If the frequency is high enough that the skin depth is much less than any
dimension of the conductor cross section and the frequency is low enough that
the conductor surface effects can be ignored, then the approximation
Rs=A*sqrt(f) holds and inductance is essentially constant with frequency.
Although I have had surprisingly good luck estimating the constant, A, using a
2-D field solver, I suspect that a field solver is the wrong tool for this. The
most obvious reason is the difficulty in discretizing the conductor with
sufficient resolution to get information on current distribution at high
frequencies. I believe that there is also a sensitivity problem here as well-
relatively large variations in the current distribution near the surface of a
conductor may have a small effect on the parameters that are actually used to
characterise the solution accuracy. Perhaps those more knowledgeable than I can
confirm or deny this.
There are transmission line analysis tools out there that are based on curve
fitting procedures. I believe that the mathcad program that appeared on the list
falls into this category. These tools are limited to certain very specific
geometries, but for these geometries they can give fairly accurate estimates of
attenuation due to Rs (and thus Rs) at high frequencies. Some implementations
also have corrections for finite conductor surface roughness.
The real difficulty occurs when you need to know the low frequency response of
the line. In this regime, Rs=sqrt(f) is a poor approximation and inductance is a
function of frequency. Often we do not care about this low frequency dispersion;
however, you need to get it right in order to transform the frequency domain
response into a reasonable time response for simulation. I believe that there is
no closed form solution for the low frequency regime in the general case. The
exact nature of the response in this region will be dependent on the geometry of
the line (again I welcome corrections from those more knowledgeable). If you
want to integrate to get a time response (or if you care about the frequency
domain response at low frequencies) you need to have the low frequency response.
This is a thorny problem because it is not obvious how you could generated this
response at arbitrary frequencies without a full geometric input and a field
solver (this has problems also).
If I understand correctly, HSPICE has implemented equations for Rs(f) (and
effectively L(f)) that are chosen to have asymptotic behaviour of Rs=A*sqrt(f)
and L=constant at high frequencies. In addition, these functions are chosen such
that they can be transformed into a real, causal time domain function. It seems
that high frequency loss will be accurately modelled by this approach provided
the constant A is chosen correctly. I do not think that there are enough degrees
of freedom to model the low frequency response for an arbitrary conductor
geometry. Is my understanding correct ? Does anyone have a feel of the
significance of this ? I have a hunch that accurate representation of the low
frequency regime is necessary for modelling long term edge settling effects.
Perhaps there are other implications as well (TL's for power systems come to
mind).
Regards,
Chris Albert
"Mellitz, Richard" <richard.mellitz@intel.com> on 08/03/99 12:57:09 PM
Please respond to si-list@silab.eng.sun.com
To: "'si-list@silab.eng.sun.com'" <si-list@silab.eng.sun.com>
cc: (bcc: Christopher Albert/Bos/Teradyne)
Subject: RE: [SI-LIST] : Proposal: Rs correlation/collaboration for W-Elem ents
Ed,
Measurements would be great! Any volunteers? Sounds like a good paper.
I'm not really disputing the accuracy of the W element model. I'm just
trying to figure out how to use them. Especially where I have thin lines and
in frequency ranges where dielectric losses may not be predominate.
I agree that the field solver is not the way to solve the accuracy issues.
However it is the starting point for the W element algorithm. So I really
want too very clearly nail down what I do with the field solver results. I'm
just starting with baby steps.
... Richard Mellitz,
Intel
-----Original Message-----
From: Dr. Edward P. Sayre [mailto:esayre@nesa.com]
Sent: Tuesday, August 03, 1999 10:15 AM
To: si-list@silab.eng.sun.com
Subject: Re: [SI-LIST] : Proposal: Rs
correlation/collaboration for W-Elements
Dear colleagues:
If we are going to check the accuracy of the .W model, I
wish to make the
suggestion that we use measurements as well for
verification. Most field
solvers do not account for losses in copper or dielectric to
as correct an
extent that I would deem acceptable. Does anybody have a set
of test board
that they would volunteer?
Dr. Dmitri Kuznetsov's comments in a recent si-list
communication see
(7/29/99) are very relevant. The behavior of the skin
effect and
dielectric formulas in the frequency domain, when
transformed back into the
time domain involve functions that have very special
properties.
The losses which are proportional to frequency, namely the
dielectric
losses can be shown to be related to the unit capacitance
and the loss
tangent of the material, Equation (8) for small losses.
See "OC-48/2.5 Gbps Interconnect Design Rules", Sayre, Chen
and Baxter;
DesignCon99 Proceedings. (Available on NESA's web site).
Use of equation
(8) together with the .W model has been found to be very
satisfactory when
the simulation results are compared to measurements.
Lastly, just as recently mentioned by Scott McMorrow, we too
have also seen
small differences with respect to the same problem solved by
two recent
versions of HSPICE. I do not think the way to resolve this
is through the
use of field solver predictions.
Sincerely,
ed sayre
However,
At 07:51 AM 8/2/99 -0700, you wrote:
>Apparently the W element model uses a pseudo-propagation
function with the
>following form.
>
>P(f)= exp{-sqrt[
(G0+f*Gd+j*2*pi*f*C)*(R0+sqrt(f)(1+j)Rs+j*2*pi*f*L) ]*len }
>
>(From HSPICE application note "Boosting Accuracy of W
Element
>for Transmission Lines with Nonzero Rs or Gd Values")
>
>Let's assume that this is valid for some conditions. It
would be nice to
>know what the assumptions are.(geometry, frequency, etc.)
We can talk about
>the validity of the above in another thread.
>
>I would like to make a proposal. I would like to know what
various field
>solvers report in regards to the above propagation
function. Let's start
>with a microstrip first (and only look at skin effect). The
geometry
>follows.
>
> Height over ground: 0.004"
> Width of conductor: 0.006"
> Thickness of conductor: 0.001"
>
> Conductivity: 0.58E8 mho/meter
>
>Let's all use the same units for Rs. Say:
> Ohms/(sqrt(Hz)*meter)
>
>Now, A colleague of mine has supplied a formula that is
used in microwave
>design. I have attached a PDF file with details. (Too tough
for text, TTFT
>:-)), I remember foobar)
>
>The answer, using the closed form formula for Rs is:
> 1.806E-03 ohms/(sqrt(Hz)*meter)
>
>If this is the magnitude of complex Rs, then Re(Rs) would
be
> 1.277E-03 ohms/(sqrt(Hz)*meter)
>
>I have received sidebar results from some of you folks, but
I don't want to
>post other people answers. However I will compile a table
of posted
>results. There are issues of complex number involved.
Remember I'm looking
>for the Rs for the above propagation formula.
>
>Step 2 will be to do same for a strip line geometry where:
>
> Height over ground: 0.005"
> Width of conductor: 0.0025"
> Thickness of conductor: 0.0005"
> Distance between ground planes: 0.0105
>
>
>It would be appreciated if we could find out what "tricks"
people are using
>to get Rs from their field solvers.
>
>Regards,
>Richard Mellitz
>Intel
>
> <<Mathcad - ms_loss_eq.pdf>>
>
>Attachment Converted: "C:\TEMP\attachments\Mathcad -
ms_loss_eq.pdf"
>
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+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+
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