Day 14 - First Order Circuits

Topics Discussed

On day 14 of the ENGR 44 course, we were introduced to first order circuits. These circuits are known as either RL or RC circuits (resistor-inductor or resistor-capacitor, respectively). These circuits contain the passive elements such as resistors, capacitors, and inductors, and one active element being the operational amplifier. We discussed several relationships to the voltage or current through a capacitor or inductor, and how we can essentially model the behavior of these elements in such a circuit. We did example problems that included deriving an equation for the voltage across a capacitor and the time constant tau associated with the equation. (Fig. 1, Fig. 2, & Fig. 3) "The time constant of a circuit is essentially the amount of time required for the response to decay by a factor of 1/e." (Mason- Lecture Notes). For capacitors, it is RC and for inductors it is L/R. We also did work with inductors in example problems and modeled the current through it as a function of time. (Fig. 4)

Fig. 1 

Fig. 2

Fig. 3

Fig. 4

Passive RC Circuit Natural Response Lab

In lab, we sought to create a simple RC circuit and observe its natural response when an input voltage is applied. We would first model the circuit in a schematic and estimate the time constant. This would allow us to create a theoretical value for how long the capacitor would take to discharge. We would use the oscilloscope trigger to determine the response of the capacitor when voltage was disconnected, or when a square wave was applied. We first created the schematic and made our predictions. (Fig. 6) Once this was done, we created the actual circuit. (Fig. 7)

Fig. 6

Fig. 7

Upon creation of the circuit, we hooked up our analog discovery device and set up an oscilloscope window to observe the specific changes occurring within the capacitor. Perhaps the most difficult aspect of this lab was properly setting up the trigger, which would ultimately determine whether or not we could get a good reading from the oscilloscope window. We found our reading. (Fig. 8) We found that our value for Tau compared to our calculated value was very close and exhibited a percent error of roughly 10%. (Fig. 9)

Fig. 8

Fig. 9

Passive RL Circuit Natural Response Lab

Although we did not necessarily get a chance to complete this in class, we recreated a similar circuit to the when mentioned above, only this time we used an inductor. We again created the schematic and made predictions for the general equaion describing the current as a function of time (Fig. 10) Prior to leaving class, we quickly assembled the circuit and ran a quick trial to observe the response of the inductor. (Fig. 11) 

Fig. 10

Fig. 11

Summary

We were able to successfully assemble and verify our models for the natural response of an RC and RL circuit today in lab. Considering our only measured and recorded trials, the RC circuit exhibited only a rough 10% error in tau when compared to our calculated value. By understanding the process by which these circuits operate, we will be able to create more complex and versatile circuits in the future.

Day 13 - Capacitors and Inductors

Topics Discussed

On day 13 of the ENGR 44 course, we were introduced to the basics of capacitors and inductors. We discussed the fundamental workings of capacitors first, noting that they work as a short before being charged and an open after being charged. We also viewed what happens when a capacitor is not aligned correctly relative to the direction of current flowing through it. (Video 1) No one wants to lose their fingers, so this demonstration was both important and entertaining, so we did it twice. We discussed how resistors must accommodate capacitors to prevent sparks from happening in circuits, and we did some example problems that involved capacitors (Fig. 1 & Fig. 2)

Fig. 1

Fig. 2

Additionally, we discussed inductors and how they work. We also discussed how we could treat them oppositely when compared to capacitors. We performed several practice problems with circuits involving inductors as well. (Fig. 3)

Fig. 3

Capacitor Voltage-Current Relations Lab

The first lab we performed in class focused on integrating a capacitor into a circuit and observing how it effected current and voltage within the circuit. We planned to use sinusoidal and triangular wave voltage inputs and see how the voltage across the capacitor would react. We predicted the shape of the resulting voltage that would occur across the capacitor given our newly acquired knowledge of how capacitors behave under such conditions. (Fig. 4) 

Fig. 4

Once we had this general idea, we created the circuit. (Fig. 5) Upon creating the circuit, we applied two separate sinusoidal waves, one with a 1k Hz frequency and the other with a 2k Hz frequency, both at an amplitude of 2V and offset of 0V. We also put in a triangular input voltage of 100 Hz frequency, 4V amplitude, and 0V offset. 

