Day 21 - Oscillator

Topics Discussed

On day 21 of the ENGR 44 course, we were introduced to op-amps that were integrated into AC circuits. We used impedance instead of resistance to find the gains in each circuit, but we utilized the same equations previously discussed for op-amps. We did a few practice problems that utilized the same methods as previously, only working now in the complex domain and using impedance. (Fig. 1)

Fig. 1

We were also introduced to oscillators, which essentially create an ac waveform in its output when supplied by a dc source. 

Op-Amp Relaxation Oscillator

Today in lab, we planned to investigate an op-amp oscillator, which would utilize a dc input and output an ac waveform. In order to do this, the Barkhausen criteria have to be met. That is, the overall gain of the oscillator must be unity or greater, therefore stating that losses must be compensated for by an amplifying device. Additionally, the overall phase shift from input to output must be zero.

Once we understood the workings of the oscillator, we proceeded to make the circuit according to the lab manual specifications. (Fig. 2)

Fig. 2

We then measured the output signal to see if we had an ac waveform. (Fig. 3) This wave showed that we were able to successfully create an op-amp oscillator.

Fig. 3

Summary

Today, we were able to successfully create an oscillator using an operational amplifier. Putting in a dc circuit, we were able to generate a waveform in the output. This alternating voltage came about due to the charging and discharging of the capacitor in the circuit, which essentially allowed the voltage to fluctuate and thus create the ac signal that we were reading. 

Day 20 - Sinusoids and Phasor Cont.

Topics Discussed

On day 20 of the ENGR 44 course, we further discussed sinusoids and phasors, integrating previous circuit analysis methods into these AC circuits.Interestingly, having an understanding of how to work in the complex domain of numbers makes these methods relatively simple. We focused on using nodal analysis, but also were introduced to mesh analysis as well. (Fig. 1)

Fig. 1

We did a few problems that focused on mesh analysis (Fig. 2). We did no lab since the lab for this day was the same as the previous.

Fig. 2

Summary

Although there was no lab to get actual hands-on learning, we were able to apply some old skills to new problems and thus develop a better understanding of how to analyze AC circuits. 

Day 19 - Sinusoids and Phasors Cont.

Topics Discussed

On day 19 of the ENGR 44 course, we had further discussion of sinusoids and phasors. Additionally, we were introduced to the concept of impedance in different elements of a circuit such as capacitors, inductors, and resistors. We were also introduced to the concept of admittance which is essentially the inverse of impedance. We did several practice problems with these ideas in mind. (Fig. 1) We also discussed the concept of leading and lagging in terms of voltage and current, and how different elements within the circuit can experience this. (Fig. 2)

Fig. 1

Fig. 2

Impedance Lab

In lab, we planned to investigate the effects of a sinusoidal voltage at a set frequency on a capacitor, resistor, and inductor in three separate circuits. We wanted to use frequency at 1kHz, 5kHz, and 10kHz. We then converted these to omega and made some calculations in reference to the responses we should get. (Fig. 3)

Fig. 3

Once the calculations were complete, we then proceeded to create each circuit. (Fig. 4,5,6) 

Fig. 4 - Inductor

Fig. 5 - Capacitor

Fig. 6 - Resistor

We then applied the three different frequencies to each circuit and observed the effects on the voltage gain across each. (Fig. 7-15)

Fig. 7 - Inductor @ 6280 Hz

Fig. 8 - Inductor @ 31 kHz

Fig. 9 - Inductor @ 62 kHz

Fig. 10 - Capacitor @ 6283 Hz

Fig. 11 - Capacitor @ 31 kHz

Fig. 12 - Capacitor @ 62 kHz

Fig. 13 - Resistor @ 6283 Hz

Fig. 14 - Resistor @ 31 kHz

Fig. 15 - Resistor @ 62 kHz

Summary

Upon completion of the lab, we were able to successfully view the effects of applying these different frequency voltages to these separate elements in each circuit.

Day 18 - Sinusoids and Phasors

Topics Discussed

On day 18 of the ENGR 44 course, we were introduced to the topics of sinusoids and phasors. Sinusoids are signals that vary according to an alternating sine or cosine function. We did several practice problems with these concepts. (Fig. 1) We also discussed using phasors which are phase shifts within these sinusoids. We practiced several methods used to add, subtract, multiply, and divide these phasors, including how to convert them from polar to rectangular coordinates and vice-versa.

