[SI-LIST] Re: S-params vs. TDR/TDT results
- From: "Yuriy Shlepnev" <shlepnev@xxxxxxxxxxxxx>
- To: <steven.d.corey@xxxxxxxxxxxxx>, <si-list@xxxxxxxxxxxxx>
- Date: Mon, 17 Sep 2007 15:50:17 -0700
Hi Steven,
Just a brief summary of S-parameters properties for a lossless
(non-dissipative) 2-port system:
|S11|=|S22|, |S12|=|S21|, phase(S11)+phase(S22)=phase(S21)+phase(S12)+-PI
If a non-dissipative 2-port is also reciprocal then phase(S21)=phase(S12),
that additionally reduces the number of independent real variables to just
3.
Similar set of restrictions can be derived for a non-dissipative and
reciprocal N-port.
Best regards,
Yuriy
Yuriy Shlepnev
Simberian Inc.
www.simberian.com
-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx] On
Behalf Of steven.d.corey@xxxxxxxxxxxxx
Sent: Monday, September 17, 2007 1:42 PM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: S-params vs. TDR/TDT results
Hi Jeff -- strictly speaking, "linear, passive" aren't sufficient to
guarantee S12 =3D=3D S21. The condition is formally "reciprocity", and = it
stands on its own, independent of the other properties. The standard
academic counterexample is the gyrator, a passive, linear device that is
non-reciprocal. Practically speaking, for the materials and geometries
we use in our interconnects, we expect them to be reciprocal.
Regarding transformations between the frequency and time domains, you point
out rightly that despite the fact that two FD magnitudes may be identical,
we don't have the full time-domain picture without phase information. If we
pay an inordinate amount of attention to one at the expense of the other, we
can set ourselves up for trouble. Often people concentrate on the magnitude
of S21 of their channels, but this is usually under the implicit assumption
that the phase is generally linear with frequency. Given a short channel,
for example, this assumption may not hold.
However, once we have both magnitude and phase, there is one-to-one
correspondence between the frequency-domain and time-domain representation
of any reasonably-behaved system. So you will never (as an engineer, that
is) find S12 and S21 equal without the corresponding pulse and step
responses also being equal. Mathematicians can find counterexamples, but
they aren't of practical interest in signal integrity.
The lossless two-port is an interesting system, as you point out, since once
we know the magnitude of one element, we know the magnitudes of all.
However, as you also point out, the phase situation is not so clear. You
may be able to come up with a reasonable relationship between the phases of
S11 and S22 by combining passivity, reciprocity, and causality. I would
expect a pretty involved mathematical effort, though, and I wouldn't expect
it to result in a purely algebraic expression. If you know one diagonal and
one off-diagonal, then you can come up with the other diagonal
algebraically.
-- Steve
------------------------------------------------------------------------
-
Steven D. Corey, Ph.D.
Principal Engineer
Tektronix - Enabling Innovation
http://www.tdasystems.com
http://www.tektronix.com
email: steven.corey@xxxxxxxxxxxxx
phone: (503) 627-6816
fax: (503) 627-2260
------------------------------------------------------------------------
-
=20
>-----Original Message-----
>From: si-list-bounce@xxxxxxxxxxxxx=20
>[mailto:si-list-bounce@xxxxxxxxxxxxx] On Behalf Of Loyer, Jeff
>Sent: Saturday, September 15, 2007 2:04 PM
>To: si-list@xxxxxxxxxxxxx
>Subject: [SI-LIST] S-params vs. TDR/TDT results
>
>Try again - different format to get rid of the "3D" characters (I hope)
>=20
>
>=20
>
>Zhenggang's question regarding "s-parameters looking the same while TDR
>shows differences" got me thinking...
>
>=20
>
>NOTE: please delete most of this message if you respond (certainly
>everything below the first horizontal line) - lots of=20 extraneous
>bits to be sending around.
>
>=20
>
>I'll use Sxy to describe s-params from port y to x, Txy for TDR/TDT
>results from port y to x, and Pxy for the corresponding pulse=20
>responses.
>
>=20
>
>Take an asymmetric system:
>
>P1 -> high_Z0_T-line -> med_Z0_T-line -> low_Z0_T-line -> P2 (lengths
>arbitrary, but probably different)
>
>=20
>
>Clearly T11 !=3D (does not equal) T22=20
>
>=20
>
>But, for both the lossy and loss-less cases, S21 =3D=3D S12 (for =
linear,
>passive networks). Similarly, T21 =3D=3D T12.
>
>=20
>
>For the lossless case, |S11|^2 + |S21|^2 =3D 1 (Hall's book, page 305;
>=
I
>think he's got a typo when citing S12, though it's a moot point)
>
>=20
>
>A little algebra (with S21 =3D=3D S12) shows that, for the lossless =
case,
>|S11|^2 =3D=3D |S22|^2.
>
>=20
>
>A little logic says that |S11| =3D |S22|
>
>=20
>
>Point (1): It's very interesting (non-intuitive) to me that |S11| =3D
>|S22| for any loss-less, linear, passive system, since T11 !=3D T22. I
>believe this is the point that Zhenggang was focused on, since we
>typically focus on only the S-parameter magnitudes.
>
>=20
>
>Question (1): is there any constant (or predictable) relationship
>between T11 & T22?
>
>=20
>
>Point (2): This implies that the only difference you'll see in VNA
>simulations of a loss-less system is in the phase of S11, S22, even
>though TDR looks markedly different.
>
>=20
>
>For a lossy system, |S11|^2 + |S22|^2 =3D Ps, where Ps is less=20 than
>1, and represents the portion of total Power dissipated in the system
>(conductor, dielectric loss for T-lines). I know that Ps is=20 going
>to be a function of frequency (losses are frequency-dependent), but=20
>I wondered if |S11| would also equal |S22| for a lossy system (since
>S21 =3D=3D =
S12).
>
>=20
>
>Question (2): does |S11| =3D=3D |S22| for a lossy, linear, passive =
system?
>
>=20
>
>I also wondered if, since S21 =3D=3D S12 for this system, what about =
the
>corresponding pulse responses (P21 and P12). Would they also be
>identical? It seemed like they had to be if S21 =3D=3D S12, but I =
wouldn't
>bet my life on it... What about the loss-less case?
>
>=20
>
>Question (3): does P21 =3D=3D P12 for a loss-less, linear, passive =
system?
>
>Question (4): does P21 =3D=3D P12 for a lossy, linear, passive system?
>
>=20
>
>For potential bragging rights, please take a minute to answer all the
>questions above for yourself before going on...
>
>=20
>
>So, I put together some simulations to measure TDR/TDT, VNA, and pulse
>responses (200ps, representing 5Gt/s system) of the loss-less and lossy
>case. The Hspice decks are included below, if you want to try them
>(absolutely no guarantees whatsoever). They show that |S11| =3D=3D =
|S22|
>for the loss-less case (the phases are very different).
>
>=20
>
>The answers I found were:
>
>Question (2): does |S11| =3D=3D |S22| for a lossy, linear, passive =
system?
>
>No, not for the lossy case. Interestingly, I think this implies that
>the ratio of power dissipated by port 1 to that dissipated by=20 the
>system is changing, depending on the circuit topology. In retrospect,
>that makes sense since, if you saw a huge discontinuity first, lots=20
>of energy would be reflected into (and absorbed by) the load. For a
>small discontinuity (say a very lossy 50 ohm T-line), lots of energy
>might be absorbed by that first T-line segment before any got reflected
>to be absorbed by port 1.
>
>=20
>
>Question (3): does P21 =3D=3D P12 (Pulse responses) for a=20 loss-less,
>linear, passive system?
>
>Yes.
>
>=20
>
>Question (4): does P21 =3D=3D P12 for a lossy, linear, passive system?
>
>Yes.
>
>=20
>
>I don't have an answer for Question (1). Anybody else?
>
>=20
>
>Some interesting stuff that was new to me, and thought I'd share. I
>welcome your thoughts (or let me know if you see that somehow I've
>erred), or if you know of interesting articles that have already
>explored this.
>
>=20
>
>I think it has interesting implications when we have to simulate both
>Northbound and Southbound directions of a topology. Is there=20 any
>point, if the pulse responses are the same? Maybe not, if the
>driver/receiver models are the same (or am I missing something else?).
>I'm not sure what adding crosstalk will do to this scenario - I suspect
>it will have a significant impact. And packages are often different
>for the two directions, Tx vs. Rx. But, if they weren't (running some
>generic simulations...)?
>
>=20
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