[SI-LIST] Re: Can L12 ever exceed L1 or L2 ??
- From: Ihsan Erdin <erdinih@xxxxxxxxx>
- To: edward.k.chan@xxxxxxxxx
- Date: Sun, 30 Oct 2005 22:51:32 -0500
Edward,
I must admit your example is challenging to the conventional reasoning. But
before getting to it, let's discuss the validity of the realizability
condition on the inductance matrix. By applying the positive definetness
test to the L matrix only you're reducing the transmission line equations to
dV/dz+jwLI=0. Here, I assume a wave propagation along the z-direction on a
lossless line. For (voltage and current) wave propagation on a line, we need
the second equation: dI/dz+jwCV=0. As such, the positive definetness test
must be applied to [0 jwL;jwC 0] matrix, with L and C being the submatrices
of the multiconductor transmission line system. I have neither time or
appetite to prove (or disprove) if L12>L11 is possible for this augmented
matrix under the condition that it's positive definite in the whole
frequency range. But I don't think that matters either because from your
argument I understand you are trying to find a freaky case where the usual
L12<L11 inequality could be broken.
So let's get to your example with the coils. I agree that the mutual
inductance will come out larger than the self inductance of the single turn
coil. But there seems to be a very clever "cheating" in that case. Because,
I can raise the counter-argument that the 2nd turn on the 2nd coil could be
very well taken as the 3rd conductor in the system. That reduces the case
L12+L13>L11, which is fair.
So, I'm still holding to the grounds that the mutual inductance could be
equal to the self at its best where the two conductors should occupy the
same place in space, which is a contradiction in itself. Thus, L12<L11
should hold no matter what...
Ihsan
On 10/30/05, Chan, Edward K <edward.k.chan@xxxxxxxxx> wrote:
>
> I've been thinking about this problem for a while too, and here's what I
> think I figured out:
>
> The requirement for an inductance matrix to be physical is that it is
> positive definite. Thus L11 * L22 - L12 * L12 > 0. This is similar to
> other equations described by others previously.
>
> Hence, it is possible that L12 is larger than the smaller of L11, L22.
>
> So what physical system exhibits this --> essentially transformers:
>
> Imagine two concentric coils with identical loop area placed infinitely
> close to each other. Then L12 is very close to L11 (and L22).=20
>
> Now we double the windings of the second coil. If we integrate the flux
> over the area of the first coil (single winding), we find that the flux
> from the second coil (two windings) is larger than the flux from itself
> (first coil with one winding). Therefore, L12 is larger than L11.
>
> Now, what structures on typical PCBs exhibit this behavior?
>
> Imagine a microstrip line over a ground plane has a certain L11 and L22.
> Now put a ground plane over this microstrip to create a stripline. With
> the top and bottom planes tied together at the near and far ends, we
> will get a 2x2 inductance matrix. When the signal trace is narrower
> (~<0.5X) than the ground planes, the additional ground plane will reduce
> the effective inductance of the ground plane such that it is less than
> the mutual inductance between the signal and ground.=20
>
> I haven't investigated all the conditions under which these observations
> are true, but I believe L12 can exceed L11 or L22.
>
> I have simulated the coils and the stripline in Fasthenry, and can
> provide the simple input decks to anyone interested.
>
> Edward Chan
> Intel
>
> -----Original Message-----
> From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
> On Behalf Of Zhou, Xingling (Mick)
> Sent: Saturday, October 29, 2005 6:44 AM
> To: eric@xxxxxxxxxxxxxxx; si-list@xxxxxxxxxxxxx; ray.anderson@xxxxxxxxxx
> Cc: susan@xxxxxxxxxxxxxxx
> Subject: [SI-LIST] Re: Can L12 ever exceed L1 or L2 ??
>
> Eric,
>
> Seems we have talked about signal lines a lot. Those lines are self-L
> dominated as we know. I don't remember any physics law prohibits
> L12>L1,or L2 other than K<=3D3D1 and some intuitive arguments. One =
> special
> case can break the statement if we do not have solid foundation. If a
> law is pointed out or newly proven in general, the argument will be much
> easier. If not, it may be considered as a conjecture based on some
> observations at most.=3D20
>
> How about when planes are involved? For example, for a package, L_vss is
> generally low referring to PCB GND. However, there are signals/planes
> that couple with VSS strongly.=3D20
>
> Even for regular lines, is it possible to construct a case that breaks
> the statement? For example, when line loops cross each other in complex
> ways in 3D (not as simple as we have in regular designs).=3D20
>
> Of course, numerical problems are always questionable. This is why the
> question confuses many of us. Is it because of the numerical errors or
> physically possible in some cases? Or in any case, we should check the
> numerical problems and force the vendor to satisfy us.
