Basic Electrical Engineering Series
2 Port Networks
A linear electrical network consisting of and input port and output port is referred to as a 2-port network (or sometimes 4 terminal network). The relationship between the input source and output load can be described a pair of equations and represented as a matrix. Two classes of such matrices for two port networks exist: wave and circuit. The matrix elements or parameters are designated as follows.
1) Wave: S,T and U parameters
2) Circuit: Z ,Y ,H,G and ABCD parameters
We consider now the transmission matrices consisting of the ABCD parameters.
Transmission Matrix – ABCD Parameters
Consider the two port network circuit below.
General two port network
The equations representing this circuit are as follows:
V1 = AV2 + B I2
I1 = CV2 + D I2
The matrix containing the ABCD parameters is the transmission matrix of the circuit.
Consider now some important properties of transmission matrices.
Cascading Circuits
To cascade two circuits in series, the transmission matrices are simply multiplied to give the overall transmission matrix.
Parallel Circuits
For circuits in parallel
Circuit Properties
A Symmetrical Network means that any circuit connected to the input port giving rise to a property at the output port will give an identical property at the input port when connected to the output port.
For a symmetrical network
A = D and AD – BC =1
The impedance Z1 seen at port 1 with port 2 loaded Z2
The impedance Z2 seen at port 2 with port 1 loaded Z1
The forward circuit transfer function between source and load is:
The open circuit forward voltage gain is: 1/A
The short circuit forward voltage gain is: -1/D
The open circuit foreward transfer impedance is: 1/C
The short circuit forward transfer admittance is: -1/B
Iterative, Image and Characteristic Impedance
For any 2 port network, it is always possible to find an impedance which when terminating the circuit will give an impedance at the other port equal to this impedance. This is known as an Iterative Impedance. Hence there will be an iterative impedance looking into port 1 - Zit1 and an iterative impedance looking into port 2 – Zit2. If the network is symmetrical, then Zit1 = Zit2 = Zit.
For a network exited by a generator with source impedance Zs, maximum power will be transmitted to the network if the input impedance is equal to the complex conjugate of the source impedance Zs. Thus Zs = Conj(Zin) – this is known as the image impedance of Zs.
Similarly the network will deliver maximum power to the load ZL if the output impedance of the network is the complex conjugate of the load impedance. Thus ZL = Conj(Zout) this is known as the image impedance of ZL.
If a network is loaded by an image impedance and at the same time is exited by a generator with an image impedance, then the network is said to be working between its image impedances.
For a symmetrical network, if the iterative impedance Zit is equal to the image impedance of both the source and load impedance, then the iterative impedance is known as the characteristic impedance, Zo, of the network.
From the above we get for a symmetrical network, where A = D :
Basic Circuits – Shunt Admittance
Basic Circuits – Series Impedance
From these two basic circuits any network may be found using the rules of cascading and parallel circuits.
Transmission Matrix – ABCD-1 Parameters
If
then
In general, inverting the matix by dividing the adjoint matrix by its determinent,
We get
For a symmetrical network A = D and AD – BC=1, therefore
to be continued