DC Network Theorems - Maxwell’s Loop Current Method
DC Network Theorems – Maxwell’s Loop Current Method
This method which is particularly well-suited to coupled circuit solutions employs a system of loop or mesh currents instead of branch currents (as in Kirchhoff’s laws). Here, the currents in different meshes are assigned continuous paths so that they do not split at a junction 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 be solved reduces from b by Kirchhoff’s laws to b − (j − 1) for the loop current method where b is the number of branches and j is the number of junctions in a given network.
In the above figure, it shows two batteries E1 and E2 connected in a network consisting of five resistors. Let the loop currents for the three meshes be I1, I2 and I3. It is obvious that current through R4 (when considered as a part of the first loop) is (I1 − I2) and that through RS is (I2 − I3). However, when R4 is considered part of the second loop, current through it is (I2 − I1). Similarly, when R5 is considered part of the third loop, current through it is (I3 − I2). Applying Kirchhoff’s voltage law to the three loops, we get,
E1 − I1R1 − R4 (I1 − I2) = 0
or
I1 (R1 + R4) − I2 R4 − E1 = 0 …………………………..loop 1
Similarly, − I2R2 − R5 (I2 − I3) − R4 (I2 − I1) = 0
or I2 R4 − I2 (R2 + R4 + R5) + I3R5 = 0 …………………loop 2
Also − I3R3 − E2 − R5 (I3 − I2) = 0
or
I2R5 − I3 (R3 + R5) − E2 = 0 …………………loop 3
The above three equations can be solved not only to find loop currents but branch currents as
well.
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