# RE: [SI-LIST] : Re: approximations for partial self inductance

From: [email protected]
Date: Wed Mar 14 2001 - 14:56:16 PST

We need to be careful in our comments. People seem to be mixing situations.
This is dangerous when the starting point is a discussion of approximations.
The original question was about vias for decoupling, not signal vias. Yes
Jian, you are quite correct that there are many cases where a capacitor
models the effect of a via on a SIGNAL better than an inductor.

I have never seen a via through a multilayer board that looked like a long
straight round wire; a via has pads on every layer which add capacitance.
So let me highlight two key parts of Brian's comments

> >>When ***d>>r***, the external partial
> >>self-inductance of a straight round wire is
> >>L=5.08d*{ln(2d/r +/- whatever
> >>This formula **should not be used** for vias because it assumes that
> >>the length is much greater than the diameter.

For DECOUPLING, considering a via alone is risky; one should consider the
LOOP defined by the Power/Ground via pairs.

Aubrey Sparkman
Signal Integrity
[email protected]
(512) 723-3592

> -----Original Message-----
> From: Jian X. Zheng [mailto:[email protected]]
> Sent: Wednesday, March 14, 2001 1:44 PM
> To: Howard Johnson; [email protected]
> Subject: RE: [SI-LIST] : Re: approximations for partial self
> inductance
>
>
> Hi, Howard:
>
> I like your comment on via inductance.
>
> -------------------------------------------------------
> For a signal which pops from one side of the
> plane, through a via, to the opposite side
> of the same plane (i.e., the return current
> doesn't have to jump planes), the via
> inductance is very, very low. This is a best-case
> scenario. I don't know a good way to make this
> calculation except with a true 3-D E&M field solver.
> -------------------------------------------------------
>
> Many people consider a via can be approximated by an
> inductance. However, I
> believe it might not be a very good approximation. It does
> seem to me the
> via inductance can be quite low and there might be
> significant capacitance
> involved. I can use our full wave electromagnetic simulator
> IE3D to confirm
> it in seconds or minutes.
>
> I published a paper on the IEEE Trans. on Antennas and
> Propag. vol. 39, No.1
> January. The paper is on a coaxial fed probe (or via). The
> structure is not
> exactly the same as in the multiple layered board. However, it is very
> similar. Accurate L and C can be obtained analytically. It is quite
> complicated and I would not put it here. The interesting
> thing is that the L
> and C involved have the frequency dependency of log(f).
> Interested users can
> find the formulas from the literature. Thanks!
>
> --------------------------------------------------------------
> ---------
> Jian-X. Zheng, Ph.D
> Zeland Software, Inc., 48890 Milmont Drive, 105D, Fremont, CA
> 94538, U.S.A.
> Tel: 510-623-7162, Fax: 510-623-7135, Web: http://www.zeland.com
> ---------------------------------------------------------------------
>
> > -----Original Message-----
> > From: [email protected]
> > [mailto:[email protected]]On Behalf Of Howard Johnson
> > Sent: Wednesday, March 14, 2001 10:22 AM
> > To: [email protected]
> > Subject: [SI-LIST] : Re: approximations for partial self inductance
> >
> >
> > Dear Itzhak Hirshtal and Brian Young,
> >
> > The difficulties with approximating the inductance
> > of a via are even worse than you
> > may have suspected. Both approximations are flawed whether
> > you use +1 or -3/4, (or, as I have also seen, -1).
> >
> > The issue of the exact constant (1, -3/4, or something
> > the path of returning signal current. (Current always
> > makes a loop; when signal current traverses the via,
> > a returning signal current flows SOMEWHERE in
> > the opposite direction.). It is a principle
> > of Maxwell's equations that high-speed returning signal
> > current will flow in whatever path produces the
> > least overall inductance.
> >
> > Let's do an example involving a signal via that
> > dives down through a thick, multi-layer board.
> > If the signal in question changes reference
> > planes as it traverses the via,
> > then the returning signal current will also have to
> > change planes, meaning that the returning signal
> > current will flow through one or more vias (often
> > leading to bypass capacitors) as it moves from
> > plane to plane. For example, if the signal starts
> > out on the top layer, the returning signal current
> > is flowing on the nearest reference plane (call it
> > layer 2). If the via conducts the signal current
> > down to the bottom layer (16), then the returning
> > signal current at that point must be flowing on
> > the nearest (bottom-most) reference plane, call it 15.
> > Somehow the returning signal current has to hop from
> > reference plane 2 to reference plane 15 in the
> > vicinity of the via.
> >
> > If you examine the space between the planes, the
> > magnetic fields within are created partly by
> > the signal current, and in equal measure (but in
> > differnt locations) by the returniing signal
> > current, which flows on different vias. The
> > total magnetic flux between the outgoing and
> > returning vias defines the inductance.
> > Specifically, to calculate the effective
> > inductance of via (A), you must first specify the
> > location of the return path, via (B), and then
> > calculate the total magnetic flux in the area
> > between the two vias. The total magnetic flux
> > generated by a signal current of one amp, in units
> > of webers, equals the inductance.
> > In the case of more complex return-path
> > configurations, other considerations apply.
> > I think at this point that the following
> > formulii for the effective series inductance
> > of a via are pretty good:
> >
> > For a signal which pops from one side of the
> > plane, through a via, to the opposite side
> > of the same plane (i.e., the return current
> > doesn't have to jump planes), the via
> > inductance is very, very low. This is a best-case
> > scenario. I don't know a good way to make this
> > calculation except with a true 3-D E&M field solver.
> >
> > For a signal which first uses reference-plane A,
> > and then changes (through a via) to use
> > reference-plane B, I'll do several examples. In
> > all cases the separation between reference planes
> > is H. (It doesn't matter if there are other
> > unused reference planes in the way, only the
> > spacing between the two reference planes A and B
> > matter).
> >
> > If the return current is carried mainly on one nearby
> > via, where the spacing from signal via to return via
> > is S and the via diameter is D:
> >
> > L = 5.08*H*(2*ln(2*S/D)) [1]
> >
> > If the return current is carried mainly on two vias
> > equally spaced on either side of the signal via,
> > where the spacing from signal via to either return via
> > is S and the via diameter is D:
> >
> > L = 5.08*H*(1.5*ln(2*S/D) + 0.5*ln(2)) [2]
> >
> >
> > If the return current is carried mainly on four vias
> > equally spaced in a square pattern on four sides
> > of the signal via, where the spacing from signal via
> > to any return via is S and the via diameter is D:
> >
> > L = 5.08*H*(1.25*ln(2*S/D) + 0.25*ln(2)) [3]
> >
> > If the return current is carried mainly on a
> > coaxial return path completely encircling the signal
> > via, where the spacing from signal via
> > to the return path is S and the via diameter is D:
> >
> > L = 5.08*H*(ln(2*S/D)) [4]
> >
> > The last formula I hope you will recognize as the
> > inductance of a short section of coaxial cable with
> > length H and outer diameter 2*S. I hope this
> > recognition will lend credence to the idea that
> > the position of the returning current path is
> > an important variable in the problem.
> >
> > My earlier formula was a gross approximation which
> > ignored the position of the returning current path,
> > and omission which I greatly regret. It made the
> > crude assumption that the return path was approximately
> > coaxial and located at a distance S=5.43*H. As you
> > note, when the inductance really matters a
> > more accurate approximation is needed.
> >
> > To obtain a result as low as 5.08*H*(ln(2*S/D)-1)
> > you would have to assume the return path were coaxial
> > and located at a ridiculously small separation of
> > S=.735*H, or that the return path were a single via
> > located at some even closer distance.
> >
> > On my web site http://signalintegrity.com under "articles"
> > there is a write-up about calculating the inductance of
> > a bypass capacitor that includes the above formulas for
> > vias, as well as some handy ways to estimate the
> > inductance of the capacitor body.
> >
> > By the way, if you find a flaw in THIS write-up,
> > please let me know.
> >
> > Best regards,
> > Dr. Howard Johnson
> >
> >
> >
> >
> >
> >
> > >>On the two versions of the equation, it looks to me like
> the version
> > >>in Johnson's book has a typo. When d>>r, the external partial
> > >>self-inductance of a straight round wire is
> > >>
> > >>L=5.08d*{ln(2d/r)-1}nH,
> > >>
> > >>where d is the length in inches, and r is the radius in inches.
> > >>The external inductance is a good approximation at high
