Ideal vs. lossy transmission line model question

[daku...@gmail.com]

Could some electronics guru please clarify
a few subtle questions regarding lumped
parameter model of transmission lines ?

The simple loss-less lumped parameter model
consists of an ideal capacitor-inductor pair
per unit length.

OTOH, the lossy model includes a series
resistance with the ideal inductor and an
dielectric conductance in parallel with
the ideal capacitor.

Similarly, the non-ideal capacitor model
includes a series resistance and a series
inductance with the ideal capacitor. The
non-ideal inductor model includes parasitic
capacitance and resistance values.

So, the question is: if one were to replace
the ideal capacitor/indcutor pair in the
lossless transmission line model with a
non-ideal capacitor/inductor pair, then
would this new model effectively re-create
the lossy transmission line model ? I have
done some SPICE modelling on this idea,
and the results look encouraging. What do
you gurus feel ?

Any hints/suggestions would be greatly
appreciated. Thanks in advance.

[RobertMacy]

Didn't understand a lot of what you said, except 'ideal' rarely represents
what I want.

Don't forget the resistance of the shield. It can have more effect than
you might think.

I like using a very small 'lumped' model segment used ovre and over to
form the whole length. You just have to watch that the smallest segment is
at least 10 times your highest frequency of interest to maintain a hope of
phase accuracy.

Although the resulting model has an upper frequency limit, you can use the
model to represent DC feed, include skin effects in BOTH the shield and
the conductor, etc etc. So you end up with a fairly good model that
represents the dispersion of your cable. if you add an external 377 ohm Zo
line, you can even start to explore radiation from your shield as you
actually use the model in an overall circuit. I've gotten such to get
close to the NEC predictions for single ended radiators.

[Allan Herriman]

You might like to read this s.e.d. thread from last century:
https://groups.google.com/d/msg/sci.electronics.design/uvkEtEJgDwM/etiIa79Rq5YJ

To answer your question: It wouldn't be a great wideband model, but
might meet your accuracy goals at low frequencies.

The ESL of the capacitor would cause the shunt impedance to rise
at frequencies above the SRF of the capacitor. Real cables don't do that.

Regards,
Allan

[John Larkin]

On Fri, 27 Feb 2015 19:54:30 -0800 (PST), daku...@gmail.com wrote:

>Could some electronics guru please clarify
>a few subtle questions regarding lumped
>parameter model of transmission lines ?
>
>The simple loss-less lumped parameter model
>consists of an ideal capacitor-inductor pair
>per unit length.

Yes. It is an OK model at frequencies well below the LC cutoff. Its
step response is very ringy, which is unrealistic.

>
>OTOH, the lossy model includes a series
>resistance with the ideal inductor and an
>dielectric conductance in parallel with
>the ideal capacitor.
>
>Similarly, the non-ideal capacitor model
>includes a series resistance and a series
>inductance with the ideal capacitor. The
>non-ideal inductor model includes parasitic
>capacitance and resistance values.
>
>So, the question is: if one were to replace
>the ideal capacitor/indcutor pair in the
>lossless transmission line model with a
>non-ideal capacitor/inductor pair, then
>would this new model effectively re-create
>the lossy transmission line model ? I have
>done some SPICE modelling on this idea,
>and the results look encouraging. What do
>you gurus feel ?
>

Heaviside derived the "telegraphers equation" around 1880. People who
didn't believe in inductance kept building telegraph systems,
including expensive undersea cables, that didn't work at the expected
speeds.

https://en.wikipedia.org/wiki/Telegrapher%27s_equations

He also invented the loading coil.

In most cases, the serious lossy element is the series resistance part
of the inductor. Shunt conductance and capacitor losses are usually
minor.

The bummer with the series loss is that it involves skin effect, and
the resistance increases with frequency. So a step response in, say a
coaxial cable, has a horrible long slow drool, clearly visible on an
oscilloscope.

PCB traces have a similar drool.

https://dl.dropboxusercontent.com/u/53724080/TDR/Chimera_TDR/TDT_risetime.JPG


I think we had a thread a while back on modeling a transmission line
that includes skin effect. You might search for that.

Bottom line, adding the series Rs to an L-C string makes it more like
a real-life lossy line, but doesn't replicate the skin effect.

I used to use ECA, a nice DOS, text-netlist simulator. It didn't have
a delay line part. I wrote a Basic program to generate the netlist for
LC delay lines. The bummer is that the number of LC sections goes up
as the square of the delay/risetime ratio.

(I also paid for Electronics Workbench, and their delay line model was
broken. Got my money back.)



--

John Larkin Highland Technology, Inc
picosecond timing laser drivers and controllers

jlarkin att highlandtechnology dott com
http://www.highlandtechnology.com

[John S]

IIRC, LTSpice has a lossy line model.

[Allan Herriman]

On Sat, 28 Feb 2015 09:13:28 -0800, John Larkin wrote:

[snip]

> In most cases, the serious lossy element is the series resistance part
> of the inductor. Shunt conductance and capacitor losses are usually
> minor.

At a high enough frequency with a practical (i.e. non-air) dielectric,
the shunt conductance will dominate the loss.

This matters e.g. when running 10Gb/s NRZ across an FR4 PCB.
I guess that doesn't meet your "In most cases" qualification :)

Regards,
Allan

[Allan Herriman]

If you're just interested in Spice simulation (rather than actually
building a lumped model out of physical components), perhaps this
series of articles by old s.e.d. poster Roy McCammon might help:

http://www.edn.com/electronics-blogs/anablog/4311804/Improved-Spice-model-of-a-transmission-line

I haven't tried it, but it would seem to avoid some of the pitfalls
of the usual lumped model.

Regards,
Allan

[John Larkin]

On 01 Mar 2015 00:16:55 GMT, Allan Herriman

I specifically mentioned telegraph lines and coax, both dominated by
copper loss. There's a reason why the phone company thinks that a
twisted pair is 600 ohms.

Microwave laminates have lower losses, but they also generally have
lower Er, so allow wider traces hence less copper loss. And the good
laminates are sold with rolled or otherwise fine-finish shiny copper,
not the black oxide crud usually used to stick copper to FR4. Copper
matters for fast signals on PCBs, too.

In something like a twisted pair or coax, copper skin loss is
dominant. That's plainly visible on an oscilloscope.

[RobertMacy]

On Sat, 28 Feb 2015 17:32:49 -0700, Allan Herriman
<allanh...@hotmail.com> wrote:

>> ....snip....

> If you're just interested in Spice simulation (rather than actually
> building a lumped model out of physical components), perhaps this
> series of articles by old s.e.d. poster Roy McCammon might help:
>
> http://www.edn.com/electronics-blogs/anablog/4311804/Improved-Spice-model-of-a-transmission-line
>
> I haven't tried it, but it would seem to avoid some of the pitfalls
> of the usual lumped model.
>
> Regards,
> Allan

Allan,

Thanks for posting that URL.

The article is circa 2011, so will post comment here instead: Roy mentions
non-noise sources of variations. But, NEVER mentions a true bane of all
cable manufacturers. Triboelectric effect. If the cable moves wind,
thermal flexing, whatever; the triboelectric effect will generate more
noise than one would think possible. The Manufacturers can reduce effect
depending on how 'tight' they can wrap that insulation around the
conductors and the material selection they use with teflon being VERY
energetic. Not sure, but expect to get worse with aging.

Security Industry purposely use this effect to make 'sensor' cables. I
once took a foot long piece such cable on the bench, put a scope probe on
the center and shield, tapped the cable in the middle with the handle of a
screw driver to watch more than 8Vpp appear on the scope trace! Now THAT's
energetic!

[RobertMacy]

On Sat, 28 Feb 2015 10:13:28 -0700, John Larkin
<jla...@highlandtechnology.com> wrote:

>> ...snip...

>Yes. It is an OK model at frequencies well below the LC cutoff. Its
> step response is very ringy, which is unrealistic.

Use a LOT of tiny lumped models in series and get decent bandwidth. uh,
only ringy *if* the input signals exceed your upper frequency range.

It's pretty easy to include skin effect with a lumped model WITHOUT
slowing down the analyses for either .ac or .tran

In other words, do not use laplace equations, which will take such to a
crawl.

[John Larkin]

In FR4 PC boards, one can see (with a fast TDR) what is almost
certainly the variation in dielectric constant caused by the fiberglas
weave.

