Series RLC Circuit Response

Today we started second order circuits. Second order circuits are RLC circuits or a simplified version of them, LC circuits. The first step is finding boundary values. i(0+), v(0+), di(0+)/dt, dv(0+)/dt, i(∞), v(∞) are examples of boundary values. We can then use a second order differential equation to solve for a second order circuit. The solutions to the characteristic equation, s1 and s2, are called natural frequencies, in nepers per second (Np/s). s1 = -α + sqrt(α^2 - ω0^2), s2 = -α - sqrt(α^2 - ω0^2). α is called the never frequency or damping factor. ω0 is the resonant frequency or the undamped natural frequency.

There are three types of solutions

1. If α > ω 0 , we have the overdamped case.
2. If α = ω 0 , we have the critically damped case.
3. If α < ω 0 , we have the underdamped case.

For overdamped case, 

i(t)=A1 es1t +A2 es2t 

For critically damped case, 

i(t)=(A2 +A1 t)e−αt 

For underdamped case,
i(t)=e−αt(B1 cosωd t+B2 sinωd t)
ωd is called the damped natural frequency, ωd = sqrt(ω0^2 - α^2)

Then we did a lab on an RLC circuit in series. Here is a schematic of the circuit

Here is the square wave we applied to the circuit

 We were able to observe a natural response as following

We compared the oscillation frequency from the graph to the theoretical value.

	Experimental	Theoretical	Percent Error
Oscillation Frequency (Hz)	101341.6985	99998.74999	1.34%

 

Summary: RLC circuits can be represented by using second order differential equation. Similar to damping oscillation in mechanics, RLC also have three different cases of oscillation. Our lab demonstrates the pattern of an underdamped case where the damping factor is smaller than undamped natural frequency.