> frequencies
> > >>where the skin effect shields the internal metal of the wire. At
> > >>low frequencies, the internal self-inductance needs to be
> > >>added to the external partial self-inductance to obtain
> > >>
> > >>L=5.08d*{ln(2d/r)-3/4}nH,
> > >>
> > >>which is the formula from Gover, as Eric pointed out.
> > >>
> > >>It seems that Johnson's book has the first
> (high-frequency) version
> > >>with a sign error on the 1 because he has
> > >>
> > >>L=5.08h*{ln(4h/d)+1}nH,
> > >>
> > >>where h is the length in inches, and d is the diameter in inches.
> > >>
> > >>
> > >>This formula should not be used for vias because it assumes that
> > >>the length is much greater than the diameter. To compute partial
> > >>self-inductance for vias, you should use the more complex formula
> > >>that does not have this assumption built in. The correct formula
> > >>is (5.49) from my book. This is the external partial
> self-inductance,
> > >>so if you want the low frequency inductance, you need to add the
> > >>internal inductance from (5.45).
> > >>
> > >>Finally, Grover does not actually derive much in his book. If you
> > >>are interested, the round wire formula above and many others are
> > >>derived in my book.
> > >>
> > >>Regards,
> > >>Brian Young
> > >>
> > >>
> > >>Eric Bogatin wrote:
> > >>>
> > >>> Itzhak-
> > >>>
> > >>> you asked the question about the difference in the
> approximations
> > >>> for the partial self inductance of a via that were given by
> > >>> myself and Howard Johnson. I wanted to provide some
> > >>> clarification. You wrote:
> > >>>
> > >>> (4) While calculating vias inductance, I've encountered
> 2 similar
> > >>> but
> > >>> different equations for this parameter. One is given by Mr. H.
> > >>> Johnson
> > >>> in his famous book (page 259), as follows:
> > >>>
> > >>> L=5d*{ln(2d/r)+1}nH.
> > >>>
> > >>> The other is given by Mr. Bogatin in one of his
> articles, and is:
> > >>>
> > >>> L=5d*{ln(2d/r)-3/4}nH.
> > >>>
> > >>> Can somwone explain the reason for the difference, or who is
> > >>> right? The
> > >>> difference starts to be quite critical when dealing with u-Vias!
> > >>>
> > >>> The approximation is for the partial self inductance of a round,
> > >>> solid rod, of radius, r and length d. The length is in units of
> > >>> inches, while the inductance is in units of nH.
> > >>>
> > >>> This is the approximation that was originally derived by Fred
> > >>> Grover, in his classic book, Inductance Calculations",
> in 1946. I
> > >>> just re-checked the one I offered, and it is correctly
> reproduced
> > >>> above. It is listed on page 35, eq 7, of his book. I
> think it has
> > >>> since been reprinted as a Dover Book.
> > >>>
> > >>> Keep in mind two things when using this approximation:
> 1st, it is
> > >>> an approximation. Grover says it is good to about 2%. I have
> > >>> found good agreement to better than 5% for wire bond structures.
> > >>> Approximations are wonderful tools to assist you in exploring
> > >>> design space, run in a spread sheet and play what-if trade offs.
> > >>> They give you good answers and let you see the geometry and
> > >>> materials trade offs. However, they are APPROXIMATIONS. You
> > >>> should never use an approximation in a situation where the
> > >>> accuracy of the answer may cost you significant time
> and expense.
> > >>> You should be using a 3D field solver that you have confidence
> > >>> in. One of the second order effects in this approximation, for
> > >>> example, is that it includes the "internal" self inductance. As
> > >>> the skin depth gets to be comparable to the geometrical cross
> > >>> section, the partial self inductance will decrease and reach a
> > >>> constant value when all the current is in the outer surface.
> > >>>
> > >>> The second thing to keep in mind when using this
> approximation is
> > >>> that it is for the PARTIAL self inductance of the via, under the
> > >>> assumptions of uniform current flow down the long axis. If you
> > >>> are using it in a situation where the length of the structure is
> > >>> comparable to the diameter, ie, d ~ 2r, the current distribution
> > >>> through the structure may not be even close to parallel to the
> > >>> long axis. Further, the actual loop inductance, which is what
> > >>> matters in a real circuit, is probably dominated by other
> > >>> elements than this small, squat element. The partial self
> > >>> inductance may depend strongly on the proximity of other
> > >>> conductors and how it affects the current flow through this via.
> > >>> If you are in a regime where worrying about the presence of the
