Day 17 (Day19 on notes)

Today we learned about second order systems in parallel and did a RLC Circuit Response Lab.

Above is an example of a RLC circuit in circuit as review for what we did in day 16.

Above is an example of a RLC circuit in parallel. It should be noted that the equation for alpha changed and the rest is the same -> alpha=1/(2RC).

RLC Circuit Response Lab:
the purpose of this lab is to model and test a RLC circuit in parallel. On our pre-lab, we calculated values for omega and alpha and found that our circuit will be underdamped. We will be comparing these theoretical values with measured values. A 2V 500Hz square wave will be the voltage input.

Above is a picture of our circuit.


Above we have the output voltage were we can see that our circuit is very close to a criticllydamped curve which, based on our alpha and omega values, seems unreasonable.

We attempted to do a manually curve fit (not great) to find alpha, then used the period of the graph to find omega. As we can see we have a huge percent error which indicates something in our circuit is not working properly. We made sure to include actual measured resistances in the circuit for our theoretical circuit but did not change much. After much tinkering, we believe some faulty component was causing our circuit to behave differently than we predicted.

Day 16 (Day18 on notes)

Today we learned about second order systems - RLC series circuits and performed a Series RLC Circuit Step Response Lab.

Above we have a RLC circuit in series and derived a characteristic equation based on the second order differential. 

We then let alpha=R/(2L) and omega=sqrt( 1/(LC) ) from our characteristic equation. Based on these individual values, we can determine the response of the RLC series circuit as shown in the bottom of the picture.

Here is a small summary of the equations for the overdamped, critically damped, and underdamped responses along with graphs.

Above we worked on a series RLC example.

Series RLC Circuit Step Response Lab:
The purpose of this lab is to

We had to build a RLC circuit with the values listed above and measure its voltage across the capacitor when a 500Hz 2V square wave for input voltage. We see that our circuit will be underdamped.
On the bottom we had to find a value for C that would make our circuit critically damped but we didn't have components in class to do so.

Above is our circuit.

In the fist picture we have our input voltage and the second picture is the output voltage.

With an exponential curve fit, we get V(t)=0.43*exp(-2340t). This was done manually so we are unsure if the curve fit was done correctly which would explain why our alpha is significantly smaller than our theoretical value.

Day 11

Today we went over inverting, non-inverting, summing, and difference amplifiers. Also, we did labs on summing and difference amplifiers.

Here we took a look at the effect of waves when saturation in the amplifiers occurs. We found a saturation voltage output of around 4.2V and -3.5V with the op amps used in class.

Above is an example problem of an inverting amplifier where we used nodal analysis to find Vo.

Above is a derivation to find Vo on a non-inverting amplifier in terms of Vi, Rf and Ri.

Summing Amplifier Lab:

The purpose of this was to compare measured output voltages with theoretical output voltages.

For our pre-lab, we were to find a ratio of resistance values Rf/Ri such that we wont reach saturation voltages. We picked 6.8kOhms for Ri and 3.9kOhms for Rf (Ri=R1=R2 and Rf=R3). The equation left of our table is the equation that was derived for Vo.

Above is our circuit set up. Unfortunately we did not take a picture with the waveform generators wires hooked up.

Based on our results, we can say that our derivation to determine the output voltage is goo since most of the voltages were at a small percent error. We believe any error in are measurements is due to treating the op amp as ideal and resistors to have exact resistances throughout the experiment.

The purpose of the Difference Op Amp Lab is to compare theoretical output voltages with measured values.

 
Is the result of the derivation for output voltage on a difference amplifier in terms of input voltages and the four resistors. But when all resistors are equal to each other, the the output voltage is just Vo=V2-V1. We will be testing this case in our lab. 

Above is a picture of our circuit and measurements made for the lab.

Day 15 (Day17 on notes)

Today we went over op amps in RC circuits, Did an Inverting Differentiator Lab, and learned about singularity functions.

The differentiator op amp is an op amp circuit where the output voltage is proportional to the rate of change of the input voltage. Also, any noise can result in high or low saturation.

