# Re: [SI-LIST] : Oscillation in lumped circuits and transmission lines

Steve Corey ([email protected])
Fri, 05 Feb 1999 09:05:19 -0800

This is an interesting question, if you have a stomach for semantics... In my opinion, although the two oscillations to which you refer are distinct, they are definitely related. System/circuit/network theory people use
"oscillation" to refer to an underdamped sinusoidal oscillation, whereas signal integrity designers use the term to refer to any nonmonotonic signal transition. In fact, the oscillation seen on an unmatched distributed
transmission line excited by a step is a sum of an infinite number of sinusoidal oscillations.

An earlier reply alluded to the fact that the time constant of an LC circuit is sqrt(LC), which is true for the single LC lump. In fact, this value will coincide with the delay of the transmission line. However, when such a
circuit is divided into n lumped segments, there will now be n modes of oscillation, each at its own frequency. (caveat -- a related point -- modes of oscillation of a network are distinct from, but related to, modes of
transmission of a T-line system.) As you allow n to approach infinity, the sum of the sinusoidal oscillations converges into a distributed transmission line response, in the spirit of Fourier theory.

In the awkward in-between stage, for n less than infinity, you get non-physical ringing, the same way you get it from a truncated fourier series, and in JPEG compression, for that matter, because the higher order modes of
oscillation are not present. However, this brings up the key point as it applies to modeling interconnects with lumped elements: in some cases the system is excited with a low enough frequency input that the higher order modes
are not excited anyway. The nonphysical ringing is not seen, and the distributed and lumped models behave the same, so either model is valid. Returning to the compressed image analogy, if the compression-induced errors are too
small to resolve with your eye, then as far as you're concerned there are no errors.

Hope this helps,

-- Steve

-------------------------------------------
Steven D. Corey, Ph.D.
Time Domain Analysis Systems, Inc.
"The Interconnect Modeling Company."
http://www.tdasystems.com

email: [email protected]
phone/fax: (206) 527-1849
-------------------------------------------

Arani Sinha wrote:

> Hi,
>
> I have the following question.
>
> We can model an interconnect as either a lumped circuit or a
> transmission line. By means of lumped modeling, we can say that
> it has an oscillatory response if its damping factor is less
> than 1. By means of transmission line modeling, we can say that
> it has an oscillatory response if the signal reflection
> co-efficients at source and load satisfy certain conditions.
>
> My question is whether oscillation in a lumped circuit and
> signal reflection in a transmission line are actually the same
> phenomenon. If so, there should be a correlation between
> conditions for oscillation in a lumped circuit and those for
> oscillation in a transmission line.
>
> After many discussions and much thought, I have not been able
> to determine a correlation. I am also ambivalent about whether
> they are the same phenomenon.
>
> I understand that the damping factor in a lumped circuit is
> equivalent to the attenuation constant in a transmission line
> and that condition of no reflection is equivalent to the
> maximum power transfer theorem.
>
> I will really appreciate help in this regard.
>
> Thanks,
>
> Arani
>
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