Maxwell's Loop Curent Method

This method which is particularly well-suited to coupled circuit soluions employs a system of
loop or mesh currents instead of branch currents (as in Kirchhoffs laws). Here, the currents in
different meshes are assigned continuous paths so that they do not split at a juncion into branch
currents. This method eliminates a great deal of tedious work involved in the branch-current method
and is best suited when energy sources are voltage sources rather than current sources. Basically,
this method consists of writing loop voltage equations by Kirchhoff's voltage law in terms of unknown
loop currents. As will be seen later, the number of independent equations to ~
reducesfromb by Kirchhoffs lawsto b - (j - 1)forthe loopcurrentmethodwhereb is the number
of branches andj is the number of junctions in a given network connected in a network consisting of five
resistors. Let the l~op currents for the
three meshes be II' 12and 13' It is obvi-
E ous that current through R4 (when con-
2 sidered as a part of the first loop) is (/) -
12)and that through Rs is (12-13), However,
when R4 is considered part of the
second loop, current through it is (/2 -
H I). Similarly, whe Rs is considered part
of the third loop, current through it is (13
-12), Applying Kirchhoff's voltage law
to the three loops, Consider the network of Fig. 2.52, which contains
resistances and independent voltage sources and has three
meshes. Let the three mesh currents be designated as II' 12
and 13and all the three may be assumbed to flow in the clockwise
direcion for obtaining symmetry in mesh equationsmesh (i) which equals the sum of all resistance in mesh (i). Similarly, the second item in the first
row represents the mutual resistance between meshes (i) and (ii) i.e. the sum of the resistances
common to mesh (i) and (ii). Similarly, the third item in the first row represents the mutual-resistance
of the mesh (i) and mesh (ii).
The item E.. in general, represents the algebraic sum of the voltages of all the voltage sources
acting around mesh (i). Similar is the case with E2 and E3' The sign of the e.m.fs is the same as
discussed in Art. 2.3 fe. while going along the current, if we pass from negetive to the positive
terminal of a battery, then its e.m.f. is taken postive. If it is the other way around, then battery e.mf.
is taken negative.
In general, let
RII =self-resistance of mesh (i)
R22 =self-resistance of mesh (ii) i.e. sum of all resistances in mesh (ii)
R33 =Self-resistance of mesh (iii) i.e. sum of all resistances in mesh (iii)
RI2 =R21 =- [Sum of all the resistances common to meshes (i) and (ii)] *
R23 =R32 =- [Sum of all the resistances common to meshes (ii) and (iii)]*
The above equaitons can be written in a more compact form as [Rm][1m]= [Em]' It is known as
Ohm's law in matrix form.
In the end, it may be pointed out that the directions of mesh currents can be selected arbitrarily.
If we assume each mesh current to flow in the clockwise direction, then
(i) All self-resistances will always be postive and (ii) all mutual resistances will always be
negative. We will adapt this sign convention in the solved examples to follow.
The above main advatage of the generalized form of all mesh equations is that they can be easily
remebered because of their symmetry. Moreover, for any given network, these can be written by
inspection and then solved by the use of determinants. It eliminates the tedium of deriving simultaneous
equations.