DAY 17: Natural Response RLC Series and Parallel Circuits/ Series RLC Circuit Step Response Lab

Part 1: Natural Response RLC Serires and Parallel

Today, we discuss about the RLC circuit series and parallel. We expect to deal with the linear second order differential equation of RLC circuit. Then, we do the Series RLC circuit step response lab to verify three situations in the RLC circuit when a>w, a=w, a<w, which a is neper frequency or damping factor, w is resonant frequency. 

We practice to find the boundary values, initial values and final values. They are very important to get the general solution of the differential equation in RLC circuit. There are two key factors mentioned in the lecture note to find the boundary values.

 First—as always in circuit analysis—we must carefully handle the polarity of voltage v(t) across the capacitor and the direction of the current i(t) through the inductor.  Keep in mind that v and i are defined strictly according to the passive sign convention. One should carefully observe how these are defined and apply them accordingly.

 Second, keep in mind that the capacitor voltage is always continuous so that

v(0+) = v(0−)

                  and the inductor current is always continuous so that

i(0+) = i(0−)

where t = 0− denotes the time just before a switching event and t = 0+ is the time just after the switching event, assuming that the switching event takes place at t = 0. Thus, in finding initial conditions, we first focus on those variables that cannot change abruptly, capacitor voltage and inductor current, by applying the equations above.

Then, we analyzed the RLC series, and deal with the linear second order differential equation (DE). For the series RLC, the neper frequency is a = R/(2L), the resonant frequency is w = 1/sqrt(LC). We yields the quadratic equation and get the general solution for s.  

There are three cases for the DE. Respectively, we have three situation for the RLC cirtuit, overdamped (a>w, 2 real roots), critical damped (a=w, 1 real roots), underdamped (complex roots). 

a>w: i= A1*e^[(-s1)*t] + A2*e^[(-s2)*t]

a=w : i = (D1*t+D2)*e^[(-a)*t]

a<w, wd = sqrt(w^2-a^2): damping frequency, i = B1*e^(-at)*cos(wd*t)+B2*e^(-at)*sin(wd*t)

Part 2:  Series RLC Circuit Step Response Lab

Purpose: This lab is aim to verify three situations of the RLC circuit in series, overdamped, underdamped, critical damped. We use the waveform generator to create the step response voltage, and use the oscilloscope to gain the behave of the voltage input and voltage output which is across the capacitor. 

Pre-lab:

The first experiment, we use the R=0.9Ohm, the capacitor = 470uF, the inductor = 1uH, so we calculated the time constant =1.08s, a= 450*10^3 rad/s, w= 461266 rad/s, a<w => underdamped, wd=101322 rad/s

The waveform generator is square, f=1kHz, A=2V, offset =2V.

The set-up for this RLC circuit in series.

The underdamped RLC circuit. Vc = -VL - Vr = Ldi/dt - Ri = A1*e^[(-s1)*t] + A2*e^[(-s2)*t].

wd=2*pi*f = 2*pi/T, T=0.045 => wd = 139626 rad/s, percent error = (139626 - 101322)/139626=27.4%

We get the overdamped circuit by making a > w. We use R=22Ohm to gain this effect. As our calculation, a=11*10^6, w=461260

The graph expressed the overdamped situation in the RLC series

Next, we discuss about the RLC circuit in parallel. In parallel circuit, a= 1/(2RC), w=1/sqrt(LC).

We apply the initial values to solve for the constant in the general solution. V(0) is initial voltage across capacitor, dv/dt (0) = 1/C*ic(0)=v'. Find ic by KCL, ic=iR+iL, and solve for A1, A2. 

Conclusion: 

We discussed RLC in both series and paralellel with no power supplly. After analyzing we came up with the 2nd order differential equation with 3 possible situations: overdamped, underdamped, and critical damped. We also did the lab to verify these situations. The result shows the reality meets with our calculation. We observe the underdamped effect when the neper frequency is less than the resonant frequency, and the overdamped effect when the neper frequency is greater than the resonant frequency.