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Introduction to Graph Theory

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Introduction

These notes are primarily a digression to provide general background remarks. The subject is an

efficient procedure for the determination of voltages and currents of a given network. A network

comprised of B branches involves 2B unknowns, i.e., each of the branch voltages and currents.

However the branch volt-ampere relations of the network, presumed to be known, relate the current and

the voltage of each branch,. Hence a calculation of either B currents or B voltages (or some

combination of B voltages and currents), and then substitution in the B branch volt-ampere relations,

provides all the voltages and currents.

In general however neither the B branch voltages nor the B branch currents are independent, i.e., some

of the B voltage variables for example can be expressed as a combination of other voltages using KVL,

and some of the branch currents can be related using KCL. Hence there generally are fewer than B

independent unknowns. In the following notes we determine the minimum number of independent

variables for a network analysis, the relationship between the independent and dependent variables, and

efficient methods of obtaining independent equations to determine the variables. In doing so we make

use of the mathematics of Graph Theory.

Graph Theory

A circuit graph is a description of the just the topology of the circuit, with details of the circuit elements

suppressed. The graph contains branches and nodes. A branch is a curve drawn between two nodes to

indicate an electrical connection between the nodes.

A directed graph is one for which a polarity marking is assigned

to all branches (usually an arrow) to distinguish between

movement from node A to B and the converse movement from

B to A.

A connected graph is one in which there is a continuous path

through all the branches (any of which may be traversed more

than once) which touches all the nodes. A graph that is not

connected in effect has completely separate parts, and for our

purposes is more conveniently considered to be two (or more)

independent graphs.

Choosing Independent Current Variables:

Given a network graph with B branches and N nodes select a tree, any one will do for the present

purpose. Remove all the link branches so that, by definition, there are no loops formed by the remaining

tree branches. It follows from the absence of any closed paths that all the branch currents become zero.

Hence by 'controlling' just the link currents all the branch currents can be controlled. This control would

not exist in general using fewer than all the link branches because a loop would be left over; depending

on the nature of the circuit elements branches making up the loop current could circulate around the

loop. Using more than the link branches is not necessary. Hence it should be possible to express all the

branch currents in terms of just the link currents, i.e., there are B-N+1 independent current variables, and

link currents provide one such set of independent variables.