.Comparing Wave Reflection in Strings and Wires

[Jdo300]

Hi I have a question about pulses in a wire. If I have a single wire of length X and I send a pulse through the wire, when the pulse gets to the end of the wire, does it reflect back with the opposite phase or in phase when the original pulse?

I'm trying to visually compare the behavior with a pulse that travels along a string that has one end tied to something. I know that if one end of the string is rigidly fixed to something that the wave will reflect back on the opposite side of the string that it came on. And if the string is fixed to a soft boundary that the wave can reflect back on the same side of the string that it came on. (Here’s my reference: http://www.kettering.edu/~drussell/Demos/reflect/reflect.html )

Is there an electrical analog of the two cases with the string wave?

Thanks,
Jason O

 

[Claude Bile]

Yes, if the voltage is fixed, the reflection will have its sign reversed (i.e. will reflect back on the opposite side of the string, as you put it). If the voltage is floating, the reflection will have the same sign as the incident pulse.

Claude.

 

[Manchot]

As a matter of fact, there are analogs for this all over the place in physics, electromagnetism included. Basically, any one-dimensional system which satisfies the wave equation will exhibit similar properties. An electrical analog would be a transmission line (e.g., two long metallic parallel plates). At low frequencies, the position-dependence of the electric field can be neglected, but at microwave frequencies, the voltage waves have a position-dependent phase and magnitude. Every one-dimensional system that obeys the wave equation has solutions that can be decomposed into a superposition of forward-travelling waves and reverse-travelling waves. If the line is terminated with a simple linear device, the reverse-traveling wave is proportional to the forward-travelling wave, with a constant of proportionality called the reflection coefficient. As it turns out, if you terminate the line with a short (forcing the voltage to zero at the end), the reflected coefficient will be -1, and the reflected wave will be the inverse of the incident wave. This is analogous to the hard boundary. If you terminate the line with an open (forcing the current to zero at the end), the reflection coefficient will be 1, and the reflected wave will be identical to the incident wave.

 

[Jdo300]

Thank you for your help. Ok, so to make sure that I got this correctly. Let's say that I have a piece of wire with a pulse generator on one end and nothing connected on the other end. If I pulse this wire, when the pulse reaches the end, it will have the same sign as the original one?

Thanks,
Jason O

 

[NoTime]

It depends on the length of the wire.

 

[Jdo300]

It depends on the length of the wire.

Okay, that makes sense. How does one predict what length wire you'd need to get a full in-phase reflection back to the source on this wire? I'm assuming that this can't be treated as a simple, continuous wave since the input is a discrete pulse. How would I determine the best conditions for in-phase reflection if I know the pulse width, pulse frequency and length of wire?

Thanks,
Jason O

 

[Claude Bile]

Thank you for your help. Ok, so to make sure that I got this correctly. Let's say that I have a piece of wire with a pulse generator on one end and nothing connected on the other end. If I pulse this wire, when the pulse reaches the end, it will have the same sign as the original one?

Thanks,
Jason O

Absolutely correct.

NoTime, I'm not sure where you got your information from, but the length of the wire has no bearing at all. You should get an in-phase reflection no matter what the length of the cable is, provided it is terminated with a near-infinite impedance (i.e. a few hundred Megaohms or more).

Claude.

 

[Jdo300]

Thanks for clearing that up for me. I have a couple more questions:

1. What could I put on the end of the wire that would give it a high impedence? Or would it work okay with the wire just haging there with nothing on it?.

2. If I were to wrap this wire into a coil, would there be any interference between the turns? If so, could I assume that the interference would always be constructive or is it more complex?

Thanks,
Jason O

 

[Claude Bile]

1. Leaving the wire hanging will probably be good enough.

2. As long as there is an insulating layer, there should be no interference if your signals are at a reasonable level (i.e. around a volt or less). Coiling the wire will change the capacitance and inductance of the wire per unit length, this however should not matter for your purposes.

Claude.

 

[NoTime]

Absolutely correct.

