[hsdd] High-Speed Digital Design Newsletter


                    VIA INDUCTANCE

                           

HIGH-SPEED DIGITAL DESIGN    --  online newsletter  --
Vol. 6  Issue 04
March 15, 2003


  I am receiving an increasing flurry of requests for
  information about via inductance. Presumably, those
  of you asking for such data have reached that level
  of speed where the inductance of a via becomes quite
  significant.
  
  And what speed is that? Assuming an ordinary pcb
  thickness of 63 mils, the inductance of an ordinary
  via works out to only a few nanohenries, something
  that becomes increasingly significant at data rates
  in excess of 1 Gb/s.
  
  Some terrific pictures from my new book, "High-Speed
  Signal Propagation: Advanced Black Magic", showing
  the path of returning signal current and its impact
  on via inductance, are replicated in the web version
  of this newsletter. This note discusses the via-
  inductance formulas presented in that book and
  relates them to other formulas and concepts you may
  have seen.
  
  By the way, if you are enjoying my new book (or my
  previous title, "High-Speed Digital Design: A
  Handbook of Black Magic") please take a moment to
  visit www.barnesandnoble.com or www.amazon.com and
  enter some reader comments. That's one way you can
  help other engineers to better understand the
  relavance and importance (as you see it) of these
  texts.
  
______________________________________________________

VIA INDUCTANCE

Rylan Luke of Rymar Engineering writes:

  First, thanks both for your book, and the Mathcad
  files on your web site.
  
  My question is about the parasitic inductance of
  vias. Shouldn't the effective length of the via be
  considered? For a recent 6 layer board, 63 mils
  total thickness, I calculated the inductance for a
  signal via connecting the two outer layers. The
  inductance of this via was 1.2nH. The inductance of
  a similar structure going only between the top layer
  trace and the first ground plane was markedly less,
  presumably due to the much smaller length of that
  via.
  
  Is this a correct analysis?
  
______________________________________________________

Dr. Johnson replies:

  You are correct that the length of the via affects
  its inductance. Not only the length of the via, but
  also the shape and proximity of the return-current
  path determines the inductance.
  
  For example, let's work with a four-layer board
  having signal traces on layers 1 and 4 and solid
  reference planes on layers 2 and 3. Consider a
  signal via that dives through this board from top to
  bottom (1 to 4). Figure 1 illustrates this
  configuration, showing how at every point where high-
  frequency signal current flows, an equal and
  opposite current flows on the solid reference plane
  adjacent to the signal conductor.
  
  
  
   -sig trace-->-----via--                   layer 1
  
  --<--return------   ||  ------------------ layer 2
  
                      ||
  
  -----------------   ||  ---<--return------ layer 3
  
                    --via-----sig trace-->-  layer 4
  
  
  
  Figure 1--Near the via, returning signal current
  originating on layer 2 must somehow leap to layer 3
  ("High-Speed Signal Propagation", Fig. 5.33, p. 353)
  
  
  
  Prior to encountering the via, returning signal
  current associated with your layer-1 signal trace
  flows on the top surface of layer 2, right
  underneath the signal trace on layer 1. The skin
  depth of copper at high frequencies being much less
  than the thickness of a typical reference plane, the
  return current actually flows only on the top
  surface of solid layer 2; it does not penetrate
  through that plane.
  
  After traversing the signal via, your signal current
  appears on a trace located on layer 4. The return
  current at this point needs to flow on the bottom
  surface of layer 3 (the nearest surface to layer 4).
  How does the returning signal current get from layer
  2 to layer 3?  It can do so through vias IF THE
  PLANES CARRY THE SAME VOLTAGE, or through bypass
  capacitors if they do not.
  
  Doesn't the returning signal current just pop
  between the planes through the parasitic capacitance
  of the planes themselves, you might ask? A complete
  answer to that question will have to wait for the
  next newsletter. The short answer is that in typical
  cases where the planes are widely spaced (as in a
  stripline cavity) the current seeks out the nearest
  metallic connection, such as a via or bypass
  capacitor, and follows that path.
  
  I brought up the subject of returning signal
  currents because in order to calculate the
  inductance of a signal via you absolutely have to
  know where the returning signal current flows. This
  need arises because both the signal current and the
  returning signal current create magnetic fields.
  Inductance being nothing more than the temporary
  storage of energy in the form of a magnetic field,
  both fields must be calculated to determine the
  total inductance.
  
  Next let's get down to the business of calculating
  the inductance of a signal via given a known pattern
  of inter-plane connections.
  
