Results 1 to 4 of 4

Thread: Maximum Percentage error

  1. #1
    Newbie
    Joined
    Feb 2009
    Posts
    4

    Question Maximum Percentage error

    The resonant frequency of an oscillation in electrical circuits is given by the formula f = 1/2pLC. If the error in measuring L is 4%, and that measuring C is 2%, calculate the maximum percentage error in calculating f?

    Many thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    849
    Hello, Stazzer5!

    I think I've solved it.
    Someone please check my reasoning and my work.


    The resonant frequency of an oscillation in electrical circuits
    . . is given by the formula: .$\displaystyle f \:=\:\tfrac{\pi}{2}\sqrt{LC}$

    If the error in measuring $\displaystyle L$ is 4%, and that measuring $\displaystyle C$ is 2%,
    . . calculate the maximum percentage error in calculating $\displaystyle f$?
    We are given: . $\displaystyle \begin{Bmatrix}\dfrac{dL}{L}\:=\: 0.04 \\ \\[-3mm]\dfrac{dC}{C} \:=\: 0.02 \end{Bmatrix}$ . [1]


    We have: .$\displaystyle f \:=\:\tfrac{\pi}{2}(LC)^{\frac{1}{2}}$

    Take differentials: .$\displaystyle df \;=\;\tfrac{\pi}{2}\cdot\tfrac{1}{2}(LC)^{-\frac{1}{2}}(L\,dC + C\,dL) \;=\;\frac{\pi(L\,dC + C\,dL)}{4\sqrt{LC}} $

    Divide by $\displaystyle f\!:\;\;\frac{df}{f} \;=\;\frac{\frac{\pi(L\,dC + C\,dL)}{4\sqrt{LC}}}{\frac{\pi}{2}\sqrt{LC}} \;=\;\frac{L\,dC + C\,dL}{2LC} \;=\;\frac{L\,dC}{2LC} + \frac{C\,dL}{2LC}
    $

    . . Hence: .$\displaystyle \frac{df}{f} \;=\;\frac{1}{2}\left(\frac{dC}{C} + \frac{dL}{L}\right)$


    Substitute [1]: .$\displaystyle \frac{df}{f} \;=\;\tfrac{1}{2}(0.04 + 0.02) \;=\;0.03$


    Therefore, the maximum percentage error in calculating $\displaystyle f$ is $\displaystyle \boxed{3\%}$

    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello everyone

    I don't know enough about electrical circuits to say for sure, but, according to this Wikipedia article, it looks as though the formula should be:
    $\displaystyle f= \frac{1}{2\pi\sqrt{LC}}$
    It's easiest to deal with percentage errors with this type of equation if you take logs before differentiating:

    So, writing $\displaystyle f$ as:
    $\displaystyle f=(2\pi)^{-1}L^{-\tfrac12}C^{-\tfrac12}$
    we get:
    $\displaystyle \ln(f) = -\ln(2\pi)-\tfrac12\ln(L) - \tfrac12\ln(C)$
    Then on differentiating we get:
    $\displaystyle \frac{df}{f}=-\frac12\left(\frac{dL}{L}+\frac{dC}{C}\right)$
    You can then substitute in the values for $\displaystyle \frac{dL}{L}$ and $\displaystyle \frac{dC}{C}$ as Soroban has done. (You get the same answer, 3%, but with a negative sign, indicating that if $\displaystyle L$ and $\displaystyle C$ are larger than the true value, $\displaystyle f$ will be smaller.)

    Note that any formula where you have powers of a number of variables can be dealt with in this way:
    $\displaystyle y = k\,a^p\,b^q\,c^r...$

    $\displaystyle \Rightarrow \ln(y) = \ln(k) + p\ln(a)+q\ln(b)+r\ln(c) + ...$

    $\displaystyle \Rightarrow \frac{dy}{y}= p\frac{da}{a}+q\frac{db}{b}+r\frac{dc}{c}+ ...$
    Grandad
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2009
    Posts
    4
    hi there,

    Thank you very much for your help.

    stazzer5
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Aug 11th 2010, 01:02 AM
  2. Finding maximum percentage error
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Dec 12th 2009, 05:56 PM
  3. Maximum error and percentage
    Posted in the Algebra Forum
    Replies: 2
    Last Post: May 6th 2009, 01:00 PM
  4. maximum error percentage
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Sep 5th 2008, 04:22 AM
  5. maximum percentage error
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 5th 2008, 12:49 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum