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halleff

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- 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

The author (Prof. Anant Agarwal) of this book also states in this MIT lecture (timestamped to relevant quote) that

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.

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?**