Signals with Multiple Frequency Components

Signals with Multiple Frequency Components

Overview: 

In this lab, we predicted the magnitude response of an electrical circuit when an input signal is applied to it. Then, we actually applied the signals, and compared the circuit's response to our predicted response. The types of signals that applied were a signal composed of multiple sin waves with different frequencies, and a sinusoidal wave with a time varying frequency (also known as a sinusoidal sweep).

Design:
We designed the following circuit to execute the measurements:

Both R's were set to 1300 Ohms

^^^^^ C was set to 100 nF ^^^^

Based on simple nodal analysis, we find that the magnitude response (the ratio of amplitude of the output sinusoid to the input sinusoid) is equal to:

H(w)=(Vout/Vin)=[1/(2+(S/(10^5))]

This implies that as the frequency fluctuates with time, so does the magnitude response of the circuit: If the frequency is low, the capacitor acts as an open circuit, which results in an output that is half of Vin. When the frequency is high, the capacitor acts as a short, resulting in a Vout of zero.

Construction and Execution:

 ^^^^ Resistances of both R values ^^^^

 ^^^^ Measured capacitance of C ^^^^

Completed Circuit from side ^^ and above >>

^^^^ 500 Hz Vout ^^^^

 ^^^^ 1000 Hz Vout ^^^^

 ^^^^ 10KHz Vout ^^^^

The yellow line of the graphs is the circuit's Vout. As the frequency increases, the Vout amplitude decreases accordingly. This is what we predicted.

^^^^ Sweep circuit response ^^^^

The yellow line is once again the circuit's Vout. Once again, as the frequency of the input wave increases, the Vout amplitude decreases. This confirms our suspicions and calculations.

Analysis: 

As the oscilloscope outputs suggest, our predictions were correct. As the frequency of the input wave increases, the Vout amplitude progressively drops to zero. Also, the output of a signal with multiple sinusoidal frequencies successfully filtered out the high frequency portion of the wave, leaving only the low frequencies of the input wave. 

Conclusion:

Filters like these are incredibly useful for transmission of data, as they can easily decode things like radio waves. Filters of all sorts can be used to filter out unwanted frequencies. 

Apparent Power and Power Factor

Apparent Power and Power Factor

Overview:

In this lab, we will examine the effect of power factors on AC circuits. By changing an inductive load to being balanced out with a capacitor (resulting in a larger power factor), we hope to observe a larger power factor, and in turn a larger apparent power delivered to the load.

Design:
We designed the following circuit to execute the experiment:

^^^^^ L was set to 1.023 milliHenrys ^^^^

RT was set to 10.1 ohms and remained constant

RL varied throughout the experiment

RT represented the resistance of the transmission lines, so it remained constant. RL represented the inductance of the load, and also remained constant. RT varied with each different "load" that we applied. We applied a total of three circuits, with RL's of 10.3 Ohms, 46.9 Ohms, and 100.1 Ohms. For each load we calculated the RMS current delivered by the source, the RMS load voltage, the average power delivered to the load, the apparent power delivered to the load, the load's power factor, the average power dissipated by RT, and the ratio of the average power dissipated by RT to the average power delivered to RL when we applied a sinusoidal signal of amplitude 1V and frequency of 5000 Hz. The following are our calculation results:

^^^^ The columns are listed using the resistor's nominal values ^^^^

As the table suggests, the power factor increases as the load resistance increases, resulting in a larger amount of the power being delivered to the load.

The goal was to execute those measurements, and then compare our results to our calculations. After that, we planned on connecting a 1 microFarad capacitor in parallel with the load, to compare the results.

Construction and Execution:

^^^^ 10 Ohm load resistance (LEFT) and 10 Ohms transmission line resistance (RIGHT) ^^^^

 ^^^^ 47 and 100 Ohms loads (respectively) ^^^^

 ^^^^ 1 nF conductor measurement and 1 mH inductor measurement (respectively) ^^^^

 ^^^^ Internal resistance of inductor and internal resistance of voltmeter (respectively) ^^^^

The .4 Ohms was subtracted from the internal resistance of the inductor's 2.2 Ohm measurement, resulting in a true internal inductor resistance of 1.8 Ohms

^^^^ The constructed circuit (without the parallel capacitor) ^^^^

Applying a 5000 Hz sinusoidal wave input with amplitude of 1V yielded the following results for each load:

