[SI-LIST] Re: S-params vs. TDR/TDT results
- From: steven.d.corey@xxxxxxxxxxxxx
- To: <si-list@xxxxxxxxxxxxx>
- Date: Mon, 17 Sep 2007 13:41:44 -0700
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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