{"id":3608,"date":"2014-04-30T03:32:07","date_gmt":"2014-04-30T07:32:07","guid":{"rendered":"http:\/\/pages.vassar.edu\/magnes\/?p=3608"},"modified":"2014-04-30T05:00:17","modified_gmt":"2014-04-30T09:00:17","slug":"final-data-rlc-circuits","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2014\/04\/30\/final-data-rlc-circuits\/","title":{"rendered":"Final Data- RLC Circuits"},"content":{"rendered":"

Overview<\/strong><\/p>\n

After talking with Professor Magnes after my preliminary results, I decided to first look at an RLC circuit with no voltage source, which is much simpler to model. Doing this allowed me to better understand the mechanism behind the circuitry and gain more experience with Mathematica as well.<\/p>\n

After doing so, I returned to my progress with an RLC circuit with a voltage source and determined that the odd structural variations with the long-term structure were due to computational limitations of Mathematica (again, thanks to Professor Magnes for the suggestion). Since I had only developed the approximate solution, I went on to graphically solve for the full solution and see what the damping terms did to the solution.<\/p>\n

Note: I included the findings from my preliminary data in this final data post (so it may seem redundant for a portion in the middle) since it was used in my final results as well. I figured it helps to have everything in one place for the reader.<\/p>\n

RLC Circuit with no voltage source<\/strong><\/span><\/p>\n

Circuit Diagram and Mathematics<\/strong><\/p>\n

The first circuit we will examine is an RLC circuit with no voltage source. The initial potential difference is stored on the capacitor, but can not be recharged due to the lack of a voltage source. The three main components of this circuit are a resistor (R), an inductor (L), and a capacitor (C).<\/p>\n

\"1\"<\/a>[Figure 1]<\/p>\n

The differential equation that governs this circuit is<\/p>\n

\"LI''(t)+RI'(t)+\frac{1}{C}I(t)= 0\" [1]<\/p>\n

where L is the inductance, R is the resistance, C is the capacitance, I is the current through the circuit (which is the same at all places at one time since this is a series circuit), and t is the time.<\/p>\n

Solving the Differential Equation<\/strong><\/p>\n

We can set arbitrary values for R, L, and C since they depend on the specific pieces of equipment themselves:<\/p>\n