Day 18 2nd Order Circuitsb

We started the day with talking about boundary values. A couple of simple things to know is that v(0+) = v(0-) and i(0+) = i(0-). So it is important to start with how the circuit behaves initially and how it behaves in the end. Inductors act like shorts, Capacitors act like open in DC setting after a long time. In order to understand this we did an example. White Board work below.

White board work: We took out the .25H inductor, and .1F capacitors, and drew the equivalent circuit of t approach infinity. V = IR, V = 4.

We then did the KVL.

White board work: Using KVL. di/dt = 0.

We then did some algebra with the properties of inductors. We end up with a second order.

White board work: algebra work to end up with i = Ae^st. and we try to find S.

More white board work continued.

White board work: Continued to find S.

More White board work continued.

White board work: Found S used square root to find s in terms on alpha and omeganot.

More white board work continued. Depending on alpha or omega bigger the circuit damps differently. OverDamp = alpha>omega C > 4L/R^2, Critically Damp alpha = omega, UnderDamp alpha<omega, used omegad = sqrt(omega^2-alpha^2), time constan is 1/alpha and T = 2pi/omegad

We then moved onto source free RLC circuit. Alpha = R/2L, Omega = 1/sqrt(LC)

White board work: circuit drawn with numbers.

More white board continued.

White board work: S = -alpha +- sqrt(alpha^2-omega^2). Alpha >omega this OD.

We then move on the lab of the day: Series RLC Circuit Step Response
Prelab: Part1 White board work and circuit and equations.

White board work: drew circuit, and second order equation

White board work: Found alpha and omega, alpha = 500, omgea = 4.61e3, alpha<omega UD. Damping ratio .108.

White board continued for part 2.

White boar work: finding R, R = 9.22 ohms.

White board work: Inductor resistance is 2.4 ohms, R = 1 (4.5)real, C = .47uf, L = 1mH

Picture of Part 1 circuit.

Picture of the Vin and Vout of part 1 circuit.

Picture of data of Vin and Vout. Vin = 15.9 mV, and Vout = 16.1 mV, rise times of 2.6us. and 2.2us, Freq = 124khz.

Picture of part 1 of the lab. Delta X = 120us.

Picture of data for part2 lab. Vout = 5.3 mV, Vin = 1V. Risetime 1.9us and 0s.

Picture of part 2 of the lab, the new R is 9.22 ohms, or closes to it. to make it CD.

Picture of part 2 of the lab. We noticed that Vin and Vout are 180 out of phase.

Picutre of the delta X = 40 us.

Picture of more data for part 2, clearer.

Picture of data for part 2 lab, Vin 7.8 mV, Vout = 18.8 mV, ristime = 15 ns. Freq = 4.3 Mhz

In conclusion of the lab:
The lab showed the oscillation between the inductor and capacitor. It made sense of the OD portion that the Vout was dying out. For part 2 Vin and Vout were 180 out of phase. There were some difficulty find the right R for the second part of the lab sense the inductor had resistance and the 1 ohm resistor has great uncertainty which made it 4.5 ohms. The gain was 1 for part 2 as shown in the pictures. Finally there was unexpected overshoot and the DC gain since we did not have ideal parts.

We then move onto to talk about source free parallel RLC circuits.
They also have a second order equation. Alpha and omega are different. We tired an example on the white board.

White board work: S1 and S2 the same as before however alpha = 1/2RC and omega = 1/Sqrt(LC)

White board work continued.

White board work: continued we found S in terms of A1 and A2, did some plug ins and found A1 and then found A2. A1 = -.208, A2 = 5.2

In summary we learned about boundary values and the rules associated with them. Did an example of it with making inductors short and capacitors open. Vinital = V after and same with current. Then found values of S with alpha and omega. There is three types OD,CD,UD, with alpha>omega, alpha = omega, alpha<omega. CD has omegad = sqrt(omega^2-alpha^2). There are two types of RLC series and parallel. We did a lab on Series RLC circuit. Found the CD resistance for part 2, 9.22 ohms. We concluded that the pictured taken made sense for the OD circuit and CD circuit, however there was unexpected overshoot with the non ideal parts used. Lastly we talked about spark plugs and how they useful.