Fig. 5

In the measuring oscilloscope window, we got a reading for the input voltage and capacitor voltage. Additionally, we used the math channel to calculate the corresponding current going through the capacitor. Fig. 6 & 7 show our acquired graphs.

Fig. 6 - 1kHz Sine Wave

Fig. 7 - 2kHz Sine Wave

Inductor Voltage-Current Relations

For the second lab conducted in class, we used the exact same circuit created in the first lab, only this time instead of using a capacitor we used an inductor. Our measurements would all work the same way, that is in using the math channel on the oscilloscope to again calculate the current through the inductor using the measured voltage. We first created the circuit. (Fig. 8)

Fig. 8

We input the exact same sinusoidal waves and triangular waves into the inductor circuit, and we yielded the voltage and current that corresponded through the inductor itself. (Fig. 9, Fig. 10, & Fig. 11)

Fig. 9 - 1kHz Sine Wave

Fig. 10 - 2kHz Sine Wave

Fig. 11 - 100Hz Triangular Wave

Summary 

Upon completion of the labs, we were able to verify the predicted and derived relationships of capacitors and inductors to current and voltage respectively. Although not perfect, our results gave us information that was good enough to give us aggreement between theory and actual practice. These basic circuits and demonstrations will allow us to properly utilize these elements in future circuits. Additionally, we were introduced to the math channel in the oscilloscope window of our analog discovery device. This is a powerful tool, in that it allows us to construct real time graphs for several other circuit readings that could be obtained from measuring only a select few parts of the circuit. 

Day 12 - Cascade Op-Amp Circuits

Topics Discussed

On day 12 of the ENGR 44 course, we discussed cascading operational amplifiers. This would essentially allow us to combine a series of amplifiers together in one circuit in order to achieve far more adjustability and precision in a circuit. What we essentially uncovered is the fact that these cascaded circuits could be analyzed the same way as previous and more simple op-amp circuits. We simply had to take each op-amp as a separate piece of the circuit, recognizing that the output voltage of one op-amp could be the input of the other. We also had a brief introduction to DAC (Digital to Analog Converters).

Temperature Measurement System Design Lab

In the temperature measurement system design lab, we were required to design a circuit using a thermistor, a wheatstone bridge, and a difference amplifier to gain an output voltage that was temperature dependent. We had created a circuit similar to this in a previous class meeting, however, in this case we wanted more precision in terms of how sensitive the thermistor would be to the slightest temperature change. Our goal was to have an output voltage that would increase in the positive direction by at least 2V. We created a schematic for the circuit (Fig. 1).

Fig. 1

Firstly, we had to create a wheatstone bridge which would have 4 resistors of the same value. However, we found that in our class-setting, this was unachievable due to the lack of extremely precise resistors which would be required. Therefore, we included a potentiometer in our wheatstone bridge that would allow us to essentially "dial-in" and achieve an exact resistor value that would give us a zero voltage difference across the two ends of the bridge. We found that our room temperature resistance for our thermistor was 10.7k ohms and the body temperature resistance was 6.6k ohms. Once this was completed, we created the circuit (Fig. 2).

Fig. 2

We found that our output voltage became extremely sensitive, and we had a range of greater than 2V. A video demonstration of the entire circuit working is shown below.

Summary
We were able to successfully create the difference amplifier that would be dependent on the temperature applied to the thermistor in this lab. One of the fundamental aspects of this lab was balancing the wheatstone bridge, in that it was an imperfect piece of equipment that we were able to dial in to get precise values. This process of creating this circuit showed us the capabilities of using op amps in practical designs to get results that are precise and responsive. 

