3/8/16 Day 5: Nodal Analysis and Mesh Analysis

Today in lecture we learned more about nodal analysis. We learned about supernodes, or two non reference connected nodes that only contain a voltage source between them. We learned about how to take into account a supernode in node analysis. Once we learned about supernodes, we then solved for the voltage at a node in a circuit containing a supernode. Next, we performed the "Node Analysis" experiment, which will be described in more detail in the "lab" section. An overview of this lab is that the voltages across two resistors in a circuit were solved for using nodal analysis. The voltages were then determined experimentally and compared to the theoretical (i.e. the circuit was built). After the experiment we learned about a new technique called Mesh Analysis; it is very similar to Kirchhoff's voltage laws, except that the current is solved for using those techniques instead of voltages. A restriction to mesh analysis is that the circuit must be planar, so that each loop is separate (i.e. there are no loops inside another loop). After learning about mesh analysis we then solved for currents in a planar circuit using the technique. We then reviewed Cramer's Rule in order to help us solve those linear system of equations obtained from all of those techniques, followed by learning about how to use an application called Every Circuit, which uses nodal and mesh analysis to solve for voltages and currents in the circuits created within the app. Lastly, we "reviewed" the sign convention for determining the resistance of a resistor using the color bands on the resistor.

LECTURE:

In the problem seen in the above picture, the objective was to first identify the supernode in the circuit. Then, it was to solve for the voltage at the node to the right of the supernode (Vx) using nodal analysis. The lower portion of the above picture and the picture to the right show the solving of this problem. First the KCL laws were determined, followed by using Ohm's law to make the unknown currents in terms of voltages at the nodes. These expressions were then substituted into the KCL equations, and once the linear system of equations was obtained the voltages were solved for at the non-reference nodes using Gaussian elimination. It was found that Vx is equal to 30 V.

In the above circuit, the objective was to determine the currents running through the circuit. This was achieved using mesh analysis, where KVL was used to obtain a system of linear equations in terms of the currents. Then, using Gaussian elimination the currents were solved for.

 For the last part of class we reviewed the color band convention for resistors and how to identify the resistance of those resistors using the order of the colors. In a four-band resistor, the first two bands are used to display the significant figures of the value. The third band provides the multiplier of the resistance, and the fourth band provides the tolerance, which is the standard deviation of the resistance in percentage form. Each color represents an integer. For example, black represents 0, brown represents 1, gray represents 8 and white represents 9. The other integers (2 through 7) are represented by the colors of visible light in the EM spectrum (or the colors of a rainbow), in the specific order of ROYGBV. As seen, indigo is excluded from the spectrum. The tolerance band is usually colored gold or silver; gold represents a 5% deviation and silver represents a 10% deviation.

In a five-band resistor, the first three bands are for significant figures, followed by the multiplier band and tolerance band. as an example, as seen in the first problem for the higher picture, a band order or red black orange represents a 20 kiloohm resistor (red and black represent the significant figures 2 and 0, whereas the orange band represents the order of kilo for the resistance). The convention was then put to the test even more with other color band examples.

LAB (NODAL ANALYSIS):

Purpose:

The purpose of this lab was to determine the voltages across the two resistors in a circuit using nodal analysis. These values were then tested by actually building the circuit and then measuring the voltages across those resistors. The measured values were than compared to the actual values.

Prelab:

The pre-lab of this experiment consisted of solving for the voltages across the two resistors in the above circuit using nodal analysis. The voltage across the 22 kohm resistor was calculated as 4.42 V, whereas the voltage drop across the 6.8 kohm resistor was found to be 2.42 V. Then, because the resistors we worked with had a tolerance of 10%, the standard deviations were calculated for the voltages. This was done by using the same equation containing V3 above, but changing the resistor values to +/- 10%. The standard deviation of V1 was found to be +/- 0.48 V and that of V2 was found to be  +/- 0.88 V. This was done using Wolfram Alpha due to solving the 27 different situations by hand for each would be very tedious.

Apparatus:

The equipment for this experiment consisted of the usual: an analog discovery voltage source, a digital multimeter, to serve as an ohmmeter and voltmeter, two resistors of the theoretical resistance values seen above, a breadboard, alligator clips, a bunch of thin wires to allow connection to the breadboard and a laptop with Waveforms to control the analog discovery.

Procedure:

The circuit seen in the schematic of the prelab was built, and is seen in the picture above. The positive 5 V gate (red wire) was connected to the 10 kohm resistor. The 10 kohm resistor then split to the 22 kohm and 6.8 kohm resistors. The negative 5 V gate (white wire) was then connected to the other end of the 22 kohm resistor. Lastly, the gate for the waveform voltage (-3 V), provided by the yellow wire, was connected to the other end of the 6.8 kohm resistor. The whole circuit was then connected together in the internal compartment of the analog discovery box.

The true resistances of the resistors was then measured using an ohmmeter. Then, power was run through the circuit, and a voltmeter was used in parallel to the 22 kohm and 6.8 kohm resistors to measure the voltage drop across them. This is seen in the picture in the "Apparatus" section.

All of this data was then recorded, as seen in the table below in the "Data" section.

Data:

The experimental values for the 10 kohm, 22 kohm and 6.8 kohm resistor are 9.94, 21.86 and 6.81 kohms, respectively. The measured values for the voltages across the 6.8 kohm and 22 kohm resistors are, respectively, 2.43 V and 4.41 V. The percent differences between the measured and theoretical values for V1 and V2 was found to be 0.413% and 0.226%, respectively.


Conclusion:

As can be seen by the low percent differences, nodal analysis was confirmed to be a legitimate technique for finding voltages in a circuit. In addition, it shows that our experiment was set up and run correctly, and our methods were proper for this situation.

Post-lab:

the post-lab of this experiment was to just create the circuit of this experiment in Every Circuit, and find the voltages across the resistors using this software. A picture of the schematic from this application is posted below:

As can be seen, the voltage across the 21.8 kohm resistor is 5V - 0.569 V = 4.431 V. Likewise, the voltage across the 6.81 kohm resistor is 3 V - 0.569 V = 2.431 V. As can be seen, these values are also very close to our experimental and calculated values. Ideally, they would be equal to the calculated values, but they are not due to the slight change in resistances as well as rounding errors. They are also not exactly equivalent to the experimental values because of the neglected resistance in the wires and the slightly different resistance values of the resistors.