Coaxial Cable Impedance: Why Length Doesn't Affect It | Explained with Model

[antonantal]

Why is the coaxial cable impedance independent of it's length? I am considering the model shown in the attached picture. Since the resistance, capacitance and inductance are all dependent of the length of the cable in one way or another how come the impedance is not? I also can't find any websites which explain this. All they do is state it.

 

[dlgoff]

http://en.wikipedia.org/wiki/Characteristic_impedance"

Regards

 

[waht]

Take a infinitesimal slice of a coaxial cable and you still have a little bit of inductance of the center conductor and capacitance between the outer shield. The ratio of these will still be the same.

The catch is if you increase the length let's say by a factor of K, the inductance and capacitance will also increase by K,

so the ratio could be K*L / K * C

K's cancels, you still get L/C just take the root for the impadance.

This impedance thing can be thought of the same thing as the sine or cosine operations of a triangle about some about theta.

If theta is the same (impedance) the ratio of sides will always be the same no matter how big the triangle is.

Hope that helps.

 

[antonantal]

Take a infinitesimal slice of a coaxial cable and you still have a little bit of inductance of the center conductor and capacitance between the outer shield. The ratio of these will still be the same.

The catch is if you increase the length let's say by a factor of K, the inductance and capacitance will also increase by K,

so the ratio could be K*L / K * C

K's cancels, you still get L/C just take the root for the impadance.

This impedance thing can be thought of the same thing as the sine or cosine operations of a triangle about some about theta.

If theta is the same (impedance) the ratio of sides will always be the same no matter how big the triangle is.

Hope that helps.

Thanks. That helped. The thing I didn't know is that the parasitic inductance varies linearly with the length. Now it's clear that since both the parasitic inductance and the capacitance of the cable vary linearly with it's length, their ratio will be independent of the length.

I still don't understand why can we neglect the resistance of the cable. Although it's small I guess it could reach a few ohms if taken a length of severals tens of meters. Also the leakage conductance of the dielectric. How about that?

 

[berkeman]

I still don't understand why can we neglect the resistance of the cable. Although it's small I guess it could reach a few ohms if taken a length of severals tens of meters. Also the leakage conductance of the dielectric. How about that?

You can't neglect the losses in the general case. Take a look at the general Zo equation at the wikipedia link that dlgoff posted. If you leave the lossy terms in, you get a Zo that is dependent on frequency. The Zo will typically rise for frequencies below 1MHz or so, and this characteristic can be important if you are designing a termination for a transmission line that is operating across a broad band below 1MHz.

Also, the loss in real transmission lines results in the attenuation of the signal, which is important when you are designing the loss budget for a long transmission line.

 

[antonantal]

Thanks berkeman. Keep up with the explanations with practical insight!

 

[Corneo]

Quick question berkeman,

How do people deal with losses? Surely you can operate at a lower frequency to reduce loss, use better conductors, etc. Can you treat loss like noise and put repeaters along the line to clean up and boost the signal?

 

[berkeman]

Quick question berkeman,

How do people deal with losses? Surely you can operate at a lower frequency to reduce loss, use better conductors, etc. Can you treat loss like noise and put repeaters along the line to clean up and boost the signal?

Yeah, higher frequencies in coax generally have more loss. You are ultimately limited by the minimum signal-to-noise ratio (SNR) that you can live with for your application, and then based on that, you chose your transmit power, decide on if you need powered repeaters, and chose the bandwidth (filtering), encoding and receiver noise level that you can live with.

One of the best handles you have to improve your SNR is to reduce the bandwidth of the signal. That doesn't necessarily mean moving to a lower frequency, since you can modulate your information up in the frequency band. There are often reasons (transformer magnetics mainly) to pick a certain frequency band for your information.

If you get a chance to take a communication theory class, I would highly recommend it. Very interesting, mathematical and very practical stuff. The textbook I had back in grad school was "Introduction to Communication Systems" by Stremler.