electrical elements in different ways.

4. Mathematical Models of Electrical Components
C. Three Terminal and two-port Circuit Elements
C.1 TRANSISTOR e b c 1 2
Ebers-Moll Equations

5. Mathematical Models of Electrical Components
C. Three Terminal and two-port Circuit Elements
C.2 IDEAL TRANSFORMER a c b 1 2

6. Mathematical Models of Electrical Components
C. Three Terminal and two-port Circuit Elements
C.2 IDEAL TRANSFORMER c d 2 a b 1

7. Mathematical Models of Electrical Components
C. Three Terminal and two-port Circuit Elements
C.3 IDEAL GYRATOR a c b 1 2

8. Mathematical Models of Electrical Components
C. Three Terminal and two-port Circuit Elements
C.3 IDEAL GYRATOR c d 2 a b 1

9. Mathematical Models of Electrical Components
Representation of Ideal Transformer with Dependent Sources c d 2 a b 1

10. Mathematical Models of Electrical Components
Representation of Ideal Transformer with Dependent Sources i1 + v1 -

11. Mathematical Models of Electrical Components
Representation of Ideal Transformer with Dependent Sources

12. Mathematical Models of Electrical Components
Representation of Ideal Gyrator with Dependent Sources

13. Mathematical Models of Electrical Components
Representation of Ideal Gyrator with Dependent Sources

14. Mathematical Models of Electrical Components
Operational Amplifier a1 1 2 a2 a3 a0 3

15. Mathematical Models of Electrical Components
Operational Amplifier a1 1 2 a2 a3 a0 3
A: open loop gain, very big!

16. Mathematical Models of Electrical Components
Operational Amplifier a1 1 a3 a0 3
A: open loop gain, very big!

17. Mathematical Models of Electrical Components
Representation of OP-AMP with Nullator and Norator.

18. Analysis of Circuits Containing Multi-terminal Components
The terminal equations of resistors are
The terminal equations of multi-terminal components are similar to two-terminal components but the coefficient matrices are full.

19. Analysis of Circuits Containing Multi-terminal Components
where
vb : branch voltages
vc : Chord voltages
ib : branch currents
ic : Chord currents

20. Analysis of Circuits Containing Multi-terminal Components
(A) Branch Voltages Method
By using the terminal equations of the multiterminal components, the above equation can be written as

21. Analysis of Circuits Containing Multi-terminal Components
On the other, the chord voltages can be written in terms of branch voltages by using the fundamental circuit equations.
If Eq.(2) is substituted into (1) and the known quantities are collected on the right hand side then the following equation is obtained:

22. Analysis of Circuits Containing Multi-terminal Components

23. Analysis of Circuits Containing Multi-terminal Components
Example : In the following figure, the circuit contains a 3-terminal component. The terminal equation of the 3-terminal component is:
Using the branch voltages method, obtain the circuit equations 1 2 a b c IS VS
(Ra)
(Rb) 1 2 a b c

24. Analysis of Circuits Containing Multi-terminal Components
Example : In the following figure, the circuit contains a 3-terminal component. The terminal equation of the 3-terminal component is:
Using the branch voltages method, obtain the circuit equations 1 2 a b c IS VS
(Ra)
(Rb) 1 2 a b c

25. Analysis of Circuits Containing Multi-terminal Components
The fundamental cut-set equations for tree branches 1 and 2: 1 2 a b c IS VS
(Ra)
(Rb)
The terminal equations of the resistors:
Subst. of Eqs.(3) and (1) into (2) yields:

26. Analysis of Circuits Containing Multi-terminal Components
va and vb can be expressed in terms of branch voltages using fundamental circuit equations.
Subst. of Eq.(5) into (4) gives:

27. Analysis of Circuits Containing Multi-terminal Components
Example : In the following figure, the circuit contains a 2-port gyrator and a 3-terminal voltage controlled current source. The terminal equations of these components are:
Using the branch voltages method, obtain the circuit equations

28. Analysis of Circuits Containing Multi-terminal Components1 2 3 4 Vs
(Ra)
(Rb)

29. Analysis of Circuits Containing Multi-terminal Components1 2 3 4 Vs
(Ra)
(Rb)
The terminal equations of the resistors:
Subst. of Eqs.(3) and (1) into (2) yields:

30. Analysis of Circuits Containing Multi-terminal Components
va , vb and v3 can be expressed in terms of branch voltages using
fundamental circuit equations.
Subst. of Eq.(5) into (4) gives: