EE 529 Circuit and Systems Analysis Lecture 6

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

Mathematical Models of Electrical Components A. Nullator a b

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.

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

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

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

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

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

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

Mathematical Models of Electrical Components Representation of Ideal Transformer with Dependent Sources i 1 + v 1 -

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

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

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

Mathematical Models of Electrical Components Operational Amplifier a 1 1 a 2 a 3 2 a 0 3

Mathematical Models of Electrical Components Operational Amplifier a 1 1 a 2 a 3 2 3 a 0 A: open loop gain, very big!

Mathematical Models of Electrical Components Operational Amplifier a 3 a 1 1 3 a 0 A: open loop gain, very big!

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

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

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

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

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

Analysis of Circuits Containing Multi-terminal Components

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

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

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

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

Analysis of Circuits Containing Multi-terminal Components n 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: n Using the branch voltages method, obtain the circuit equations

Analysis of Circuits Containing Multi-terminal Components (Rb) 3 (Ra) 1 2 Vs 4

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

Analysis of Circuits Containing Multi-terminal Components nva , vb and v 3 can be expressed in terms of branch voltages using fundamental circuit equations. • Subst. of Eq. (5) into (4) gives: