Day 8 - Thevenin's Theorem

Topics Discussed

On day 8 of the ENGR 44 course, we were introduced to the Thevenin's Theorem, which essentially is a theorem/application that allows us break down a rather large circuit into a simple single voltage source and single resistor, of which we call Thevenin voltage and Thevenin resistance respectively. After being introduced to the theorem, we applied it to numerous practice problems in class. (Fig. 1)

Fig. 1

We also touched base on some calculations involving finding the maximum power that could be delivered to some element attached to the Thevenin circuit diagram (Fig. 2)

Fig. 2

Thevenin's Theorem Lab

In the Thevenin's Theorem Lab, we sought to verify Thevenin's Theorem and see whether our not the Thevenin Resistance and Voltage would yield results similar to those calculated. To begin, we drew a diagram of the circuit and proceeded to find the Thevenin resistance and voltage. (Fig. 3) 

Fig. 3

Once we had our values calculated, we proceed to create the circuit without the voltage sources. We would then measure the resistance of the entire circuit to verify whether or not the Thevenin resistance was true. We then input the voltage sources and attached a potentiometer to the circuit. (Fig. 4) We compared our measured values to our calculated values. (Fig. 5)

Fig. 4

Fig.5

We then proceeded to vary the resistance within the potentiometer to create a graph with the said data points reflecting the power vs resistance and yielded a decent graph that illustrated an upward slope. (Fig. 6)

Fig. 6

Summary

In this lab, we were able to verify Thevenin's Theorom and ultimately see how it could be put into practical use. By finding the Thevenin resistance and voltage, we could essentially find the maximum power that could be dissipated across some device, which in our case was a potentiometer. 

Day 7 - EveryCircuit, Linear Circuit, and Superposition

Topics Discussed

On day 7 of the ENGR 44 course, we discussed the topics of superposition and source transformation. These new tools would allow us to efficiently analyze circuits that may at first glance seem complex. We did practice problems that allowed us to exercise these new techniques (Fig. 1).

Fig. 1

Additionally, we were granted an EveryCircuit access code and practice using the program to double check calculations for various voltages, currents, and resistances through different elements in a circuit. We constructed our own circuits and were able to gain useful values for all the elements within the circuit (Fig. 2).

Fig. 2

Superposition II Lab

In the superposition II lab, we sought to test whether our not removing voltage sources would yield the same results for current and voltage as we calculated using the superposition technique of analyzing a circuit. First, we drew a diagram of our circuit and found the corresponding voltages of the 6.8k ohm resistor as we continuously changed which power source we were applying. (Fig. 3)

Fig. 3

Upon making our calculations, we proceeded to construct the circuit using a breadboard, analog discovery, wires, and the corresponding resistors. We then measured the voltage of the 6.8k ohm resistor and recorded our data. We compared the results found in the lab with those we had predicted, and found each result to be within 7% error (Fig. 4).

Fig. 4

Summary

In this lab, we were able to verify the technique of superposition and its practical application when doing real circuit analysis. We found that when disregarding some voltage sources, we were able to easily calculate the voltage across certain resistors. If each source was taken into account individually, we could then combine our calculated values to yield the true total voltage across a certain resistor. Our percent error was within 7% in all cases, which allowed us to verify the newly taught concept of superposition. 

Day 6 - Mesh Analysis and Transistors

Topics Discussed

On day 6 of the ENGR 44 course, we further discussed the advantages of mesh analysis. We continued to do example problems to reiterate how to apply these techniques to circuit analysis (Fig. 1). Additionally, we also developed a new skill using mesh analysis in which we could create a matrix by simply looking at the circuits elements (Fig. 2).

Fig. 1

Fig. 2

We were also introduced to transistors and the difference between NPN transistors and PNP transistors, and how each differ in terms of the current they direct. We discussed techniques of how to analyze circuits which contained a transistor element, and we solved group practice problems (Fig. 3). We also performed a demo with the analog discovery system in which a current or voltage could be provided at an alternating rate. We formed graphs from the information received (Fig. 4 & 5).

Fig. 3

Fig. 4

Fig. 5

Mesh Analysis 2 Lab

In this lab, we sought to use mesh analysis to predict the values of voltage and current across certain elements within the circuit. By using the analog discovery device, we should be able to build our theoretical circuit and test to see if mesh analysis yields us the true values we will measure in the lab. 

To begin, we first drew a diagram of the circuit and used mesh analysis to find the voltage across our 6.8k ohm resistor, and current through our 10k ohm resistor (Fig. 6).

Fig. 6

Once we found these values we built our circuit (Fig. 7). We measured our values for current and voltage across the desired elements and recorded our data as well as percent error. (Fig. 8)

Fig. 7

Fig. 8

Summary

In this lab, we were able to verify the true application of mesh analysis on an actual circuit. Our value for voltage had only a 1.6% error, while our current had a 21.7% Error. The largest source of error accounted in the current may be in part due to the real values of our resistors. Our 4.7k ohm resistor had the biggest difference than what we had intended, and this may have had a large impact on how much current went through the 10k ohm resistor. Overall, our lab successfully verified the usefulness of mesh analysis. 

Day 5 - Mesh Analysis

Topics Discussed

On day 5 of the ENGR 44 course, we further discussed the advantages and disadvantages of using nodal analysis when working with a circuit. Additionally, we were introduced to another powerful tool used in circuit analysis known as nodal analysis. To illustrate the advantage of mesh analysis over nodal analysis, we analyzed a circuit in the beginning of class using nodal analysis. After being introduced to mesh analysis, we used the new method on the same problem as earlier (Fig. 1). We also did work with other problems to gain more practice with the new method (Fig. 2). We found that the complexity of the problem greatly decreased and that we could now handle a wider range of circuits.

Fig. 1

Fig. 2

Nodal Analysis Lab

The goal of the nodal analysis lab was to find the voltage across a certain resistor first by means of nodal analysis, and second experimentally by means of actual measurement with a voltmeter. To begin, we drew a diagram of the circuit with which we would deal with and calculated the corresponding voltages across certain elements in the circuit. Due to availability, we changed one of the resistors to a 22k ohm resistor.

After calculating the theoretical voltage across the 22k and 6.8k ohm resistors, we proceeded to create the circuit using a breadboard and Analog Discovery System (Fig. 3). With multiple voltage sources, we were careful to make the circuit run properly.

Fig. 3

We measured the voltages across the resistors and compared them with the theoretical voltages and found that we were less than 1% off for both. (Fig. 4)

Fig. 4

Summary

This lab illustrated the versatility and usefulness of nodal analysis. By analyzing the junction between wires in a circuit, we were able to calculate accurate results for the voltages across multiple elements in our circuit that were within 1% of the actual values measured. This lab verified this powerful method and will allow us to handle more complex problems in the future. 

Day 4 - Circuit Analysis

Topics Discussed

In class, we discussed two new topics essential to doing a proper circuit analysis that would not require us to create a multitude of equations with a multitude of unknowns . We discussed nodal analysis and super nodes in particular. Nodal analysis would allow us to place a voltage on specific nodes in order to reduce the number of equations needed to find various elements within the circuit. Super nodes involved two nodes which had a voltage source.

On day 4 of the ENGR 44 course, we took a quiz involving using KVL and KCL to find a certain voltage across a resistor in terms of other given variables. The quiz was meant to illustrate the difficulty of using such methods to solve for unknown values in a circuit. This provided basis for the discussion of breaking a circuit down into simpler components that allow for less equations to be needed (order reduction). We then went into lab rather than a class discussion on new topics.

Temperature Measurement System Lab

In lab, we attempted to create a circuit that would include a thermostat to change an output voltage by at least 0.5 V. We were provided the thermostat, and we were set to design a circuit that would accommodate three criteria: a required 5 V input voltage, an output voltage that varied at least 0.5 V in a range of 25°C (Room Temperature) to 37° C (Body Temperature), and the output voltage would have to increase as the temperature increased. 

We first designed a circuit that would properly adhere to all the specifications required of the lab. (Fig. 1) We found the ideal resistance we would need in or circuit in order to have the best range of resistance as our thermostat changed temperature. (Fig. 2) By taking the derivative of our voltage divider equation, our ideal resistance was found to be 8770 ohms. However, in the lab we 8200 ohms was the closest resistor available, and was thus used. Ideally, our thermostat would operate at 11000 ohms and 7000 ohms at room and body temperature respectively. 

