In this post, I will draw conclusions from my previous final data post about both RLC circuits that I have modeled.
RLC Circuit- No Voltage Source
This RLC circuit [Figure 1] proved to be an interesting demonstration of the current in a circuit without a voltage source. The initial current running through the circuit is provided by the charged capacitor. However, this initial current undergoes damping due to the resistor in place, and the current running through the circuit pretty approaches zero pretty quickly.
My model of this circuit verifies this idea since it shows an exponential decay of the current as a function of time [Figure 2]. After about 4 seconds, there is no longer an active current running through the circuit due to the resistor. My Mathematica notebook is easily set up for changing the values of the different components, and one can easily change them to see what effect this has on the circuit.
RLC Circuit- AC Voltage Source
The second RLC Circuit that I modeled was identical to the one above, except that it had an alternating current voltage source as well [Figure 3]. This allowed it to continue to have a current present despite the effects of the resistor.
Initially, I approximated the solution to the differential equation governing the circuit by ignoring the damping terms. This resulted in a an extremely good approximation, as the damping terms should only effect the circuit’s current flow in its first few seconds. After some computational hiccups that inaccurately displayed long-term variations in the current, I went on to graphically solve the complete form of the equation and graphed it to prove that the two damping terms only had a small effect in the first few seconds of the circuits behavior[Figure 4].
Given the chance to work more on this project, I would develop manipulable animations within Mathematica that would allow one to change a given variable (ex. resistance, inductance, capacitance). This would allow the reader to easily see the effects that the different components have on the circuit. This would also be helpful from an educational standpoint as an applet to someone wanting to learn more about RLC circuits, or circuitry in general.
On a final note, I managed to learn a lot more about Mathematica and its differential equation solving capabilities. Solving the equations for these circuits by hand would have been an extremely length procedure, but once I provided Mathematica with the commands in the correct syntax, it solved them much faster than any human could. This project helped me appreciate the uses of computational tools in physics, and I am very glad that they exist.