Signal Speed Below the Speed of Light for Circuit Abstraction

[halleff]

TL;DR Summary

I'm aware of the requirement for the lumped circuit abstraction that the wavelength of the highest frequency signal component be much larger than the circuit; I'm confused about the limitation that the speed of the signal waves themselves be less than the speed of light

I've read a couple of other topics on the Physics Forums about the lumped circuit abstraction requiring that signal timescales (i.e. the period of the highest frequency signal component) be much longer than the propagation delay of the signals though the circuit and that the wavelength should be much longer than the largest dimension of the circuit. This makes sense to me since in order for the voltages and currents to be the same at every point in an element at a given time you need the signal to propagate essentially instantaneously and have the wavelength be much larger than both the circuit and the elements.

However, in the book Foundations of Analog and Digital Circuits by Agarwal and Lang there is this additional comment: "If we are interested in signal speeds that are comparable to the speed of electromagnetic waves, then the lumped matter discipline is violated, and therefore we cannot use the lumped circuit abstraction" (emphasis mine).

The author (Prof. Anant Agarwal) of this book also states in this MIT lecture (timestamped to relevant quote) that an assumption for the lumped circuit model to hold is that all signal speeds of interest are much slower than the speed of light.

This is confusing to me since while I understand the need for the wavelength to be much larger than the elements and the circuit, I don't see why the propagation speed of the signal itself couldn't be close to the speed of light. I might also be getting confused since my understanding is that all EM waves of any frequency propagate at the same speed (through the same material) their propagation rate should be an issue separate from their frequency and wavelength.

For example, suppose that you have a circuit with largest dimension $d$. The material of the conductors in this circuit is such that the speed of propagation of electromagnetic waves is the speed of light $c$ (or very close to it). You know that the wavelength $\lambda$ of the signal should be greater than $d$, so pick a signal frequency which will result in the signal having $\lambda = 10d$, as an example.

Then,

$$
\lambda f = v \to f = v/ \lambda \to f = \frac{c}{10d}
$$

Then if for example $d$ is 1 cm, $f$ will be approximately 3 GHz. So now the signal period is about 333 picoseconds and the propagation delay is about 33 picoseconds. This seems to fulfill the requirements for the lumped circuit model, but apparently I'm missing something.

So my question is, why is there a requirement that the speed of signal propagation be less than the speed of light if you could have the signal travel at the speed of light and fulfill the requirement that the wavelength is much larger than the circuit dimensions?

 

[essenmein]

So my question is, why is there a requirement that the speed of signal propagation be less than the speed of light if you could have the signal travel at the speed of light and fulfill the requirement that the wavelength is much larger than the circuit dimensions?

Its not a requirement, it is an outcome!

Speed of light changes depending on the medium its traveling through.

Eg Cherenkov radiation (https://en.wikipedia.org/wiki/Cherenkov_radiation)

Now when an electric signal travels down a wire, the dielectric, ie capacitance is what slows it down.

In RF world this is know as velocity factor: https://en.wikipedia.org/wiki/Velocity_factor

 

[berkeman]

The author (Prof. Anant Agarwal) of this book also states in this MIT lecture (timestamped to relevant quote) that an assumption for the lumped circuit model to hold is that all signal speeds of interest are much slower than the speed of light.

I only listened to a little bit of the lecture right at the timestamp, and initially he is saying that the "signal transitions" are finite speed (like the low-to-high transition of a logic signal), which is fine. Then he does allude to the propagation speed of the signals, which doesn't make sense to me.

My advice right now would be to ignore this statement and just keep going. Maybe it comes up in some specialized circuit situation, but I'm not able to think of any at the moment.

 

[essenmein]

The propagation speed of an electrical signal down a wire is slower than the speed of light in vacuum, how much slower depends on the dielectric, high quality transmission/ladder lines can get ~99% c, average cables can be as low as 40% c.

This impacts the wavelength criteria for determining when a transmission line analysis is needed as well as other things like prop delays etc. Reducing the speed by nearly half, also reduces the wavelength by the same proportion, ie if you use c, you might think you do not need a TL, but add the velocity factor, all of a sudden that number is different.

Velocity factor is important and should not be ignored in high speed system design.

 

[halleff]

This impacts the wavelength criteria for determining when a transmission line analysis is needed as well as other things like prop delays etc. Reducing the speed by nearly half, also reduces the wavelength by the same proportion, ie if you use c, you might think you do not need a TL, but add the velocity factor, all of a sudden that number is different.

I think I get this, but this would suggest that getting signals closer to the speed of light is actually beneficial, which doesn't seem to be what they're saying in the book and lecture.

I only listened to a little bit of the lecture right at the timestamp, and initially he is saying that the "signal transitions" are finite speed (like the low-to-high transition of a logic signal), which is fine. Then he does allude to the propagation speed of the signals, which doesn't make sense to me.

It's very confusing for me since I haven't found any source saying the same thing and no one else seems to have commented about this specific thing from the book/lecture.