Fig. 1

Fig. 2

We measured the actual resistance of our thermostat at both room and body temperature and calculated their percent difference to our ideal values. Additionally, we measured the actual resistance of our resistor and found it to be 8260 ohms rather than 8200 ohms. We created our circuit and attached it to our Analog Device to give it power and measure voltage across different elements within it. (Fig. 3)

Fig. 3

Our circuit functioned properly and measured a different voltage value across our resistor at two different temperatures. (Video & Fig. 4)

Fig. 4

Summary
In this lab, we properly constructed a circuit under set conditions that would cause it to have a multitude of different voltage outputs across different temperatures. Our values for voltage output varied by 0.6 V, and our voltage increase as temperature increased. We yielded a percent difference of less than 5% in terms of our output voltage from what we had predicted. Creating this circuit reinforced the concept of voltage dividers and varying resistance elements. We designed the circuit ourselves and successfully created our required thermally dependent circuit.

Day 3 - Voltage Dividers, Parallel vs. In series

Topics Discussed

A demonstration of completing a circuit with a hot dog was shown in the beginning of class. A 120 V potential difference was placed across the hot dog, and the hot dog acted as a wire in the circuit. It was predicted that the hot dog would slowly cook, and this is indeed the case. LED lights were placed across the hot dog, some parallel and some perpendicular to the hot dog. (Fig. 1) It was predicted that the LEDs parallel to the hot dog would light up, since they were in the direction of the potential difference, while those perpendicular would experience no potential difference and thus remain off.

Fig. 1

Circuits of varying complexities were introduced in class. More practice with dependent sources was done, as well as an introduction to voltage dividers that involved an abundance of in class group work. The equation of voltage dividers and resistors was derived, and more examples reinforced the concept. (Fig. 1 & Fig. 2)

Fig. 2
A circuit with an LED is shown, instructing us to find the range of possible resistors

Fig. 3
A circuit with a current controlled current source (CCCS) is shown.

Dusk-to-Dawn Light Lab

In the dusk-to-dawn light lab, our goal was to create a circuit which would cause an LED light to turn off and on as its surroundings became more and less luminous. A BJT (Bipolar Junction Transistor) was used as a switch in this lab, and a photocell was used as the device which would react to how luminous our environment was, and would obtain a certain resistance accordingly. This varying resistance is what would allow, or not allow, current to pass through the LED light. To create the circuit, it was first determined how voltage would be affected when the photocell obtained different resistances (Fig. 4).

Fig. 4

We first tested the photocell to see the varying resistances we would get at high and low light situations. We found that at its highest level of resistance under direct flash, the photocell was capable of up to 60k ohms as read by our DMM. At the lowest level of light, we measured a resistance of roughly 100 ohms. 

We proceeded to create the circuit, following the diagram given in the lab manual. Using a bread board, BJT, photocell, 10k ohm resistor, and wires as needed, our circuit was operational. (Fig. 5 & Fig. 6) 

Fig. 5

Fig. 6

As intended, the led light turned on in low light environments, and off in high level light environments. We had calculated the voltage across the photocell (potentiometer) to be between .05V and 4.3V at the lowest and highest resistance respectively. (Fig. 7) We then measured the voltage drop across the photocell using our analog device. (Fig. 8)

Fig. 7

Fig. 8

Video demonstration: 

Summary
Having made the circuit work under the expected conditions, the voltages measured across the photocell were much different then we had initially calculated. As seen in Fig. 8, the voltages ranged from 0.32V in the light, and only 0.72V in the dark. Although this was initially surprising, a possible explanation for this difference may lie when accounting for a possible lower resistance across the photocell then initially expected. This lower resistance would in turn cause a lower voltage reading across the photocell. 

Day 2 - Resistance, Conductance, Branches, Loops, etc.

