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.

Day 17 1st Order Circuits contc

We started the day with talking about first order op amp circuits. Capacitors and inductors apply three useful properties in  electric circuits. Caps store energy, Caps oppose abrupt changes in voltage, Inductors oppose abrupt changes on current, this is useful for spark and arc suppression and for changing pulsing dc voltage into a smooth dc voltage. Lastly they both are frequency dependent, which in turns make them frequency discrimination for ac applications.

We were asked to show the algebra of an Integrator Op-amp. White board work below.

White Board work of an Integrator Op-amp. use Ir = Ic and Ic = -CdVo/dt. Gain is -1/RC and integration of Vin which is -Vin/Tao.

Next we were asked to another example with values and show the signal going in of the Op-amp.

White Board work: We take an input signal and take the derivative and negative 1.

White board work of the Differentiator. Not the same integrator, since output is -RC(derivative) and not -1/RC(integrate)

White board work: showing the output real. It is the gain(-1) times the derivative of input signal.

Afterwards we move onto the lab of the day: Inverting Differnetiator.
Prelab: Is done on the white board. C1 = .1uF, R = 1M ohm, RC = -1
Vout = -1/RC dVin/dt Vin = Acos(wt) w = 2f,  Vout = 1/(1.04)(w)sin(wt)

White Board work: Showing our circuit with values on the Op amp. Note R of 10M is 9.6M

White board work of voltages in at the different freq. 2khz,1khz and 500 hz.

White board work: The lab asked for 1V offset but we used 5mV,5mV and 2mV instead, don't remember why. The order is 2khz, 1khz, and 500 hz.

We then took the amplitude reading of the values. Channel 2 is Vin, Channel 1 is Vout.

White Board work: 500hz amp in 2.7mV, 2khz amp in 2.68mV, and 1khz amp in 2.69mV. Their respected Amplitude out are: 3.3V,2.56V and 3.48V. This is the tablet for post lab. Percent error is further down. There are difference between expected and measured, the differences believe are because of using the wrong formula and/or not the right input voltage/different RC.

Lastly our comments on the lab, Vout is Pi/2 out of phase of Vin and if frequency goes up Amplitude goes down.

White board work: Our conclusion of the lab, Vout is pi/2 out of phase and frequency goes up Amplitude goes down.

Here are the pictures required of the post lab: Pics of different frequency 2khz,1khz, and 500hz

Picture of the 1khz signal, yellow is output signal, cosine, blue is input signal cosine plus phase shift.

Picture of the data from the 1khz signal: Vout =3.2 V, Vin = 2.68 mV

Picture of 2khz signal, Yellow is Vout, Blue is Vin. Ch1, Ch2.

Picture of the 2khz signal, Yellow is cosine, blue is cosine +phaseshift. Vin, Vout.

Picture of the 2khz data: Vout = 1.6V, Vin= 2.68 mV

Picture of 500hz signal. blue is Vin, Yellow is Vin.

Picture of the 500 hz signal: Yellow is Vout, Blue is Vin.

Picture of Data for 500hz: Vout = 3.3V, Vin = 2.68 mV.

Here is a picture of the circuit used.

Picture of the circuit used: Op amp, .1uF and 10M (9.6 M real) resistor.

For the conclusion of the lab: POST LAB
5*cos(2*pi*f) w1 = 12566,w2 = 6283, w3 = 3141 1/RC = .96 Vin 2.68mV
-1/RC *Vin*w1*sin(w1) = 2.68 V, real is 2.56 %diff = 3.7%
-1/RC*Vin*w2*sin(w2) = 4.7 V, real is 3.48 %dif = 25%
-1/RC*Vin*w3*sin(w3) =4.5 V, real is 3.3 %dif = 26.6%

I do not feel confident is these percent error's,  I tried using different input voltages and the numbers calculated do not add up, I do believe we recorded the right numbers. I think I must be using the wrong formula :(

 We then moved onto switching functions! We started with the delta and what it represents. An area of 1.

White board work of delta, delta = 1!

We then tried an example of using switching formula.

White board work of the switching example. Ic = CdVct/dt. The derivative of u is delta the derivative of rt is u.

 Vc at 5 at0.

More white board work shown.

White board work: Ic, the impulse is .6 at .5.

Next we move onto the step response of RC circuit.White board work below.

White Board work of the step respond. V/R times e of -t/tao times the step respond.

We then tried another example by using what we learned and did on the white board.

White board work: Vs = 5u, Vintial = 0, Tao = 5. Tao is Req times C.

More white board work shown. The final formula needed. Vinf + [Vnot - Vinf]e to the -t/tao.

White board work with the equation in standard form.

White board work with numbers plugged in.

White board work: Cdv/dt - .5(-5/4)(-1/5)e^-t/Tao.

