In this project, I plan to study the relevant differential equations that govern RLC circuits and use Mathematica to solve them for values that are useful. The general equation governing a basic RLC circuit with a capacitor, voltage, resistor, and inductor in series, in that order is:
[Equation 1] (UBC- Source 4)
which, when going through a series of substitutions, becomes:
[Equation 2] (UBC- Source 4)
Equation 1 has six variables: L (inductance), R (resistance), C (capacitance), V (voltage), Q (charge), and I (current). When a circuit like this is set up in the lab, the values that are known are L, R, C, and V because they directly depend on the components of the circuit. Once the differential equation  is solved, values for current (I) and charge (Q) can be determined. I will use Mathematica to solve for the general solution to this differential equation , which is a second-order differential equation. Once I have the general solution, I will vary the initial conditions to determine the effect of different circuit components on the overall properties of the circuit. Following this, I will develop more series RLC circuits with components that are set up differently in terms of their component structure.
Timeline (Weeks 1-5):
Week 1: I will begin by reviewing basic differential equation solving techniques for first and second order differential equations. I will also study the differential equation solving capabilities of Mathematica and review the techniques for solving second-order differential equations as they apply to RLC circuits.
Week 2: I plan on solving for the general solution to Equation 2 above (using Mathematica). I will vary different initial conditions and create graphs that visualize the changes that occur. I will also have a visual representation of the circuit.
Week 3: Develop another series circuit or study a previously built one. Determine the general equation for it and begin the solution process.
Week 4: Finalize the solution for the second circuit. Develop graphs for visualization purposes.
Week 5: I will finalize my project by proofreading all the components and making sure everything is presentable. I will also provide constructive criticism to my peers on their projects.
1. Mathematica Cookbook by Sal Mangano
2. Electronic Circuit Analysis for Scientists by James A. McCray and Thomas A. Cahill
3. Dynamical Systems with Applications using Mathematica by Stephen Lynch
4. The RLC Circuit- University of British Columbia- http://www.math.ubc.ca/~feldman/m121/RLC.pdf
5. Class Notes- Mathematics 228 (Methods of Applied Mathematics) taught by Matthew Miller
I am looking forward to the continued development of your project. Remember to explain all abbreviations, variables and substitutions. It is fine to skip some calculations algebra as long as you explain what is being substituted. A circuit diagram would really help. Also, remember that your work must be original. How is your work going beyond the existing solution, which cited?