Nodal Analysis Lab

Objective:
The goal is to review nodal voltage analysis for which will helps us find a system of linear equations. From the system of equations, we can use the the idea of Ax=B and use Cramer's Rule or use other methods of finding the voltages such using the reduced row echelon. From this idea, we will build a circuit for our lab containing multiple sources for which we can apply nodal voltage analysis. Furthermore, we covered mesh analysis which is another way of doing circuit analysis. The idea is to
define a mesh when a loop with loops that go clockwise because of convention so that results are all the same for the classroom.

Group Practice:
1. We apply nodal voltage analysis for Figure 1 as seen below in order to acquire all the voltages at the nodes. Figure 1 does not give the correct answers since we did not have correct voltage direction. Figure 2 shows how to set the equations correctly and may find the unknown voltages at the nodes using Kramer's rule or Gaussian elimination.

Figure 1. The images shows a circuit where nodal voltage analysis is used. 

Figure 2. The image shows the correct way to set up a system of linear equations for the first practice problem worked in class. 

Lab Procedures and Results:
1. The focus of the lab is to use nodal analysis for Figure 3 to predict the circuit behavior before we build and test the circuit with what we have available in class. We will then compare our experimental results with our theoretical results to find the percent error.

2. We acquired a theoretical voltage of V1 = 2.42V and V2 = 4.4241V by using nodal voltage analysis in Figure 4. Note: We used a 20K resistor at V2 since it was the only one available and near to the value.

Figure 3. A circuit that is used to predict and then test for the voltages at V1 and V2. 

Figure 4. Our predicted results for V1 and V2

3. The next step is to physically build the circuit as seen in Figure 4 by following the circuit schematic in Figure 3. It is crucial to point out that we did not use 20K resistor since the value of that resistor was not available to us. Instead, we used 22k resistor. We can see that the circuit has parallel resistors and voltage sources for which we included in our build. We note that the red wire represents the +5V, white as -5V, and yellow as -3V from our analog discovery box. The -3V was programmed through the waveform tab while the +5V and -5V are automatically outputting voltages. 

Our exact values for resistors are 9.8K, 21.8K, and 6.72K.

Figure 5. Actual circuit testing and connection as seen from Figure 3. 

3. The experimental measurements for V1 = 2.43+/-.05V as seen in Figure 6 and V2 = 4.39V+/-.05 as seen in Figure 7. By comparing our theoretical values of  V1 = 2.42V and V2 = 4.4241V and experimental values of V1 = 2.43V and V2 = 4.39, we can see that our results were fairly the same. 

Figure 6. A measurement of 2.43V for V1.

Figure 7. A measurement of voltage of 4.39V for V2. 

4. Finally, we calculate the percent error for each voltage where % error = abs((experimental-theoretical)/theoretical)X100. The percent error for V1 is % error = | (2.43-2.42)/2.42 |*100= .41%

The percent error for V2 is % error = | (4.39-4.4241)/4.4241 |*100 = .77%

Summary of Lab and Overall Learning Outcome:

We can conclude that our theoretical values are accurate to our experimental measurements which means that the nodal analysis holds true to its name. Though, our true resistance measurements were not the same as the theoretical values of the resistors, we found that there is a percent error between them where V1 had .41% error and V2 a .77% error. Since the percent error is below 1%, we are satisfied with the idea of using nodal voltage analysis for calculating for the unknown voltages at any node.