Bode Plots

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24 - Signals with Multiple Frequency Components

 INTRO: Today we covered the topic of signals with multiple frequency components. In other words, this is something one would see when you have several input signals in your circuit. An example is if you were to have several different sinusoidal input signals feeding into your circuit, as opposed to the circuit we have been working with that just have one.

 Below Professor Mason did some examples of what we would be learning for the day.

Below we cover the topic of transfer functions. Voltage gain,current gain, etc., are all common function

 

LAB: SIGNALS WITH MULTIPLE FREQUENCY COMPONENTS

In today's lab we built a circuit with multiple input signals. Just like described above, this circuit has multiple frequency components. We will then measure circuit values across our circuit.

 Below is a picture of the circuit schematic that we will be building for this lab. We are building a low-pass filter, which will eliminate high-frequency signals from passing through it.

Below are some values we calculated for our circuit.


Below is a picture of our low-pass filter circuit that we built. Per the schematic 2 pictures above, we have two resistors and a capacitor in our relatively simple circuit.


For this circuit we apply a custom input wave. The wave is represented by the function: 20(sin(1000πt)+sin(2000πt)+sin(20,000πt)),  and then a sinusoidal sweep function.
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Here is our wave at 500HZ


 Here is our wave at 1000HZ
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Here is our wave at 10KHZ
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Here is the second part of our lab. As mentioned above, the second waveform that we run is a sweep function. Analyzing the wave, it can be seen that the blue wave gets more suppressed as time passes in the circuit.


CONCLUSION
In today's lab we dealt with circuits with multiple frequency components. The frequency response of a circuit is is the variation in its' behavior with a varying input signal frequency. We also discussed transfer functions and some common types that we have already used through this course. At times like these I reminisce about my time in the enchanted land of Fantastica. My main questions arising out of the completion of this lab is why didn't Bastian take the Rock Monster to fight the Darkness? Why did Atreyu take Artax through the Swamp of Sadness when he CLEARLY could have taken him over the forest of Fantasies!? I took this class in hopes that these questions would be answered, but alas, I must extend my search further to answer these pertinent life questions.

23 - Apparent Power and Power Factor

INTRO: Today we continued our lecture on AC power by moving onto the topics of apparent power and power factor. Due to the non-constant nature of AC current, there are several power measurements used for an AC system. We will add these two new quantities to the list.


Pictured at the bottom of the page in the picture are the definitions for apparent power and the power factor. The apparent power is the combination of true and reactive power in a circuit. The power factor is the ratio of the average power to the apparent power.


Below is a visual representation of the apparent power. It is the combination of the reactive and real power in a circuit.




LAB: Apparent Power and Power Factor
In this lab we will use work around the apparent power and power factor when working with a circuit. Mainly we will deal with power transmissions to our AC circuit and also deal with the delivery efficiency of the power.


Below is a schematic we will build during this lab. It is a fairly simple lab, however it will be perfect for what we need.


Below is the circuit we built. We used a 10, 47, and 100 ohm resistor. When we measured the actual resistance of the resistors with our multi-meter they were relatively close to the given values.


Below is the table of several values we found when performing adjustments to our circuit. We found our Vrms, as well as thevenin equivalent resistence, inductance, and several other values. AS the table shows, we performed operations with our circuit with the three different resistors.


Below is our graph when we have the 10 ohm resistor. We find the magnitudes and phase angles by the graph quantities.


Below is our graph when we have the 47 ohm resistor. We find the magnitudes and phase angles by the graph quantities.


Below is our graph when we have the 100 ohm resistor. We find the magnitudes and phase angles by the graph quantities.

22 - AC Power - Average and Max

INTRO: Today in class we continued our discussion on AC circuits. An integral part to electronics is the power supplied and absorbed by circuit elements. Due to the alternating current, power calculations have different equation structures then direct current equations. 

Below are the equations for power for AC circuits. Due to the alternating current, their is instantaneous and average power per circuit element. The average power is the quantity that is typically used. Also the effect/rms value is the DC currrent or voltage needed to create the same power quantity in a DC circuit.

In the picture below Professor Mason connected two bulbs, one with DC current and the other with AC current. The bulbs with DC current was the bulb that shown brighter, like the face of a child opening a copy of the ultra-super-mega packed director, Arnold cut edition of Jingle all the way.

When the Professor turned up the voltage to ~1.5 times on the AC bulb, the light was relatively the same.

CONCLUSION: Today we worked with Power in AC circuits. Per the equations, which were backed up by a physical example, power equations are different for AC and DC. We also discussed the Effect/RMS value, which is the DC equivalent current or voltage necessary to mimic the same power in an AC circuit. Additionally, we did not perform any labs today, as we ran out of time.

21 - Inverting Voltage Amp & Op - Amp Relaxation Oscillator

INTRO: Today we return back to the familiar category of Op-amps, however this will be with an AC voltage source. Op-amps of course are circuit elements that amplify the input signals feeding into the amplifier. These are ideal devices because they allow a large amount of current/voltage to be changed with simply making changes to the much smaller input voltage/current feeding into the amplifier. The all-familiar guitar amplifier is a common example that many people have experience with.


