[Imperial]

  Awesome!

Imperial

[andy_c]

Anyway, when I get a chance I'll take your example above (which was perfect, thanks), replace my equivalent circuit with a long series of LC elements, and see if I believe you. 

One good (and free) way to do circuit analysis with transmission lines is to use the freeware LTspice from Linear technology.  It is available here:

http://www.linear.com/designtools/software/

It has a model called LTRA, which is a direct implementation of the Telegrapher's Equations for transmission lines, described here:

http://en.wikipedia.org/wiki/Telegrapher%27s_equations

It's restricted to the case for which R, L, C and G are independent of frequency, so it can't handle e.g. the skin effect.

Of course, the easiest fix for this problem is just what Frank said - run zip cord to the woofer.

[jneutron]

So why not just go back to Radio Shack and buy some 25 cents per foot 14 gauge standard low capacitance speaker wire?  The object is the music, the best music possible for the money.
Frank Van Alstine

And 14 guage zip is what you call the best for your money?  We're talking high-end audio here dude..#14 zip's gonna run 5 milliohms per foot, it's gonna be 100-200 ohms impedance..woofer speed is gonna be compromised at least 2-5 uSec delay w/r to the mids...it's gonna spray magnetic field all over creation...anarchy I tell ya....what's the world comin to??

The smart money's in....superconductivity and pre-charged guarded inverted quantum dielectrics..  Pre-charging the dielectric opposite of what they'd expect us to do, so that the cable pulls the signal along at superluminal velocities..

Actually, I'd go with #12 or #10..it'd be different if the load was full bandwidth..

Cheers, John

ps... I just know that garbage is gonna end up on some vendor's marketing blurb, I just know it...

[Imperial]

Lithium phosphide + sodium phosphide ... that could do the trick.

 


You just make the conductor of  indium + gold...  then you make the sheath of Cyclic carbonate-modified siloxane that can retain a charge and be shaped in many ways, and even recharged or, you know..
Now the indium + gold is actually a magnetic alloy, so it will have a field..
When you play with the cable it will charge its dielectric sheath...

You make lots and lots of very small transformers and chain them up!!! of course you have covered the ends of the micro transformers
with 1256/1212 compound..

Eh?

 

Imperial

[jneutron]

Lithium phosphide + sodium phosphide ... that could do the trick.

 


You just make the conductor if  indium + gold...  then you make the sheath of Cyclic carbonate-modified siloxane that can retain a charge and be shaped in many ways, and even recharged or, you know..
Now the indium + gold is actually a magnetic alloy, so it will have a field..
When you play with the cable it will charge its dielectric sheath...

You make lots and lots of very small transformers and chain them up!!! That's your conductor...

Eh?

 

Imperial

Now I remember why I hated chemistry so much..I bet you can't even pronounce all that stuff..

Three things wrong with your theory.

1.  Cyclic carbonate-modified siloxane is a food additive for twinkies..(or was it the package??  would it matter?)
2.  ""When you play with the cable it will charge its dielectric sheath... "" There's just something wrong with that statement..on so many levels..
3.  The Rocketts only perform two shows on Saturdays..(anybody know where that's from?)

John

[Imperial]

Right, never mind. I'll keep the chemistry out of  cable debates.. 
The stuff anyway will conduct a charge, let's just say that.
There are a lot of names and obviously I shouldn't play havoc here with chemistry.. 

It's just ... you talked about signal traveling faster than light, so I thought you were making a joke...
I just wrote my cable post as a joke too..

Lithium phosphide + sodium phosphide is a drug against delusional states in the brain.. So I thought that was understood as
a warning from me that my cable construct would be a bit silly... sort of.. - again I should have stated that more precise.

But obviously you did not joke? Signal faster than light? Is that possible?

Imperial

[jneutron]

Uh oh...my spidy sense is tingling....

Somebody on the internet at some audio forum is invoking the essex echo skin effect article by MH..

Sigh..here we go again...

John

[jneutron]

Right, never mind. I'll keep the chemistry out of  cable debates.. 
The stuff anyway will conduct a charge, let's just say that.
There are a lot of names and obviously I shouldn't play havoc here with chemistry.. 

It's just ... you talked about signal traveling faster than light, so I thought you were making a joke...
I just wrote a joke too.. (Lithium phosphide + sodium phosphide is a drug against delusional states in the brain..)

But obviously you did not joke? Signal faster than light? Is that possible?

Imperial

Absolutely nothing in that post was legitimate.. except where I agree with Frank.

I first thought you meant some kinda drug stuff, but wiki and google were not good to me in that regard.

I like reading scifi, so I believe that someday faster than light will be shown as possible.  But what we know of the universe says NO, it is not possible.

Sometimes "geek humor" is just an oxymoron..

Cheers, John

[Imperial]

Hehe.. you give lithium to people who hears voices...
You wrote some posts above that the voices had stopped talking..  aa

Ok. We can't know it all, can we..

  I guess I should speak more accurate, and not try a funny side 'trip'

Sorry jneutron, I'm a evil 'git sometimes.. 

