Stupid Math Error

Discussion in 'Math' started by Wendy, Feb 25, 2014.

  1. Wendy

    Thread Starter Moderator

    Mar 24, 2008
    20,766
    2,536
    I am having a massive senior moment, all day long.

    I want to roll an inductor 1.27uH. I have chosen a drill bit 0.228" in diameter, and I will use 24 Ga enamelled wire.

    According to this page, the wire is 0.0201" in diameter.

    I am using the formula from this page,

    L=(N^2 * r^2) / (9 r + 10 l)

    where

    L = micro Henrys
    N = Number of turns in the inductor
    r = mean radius of inductor
    l = Length of inductor

    I have empirically solved this to be 21 turns. However I have a hard head, and would like to do this with math. So I redefine some of the variables to make tracking easier.

    Redefine:

    x = N
    g = 24Ga = 0.02010
    i = l, which also = g x

    So, the formula becomes

    L = (x^2 * r^2) / (9 r + 10 i)

    here is where it gets interesting

    9 r + 10 i = (x^2 * r^2) / L
    9 r + (10 g x) = (r^2 / L) x^2
    0 = (r^2 / L) x^2 - (10 g) x - 9 r

    This is the weakest step for me, where I feel I have most likely made the error. I have translated this into a quadratic equation.

    OK, the variable numbers are:

    x = Number of turns
    r = Mean radius = (dia of core * dia of wire) / 2 = 0.124
    g = .0201
    L = 1.27

    Using the quadratic equation

    x = ( -b +/- sqr( b^2 - (4 a c)) / (2 a)

    where

    0 = a x^2 + b x + c

    I get x = 2.5, which is a non-starter. I know I'm messing up somewhere. Help?
     
    Last edited: Feb 25, 2014
  2. WBahn

    Moderator

    Mar 31, 2012
    17,765
    4,801
    Your problem most likely stems from sloppy units in the original equation.

    What is the length of the inductor in? cm? or m? or inches?

    Same for the radius?

    If we know what the units are then we can figure out what the units on the coefficients need to be and then we can track them through.
     
  3. panic mode

    Senior Member

    Oct 10, 2011
    1,321
    304
    quadratic equation has two solutions.
    since it is late, i did not go as far as evaluating it though, so i cheated :)
    just opened excel and typed in formula and all known parameters, plugged in 3 numbers to get to N=21 which produces 1.27053uH
     
  4. panic mode

    Senior Member

    Oct 10, 2011
    1,321
    304
    ok without cheating i get solutions N=-4.38891 and N=20.99388
    of course negative one is to be discarded so N=21 stands.
    i think you simply made a typo somewhere while using quadratic equation (calculator...?).
    what you posted looks good, i got
    a=0.012117
    b=-0.2012
    c=-1.11645

    then two terms to be added/subtracted are
    -b/(2a) = 8.302486
    +/-sqrt(b^2 - 4ac)/(2a)= +/-12.6914

    which produces mentioned sum and difference of -4.38891 and +20.99388
     
    Last edited: Feb 26, 2014
  5. Wendy

    Thread Starter Moderator

    Mar 24, 2008
    20,766
    2,536
    If you had referenced the link provided (the source for the equation) you would have noted all dimensions are in inches. Given I am using a standard set of dimensions that would not explain a fundimental math error,
     
  6. Wendy

    Thread Starter Moderator

    Mar 24, 2008
    20,766
    2,536
    The 21 agrees with my empirical solution. About to step out so I will go through the math again later.
     
  7. t06afre

    AAC Fanatic!

    May 11, 2009
    5,939
    1,222
Loading...