**From:** Jian X. Zheng (*[email protected]*)

**Date:** Wed Mar 14 2001 - 11:44:02 PST

**Next message:**[email protected]: "RE: [SI-LIST] : Re: approximations for partial self inductance"**Previous message:**Dunbar, Tony: "RE: [SI-LIST] : noise budgeting"**In reply to:**Howard Johnson: "[SI-LIST] : Re: approximations for partial self inductance"**Next in thread:**Sainath Nimmagadda: "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
*

*> else) depends critically on your assumption about
*

*> 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
*

*> >>> pads so
*

*> >>> 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] *
*

*> >>***************************************************************
*

*> >>
*

*> >>**** To unsubscribe from si-list or si-list-digest: send e-mail to
*

*> >>[email protected] In the BODY of message put: UNSUBSCRIBE
*

*> >>si-list or UNSUBSCRIBE si-list-digest, for more help, put HELP.
*

*> >>si-list archives are accessible at http://www.qsl.net/wb6tpu
*

*> >>****
*

*> >>
*

*> >>
*

*> >_________________________________________________
*

*> >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
*

*> >
*

*>
*

*> **** To unsubscribe from si-list or si-list-digest: send e-mail to
*

*> [email protected] In the BODY of message put: UNSUBSCRIBE
*

*> si-list or UNSUBSCRIBE si-list-digest, for more help, put HELP.
*

*> si-list archives are accessible at http://www.qsl.net/wb6tpu
*

*> ****
*

*>
*

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****

**Next message:**[email protected]: "RE: [SI-LIST] : Re: approximations for partial self inductance"**Previous message:**Dunbar, Tony: "RE: [SI-LIST] : noise budgeting"**In reply to:**Howard Johnson: "[SI-LIST] : Re: approximations for partial self inductance"**Next in thread:**Sainath Nimmagadda: "Re: [SI-LIST] : Re: approximations for partial self inductance"

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