In electrical engineering and electronics, a network is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values; however, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.
Definitions ComponentA device with two or more terminals into which, or out of which, current may flow.NodeA 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.BranchThe component(s) joining two nodes.MeshA group of branches within a network joined so as to form a complete loop such that there is no other loop inside it.PortTwo terminals where the current into one is identical to the current out of the other.CircuitA 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. cites to justify this definition.Sidney Darlington follows Belevitch but notes there are now also many colloquial uses of "network".Transfer functionThe 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 functionFor 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).
Equivalent circuits
A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing physical 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 into 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. Analysis of a circuit consists 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 implies for all (real) values of , then with respect to terminals and , 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.
Impedances in series and in parallel
Some two terminal network of impedances can eventually be reduced to a single impedance by successive applications of impedances in series or impedances in parallel.
Impedances in series: Impedances in parallel: The above simplified for only two impedances in parallel:
Delta-wye transformation
A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An -terminal network can, at best, be reduced to impedances (at worst ). For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or 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 nodes 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. For some networks the extension of Y-Δ to star-polygon transformations may also be 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.
Delta-to-star transformation equations
Star-to-delta transformation equations
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 resistors to nodes can be replaced by resistors interconnecting the remaining nodes. The resistance between any two nodes is given by:
For a star-to-delta () this reduces to:
For a series reduction () this reduces to:
For a dangling resistor () it results in the elimination of the resistor because .
Source transformation
A generator with an internal impedance (i.e. 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 and must be identical for both networks. Thus,
or Norton's theorem states that any two-terminal linear network can be reduced to an ideal current generator and a parallel impedance. Thévenin's theorem states that any two-terminal linear network can be reduced to an ideal voltage generator plus a series impedance.
Simple networks Some very simple networks can be analysed without the need to apply the more systematic approaches.
Voltage division of series components
Consider n impedances that are connected in series. The voltage across any impedance is
Current division of parallel components
Consider n admittances that are connected in parallel. The current through any admittance is
for
Special case: Current division of two parallel components
Nodal analysis
Nodal analysis uses the concept of a node voltage and considers the node voltages to be the unknown variables. For all nodes, except a chosen reference node, the node voltage is defined as the voltage drop from the node to the reference node. Therefore, there are N-1 node voltages for a circuit with N nodes.
In principle, nodal analysis uses Kirchhoff's current law (KCL) at N-1 nodes to get N-1 independent equations. Since equations generated with KCL are in terms of currents going in and out of nodes, these currents, if their values are not known, need to be represented by the unknown variables (node voltages). For some elements (such as resistors and capacitors) getting the element currents in terms of node voltages is trivial.
For some common elements where this is not possible, specialized methods are developed. For example, a concept called supernode is used for circuits with independent voltage sources.
Label all nodes in the circuit. Arbitrarily select any node as reference. 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. Write a KCL equation for every node except the reference. Solve the resulting system of equations.
Mesh analysis
Mesh — a loop that does not contain an inner loop.
Count the number of “window panes” in the circuit. Assign a mesh current to each window pane. Write a Kirchhoff's voltage law (KVL) equation for every mesh whose current is unknown. Solve the resulting equations
Superposition
In this method, the effect of each generator in turn is calculated. All the generators other than the one being considered are removed and 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. Therefore, the method cannot be used if non-linear components are present. Superposition of powers cannot be used to find total power consumed by elements even in linear circuits. Power varies according to the square of total voltage or current and the square of the sum is not generally equal to the sum of the squares. Total power in an element can be found by applying superposition to the voltages and current independently and then calculating power from the total voltage and current.
Choice of method Choice of method 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. 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. 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 can only be used with networks which can be drawn as a planar network, that is, with no crossing components. 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. Effective medium approximations: For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible. Instead, the effective resistance and current distribution properties can be modelled in terms of graph measures and geometrical properties of networks.
Transfer function