Solution to a second order LCR differential equation

[q5101997]

I have built an IR proximity sensor for a mouse trap and have come up with a series LCR circuit with a 2nd order differential equation of the form:
LCv'' + RCv' + v = Vs

where: Vs is a stepped voltage at t=0
and: v is the voltage accross the capacitor
and: v'' and v' are the first and second derivitives respectivly

A solution is v = Ae^(αt)cos(βt) + Be^(αt)sin(βt) + Vs

where: α = -R/2L rad/sec
and: β = √(1/LC - R²/4L²) rad/sec

The wave form is a phase shifted sine wave with exponential ring down because the determinant is complex.

What are A and B?
Using initial conditions for both v(t=0) and i(t=0) = Cv'(t=0) I have come up with:

A = Vi - Vs and
B = Ii/Cβ - (Vi - Vs)α/β

where Ii is the current through the circuit at t=0 (initial current) and Vi is the voltage accross the capacitor at t=0 (initial voltage).

I've noticed that the oscillating frequency, β, induces an impedence accross C of 1/Cβ so that the initial current gives an initial voltage of Vi = Ii/βC = Ii.Xc

As of yet, I have not found a concise sollution on the web.

Does anybody have the correct sollution to an underdamped series LCR circuit?

 

[MrAl]

Hi,


Not to nitpick or anything, but since when do we say -R/2L is in units of rads/sec?

Here is an expression for the voltage across the capacitor at time t for the underdamped
case with initial conditions all zero and a step input:

Vc(t)=1-e^(-a*t)*(a/W*sin(W*t)+cos(W*t))

with
a=R/(2*L)
and
W=sqrt(1/(L*C)-a^2)

 

[q5101997]

-R/2L is a frequency just as 2L/R is a period.

-R/2L is a frequency just as 2L/R is a period.

Regs Q

 

[MrAl]

-R/2L is a frequency just as 2L/R is a period.

Regs Q

Hi,

Oh, i am not saying that it can not be called a frequency, just that its
units are not 'measured' in rads/sec, that's all.
Omega is in rads/sec, but Alpha is not, that's all.

 

[q5101997]

I have worked out the units of α (the exp constant) and β(the sin constant) and have come up with units of hertz?
α = -R/2L
v(L) = L.di(L)/dt
L = v(L).dt/di(L) units are V.s/A
R = V/I units are V/A
units for α are therefore (V/A)/(V.s/A) = (1/s) = Hz
where s is time in seconds
i(C) = C.dv(C)/dt
C = i(C).dt/dv(C) units are A.s/V
β = √(1/L.C - R²/4L²)
units for β are therefore √(1/[(V.s/A)(A.s/V)] - (V/A)²/(V.s/A)²) = √(1/s² - 1/s²) = 1/s = Hz
Well, α and β are both in the same units of hertz. I was right and wrong but now I'me confused. Are they rad/sec or cycle/sec?

 

[MrAl]

Hi again,


Nepers per second.

Please dont get too worried over this, but the correct units are as follows:

For say:

e^(5t)

the dimensions of 5t are "Nepers" and the 5 alone is the neper frequency
in nepers per second.

The neper was named after Napier (probably stuff on the web about him).

 

[q5101997]

OK, I wont lose sleep, but back to units. I get a buzz out of digging into fundamentals, I dont like taking things at face value much. Regarding the ringing frquency, β, for now, I whipped up a series cct using a 68mh inductor having an inherent 190Ω series resistance, and a 10nF cap. I switched it from an oscillator using a low resistance MOSFET as the swithing element. I got a nice sine wave with little decay and measured the period as T = 170uS, giving a frequency of 1/T = f = 5,880 Hz. The radian frequency is therefore w = 2.pi.f = 36,900 rad/sec. Now β = √(1/LC - R²/4L) = 38,300 (units of something) ≈ w so β must be in units of rad/sec (accounting for errors measuring T and component tolerances). OK, the proof there is in the pudding (as well as being stated in most texts), but this leads me to ask further. Why does pi seem to apear out of knowhere, for no particular reason? There must be a deeper physical (or mathematical) reason for this, which I can in no way elude to. I will get back to the decay period issue at a later date.

PS A pi that looks like one with wiggely legs, not π, seems to have been monumentally dropped from our list of symbols.