RE: [SI-LIST] : Oscillation in lumped circuits and

Vadim Heyfitch (vadim.heyfitch@intel.com)
Tue, 16 Feb 1999 12:39:07 -0800

Arani,

those phenomena are two disparate animals. They can be easily confused
because they both include SQRT(L*C).

Consider a ladder model of lossless transmission line. A resonance
frequency for each step of the ladder is

w=1/SQRT(L*dx*C*dx)=1/(dx*sqrt(L*C)),

where w is in rad/sec, L and C are inductance and capacitance per unit
length and dx is the length represented by this step of the ladder.
(This
frequency incidentally coincides with the cutoff frequency of the
low-pass
filter which this ladder step is. Here I will not go into how one
should
choose the number of such ladder steps per unit length depending on the
desired bandwidth of the tline model. This is a separate issue.)

In this formula, the resonance frequency is inversely proportional to
the
dx ( as well as to the Tpd=sqrt(L*C).) The choice of dx is yours. How
long
does it take for one step of this LC ladder to discharge itself by
charging up the adjacent step?? (i.e. to transmit the signal down the
line
to the next segment of this chain). It takes about t~1/w=dx*sqrt(L*C).
The
capacitor there discharges into the next inductor, which in turn
discharges into the next cap, and so on. If dx=1 unit length, then
t=sqrt(L*C)=Tpd. If dx=0.5, i.e. you have two ladder steps per unit
length, then t=0.5*Tpd is the delay through each step. Follow the
reasoning here?

Reflections on the other hand, have to do with impedance mismatch. On a
perfectly terminated line there are none. However, that same (perfectly
terminated) line can be modeled with a ladder, each step of which has a
resonance frequency as described above.

The condition of perfect termination has to do with the resonant
frequency
of an LC ladder step.

The last in the chain LC step can discharge into the terminating
resistor
in time R*C*dx. While this is happening, it is supplied with more charge
from its LC neighbor from the other (than Rterm) side. It takes
t~1/w=dx*sqrt(L*C). If you equate these two times,

R*C*dx=sqrt(L*C)*dx,

and solve this for R, you get R=sqrt(L/C). Recognize it? This tell us
that
Rterm has to be equal the line impedance Z in order to have perfect
termination. We arrived to this relation by starting with a resonance
frequency of an elementary step of the ladder model. (Truthfully, I did
not know of this derivation when I began to write in response to your
post.)

On a line with mismatched termination, the last LC step cannot discharge
into Rterm (on its left)at the same rate it's charged up by its LC
neighbor (on the right). That creates a reflected wave. That's how
reflections are related to LC resonance of a ladder tline model. They
are
not directly related, as you can see, to the time a reflected wave
propagates to the other end.

Vadim Heyfitch
Intel Corp.


> -----Original Message-----
> From: Arani Sinha [SMTP:sinha@poisson.usc.edu]
> Sent: Friday, February 05, 1999 12:02 AM
> To: si-list@silab.Eng.Sun.COM
> Cc: sinha@poisson.usc.edu
> Subject: [SI-LIST] : Oscillation in lumped circuits and transmission
> lines
>
> 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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