Re: Formula for self-inductance?

On 24 mar, 10:02, "Benj" <bjac...@xxxxxxxxxxx> wrote:
Here's another simple question.
We all know that self-inductance is basically determined only by the
geometry of conductors in space. [Except for a few material
properties etc.]

So what I'm looking for is an elemental equation that one could say
approximate on a computer to find the self-inductance of various
geometries. For example one can integrate over a current source using
Biot-Savart to find the magnetic field in space about that source
(wire). One can use the Neumann formula to find mutual inductance for
various geometries. But self-inductance ends up with a problem because
of the r [separation between current source and induced emf] which
goes to zero as one tries to integrate close to the given current
element causing the inductance to become infinite!

A very simple case is that of the self-inductance of a straight wire!
Two ways this is often calculated is using flux linkages. One way is
to start with a coaxial cable and then let the outer pipe go to a
large diameter leaving the center conductor alone. Another is to start
with a wire loop and let the loop diameter go to a very large value so
as to approximate as straight wire. BOTH methods fail. One finds that
not only does the self-inductance go to infinity [as expected] but the
self-inductance PER UNIT LENGTH also become infinite. Clearly this is
wrong.

Straight wires do indeed have finite self-inductance per unit length!
Plus we know that the self-inductance depends upon the diameter of the
wire, becoming very large as the wire diameter becomes very small.

So for starters just what kind of formula would be valid to compute
the self-inductance of a length of straight wire of some given cross-
sectional geometry [Start with round, but really should be able to use
any cross-sectional geometry]? Needing a computer to calculate the
answer is OK, I'm just looking for a CONCEPTUAL formulation that
connects geometry with self-inductance! Sort of like the Neumann
formula but for self-inductance.

Thanks.

PS. And here's another gem: Note that for a thin straight wire by Biot-
Savart a given current element along the length of that wire cannot
induce a B field back into the wire because the sin of the angle is
zero and hence B=0! Makes you wonder just what is causing the
induction giving rise to self-inductance!

Hi Benj,

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.

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).

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).

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.

Best Regards,

Wim


.

Relevant Pages

  • Formula for self-inductance?
    ... We all know that self-inductance is basically determined only by the ... geometry of conductors in space. ... A very simple case is that of the self-inductance of a straight wire! ...
    (sci.physics.electromag)
  • Re: Inductance of wire
    ... there is a relatively simple formula for the mutual inductance ... This is actually a self-inductance calculator for round wire. ... The approaches for long lines do recognise that somewhere there is a return conductor and a flux linkage approach is used- this is valid for multiconductor lines as long as the sum of the currents is 0. ...
    (sci.physics.electromag)
  • Re: Inductance of wire
    ... there is a relatively simple formula for the mutual inductance ... This is actually a self-inductance calculator for round wire. ... inductance of straight wire per unit length does indeed exist. ...
    (sci.physics.electromag)
  • Re: Inductance of wire
    ... See the classic Grover book "Inductance ... Hence "r" above for round wire simply becomes the radius ... evenly spaced on a circle so the mutual inductance of each conductor ... The basic idea is that inductance of a geometry can be calculated by ...
    (sci.physics.electromag)
  • Re: Hum, Ground Loops and Safety
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