standing wave coil frequency

[Montec]

Hello all
After reading about "slow wave" traveling wave tubes for high frequency amplification I came up with the following equations.

f = (.9c/Lc)sin(acrtan(P/D))  where

     .9c is the speed of light within a insulated copper wire. This will vary because the speed is influenced by changes in the permittivity around the wire. Your hand, etc.

     Lc = length of the coil (or a toroid's major radius circumference)

     P = Pitch of coil - distance between adjacent coils Controls the "slow wave" speed in the coil.

     D = Diameter of coil

Lw/N = P/Sin(a)   where

     Lw = length of straight wire used to make coil

     N = number of turns on the coil

     a = arctan(P/D)

Sin(a) = Lc/Lw

Lc = (lamda)sin(a) where "lamda" is the "free space" wavelength.

Have fun

[Montec]

Hello all
As a corollary to the above post I looked into skin effect. The general rule of thumb is that that the skin effect should be more than 1/4 of the wire diameter. This works out for copper to be:

    f = 0.06987/d²  where "d" is the diameter of the wire in meters (m)

So for 20 Gage (0.8128mm) wire the maximum frequency is approximately 105KHz and

    for 48 Gage (0.0315mm) wire the maximum frequency is approximately 70MHz.

Looks like Litz wire is the way to go when driving a circuit at higher frequencies to avoid higher resistance and heating problems.

[stevenfrank38]

lol
by the way you do a outstanding job
I think This kinds of work is very impotent for refereeing the mind
And you have done it well
I think you should do this kinds of work more and more
thank you very much

[MrMag]

I agree, thank-you Montec

[Montec]

Hello all
The approximate inductance (L) of a toroidal coil can be calculated by the following.

   L = μ_oμ_rr²N²/D

      μ_o = permeability of space = 0.000001256636  or 4Ï€^-7 (4"pi"^-7)

      μ_r = relative permeability of toroidal core  (air = 1, other ferromagnetic cores will be based on core alloy)

      r = radius of toroid minor axis (what you wind the wire around) plus 1/2 the wire diameter , both in (m)

      N = number of turns (wraps around minor axis)

      D = diameter of major axis (distance across toroid, center to center of minor axis) in (m)

  A formula derived from the above is:

     L = μ_oμ_rNA/P  (Somewhat more useful)

        A = area across the minor axis in (m²) (Ï€r² of coil cross section)

        P = pitch of coil winding (distance between wraps at major axis center line) in (m)

This shows that coil inductance is directly related to the number of turns and area within the turn. But is indirectly related (inverse) to the pitch of the coil. A multi- layer coil increases "N"  and "r" is then an averaged layer-radius of the coil.

The inductance of a coil allows the calculation of reactance (resistance) at a desired operating frequency. So a high frequency transformer needs less wire to act as resistance at an operating frequency. This keeps the current flow within the specs for the wire.

@ stevenfrank38, MrMag
  Thanks for the encouragement.