We then move onto the Step Respond of a RL circuit. i = in+if. In = Ae^-t/Tao, Tao =  L/R. We tried an example shown on the white board.

White Board work. We needed to find Inot, then Req for Tao. Then we used node voltage to find Iinf.

We came to the conclusion formula of i(t) = 5/6(1+e^-t)A

In summary we started with op amps. One being the integrator and the other being the Differentor. The integrator integrated Vin, and the differenetor differentiated Vin by the op amp. We then moved onto the lab of the which was the differentor. We found our numbers to not match the expected values with percent different errors up to 26% but we believe that is due to not following the instruction in the lab with the right input voltages are the right resistors. We do think we got resonalbe data. It seemed from the pictured that the Vout was 90degress out of phase from the input which makes sense and the higher the frequency the lower the amplitude was recorded. Lastingly for the day we talked and derived step response logic for both the RC and RL circuit. For RC the initial capacitor voltage is 0, final is Vinf and then find Tao. For RL the initial inductor current is 0, final current if Iinf and then find Tao. last thing we talked about was delay circuits and relays.

D13 - 1st Order Circuits

We started the class with what questions would be on the day of celebrations.

The Six question type to be on the day of celebrations.

We then move onto finishing our talk of inductors. Inductors in parallel L1*L2/(L1+L2), in series L1+L2. We did a example below.

White Board work of inductors in series and parallel, in series they just add, while in parallel they L1L2/(L1+l2).

We then further discussed the difference in capacitors and inductors.

White Board knowledge, inductor for i integrate, while i for capacitors take derivative.

We then did some first order derivations to show that inductors behave first order like.

White board work for finding Vo using first order.

More White Boar work for finding Tao of the capacitor. We took Tao = RC and found after 5 Tao there is nothing left in capacitor.

We continued to explore the current by using V = IR. Power = I^2R

White Board work using Tao of a capacitor, note that 2*tao and everything else is the same for power.

We then move onto  source free examples.

White board work for V. We used source free and then Tao for RC and got Vc of cap and Ic of cap.

More white board work and we figured the time constants of Tao range form nanoseconds to hundred seconds.

We then move onto the first lab of the day:
Passive RC circuit Natural Response.
Prelab:
R1 = 971 ohms, R2 = 2.16 k ohms, C = 22uF. After 5 tao's Vc should be near zero. Tao is 15.

Picture of the Passive Capacitor circuit.

Picture of capacitor discharging by hand feeding in and out, after 5 Tao's its close to zero. 1 Tao = 15ms.

Another picture showing Tao and voltage at those points, starts at 3.4 V, after 15ms (1 Tao) drops to 2.67 V, and after 5 Tao (75ms) V drops to 0.891 V

Another picture of the capacitor discharging with hand feeding and pulling out.

Picture of waveform feeding the capacitor. Note the drops look the same as the disconnect above.

Another picture showing the amplitude 1.75 V and maximum of 3.45 V.

Picture below with trigger of one of the humps shown above.

Close up picture of one of the humps, same as the disconnect discharge.

Picture of the hump data, same amp and max as before, since it the same trigger data.

Picture of more data, however note the Tao is the same of 15ms and after 5 Taos (75ms) V is near zero. Just like the previous example.

We then move onto a source free inductor example with inductors with a switch component added to it.

White Board work, inductors act like a short in DC settings. Tao = L/R.

We then move onto the second lab of the day:
Passive RL circuit Natural Respond
Prelab:
Same resistance values of before, r1 = 971 ohms, r2 = 2.16 k ohms however L = 1mH. Tao is 1.7us.

Picture of the inductor circuit.

Picture of the fast disconnect and reconnect on the inductor.

Picture of the data from the quick disconnect reconnect. Amp = 1.2V max = 2.4 V.

Second part we assigned square wave to the inductor.

Picture of square wave feeding into inductor. Amp at 2.5 V offset at 2.5 V we changed to frequency to making feeble to see with diligent.

Another picture of data from the squarewave max and amp the same.

White board work showing how we got Tao to equal 1.7us. We noted we need to change the frequency higher then the lab requested since our time constant was so small, we ended up using 1k hz.

In summary we talked about inductors and how they behave in series and parallel. And then did first order problems with capacitors and inductors. We then did two labs showing how reliable our Tao constants calculated are close to in actual circuits. The first being in a capacitors which match our time constant perfectly, 15 ms and after 5 Taos zero Volts which matched the 75ms. For the inductor lab we found tao to be 1.7us which very small so we had the option to change R or change frequency in the circuit to get reliable data out of the oscilloscopes, we chose the frequency change to 1k hz. We found our Tao matched the one observed in the oscilloscope, however the picture can't be found :(
Lastly we talked about source free problems on which capacitors are open, and inductors are shorts.