# Lab 3: Mesh and Nodal Analysis

## 1. Objectives

By the end of this laboratory session students will be able to:

- Analyze circuits using mesh analysis and experimentally confirm their prediction
- Analyze circuits using nodal analysis and experimentally confirm their prediction

## 2. Background

Mesh analysis is based on KVL and is used to solve circuits by finding the unknown currents in each mesh. The steps for mesh analysis are:

- Determine if the circuit is a planar circuit.
- Count the number of meshes \(M\).
- Label each of the \(M\) mesh currents.
- Write a KVL equation around each mesh.
- Express any additional unknowns such as voltages or currents other than mesh currents in terms of appropriate mesh currents.
- Organize the equations.
- Solve the system of equations for the mesh currents (M of them).

Nodal analysis is based on KCL and is used to solve circuits by finding the unknown voltages at each node. The steps for nodal analysis are:

- Count the number of nodes \(N\).
- Designate a reference node.
- Label the nodal voltages (\(N-1\) of them).
- Write a KCL equation for each of the nonreference nodes.
- Express any additional unknowns such as currents or voltages other than nodal voltages in terms of appropriate nodal voltages.
- Organize the equations.
- Solve the system of equations for the nodal voltages (\(N-1\) of them).

## 3. Laboratory Equipment

- Resistors
- DC Power supply
- DMM
- Oscilloscope
- Multisim

## 4. Procedure

### 4.1. Mesh Analysis

- Gather the resistors listed in Figure 1.
- Measure and record each resistor value using the multi-meter. Be sure to note the color of the fourth band.
- Construct the circuit in Figure 1.
Measure and record the voltages across \(R_1\), \(R_2\), and \(R_5\). From these voltage measurements and your measured resistor values, calculate the current passing through \(R_1\), \(R_2\), and \(R_5\).

Derive the mesh equations for this circuit and solve for \(I_1\), \(I_2\), and \(I_3\). Use the measured resistor values instead of the values listed in Figure 1.

Construct this circuit in Multisim and verify your calculations. Use either interactive mode or DC operating point analysis. Use the measured resistor values. Capture the screen showing the simulated current values. (Refer to section 2-7 in the “Introducing Multisim” handout for help with Multisim.)

Change the resistor values in your Multisim simulation to the ideal values shown in Figure 1 and run the simulation again. Capture the screen showing the simulated current values.

### 4.2. Nodal Analysis

- Rearrange your circuit on the breadboard so that it looks like the schematic in Figure 2.
- Measure and record the voltage values for nodes \(V_0\), \(V_1\), and \(V_2\).
- Derive the nodal equations for this circuit and solve for \(V_1\) and \(V_2\). Use the measured resistor values.
- Construct this circuit in Multisim and verify your calculations. Use the measured resistor values. Capture the screen showing the simulated nodal voltages.
- Change the resistor values in your Multisim simulation to the ideal values shown in Figure 2 and run the simulation again. Capture the screen showing the simulated nodal voltages.

## 5. Results

Calculate the percent difference between each measured and ideal resistor value. Create a table with three columns and include the data showing your measured resistor values, the ideal resistor values, and the percent error. Is the percent error less than the rated tolerance? Percent error is calculated with the formula

$$\frac{|\mathrm{measured}-\mathrm{ideal}|}{\mathrm{ideal}} \times 100$$

Report the voltages measured in step 4 of Section 4.1.

Create a table that compares the calculated and measured current values from steps 4 and 6 in Section 4.1. Include a column in your table that shows the percent error for each current.

Include the screenshots from steps 6 and 7 of Section 4.1.

Create a table that compares the calculated and measured nodal voltages for the circuit in Figure 2. Include a column that shows the percent error.

Include a screenshot from steps 4 and 5 of Section 4.2.