Cable Impedance?

[olsenn]

This is a question that by all rights I should already know the answer to, but I don't. What is meant by cable impedance? First off, I understand that copper wire isn't a superconductor and there will be some resistance to the wire; however, I often hear of coax being 50-ohms etc. and as far as I'm aware, the impedance (at least at the frequencies which the cable was meant for) is more like 0.001-ohms. The resistance between the outer shell (ground) and the pin inside is practically infinite (they aren't connected). So where does 50-ohms come in?

[mcinque]

I think that you can read that value by measuring the resistance between the two leads of the cable (inner core) on every 100 meters of cable. At least, this is what I've always read in the cable datasheets. Maybe I'm wrong?

EDIT: yes, I'm wrong.

http://www.epanorama.net/documents/wiring/cable_impedance.html

"The length has nothing to do with a coaxial cable impedance. Characteristic impedance is determined by the size and spacing of the conductors and the type of dielectric used between them.[...]"

[madires]

For a short introduction please see http://en.wikipedia.org/wiki/Transmission_line, it's a quite complex topic.

[c4757p]

It's not really that complicated. The simplified version is that the characteristic impedance is what you see at the input to a transmission line before your applied signal has the chance to propagate to the other end and experience whatever effects are there. It is what you would see as the "true" impedance of an infinitely long transmission line.

If you could take a "resistance" measurement in the time between applying the test current and two times the propagation delay after then, you'd see the characteristic impedance.

[w2aew]

Another way to think about it is this:  The wire has inductance distributed evenly throughout its length.  And, it has capacitance between the two conductors distributed throughout its length.  This distributed inductance and capacitance is what gives rise to the impedance of the cable.  A step voltage change at the input will cause a current to flow through the inductance, charging the capacitance. The relationship between voltage and current during this time is 50ohms.

I go into a little more detail in this video I did last year...

[JackOfVA]

The only thing I would add to W2AEW's response is that the impedance of a coaxial cable or twisted pair transmission line is not constant. At low frequencies (think audio or a bit above audio) cable impedance is not the same as it is when measured at radio frequencies (few hundred KHz and up.)

There are several reasons for the change that I won't go into here, and for many purposes the low frequency change in cable impedance can be safely neglected.

[c4757p]

Damn! Beat me to it. Oh well... I did it differently, so here's my explanation.

If you could take a "resistance" measurement in the time between applying the test current and two times the propagation delay after then, you'd see the characteristic impedance.

I did precisely this. As you can see in the measurement setup diagram, I applied a "fast" (~3ns rise) pulse to a short-circuited piece of 75 ohm coax (about a meter long) through a 100 ohm sense resistor, measuring the voltage on each side of the resistor.

Computing the impedance from this is as easy as Ohm's law. 0.51V across the 100.3 ohm sense resistor gives 5.08mA. With 0.38V across the coax at that moment, this is an instantaneous impedance of 380mV/5.08mA = 74.8 ohms I'm actually surprised by how accurate that is.

After double the propagation delay, the impedance drops to 0 (the short circuit). This can be verified as well, though I didn't think to take proper measurements at the time. A typical propagation velocity for cheap RG-59 could be about 200 Mm/s, or 5 ns/m. A quick glance at the scope screen shows the peak to be about 10ns wide, consistent with 2x delay on a 1m cable.

[ivaylo]

If you could take a "resistance" measurement in the time between applying the test current and two times the propagation delay after then, you'd see the characteristic impedance. 

Can you please explain what is this based on (formula, law)? Why exactly two times the propagation delay? And is the 1m length of the 75ohm cable selected to match this or something?

Also what is the effect of the 50ohm tees (esp. the one on the right hand side where the 75ohm cable is)?

After double the propagation delay, the impedance drops to 0 (the short circuit). This can be verified as well, 

And what is the setup for this?

Thanks... Too many questions 

[c4757p]

It's two times the propagation delay because the signal has to travel two times the cable length - down to the short, then the effects of the short have to propagate back.

The 1m length was chosen for being the nearest piece of coax I had lying around

The effect of the 50 ohm tees is minimal. It will cause a small impedance mismatch and a little bit of reflection, but not enough to skew the measurement. I tried 75 ohm and 50 ohm cable and a 200 ohm pair and couldn't get anything nasty out of that.

The setup is the same - look at the traces in that picture. The voltage across the shorted coax drops to zero after some ringing, meaning the impedance has dropped to zero (V=0, so V/I = R = 0).

Be warned that this is a bit of a fiddly experiment to do - the signal generator needs to output a clean pulse with a rise time a good bit faster than the propagation delay, or else it's hard to find a good spot to take the measurement from. The pulse generator I used here was barely good enough. And if the rise time is too fast, the exact way it's set up becomes critical or else it will be set into a ringing fit. (I tried a Jim Williams pulse generator and couldn't get anything coherent. I suspect much of the problem was related to the gross impedance mismatch around the sense resistor.)

Also, the resistor should have somewhat low inductance. I used metal film. Don't use wirewound or anything silly like that.

[Jeff1946]

A way I think about it.  Suppose you had an infinite length cable.  Send a pulse down it and it never returns.  Now if you cut the cable and put a resistor across it, the impedance is the value of the resistor that makes it behave as if it was still infinite. Yes this is overly simple.   A nice video showing the effect of different kinds of terminations with strings is at:

http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html

Note a hard boundary is like a shorted termination and a soft one like no termination.  Hope this helps.  Also note the effect of discontinuities.  This is why you should drive a cable with a source having the same impedance as the cable.

[jahonen]

If one could have an infinite lossless cable (or at least a cable with no resistance), then the impedance could be indeed measured with a DMM, since the reflected wave would never get to the DMM end.

Regards,
Janne

[MacAttak]

This chapter goes into a fair bit of detail: http://www.allaboutcircuits.com/vol_2/chpt_14/1.html

(be sure to click through the "Next Page" links at the end of each page to read the entire chapter contents)

[Lightages]

You can calculate the impedance with this:

http://www1.sphere.ne.jp/i-lab/ilab/tool/cx_line_e.htm

This one is handy to for two wire impedance.
http://www.eeweb.com/toolbox/coax/

[ivaylo]

@c4757p Thanks for explaining it. I had somewhat working knowledge on the subject, just haven't seen this way of doing it.

Love the old at&t video:

[c4757p]

Yes, that video is great