Imperial

[jneutron]

Hehe.. you give lithium to people who hears voices...
You wrote some posts above that the voices had stopped talking..  aa

Ok. We can't know it all, can we..

  I guess I should speak more accurate, and not try a funny side 'trip'

Sorry jneutron, I'm a evil 'git sometimes.. 

Imperial

If you stop the humor, I will slap you upside the head via the internet...(it's possible, you know.)

I deserved the comment about the lithium, but did not get it at first.....You've no idea how stupid I felt when the hammer landed on me..I figure it's  better to inform all....that stupid thinks like that are well, just stupid, and hope others learn from my mistake..

Cheers, John

[JoshK]

I always wanted to try out thick gauge romex on the woofers and the silver plated copper stranded wire I like to use on the high freq side.  I'd think the romex on woofers and subs should work well.   And since I am renovating my house and have a lot of excessive lying around it is basically free.

[opaqueice]

Here's a good example:

OK, I had a chance to sit down and work this out.  First off, you're right that when the load impedance matches the line impedance, the amp just sees the load.  But that follows also from the simple equivalent circuit analysis (which I hadn't noticed before), and in general I was correct that one can always use the equivalent circuit for audio frequency signals in speaker cables.

How do I know?  Well, I worked out the equation describing signal propagation down a long chain of LC elements, which turns out to be exactly the telegrapher's equations (which I'd never heard of before - thanks, andy_c).  I ignored the resistance of the cable, and then those equations are pretty nice and very familiar, they're just 1-dimensional wave equations, and so the general solution for the voltage is V = f(wt - kx) + g(wt + kx), and for current it's I = (1/Z0)(f(wt-kx) - g(wt + kx)), where Z0 is the characteristic impedance of the cable and f and g are completely arbitrary functions which get fixed by the boundary conditions.  So you can see right away that if you attach a load with impedance Z0 at x=l and a voltage source at x=0, the solution is g(x) = 0 and hence V = I Z0, just as you said.

Now, let's compare that to the equivalent circuit (where the voltage source is connected across the capacitor of a standard series RLC circuit).  When I work that out I find an expression for Z which I don't want to type here because it would be hard to read, but it has the property that it agrees almost perfectly with the results of the exact solution so long as omega^2*l^2*L*C is small, where omega is the driving frequency and l is the length of the cable.  But if we take your numbers from above, that just means the cable should be less than a few kilometers long (just as I said originally).  And moreover the impedance has precisely the characteristics Daryl said (which is why it made sense to me in the first place).

So to summarize:  for analogue audio signals traveling down speaker cables less than kilometers in length one can use a plain old RLC equivalent circuit.  Transmission line theory, while technically more precise, is unnecessary and total overkill. 

And this claim:

If one were to be rigorous in the application of lumped elements, the line has to be modelled as a long string of LC's..  Otherwise, the models fall apart when the load matches the line.

is wrong.  The equivalent circuit model works fine in that case so long as the frequencies in question are sub-RF.

[denjo]

S
This all kind of sounds like "how to make my Rolex keep exact time."  The answer of course is buy a Timex.

Not that I am knocking Rolexes, I have one, its jewelery, my only piece of jewelery, but I don't confuse it with WWV time.  Now I am going to hear bad things from those who "know" that Rolexes aren't really the best of jewelery.  I don't care. 

Best regards,

Frank Van Alstine

Frank
Rolexes are great jewelry! They are built like the proverbial tank and hold their value well - much like your gear!

Best Regards
Dennis

[andy_c]

Now, let's compare that to the equivalent circuit (where the voltage source is connected across the capacitor of a standard series RLC circuit).  When I work that out I find an expression for Z which I don't want to type here because it would be hard to read, but it has the property that it agrees almost perfectly with the results of the exact solution so long as omega^2*l^2*L*C is small, where omega is the driving frequency and l is the length of the cable.  But if we take your numbers from above, that just means the cable should be less than a few kilometers long (just as I said originally).  And moreover the impedance has precisely the characteristics Daryl said (which is why it made sense to me in the first place).

So to summarize:  for analogue audio signals traveling down speaker cables less than kilometers in length one can use a plain old RLC equivalent circuit.  Transmission line theory, while technically more precise, is unnecessary and total overkill.

Was your analysis based on the assumption that the load impedance matches the characteristic impedance of the line?  That is almost never true in audio (except for S/PDIF which could technically be considered RF).  In order to show that simple two-element lumped circuit analysis is sufficient for any load impedance, that's a more complex problem.

[PLMONROE]

Gad, what did I start? I'm getting a headache. 

Paul

[opaqueice]

Was your analysis based on the assumption that the load impedance matches the characteristic impedance of the line? 

No - although I checked that case explicitly, as well as load imp going to zero and infinity.  I did assume when I did the equivalent circuit analysis that the load was purely resistive, but only because I was feeling lazy.  I can easily go back and check the general case.

That is almost never true in audio (except for S/PDIF which could technically be considered RF).  In order to show that simple two-element lumped circuit analysis is sufficient for any load impedance, that's a more complex problem.

I don't see why it's so complex.  I have an exact equation (in the limit we can ignore the resistance of the cable) which follows directly from the telegrapher's equations plus the correct boundary conditions for the load and source.  That equation can be Taylor expanded in the limit that omega^2 LC is small.  It's slightly tricky in that you have to expand past first order, but it looks like the result agrees with the simple circuit to the same order.  The corrections are of order omega^2 LC, which is tiny for speaker cables.

Physically it all seems to make sense - it's just because in this limit the voltage and current barely depend on position at all (since the wavelength is much longer than the cable length).  So you ought to be able to replace the continuous cable with a single group of circuit elements, and apparently you can.

[jneutron]

OK, I had a chance to sit down and work this out.  First off, you're right that when the load impedance matches the line impedance, the amp just sees the load.  But that follows also from the simple equivalent circuit analysis (which I hadn't noticed before), and in general I was correct that one can always use the equivalent circuit for audio frequency signals in speaker cables.

Nice.

An RLC model runs flat from DC to daylight?   How did you model that?

I see you are also running into the chicken and the egg problem..which is first, the C or the L...

So to summarize:  for analogue audio signals traveling down speaker cables less than kilometers in length one can use a plain old RLC equivalent circuit.  Transmission line theory, while technically more precise, is unnecessary and total overkill. 
And this claim:

If one were to be rigorous in the application of lumped elements, the line has to be modelled as a long string of LC's..  Otherwise, the models fall apart when the load matches the line.

is wrong.  The equivalent circuit model works fine in that case so long as the frequencies in question are sub-RF.

Model the settling time by varying the line to load ratio.  It must produce a cusp minima at unity....zero if you offset the transit time.

Nice work..

Cheers, John

[opaqueice]

Nice.

An RLC model runs flat from DC to daylight?   How did you model that?

If daylight means infinite frequency, it isn't - it's just almost flat from DC to omega^2 L C of order 1.

Actually you can get that from dimensional analysis - at DC the capacitor and inductor are both irrelevant, so the circuit just sees the load.  The corrections to that must scale as some power of omega, each of which has to be multiplied by (LC)^(1/2) by dimensional analysis...  so the only question is which power.  Turns out matching the load to LC makes the correction quadratic (the linear term cancels).

I see you are also running into the chicken and the egg problem..which is first, the C or the L...

Looks like you have to put the C first or it won't work.  Not sure what to say about that.

Model the settling time by varying the line to load ratio.  It must produce a cusp minima at unity....zero if you offset the transit time.

You'll have to translate that into physics - or English, take your pick  .

[andy_c]

I have an exact equation (in the limit we can ignore the resistance of the cable) which follows directly from the telegrapher's equations plus the correct boundary conditions for the load and source.  That equation can be Taylor expanded in the limit that omega^2 LC is small.  It's slightly tricky in that you have to expand past first order, but it looks like the result agrees with the simple circuit to the same order.  The corrections are of order omega^2 LC, which is tiny for speaker cables.

I'd be interested in looking at the math, as I haven't looked at the problem in that way myself.  Could you scan it?  It's always nice to see how others approach a problem, especially someone taking a fresh approach.  A Physics person is likely to look at it in a more fundamental way than a typical engineer might.

Also, there was one thing I carelessly left out.  This whole discussion started on the subject of amplifier instability due to the load impedance seen by the amplifier.  It becomes necessary to look at this impedance at the frequency the amplifier tends to oscillate.  That frequency is the so-called unity loop gain frequency of the amp.  This varies somewhat with design, specifically the amount of feedback used.  The lowest I've seen is the DIY Leach amp, which is around 400 kHz.  A typical design is 1 MHz.  An aggressive high feedback design might be 2 MHz or somewhat higher.  So although it's an audio application, the frequencies we're interested in regarding stability are much higher than that.

A similar situation occurs with preamps that use op-amps or discrete feedback amplifiers for which we want to look at stability with a long interconnect.  One extreme case might be the AD797 op-amp.  It has a 100 MHz gain-bandwidth product.  So if it is configured as a unity-gain buffer, its unity loop gain frequency will be around 100 MHz.  Definitely not an audio frequency problem in that case.

Another point worth mentioning is that as soon as a non-zero R is assumed for the transmission line, it fouls up the concept of impedance matching at low frequencies.  In this case, the characteristic impedance is complex, given by Z0=sqrt((R+j*omega*L)/(G+j*omega*C)).  At low frequencies, even though R may be small, R, together with omega*C in the denominator, dominates the expression.  In practice, this isn't really a problem as impedance matching is almost never done in the audio range anyway, but it is an oddity with the math.

[JoshK]

I am pleased at the direction of the thread.

One point that may be helpful for others reading along who aren't familiar with the subject matter is to define terms that you use that aren't everyday terms.  Not needed for opaq'nce or jneutron but for the rest of us.