  Assume one signal via and one inter-plane connection
  via, with the following parameters:
  
  s - separation between vias, center-to-center
  h - separation between planes 2 and 3 (it doesn't
  matter if there are other unused reference planes in
  the way, all that matters is the aggregate spacing
  between the outermost reference planes traversed by
  the signal)
  r - radius of the via holes
  
  Inductance is calculated for this configuration by
  assuming a current of 1 Amp flowing through the
  signal via and the same 1 Amp returning on the inter-
  plane connection. From this pattern of currents, one
  then calculates the magnetic field at all points
  inside the stripline cavity (between the planes).
  From this known magnetic field one then integrates,
  in the region between the vias, the total magnetic
  flux penetrating through a flat 2-D surface held
  vertically (like a curtain) and stretched from one
  via to the other, covering the entire stripline
  cavity from top to bottom. The total flux
  penetrating this 2-D surface equals the inductance.
  The procedure sounds laborious, I know, but it is
  how inductance calculations are done.
  
  Step 1: The magnetic lines of force due to the
  signal current form a set of concentric circles
  centered on the signal via. The field strength is
  constant in the z-axis direction (parallel to the
  via). The field falls off with 1/x as you move away
  from the via, with intensity
  
     B1(x)=u/(2*pi*x)
     
  where u=4*pi*1E-7 H/m is the magnetic permeability
  of free space, and x is the radial distance in
  meters away from the signal via.
  
  Step 2: The magnetic lines of force due to the
  return current have the same profile, but centered
  around the position s of the return-current
  connection.
  
     B2(x)=u/(2*pi*(x-s))
     
  Step 3: Integrate the magnetic field between the two
  vias. This step assumes that the signal and return
  vias are sufficiently far apart that the existence
  of each via does not noticeably perturb the magnetic
  field from the other.
  
  The symmetrical arrangement of the vias suggests
  that the signal and return vias will contribute
  equal amounts of flux -- therefore we can just
  calculate the flux from the signal via and double it
  to get the result.
  
  This is a two-dimensional integration. The first
  integration (dx) goes horizontally across the
  curtain between the two vias. The second integration
  works vertically from floor to ceiling (z=0 to z=h).
  Since the field is constant in the z dimension this
  second integration merely multiplies the first
  result by a factor of h.
  
     L1 = h * 2 * integral [from r to s-r] of B1(x)dx
     
  As the integrand in this expression involves 1/x,
  the integrated result involves logarithms. The 
  inductance represented by the combination of one
  signal via and one return via at distance s
  equals:
    
     L1 = [u/(2*pi)]*2*h*ln(s/r)
 
  In metric mks units (h, s, and r in meters, L in 
  Henries), the constant (u/(2*pi)) works out to 
  2E-07 H/m (assuming a non-magnetic dielectric).
  
  In English units, (h, s, and r in inches, L in nH) 
  the constant (u/(2*pi)) equals 5.08 nH/in.
  
  Other examples are worked similarly. 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 radius is r (with s, r and h in inches, L2 in nH):
  
     L2 = 5.08*h*(1.5*ln(s/r) + 0.5*ln(2)) nH    [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 radius
  is r (with s, r and h in inches, L4 in nH):
  
     L4 = 5.08*h*(1.25*ln(s/r) + 0.25*ln(2)) nH  [4]
     
  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 radius is r (with s, r and h in 
  inches, Lcoax in nH):
  
     Lcoax = 5.08*h*(ln(s/r))     nH          [coax]
     
  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.
  
  Formula [7.9], page 259 in High-Speed Digital
  Design, lists the inductance of a via (with h
  and diameter d=2r in inches, L? in nH):
  
     L? = 5.08*h*(ln(4*h/d)+1)     nH           [?a]
     
  This formula for L? is a gross approximation that
  glosses over the position of the returning current
  path, a simplification I greatly regret not making
  more clear in the book. It makes the crude
  assumption that the return path is approximately
  coaxial and located at a distance s=2*e*h, where e
  is the base used for natural logarithms. When the
  inductance really matters, a more accurate
  approximation is needed.
  
  To see how formula [7.9] was derived, substitute 2*r
  for the diameter d, and incorporate the additive
  factor of 1 inside the logarithm as a multiplicative
  factor of e:
  
     L? = 5.08*h*(ln(2*e*h/r))      nH          [?b]
     
  Comparing this to the formula [coax] shows the
  unwritten assumption present within formula [7.9],
  namely that the return path is coaxial and located
  at a distance of approximately s=2*e*h away from the
  signal via.
  
  If you are working with a 0.063-in. thick board with
  eight-to-ten layers, the outer two solid planes are
  probably about 0.050 in. apart. This height times 2e
  would be .2718 in.  If you have a bunch of return
  paths about this far away, that is the case
  approximation [7.9] was intended to cover.
  
  If you hold the position of the return paths
  constant and vary the inter-plane separation, the
  inductance changes linearly with inter-plane height
  (which equates roughly to via length).
  