NoTime, I'm not sure where you got your information from, but the length of the wire has no bearing at all. You should get an in-phase reflection no matter what the length of the cable is, provided it is terminated with a near-infinite impedance (i.e. a few hundred Megaohms or more).

Claude.

Perhaps you are referring to something else.
Terminated(shorted) or non terminated(open) transmission line traps work equally well, but the length is different.
A transmission line trap works by refecting a pulse out of phase, assuming repetitive pulses.
The length is determined by frequency and propagation speed.
In the case where the transmission line is perfectly terminated with its characteristic impedance there is no reflection.

A straight wire would just be a special case of transmission line.

 

[Jdo300]

Hi Claudius,

Two more questions:

1. If we assume that the wire is insulated (enameled or rubber insulation) but the voltages are high enough for the adjacent windings to start affecting each other, what would the outcome be?

2. With the added capacitance in the equation, does that generally positively or adversely affect the traveling pulses in the line? Or would it have no affect at all?

Thanks,
Jason O

 

[Claude Bile]

Perhaps you are referring to something else.

Perhaps, I'm not an electronics expert, I'm going from the mechanical analogue, that is a wave on a string, fixed or free at one end. Saying that the length of the line matters is like saying the length of the string matters in the mechanical analogue, which it doesn't.

I'm not really familiar with these traps you refer too, but I'm guessing they are some way of coping with reflections in a transmission line without having to match the impedance of the wire. I'm not sure this is what the OP was looking for, it sounded like he wanted to observe the reflected pulse, not eliminate it.

Jason, wrt 1. I'm not 100% sure what the effect will be, that is why I suggested you keep the voltage down. Keeping it below 1V is probably over-cautious on reflection, you could probably use 10V without introducing unwanted effects.

As for 2. the added capacitance changes the impedance of the wire, which would be important if you were trying to impedance-match to eliminate reflections, however since you are not, then I don't think it should matter.

There is a risk that high-frequency components may transmit across loops, the same way an AC current passes through a capacitor, the obvious solution to this is to not have high frequency components present, i.e. use a broad pulse.

Claude.

 

[Jdo300]

I'm not really familiar with these traps you refer too, but I'm guessing they are some way of coping with reflections in a transmission line without having to match the impedance of the wire. I'm not sure this is what the OP was looking for, it sounded like he wanted to observe the reflected pulse, not eliminate it.

You are right. This might sound a bit contrary but I am trying to maximize the positive reflections that occur in the transmission line.

Jason, wrt 1. I'm not 100% sure what the effect will be, that is why I suggested you keep the voltage down. Keeping it below 1V is probably over-cautious on reflection, you could probably use 10V without introducing unwanted effects.

I am not sure yet that the mixing of voltage signals will not produce the effect I am looking for (positive reinforcement of the pulse waves traveling down the line). But for the expriments I am doing, I am working with voltages anywhere from 20-200 volts (electrostatic).

As for 2. the added capacitance changes the impedance of the wire, which would be important if you were trying to impedance-match to eliminate reflections, however since you are not, then I don't think it should matter.

If I want to create a condition where maximum positive reflections occur, do I want a lot of impedence or as little impedence as possibe? Or does this all depend on the geometry of the problem?

There is a risk that high-frequency components may transmit across loops, the same way an AC current passes through a capacitor, the obvious solution to this is to not have high frequency components present, i.e. use a broad pulse.

Again, this makes good sense, but I am not sure that this effect won't have favorable consequences. I am basically interested in understanding what happens when you mix the worse possible combination of pulses into the wire at a (relatively) high frequency.

Thanks,
Jason O

 

[NoTime]

Perhaps, I'm not an electronics expert, I'm going from the mechanical analogue, that is a wave on a string, fixed or free at one end. Saying that the length of the line matters is like saying the length of the string matters in the mechanical analogue, which it doesn't.

Doesn't the length, mass and tension of a string have an effect on the resonant frequency of the string?
Or how fast a single pulse will travel to the end of the string before being reflected?
These properties have analogues in the transmission line.
If you tap a vibrating string at the proper times won't the tap cancel the energy in the mechanical system, causing the string to stop vibrating. Or conversely taping at a slightly different relative time cause the vibration to increase?
Length would seem to be quite important even in the mechanical system.

