Thanks Ray for the results from ApsimRLGC. I have attempted to explain below why
the results at 1MHz seem to be at variance from the values at other frequencies.
It was also mentioned that at 1 MHz the skin depth is more than 2 mils and one
should not see much skin effect in a 1 mil thick conductor. I have attempted to
clarify this as well. Let me first reproduce Ray's results with more
information.
valuated @ Rs RS (ohms/m) RS+RDC
(ohms/m)
----------- -------------------------------- -----------
---------------
@1MHz 1.4874E-3 Ohms/(sqrt(Hz)*meter) 1.4874
6.8340
@100MHz 1.0933E-3 Ohms/(sqrt(Hz)*meter) 10.933 16.280
@1GHz 1.2272E-3 Ohms/(sqrt(Hz)*meter) 38.807 44.154
@5GHz 1.3207E-3 Ohms/(sqrt(Hz)*meter) 93.387 98.734
The value of Rs depends on the current distribution across the cross-section of
the conductor. This in turn depends on
1. The variation across the width of the conductor. The width here is 6 mils and
even at 1 MHz, there will be variation across the width.
2. The variation across the height of the conductor.
3. The current distribution across the width and height of the ground plane. The
ground plane can contribute as much as 25% of the total resistance and this also
needs to be accounted for. At low frequencies the current flows all across the
ground plane. At high frequencies, it flows directly below the trace or
conductor. A frequency of 1 MHz is in the transition region.
A recent paper in IEE Transactions on Microwave Theroy and Techniques (July
1999, p. 979) titled "Internal Impedance of Conductors of Rectangular
Cross-Section" by G. Antonini, A. Orlandi, and C. R. Paul has excellent figures
of current distribution at different frequencies for a microstrip. It also makes
the point that sqrt (f) variation at high frequencies is valid only for circular
conductors. Also, the high frequency internal inductive reactance is not equal
to the high frequency resistance for rectangular conductors.
Coming back to my original point, the reason the value of Rs
(ohms/(sqrt(Hz)*meter) at 1 MHz seems high can be attributed to the following:
1. The sqrt (f) variation is not valid for rectangular conductors and is not
valid at frequencies where skin effect is not well established (as at 1 MHz for
this example).
2. In the range 100 KHz - 10 MHz, a very big contributor to the variation in
RTOTAL (i.e the derivative of RTOTAL with respect to frequency) is the reduction
of the width of the current flow in the ground plane. This again, is not
necessarily a sqrt (f) variation.
3. ApsimRLGC calculates the total R (RTOTAL) and then subtracts out the DC R
(RDC) to get Rs. At 1 MHz, RTOTAL is close to RDC and this can lead to greater
errors when Rs is separated out and divided by frequency.
On a related issue, Dimitri has rightly pointed out that one cannot arbitrarily
assign a value to Rs without suitably changing the inductance. Otherwise some
fundamental principles such as causality may be violated. ApsimRLGC was
originally designed to work with ApsimSPICE and its output is compatible with
ApsimSPICE. ApsimSPICE avoids the problem of violating these principles by using
physical models for its transmission line simulation.
I think this email is long enough, espeically considering the increased traffic
in the SI reflector, and I would be happy to supply more information on any of
this offline to anybody who may be interested
--************************************************************************ Raj Raghuram Applied Simulation Technology 1641 N. First Street, Suite 170 San Jose, CA-95112. Tel: (408) 436-9070 ext.101 Fax: (408) 436-9078 e-mail: raghu@apsimtech.com ************************************************************************
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