Below is a voltage amplifier. Just like stating above, this device is capable of controlling the voltage/current through manipulating the smaller voltage/current that is feeding into it.



LAB: INVERTING VOLTAGE AMPLIFIER
In this lab we will build a an inverting voltage amplifier. We will do so with an AC source, resistors, an op-amp, and a capacitor. Like the prior labs, we will calculate circuit values prior and compare them to the experimental values at the end of the lab.


Below is the  circuit diagram that we will build. It is very similar to our prior op-amp setups, but this one has a capacitor in the top branch.


Below we completed the pre-lab for the lab. We calculated values using given values of frequency, resistance, and capacitance. We completed calculations for amplitude gain, as well as the phase change for all three of the frequencies we ran the circuit at.

Here we set up the diagram according according to the above schematic. We have our op-amp, resistors, as well as the capacitor hooked into our breadboard. We also have an analog discovery connected at the designated points so we can send a varying voltage through the circuit.


Here we had an input voltage of 100HZ


Here we had an input voltage of 1KHZ


Here we had an input voltage of 5KHZ


Conclusion: We then measured the difference between our calculated and measured amplitude gain as well as the phase shift. The percent difference between the amplitude gain was 3.5%, while the phase shift percent difference was 7.8%.


LAB: OP-AMP RELAXATION OSCILLATOR
In this lab we need to design an oscillating circuit that oscillates.. We will do so by using an op-amp, capacitor, and a resistor.


The following schematic is what we will model our circuit after. It is a bit more complex then the prior lab, however we are still using familiar circuit elements.

Below is the pre-lab where we calculated circuit element values. Since we know we want a frequency of 99Hz, we calculate a required resistance of 8371 ohms.

Below is a picture of our circuit, which is modeled after the aforementioned diagram. Also, the output of the circuit is below.



Conclusion: Plugging in numbers from the measured values, our percent error of the measured vs. theoretical frequency was 1.9%. I would say this is within the realm of acceptable error. Also, we learned how to construct a op-amp relaxation oscillator. I've been waiting my whole life for this moment. This is better then the time I met Falcor and saved the Princess, the Rock Monster, and Atreyu from the nothing. But alas, that is a story for another time.

20: Phasors - Passive RL Circuit Response

INTRO: Today we were reintroduced to the familiar concept of phasors. Phasors were introduced to us in the prior 4B class as a way to simplify working with circuits with oscillating voltages. In other words, phasors are synonymous with working with AC circuits. They are a complex number that is composed of a real and imaginary part.


Below is visual representation of the phasor as cartesian axes. The x-axis represents the real axis, while the y axis is the imaginary axis.
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Below we began working with converting values from the time-domain to the polar/phasor domain. Using relationships we rewrite the voltage and then find the current doing mathematical operations in the phasor domain.

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Below is another representation of circuit values in the phasor domain. We do the opposite in this exercise as we go from the phasor domain to the time domain. While doing multiplication and division is simpler in the phasor domain, doing addition and subtraction is easier in the time domain.


LAB: PHASOR
In today's lab we built a RL circuit and sent a sinusoidal input and observed the circuit response. However, instead of working in the time domain to describe and evaluate the circuit we will do so in the phasor domain.


Below is a visual representation of what we will be doing in this lab.


Below is a pre-lab where we calculated circuit values in the phasor domain. For three different frequencies we calculated the gain, as well as the accompanying phase change. As to be expected, we will be comparing these to the experimental values at the end of the lab.


As stated above, the circuit was composed of a resistor as well as an inductor. We hooked it up so we could also send a voltage through the circuit when need be.


Below are the graphs of the RL circuit during the three different scenarios.




To conclude our lab I made a graph summarizing the calculated and experimental data. As the table shows, the calculated and experimental are reasonably close, resulting in a percent error less then 6% in every category.

19 - Impedance

INTRO: Today we focused on a familiar concept of Impedance. Impedance (Z) is the ratio of the phasor voltage to the phasor current. In more familiar terms, the impedance of a circuit is the circuits resistance to alternating current. Impedance of a circuit can alternatively be calculated by summing the resistance, capacitance, and inductance in a circuit.


Below is a simple circuit with a voltage source, and impedances. Ignore the circuit values in non-time domain values,since it is unimportant for now.



 LAB: IMPEDANCE
 In this lab we will build a circuit and calculate the impedance of three elements; a resistor, capacitor, and inductor. We will then check our experimental values against our calculated ones.

 Below we drew and calculated the impedance values for our three circuits. The circuits are a normal resistive circuit, a RL circuit, and then a RC circuit. As seen as the pictures below, we list the calculated values for the impedance.


Resistive Circuit and accompanying graph at 1k frequency



 RC circuit and accompanying graph at 1k frequency



 RL and accompanying graph at 1k frequency



In the chart below we charted the values from all three of our circuits. Although just one graph per circuit is listed in the situations above, we ran each circuit three times at 1,5,10 khz. The experimental values were relatively close to our calculated values.