# 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:

**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:

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

**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:

#### Answer

**b)** Calculate the voltages across each resistor:

#### Answer

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

#### Answer

**b)** Calculate the voltages across each resistor: