[SI-LIST] Transmission Line Causality
- From: "Dmitriev-Zdorov, Vladimir" <vladimir_dmitriev-zdorov@xxxxxxxxxx>
- To: <si-list@xxxxxxxxxxxxx>
- Date: Wed, 25 Apr 2007 09:22:00 -0700
Hi Sam,
The issue you are rising is very important and I believe it is not
receiving enough attention. BTW, very similar problem exists for
S-parameters data and this of course converge in the fact that very
often we use S-parameters to describe same objects we represent as
transmission (T-) lines.
I know two quick tests for T-lines causality.
1. Chopped line. You choose a certain type of lossy line and describe a
long segment of it (e.g. 1 meter with typical PCB PUL matrices). Perform
transient simulation by finding response to a step or square pulse.
Then, represent the same line as a series connection of N pieces each
1/N meter long, with same parameters. Perform similar analysis, compare
the results. Ideally, they should be identical. Try the case of large
dielectric loss.
2. Comparison between transient and AC analysis. Apply sine input to the
lossy line, simulate long enough to see the magnitude settled. Repeat
this for different frequencies of the sine input. Then, run AC analysis
and compare the response magnitudes you see in AC to those you measured
in transient. The test may bring surprises especially for lines with
considerable dielectric losses (frequency dependent) and high enough
frequencies. Of course, when simulating in time domain, make sure the
step is small enough not to bring in LTE.
More general observation.
Model non-causality is a violation of cause-effect relations. Due to
sequential manner of time domain analysis algorithms, we can never see
that the model responds prior to receiving the input. However, this does
not mean that it corresponds to a causal representation in frequency
domain. In most cases, the core of lossy T-line model is its frequency
response (not voltage or current but characteristic admittance and
delay-less propagation operator). The models differ in how they (a) form
those frequency-dependent matrices and (b) how they convert those matrix
operators into time domain responses when performing transient analysis.
Hence, we first need to make sure that those internal frequency domain
responses are causal. Typically, we can output them (or others, closely
related to them) from AC analysis. After you get the responses in
frequency domain, they can be investigated for causality. Any causal
response is a complex function of frequency where real and imaginary
parts obey Kronig Kramers relations (derived from Hilbert transforms).
These relations are not easy to check though. One practical way to do so
is e.g. applying Vector Fitting, or transforming them into touchstone
files and applying ELDO simulation algorithms. The key point is that
causal dependences can be fitted VERY accurately, non-causal - cannot.
Of course, you can also right your own IFFT procedure to find the time
domain response and see if it has something at t<0. However, make sure
that your frequency response is not 'truncated' otherwise this IFFT will
always give you nonzero output for negative time.
> I notice one thing that the G matrix is exactly linear > > > with
frequency and C is almost constant which is beacuse permitvitty is > not
following frequency dependence and Hilbert realtions. What is the proper
way to correct the data without changing other information in result.
Agree. Any matrix-value function (like L,C, etc.) must be causal in
order to form causal PUL Z or Y matrices. If G is linear (real part!)
then there should also be defined some behavior for imaginary part, not
just left zero, otherwise the model is non-causal. The common error is
that they provide only real part that describes the changing with
frequency inductance or capacitance or conductance and forget about
imaginary part.
A good transmission line algorithm must be able to analyze causality of
the tabulated matrix data and if necessary, restore causality, e.g. by
restoring missing imaginary parts.
Vladimir
Msg: #7 in digest
Date: Tue, 24 Apr 2007 21:26:06 -0400 (EDT)
From: Sam Sam <si.rules@xxxxxxxx>
Subject: [SI-LIST] Transmission Line Causality
Hi,
=20
=20
I would like to raise a discussion on testing causality of a frequency
dependent lossy transmission line model. I have two concerns. One is
regarding visually checking causality and the other mathematically
checking the self-consistency of matrix data. Visually testing causality
means applying an input pulse, you get a non-zero signal before the
minimum propagation time of the line. But what is the minimum
propagation time. Is it d/c where d is length of line and c equal to
speed of light/sqrt(Er). I also found that there is another way of
visually testing using pulse and impulse response.=20
=20
=20
1. Apply a unit step function to input of the line.
2 Record the time required for the output pulse to reach 10% of the
final value compared to the reference time determined by the time of the
launched step -this time is the minimum propagation time
3. Measure the magnitude of the impulse response at the minimum
propagation time calculated. If the value at this time or at any
preceding time exceeds 5% of the final value (unity) it is not causal.
=20
But I really dont see how this method works. When i use the same edge
rate for pulse and impulse responses, they seem to rise at same time. Or
may be i am not seeing something here.=20
=20
Secondly, to mathematically verify all i am doing is using hilbert
transforms. Reconstructing the imaginary part of Z from the real part of
Z where Z =3D R+j*w*L.And then trying to match the original L data =
versus
freq and the reconstructed L datawith freq. The error was more than 6%
and hence i concluded non-causal whcih is also verified by my visual
test 1 ( i dnt understand visual test2). However when i used the
reconstructed L data to perform my simulation, the result is crappy. it
still isnt causal, i am puzzled...
=20
What is the efficient way to enforce causality on RLGC data obtained
from field solvers. I notice one thing that the G matrix is exactly
linear with frequency and C is almost constant which is beacuse
permitvitty is not following frequency dependence and Hilbert realtions.
What is the proper way to correct the data without changing other
information in result.
=20
=20
Sam
=20
------------------------------------------------------------------
To unsubscribe from si-list:
si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
or to administer your membership from a web page, go to:
//www.freelists.org/webpage/si-list
For help:
si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field
List technical documents are available at:
http://www.si-list.net
List archives are viewable at:
//www.freelists.org/archives/si-list
or at our remote archives:
http://groups.yahoo.com/group/si-list/messages
Old (prior to June 6, 2001) list archives are viewable at:
http://www.qsl.net/wb6tpu
Other related posts:
