ECE 220 Network Analysis I

Lesson 5. The Mesh Current Method

Mesh Current Theory and Practice

The mesh current method simplifies and shortens the solution of problems by allowing you to write only Kirchhoff's Voltage Law. Kirchhoff's Current Law is implicit. Consider the circuit below.

Using branch currents, we can write KVL:
-V_S + i₁R₁ + i₃R₃ = 0
-i₃R₃ + i₂R₂ = 0

We can also write KCL at the top node, which gives:
-i₁ + i₂ + i₃ = 0

Which gives:
i₃ = i₁ - i₂

Substituting into KVL gives:
-V_S + i₁R₁ + (i₁ - i₂)R₃ = 0
(i₂ - i₁)R₃ + i₂R₂ = 0

With the mesh current method, we skip several steps, and just write the two equations above. i₁ and i₂ are then considered to be mesh currents rather than branch currents. Nevertheless, if an element is not shared by two meshes, such as R₁, the mesh current is the same as the branch current.

Now consider a more complicated circuit.

We want to write mesh current equations for this circuit, so we define currents in the three meshes. The top and right meshes are labeled I₁ and I₂, respectively. The left mesh can be labeled 6 A, since we already know the current. We also define a voltage across the 4 A source. Next, we write KVL equations for the two unknown mesh currents. The convention for doing this is to put sources on the left, and loads on the right.

5 - V_I = 2(I₁ - 6)
V_I = 3(I₂ - 6) + 7I₂

We need a third equation, one relating 4 A to the mesh currents. Observe that the branch current in the 4 A source is made up of two mesh currents, I₂ in the direction of the 4 A current, and I₁ opposite to the 4 A current. Therefore:

4 = I₂ - I₁

With three equations and three unknowns, we can solve for all the unknowns.

Supermesh

Another solution technique uses the concept of a "supermesh", and saves the writing of one equation. The circuit below is the same as before, but it is labeled differently.

The 4 A source is unlabeled (it still has a voltage, but we don't have to know what it is). The supermesh is shown by the red line. We write KVL around this supermesh as follows:

5 = 7I₂ + 3(I₂ - 6) + 2(I₁ - 6)

We still need the equation for the current in the 4 A source:

4 = I₂ - I₁

Now we have two equations and two unknowns. Use either solution method you wish.

Before going on to the homework, you should complete Tutorial 5 on the mesh current method.

Homework Problems

1. Use the mesh current method to write enough equations to solve for the mesh currents. Do not solve.
   
2. E1 = 195 V, E2 = 20 V, R1 = 5 Ω, R2 = 40 Ω, R3 = 20 Ω, R4 = 50 Ω, R5 = 80 Ω.  Use the mesh current method to find I. The answer is in integer.
   
3. Ix = 5 A, Iy = 7 A, E = 76 V, R1 = 12 Ω, R2 = 20 Ω, R3 = 8 Ω.  Use the mesh current method to find V. The answer is in integer.
   
4. Is = 1.5I, E1 = 88 V, E2 = 52 V, R1 = 7 Ω, R2 = 3 Ω, R3 = 10 Ω, R4 = 30 Ω.  Use the mesh current method to find I. The answer is in integer.