Day 11 - Op-Amps #2

Topics Discussed

On day 11 of the ENGR 44 Course, we further discussed operational amplifiers. We discussed and did practice problems pertaining to non-inverting amplifiers, summing amplifiers, and difference amplifiers. In every instance, we were dealing with an op-amp that was ideal, and our derivation of the relationship between input and output voltage all came from our previously learned methods of circuit analysis. We discussed how different ranges of voltages would yield us different output values when paired different gain values. (Fig. 1) We also did practice problems with non-inverting, summing, and difference op-amps (Fig. 2 and 3)

Fig. 1

Fig. 2

Fig. 3

Summing Operational Amplifier Lab

In the first part of the lab, we sought to create a summing operational amplifier circuit. We did this with two separate voltage sources that would have corresponding resistors of the same value. We chose the same values in order to gain a result that was an actual sum of the voltage supplies. We drew up the schematic for the circuit (Fig. 4) and proceeded to build it after (Fig. 5).

Fig. 4

Fig. 5

We found that our output voltage was again limited to the voltage range used to power the operational amplifier. We yielded a series of results and plotted a graph of the results (Fig. 6).

Fig. 6

Difference Operational Amplifier Lab

In the second part of the lab, we sought to create a difference amplifier circuit. We did this again with two separate voltage sources who in this case would be applied to the separate positive and negative input terminals of the operational amplifier. We again created a schematic for the circuit (Fig. 7) and constructed the circuit after (Fig. 8).

Fig. 7

Fig. 8

We again yielded results that showed that supported our initial theory of how the circuit should behave. We created a data table of the two separate input voltages and the output voltage.

Summary

In both labs, we were able to successfully create summing and difference amplifiers. We found, however, that the limits of our output voltage were not exactly equivalent to the 5V/-5V power supplied to the op-amp. Instead, we gained values such as 4.29V or -3.50V. This is because not all the power being applied to the op-amp is going to the out put voltage. Because it is not a perfect system, voltage is dropped at some point or another, preventing us from observing complete saturation at 5V or -5V.

Day 10 - Op-Amps

Topics Discussed
On day 10 of the ENGR 44 course, we were introduced to the concept of operational amplifiers. In the context of the classroom, we referred to voltage controlled voltage sources that essentially amplified some in-voltage to be adjusted mathematically to a new out-voltage. We discussed the process by which we could redraw an equivalent circuit from the operational amplifier. After the initial introduction into the topic, we performed a series of problems which focused on determining the relationship between the input voltage and output voltage (Fig. 1). We discussed the linear range of an op-amp as well, noting that any output voltage could only vary between the values of the positive and negative power supplies of the op-amp (voltages).

Fig. 1

Inverting Voltage Amplifier Lab
In lab, we created an op-amp based circuit using a general model idea from previous practice problems. For a certain circuit we worked on previously, we found that the out-voltage could be related to the ratio of the input resistor to the parallel op-amp resistor multiplied by the in-voltage. We wanted to get the ratio close to 2, so we decided to use a 4.7k ohm and a 2.2k ohm resistor respectively. We created the schematic for the circuit as well as the corresponding resistance values we would be using. (Fig. 2)

Fig. 2

We proceeded to actually create the circuit. (Fig. 3) We then measured the corresponding input and output voltages of the op-amp, seeking a gain of negative two across the voltage values. We also wanted to find the saturation points of the op-amp, where the voltage would not go above or below. We acquired our values and created a data table and graph. (Fig. 4)

Fig. 3

Fig. 4

Summary
In this lab, we were able to construct a circuit using an OP27 op-amp and acquire values for input and output voltages for the circuit. We constructed a graph of the relationship between the input and output voltages and found that the relationship was linear up to the points of saturation. This led us to conclude that our gain was in fact what we had hoped for. In the future, we will now be able to use these inverting amplifiers more effectively and with better understanding.