Thevenin's Theorem

In this lab, we are introduced to the fourth way to analyze a circuit. The first one we covered is nodal analysis. The second one is mesh analysis. The third one is superposition. Thevenin's theorem is not similar to any of the above three methods. It neither does divide-and-concur strategy to the circuit calculation by looking at nodes or meshes, nor removes the power sources to simplify each case. The way it works is to replace everything in the circuit with a simple voltage source and a equivalent resistor except for the load.

Here is the circuit graph

In the first picture, we showed our calculation the Rth by removing all the power sources in the same way as superposition method. The we can get the equivalent of the resistors that are left over.

In the second picture, we found the Vth across the load by using mesh analysis. we found the current of each mesh and use them to calculate the potential difference across the load. The theoretical values are listed on the board.

Then we did an experiment based on this circuit. We measured the resistance of the resistors we would use.

After we constructed the circuit, we measured the Rth and Vth

We compared the experimental data and the theoretical data

	Experimental	Theoretical	Percent Error
Thevenin Voltage (V)	-0.46	-0.46	0.76%
Thevenin Resistance (kΩ)	7.61	7.70	1.15%

It turns out that the lab results have very small percent error. We can say that it supports the Thevenin's theorem. 

Summary:

Thevenin's theorem is a very convenient way of studying circuit when only one element of the circuit is considered as a variable. It works the best when the value of a resistor is changing. It might take more time to calculate once, but it does not require any re-calculation even if the value of the load is changed. It is inevitable to redo the calculation if we use mesh analysis or node analysis. The disadvantage is that dependent sources will make the Thevenin theorem a lot more complicated. An extreme case would be that all sources are dependent sources. The Thevenin voltage would be very hard to calculate. Even if it does not work the best for every single problem, it is a useful technique overall.

Superposition II

In this lab, we are introduced to the third circuit analysis method, superposition method. It considers the current through or the voltage across an element to be a combination effect of every single source. Its essence is to turn on one power source at a time to calculate the current of voltage while the others are kept off. Then it switches to a another power source until every one of them is turned on once. The actual current or voltage will be the mathematical addition of the current or voltage of each case.

Here is the circuit graph

The circuit on the top is the actual circuit before the use of superposition method. The one at the bottom left is the configuration of the circuit with the 5V source replaced by a short circuit. The one at the bottom right is the configuration of the circuit with the 3V source replaced by a short circuit. The theoretical voltages for each case is 0.708V and 1.99V, respectively. The combined voltage is 2.70V.

We constructed the circuit based on the two cases.

3V on, 5V off

3V off, 5V on

Both 3V and 5V on

We also measured the resistance of the resistors under different cases

We put all the data in a table and compared the experimental values and theoretical values

	Experimental	Theoretical	Percent Error
voltage from 3V	0.70	0.708	1.13%
voltage from 5V	1.99	1.987	0.15%
voltage from both	2.70	2.695	0.19%

Theoretical Resistor (k Ω)	Measured Resistor (kΩ)
1	0.96	± .01kΩ
4.7	4.65	± .01kΩ
10	9.95	± .01kΩ
6.8	6.77	± .01kΩ
22	21.2	*±.1 kΩ

The percent errors are really small. The lab result successfully verifies the validity of superposition method.
We simulated the same circuit on everyCircuit app and it shows the same result.

Summary:
Superposition method is a very useful way to analyze circuit when there are multiple power sources in different places. If removing all power sources but one can make the circuit very easy, this would be the best way to analyze. We remove current source by using open circuit and voltage source by short circuit. However, this method has a significant flaw. We have to do a calculation of the whole circuit for each turned on source. An additional power source can significantly increase the amount of work. Additionally, removing a power source does not always simplify the circuit very much when they are close to each other.

This method reminds me of the composition of waves. The way the superposition method works resembles the constructive interference and destructive interference. Current is the movement of electrons. Electrons are not only particles, but also waves. A traveling wave can mix with another traveling wave and create a combined effect in the same way as the mathematical addition of two current values.

Mesh Analysis 2 & Time-arying Signals

Mesh Analysis 2

In this lab, we used another method other than the nodal analysis to solve circuit problems. Mesh analysis uses meshes instead of nodes to form equations. A mesh is a loop that does not contain other loops. we assume every mesh has a certain current running around. For convenience, the directions of the currents are set to be either all clockwise or all counterclockwise. This method can quickly setup a coefficient matrix of all the currents. It saves the time of simplifying the equation. Whenever there is a current source separating two meshes, the two meshes can form a "super-mesh". A "super-mesh" construct one loop equation consisting of multiple currents.

Here is a circuit graph of the lab with theoretical calculations. we used mesh analysis to get the three equations and used matrix to find the solutions. the mesh currents are listed at the top right corner.

we chose four resistances and measured their experimental resistances.

This picture shows the setup of the circuit

We measured the experimental voltages and compared to the given voltages in the circuit graph.

The following table shows the theoretical data, experimental data and percent error.

	V1 (V)	V2 (V)	I1 (mA)	R1 (kΩ)	R2 (kΩ)	R3 (kΩ)	R4 (kΩ)
Experimental	4.99	-3.6	-0.362	6.62	4.57	21.6	9.6
Theoretical	5	-3.22	-0.322	6.8	4.7	22	10
Percent Error	0.20%	11.80%	12.42%	2.65%	2.77%	1.82%	4.00%

Time-varying Signals

In this lab, we used the wavegen to generate time-varying signals. In all previous labs, we have been using the DC input only. We tried sinusoidal, ramp up waves on the wavegen.

sinusoidal wave scope

sinusoidal Wavegen

RampUp Scope

RampUp Wavegen

The amplitudes, frequencies and periods from the oscilloscope perfectly match the settings in the wave gen.

