[SI-LIST] Re: Can L12 ever exceed L1 or L2 ??
- From: steve weir <weirsi@xxxxxxxxxx>
- To: Ihsan Erdin <erdinih@xxxxxxxxx>
- Date: Mon, 31 Oct 2005 22:09:21 -0800
Ihsan, yes, I agree that there is not a transmission line structure that
makes any sense with unequal N, and that in all likelihood the issue Ray
has is a tool, or tool applications issue such as too coarse of a mesh.
However, for various other purposes, people do build spiral inductors, and
other forms of planar magnetics. I had not considered those cases when
answering Ray. Both Andy and another colleague both corrected me.
Regards,
Steve.
At 11:01 PM 10/31/2005 -0500, Ihsan Erdin wrote:
>Steve,
>
>It looks like we've reached to a common ground where the only possible
>case that would make the mutual inductance greater than the self involves
>unequal number of turns between the two coils. Now, let's dig deeper this
>"turn" issue. There's always a mutual interaction among the turns which is
>accounted for with capacitance and inductance parameters. In order to
>simplify the calculations the mutual interaction is usually neglected.
>That simplification leads to the approximation of all the turns lumped
>together. In a rigorous analysis, however, each turn has to be considered
>separately and the whole structure resembles a multiconductor transmission
>line system. In this case, it is physically impossible to say that the
>mutual impedance can exceed the self.
>Having said that, I can't think of a case where that multiple turn
>configuration of a transformer could be extended to a PCB structure. So, I
>wouldn't hesitate to contact the FAE of any CAD tool which would yield
>L12>L11...
>
>Regards.
>
>Ihsan
>
>On 10/31/05, steve weir <<mailto:weirsi@xxxxxxxxxx>weirsi@xxxxxxxxxx > wrote:
>>Ishan, I think the point that Andy brought up and a very esteemed colleague
>>also pointed out off-line is that what we have been discussing holds for
>>the case where the number of turns in L1 and L2 are equal. For the case
>>where L2 has N times as many turns as L1:
>>
>>L22nominal = N^2*L11
>>
>>L12perfect = L22nominal = N^2*L11 and for N>1 L12 > L11
>>L21perfect = L11
>>
>>The point that the flux density will always be highest at the source of the
>>MMF is not diminished. The implication that Fred, Eric, yourself and I all
>>appear to strongly agree upon is that we cannot end up with a coupling
>>factor greater than 1.0. But it does point out that the mutual inductance
>>can exceed one but not all of the self inductances. So, the correct
>>expression appears to be all:
>>
>>L22 > L12,
>>L11 > L21, and
>>L11*L22 > L12*L21, and
>>
>>I believe that this satisfies Siddarth's issues as well.
>>
>>Best Regards,
>>
>>Steve.
>>
>>At 10:51 PM 10/30/2005 -0500, Ihsan Erdin wrote:
>> >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 <<mailto:edward.k.chan@xxxxxxxxx>
>> 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:
>> <mailto:si-list-bounce@xxxxxxxxxxxxx>si-list-bounce@xxxxxxxxxxxxx
>> [mailto:si-list-bounce@xxxxxxxxxxxxx]
>> > > On Behalf Of Zhou, Xingling (Mick)
>> > > Sent: Saturday, October 29, 2005 6:44 AM
>> > > To: <mailto:eric@xxxxxxxxxxxxxxx>eric@xxxxxxxxxxxxxxx;
>> <mailto:si-list@xxxxxxxxxxxxx>si-list@xxxxxxxxxxxxx;
>> <mailto:ray.anderson@xxxxxxxxxx>ray.anderson@xxxxxxxxxx
>> > > Cc: <mailto:susan@xxxxxxxxxxxxxxx>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:
>> <mailto:si-list-bounce@xxxxxxxxxxxxx>si-list-bounce@xxxxxxxxxxxxx
>> [mailto: si-list-bounce@xxxxxxxxxxxxx]
>> > > On Behalf Of Eric Bogatin
>> > > Sent: Saturday, October 29, 2005 8:58 AM
>> > > To: <mailto:si-list@xxxxxxxxxxxxx>si-list@xxxxxxxxxxxxx;
>> <mailto:ray.anderson@xxxxxxxxxx>ray.anderson@xxxxxxxxxx
>> > > Cc: <mailto:eric@xxxxxxxxxxxxxxx>eric@xxxxxxxxxxxxxxx;
>> <mailto:susan@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
>> > > <mailto:e:eric@xxxxxxxxxxxxxxx>e:eric@xxxxxxxxxxxxxxx
>> > > www.BeTheSignal.com
>> <<http://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" <<mailto:ray.anderson@xxxxxxxxxx>
>> 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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