Topics Discussed

On day two of the ENGR 44 course, we were reintroduced to the concepts of resistance, Ohm's Law, and loops. Additionally, we reinforced old topics and reinforced new ones. This included an overview of power, as well as an introduction to nodes and branches. Specifically, we also discussed how to identify independent loops and how this could lead us to discovering the number of branches and nodes and vice-versa. We also saw a brief demonstration of voltage potential in the beginning of class, and how certain loops, even when powered, will not cause a voltage change to, say, a light bulb in another loop of the circuit. (Fig. 1)

Fig. 1
It was found that attaching the lower battery had no effect on the light bulbs.

In class, we discussed what made a material have a resistance and concluded that it was based on a combination of material type, length, and cross-sectional area. We found that resistivity varied directly with material type and the corresponding coefficient of resistivity, that length varied directly as well, and that cross-sectional area varied inversely. We also discussed the topic of "hot" and "cold" resistance, and completed a problem involving a tungsten filament light bulb which utilized the concept (Fig. 2)

Fig. 3

Dependent Sources and MOSFETS

Fig. 4

In lab, we attempted to construct a circuit that had a dependent source of current. This dependent source was achieved by use of a MOSFET (Metal Oxide Semiconductor Field Effect Transistor), which essentially would act as a voltage controlled current source (VCCS). We used a 130 ohm resistor in the circuit to offset the current that would pass through the MOSFET, ensuring no damage would happen to it or to the circuit. We used a 4.5 V voltage source, and the MOSFET would be used to regulate how much current would go through the circuit. The circuit can be seen in Fig. 4. 

Using the analog device to control the current, we recorded the corresponding voltages at the gate of the MOSFET to the current with which we were measuring on our DMM. A table and graph show our results in Fig. 5. 

Fig. 5

Fig. 5

Summary

In this lab, we found that there is an initial baseline where the MOSFET will essentially not allow any current into the circuit. As we increased voltage, we uncovered a range in which the MOSFET would greatly change voltage. Although not recorded, it was found that at roughly 5 V, the MOSFET would again level out and not show any change in current. Using the slope of the data we created, although limited, gave us a value for "g", which is essentially the parameter of our VCCS. "g" was found to equal 10.4 V/mA. Unfortunately, our graph did not fully verify the leveling out at the upper range of voltage for the MOSFET. 

Day 1 - Introduction and Review of Power, Voltage, Current, Charge, and Energy in Circuits

Topics Discussed

Day one of the Electrical Engineering (ENGR 44) Course introduced and included a review of the many elements learned in previous physics courses (PHYS 4B) in regards to electric current. Among these topics were charge, current, voltage, power, and energy. Additionally, a brief demonstration of breadboards was included which emphasized open-circuits and a short-circuits.

The fundamentals of electricity and circuits were introduced on Day 1. Charge and current were discussed first, as charge is considered the fundamental building block to understanding and "explaining all electric phenomena" (Mason). With this understanding of charge, we derive an understanding of current,  which is essentially a description of the mobility or change of charge over time. Charge is measured in coulombs, with the standard relating to the charge of an electron (1e=-1.602×10^-19C or 1C=6.24×10¹⁸ e) Two different types of currents were also briefly discussed, such as direct current (dc) and alternating current (ac). Direct current is simply a current that remains constant over a set period of time, while alternating current is a current that varies over time in a sinusoidal fashion. Power, energy, and voltage were introduced after, illustrating methods of how graphs could be derived by knowing the behavior of certain elements of a circuit.

Many of the prior relationships learned in previous physics courses were reinforced in the beginning of class. This included the relationship of current to charge, voltage to work and charge, and power to energy.  These relationships would intersect in various situations, and several problems and questions were presented in class that illustrated this merging of concepts. (See Fig. 1)

Fig. 1
In this problem, the graph of current and voltage as functions of time were given. With these graphs, the total energy absorbed by the device was to be found. In order to properly solve the problem, the relationship between energy, voltage, and current had to be derived. A necessary relationship that was noted was that of power to energy, in which power is the change in energy over a period of time. Rearranging this relationship would lead us to understanding that the energy function is equivalent to the integral of power multiplied by time (dt). We could then recall that power is also defined as the product of current and voltage, and by substitution we can derive the equation presented in the lower right corner of Fig. 1. The calculation to find the exact value for the problem was not recorded.