The use of the word polarity is somewhat questionable.
Mostly you would talk phase angle.
The only place polarity might be useful, is to note that the reflection travels in the opposite direction from the original pulse.
Thus polarity of the reflection is always negative.

 

[NoTime]

Hi Claudius,

Two more questions:

1. If we assume that the wire is insulated (enameled or rubber insulation) but the voltages are high enough for the adjacent windings to start affecting each other, what would the outcome be?

2. With the added capacitance in the equation, does that generally positively or adversely affect the traveling pulses in the line? Or would it have no affect at all?

Thanks,
Jason O

1) Any practical device is a mixture of R L C.
A coil or even a straight wire is going to have some capacitance associated with it.
Voltage below insulation breakdown will not be a major factor, but nothing is a perfect insulator, so some leakage could occur at high voltages.

2) added distributed capacitance will affect propagation times.


Your coiled wire could be considered an antenna with a loading coil.

The energy of a reflected pulse will be lower by resistive losses in the transmision line.
Also for a pulse, rather than a sine wave, real transmision lines have slightly different characteristics at different frequencies.
Thus the pulse will be somewhat distorted.

I've read a lot of your posts.
And you seem to know a lot more than your questions imply.
Why do I keep thinking you are looking for a hole in reality?
I don't have any problem with you trying though

 

[Jdo300]

1) Any practical device is a mixture of R L C.
A coil or even a straight wire is going to have some capacitance associated with it.
Voltage below insulation breakdown will not be a major factor, but nothing is a perfect insulator, so some leakage could occur at high voltages.

2) added distributed capacitance will affect propagation times.


Your coiled wire could be considered an antenna with a loading coil.

The energy of a reflected pulse will be lower by resistive losses in the transmision line.
Also for a pulse, rather than a sine wave, real transmision lines have slightly different characteristics at different frequencies.
Thus the pulse will be somewhat distorted.

I've read a lot of your posts.
And you seem to know a lot more than your questions imply.
Why do I keep thinking you are looking for a hole in reality?
I don't have any problem with you trying though

No holes in reality here, just looking at the same old stuff from a different perspective .

I am asking about the inductance and capacitance of the transmission line because I programmed a basic "soliton simulator" in Microsoft Excel that displays a graph of simple solitary waves (using a variation the function Sech(x)^2) and I want to incorporate the real world conditions that would dissipate the waves over time. Right now, they just elastically reflect back and forth on the graph and you can see how they combine (and cancel if a negative reflection combines with a positive incoming wave). The program can be used to simulate up to 16 waves on the same graph. I attached it below for all to look at. Again, it's very basic and I made it mainly to visualize the interactions. Hardly real-world but I am hoping to make it more realistic.

Thanks,
Jason O

 

[Claude Bile]

Doesn't the length, mass and tension of a string have an effect on the resonant frequency of the string?
Or how fast a single pulse will travel to the end of the string before being reflected?

Insofar as getting a pulse to reflect is concerned, the resonant properties of the string/cable is somewhat irrelevant. As is the velocity of the pulse as it travels down the string/cable. If the end of the string/cable is perfectly fixed, then the reflection coefficient will always be 1.

Jason - The formula for the reflection coefficient (i.e. signal in/signal reflected) is dependant on the impedance of the wire divided by the impedance of the terminus. If the impedance of the terminus is infinite (or at least, very large), then the ratio tends to zero, irrespective of what the impedance of the cable is.

A final comment. There comes a point where theory will only take you so far, eventually you have to just sit down and perform the experiment and observe what happens!

EDIT: I just saw your last post - Have you incorporated an attenuation term in your simulation? I know in the case of optical solitons, this tends to be the thing that kills them off.

Claude.

