TWO PORT NETWORK ANALYSIS

Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this. However, for the most part, they assume that the components of the network are all linear. The methods described in this article are only applicable to linear network analysis except where explicitly stated.

Component
A device with two or more terminals into which, or out of which, charge may flow.
Node
A point at which terminals of more than two components are joined. A conductor with a substantially zero resistance is considered to be a node for the purpose of analysis.
Branch
The component(s) joining two nodes.
Mesh
A group of branches within a network joined so as to form a complete loop.
Port
Two terminals where the current into one is identical to the current out of the other.
Circuit
A current from one terminal of a generator, through load component(s) and back into the other terminal. A circuit is, in this sense, a one-port network and is a trivial case to analyse. If there is any connection to any other circuits then a non-trivial network has been formed and at least two ports must exist. Often, "circuit" and "network" are used interchangeably, but many analysts reserve "network" to mean an idealised model consisting of ideal components.[1]
Transfer function
The relationship of the currents and/or voltages between two ports. Most often, an input port and an output port are discussed and the transfer function is described as gain or attenuation.
Component transfer function
For a two-terminal component (i.e. one-port component), the current and voltage are taken as the input and output and the transfer function will have units of impedance or admittance (it is usually a matter of arbitrary convenience whether voltage or current is considered the input). A three (or more) terminal component effectively has two (or more) ports and the transfer function cannot be expressed as a single impedance. The usual approach is to express the transfer function as a matrix of parameters. These parameters can be impedances, but there is a large number of other approaches, see two-port network.
[edit] Equivalent circuits

Main article: equivalent impedance transforms
A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing the actual components with other notional components that have the same effect. A particular technique might directly reduce the number of components, for instance by combining impedances in series. On the other hand it might merely change the form in to one in which the components can be reduced in a later operation. For instance, one might transform a voltage generator into a current generator using Norton's theorem in order to be able to later combine the internal resistance of the generator with a parallel impedance load.
A resistive circuit is a circuit containing only resistors, ideal current sources, and ideal voltage sources. If the sources are constant (DC) sources, the result is a DC circuit. The analysis of a circuit refers to the process of solving for the voltages and currents present in the circuit. The solution principles outlined here also apply to phasor analysis of AC circuits.
Two circuits are said to be equivalent with respect to a pair of terminals if the voltage across the terminals and current through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network.
If V2 = V1 implies I2 = I1 for all (real) values of V1, then with respect to terminals ab and xy, circuit 1 and circuit 2 are equivalent.
The above is a sufficient definition for a one-port network. For more than one port, then it must be defined that the currents and voltages between all pairs of corresponding ports must bear the same relationship. For instance, star and delta networks are effectively three port networks and hence require three simultaneous equations to fully specify their equivalence.
[edit] Impedances in series and in parallel
Any two terminal network of impedances can eventually be reduced to a single impedance by successive applications of impendances in series or impendances in parallel.
Impedances in series:
Impedances in parallel:
The above simplified for only two impedances in parallel:
[edit] Delta-wye transformation
Main article: Y-Δ transform

A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An n-terminal network can, at best, be reduced to n impedances. For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or a four node star (Y) network. These two networks are equivalent and the transformations between them are given below. A general network with an arbitrary number of terminals cannot be reduced to the minimum number of impedances using only series and parallel combinations. In general, Y-Δ and Δ-Y transformations must also be used. It can be shown that this is sufficient to find the minimal network for any arbitrary network with successive applications of series, parallel, Y-Δ and Δ-Y; no more complex transformations are required.
For equivalence, the impedances between any pair of terminals must be the same for both networks, resulting in a set of three simultaneous equations. The equations below are expressed as resistances but apply equally to the general case with impedances.
[edit] Delta-to-star transformation equations
[edit] Star-to-delta transformation equations
[edit] General form of network node elimination
The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm. Any node connected by N resistors (R1 .. RN) to nodes 1 .. N can be replaced by resistors interconnecting the remaining N nodes. The resistance between any two nodes x and y is given by:
For a star-to-delta (N = 3) this reduces to:
For a series reduction (N = 2) this reduces to:
For a dangling resistor (N = 1) it results in the elimination of the resistor because .
[edit] Source transformation


A generator with an internal impedance (ie non-ideal generator) can be represented as either an ideal voltage generator or an ideal current generator plus the impedance. These two forms are equivalent and the transformations are given below. If the two networks are equivalent with respect to terminals ab, then V and I must be identical for both networks. Thus,
or
Norton's theorem states that any two-terminal network can be reduced to an ideal current generator and a parallel impedance.
Thévenin's theorem states that any two-terminal network can be reduced to an ideal voltage generator plus a series impedance.
[edit] Simple networks
Some very simple networks can be analysed without the need to apply the more systematic approaches.
[edit] Voltage division of series components
Main article: voltage division
Consider n impedances that are connected in series. The voltage Vi across any impedance Zi is
[edit] Current division of parallel components
Main article: current division
Consider n impedances that are connected in parallel. The current Ii through any impedance Zi is
for i = 1,2,...,n.
[edit] Special case: Current division of two parallel components
[edit] Nodal analysis
Main article: nodal analysis
1. Label all nodes in the circuit. Arbitrarily select any node as reference.
2. Define a voltage variable from every remaining node to the reference. These voltage variables must be defined as voltage rises with respect to the reference node.
3. Write a KCL equation for every node except the reference.
4. Solve the resulting system of equations.
[edit] Mesh analysis
Main article: mesh analysis
Mesh — a loop that does not contain an inner loop.
1. Count the number of “window panes” in the circuit. Assign a mesh current to each window pane.
2. Write a KVL equation for every mesh whose current is unknown.
3. Solve the resulting equations
[edit] Superposition
Main article: Superposition theorem
In this method, the effect of each generator in turn is calculated. All the generators other than the one being considered are removed; either short-circuited in the case of voltage generators, or open circuited in the case of current generators. The total current through, or the total voltage across, a particular branch is then calculated by summing all the individual currents or voltages.
There is an underlying assumption to this method that the total current or voltage is a linear superposition of its parts. The method cannot, therefore, be used if non-linear components are present. Note that mesh analysis and node analysis also implicitly use superposition so these too, are only applicable to linear circuits.
[edit] Choice of method
Choice of method[2] is to some extent a matter of taste. If the network is particularly simple or only a specific current or voltage is required then ad-hoc application of some simple equivalent circuits may yield the answer without recourse to the more systematic methods.
Superposition is possibly the most conceptually simple method but rapidly leads to a large number of equations and messy impedance combinations as the network becomes larger.
Nodal analysis: The number of voltage variables, and hence simultaneous equations to solve, equals the number of nodes minus one. Every voltage source connected to the reference node reduces the number of unknowns (and equations) by one. Nodal analysis is thus best for voltage sources.
Mesh analysis: The number of current variables, and hence simultaneous equations to solve, equals the number of meshes. Every current source in a mesh reduces the number of unknowns by one. Mesh analysis is thus best for current sources. Mesh analysis, however, cannot be used with networks which cannot be drawn as a planar network, that is, with no crossing components.[3]
[edit] Transfer function

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