# Mesh Current Analysis

Mesh Analysis (aka Mesh Current Analysis) is a systematic method for determining all of the branch currents in a circuit where KVL is applied to each mesh. The generated equations are known as mesh equations.

Recall that a mesh is a special kind of loop which contains no other loops, i.e. they do not contain smaller loops within them. A good way to think of a mesh is that of an open window in the circuit, using this analogy it is rather straightforward to identify the loops by inspection. This is an important point to be aware of for mesh analysis.

For mesh analysis we encounter the concept of an imaginary loop current which flows around loops. This is best illustrated with an example:

In the circuit above we have defined two loop currents: I1 and I2 which flow around the two meshes in the circuit. We can also say that the current flowing through R3 is I1+I2 according to the defined current directions.

The reason why the curent can be algebraically summed is that a resistor is a linear circuit element and the current flowing through it is the sum of the currents flowing through the element produced by each power source. This is formally known as Superposition which we will look into further in another section :)

The steps for mesh analysis are as follows:

1. Identify all of the meshes(windows) of the circuit and assign loop currents with voltage polarities assigned to each resistor.

2. Apply KVL to each of the meshes.

3. Solve the equations to determine the mesh currents.

### Examples:

1. a) Calculate all of the mesh currents in the following circuit:

The first step is to highlight all of the meshes in the circuit and assign loop currents and voltages to the circuit: We can then write a mesh equation (Perform KVL) for the mesh containing loop current I1: We can then write the other mesh equation: From here we have two equations with two unknowns. Multiplying [2] by 3: We can then sum [1] with [2'] to eliminate I1 and solve for I2: Substituting this back into [1] to determine I1: And finally we can determine I3 with KCL:

b) Calculate the voltages across each resistor:

As we now know all of the currents we can simply apply Ohm's law to each resistor to determine the voltage drops:

2. a) Calculate all of the mesh currents in the following circuit:

The first step is to highlight all of the meshes in the circuit and assign loop currents and voltages to the circuit: We can then write a mesh equation (Perform KVL) for the mesh containing loop current I1: Mesh equation for mesh 2: Mesh equation for mesh 3: We now have three equations with three unknowns which we can solve to determine the unknown currents. We can multiply [2] by 3 and then sum this with [3] to eliminate I3 We can then add [1] with this to eliminate I1 to solve for I2: Substituting this into[1] to solve for I1: We can then solve for I3 by substituting I2 into [3]:

b) Calculate the voltages across each resistor:

### Questions:

1. a) Calculate all of the mesh currents in the following circuit:

I1=2.857mA, I2=1.4286mA

b) Calculate the voltages across each resistor:

VR1=5.714V, VR2=5.714V, VR3=4.286V

2. a) Calculate all of the mesh currents in the following circuit: