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

Rajkumar C ([email protected])
Wed, 17 Feb 1999 18:20:21 +0530

A Very good explanation. Thanks a lot.

Regards,
Rajkumar,
Texas Instruments.

> 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.
>
> Intel Corp.
>
>
> > -----Original Message-----
> > From: Arani Sinha [SMTP:[email protected]]
> > Sent: Friday, February 05, 1999 12:02 AM
> > To: [email protected]
> > Cc: [email protected]
> > 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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