[SI-LIST] S-params vs. TDR/TDT results
- From: "Dmitriev-Zdorov, Vladimir" <vladimir_dmitriev-zdorov@xxxxxxxxxx>
- To: <si-list@xxxxxxxxxxxxx>
- Date: Mon, 17 Sep 2007 11:51:19 -0700
High Jeff,
> 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.
This is indeed true but only for 2-port loss-less model.
Your question (1) (about being able to restore timing relations in T11
and T22 since the modules of the S11 and S22 are same) is largely
similar to:
- Is it possible to restore complex-value S11 and S22 from |S11| or
|S22|?
Because, once we restored complex value functions, the timing responses
are defined uniquely.
The causal complex value functions can be restored uniquely if they are
'miminal phase'. That is, of the pole zero representation of such
functions implies that all zeros are located in the left half plane:
=20
S11 =3D [(s-sz1)(s-sz2)...]/[(s-sp1)(s-sp2)...]
Of course, poles must be on the left too, otherwise your model will not
be stable. Then, by using something like Kramer-Kronig relations, you
can restore phase from the magnitude (or, imaginary part from real).
Zeros are sz1, sz2 ... If you know that all real(szk) >=3D 0, k=3D1...N =
then
your dependence is minimal phase and can be uniquely restored from
|S11|.
If not, then you have some ambiguity.=20
We also know that non-minimal phase dependence can be represented as a
product of the minimal phase one and the purely phase rotating block:
S11 =3D [(s-sz1)(s-sz2)...]/[(s-sp1)(s-sp2)...] * [(s+szk)/(s-szk)]
Here, the first multiplier is minimal phase and the second multiplier
rotates phase while having magnitude =3D=3D 1.
If you have a representation of your dependence with finite number of
poles and residues, then theoretically you can fine all the combinations
when some of your zeros are swapped into right half of the complex
plane. The number of such combinations is finite but may be large. Of
course, the problem gets more complicated if you don't know the #
poles/zeros of if you may have multiple poles/zeros in the dependence.
Vladimir
Msg: #6 in digest
Subject: [SI-LIST] S-params vs. TDR/TDT results
Date: Sat, 15 Sep 2007 14:03:42 -0700
From: "Loyer, Jeff" <jeff.loyer@xxxxxxxxx>
..........=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.
Question (1): is there any constant (or predictable) relationship
between T11 & T22?
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