[DecadentLinuxUserNumeroUno]

On Sun, 01 Mar 2015 09:20:05 -0800, John Larkin wrote:

> On Sun, 01 Mar 2015 03:39:23 -0700, RobertMacy <robert...@gmail.com>
> wrote:

snip

>>
>>The article is circa 2011, so will post comment here instead: Roy
>>mentions non-noise sources of variations. But, NEVER mentions a true
>>bane of all cable manufacturers. Triboelectric effect. If the cable
>>moves wind, thermal flexing, whatever; the triboelectric effect will
>>generate more noise than one would think possible. The Manufacturers can
>>reduce effect depending on how 'tight' they can wrap that insulation
>>around the conductors and the material selection they use with teflon
>>being VERY energetic. Not sure, but expect to get worse with aging.
>>
>>Security Industry purposely use this effect to make 'sensor' cables. I
>>once took a foot long piece such cable on the bench, put a scope probe
>>on the center and shield, tapped the cable in the middle with the handle
>>of a screw driver to watch more than 8Vpp appear on the scope trace! Now
>>THAT's energetic!
>
> In FR4 PC boards, one can see (with a fast TDR) what is almost certainly
> the variation in dielectric constant caused by the fiberglas weave.

Try G10.

[John Larkin]

FR4 is just fire-retardant G10. Both are fiberglass/epoxy laminates.
Neither is very well controlled as regards Er or very high frequency
behavior.

http://g10fr4.com/

[Jeff Liebermann]

On Sat, 28 Feb 2015 12:43:11 -0600, John S <Sop...@invalid.org>
wrote:

>IIRC, LTSpice has a lossy line model.

O Lossy Transmission Line
<http://ltwiki.org/index.php5?title=O_Lossy_Transmission_Line>

T Lossless Transmission Line
<http://ltwiki.org/index.php5?title=T_Lossless_Transmission_Line>

O-device (Lossy Transmission Line) and T-device (Lossless Transmission
Line) modeling issues
<http://ltwiki.org/?title=O-device_%28Lossy_Transmission_Line%29_and_T-device_%28Lossless_Transmission_Line%29_modelling_issues>

Modeling and Simulation of Nonlinear Transmission Lines by Frank
Crowne, Army Research Laboratory
<http://ltwiki.org/images/8/8d/Modeling_Transmission_Lines.pdf>


--
Jeff Liebermann je...@cruzio.com
150 Felker St #D http://www.LearnByDestroying.com
Santa Cruz CA 95060 http://802.11junk.com
Skype: JeffLiebermann AE6KS 831-336-2558

[RobertMacy]

On Sun, 01 Mar 2015 10:20:05 -0700, John Larkin
<jla...@highlandtechnology.com> wrote:

>> .....snip....

> In FR4 PC boards, one can see (with a fast TDR) what is almost
> certainly the variation in dielectric constant caused by the fiberglas
> weave.
>
>

didn't know that, but makes sense since the epoxy used has a different Er
than glass fibre.

[daku...@gmail.com]

Thanks for the reference. I had seen it a few
years ago, but did not pay too close attention
to it. This time, I examined it carefully, and
some peculiar issues crop up.

The basic scheme the author has followed is to
use the frequency plane(s = jw) expressions for
conductance, impedance, inductance, resistance
and propagation constant and then take the
inverse transform of these quantities in the
SPICE code. This is perfectly fine in theory,
because any complex expression may be split up
in a partial fractions expansion, and then the
inverse Laplace transform of each of these can
parts may be obtained.
The problem starts when one considers the s=plane
expressions for impedance and propagation
constant -- BOTH have square roots. And as far
as I could see from my trusty copy of Abramovich
and Stegun, BOTH forward and backward Laplace
transforms for expressions with positive
fractional exponents do not exist !!!
For example, the s-plane expression for the
frequency dependent resistance is:
R(w) = Rdc(1 + (w/Wr)^2))^0.25
So how is the LTSpice engine going to evaluate
the inverse transform for this expression ?
In my humble opinion, a frequency plane solution
with a simple C/C++ module with the input signal
frequency being increased incrementally would provide a lot better solution -- I await each of
your comments on this.

[RobertMacy]

On Fri, 06 Mar 2015 00:37:19 -0700, <daku...@gmail.com> wrote:

>> ....snip....

The last time I used a SPICE model with a 'frequency' term and tried to do
a really useful analysis, like try to observe the expected change to the
digital square wave, or obtain a useful set of 'eye patterns' for Error
Rate Detection values; the analysis went from step, step, ..step..,
....step....., to predicting it might not finish in my lifetime, Stopping
the analysis before anywhere nearly completed, it added insult to injury,
the results had wild variations of error. so...

I now make my OWN transmission line models using lumped models in small
enough sections the error at maximum frequency [as caused by minimum
risetime] PLUS, and this has been illuminating in understanding EMC
emanations off a cable, it is possible to include 'free space' and
actually estimate radiation from circuitry in a system that is not
properly done.

How to do Skin effect? I found that around 5 sets of elements, configured
like eddy current models, inductor parallel with resistor feeding parallel
inductor, etc can be made to pretty accurately 'curve fit' the resistance
vs frequency, and a few more terms will even yield fairly accurate 'phase'
shift from skin effects. [Note technique pretty accurately models those
lossy RF Beads, somewhere I have a set of models for commercially
available parts that are good to 1GHz, some beyond.] The advanatage of
keeping the model frequency independent is that the model can be used for
either .ac or .tran analyses. And, not take several days to RUN.

Now, applied to transission lines, the model has conductor inductance and
loss, return path inductance and loss [usually left out of lossy models],
capacitance between conductor and return path, dielectric loss, AND the
coupling [also left out of most models] which makes coax and twisted pair
so desirable to use. At least with such a model you KNOW what's inside it.

Also, you can really get to 'see' the dispersion in a cable. put in a step
and watch the value step then 'slide' up to where it's supposed to go.

The FREE PC Tools to create these models: femm 4.2; octave [Matlab clone];
and LTspice. [Of note, Mike Engelhardt, creator of LTspice, placed inside
LTspice an 'array' function, which Alex Bordodynov has used to create
incredible transmission line models. The array function makes it easy to
have a very simple schematic containing a LOT of sections, 250 to more
than 1000 sections with the schematic showing only a single little
transmission line symbol. And, again since the model has NO frequency term
it is easy to do either .ac or .tran analyses.

[daku...@gmail.com]

I am wholly in favor of the infinitesimal
lumped element model, but have some questions
about actual the values of these lumped capacitors/conductors/inductors.
Specifically, so far I have found that the
published values for resistance per unit
length, capacitance per unit length etc.,
use the unit of length as either kilo-foot
or kilometer. Resistance is directly
proportional to length, so the resistance
of e.g., a 0.5 centimeter unit length
transmission line can be easily computed.
But what about shunt conductance per unit
length and more importantly inductance per
unit length ?
In addition, each of these parameters have
frequency dependencies, but they can be
tackled.
Any hints/suggestions would be very helpful.

[RobertMacy]

On Sun, 08 Mar 2015 21:44:38 -0700, <daku...@gmail.com> wrote:

>> ....snip...

> I am wholly in favor of the infinitesimal
> lumped element model, but have some questions
> about actual the values of these lumped capacitors/conductors/inductors.
> Specifically, so far I have found that the
> published values for resistance per unit
> length, capacitance per unit length etc.,
> use the unit of length as either kilo-foot
> or kilometer. Resistance is directly
> proportional to length, so the resistance
> of e.g., a 0.5 centimeter unit length
> transmission line can be easily computed.
> But what about shunt conductance per unit
> length and more importantly inductance per
> unit length ?
> In addition, each of these parameters have
> frequency dependencies, but they can be
> tackled.
> Any hints/suggestions would be very helpful.

Just as resistance per unit length gets divided down, so does 'reactance'
per unit length. At a specific frequency, they are handled the same.
Reactance just has a j term multiplied times it where j is sqrt(-1). +j is
inductance and -j is capacitance, otherwise the reactance term is handled
identically. Then to make sense of THAT, reactance per unit length, most
people knowing the frequency, remove the frequency term, (actually radians
term, 2pif,) and refer to reactance as either inductance or capacitance.
Sadly, using terms like inductance and capacitance is misleading just
because it implies NO change with frequency. and as you've seen even
resistance is not the way to think, rather think in terms of 'loss' at a
frequency, so ALL the cables' terms ultimately are a function of frequency.

The better way to think is NOT R, L, and C but think in terms of Loss,
Inductive Reactance, and Capacitive Reactance AT each specific frequency
over the band of interest.

[daku...@gmail.com]

I totally agree with you - I had been emphasizing
the frequency dependency of each of the parameters
all along. However, each of the capacitance,
conductance, inductance and resistance per unit
length, require a DC value as a basis/starting point
for the analysis, and I had some doubts, which are
now a lot clearer.

[o pere o]

Some years ago we had reasonable success modelling lossy and dispersive
transmission lines. The base was modeling the two-port with some
controlled sources plus some equivalent lumped networks. One of them was
used to model the desired Zo(s)=sqrt((R+Ls)/(G+Cs)). The other, the
propagation function F(s)=exp(-theta(s)), with
theta(s)=l*sqrt((R+Ls)(G+Cs)) was modeled with an ideal transmission
line together with a shaping network.

The procedure to find the lumped equivalent used multipoint Padé
approximation. The usual Padé approximants try to fit the first terms of
the Taylor series at s=0. In our approach, we took a bilinear
transformation from s to z plane

s=so*(1-z)/(1+z) with so = sqrt(RG/LC)

and then made a multipoint Padé approximation, fitting some terms around
z=0, and fixing the first series term at z=1 and z=-1 (lastly this
corresponds to fitting to f=inf and f=0 and the intermediate frequency so).

This performed much better than the previous techniques without having
to do any kind of optimization, i.e. the procedure was explicit.

You may google the title "An explicit method for modeling lossy and
dispersive transmission lines"

Pere

Pere

[RobertMacy]

On Thu, 12 Mar 2015 02:36:45 -0700, o pere o <m...@somewhere.net> wrote:

>> ...snip....

>
> Some years ago we had reasonable success modelling lossy and dispersive
> transmission lines. The base was modeling the two-port with some
> controlled sources plus some equivalent lumped networks. One of them was
> used to model the desired Zo(s)=sqrt((R+Ls)/(G+Cs)). The other, the
> propagation function F(s)=exp(-theta(s)), with
> theta(s)=l*sqrt((R+Ls)(G+Cs)) was modeled with an ideal transmission
> line together with a shaping network.
>
> The procedure to find the lumped equivalent used multipoint Padé
> approximation. The usual Padé approximants try to fit the first terms of
> the Taylor series at s=0. In our approach, we took a bilinear
> transformation from s to z plane
>
> s=so*(1-z)/(1+z) with so = sqrt(RG/LC)
>
> and then made a multipoint Padé approximation, fitting some terms around
> z=0, and fixing the first series term at z=1 and z=-1 (lastly this
> corresponds to fitting to f=inf and f=0 and the intermediate frequency
> so).
>
> This performed much better than the previous techniques without having
> to do any kind of optimization, i.e. the procedure was explicit.
>
> You may google the title "An explicit method for modeling lossy and
> dispersive transmission lines"
>
> Pere

Fascinating approach!

As many of you know, google is no longer my friend in searches, in between
yielding high quantity of rubbish; it now deems me inappropriate to work
with and either bogs down, or worse, hangs our PC's! This translates into
the PC is completely locked up while doing a search so NOTHING else can be
done!!! arrrrrgggg! It often takes an hour and a half to get a 20kB paper!

Pere, is it possible for you to directly send me a copy of your paper to
the above email address?

Robert

[o pere o]

I have posted it here (in case anyone else is interested)
http://circuit.epsem.upc.edu/public/sed/an-explicit-method-for-modeling-lossy-and-dispersive-transmission-lines/at_download/file


An e-mail copy should also reach you...

Pere

[Phil Hobbs]

I've often made nearly minimax (equiripple error) approximations using
Chebyshev techniques. For a complicated function, you take a whole
bunch of samples at Chebyshev abscissae (i.e. you warp the X axis by the
derivative of the arcsin and then take equally spaced samples) and run
an FFT, which gives you the Chebyshev polynomial expansion of the
original function to whatever order you like.

Then you make that into a ratio of two Chebyshev series by using the
orthogonality relationship for Chebyshev polynomials, which gives a very
cute and simple recursion formula that's easy to automate. With an M/N
order rational function, you can match the Chebyshev coefficients up to
order M+N.

It's usually very close to the true minimax approximation, and no
iteration is needed.

This technique isn't at all original--I found it in a numerical analysis
book from the '70s--but it's easy to do and it works like the bomb.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

[RobertMacy]

On Thu, 12 Mar 2015 08:39:40 -0700, o pere o <m...@somewhere.net> wrote:

>> ...snip....

>> Robert
>
> I have posted it here (in case anyone else is interested)
> http://circuit.epsem.upc.edu/public/sed/an-explicit-method-for-modeling-lossy-and-dispersive-transmission-lines/at_download/file
> An e-mail copy should also reach you...
>
> Pere

Thanks, got it! and sent confirmation reply.

[daku...@gmail.com]

Thanks a lot for the link.