![\"1465](\"http:\/\/pages.vassar.edu\/magnes\/files\/2014\/05\/1465-nF-Capacitor.png\")
<\/a>Figure 1<\/p><\/div>\n
<\/a>Figure 2<\/p><\/div>\n
<\/a>Figure 3<\/p><\/div>\n
For the 1465, 1007, and 47 nF capacitors the measured frequency of oscillation was\u00a0128,\u00a0155, and\u00a0730 Hz, respectively.<\/p>\n
Voltage As a Function Of Time \u00a0and Frequency of Oscillation<\/h3>\n
It is helpful to layer these three plots on top of each other (Figure 4). In doing so we can readily compare the curves to each other.<\/p>\n
<\/a>Figure 4<\/p><\/div>\n
From Figure 4 it is evident that the higher the capacitor value, the lower the frequency of oscillation. This is not surprising, given the equation that the frequency and inductor*capacitor product are inversely related:\u00a0.<\/p>\n
How are frequency and capacitance, or frequency and inductance related? I recorded the frequency of oscillation in several different LC circuits to try and illustrate this relationship. For three different inductor values I discharged five different capacitor values. The resulting plot (Figure 5) is below:<\/p>\n
<\/a>Figure 5<\/p><\/div>\n
Because it is harder to see the shape of the blue line in Figure 5, I plotted it separately (Figure 6):<\/p>\n
<\/a>Figure 6<\/p><\/div>\n
Figures 5 and 6 experimentally illustrate the inverse square root relationship between frequency and capacitance\/inductance.<\/p>\n
I then plotted the frequency of oscillation with respect to inductor value (Figure 7):<\/p>\n
<\/a>Figure 7<\/p><\/div>\n
\u00a0Exponential Rate Of Decay<\/h3>\n
If you take a look at Figure 4 again, notice that\u00a0despite changing capacitor values, the exponential decay of each curve is relatively the same.\u00a0The decay for each curve is the same because the decay coefficient\u00a0\u03b2 is a function of resistance and inductance, not capacitance(). In Equation (1) you can see the decay term of the voltage, e^(-\u03b2t):<\/p>\n
\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (1)<\/p>\n
In order to measure the decay constant I used Origin Pro to fit an exponential curve to the peaks of the voltage plots. The formula used for this function fit was as follows:<\/p>\n
<\/p>\n
Where R0 is the decay coefficient, -\u03b2<\/p>\n
Figure 8 below is an example of this function fit:<\/p>\n
Figure 8<\/p><\/div>\n
I fit an exponential function to the 996 mH and 1007 nF, 47 nF oscillations as well, resulting in the table below (Figure 9):<\/p>\n
<\/a>Figure 9<\/p><\/div>\n
Theoretically the \u03b2 value should be the same for all of them.<\/p>\n
In order to explore how the decay coefficient\u00a0\u03b2 changes I needed to change the inductor values. Because the voltage curves were more manageable at greater LC values, I used the largest capacitor I had and varied the inductor value. \u00a0The results were as follows (Figures 10-11):<\/p>\n