^^^^ 10.3 RL with no capacitor ^^^^

^^^^ 46.9 RL with no capacitor ^^^^

^^^^ 100.1 Ohm RL with no capacitor ^^^^

We then added the parallel capacitor measured earlier, and repeated the process:

^^^^ Complete circuit with parallel capacitor to increase power factor ^^^^

^^^^ 10.3 Ohm RL with capacitor ^^^^

^^^^ 46.9 Ohm RL with capacitor ^^^^

^^^^ 100.3 Ohm RL with capacitor ^^^^

Analysis: 

As the oscilloscope outputs suggest, our calculations were correct. Not only does increasing the resistance load increase the power factor, but adding a capacitor in parallel results in a larger power factor as well. 

Conclusion:

The average power delivered by a source is not necessarily the average power received by the load. To determine the average power delivered to an object, complex analysis is necessary. Once that has been calculated, adding additional inductance or capacitance to create an increased power factor will result in a more efficient power delivery. The results of an increased power factor are an increase in apparent power being delivered to the load, a decrease in power being "dissipated" within the imaginary spectrum of the power band. 

Inverting Voltage Amplifier / Op Amp Relaxation Oscillator Labs

Inverting Voltage Amplifier Lab

Overview:

In this lab, we measured the gain and phase responses of an inverting voltage amplifier circuit, and compared the measurements with expected values.

Design:

We designed the following circuit to conduct the lab:

^^^^ C is the capacitor on top ^^^^

The resistors 'R' were both set to 10K ohms

The capacitor 'C' was set to .1 microFarads

The op amp used was an OP27

Based on the fact that the input voltage and the output voltage are complex exponentials, we can 

Calculate the circuit's input/output relationship using the equation:

Vout/Vin =   1/(1+j*omega*R*C)

Calculate the circuit's amplitude gain using the equation:

| Vout/Vin | =   -1/sqrt(1+(j*omega*R*C)^2)

and finally calculate the phase shift relationship between input and output using the equation:

angle(Vout) - angle(Vin) = 180 - arctan(1/(omega*R*C))

Construction and Execution:
We began by measuring the resistance and capacitance of the elements. A few sample pictures are shown below:

^^^^ (Left) Measured resistance of left resistor and (Right) measured resistance of top resistor ^^^^

Measured values of elements:

R left: 9.85 Ohms

R top: 9.94 Ohms

C: .099 microFarads

Naturally, the slight differences in measured values from ideal values would propagate to a huge level of uncertainty, so we had to do all of our calculations twice. Once before measuring the values, and once after measuring the values, to recieve more accurate predictions.

We calculated gain and phase shift values at three different frequencies: 100 Hz, 1 KHZ, and 5 KHZ. The table of calculated values are below, with the 'updated' (and more accurate) values being on the right:

We then constructed the circuit, and subjected it to a sinusoidal input voltage of varying frequency (as mentioned above) and amplitude of 2V. Below is a snapshot of it:

Results:

Beginning with 100 Hz, these are our results:

Measured gain: 1.48/2 = .74
Expected gain: .847
% difference in gain: 12.6 % (yikes)

Measured phase shift: 31.50 degrees
Expected phase shift: 32.01 degrees
% difference in phase shift: 1.60 % (sweet)


Now, 1 KHz:

Measured gain: 0.32/2 = .16
Expected gain: .16
% difference in gain: 0 % (holy crap)

Measured phase shift: 82.03 degrees
Expected phase shift: 80.82 degrees
% difference in phase shift: 1.50 % (sweet)

Now 5 KHz:

Measured gain: .08/2 = .04
Expected gain: .032
% difference in gain: 25 % (yikes)

Measured phase shift: 87.76 degrees
Expected phase shift: 88.15 degrees
% difference in phase shift: .45 % (sweet)

Analysis:
Overall, the values were all pretty much perfect other than the gains. A possible source of error though, is that we didn't record the exact values, so the analysis was based on simply eyeballing the graph. We wont make that mistake again. Obviously, the method is accurate though.

Op Amp Relaxation Oscillator Lab

Overview:

In this lab, we designed an Op Amp Relaxation Oscillator having a frequency of 921 Hz.

Design:

We designed the following circuit to execute the objective:

**** Ignore the wires going up to the top left of the schematic and out to the right of the schematic. Also, ignore the inverted poles for the +/- 5 V supplies. ****

R was 10K ohms

R1 was 1K ohms

Ca was 1 microFarad

R2 was our controlling variable, creating the oscillation frequency of choice (in our case 921 Hz)

Using the equations:

T = 2RC ln {(1+beta)/(1-beta)}

-and-

beta = R1/(R1+R2)

We were able to calculate the necessary value for R2.

