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Re: Calculating inductance...

>Message-ID: <199605191901.NAA02699-at-poodle.pupman-dot-com>
>Subject: Re: Calculating inductance...
>From sroys-at-umabnet.ab.umd.eduSun May 19 12:48:09 1996
>Date: Sun, 19 May 1996 13:24:38 -0400 (EDT)
>From: Steve Roys <sroys-at-umabnet.ab.umd.edu>
>To: tesla-at-pupman-dot-com


   I have been watching this thread for a while now.  I have a few
   comments.

   Steve writes,

>On Fri, 17 May 1996, Tesla List wrote:
>
>> Z is AC current? (as in V = IR)
>
>Z is the complex impedence or reactance - the analog of resistance for 
>inductors and capacitors.  The impedence of a inductor or capacitor is 
>frequency dependent.
>
>For an inductor, Z = 2*pi*f*L as mentioned, f = frequence in Hz, L = 
>inductance in H.  For a capacitor, Z = 1/(2 * pi * f * C) where f = Hz as 
>before and C = capacitance in F.  Just like a resistor, V = I*Z


   True, but what you have given here are the formulae for the
   **reactance** of an inductor and capacitor.

   Impedance is the *broader* term which includes resistance and
   reactance.  The symbol for reactance is X.  Xl is inductive
   reactance (where the voltage leads the current by 90 degrees), and
   Xc is capacitive reactance (where the current leads the voltage by
   90 degrees).

   Impedance of an inductor is, in complex polar notation is
   R +jXl where R is the series DC resistance in OHMS of the 
   coil and Xl is the reactance as you have described it, namely

         Xl = 2 * PI * F * L  

   If F is in Hertz and L is in Henrys, then Xl will also be in Ohms,
   but these are reactive Ohms and they add at 90 degrees to DC ohms. 
   In complex polar notation "j" is the square root of minus 1, and it
   serves in the mathematics to keep the reactive components and the
   resistive components from being incorrectly added linearly.  They
   must be added in quadrature.

   Because normal resistive ohms and reactive ohms add at ninety
   degrees, the *magnitude* of the impedance of a coil is the square
   root of the sum of the squares of the resistance and the reactance.

         Z = SQRT ((R * R) + (Xl * Xl))

   Since DC ohms and reactive ohms act at 90 degrees to each other it
   is obvious that this formula is just an adaptation of Pythagoras'
   theorem for finding the hypotenuse of a triangle.

   Incidentally the Q of a coil is Xl divided by R.  So for a good
   coil you want a low R.

   It should be noted that R is a function of frequency because as the
   frequency goes up, the current does NOT travel through the whole
   cross-sectional area of the coil, rather it travels along the
   surface, with the depth of the current decreasing as the  frequency
   increases.  The resistance of this "tube" comprising the surface
   few millimeters of the coil is higher than the resistance of the
   entire solid-conductor coil.  This is why silver-coated coils have
   a slightly better Q, because clean, bright silver lowers the
   surface resistance slightly.


   The expression for the impedance of a series resistive-capacitive
   circuit is
 
              Z = R - jXc

   where Xc  =  1 / (2 * PI * F * C)

   If F is in Hertz and C is in Farads, then Xc is in reactive Ohms.
   Note the sign of the "j" term.  It is negative.
   

   The same Pythagorean-derived formula for the magnitude of the
   impedance of a series resistance-capacitance circuit applies as for
   a series resistance-inductance circuit namely the square root of the
   sum of the squares of the resistance and the reactance.



   I think all you Tesla Coil builders would do good to deal in
   complex polar calculations.  The technique is quite beautiful and
   easy to understand (I learned it in Grade 11 Electronics class).
   But you do need to know a little about the sine, cosine, and
   tangent trigonometric functions.

   Let me know if I should continue with a discussion of parallel
   resistors and series and parallel tank circuits.  There are a lot
   of engineers in this group and I don't want to bore you.  Perhaps I
   should take it off line.  In my sparse off-line time, I'm working
   on such a document for the USA-TESLA list and I could use some
   specific questions about simple RLC circuits to find out what
   ground I have to cover.

 Fred W. Bach ,    Operations Group        | Internet: music-at-triumf.ca
 TRIUMF (TRI-University Meson Facility)    | Voice:  604-222-1047 loc 6327/7333
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 University of British Columbia, Vancouver, B.C., CANADA   V6T 2A3
 "Accuracy is important. Details can mean the difference between life & death."
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 They do NOT necessarily reflect the views of my employer or fellow workers.