N-Terminal Resistors d1d1 d2d2 d3d3 + __ + v1v1 v2v2 i1i1 i2i2 3-terminal or 2-port + _ _ + v1v1 v2v2 i1i1 i2i2 3-terminal 2-port d1d1 d2d2 + __ + v1v1.

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N-Terminal Resistors d1d1 d2d2 d3d3 + __ + v1v1 v2v2 i1i1 i2i2 3-terminal or 2-port + _ _ + v1v1 v2v2 i1i1 i2i2 3-terminal 2-port d1d1 d2d2 + __ + v1v1 v2v2 i1i1 i2i2 Terminal variables for a 3-terminal: v 1,v 2, i 1, i 2 Terminal variables for a 2-port: v 1,v 2, i 1, i 2

Different Representations For a 1-port(2-terminal) there are two representations For a 2-port(3-terminal) there are representations Why? Current-controlled: Voltage-controlled: Hybrid-1: Hybrid-2: Transmission-1: Transmission-2:

An Example Voltage-controlled Hybrid-1 L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

Hybrid-2 Transmission-1 Transmission-2 L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

If functions are linear then they act like matrices: Six Representations for Linear 2-ports + _ _ + v1v1 v2v2 i1i1 i2i2 N Current-controlled: + _ _ + v1v1 v2v2 i1i1 i2i2 N Voltage-controlled: + _ + _

+ _ _ + v1v1 v2v2 i1i1 i2i2 N Hybrid-1: + _ + _ _ + v1v1 v2v2 i1i1 i2i2 N Hybrid-2 _ + Transmission-1Transmission-2

How To Obtain A Representation For Linear 2-ports There are two ways: 1) All in one go 2) Each parameter seperately 1) All in one go: Connect appropriate sources to the ports according to the type of the representation. Write KCL and KVL equations and element equations for the elements in 2-port. Use these to find the representation. EXAMPLE: Find the current-controlled representation for the following 2-port:

For instance, for finding r _ _ + v1v1 v2v2 i1i1 i 2 =0 2) Each parameter seperately: Check the representation and express the parameter in terms of port variables. From this you will see that you need to connect a voltage or current source to one of the ports and the other port needs to be left either as a short-circuit or as an open circuit. Use this circuit to find the parameter, again using KVL,KCL and element equations. For finding g

L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits”, Mc.Graw Hill, 1987, New York EXAMPLE: Find hybrid-2 representation for the following 2-port:

Voltage controlled current source + _ v 1 =0 i1i1 _ + v2v2 i2i2 + _ rmi1rmi1 Current controlled voltage source + _ v1v1 _ + v2v2 i2i2 gmv1gmv1 i 1 =0 Some Linear 2-ports

Current controlled current source + _ v 1 =0 i1i1 _ + v2v2 i2i2 αi1αi1 Voltage controlled voltage source + _ v1v1 _ + v2v2 i2i2 + _ µv 1 i 1 =0

Ideal transformator n:1 Gyrator + _ v1v1 i1i1 _ + v2v2 i2i2 G

Obtain the element equations for the following 2-port! L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits”, Mc.Graw Hill, 1987, New York

_ _ _ 3-terminal element V 12 V 23 V 13 İ 1 (t) İ 2 (t) İ 3 (t) Figure Figure 2 The element equation for the element in Figure 1 is given below: a)Find the element equation in terms of the quantities given in the element graph in Figure 2. b) Construct a 3-terminal element that has this element equation using only 2-terminal linear resistors and some dependent sources.