R2 = 35,413 Ohms.

Below are our calculations:

Construction and Execution:

With the calculations completed, we tested our circuit virtually in EveryCircuit:

^^^^ SMASHING SUCCESS!!! ^^^^

We then constructed the circuit as seen below. *Note* R2 was a rather unique resistance, and we had very limited selection of resistors, so we threw several in series to achieve the desired result (Net resistance equaled 35K ohms)

Results:
After supplying power to the voltage rails, these were our results:

^^^^ Voltage across the capacitor ^^^^

^^^^ Voltage out of the Op Amp ^^^^

Measured Values:

T = 1.09 ms

Therefore the measured frequency is 917.4 Hz

desired frequency is 921 Hz

% error: .39%

Analysis:

Things went great; I feel like a million bucks. With a percent error of a marginal .39%, the method of creating relaxed oscillators via Op Amps is flawless.

Phasors: Passive RL Circuit Response Lab

Phasors: Passive RL Circuit Response Lab

Overview: 

In this lab, we measured the gain and phase responses of a passive RL circuit, and compared the measurements with expected values.

Design:
We designed the following circuit to execute the measurements:

R was set to 1.1 ohms

^^^^^ L was set to 1 microHenry ^^^^

Based on the fact that the input voltage and current are complex exponentials, we can 

Calculate the circuit's input/output relationship (the amplitude gain) using the equation:

I/V = Amplitude gain =  1/(R+j*omega*L)

Calculate the phase shift relationship between input and output using the equation:

phi-omega = -arctan((omega*L)/R)

And finally, calculate the cutoff frequency of the circuit using the equation: 

omega cutoff = R/L

We wanted to analyze the circuit at three different frequencies: cutoff frequency,  cutoff frequency/10, and cutoff frequency*10

Below are our calculations for the expected phase shifts and amplitude gains:

Calculated values:

Cutoff Frequency = 1.1*10^6 Hz

Phase Shift:
Cutoff Frequency = -45 degrees
Cutoff Frequency/10 = -5.71 degrees
Cutoff Frequency *10 = -84.29 degrees

Amplitude Gain:
Cutoff Frequency = 0.6482
Cutoff Frequency/10 = 0.9046
Cutoff Frequency*10 = 0.0905

These gains make sense, as a higher frequency will cause a much larger impedance response from the inductor, while a lower frequency will cause the inductor to act more like a short.

Construction and Execution:

 ^^^^ The resistor was measured to be approximately 1.35 Ohms ^^^^

This much of a variation unfortunately called for all new calculations of phase shifts and amplitude gains:

 ^^^^ Recalculated values, accommodating for the change in resistance ^^^^

Updated calculated values:

Cutoff Frequency = 1.1*10^6 Hz

Phase Shift:
Cutoff Frequency = -39.17 degrees
Cutoff Frequency/10 = -4.66 degrees
Cutoff Frequency *10 = -83.00 degrees

Amplitude Gain:
Cutoff Frequency = 0.5742
Cutoff Frequency/10 = 0.7383
Cutoff Frequency*10 = 0.0902

We then constructed the circuit as illustrated previously in the schematic:

Results:

Using a 1 volt amplitude sinusoidal signal, these were our results:

The screenshot above shows the oscilloscope output of our circuit. 

Analysis:

Unfortunately, due to limitations of our equipment, we were only able to record data at 15 KHz. This is 1/11.7 of our cutoff frequency, so the data is quite skewed. That being said, the oscilloscope graph is quite true to out expectations at that frequency. As you can see, the voltage across the inductor (blue) is TINY in comparison to the voltage across the resistor (notice the differences in scale). Overall, the general behavior of an inductor has been shown: 

An inductor in a circuit that is oscillating at below its cutoff frequency will only marginally affect the circuit, as its impedance will be minuscule in comparison to the impedance of the other resistance.

An inductor in a circuit that is oscillating at its cutoff frequency will significantly affect the circuit, as its impedance will be at least equal to that of the other resistance.

And an inductor in a circuit that is oscillating at above its cutoff frequency will drastically affect the circuit, as its impedance will be massive in comparison to the other resistance.