>
> Finding out when the statement is true (if not always) is also very
> helpful.
>
> Just some wild thoughts.
>
> Thanks.
>
> Mick
>
> -----Original Message-----
> From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
> On Behalf Of Eric Bogatin
> Sent: Saturday, October 29, 2005 8:58 AM
> To: si-list@xxxxxxxxxxxxx; ray.anderson@xxxxxxxxxx
> Cc: eric@xxxxxxxxxxxxxxx; susan@xxxxxxxxxxxxxxx
> Subject: [SI-LIST] Can L12 ever exceed L1 or L2 ??
>
> Ray-
> I completely agree with Steve Weir's comment that it is not
> physically possible for L12 > L11, whether we use these
> terms as loop inductance elements or partial inductance
> elements.
>
> The self inductance is the number of loops of magnetic field
> lines that surround one conductor per amp of its current.
> The mutual inductance is the number of loops of magnetic
> field lines that surrounds both conductor, per amp of
> current in one. All the mutual field lines from one
> conductor must also, by definition surround its own
> conductor and be part of its self inductance. This is true
> for loop inductance or partial inductance.
>
> The real question is why your vendor supplied you with
> matrix elements where you got L21 =3D3D 3 x L11. I've gotten
> similar comments from other end users. The answer is that
> when the vendor ran their field solver, they did not use a
> fine enough mesh. If you have radically different conductor
> geometries, like a short, wide conductor and a long,
> meandering trace, you will often see on the first pass
> calculation of the field solver that the partial self
> inductance of the short trace is less than the partial
> mutual between them.=3D20
>
> Your vendor needs to refine their mesh. When the mesh is
> refined so that the matrix elements do not change by more
> than about 1% for a 10-20% increase in mesh elements in the
> high field regions, the mesh is refined enough.
>
> Perhaps if your vendor were to read chapter 6 in my book
> Signal Integrity-Simplified, they would have a better chance
> of providing you more accurate and meaningful models.
>
> Hope this helps.
>
> --eric
>
> ***************************************
> Eric Bogatin
> Bogatin Enterprises
> OnLine Lectures on Signal Integrity
> 26235 w 110th terr
> Olathe, KS 66061
> v:913-393-1305
> cell: 913-424-4333
> f:913-393-0929
> e:eric@xxxxxxxxxxxxxxx
> www.BeTheSignal.com <http://www.BeTheSignal.com> <http://www.BogEnt.com
> >=3D20
>
> Signal Integrity- Simplified
> published by Prentice Hall
> *****************************************
>
> Msg: #7 in digest
> Subject: [SI-LIST] Can L12 ever exceed L1 or L2 ??
> Date: Fri, 28 Oct 2005 12:21:06 -0700
> From: "Ray Anderson" <ray.anderson@xxxxxxxxxx>
>
> A question for the E&M gurus on the list:
> =3D20
>
> Are there any conditions (pathological or not) that the
> mutual inductance between two conductors can be greater than
> the self inductance of either one?
>
> =3D20
>
> Since L12 =3D3D k * sqrt(L1*L2) and the coupling factor k must
> be -1 < k < 1 then this would seem to imply that L12 must
> be less than or equal to geometric mean of the self
> inductances. However this leaves open the possibility the Lm
> could be > than one or the other which flies in the face of
> the commonly made assertion that Lm must be less than
> either. It seems that there must be other qualifying
> statements to made regarding the relationship to the self
> and mutual inductances.
>
> =3D20
>
> Going back to some of the basic fundamental relationships
> (such as Grover's formulas) I can convince myself that for
> circular conductors
> L12 must be less than or equal to either L1 or L2, but how
> about the mutual coupling between some other structures say
> a signal trace and large planar structure that isn't
> intended to be be a signal return path but very well may be
> ?
>
> =3D20
>
> I've got a field solver reporting Lm being 2 to 3 times
> Lself on one particular problem. I'm trying to determine if
> the solver is having a difficult time dealing with the
> particular geometries involved or if it is indeed possible
> despite the common wisdom to the contrary.
>
> =3D20
>
> Any comments one way or the other are appreciated.
>
> =3D20
>
> Regards,
>
> =3D20
>
> -Ray
>
> =3D20
>
> =3D20
>
> Raymond Anderson
>
> Senior Signal Integrity Staff Engineer
>
> Product Technology Department
>
> Advanced Package R&D
>
> Xilinx Inc.
>
>
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