Network analysis (electrical circuits)

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

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 V 2 = V 1 {\displaystyle V_{2}=V_{1}} implies I 2 = I 1 {\displaystyle I_{2}=I_{1}} 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.

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: Z e q = Z 1 + Z 2 + ⋯ + Z n . {\displaystyle Z_{\mathrm {eq} }=Z_{1}+Z_{2}+\,\cdots \,+Z_{n}.}

Impedances in parallel: 1 Z e q = 1 Z 1 + 1 Z 2 + ⋯ + 1 Z n . {\displaystyle {\frac {1}{Z_{\mathrm {eq} }}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\,\cdots \,+{\frac {1}{Z_{n}}}.}

The above simplified for only two impedances in parallel: Z e q = Z 1 Z 2 Z 1 + Z 2 . {\displaystyle Z_{\mathrm {eq} }={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}.}

Delta-wye transformation

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 (at worst ( n 2 ) {\displaystyle {\tbinom {n}{2}}} ). 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

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