Impedance Lab

Impedance Lab

Overview:

In this lab we measured impedances of resistors, capacitors, and inductors. We then compared them to their expected values.

Design:

We designed three circuits to experimentally measure the impedance of the elements:

^^^^ (Left) The circuit to measure the impedance of a resistor and (Right) the circuit to measure the impedance of an inductor ^^^^

^^^^ The circuit to measure the impedance of a capacitor ^^^^

The resistors impedance was simply equal to its resistance, while the impedance of the inductor was equal to j*omega*L, and the capacitor's impedance was equal to 1/(j*omega*C). By using these equations, it became clear that the current for the capacitor would lead the voltage by 90 degrees, and the current for the inductor would follow the voltage by 90 degrees. Obviously the current through the resistor is in phase with the voltage. 

Construction and Execution:

^^^^ (Left) Resistance of the 47 Ohm resistor and (Right) the 100 ohm resistor ^^^^

^^^^ Measured capacitance of the the capacitor ^^^^

R47: 48.7 Ohms

R100: 100.0 Ohms

C.1 microF: .093 microF

L: 1 microH

We then constructed the first circuit with the two resistors, and measured the voltages across each element and each of the three frequencies. We also added a mathematic channel (red) which illustrated the current.

^^^^ The dual resistor circuit, as viewed from the side ^^^^

 ^^^^ Output for circuit with input frequency of 5 KHz ^^^^

The voltage across the 100 ohm resistor is in blue, and has an amplitude of 1.35 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 0.54 V.

The current is in red, and has an amplitude of 14.6 mA.

The voltage for the resistor is in phase with the current.

These values are consistent with our expectations to +/- 4.9 %

 ^^^^ Output for circuit with input frequency of 10 KHz ^^^^

The voltage across the 100 ohm resistor is in blue, and has an amplitude of 1.35 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 0.54 V.

The current is in red, and has an amplitude of 14.6 mA.

The voltage for the resistor is in phase with the current.

These values are consistent with our expectations to +/- 4.9 %

 ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the 100 ohm resistor is in blue, and has an amplitude of 1.35 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 0.54 V.

The current is in red, and has an amplitude of 14.6 mA.

The voltage for the resistor is in phase with the current.

These values are consistent with our expectations to +/- 4.9 %

We then went on to testing the inductor:

  ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the inductor is in blue, and has an amplitude of .269 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.90 V.

The current is in red, and has an amplitude of 39.1 mA.

The voltage for the inductor leads the current by 90 degrees.

The inductors impedance was measured as 53.7 ohms with a voltage gain of .169 Volts.

  ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the inductor is in blue, and has an amplitude of 1.07 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.50 V.

The current is in red, and has an amplitude of 34.2 mA.

The voltage for the inductor leads the current by 90 degrees.

The inductors impedance was measured as 78.5 ohms with a voltage gain of .57 Volts.

  ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the inductor is in blue, and has an amplitude of 1.56 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.18 V.

The current is in red, and has an amplitude of 24.4 mA.

The voltage for the inductor leads the current by 90 degrees.

The inductors impedance was measured as 109.1 ohms with a voltage gain of .74 Volts.

Finally, we went on to testing the capacitor:

   ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the capacitor is in blue, and has an amplitude of 1.61 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.05 V.

The current is in red, and has an amplitude of 19.5 mA.

The voltage for the capacitor follows the current by 90 degrees.

The capacitors impedance was measured as 119.3 ohms with a voltage gain of .66 Volts.

 ^^^^ Output for circuit with input frequency of 5 KHz ^^^^

The voltage across the capacitor is in blue, and has an amplitude of .58 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.75 V.

The current is in red, and has an amplitude of 34.2 mA.

The voltage for the capacitor follows the current by 90 degrees.

The capacitors impedance was measured as 61.3 ohms with a voltage gain of .33 Volts.

^^^^ Output for circuit with input frequency of 10 KHz ^^^^

The voltage across the capacitor is in blue, and has an amplitude of .34 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.82 V.

The current is in red, and has an amplitude of 39.1 mA.

The voltage for the capacitor follows the current by 90 degrees.

The capacitors impedance was measured as 55.8 ohms with a voltage gain of .16 Volts.

Analysis:

All of the measurements came out to being incredibly close to our theoretical values. This shows that the calculation of impedances as imaginary parts of the voltage and current is a valid method of calculation. Additionally, it defends the theory of phasors (which is essentially the same thing).