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.