[SI-LIST] Re: Even mode, common mode, and mode conversion

From: Steve Corey <steve@xxxxxxxxxxxxxx>
To: si-list@xxxxxxxxxxxxx
Date: Fri, 01 Mar 2002 10:39:08 -0800
Mary -- you're right, "signal" can be a bit ambiguous.  Usually it's 
just some system variable or a weighted sum of system variables which 
are of application-specific interest.  Terminal voltages are seemingly 
the most common signals in digital design. I prefer to think about 
things in terms of excitations and responses, since the response covers 
all the system variables, whether voltage and current, E and H, or 
whatever else, and we are then free to call any or all of them our "signal".

I have a feeling we may be hung up on the word "independent", since I 
think there may be as many as three uses floating around.  One sense is 
the independence of excitation response pairs, where the response to the 
sum of two excitations is the sum of the individual responses.  The 
other is independence of variables -- whether they are coupled, whether 
changing one causes another to change.  The third is whether a set of 
quantities are linearly independent of each other, i.e., whether it is 
possible to form one as a weighted sum of the others.  Looking at each 
one separately:

In a linear system, all excitation response pairs propagate 
independently, as if no others existed, which allows us to use 
superposition and many other tools, as I'm sure we agree.  This is why 
we can solve separately for the different modal responses and add them 
together afterward to determine the aggregate result.

In a coupled transmission line system, we would expect that for the most 
part, placing a voltage on one line would cause a voltage to appear on 
another line as well, so clearly the individual terminal variables are 
dependent on one another.

In general transmission line analysis, each of the individual modal 
variables (let's say voltages) is linearly independent of each of the 
others IF each has its own propagation constant.  This means that in the 
case of homogeneous media and no conductor losses, because the modes 
share the same propagation constant, the modal voltages are linearly 
dependent on one another, and one can be computed by scaling another. 
As soon as you introduce conductor loss or inhomogeneous media, one mode 
can't be computed linearly from another.

To summarize, before we lose too many readers to the snooze factor:

If media is homogeneous and there are no conductor losses, you can 
choose to define your modes in any number of ways -- there's not a 
unique set of eigenmodes.  (Mathematically, there's not a unique set of 
eigenvectors.)  How you choose to define them will affect the modal 
impedances, but each mode will still have a single propagation constant, 
making analysis easy. You have stated this in an earlier post, and as we 
discussed at the time, it only applies to this particular case.

If the media is inhomogeneous and/or there are conductor losses, you 
have only one good option for how you define your modes -- the 
eigenmodes.  You are at the mercy of the system, since you can't do 
anything to change its eigenvectors unless you change the system itself. 
  If you choose the eigenmodes of the system, each mode will propagate 
(on all conductors) with a single propagation constant, likely 
simplifying analysis and intuitive understanding.  If you choose just 
any set of "modes", each one will have up to N propagation constants, so 
you may as well just analyze the system without any modal decomposition 
at all.

   -- Steve

Mary wrote:

> Steve,
> 
> Thanks for the explanation. Everything you said seems reasonable,
> but I'm still confused. The discussion so far has been about
> even and odd mode "signals". By "signal" do we mean the signal
> voltage, the signal current or something else? Are you saying
> that the even and odd modes of the "signal" will propagate
> independently while the signal voltages and currents will 
> not propagate independently?
> 
> Mary
> 
> -----Original Message-----
> From: si-list-bounce@xxxxxxxxxxxxx
> [mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of Steve Corey
> Sent: Thursday, February 28, 2002 9:41 AM
> To: si-list@xxxxxxxxxxxxx
> Subject: [SI-LIST] Re: Even mode, common mode, and mode conversion
> 
> 
> 
> Mary -- thanks for bringing me back to the earlier question.  Because 
> each mode has its own characteristic impedance, each has a different 
> relationship between its voltage and current waves.  As a result, the 
> responses to the even and odd modes do not add up to zero on the 
> quiescent conductor, and this is where the crosstalk comes from. 
> Because modal analysis takes all the coupling into account, it properly 
> predicts crosstalk.  And of course, if there's negligible coupling, the 
> modal impedances are effectively the same, and the modal 
> responses essentially cancel out for zero crosstalk.
> 
> I also want to emphasize as a more general point that any set of 
> inputs/responses, eigenmodes or not, have to propagate independently 
> from one another, due to linearity. Otherwise the system is nonlinear 
> and superposition does not hold, and all of our jobs just got a lot harder.
> 
> Finally, I was looking over my previous post (I suppose I should have 
> done that before sending it out...) and I spotted a couple of errors 
> which I would like to correct.  First, as I stated above, in the 
> presence of coupling, each mode has its own unique characteristic 
> impedance.  Second, if the dielectric is homogeneous -- even if it is 
> lossy -- (as long as the conductors are lossless) then every mode shares 
> the same propagation constant.  Inhomogeneity and/or lossy conductors 
> cause splitting of the modal propagation constants.  Sorry if I created 
> any confusion there.
> 
>    -- Steve
> 
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-- 
-------------------------------------------
Steven D. Corey, Ph.D.
Time Domain Analysis Systems, Inc.
"The Interconnect Modeling Company."
http://www.tdasystems.com

email: steve@xxxxxxxxxxxxxx
phone: (503) 246-2272
fax:   (503) 246-2282
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[SI-LIST] Re: Even mode, common mode, and mode conversion

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