Summary: The mesh analysis is another alternative to Kirchoff's Laws. I believe that it applies to more circuits that nodal analysis. It handles both the voltage source and current source. It also helps to set up the coefficient matrix very fast while nodal analysis still requires one extra step to simply the equations. It is a very useful method and it should be the first choice for most circuit problems.

The time-varying signals allow us to test circuit with an alternating voltage. It is a better way than using the power supply box knob to adjust the voltage. We are going to use this technique in the following experiments.

Nodal Analysis

In this lab, we used nodal analysis to calculate and analyze a circuit. It works well when the circuit does not have a lot of nodes. The following pictures show the circuit we worked on, which only consists of 5 nodes. by setting one of them to ground voltage, four nodes out of the five nodes have a known voltage. This method significantly reduces the number of variables. The only unknown is the node indicated with V1 in the following graph.

The theoretical voltages are listed on the left side of the first picture. The experimental resistance and voltage values are marked on the circuit graph in the second picture. We made a table to calculate the percent difference between theoretical voltage and experimental voltage for each resistor.

	Theoretical Values	Experimental Values	% difference
V across 10 K ohm	5.53	5.52	0.18%
V across 22 K ohm	4.47	4.37	2.28%
V across 6.8 K Ohm	2.47	2.42	2.06%

Summary: The nodal analysis provides an alternative solution to complex circuit problems. Instead of using Kirchoff's Laws for every single circuit, nodal analysis enables us to find the unknowns in a faster way in some circuits. Circuits with many nodes and voltage sources are very suitable for the nodal analysis method. This experiment gives me a visualization of the nodal analysis method.

Temperature Measurement System

This lab uses a NTC 10K @ 25ºC thermistor to measure the temperature. The thermistor has a linear relation between the temperature and its resistance within a certain range. For this specific thermistor in the lab, the resistance is 11.18 kΩ at 25 ºC and 6.65 kΩ at 37 ºC.

The calculation shows the predicted output voltage at 25 ºC and 37 ºC.  The theoretical resistor we need in the circuit is 4.633 kΩ. In order to have the output voltage increase by a minimum of 0.5V over a temperature range of 25 ºC to 37 ºC, we picked a resistor of 4.7 kΩ to connect in series. It guarantees the voltage range out of all the resistor selections available.

 Here is the setup of the circuit:

Here are two videos showing the change of voltage when the temperature increases. The temperature rises from 1.43V to 1.98V. It creates a voltage difference of 0.55V. The theoretical value is 0.51V. The percent difference between the experimental drop and the theoretical drop is 7.27%.

Summary: This experiment shows that a thermistor can be used to control the output voltage by connecting them in series. A voltage divider circuit with a thermistor can use the temperature as a variable to control how much voltage is delivered to a specific load. By changing the resistance and power supply, thermistor can cover various range of voltage when it is in a certain temperature interval. Also, by measuring the voltage of the load, we can calculate the current temperature since the thermistor has a fixed value of resistance corresponding to a certain temperature.

Dusk-to-Dawn Light

This lab creates a circuit that operates as a "dusk-to-dawn" light. It turns a light on when the ambient light level goes below a certain level.
Circuit

BJT is a current controlled current source. The collector–emitter current can be viewed as being controlled by the base–emitter current (current control), or by the base–emitter voltage (voltage control). 

The calculation shows that the voltage of the photocell is 5/3 V when the resistance is 5K ohms. The voltage is 10/3 V when the resistance is 20K ohms. 

There is a 10K Ohms resistance connected in the circuit.

When the photocell does not receive enough light, the LED will be turned on. Likewise, if the photocell receives enough light, the LED remains off. Here is a failed trial that shows exactly the opposite due to the negative voltage mistake.

After we fixed the voltage problem, it works in the intended way.

Summary:

The photocell works as a light-sensitive resistor. It has high resistance when it received limited light, but the resistance drops as it receives more light. By using the Bipolar Junction Transistor in the circuit, it can operate as a trigger for dusk-to-dawn light.

Dependent Sources and MOSFETs & Resistors and Ohms Law

Fundamental Theorem of Topology: b = L + N -1
b: number of branches
L: number of loops
N: number of nodes

Dependent Sources and MOSFETs

Circuit Graph

Setup

By applying different gate voltage, the drain current goes from 0 to a steady value. 

According to the table and the graph, the threshold voltage for ZVN2110 MOSFET is 1.55V. The maximum current is about 44mA. Between the threshold voltage and the voltage of maximum current, the graph is straight. The rate of change g is about 120mA/V, which is also the slope of the straight part of the graph.

Summary: The MOSFET works as a Voltage Controlled Current Source. By applying different gate voltage, a specific drain current can be delivered. It can also be used for a trigger since it does not deliver any current when the voltage is below the threshold.

Resistors and Ohms Law

This lab measures the resistance of an unknown resistor by applying different voltages. Ohm's Law is verified by the relation between voltage and current.

Circuit Graph

Setup

We applied 6 different voltages and measured both voltage and current.

The graph shows that current is linearly related to voltage. It verifies the Ohm's Law. The slope of the graph is the reciprocal of resistance. there is a relatively law percent error of -3.18%. The equation of the best fit curve is y = 0.001x.

Summary: Ohm's Law shows that current is linearly related to voltage and the ratio is the reciprocal of resistance. This is the law that connects the three basic elements of circuit, voltage, current and resistance together.