082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

1.8 Summation notation 47
Engineeringapplication1.8
Kirchhoff’scurrentlaw
Kirchhoff’scurrentlaw,oftenabbreviatedtoKCL,providesoneofthefundamental
equationsforanalysingelectricalcircuits.Thelawstatesthatthesumofthecurrents
flowing out of any junction, or node, in a circuit must equal the sum of the currents
flowing into it.
This principle is intuitive as it has a direct analogy with fluid flow in connected
water pipes. Currents flowing into a junction are considered positive; those flowing
out of a junction are negative. It is then valid to say that the sum of the currents
at a junction is zero. If there are N currents at the junction, denoted I 1
,I 2
,...,I N
,
then
I 1
+I 2
+I 3
+···+I N−1
+I N
=0
This can be expressed using the summation notation as
N∑
I k
=0
k=1
HereI k
means ‘the current,I, in branchk’. The first equation can be produced from
the summation notation by first substitutingk = 1, thenk = 2, right up tok = N.
Theexpressionbelowthesummationsymboltellsyouwheretostartandthevariable
to be substituted, and the number above the summation symbol indicates where to
stopcounting.Summationnotationisaverycompactandprecisewayofexpressing
KCL forany number ofcurrents atanode.
Consider the node shown inFigure 1.5.
Branch 1 Branch 3
1 A
2 A
3 A 2 A Figure1.5
Acircuit node with four separate branches.The currents are
Branch 2 Branch 4 given in amperes(oramps,A).
It can be seen that the total current flowing into the node is 1 + 3 = 4 amps. The
current flowing outof the node is2 +2 = 4 amps. Clearly,
Total current flowing into node =total currentflowing outof node
Alternatively, using the summation formof KCL wehave
4∑
I k
=I 1
+I 2
+I 3
+I 4
=0=1+3−2−2
k=1
Notethatforcurrentsflowingoutofthenodeanegativesignisusedandforcurrents
flowing into the node a positive sign is used. This is equivalent to considering the
currents separately as inward and outward flowing currents and equating the two.
Suppose for a moment that we did not know the current in branch 4 and,
furthermore, itwas notlabelled with anarrow toshow the direction of current flow.
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