Loop current method

Using this method, the number of equations is reduced, and the equations for the first law are eliminated.
Kirchhoff's gamble. The concept of a loop current is introduced (this is a virtual concept), equations are compiled according to the second Kirchhoff law.

Consider our example pic. one



Pic.1
Contour currents are designated Im, In, Il, their directions are given, as shown in fig.

Algorithm of the decision:
1. we write the real currents through the contour: along the external branches I1 = Im, I3 = Il, I4 = In and along the adjacent branches I2 = Im - In, I5 = In - Il

2. Create equations according to the second Kirchhoff law, as there are three contours, therefore there will be three equations:

for the first circuit Im • (R1 + R2) - In • R2 = E1 - E2, the “-” sign in front of In is placed because this current is directed against Im

for the second circuit - Im · R2 + (R2 + R4 + R5) • In - Il • R5 = E2

for the third circuit - In • R5 + (R3 + R5) • Il = E3

3. Solving the resulting system of equations, we find the contour currents

4. Knowing the contour currents, we determine the actual currents of the circuit (see paragraph 1.