Mesh Analysis

Mesh Analysis

       Another way of simplifying the complete set of Kirchhoff’s equations is the mesh or loop current method. Note: Mesh current method is only applicable for “planar” circuits.
       There are 3 steps to consider in mesh analysis:
1.) Assign a mesh current to each mesh.
2.) Apply Kirchhoff’s voltage law (KVL) around each mesh, in the same direction as the mesh current so loop every mesh in the circuit .For example:

 

There are five mesh in this circuit.

3.) Solve the resulting loop equations for the mesh currents.

Example:

 

    Solve for the current through the 5 ohm resistor and the current through the 18V source using Mesh-Current Analysis.

Now write KVL equations for each loop.
KVL for i1:
-18V + 5(i1-i2) + 4(i1-i3) + 1(i1) = 0
then gather terms:
10i1 - 5i2 - 4 i3 -18V = 0
Note that the i1 term is positive, and all other current terms are negative (because they are all clockwise, all other panes will contribute a negative term). Let's do the other two panes with terms gathered up directly (write the total resistance of the loop multiplied by the mesh current that goes through that total resistance):
KVL for i2:
-5i1 + 10i2 - 3i3 - 12 = 0
KVL for i3:
-4i1 -3i2 +9i3 = 0
Now solve the three equations in three unknowns:
i1 = 7.02A
i2 = 6.28A
i3 = 5.21A     The current through the 5 ohm resistor is just i1 - i2, or 0.74A. The current through the 18V is i1, or 7.02A.