Ideal vs. lossy transmission line model question

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.

You might like to read this s.e.d. thread from last century:

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

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

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.

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.

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.)

IIRC, LTSpice has a lossy line model.

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

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:

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

Regards, Allan

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.

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!

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.

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.

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

O Lossy Transmission Line

T Lossless Transmission Line

O-device (Lossy Transmission Line) and T-device (Lossless Transmission Line) modeling issues

Modeling and Simulation of Nonlinear Transmission Lines by Frank Crowne, Army Research Laboratory

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

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.

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.

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.

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.