Day 19 2nd Order Circuits Contb

We started with talking about the Qi wireless charger which had two modes of charging, Resonant and Inductive. There is a trade off between each mode, loosely coupled system trade off larger distance at the cost of lower power transfer efficiency and higher EMF. Tightly coupled system produce less heat which is favorable for heat budgeted devices like cell phones.

We then move onto the real gritty stuff of continued the work on Series and Parallel RLC. We started with the step respond of a series RLC circuit. It is a second order differential, with i = Cdv/dt. Vt = Vn + Vf. In natural response Vs = 0. Vn can be OD,CD,UD, Vf = Vinf = Vs.
 OD = A1e^s1t+A2e^s2t
 CD = (A1+A2t)e^-alpha*t
 UD = (A1coswdt + A2sinwdt)e^-alpha*t
The complete version then is:
 OD = Vs + A1e^s1t+A2e^s2t
 CD = Vs + (A1+A2t)e^-alpha*t
 UD = Vs + (A1coswdt + A2sinwdt)e^-alpha*t

A1 and A2 are from V(0) and dV(0)/dt. V is Cap, I is current of inductor.

Knowing this we try an example.

White board work: Using above information and numbers given.

More white board work continued.

White board work: more white board work we found di/dt in terms of A2 = -10. S = j8

More white board work continued.

White board work continued, found A1 = 20V used wd since it was a O

We then move onto the step response of a parallel RLC circuit. We use current for parallel.
I(t) = In + If. . In can be OD,CD,UD, If = Is.
 OD =Is +  A1e^s1t+A2e^s2t
 CD = Is + (A1+A2t)e^-alpha*t
 UD = Is + (A1coswdt + A2sinwdt)e^-alpha*t
We tried an example of our own with the knowledge attain above.

White board work: S1 = -1+j S2 = -1-j, UD. If = Is  = 5A

More white board work continued.

White board worked continued: A2 = -5

We then move onto the lab of the day: RLC circuit response.
Prelab: We first did the lab in every-circuit. Then we wrote the second order diff on the white board. Then estimated damp ratio. And lastly rise time and frequency on white board.

Picture of the everycircuit schematic and working, min -20 mV max 62.3 mV and freq 1khz

White board work of prelab it has the second order diff, (kind of cut off but is there).

White board work of the second order.

More white board prelab work.

White board work: Has alot of information has Vo at 42mV. A1 = 42. alpha = 500, w0 = 10,000, UD. Damping ratio = .05.

Picture of the time respond in the every-circuit app.

Picture of the time respond of every-circuit, 10 ms.

Picture of real circuit.

Picture of the real circuit, none of that fake stuff.

Picture of the real stuff data.

Picture of the step respond.

More picture data.

Picture of data of step respond freq 200 hz, Vin = 40 mV Vout = 2 volts. Matches every-circuit and our white board work very closely with Vin 42 mV. As percent diff  it is = 4.7 % error.

More pictures of real, cleaner version!

Picture of the real but prettier thanks to Edgar. We will not name the original maker.

White board work for POST lab:
We found wd to be 9987, r1 real to be 47.4 ohms. A1 - 42, A2 = 6.3

White board work showing the used second order diff and the steps needed to find A1 and A2.

More white board work, found di/dt and t of rise time = 10us. Which matched was close to every-circuit but was off by a factor of 10 to get percent error of = 1000%. We think we lost a exponent somewhere in our math but think we got it correctly.

In conclusion of the lab: We found our numbers to match every-circuit and the data from diligent. We are off by a factor 100 since picture data says 1us and we got 10us and every-circuit says 10ms. We are close but far away. Our percent error for A1 and A2 are good with Vin being 42 mV and recorded at 40 mV our percent error is 4.7%. The damping ratio is .05 and we got .5 again out numbers are off by a factor so somewhere in the math we lost an exponent but our percent error would be 0 but in this case it is over 1000%. We also think we may be off because real has non-ideal components while every-circuit had ideal components.

The text file for this contiues on with talking general second order circuts like op amps and smoothing digital signals but I dont have pictures for that.

In summary of the day we learned more about step response circuit and did a lab on it and briefly talked about second order op amps. We learned that series use V for step response and parallel use I for step response but the equations of OD,CD, UD are the same for both just need to swamp out V for I. Vf = Vinf, If = Iinf.