Lossless Transmission Line Length vs Rise Time

Question

I am trying to answer the question - Does a signal's rise time increase with the length of a lossless transmission line? - using LTspice simulation. Some references say that lossless transmission line does not degrade the signal. Does it mean the rise time remains the same? Intuitively, it seems to me that the rise time should increase as the length increases.

I tried two cases. Case-1 is with LTspice T-Line model and Case-2 is with distributed LC model. In each case, I simulated with four cascaded T-Line models, each having 250ps delay, to give a total of 1ns delay. In Case-2, each "T_100" component has 100 LC segments (L = 0.125nH and C = 0.05pF). Rise time was set to 10ps.

Case-1 results in signal being delayed by 250ps per T-line with no appreciable change in rise time. While Case-2 results in significant increase in rise time along with 250ps delay per T-line.

I can't seem to figure out which one is correct. Attached are the LTspice schematics I used.

Answer 1

They are both correct.

If you model the transmission line as ideal, having infinite bandwidth, then a step will go in and come out with the same risetime.

If you model the transmission line as having a finite bandwidth (by using an LC model for it, aka a Low Pass Filter), then a step (which has an infinite frequency spectrum) will lose the high frequencies and look more rounded as it emerges. The more filter stages you pass it through, the more you attenuate the high frequencies, and the more rounded it will appear.

Run a frequency response for your cascaded transmission lines, and your cascaded low pass filters, and compare them.

Answer 2

What you're describing is a known, old problem. You are using the ideal translmission line (tline), which should output no overshoot, no matter what, yet they clearly exist. This has nothing to do with the tline, but with the way LTspice, and SPICE, in general, work: the timestep is variable and it is halvened, or doubled, depending on the speed of the simulation, which is dictated by the rates of change (the derivatives) -- if it's smooth, the simulatino flies, if not, it has to reduce the timestep to calculate the sharp(er) variations. If these are too sharp (i.e. discontinuities), the solver may not be able to halven the timestep anymore and it will throw "timestep too small" errors.

In this case, the circuit is simple, with only one sharp rise, which propagates. This means that the solver would normally "fly" through the simulation, but it has to break around the edges. And here, the default, compressed waveform display comes into play: it is by default, compressed to only 300 points, and thus it has to connect the dots somehow. This, coupled with the relative speed of the simple circuit, causes some points to be miscalculated, and the errors propagate. If you'd prolonge the chain of transmission lines, you'd see more and more distortion.

The solution: force a timestep. Simply add .tran 0 2.5n 0 1p. This forces LTspice to no longer fly through the rest of the circuit more than the imposed timestep, which causes tighter tolerances, thus more accurate results.

It's always good to remember that, no matter the simulator, they are all limited by the machine precision and, usually, they use float unless forced to double. In addition, the tolerances (abstol, reltol, etc) are even more relaxed, in order to not force you to wait eons for simulating a simple transmission line propagation test.