[SI-LIST] Re: How to calculate the resistance and inductance of vias

• From: "Robert Friedman" <rpf1066@xxxxxxxxx>
• To: <Ken.Cantrell@xxxxxxxxxxx>, <Andrew.Ingraham@xxxxxx>,"si-list" <si-list@xxxxxxxxxxxxx>
• Date: Mon, 6 Jan 2003 12:17:06 -0600

```Zhangkun

The following train of thought about inductance is true and is from
publication number 9 on the Gigatest lab website - www.gigatest.com.  It
only takes a few minutes to register to get this for free.  If you ever get
the chance to here Mr. Eric Bogatin in a lecture it is well worth doing so.

In regards to what is inductance and capacitance in general, a good place to
go is first principles.  To get the true root meaning of Inductance and
Capacitance in time-varying waveforms (like your case of solving for partial
inductance in the section of your circuit that is a via) the ultimate source
is from Maxwell's equations in Electromagnetics.  The best Electromagnetic
books are the older ones that better describe first principles.  James Clerk
Maxwell used experimental evidence that can be summerized best in 9
experiments and these experiments make a great basis to understand Maxwell's
Equations.  His thesis is written explanation, not just the equations.

A book that actually derives inductance and capacitance from first
principles is "Foundations of Microwave Engineering" by R.E. Collins,
publisher McGraw-Hill.  Circuit theory, is a subset of Electromagnetic
theory.  The concepts of Inductance, Capacitance and Resistance can be seen
in Sections 2.2 and 2.5. This is the section covering constitutive relations
of how Electrical and Magnetic fields interact with materials (or each other
in a vacuum)
(Displacement Vector)D = e(permittivity * E(Applied Electrical Field +
P(Polarization)
u(permability)H(Magnetic intensity) = B(Magnetic Field) - u(permability) *
M(magnetic dipole)
The equation of motion is used (the "spring" equation in classical
mechanics) in decribing the electric and magnetic dipole effects.

Section 2.5 discusses Energy and Power that is stored in Electric and
Magnetic fields.  Consider an oscillating circuit where there is an exchange
of energy between the Electric Field and Magentic field.  This exchange has
damping effects - resistance causing loss, like a spring in viscous heavy
oil. The equations of motion used here have mechanical equivalents to
Inductance and Capacitance.  Inductance is related to the kinetic energy or
motion of a spring which is equivalent to the magnetic energy store in the
field.  This derivation isn't limited to electromagnetic forces in a
material, this works in a vaccum or free space and it will work with
Einsteins' relativistic equations of motion. Anytime charge moves there is
current and a magnetic field related to the time change of the cuurent flow.
Capacitance, the stored energy in an electric field is related to the
potential energy in the spring.  Capacitance and Inductance are useful
concepts in transmission line design, including waveguides that do not have
ANY metallic circuit return paths in the sense of a circuit (positive and
negative ternimals with a conducting wire between the terminals).  They have
time varying electric and magnetic fields and inductance and capacitance.
Microwave waveguide filters have structures with predictable inductive and
capacitive qualities that can be combined with filter qualities just like
filters with dicrete inductors and capacitors in a circuit.

With all that there is a rule when circuit theory is used in regards to the
wavelength of a field.  Circuit theory holds for dimensions much smaller
that the electromagnetic wavelength of the frequency of the signal.  So if
the highest frequency considered is 50 Mhz, the wavelength is 6 meters.  The
dimensions of a printed circuit board is typically smaller than a tenth of a
wavelength which is is 60 cm.  Since the field strengths across the via are
constant at any one time, one can use a quasi-static approximation with very
good accuracy.

My point is that the partial series inductance and shunt capacitances of a
via at 50 MHz is relatively small ,since a via is much smaller than 60 cm or
a tenth of a wavelength.  You need a capacitor with a lot more area between
plates, or an inductor with coils to be really significant compared to a via
in a printed circuit board at 50 MHz.  At 500 MHz vias are a more
significant concern.  If you are a hunt for the approximations, look for
electromagnetics or maybe Dr Richard Feynman's physics lectures to get an
INTUITIVE sense of electromagnetic fields.  Then the approximations you find
in books like "High Speed Digital design" by Howard W Johnson and Martin
Graham  or "Foundations for Microstrip Design" by Terry Edwards become more
acceptable and easier to understand.

Best Regard - Robert Friedman

----- Original Message -----
From: "Ken Cantrell" <Ken.Cantrell@xxxxxxxxxxx>
To: <Andrew.Ingraham@xxxxxx>; "si-list" <si-list@xxxxxxxxxxxxx>
Sent: Monday, January 06, 2003 10:15 AM
Subject: [SI-LIST] Re: How to calculate the resistance and inductance of
vias

>
> Zhangkun,
> As far as I know: Andy quote -"I believe there are no closed
> form solutions to all but the simplest of inductance problems.", is the
> correct answer.  The same goes for via capacitance, and more importantly,
> via impedance.  If anyone else on the list comes up with an analytical
> solution, I'd love to hear about it.
> Ken
>
> -----Original Message-----
> From: si-list-bounce@xxxxxxxxxxxxx
> [mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of Ingraham, Andrew
> Sent: Monday, January 06, 2003 8:41 AM
> To: si-list
> Subject: [SI-LIST] Re: How to calculate the resistance and inductance of
> vias
>
>
> Greg Edlund and Rich Peyton are 100% correct ... you must look at the
> whole current path as a loop ... include return paths in the inductance
> calculation.  These formulas are for the partial inductance.  The total
> loop inductance could be much greater or much less.
>
> As to where these formulas come from ... I believe there are no closed
> form solutions to all but the simplest of inductance problems.  Thus,
> almost all inductance formulas you see, are approximations, derived
>
> Any approximation has some region over which it fits the actual data
> reasonably well, and regions where it doesn't.  Sometimes pairs of
> formulas like this one, are meant to apply over two different ranges of
> values.  Sometimes one formula was an improvement over the other.  And
> occasionally a formula is just plain wrong, but it continues to be
> quoted in publications decades later.
>
> Off-hand, I know nothing about the origins of these two formulas.  They
> might be perfectly good approximations, given their limitations.  They
> are no good without knowing their limitations.
>
> Regards,
> Andy
>
>
>
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