I regard the use of Fourier analysis and description as simply an elegant form of accountancy. Sinewaves are not an essential component of any signal; there are many other basis functions that have good book keeping properties for different purposes. Best to think of the signals as simply functions of time (but don't get stuck with "what is time?"!). Then the question of cutting off some components does not arise. Ponder - you could describe the motion of your legs while walking in terms of Fourier series, going down to maybe 1 Hz, or a wavelength of 300 km. You do not have to involve such sinewaves to be able to walk. Thus the Fourier components are not needed to describe the action.

Conversely, if one perversely does use a Fourier description, that is fine, but don't expect the artifices of the analysis to have a physical relationship to the actual hardware.

A favourite story - A centipede was asked how he knew when to move leg no. 27. After that he could never walk again.

I regard the use of Fourier analysis and description as simply an elegant form of accountancy. Sinewaves are not an essential component of any signal; there are many other basis functions that have good book keeping properties for different purposes. Best to think of the signals as simply functions of time (but don't get stuck with "what is time?"!). Then the question of cutting off some components does not arise. Ponder - you could describe the motion of your legs while walking in terms of Fourier series, going down to maybe 1 Hz, or a wavelength of 300 km. You do not have to involve such sinewaves to be able to walk. Thus the Fourier components are not needed to describe the action.

Conversely, if one perversely does use a Fourier description, that is fine, but don't expect the artifices of the analysis to have a physical relationship to the actual hardware.

A favourite story - A centipede was asked how he knew when to move leg no. 27. After that he could never walk again.

I would say that Robert's contribution misses an essential point: Sine waves are in fact distinguished as the periodic curves that realize a certain frequency with the least rise time requirements for the circuit elements. Of course, cutting off the higher fourier components means in the time domain that steep edges become less steep. Engineers should have the formulas for this ( rise times correspond to frequencies in a natural manner). The bottom line: each electronic component has its frequency response and this is effectively zero for sufficiently high frequencies (cutoff frequency). The cutoff frequency of a system is close tho the cutoff frequency of its slowest component. So, yes, the actual signals in a chip are truncated at high frequency and the reasons can be understood by inspecting the rise time limitations of the components.

@Ulrich. The question is somewhat different we are talking about low frequencies with long wavelengths, not the high frequency components.

Can you explain the physical picture of a 5 MHz clock signal inside a tiny chip, when the fundamental frequency itself is 60 metres? A square wave will contain even higher frequencies too, but they are not in question, here, because they will have shorter wavelengths.

@Derek, Robert

sorry, to a high energy physicist 'cutoff' and 'high frequency' are so directly related that I considered the high frequency behavior in spite of the different content of the question.

With the new understanding I'm even more confused about the meaning of the question. In fourier analysis the meaning of the lowest frequency is absolute, there is nothing like the 'next lower octave' only the next upper one. A lack of long term stability (say cycles of temperature rising and falling in some component) will probably be more naturally considered as noise and not as signal response. In short: a cutoff at frequencies below the ground frequency is an artificial concept based on a wrong analogy. It seems save to say that one has not to consider any influence of this kind in practical devices.

After reading Robert's contribution again, I understand that his allegory expresses just this.

So can we develop some kind of physical explanation or picture of how we should visualize what the 60 metre wavelength fundamental frequency is actually doing along a wire interconnect within the the tiny 1 cm chip? Surely we can't say this 60 metre wave is a fiction, or can we?

It is there, because we could imagine a thought experiment where we filter out everything but that remaining frequency.

But the real question is this: what is the correct way to talk about this 5 MHz fundamental? Should we say we are in the quasi-static regime and that it no longer is a valid input into the wave equation? And if so, what exactly is it and how should we interpret it?

The rising edges and the falling edges of the fundmental frequency signal traverse the tiny chip and cause rising and falling voltages in the various stuctures of the chip components. To associate a wave with the periodic passage of the edges is not clarifying and is superfluous.

Robert’s answer is clearly in the spirit of the question.

The centipede joke confirms it.

However this question can be asked to students and see their understanding of the problem.

This is an example contradicting the concept of Hadamard’s “well-posed problem”, because some wrong solutions are suggested.

The quartz central frequency of 5MHz involves lower and higher frequencies in the Fourier spectrum, but it does not require a physical length (60m) of the device to exist.

The cutoff will come not from the wavelength compared to the device size, but from the highest commutation frequency of the device.

The best proof (for students) is that before the 5MHz we were using KHz frequencies in radios sets for example and still the kilometers wavelengths “fitted in” the device.

Today we speak about GHz CPUs. The wavelengths are now of the size of CPU, so they “fit in” and the problem is closed.

@Mihai,

I see the frequences higher than 5MHz in our scenario, but not the lower ones. What is your idea about these?

If we write a traveling pulse as delta(x-ct), then a traveling train of pulses at freq of 5MHz is

sum[ delta(x-c(t-nT)) ]

where T=1/5 x 1e-6 sec.