 

[Jdo300]

Hi Claude,

You are absolutely right and I agree with you 100%. I do plan on running some simple tests to see what will happen but I wanted to see if I could do some calculations about the wire I'm going to use to get a ballpark idea of how it could behave. Like for instance, what pulse width should I use? What length of wire? Would a coil or a straight piece be better? What frequency minimum do I need to have before I can start observing the effects? How do I observe the effects??

Thanks,
Jason O

 

[Jdo300]

EDIT: I just saw your last post - Have you incorporated an attenuation term in your simulation? I know in the case of optical solitons, this tends to be the thing that kills them off.

Claude.

No actually, I haven't. Thats what I want to figure out how to do. What factors determine how the wave attenuates and how can I incorporate that into the equation that I am using to represent the soliton?

Thanks,
Jason O

 

[NoTime]

No holes in reality here, just looking at the same old stuff from a different perspective .

I am asking about the inductance and capacitance of the transmission line because I programmed a basic "soliton simulator" in Microsoft Excel that displays a graph of simple solitary waves (using a variation the function Sech(x)^2) and I want to incorporate the real world conditions that would dissipate the waves over time. Right now, they just elastically reflect back and forth on the graph and you can see how they combine (and cancel if a negative reflection combines with a positive incoming wave). The program can be used to simulate up to 16 waves on the same graph. I attached it below for all to look at. Again, it's very basic and I made it mainly to visualize the interactions. Hardly real-world but I am hoping to make it more realistic.

Thanks,
Jason O

Interesting.
Baring resistive losses as heat, radiation or external inputs there isn't any reason that I can think of why they should die out.
Superconduction comes to mind here.
How much any particular factor comes into play in the real world depends on just what you want to model.
In any event, building a meaningful simulation of any device is non trivial.

A similar topic to solitons would be the phonon.
http://en.wikipedia.org/wiki/Phonon

 

[NoTime]

Insofar as getting a pulse to reflect is concerned, the resonant properties of the string/cable is somewhat irrelevant. As is the velocity of the pulse as it travels down the string/cable. If the end of the string/cable is perfectly fixed, then the reflection coefficient will always be 1.

Yes, I mostly agree with this.
The reflection coefficient will only be 0 when the resonant properties of the string/cable and the cable end connection match.
Not when the end is perfectly unfixed.
This is easier to demonstrate in an example like the coaxial cable transmission line.
The resonant properties of the string/cable are not entirely irrelevant.
If you tie two different strings together, you now have two reflections. One from the join and the other from the endpoint.

 

[NoTime]

sorry double post

 

[Jdo300]

A similar topic to solitons would be the phonon.
http://en.wikipedia.org/wiki/Phonon

Thank you for the reference. But do you know of any good references that may have equations to predict how a soliton pulse wave will dissipate as it travels down a wire/transmission line?

If you tie two different strings together, you now have two reflections. One from the join and the other from the endpoint.

This made me think of another question. If the transmission line were a wire that was closed looped in a ring and you induced a pulse in the coil (say, using a pulse coil or something) would the pulse just travel around and around the ring until it dissipates. Or would there still be some kind of positive/negative reflection at the point where the loop ends are joined together?

Thanks,
Jason O

 

[Claude Bile]

No actually, I haven't. Thats what I want to figure out how to do. What factors determine how the wave attenuates and how can I incorporate that into the equation that I am using to represent the soliton?

Thanks,
Jason O

I can't give you a comprehensive list of the factors that cause attenuation, I'm just not familiar enough with the theory that deals with that sort of thing. What you can do however is simply introduce an exponential decay term, and adjust the decay rate to a reasonable value (i.e. treat it like an empyrical term). You can then work to show how the decay rate relates to fundamental constants if you wish.

Solitions undergoing attenuation typically reach a point where the nonlinear effects are too weak to 'keep' the shape of the solition. When this happens, the pulse behaves like an ordinary pulse.

Claude.

 

[Jdo300]

Solitions undergoing attenuation typically reach a point where the nonlinear effects are too weak to 'keep' the shape of the solition. When this happens, the pulse behaves like an ordinary pulse.

Hi Claude,

Maybe I missed something here but in an electrical sense, what is the difference between an electrical soliton and a simple pulse? Could I still produce a soliton by using something like a square wave pulse or does this require something more advanced?

