Re: Formula for self-inductance?


Wimpie wrote:
Hi Benj,

Thanks for the comments, Wimpie,

The coaxial model fails because of the assumption that the coaxial
cable has infinite length (in that case the B or H, fails off
inversely with distance), however in a finite length wire, above a
certain distance the field falls of with 1/r2. In that case, the
integral has a finite value.

The coaxial model just plain fails, period! Wires do not have to be
short to have a finite inductance PER UNIT LENGTH! Make a wire as
long as you like, the inductance per unit length does not suddenly
become infinite! Something is seriously wrong with the model!

There are many formulas around. For the inductance of a straight
finite length wire without return conductor, I use
L = 0.2*(rel. permeability)*le*(ln(4*le/D)-1), L in uH, le=length of
wire, D=diameter, all in metric units (m). Current flows at surface of
conductor (so not valid for DC).

Yes, I'm familiar with the Grover formula, however it provides
practical answers rather than what I was looking for. But notice that
even this formula has a problem! [actually more than one, but I won't
go into that now] Observe there is a certain value of le/D which gives
a zero inductance for certain lengths greater than zero. Sure, zero
length wires should have zero inductance, but this additional zero for
non-zero lengths is clearly a problem.

But actually what I was looking for was something less practical and
more based on differential elements such as a re-working of the Biot-
Savart for current densities as you suggest.

For a round wire you can use symmetry to (numerically) calculate the
B field at any distance and orientation with respect to the wire with
the Biot Savart law. Assuming 1A through the wire segment, integrating
the B-field will result in the total Flux (and inductance) surrounding
the wire (assuming wire flows at surface of wire). For DC with uniform
current distribution in the wire, the situation is more complicated.
For non-circular wire, the situation is also more complicated (FEM
required).

Yes, the problem gets complex. Even more complex given that it seems
to me that there are really two problems here. one is the calculation
of the B field from the current densities in the wire of a given
shape, but also there is the problem of induction causing some kind of
re-distribution of current in the wire as time goes on. Even with a
linear ramp current finding this new current distribution could be a
nightmare!

You are right that when r goes to zero, B goes to infinite, however,
in a real world, the current is distributed over the wire cross
section. So in fact r can never be zero. So you should dive into
mathematics to convert the Biot-Savart law into a form that can handle
current densities rather then concentrated current.

I will have to take another peer at Biot-Savart for current densities.
Nice suggestion!
Thanks for the comments!

Benj

.

Relevant Pages

  • Re: Velocity Factor and resonant frequency
    ... Consider a wire that's perpendicular to a ground plane; ... which is different than the inductance at low frequencies--per unit ... I believe it will be relatively unaffected by distance along ...
    (rec.radio.amateur.antenna)
  • Re: Why ground a machine?
    ... Technically a two wire plug is grounded. ... breaker panel and connected to a neutral bus bar. ... Home of the world's finest pinball protectorshttp://www.passionforpinball.com ... You should have a zero read. ...
    (rec.games.pinball)
  • Re: Trying to understand AC by analogy to water waves.
    ... the average actually zero? ... wire directly to ground produces a huge current. ... When you look at the voltage represented as a curve, ... neither have significant natural waves. ...
    (sci.physics)
  • Hardware id always zero = bug?
    ... (Sorry to cross post here, I posted this msg on ... I thought it was my fault, but now I found out that it displays zero ... Other workaround is to use the MAC address travelling on the wire, ...
    (microsoft.public.windows.terminal_services)
  • Hardware id always zero = bug?
    ... WTSclientHarwareId in its WTSQuerySessionInformation api call, ... I thought it was my fault, but now I found out that it displays zero ... Other workaround is to use the MAC address travelling on the wire, ...
    (microsoft.public.win2000.networking)