  If you modulate the position of the return paths at
  the same time you vary the inter-plane height,
  always keeping s=2*e*h, you get formula [7.9].
  
  Note that all the approximations I have provided so
  far have the property that they give you the
  inductance of a via and its associated return path
  only covering that part of the via embedded between
  the planes. The part of the via that sticks up above
  the planes is not included. If you want to model
  that part (and I think that at today's speeds we are
  beginning to *need* that part), check out the models
  in my article: "Parasitic Inductance of Bypass
  Capacitors", available at www.signalintegrity.com
  under "archives".
  
______________________________________________________

MORE VIA INDUCTANCE

Herbert Seidenberg at Conexant writes:

  How do you reconcile that the inductance of a via is
  proportional to its length (page 259 of High-Speed
  Digital Design) and proportional to the square of
  its length in EDN, "Parasitic Inductance of Bypass
  Capacitors", July 20, 2000?
  

Dr. Johnson replies:

  Thanks for your interest in High-Speed Digital
  Design.
  
  That's the trouble with approximations, there are so
  many to choose from.
  
  What you are seeing are two crude approximations
  that apply to two different situations.
  
  In 3-dimensional space, as you move to a distance x
  away from a magnetic dipole, the fields spread out
  in an expanding sphere (whose surface area grows
  with x-squared) so you get a field intensity that
  falls off with 1/(x-squared).
  
  In a 2-dimensional problem (such as a signal via and
  a collection of return vias, with all the fields
  trapped between two big, solid reference planes),
  the fields spread out radially from the signal (and
  return) vias in expanding cylinders whose surface
  areas grow proportional to x, so you get a field
  intensity that falls off with 1/x.
  
  In the 2-dimensional problem (vias contained within
  two reference planes), if I increase the height h
  between the planes by a factor of k (essentially
  making the board k times thicker) then each new,
  longer via generates k times the magnetic flux *all
  of which* is captured between the planes. In this
  situation inductance grows strictly in proportion to
  height.
  
  In the 3-dimensional problem (here I'm thinking
  about the partial stubs of the vias that stick up
  above the topmost reference plane), the magnetic
  fields are not trapped between two planes. The
  fields are free to spill all over the place in a
  full 3-D pattern. If you measure the total flux that
  falls under a particular trace, or under the body of
  a bypass capacitor, you are seeing an inductance
  that depends only upon a fraction of the total flux
  generated by that via.
  
  In the 3-D case when you increase the length of the
  vias two things happen: first, the vias create more
  total magnetic flux in proportion to their new
  length, and second, the percentage of that flux that
  falls beneath the associated trace (or bypass
  capacitor) goes up. The overall effect is an
  increase in inductance proportional to the square of
  the length of the part of the via that sticks up
  above the planes (which was the point of the July
  20, 2000 article).
  
  In the 3-D problem (vias sticking up above the
  reference planes) if you increase the via length to
  a ridiculous extent the space between the vias will
  eventually grow to the point where it accepts as big
  a fraction of the flux as it is ever going to get.
  Beyond this point the "increased proportion of flux"
  part of my argument evaporates, and you revert to an
  increase in inductance that scales merely in
  proportion to length, not length squared.
  
  This transition from the 3-D behavior to the 2-D
  behavior corresponds to the "long, skinny object"
  assumption often made in inductance approximations.
  That is, for an object sufficiently long and skinny
  we can ignore the "fringing fields" at the ends
  (that's the 3-D part) and just concentrate on the
  part in the middle, for which the field cross-
  section is uniform along the length of the object.
  The "long, skinny" assumption renders many problems
  more tractable. A classic example is the calculation
  of inductance between two long parallel wires. One
  normally ignores fringing fields at the ends of the
  wires in this problem, which leads to the conclusion
  that the inductance grows in direct proportion to
  the length of the wires. The via problem
  investigated in the July 20, 2000 article was so
  short that it was "all fringe" and "no middle", so
  the inductance developed a squared dependence on
  length.
  
  Lastly, please beware that while the approximation
  on page 259 produces an answer with the right order
  of magnitude, it's not particularly accurate as it
  does not take into account details of the actual
  signal return path or the configuration of reference
  planes near the via.
______________________________________________________

  Join us at our upcoming seminar in Dallas, Mar. 24-
  25, 2003. A full schedule of cities and dates
  appears at: http://sigcon.com.
  
  High-Speed Signal Propagation: Advanced Black Magic
  is here! Check for it at www.barnesandnoble.com, or
  www.amazon.com , ISBN 013084408X.

  A brand new Advanced Signal Propagation course is being
  created, and will be available this fall. If you are 
  interested in attending, please send us an email for 
  more information. hsdd@xxxxxxxxxx

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