Another important topic discussed in day one was dependent sources of current and voltage and how they apply to the real world compared to the ideal independent voltage and current sources used in previous physics classes. The four dependent types of sources included a voltage-controlled voltage source (VCVS), a current-controlled voltage source (CCVS), a voltage-controlled current source (VCCS), and a current-controlled current source (CCCS). Understanding these dependent sources grants us the ability to solve problems where circuits contain  contain fluctuating current and voltage sources. Additionally, power and a reinforcement of the conservation of energy within a circuit was also discussed. As current moves through a circuit, voltage sources can be used to determine the power absorbed or released at various points across the circuit. We can solve for unknowns by combining this idea with the law of conservation of energy. (See Fig. 2)

Fig. 2
In this problem, we are given a circuit that has an ideal voltage source of 50 volts, creating a current of 3 amperes. At multiple junctions within the circuit, we are given the corresponding currents as the wire is split, as well as the voltage experienced at multiple different points. We are instructed to find the unknown voltage of the circuit element in the center. By using the current at each specific point, as well as the voltage experienced, we are able to find the power absorbed and produced. Applying the Law of Conservation of Energy, we can find the necessary power remaining that needs to be absorbed by the unknown circuit element in the center, and thus we can find the unknown voltage of said circuit element.

Solderless Breadboards, Open-circuits, and Short-circuits Lab

Introduction

The lab performed on Day 1 was intended to allow us to become familiar with breadboards and digital multimeters (DMMs). The electrical resistance experienced by creating these circuits across different holes in the breadboard, which would be connected by jumper wires, would be measured using the DMM (used as an ohmmeter) (Fig. 3)

Fig. 3
By connecting the leads of the ohmmeter to different holes across the breadboard, we were able to measure the resistance across different potential circuits. This would allow us to develop a basic understanding of the functionality of breadboards and how to properly use them in future labs. In this figure, the resistance across holes in the same row is shown.

Procedure

In order to conduct the lab, jumper wires would be placed across several holes in the breadboard and the resistance between the holes would be measured. First, the leads of the ohmmeter would be connected to two holes in the same row on one distinct side of the central channel on the breadboard and the corresponding resistance of that circuit would be recorded (Fig. 4). Next, the leads of the ohmmeter would be connected between two holes in the same row but on opposite sides of central channel and the corresponding resistance of that circuit would be recorded (Fig. 5). Following, the leads of the ohmmeter would be connected across two arbitrary holes across the breadboard and the corresponding resistance of that circuit would be recorded (Fig. 6). Finally, with the leads of the ohmmeter in two arbitrary holes, a jumper wire will connected the rows of the arbitrary holes, and the resistance across this circuit will be measured (Fig. 7). For every circuit, it will be determined whether it is an open-circuit or short-circuit.

Fig. 5
Circuit connected by holes on same row on opposite
 sides of channel, resulting in infinite resistance. 

Fig. 4
Circuit connected by holes on same row on same side
channel, resulting in relatively low resistance.

Fig. 7
Circuit connected by jumper while on same row of corresponding
arbitrary holes, resulting in relatively low resistance.

Fig. 6
Circuit connected by arbitrary holes,
 resulting in infinite resistance

Data and Results 
Among the separate trials, it was found that in instances of low resistance (or any numerical reading of resistance) a circuit was being properly created and we would classify this as a closed circuit. If the resistance was found to be extremely high/infinite (See Fig. 8), this signified an open circuit. Data table with measurements can be found in Fig. 9.

Fig. 8
This reading occurred during the trial of two holes in the same row on
opposite sides of the central channel and the trial of two arbitrary holes.

Fig. 9

Conclusion

In this lab, we familiarized ourselves with the proper use of breadboards and what type of circuits would be created in a multitude of situations. It was found that closed circuits were created when two wires were attached to holes in the same row on the same side of the central channel. In any other  instances where a circuit was created by connecting holes that were NOT in the same row and NOT on the same side of the central channel simultaneously, open circuit conditions would be observed. However, in this instance, if a jumper cable connected the two rows of these holes separated by row or by central channel, then a short circuit would be created. Understanding how the breadboard behaves will allow us to became more efficient in more complex experiments in the future.