But this is the same as

sum[ delta(x-ct + ncT) ]

where cT = 60 m (for c in vacuum)

So the train of pulses is separated in time by T, in space by cT.

So this distance of 60 m plays exactly the same role as the time

steps of 0.2 x 1e-6 seconds. It only becomes a "wave length"

(viz. of the fundamental mode) if the (x,t)-function is written as a Fourier series.

Derek,

The smallness of the dimension of the circuit compared to the wavelength means that radiation effects do not have to be taken into account in describing the circuit. As you suggest, this is the quasi-static regime, which means that one can solve Maxwell's equations in powers of L/Lambda.

The shape of the pulses will be determined by the impedances and resistivity of its elements. Harmonics with wavelengths comparable to the dimension of the circuit will be heavily filtered (damped).

Despite its small size, the circuit will lose some energy through radiation at the fundamental wavelength and harmonics, but this should be extremely small.

Dear all

there is the risk of classical misinterpretations of certain approximations... of course 5 Mhz is equal to 60 meter wavelength and there is very little radiation (but NOT zero) in the farfield..but we also have a near field. And I often hear the argument ..this structure is much smaller than lamda/4 (and thus cannot transmit or receive anything) and then I ususally ask the the following question to the person ..do you still have an old style long wave AM radio receiverfor 300 Khz or medium wave around 1 Mhz..whats its size (ususally 20 cm) ..does it work..yes..and what the ratio between linear dimension and free space wavelength?...

Anyway all those RFID applications are typically near field devices (with rather little ration in the far field) and work VERY well...

Thus 5 Mhz certainly may play an important role in the near field...and can be seen with sensitive receivers also in the far field. It all a quation of numbers..

And referring to one of the other earlier comments : 1 hz has a wavelength of 300000 km roughly the distance earth -moon...or 7.5 timers the length of the earthcircumference..

All,

Derek is not asking about radiation. His question is simply if the electrical transport within a chip with size L can take place via wave modes with wavelengths greater than 2L. The answer is simple: the wave equation has no solutions for lambda > 2L. Therefore those wave modes do not exist. The corresponding frequency components of the transport are not waves but flow and relaxatory modes. Examples: water with fluctuating velocity flows in a pipe; water fills a cup and overflows, etc.

Laszlo, do you have a reference for no modes with wavelengths >2L? Because all the references I have say that TEM modes in coax exist for any length. There is no low frequency cut-off for these TEM modes. For example, see the EM text by Jackson on p. 358.

That interpretation is wrong (while Jackson is correct), it does not do anything with the quantization of waves in a 1-dimensional conductor. I am writing a comment article on a related bad paper; i will show it to you.

Quantization of wavelengths sounds like you are taking about standing waves only. However, this is obviously not the case for the computer chip example in the original question (above), because these are digital waveforms with rich Fourier components.

Derek,

I don't have time to argue here, see the paper that i will send you. Quantization is the fundamental base of wave states and their superposition and provides the traveling wave pockets, too, just like in solid state physics, see the intro courses there. Solid state physics of the finite-size chip. I will send you my paper commenting your one mentioned above.

The microchip clock speed measured in MHz or kHz (in the past) is not related to any wave propagation through the microchip. This cock speed is the measure of haw fast are carried out the operations in the microchip or how fast it can switch a clock circuit from "on" to "off"

Here is the paper Laszlo promised that argues no waves are present: http://vixra.org/abs/1403.0964

It appears to run counter to all emag text books! Can anyone pinpoint the exact reason why the above paper might be wrong?

I see a lot of discussion here. I will simply point out that microstrip line is a two-conductor waveguide; the dominant mode in two-conductor lines has no low-frequency cut-off; it propagates as pure mode from zero frequency to a high frequency fm ~C/a, where C is the velocity of light in the dielectric medium of the line and a is the inter-conductor separation. If the length of the line L

Coming back again in view of the discussion between Laszlo and Derek. It seems that Laszlo is confusing between wave guides (metallic pipes; rectangular cross section) and two-conductor transmission lines and also confusing between the length of the wave guide and its transverse dimension. It is true that when wave length lambda > 2a, or equivalently f < C/2a , the wave guide has no propagating mode, a being the cross-section size. Coaxial cable has no low frequency cut-off as I discussed above, but metallic wave guides do and the cut-off depends on the size (a) of the cross-section and not on the length of the guide.

1. Concerning diameter vs length we say the same as Nagendra: no cut-off versus cable diameter.

2. However we strongly differ about cut-off vs cable length. Obviously, no cut-off of energy transport; even DC currents pass. However, the cut-off of wave-based energy transport is a very different thing. Physics is very strict about what we can call wave and what we cannot. Fritz's "near field" or the "retarded non-wave potential" describes what actually it is. Assuming wave in that (quasi-static) frequency range violates some 4-5 laws of physics and implies perpetual motion machines, see here: the final, expanded version of the paper is now accepted for publication:

http://vixra.org/pdf/1403.0964v4.pdf