[SI-LIST] Re: lumped model vs distributed model
- From: Steve Corey <steve@xxxxxxxxxxxxxx>
- To: si-list@xxxxxxxxxxxxx
- Date: Tue, 04 Dec 2001 11:00:29 -0800
Hello Abe -- I may not have explained myself clearly, so thanks for your
patience. I think there are two separate issues here:
1. Across what frequency range is a model with constant RLGC parameters
accurate?
2. How many segments are necessary for a lumped model to accurately
mimic its distributed counterpart up to a certain frequency?
In answer to #1, the model can only be accurate up to the frequency at
which frequency-dependent loss becomes important. At this point, the
model starts to diverge from the behavior of the actual device, and
therefore loses accuracy. So, although the model replicates the
modeling equations more precisely as the number of segments is
increased, the accuracy is not necessarily improved, since the
underlying modeling assumption of frequency-independent RLGC is
inaccurate at high frequencies. The model accuracy ends up being
limited by the assumptions as much as by the segmentation scheme, so in
the frequency-independent case, additional segmentation past a certain
point doesn't help, and in some situations can even reduce the accuracy.
In answer to #2, if the RLGC parameters are constant with frequency, the
analysis you have provided is correct for computing the minimum number
of segments. I don't own Hall's book, but I would presume that the
analysis was done assuming constant RLGC parameters. A model with
frequency-dependent RLGC parameters, however, dissipates additional
signal energy at higher frequencies. As a result, certain segment
length restrictions get relaxed as compared to the frequency-independent
case, which is the limiting case. However, although fewer segments may
be required, it is still generally necessary to use multiple segments,
as you have correctly pointed out. (And as I alluded earlier, optimal
segmentation schemes become more complex, and are a subject of current
research.)
Finally, to make sure I answer your question directly:
Frequency-independent RLGC parameters are only valid as a low-frequency
approximation. Your analysis is valid for the frequency-independent
case, but overly restrictive regarding segment length when
high-frequency effects such as skin effect and dielectric losses are
incorporated.
Hopefully this was more clear...
-- Steve
-------------------------------------------
Steven D. Corey, Ph.D.
Time Domain Analysis Systems, Inc.
"The Interconnect Modeling Company."
http://www.tdasystems.com
email: steve@xxxxxxxxxxxxxx
phone: (503) 246-2272
fax: (503) 246-2282
-------------------------------------------
Abe Riazi wrote:
> Steve:
>
> Indeed, theoretical transmission line derivations and modeling frequently
> employ transverse electromagnetic (TEM) assumption. Fortunately, TEM
> mode is regarded sufficient estimate for transmission line propagation
> up to high frequencies.
>
> I find it difficult to accept that a transmission line model consisting of
> a large number of cascaded RLCG segments can be a good representation
> only at low frequencies. The following relationship adapted from
> S. H. Hall, et al., Page 16 and discussed in my previous post leads to
> different conclusions :
>
> Segments >= 10 * x / (Tr * v)
>
> The required number of RLCG segments varies directly with transmission line
> length and inversely with Tr . Therefore, modeling a long fast edge rate
> (wide
> bandwidth) line can demand multiple RLCG stages; whereas, a single RLCG
> lump model appears suitable only for a line exhibiting short electrical
> length.
>
> A question:
>
> Do you think above conclusions are valid for low frequency lossless (ideal)
> case, and break down for high frequency lossy (real) lines?
>
> Thank you.
>
> Abe
>
>
> ----- Original Message -----
> From: "Steve Corey" <steve@xxxxxxxxxxxxxx>
> To: <si-list@xxxxxxxxxxxxx>
> Sent: Sunday, December 02, 2001 8:11 PM
> Subject: [SI-LIST] Re: lumped model vs distributed model
>
>
>
>>Abe -- In my opinion and experience, a model with an infinite number of
>>infinitesimal segments (implying infinite transmission bandwidth) is
>>only accurate at low frequencies, since real interconnects do not behave
>>this way. Such models are based on TEM assumptions and no skin effect,
>>and they also depend on a non-physical profile of loss tangent vs.
>>frequency that varies as 1/f.
>>
>>As Ray Anderson mentioned in an earlier post, lumped element models can
>>be used to represent frequency-dependent losses such as dielectric and
>>skin-effect losses. Since a lossy lumped model itself needs to
>>attenuate high frequencies, it's not subject to such restrictive
>>constraints on segment length as those you have laid out for the
>>limiting, or frequency-independent case. However, determining how to
>>optimally segment such a model is still an area of active research.
>>
>> -- Steve
>>
>>-------------------------------------------
>>Steven D. Corey, Ph.D.
>>Time Domain Analysis Systems, Inc.
>>"The Interconnect Modeling Company."
>>http://www.tdasystems.com
>>
>>email: steve@xxxxxxxxxxxxxx
>>phone: (503) 246-2272
>>fax: (503) 246-2282
>>-------------------------------------------
>>
>>
>>Abe Riazi wrote:
>>
>>
>>>Dear All:
>>>
>>>Ideally, an infinite (though impractical) number of cascaded RLCG
>>>
> segments
>
>>>are required to construct an accurate distributed transmission line
>>>
> model.
>
>>>Stephen H. Hall, et al., "High-Speed Digital System Design A handbook
>>>of Interconnect Theory and Design Practives", on page 16, describe a
>>>
> useful
>
>>> formula for determining the number of RLCG segments sufficient for
>>>distributed modeling :
>>>
>>> Segments >= 10 * x / (Tr * v)
>>>
>>>Where Tr, x, and v represent signal rise (fall) time, transmission
>>>
> line
>
>>>length, and velocity repectively.
>>>
>>>For example, when Tr = 500 ps, x = 6 Inch (= 15.24 cm) and
>>>v = c / SQRT(Er) = 3E10/2.06 = 1.456E10 cm/sec
>>> (a substrate dielectric constant of 4.25 being assumed)
>>>
>>>Then:
>>>
>>>Minimum number of segments = 10 * 15.24 cm / ( 500E-12 sec * 1.456E10
>>>cm/sec).
>>>
>>>Minimum number ~ 21 segments.
>>>
>>>When the cross sectional geomtery (and hence charateristic impednace
>>>
> Zo )
>
>>>of the stripline (or mictrostrip) transmission line are also known,
>>>then R, L, C and G for each segment can be ascertained by dividing the
>>>total value of each parameter by the number of segments.
>>>(Example: C_segment = C_total / number of segments). These
>>>
> calculations
>
>>>are of course simpler for the lossless case where R and G are
>>>
> negligible.
>
>>>Best Regards,
>>>
>>>Abe Riazi
>>>ServerWorks
>>>
>
>
>
>
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