Day 28 - Thursday, May 28, 2015 - Signals with Multiple Frequency Components

Today we thought about what happens to circuit elements as the frequency increases and we got the following relationships: 

The graphs show what happens to the impedance of those circuit elements as the frequency is increased. For resistors, different frequency values don't affect the value of its impedance. However, inductors and capacitors act differently. As the frequency increases across an inductor, the inductor responds proportionally to changes in current, so its impedance increases. For capacitors, there are the opposite relations. As the frequency increases, the impedance across it is observed to approach zero. The explanation for this is that faster changes in current are directly associated with faster changes in the polarization of the plates on the capacitor, thereby acting as a wire with increasing frequencies. 

However, our circuits are a lot more complex and have multiple components. So, we wrote a program in Freemat that let us see the gain as a function of s which equals w times j  on a circuit. Below is a picture of the program on the right side of the picture, and a graph on the left side of the picture.  

Today's experiment is called Signals with Multiple Frequency Components and we putted in a lot of signals and observed how the circuit reacted. 

For our pre-lab, we calculated what the response of our circuit would be when we put in a signal. For this, first, we had to calculate the equivalent impedance of the circuit. Then we calculated a voltage divider to see what the voltage across the resistor on the side looks like. The goal of the experiment was to calculate the gain, so the voltage across the resistor was divided by the input signal, and we were left with our formula for our voltage divider. This formula came to be realized as shown below. Our group had a hard time doing it, but Mason gave us a hand and showed us that the voltage divider can be calculated through the formula in the picture: 

We measure the values of our resistors and capacitor as: 

So, our gain equation looks like: 

And the gain values for when the frequency equals 500 Hz, 1,000 Hz, and 10,000 Hz are: 

For Part A of the experiment, we designed a custom signal whose equation is 

This is shown on AWG 1 as we set the frequency to 500 Hz and the amplitude to 4 V.

Below is a picture of what the signals on the circuit look like. The orange wave is the input signal and the blue wave is the output signal. They measure voltage.

As it can be observed, the gain on the big wave is much bigger than the gain on the smaller waves riding on the big wave. This is seen because the amplitude of the orange waves that are riding on the big orange wave is much higher than the amplitude of the blue small waves riding on the big blue wave. This is consistent with the mathematical description derived on the prelab where it shows the frequency as part of the denominator; hence, at higher frequencies, the gain will be zero. And, at lower frequencies, the gain approaches 1/2. This can be seen by looking at the gains of our circuit for the high, middle and low frequencies. 

For Part B of the experiment we fed in a sinusoidal sweep to the same circuit as in Part A.

Below is a picture of what our our signals look like. Just like in Part A, the orange signal is the input signal, and the blue wave is the output signal. Also, the measure voltage on the circuit.

As we can see, the gain in the output signal decreases as the frequency increases. Interestingly, it does so by forming a figure similar to a curvy cone.