DAY 18: 2nd Order Circuits/ RLC Circuit Response Lab

Part 1: Step Response of a Series RLC circuit 

Today, we expanded the RLC circuit in the previous class by applying the power supply into the RLC circuit. This expansion will make the homogeneous differential equation become the non-homogeneous differential equation, with the solution x= xn + xf, xn is natural solutions with natural response when the circuit only have R,L,C; xf is the particular solution. 

We began with the series circuit and deduce the DE for the voltage across the capacitor. The DE in terms of the vc because we need to express Vs in the DE. vf(infinite) = Vs

Then we did a practice problem for the RLC circuit in series with Vs. First, we need to initial values of vc(0), dvc/dt (0), and vf(infinite) to solve for the constant coefficient. We calculate the neper frequency a = R/(2L) , and resonant frequency w=1/sqrt(LC). So, a<w => underdamped. Then, we deduced the vn, and A1, A2. 

Then we studied about the RLC circuit in parallel, We analyzed the circuit, write out the equation for KCL, try to remain the Is in the equation in order to deduce the non-homogeneous DE. Thus, iL = iLn + iLf. neper frequency a=1/(2RC), resonant frequency w =1/sqrt(LC). Comparing the a and w, we concluded the circuit was underdamped. Based on the boundary values, we got the coefficients of the solution.

Part 2: RLC Circuit Lab

Purpose: We will verify the theory of the RLC circuit with the voltage supple through this experiment. Particularly, we will verify the property of the RLC in parallel. 

iL= in+if, t>=0.

Pre-lab:

The schematic circuit is the RLC circuit in parallel with the voltage supply. We will transform the circuit from the voltage supply in series with the resistor into the current supply in parallel with the resistor. The we will observe the standard circuit for the RLC in parallel. The inductor current equals iL= in + if, which iL is the solution of the 2nd order non-homogeneous DE. 

We deduced the iL for this circuit. We found that the neper frequency is less than the resonant frequency, so the circuit is under the underdamped effect. 

We found the coefficients for the general solution of iL, then Vo = R2*iL. 

The Everycircuit for the current across the inductor, capacitor, and R2. We observed iL=iR2, and iL, iC, iR2 expressed the underdamped effect. 

The current through the Vin and current through R2.

The circuit for this experiment. RLC in parallel. 

The oscilloscope for the Vin, Vout. From our analysis, Vout=R2*iL, so it will expressed the underdamped effect, and the oscilloscope showed the same phenomena. 

The period is 0.67ms. wd = w*pi/T = 9424 rad/s . wd (calculation) = 9943 rad/s. percent error = 5.5%

Part 3: Second order of op amp circuit

With the op amp and the capacitor in a circuit, when we analyzed the circuit by applying KCL, KVL, ideal op amp assumption for simplicity, we will get the second order differential equation. One example of this circuit is described in the lecture note. 

v _o (t) = v _on + v _of = 10 + e^{− t}(A cos 2t + B sin 2t) mV

v o (t) = 10 − e^{− t}(10 cos 2t + 5 sin 2t) mV

Conclusion:

We discussed about the RLC with the power supply both in series and parallel. By applying KCL, KVL, we try to get the non-homogeneous 2nd order DE, and solve for the general solution y= yn + yp. We also did the experiment to very find the theory with a parallel RLC circuit and observed that the oscilloscope graph behaved closely to the theory. Error occurred because the inductor and capacitor is not ideal and have inner resistance. We also discussed a second order op amp circuit which combine the capacitor and the op amp in a circuit.