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Re: [femm] self inductance calculation.. need advice
In a message dated 7/6/01 10:16:25 AM Eastern Daylight Time,
corneliuspaul@xxxxxx writes:
Dear all,
I have a question about calculating self inductances in a
transformer coil in 2D models.
could you please comment on my understanding of the situation?
the standard magnetostatic method is to drive the winding in question
with a current of one ampere and let FEA calculate the total energy E.
once you have calculated the total field energy you easily compute
the inductance L by the formula:
E = 0.5 L * I^2 --> L = 2E/I^2 --> L=2E for I=1amp
so far so good...
but what happens when we model the geometry in 2D?
consider we have a core with a quadratic cross section
(width a * depth a millimeters)
and a quadratic coil wound around it
(width b * depth b millimeters).
let the coil and core be of height c millimeters.
our 2D model shows the core as a rectangle of c * a millimeters
(and infinite depth into the z direction).
the coil wires can only be modeled as their cross sections on either
side of the core, spaced b millimeters apart. They also extend into
infinite depth.
that is, in reality I do not model a coil but I model an infinetely long
transmission line with a ferrite core in between the 2 conductors.
the other 2 coil wires do not show up in the model at all: going from
left to right in front of the core and from right to left behind the core.
if I calculate the field energy of this model and compute the inductance
by the formula given above, the resulting Inductance is in Henries per
unit depth (normally Henries per meter).
the question is: how do I compute the inductance for the real coil
from this value for the inductance/meter of the transmission line model?
do I multiply by core depth a or by coil depth b?
You'd multiply by the the depth of the core. The rationale here is that flux
strongly prefers to flow in the core, rather than through the air, so you get
a reasonable approximation of the total flux multiplying by the depth of the
core.
and much more of interest: am I not missing out the field (energy) that
is generated by the other 2 coil wires that do not show up in the model at
all??
if the coil depth into z direction is much bigger than the coil width this
does not matter, but in my case the coil depth and coil width are equal
b=b.
so do I have to multiply the value by 2?
The discrepancy in inductance between a 2-D and 3-D model is typically much
smaller than a factor of 2 error. It sounds like your rationale is that
since you have a nearly square cross-section and the end-turns aren't taken
into account, you'll have about a factor of 2 error in total flux. It
doesn't work quite like that. Think about this terms of magnetic
circuits--the only parts of the wire that actually "drive" flux are those
that get encircled by the flux path. The part of the winding that actually
drives the primary, desirable part of the flux in the transformer is the
coils that passes through the "winding window" in the transformer, because
the flux flowing through the core circles this part of the winding. In
contrast, the only flux that actually encircles the end turns is leakage
flux, traveling on a highly flux reluctant path, mostly through air.
Because this end turn leakage path is highly reluctant relative to the
primary flux path (hopefully), it doesn't add much inductance to the model.
At least, this is the basic rationale in using a 2-D approximation...
If you have a model with air gaps, flux fringing effects at the axial ends of
the core can increase the flux in the "primary" path some, but this is a
different mechanism than end-turn leakage inductance. This would mostly be
an issue in machines with a relatively "shallow" axial extent and relatively
large air gaps.
this, of course becomes even more interesting if the core has a quadratic
cross section and the coil has a circular cross section.
by which factor k do I multiply then?
k = (circumference of circular coil)/2b...?
Well, you'd probably just want to multiply by the depth of the core again,
because that's where most of the flux is.
the same questions arise when calculating the flux density in the core:
my feeling is that by using the infinitely long transmission line model
I am calculating too low a flux density in the core, because I am missing
out on the flux generated by the missing wires in the model.
Well, there can be some funny issues about leakage flux penetrating into the
ends of the lamination stack causing local eddy current heating in
transformers. There are also some machines that are fundamentally 3-D in
nature. However, for machines like conventional motors and transformers,
even ones with a fairly squat profile, the 2-D model (and its associated flux
density predictions) is usually a reasonable approximation.
Dave.
--
David Meeker
http://members.aol.com/_ht_a/dcm3c