> > >>> -3/4 term is important, you probably want to use a 3D field
> > >>> solver before any design signoff. A good 3D solver will
> calculate
> > >>> the actual current distribution through the via
> structure and the
> > >>> rest of the current path.
> > >>>
> > >>> I hope this helps.
> > >>>
> > >>> If anyone is interested, I have various application
> notes related
> > >>> to approximations to inductance and general principles
> related to
> > >>> inductance posted on our web page. These are listed as app notes
> > >>> with index numbers: 33, 32, 29, 25, and 9. You can find them
> > >>> under application notes at www.gigatest.com
> > >>>
> > >>> As always, comments are welcome.
> > >>>
> > >>> --eric
> > >>>
> > >>> From: Itzhak Hirshtal [mailto:[email protected]]
> > >>> Sent: Monday, March 12, 2001 09:33
> > >>> To: si-list
> > >>> Subject: [SI-LIST] : Inductance and Decoupling
> > >>>
> > >>> Hello, all
> > >>>
> > >>> I've recently started to calculate the de-coupling needed for
> > >>> efficiently supplying the spike currents needed by high-speed
> > >>> devices.
> > >>> During this task, I've encountered several ambiguities and
> > >>> results that
> > >>> I would like to share with you and perhaps hear some (useful)
> > >>> feedback
> > >>> from you.
> > >>>
> > >>> (1) I tried to evaluate the situation for one high-pin-count
> > >>> device with
> > >>> several buses connected to it (essentially a bus bridge). Even
> > >>> calculating for just one synchronous bus (with 144 bits overall)
> > >>> I
> > >>> arrived to the result that a few Amps (maybe even 5) are drawn
> > >>> when all
> > >>> or most of this bus bits change state. I wonder what will be the
> > >>> result
> > >>> if I would calculate for an additional bus (assuming it's
> > >>> synchronous
> > >>> with the first). And what about the internal changes? They might
> > >>> be
> > >>> contributing even more than the external bus! (e.g.,
> the Motorola
> > >>> PowerPC HW manual states that 90% of the power consumption of
> > >>> this
> > >>> device is drawn internally, not externally).
> > >>>
> > >>> (2) I've also tried to calculate the inductance of the
> decoupling
> > >>> capacitors connections to the device. Even assuming a
> 40-mil wide
> > >>> 50-mil
> > >>> long trace right above a reference plane for the connection I
> > >>> have app.
> > >>> L=150-200pH. If I can't connect at least one of the capacitor
> > >>> short I might have to do a direct connection via to a reference
> > >>> plane. I
> > >>> calculated this to have more than L=1nH!
> > >>>
> > >>> (3) I assumed the calculated peak currents change at a rate
> > >>> equivalent
> > >>> to the rise time of the device's output buffers. I don't know if
> > >>> it's
> > >>> true, but this seems to me the most logical thing to do. Even if
> > >>> I take
> > >>> it to be 2ns (1 ns is closer to worst-case, I believe), I get
> > >>> the
> > >>> result that I need 40 to 50 low-ESL decoupling
> capacitors for the
> > >>> case
> > >>> where L=1nH. Only if I succeed to connect the
> capacitors directly
> > >>> and
> > >>> close enough to both GND and VDD pins (L=150-200pH) do I get the
> > >>> result
> > >>> that it is sufficient to use 4-6 decoupling capacitors.
> > >>>
> > >>> (4) While calculating vias inductance, I've encountered
> 2 similar
> > >>> but
> > >>> different equations for this parameter. One is given by Mr. H.
> > >>> Johnson
> > >>> in his famous book (page 259), as follows:
> > >>>
> > >>> L=5d*{ln(2d/r)+1}nH.
> > >>>
> > >>> The other is given by Mr. Bogatin in one of his
> articles, and is:
> > >>>
> > >>> L=5d*{ln(2d/r)-3/4}nH.
> > >>>
> > >>> Can somwone explain the reason for the difference, or who is
> > >>> right? The
> > >>> difference starts to be quite critical when dealing with u-Vias!
> > >>>
> > >>> Thanks for anyone who makes the effort to read this email.
> > >>
> > >>
> > >>--
> > >>***************************************************************
> > >>* Brian Young phone: (512) 996-6099 *
> > >>* Somerset Design Center fax: (512) 996-7434 *
> > >>* Motorola, Austin, TX [email protected] *
> > >>***************************************************************
> > >>
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> > >>
> > >>
> > >_________________________________________________
> > >Dr. Howard Johnson
> > >tel 425.556.0800 fax 425.881.6149
> > >Signal Consulting, Inc.
> > >16541 Redmond Way #264
> > >Redmond, WA 98052
> > >http://signalintegrity.com -- High-Speed Digital Design
> > >books, tools, and workshops
> > >
> >
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