Why Is My Calculated Error for Resonant Frequency So High?

[ma18]

Homework Statement

So I am calculating the error for something and I am getting really weird values.
So I know that the value for the Inductor is 24.97 +- 0.005 mH and that for the capacitor is 105.7+-0.0005 nf.

So I am finding the value for the resonant frequency

Homework Equations

f_0 = 1/(2*pi*sqrt(LC))

The Attempt at a Solution

So for the f_0 I get 3097 Hz which is very close to my experimental observations. But for the error I get:

error in LC = (2.64e-9) * sqrt((0.005/24.97)^2+(0.0005/105.7)^2) = 5.29e-13

error in LC^-.5: (19464.95)*0.5*2.64e-8/5.29e-13 = 37994

final error: 37994 * 1/(2*pi) = 6039.

Now this final value is too high, what am I doing wrong? Thanks

 

[haruspex]

error in LC = (1.03e-8) * sqrt((0.005/24.97)^2+(0.0005/105.7)^2) = 5.29e-13

I don't see where the 1.03e-8 comes from. LC is about 2.6e-6, no?

error in LC^-.5: (19464.95)*0.5*1.03e-8/5.29e-13 = 37994

What's the reason for dividing by 5.29e-13? It would be clearer if you were to write the equation in purely symbolic form, not plugging in numbers.

 

[ma18]

Sure, sorry. The 1.03e-8 is the LC value and the following values in the equation are the error divided by the value. For the second equation 19464.95 is the LC^-.5 is the value and 0.5 is the exponent and 5.29e-13 is the error in LC^-0.5.

The eqn's are:

for multiplication (z=xy): dz = z *sqrt((dx/x)^2+(dy/y)^2)

for exponents (z=x^y): dz = abs(y)*z*dx/x

 

[haruspex]

Sure, sorry. The 1.03e-8 is the LC value

Isn't LC = 24.97 * 105.7e-9?

for exponents (z=x^y): dz = abs(y)*z*dx/x

Sure, but when you plugged in the numbers you seem to have used x/dx instead of dx/x.

 

[ma18]

L is in mH so there is an extra 10^-3 factor I forgot note here. I wrote down the exponent thing wrong here but the number is right, it should be:

error in LC^-.5:
value*exponent*error_LC/LC
= (19464.95)*0.5*2.64e-8/5.29e-13 = 37994

ah there are so many numbers in my sheet I get mixed up

 

[ma18]

Alright I think I've got it, my data was just too ugly, I cleaned it up and did it another sheet.

I got:

Error in LC: 5.29E-13
LC: 2.64E-09
LC^-0.5: 1.95E+04
Error in LC^-.5: 1.95E+00
Error in final : 3.10E-01

which ironically seems a little small but whatever.

Thanks for the help!

 

[haruspex]

I find it easier to think in terms of fractional errors. (Mistakes in the calculation usually become more obvious.)
The fractional error in LC is 5.29E-13/2.64E-09 ~ 2E-4 (almost entirely owing to the error in L).
The fractional error in sqrt(LC) will be half that: 1E-4.
The fractional error in f_0 will also be 1E-4, giving an absolute error of ~ 3E3 * 1E-4 = 3E-1.
That confirms your answer.