5/3/16 - Solving RLC Circuits

Today, we learn about solving RLC circuits, which are second order differential equations.

Here we derive the formula for the output voltage and we can see that it forms a second order differential equation with all three RLC values.

Here's a problem where we are asked to find the voltage across the capacitor. We know that the instantaneous voltage is going to be the same as the resistor and the current will be the voltage over the total resistance.

We find the voltage drop and current drop per second due to the inductor and capactior.

And then finally, we know that at t = ∞, there will be no current across the capacitor and the voltage will be the same as the source.

Here we are to find the characteristic roots of the function. Since alpha is greater than omega_0, we can say that the circuit is overdamped. We then use the equations to find the first and second constants for the solution. 

Lastly, we are to solve for when all three elements are in parallel.

Alpha and omega_0 can be solve for and then used to solve for the constants.

Lastly, the lab of the day was to hook up all three elements in parallel. And since the nepper frequency is less than the resonant frequency, the circuit is underdampped. We can observe the voltage become a sinusoidal function with a exponentially declining amplitude.