Thanks,
Jason O

 

[NoTime]

This made me think of another question. If the transmission line were a wire that was closed looped in a ring and you induced a pulse in the coil (say, using a pulse coil or something) would the pulse just travel around and around the ring until it dissipates. Or would there still be some kind of positive/negative reflection at the point where the loop ends are joined together?

Generally yes, it would just keep looping around.
As to if the connection generates reflections.
That depends on how good the connection is.
A coax line tester can normally show splices and dings in the cable as well as cut cable or shorts, giving the distance to each.
Not impossible to build one that doesn't have reflection issues, but very difficult.

Attenuation shouldn't be that hard.
Just subtract a small amount from the parameter controlling amplitude of the wave and recompute for each iteration.
You can pick your choice of resistance, emission, or out of phase input to assign the subtraction to.

 

[Jdo300]

Hello All,

Thanks again for the multitude of help you have given me. I have a couple more questions to ask.

1. What is the speed of a pulse wave through copper? I know it would be c if it were traveling in a vacuum but I would expect that it would be slower in the metal? If so, then where can I go to find out what it is?

2. When the pulse wave attenuates, does the amplitude alone decrease or does the width of the wave increase also as the amplitude decreases? (I'm asking because I thought I saw an animation somewhere where a decaying wave got wider as it got shorter).

Thanks,
Jason O

 

[NoTime]

1: It's going to vary depending on just what you mean in terms of a physical construct.

2: A pulse in a transmission line will tend to spread out some. This is due to the fact that a pulse is a sum of different frequencies and that physical transmission lines don't respond the same to all frequencies.

This is some new info on phonons
http://www.sciencemag.org/cgi/content/full/314/5802/1065a?etoc

 

[Jdo300]

1: It's going to vary depending on just what you mean in terms of a physical construct.

Well, I kind of look at the pulse as a wave of compressed electrons that is propagating down the line. I know this is referred to as a longitudinal or "compression" wave. So I suppose that is the physical construct that I am looking at. Does this help?

Thanks,
Jason O

 

[NoTime]

Not really, pulse shape doesn't matter.
Different impedence transmission lines have different propagation times.
You should be able to work that out from the manufactures data.

 

[Jdo300]

Ahhh I see,

So if I know the impedance of the line I am playing with, is there any formula out there I can use to calculate the speed of the wave? Especially if I know other factors like the voltage, pulse width, and frequency (if that matters)?

Thanks,
Jason O

 

[NoTime]

Distributed capacitance and inductance are the key factors which define a transmission line (impedance).

Time to do a little more homework

 

[Jdo300]

No problem. Do you have any recomendations for a good online tutorial on transmission lines?

 

[Claude Bile]

Hi Claude,

Maybe I missed something here but in an electrical sense, what is the difference between an electrical soliton and a simple pulse? Could I still produce a soliton by using something like a square wave pulse or does this require something more advanced?

Thanks,
Jason O

A soliton is just another name for a pulse whose shape does not vary as it propagates. Pulses have a natural tendency to change their shape as they propagate due to the properties of the transmitting medium varying with frequency. Typically solitons counter this natural tendency through some non-linear effect.

The difficulty of producing a solition depends on how quickly the pulse shape changes as it propagates, which usually depends on the bandwidth of the pulse. Square waves would be a good choice in my opinion, as the behaviour of each harmonic will give you a good overall picture of how the pulse is behaving. It is also very easy to measure changes in pulse shape of a square pulse. Higher amplitudes will improve the chances of creating a soliton, as the nonlinear effects that keep the shape of the solition will be enhanced.

Claude.

 

[Claude Bile]

2. When the pulse wave attenuates, does the amplitude alone decrease or does the width of the wave increase also as the amplitude decreases? (I'm asking because I thought I saw an animation somewhere where a decaying wave got wider as it got shorter).

Pure attenuation will result in a decrease in amplitude only. The animation you saw was probably on dispersion (the spreading of the pulse in time).

Claude.