Fig. 1

Passive RL Circuit Response Lab

In lab we planned to make an RL circuit and analyze the effects on said circuit when applying a sinusoidal voltage function. Using the phase shift within our calculations, we were able to find theoretical amplitude gains that were given by a formula within the lab manual. (Fig. 2)

Fig. 2

Once we had our theoretical values, we then proceeded to create the circuit and analyzed the phase shift and gains and found that we were relatively close to our theoretical values. (Fig. 3)

Fig. 3

Summary

Upon completion of the lab, we were able to successfully analyze an RL circuit that had an alternating voltage applied to it. We were able to find the gain and phase shift across the inductor and resistor. Since AC circuits are the most common circuits we find based off of the alternating nature of circuits in our day-to-day lives, being able to properly investigate the effects on certain elements in the circuit becomes extremely useful. 

Day 17 - Second Order Circuits Cont.

Topics Discussed

On day 17 of the ENGR 44 course, we further discussed the topic of second order circuits and how to analyze different elements within the circuit. We were introduced to new techniques and again did some practice problems. (Fig. 1)

Fig. 1

RLC Circuit Response Lab

In lab, we planned to investigate how another version of an RLC circuit would respond when an alternating voltage was applied to it. We calculated values for the theoretical natural frequency and oscillation frequencies. (Fig. 2) Once these values were found, we built the circuit. (Fig. 3)

Fig. 2

Fig. 3

Summary
Despite some alteration to this circuit as opposed to previous RLC circuits, we were able to use the same methods to find several of the characteristics of the circuit, such as natural frequency, oscillation frequency, and whether the system could be considered critically, over-, or under-damped. 

Day 16 - 2nd Order Circuits

Topics Discussed

On day 16 of the ENGR 44 course, we discussed 2nd order circuits. We discussed heavily the importance of understanding the initial conditions of RLC circuits and how they can be used to determine the behavior of these circuits. We also discussed source-free circuits and the tools and equations that can be used to uncover how these circuits behaved. We did a multitude of problems focusing on whether a system was damped, over damped, or critically damped. (Fig. 1) Additionally, we did some practice with source free circuits with granted initial conditions. (Fig. 2). We further derived some expression that we would commonly use throughout these types of problem. (Fig. 3)

Fig. 1

Fig. 2

Fig. 3

Series RLC Circuit Step Response Lab

In lab today, we investigated the voltage response of an RLC circuit when inputting a square wave voltage. We intended to find the alpha, tau, natural frequency, damping ratio, and other various pieces of information relative to our circuit. We first drew up a schematic for the circuit and found the according pieces of information that would be used as our theoretical values. (Fig. 4) Once this was complete, we proceeded to create the actual circuit. (Fig. 5) 

Fig. 4

Fig. 5

Upon completion, we ran the circuit and proceed to analyze the voltage response. (Fig. 6) We drew up several experimental values and compared them to what was originally expected.

Fig. 6

Summary

Upon completion of the lab, we were drawn to one value in particular that stuck out, which was our measured/experimentally calculated tau. We had originally expected a much shorter period for a single oscillation but found one to be relatively 1/4 of a second. This may have been accounted for in the additional resistance of the circuit, which would have increased our alpha and thus decreased our tau. 

Day 15 - First Order Circuits Cont.

Topics Discussed

On day 15 of the ENGR 44 course, we further discussed first order circuits. We discussed unit step functions and how they can apply to RLC circuits and likewise. We also did a multitude of practice problems concerned with finding the voltages across different elements in terms of these unit step functions. (Fig. 1) We also derivated in class the voltage of an RC circuit. (Fig. 2) We also predicted the voltage in an RC circuit given different input voltages of various shapes.. (Fig. 3)

Fig. 1

Fig. 2

Fig. 3

Inverting Differentiator Lab

In the inverting differentiator lab, we sought to make a classic inverting differentiator using a capacitor, resistor, and op-amp. We calculated a multitude of theoretical gains when inputting different frequencies into the circuit, and we planned to use these frequencies and find an experimental gain for comparison. Sample calculations can be seen in Fig. 4.

Fig. 4

Once we had our experimental values accounted for, we proceeded to create the circuit. (Fig. 5) Upon completion of the circuit, we hooked up our circuit to our analog discovery device and began taking measurement. We tried different signals at 100 Hz, 230 Hz, and 500 Hz. (Fig. 6, Fig. 7, Fig. 8).

Fig. 5

Fig. 6 - 100 Hz

Fig. 7 - 230 Hz

Fig. 8 - 500 Hz

Once our measurements were taken, we created a table to document our theoretical versus experimental gains. We found that our highest percent error was below 8%. (Fig. 9)

Fig. 9

Summary

In today's lab, we were able to verify the theoretical gains and our perceived relationship of input to output voltage of an inverting differentiator. With a percent error of less than 8%, our predicted gains at different frequencies held true.