The integrator op amp output voltage is proportional to the integral of the input voltage. This op amp can lead to saturation fairly quick so it requires a feedback resistor to be added.

Inverting Differentiator Lab:

The purpose of this lab is to test the behavior of the differentiator op amp. We will then compare measured output voltage with theoretical values.

The first picture is the input voltage of 1V at 1kHz and the bottom picture is the output voltage.

The first picture is the input voltage of 1V at 2kHz and the bottom picture is the output voltage.

The first picture is the input voltage of 1V at 500Hz and the bottom picture is the output voltage.

Above we see the theoretical output voltage amplitudes and in pink are the measured output voltage amplitudes. The large percent error may be due to the op amp having saturation. It can be noted that as we decreased the frequency of the input voltage, the percent error decreases so it may imply that our op amp is more efficient at lower frequencies. When taking this into account, we can conclude that the output voltage is proportional to the rate of change of the input voltage.

Above is an example of circuit analysis using singularity functions to approximate the current in the
RC circuit shown. Singularity functions are functions that are either discontinuous or have discontinuous derivatives.

Above is another example of circuit analysis that uses singularity functions.

Day 14 (Day16 on notes)

Today we learned a bit more about inductors and went over 1st order circuits. We also did a Passive RC Circuit Natural Response Lab and a Passive RL Circuit Natural Response Lab. 

Above we took a look at equivalent inductance that works like ohms laws for resistors.

We did a quick example of finding the time for a capacitor at 5V to drop to .01V which we see is about 6 seconds.

Passive RC Circuit Natural Response Lab:

The objective for this lab is to measure the response of a RC circuit when charging then manually disconnecting the power source to measure the voltage across the capacitor using an oscilloscope.

Above is our circuit set up for part b of the lab.

Part a) Here we have the capacitor fully charged at 3.5V then the voltage drop is recorded as the power source is shorted. The time constant is then measured by 36.8% of the fully charged voltage (1/e of the exponential curve).


Part b) Like in part a, we use the same method to calculate the time constant.

Above are the results we got for the lab. the large percent error in part b is expected because we were not able to switch the circuit manually fast enough to decrease error. Despite the lack of an efficient switch, we had expected to see a large uncertainty value in our results.

Above are examples of first order RL circuits. We were to find current and voltages as a function of time of the inductor and its time constant.

Passive RL Circuit Natural Response Lab.:
The purpose of this lab is to measure the response of a RL circuit and manually disconnecting the power source to measure the voltage across the inductor using an oscilloscope.

Since time in class was running out, the professor set up the circuit and displayed the oscilloscope measurements on the projector. We had made a prediction of the behavior of the voltage across the inductor (in green) with a square wave voltage input. The current (in blue) was then drawn based on the green graph.

Here is the voltage graph that was measured across the inductor which agrees with our predictions.


Here I included a table of equations from today's lecture.

Day 13 (Day15 on notes)

Today We learned about capacitors and RC circuits, did a lab on capacitor voltage-current relations, and learned inductors and also did a lab on inductor voltage-current relations.

Above we did an example where we had to find the energy stored in each capacitor and the voltage across each of the capacitors.

Capacitor Voltage-current Relations Lab:

The objective of this lab is to measure the relationship between then voltage difference across a capacitor and the current passing through it in an RC circuit when applying different types of time-varying signals.

Above we predicted what the current graph as a function of time will look like at the specified voltage function graph drawn in black.

Above is a our circuit layout and we are measuring the voltage across the capacitor.

From our oscilloscope, the input voltage is the sine wave at 1kHz and 2V in blue and the current is shown in orange.

This is our measurement of a 100Hz triangular wave at 4V in blue and current in orange.

This is our measurement of a 2kHz triangular wave at 2V in blue and current in orange.

Based on our results, we see that the current across the capacitor is not the cosine wave we had predicted. Since the capacitor is not able to charge and discharge quickly, we see that result in the current graphs. We also see this same effect in the triangular voltage input but is more "square" shaped as predicted. 

Above we do equivalent capacitance in a circuit that are like ohms laws but opposite. 

Above is a useful table or questions

Above is an example where we going current and energy in an inductor based on known voltage function and inductance.