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Homework Help: Combination of uncertainties/errors when dealing with equations

  1. Jan 3, 2008 #1
    [SOLVED] Combination of uncertainties/errors when dealing with equations

    Greetings; I wasn’t sure exactly which section to place this so please move it if necessary.

    My problem is as follows:

    I have worked out an impedance for a coaxial cable, using the following formula: Z = (L/C)^1/2. Here, L is the inductance and C is the capacity per unit length. L is given by L = (μ/2π)ln(b/a). C is given by C = 2πε/[ln(b/a)]. Both b and a are distances. All other values are constants with their usual meaning.

    I have no trouble in working through these equations and combining them to find a Z value. What I’m struggling with is the ‘associated errors’ of these things. As this result is required for a lab report it is important that I do quote the overall uncertainty of my final result.

    I understand that both a and b are measured values and this means I do have an error associated with them (given by the resolution of the measuring instrument). I also understand how to combine the errors of b and a when the 2 values ‘interact’ e.g. the error combination for (b/a) can be found using the standard error formula (ΔZ/Z)^2 = (ΔA/A)^2 + (ΔB/B)^2 (I can also do it for when the natural log of the combined values is taken). However, what I find difficult is the consideration of the constant values in the equation. If I just ignore these constants and take the errors for the combinations as explained above, I find the final error outcome is many times greater in magnitude than the final answer, so I am positive I do need to reconsider this. How do I consider the constants and their relationship to the errors, as I can’t quote an error for something like π yet it does effect the equation?

    Any help would be much appreciated. All I am interested in is how to consider the constants in the equation with regards to errors (as I’ve explained I already know about standard error combinations when both values have errors). Obviously it is down to me to go through the actual numbers once I know what to do.

    Many thanks for any help,

  2. jcsd
  3. Jan 3, 2008 #2

    Dr Transport

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    Science Advisor
    Gold Member

    you need to differentiate the equation with respect to both a and b and comb ine them using propagation of uncertainties.

    Here is a link: http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html [Broken]
    Last edited by a moderator: May 3, 2017
  4. Jan 4, 2008 #3
    Ok well I've ran it through and found my impedance of 83Ω has an error of +- 0.168. Now that sounds a lot more reasonable to me, and I have to do another 4 impedances so I should be able to confirm they are realistic values.

    Many thanks for the help Dr Transport, and for a very useful link <adds to bookmarks>.
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