[SI-LIST] Re: Mixed-mode impedance matrix
- From: "Yuriy Shlepnev" <shlepnev@xxxxxxxxxxxxx>
- To: <xlzhou@xxxxxxxxx>, <si-list@xxxxxxxxxxxxx>
- Date: Fri, 30 Jun 2006 07:23:35 -0700
Mick,
All matrix transformations to get Zmm are isomorphic as long as Z is
non-singular (that is always true for non-ideal lossy structures).
Thus, it does not matter how you get it. S->Z->Zmm is fine if you start with
S. Smm->Zmm is even better if you have to start with Smm. Zmm obtained from
Smm with the Cayley transform has to be denormalized to relate regular
voltages and currents in the differential-common mode basis. It can be done
multiplying it with two diagonal matrices D from left and right D*Zmm*D, D
is the denormalization matrix with 1/sqrt(0.5*Zo) on diagonal for the
common-mode ports, and 1/sqrt(2*Zo) on diagonal for the differential-mode
ports. Your common-differential impedance is going to be multiplied simply
by 1/sqrt(Zo). It is assumed that all original ports are normalized to the
same impedance Zo.
What are the benefits of Zmm? Comparing to S-parameters, It does not contain
numerical normalization error caused by the Cayley transform. This is
obviously applicable only if you compute Zmm directly and not through Smm.
Poles and zeros are more visible with Z. In addition, some tools do not use
S-parameters internally and convert them into Y or Z - it is better to use Y
or Z directly in that case. Something else?
Yuriy
Yuriy Shlepnev
Simberian Inc.
shlepnev@xxxxxxxxxxxxx
-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx] On
Behalf Of Zhou, Xingling (Mick)
Sent: Friday, June 30, 2006 6:30 AM
To: shlepnev@xxxxxxxxxxxxx; si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: Mixed-mode impedance matrix
Yuriy,
Thanks for your clear explanation in the first part. I think it is a smart
way to do the math, S->Z->Zmm.
In the second part, Zmm includes non-pure diff/comm Zmmdc etc, what is the
reference impedance I should use? Maybe I just need to do some homework.
After all this math, the next question is: what is the benefit of Zmm?
As a different representation of the network, or a tool, it is fine.
However, when we look at it, it seems not so easy to judge a design based on
Zmm. Smm is much easier as we know. Or Smm is enough.
Regards,=20
Mick
-----Original Message-----
From: Yuriy Shlepnev [mailto:shlepnev@xxxxxxxxxxxxx]=20
Sent: Thursday, June 29, 2006 10:45 PM
To: Zhou, Xingling (Mick); si-list@xxxxxxxxxxxxx
Subject: RE: [SI-LIST] Mixed-mode impedance matrix
Hi Mick,
Conversion of terminal Z-parameters to the generalized mixed-mode form is
quite straightforward. It looks like that:=20 Zm=3DTv*Z*Tv", where Z is the
impedance matrix, Tv is the voltage transformation matrix and " is
transposition sign.
Here mixed-mode voltage Vm and current Im vectors are defined trough the
terminal voltage V and current I vectors as follows Vm=3DTv*V; =
Im=3DTi*I; where Tv and Ti are voltage and current transformation matrices.
Every row of matrix Tv contains either 1 and -1 for the differential mode or
0.5 and 0.5 for the common mode. Every row of matrix Ti contains either
0.5 and -0.5 for the differential mode or 1 and 1 for the common. All other
elements are zero. Matrices Tv and Ti related as Tv=3D(Ti^-1)".=20 Backward
transformation from Zm to Z is Z=3D(Ti")*Zm*Ti
Simple example. For a two-port structure elements of 2 by 2 matrix Tv can be
defined as Tv11=3D1.0; Tv12=3D-1.0; Tv21=3DTv22=3D0.5 = Corresponding
elements of the matrix Ti are Ti11=3D0.5; Ti12=3D-0.5; Ti21=3DTi22=3D1.0 =
This is just simple statement of the fact that differential voltage Vd and
current Id and common voltage Vc and current Ic are defined as = Vd=3DV1-V2;
Id=3D0.5*(I1-I2) Vc=3D0.5*(V1+V2); Ic=3DI1+I2 Elements of the mixed mode
matrix of two-port are
Zm11=3DZ11-Z21-Z12+Z22
Zm22=3D0.25*(Z11+Z21+Z12+Z22)
Zm12=3D0.5*(Z11-Z21+Z12-Z22)
Zm21=3D0.5*(Z11+Z21-Z12-Z22)
Mixed mode matrix defined in that way is exactly what you get transforming
mixed-mode S-parameters directly to Z with the Cayley transform:=20
Zm=3D(I+Sm)*(I-Sm)^-1, where Sm is the mixed-mode S-matrix, I is the
identity matrix. The impedance matrix Zm derived from Sm is normalized.
Assuming identical normalization impedance for all ports Zo, common mode
ports are going to be normalized to 0.5*Zo and differential to 2*Zo. It is
easy to derive a generalized expression for the case with non-identical
normalization.
I hope it helps in this scriptic form,
Yuriy=20
Yuriy Shlepnev
Simberian Inc
shlepnev@xxxxxxxxxxxxx=20
-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
On Behalf Of Zhou, Xingling (Mick)
Sent: Thursday, June 29, 2006 9:09 AM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Mixed-mode impedance matrix
Hi,gurus,
I was asked to simulate differential Z. It was obviously motivated by mixed
mode S parameter theory. So strictly speaking, I need to convert mixed-mode
S to Z. I remember somebody asked the same question last year.
As we know, the mixed-mode S parameter theory is well-establised
(generalized by Andrea Ferrero in 2005). However, I don't know any mixed
mode Z.=3D20
1. Some engineers do the so-called pure mode conversion, either diff or
common modes. If so, what do we know about the c-d/d-c mode conversion?
Something is missing here? Of course, mathematically you can do it.
2. Is it possible or meaningful to define the mixed-mode Z? We need a
reference impedance to do any S-Z converison, what is it for the d-c part?
What is the latest development in theory other than some personal practices?
I mean we need a solid foundation to do it. Any literature? I think if the
conversion can be defined, we must convert the whole S into Z, not piece by
piece (so-called pure mode conversion). Otherwise the information is
incomplete?
Thanks,
Mick
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=20
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