 

[essenmein]

I think I get this, but this would suggest that getting signals closer to the speed of light is actually beneficial, which doesn't seem to be what they're saying in the book and lecture.

I think you might have it backwards, the fact that signals are slower than the speed of light is a property of a conductor. This is neither intended or desired, it is a negative effect that must be accounted for.

Then the lumped model assumes that along a single conductor you will not experience different voltages, ie with an AC signal all the wires move the same way. Since ac signals are really EM waves (which we conveniently ignore in the lumped model), as the size of the structure gets close to the wavelength, you will see different voltages on a wire. At this point the lumped model no longer works, you have to analyse it as a transmission line. So a signal slowing less than c is a side effect of insulators being present, this side effect is wavelengths get shorter, so depending on how much slowing you experience, you will have to analyse as a transmission line sooner.

Edit: Yes, as speed goes up, it is beneficial to have velocity factors approaching 100%, less prop delay, less insertion loss etc

 

[alan123hk]

Then if for example ddd is 1 cm, fff will be approximately 3 GHz. So now the signal period is about 333 picoseconds and the propagation delay is about 33 picoseconds. This seems to fulfill the requirements for the lumped circuit model, but apparently I'm missing something.

I don't think you are missing something.
You may consider to a have more rigorous restriction, such as d=λ/50 instead of d=λ/10, but basically your reasoning and calculations are correct anyway.

So my question is, why is there a requirement that the speed of signal propagation be less than the speed of light if you could have the signal travel at the speed of light and fulfill the requirement that the wavelength is much larger than the circuit dimensions?

I believe that it may not mean to require the speed of signal propagation be less than the speed of light. In fact, nothing is faster than the speed of light.

"If we are interested in signal speeds that are comparable to the speed of electromagnetic waves, then the lumped matter discipline is violated"
"an assumption for the lumped circuit model to hold is that all signal speeds of interest are much slower than the speed of light "

I have been contemplating the above statement, I think it seems not easy to understand.

Anyway, I believe this is just another way of expression, and its meaning is similar to the requirement that the wavelength is much larger than the circuit size

Since we do not consider the propagation delay of the lumped elements R, C, L and the connecting line in the lumped circuit model, the signal timescales of interest should be much larger than the maximum propagation delay of the electromagnetic wave in the lumped circuit.

For example, if we consider a pulse width T_w, then in order to obtain a good approximation, the T_w should be much larger than the maximum propagation delay of the electromagnetic wave in the lumped circuit.

However, the above is only a comparison of timescales, not a comparison of speeds. Regarding the so-called "signal speed of interest", if it represents the signal speed of voltage change per second at a fixed position, then we cannot directly compare it with the speed of light in meters/second. I do not rule out that the signal speed can be defined in some way as being related to the size of the circuit and the timescale of the signal, so that it can be processed and compared in this case.

 

[Svein]

Rule of thumb: In electronics, the signal propagation speed is about 2c/3 - which means about 20cm/ns (or 8inches/ns).

 

[alan123hk]

The lumped circuit model has three constraints.

https://en.wikipedia.org/wiki/Lumped-element_model

Regarding the third constraint, I think that "The signal timescales must be much larger than the propagation delay of electromagnetic waves across the lumped circuit" is easier to understand than "the signal speed must be significantly lower than the speed of electromagnetic waves passing through the circuit".

The so-called "signal speed" seems a bit ambiguous and difficult to understand. So far, I cannot find a clear mathematical definition on the Internet.

Therefore, I tried to make a bold assumption that whether it can be defined as $$S = \frac {Lm} {Ts}$$, where
S = Signal Speed
Lm = largest dimension of the lumped circuit
Ts = signal timescale of interest.

For the largest dimension of the lumped circuit Lm and the signal timescale of interest Ts, in order to obtain good approximation, the signal speed S should be much lower than the speed of electromagnetic wave, namely $$S =\frac {Lm} {Ts} << C$$
Let's consider the same example, for a pulse width Tw, obviously the signal timescale of interest should be Tw, then Ts=Tw.

Assuming that Lm=10cm, Tw=10ns, then the signal speed S is equal to 0.1×10⁸ m/s, which is much lower than C= 3×10⁸ m/s, and S/C =0.033, so in this case the lumped circuit model should provide a quite good approximation.

In fact, this is the same thing as comparing the propagation delay of the electromagnetic wave across the lumped circuit to the signal timescale of interest on the lumped circuit, since

$$ \frac S C = \frac {Lm/Ts} C = \frac {Lm/C} {Ts} $$
where
$$ \frac S C = \frac { \text {Signal Speed}} {\text {speed of electromagnetic waves through the circuit}} $$
$$\frac {Lm/C} {Ts} = \frac {\text {propagation delay of electromagnetic waves through the circuit}} {\text {signal timescale of interest }} $$
Both of the two should be as small as possible.

I hope all the above information will not cause confusion or give big wrong reasoning.

Besides, please note that the speed of the electromagnetic wave passing through the circuit may be only about $\frac 2 3 C$.