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Error Arithmetics

by Johann
(Malta)











































1) The capacitive reactance of a capacitor is found by the equation , where and are uncertain quantities having errors and respectively (both of which are greater than zero). Show that the error of is given by the formula
(If you get the same formula as above but with a negative sign, it is also correct.)
Hence show that the percentage error of is always larger than both the percentage error of and that of .

u{X_C }=X_C ¡Ì(((u{¡¼f}¡½^2)/f))+ ((u{¡¼C}¡½^2)/C)

Comments for Error Arithmetics

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Apr 09, 2011
Error Arithmetics
by: Staff


The question:

by Johann
(Malta)

1) The capacitive reactance of a capacitor is found by the equation , where and are uncertain quantities having errors and respectively (both of which are greater than zero). Show that the error of is given by the formula

(If you get the same formula as above but with a negative sign, it is also correct.)

Hence show that the percentage error of is always larger than both the percentage error of and that of .


u{X_C }=X_C ¡Ì(((u{¡¼f}¡½^2)/f))+ ((u{¡¼C}¡½^2)/C)


The answer:

The notation in your equation did not display properly.

I’m not sure what you are asking.

The formula for capacitive reactance is:

Xc = 1/(2πfC)

The impedance is:

Z = √[R² + (Xc)²]

Are you measuring capacitive reactance in a lab, and calculating the error?


Let us know.




Thanks for writing.


Staff
www.solving-math-problems.com


Apr 09, 2011
Error arithemtic
by: Anonymous

Hi


No, i need to get this formula when calulating the error that gets from the calculation of Xc

Apr 10, 2011
Error Arithmetics
by: Staff


You’re not giving us much information to go on.

If you are taking various measurements and calculating the value of Xc in a lab, the actual value of Xc will fall within a range which can be calculated statistically:

Actual Value of Xc = measured/calculated value of Xc ± margin of error


Actual Value of Xc = 1/(2πfC) ± (Z value)*(Standard Deviation)/(square root of number of measurements/calculations)

Z value = confidence level

90% confidence level. Z = 1.645
95% confidence level. Z = 1.96
98% confidence level. Z = 2.33
99% confidence level. Z = 2.58

Standard Deviation = sqrt[∑(X-M)²/(n-1)]

X (this is not Xc) = individual measurement/calculation
M = mean of all measurements/calculations
n = number of measurements/calculations


Actual Value of Xc = 1/(2πfC) ± (Z value)*(Standard Deviation)/( square root of number of measurements/calculations)

Actual Value of Xc = 1/(2πfC) ± (Z)*( sqrt[∑(X-M)²/(n-1)])/sqrt(n)



Thanks for writing.


Staff
www.solving-math-problems.com


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