General two-Port networks

General two-port networks
The contents:
The distinction between one-port and two-port net works-------1
Admittance (Y) parameters------------------------------------------3
Some equivalent networks-------------------------------------------7
Impedance (Z) parameters------------------------------------------10
Hybrid (h) parameters-----------------------------------------------13
Transmission (t) parameters---------------------------------------14
The distinction between one-port and two-port networks
A pair of terminals at which the signal may enter or leave the network is called a port, and a network having only one such pair of terminals is called one-port network.
No connection may be made to any other nodes internal to the one-port. Therefore, must be equal to in the one-port.
When more than one pair of terminals is present, the network is known as a multiport network. The two port network is shown as in the picture:
The currents in two leads making up each port must be equal. Therefore,
 = and = .
The special methods analysis which have been developed for two port networks emphasize the current and voltage relationships at the terminals of the networks and suppress the specific nature of the currents and voltages within the networks.
One and two port networks are accomplished best by using a generalized network notation and the abbreviated nomenclature for determinants. If we write a set of loop equations for a passive network,
 Then the coefficient of each current will be an impedance(s), and the circuit determinant is:
Suppose that one port network is composed entirely of passive elements and dependents sources. Linearity is also assumed.
Or more concisely,
The admittance (y) parameters
We will consider the two port network as shown as in the picture:
We may begin with the set of equations:
Where the (y) are no more than proportionality constants, it should be clearly that their dimensions must be an or S. Therefore they are called admittance (y) parameters.
Where each admittance parameters is defined by:
For example:
Now, for the one port network, to find the input admittance we will use the set of loop equations:
We set = and any device for which = is called a bilateral element, and a circuit which contains only bilateral elements is called a bilateral circuit. An important property of a bilateral two port is:
 =
Some property theorem:
In any passive linear bilateral network, if the single voltages source in branch x produces the current response in branch y, then the removal of the voltages source from branch x and its insertion in branch y will produce the current response in branch x.
It means that the interchange of an ideal voltage source and an ideal ammeter in any passive, linear, bilateral circuit will not change the ammeter reading.
In any passive linear bilateral network, if the single current source between nodes x and produces the voltage response between nodes y and, then the removal of the current source from nodes x and and its insertion between nodes y and will produce the voltage response between nodes x and.
In other words, the interchange of an ideal current source and ideal voltmeter in any passive linear bilateral circuit will not change the voltmeter reading.
Some equivalent networks
Two basic equations which determine the short-circuit admittance parameters 
When we both add and subtract, we will have other equations:
Here are some equivalent two-port networks:
The three-terminal networks and the three-terminal Y networks
They are equivalent is that the six impedances satisfy the conditions of the Y-△ transformation.
One network may be replaced by other if certain specific relationships between the impedance are satisfied, and these interrelationships may be established by use of the y parameters. We find that:
The performance of an amplifier is often described by giving a few specific values. We calculate four of these values for terminations. We shall define and evaluate the voltage gain, the current gain, the power gain, and the input impedance.
The parallel connection of two two-port networks
We can connect two two-port networks if both inputs and outputs have the same reference node, then [y] = [] + []
Impedance (Z) parameters
The concept of two-port parameters has been introduced in terms of the short circuit admittance parameters. There are other sets of parameters. However; each set is associated with a particular class of networks for which its use provides the simplest analysis. 
We begin with two equations:
The most informative description of the Z parameters is obtained by setting each of the currents equal to zero. Thus:
The Z parameters are known as the open circuit impedance parameters.
For example:
For two two-port network
V= + = + = I( + )= z× I
With I= = 
Hybrid parameters
The use of the hybrid parameters is well suited to transistor circuit, because there parameters are among the most convenient to measure experimentally for a transistor.
The hybrid parameters are defined by writing the pair of equations relating , , , :
Transmission parameters 
The last two port parameters that we shall consider are called the transmission (t) parameters the ABCD parameters. They are defined by:
Other widely used nomenclature for this set of parameters is 
For two port network: