Characteristic Impedance of a Transmission Line

[KasraMohammad]

So we are told in our class that the characteristic impedance of a coaxial cable is impractical to be at a value like 500Ohms. Why is that?

The assumption is that the transmission line is lossless and that the dielectic constant (E_r) is 1

 

[yungman]

You cannot have characteristic impedance larger than the free air impedance of 377Ω. That is even if you have a wire in free space great distance from ground, Earth or anything, the impedance is only 377Ω. Anything that has ground or shield or anything closer by is going to be lower. Any coax line impedance is

$$Z_0=\sqrt{ \frac{R+jωL}{G+jωC}}$$

And is going to be a lot lower than than free air.

 

[skeptic2]

There are a few transmission lines over 500 ohms but appears they are used as delay lines.

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1697084&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F10933%2F35760%2F01697084.pdf%3Farnumber%3D1697084

http://www7.taosnet.com/ebear/coaxlist.html

 

[KasraMohammad]

Ok I can sit well with that explanation. So a transmission line cannot have an impedance larger than the impedance of free air, which is at 377. So how come resistors are able to bypass this law yet transmission lines cannot? Does it have to do with material limitation in the construction of a transmission line??

 

[yungman]

Why delay lines can have higher than 377Ω. I can't login to IEEE. delay line is just tx line!

 

[yungman]

Ok I can sit well with that explanation. So a transmission line cannot have an impedance larger than the impedance of free air, which is at 377. So how come resistors are able to bypass this law yet transmission lines cannot? Does it have to do with material limitation in the construction of a transmission line??

Transmission line is really a wave guide. Signal is really an EM wave. Any varying signal is actually EM wave. The voltage and current you measure is only the consequence of the boundary condition of the EM wave. EE tend to use the voltage and current result from the EM wave propagation to do calculation.

 

[KasraMohammad]

Transmission line is really a wave guide. Signal is really an EM wave. Any varying signal is actually EM wave. The voltage and current you measure is only the consequence of the boundary condition of the EM wave. EE tend to use the voltage and current result from the EM wave propagation to do calculation.

Hmm...I don't quite grasp the message here. I apologize for that. So what i understand from it is that the transmission line wave is the EM wave running through the medium(coaxial cable, wire, etc.) and the IV characterisitcs we see in circuit analysis are bound by the EM wave propagation parameters. And so in transmission lines, the impedance cannot, in almost all cases, exceed the impedance of free space. However, impedance in resistors can and are typically higher than 500Ω. Why this is possible for resistors yet not for the transmission line itself I don't quite understand.

 

[yungman]

Hmm...I don't quite grasp the message here. I apologize for that. So what i understand from it is that the transmission line wave is the EM wave running through the medium(coaxial cable, wire, etc.) and the IV characterisitcs we see in circuit analysis are bound by the EM wave propagation parameters. And so in transmission lines, the impedance cannot, in almost all cases, exceed the impedance of free space. However, impedance in resistors can and are typically higher than 500Ω. Why this is possible for resistors yet not for the transmission line itself I don't quite understand.

I am not sure about what Skeptic2 referred to, but yes, it is the EM wave that move through the transmission line. About the resistor, that's the reason if people terminate the tx line with higher impedance than the tx line, you can see voltage jump up. Just like you put your 500Ω resistor at the end of a 50Ω coax line, you'll see a step jump in voltage because the current at the end point see a higher impedance and the voltage jump.

This is getting into transmission line theory where we treat the voltage and current at every point as time dependent signal and we use phasor representation. When you work with transmission lines, you work in propagation environment where signal at each point of the tx line is different and is time dependent. Think of it this way, EM wave takes time to travel from one end of the tx line to the other, the consequence voltage and current at every point along the line is different as it take time for the signal at one point of the line to move to the second point.

After you study EM and get into RF tx lines, you will understand this.

 

[KasraMohammad]

I see what your saying. The higher the impedance in the resistor of the load, the more of a jump the voltages gets because the signal is reversed back towards the source, causing either a constructive or destructive wave interference. I'm sure i'll get a more intuitive understanding of this and why the tx line impedance cannot exceed the free air impedance after I finish my current course (Upper Division Undergraduate Electromagnetism)

 

[skeptic2]

The abstract of the IEEE reference says:
Abstract

A cable with an impedance of the order of 1000 ohms is described. It resembles the usual flexible concentric cable with a 3/8-inch outside diameter, but its inner conductor is a single-layer coil continuously wound on a flexible core of 0.110-inch diameter. The cable is suitable for video connections from chassis to chassis and to remote indicators.

The second reference shows all known cables and their impedances such as:

Cable Imped Max Oper O.D.
Type (Ohms) Volts Inches Remarks
----------------------------------------------------------------------------
RG-65A/U 950 1,000 .405 High impedance delay line, video cable
RG-185/U 2000 -- .282 Delay cable
RG-186/U 1000 -- .405 Delay cable
RG-266/U 1530 4,000 .400 Delay cable

 

[yungman]

I saw the abstract and the table on impedance in the second link, but what is the theory behind it?

Only thing I know so far that some point in the tx line is high impedance only through standing wave due to mismatch at the termination particular in open or shorted termination where at some length the impedance goes to infinity. eg. at λ/4 of a shorted tx line.

 

[davenn]

You cannot have characteristic impedance larger than the free air impedance of 377Ω. That is even if you have a wire in free space great distance from ground, Earth or anything, the impedance is only 377Ω. Anything that has ground or shield or anything closer by is going to be lower. Any coax line impedance is

$$Z_0=\sqrt{ \frac{R+jωL}{G+jωC}}$$

And is going to be a lot lower than than free air.

ummm that goes against practice, unless your whole statement and formula relate soley to coax cable ??
but your second sentence suggests not ?

RF ladderline is readily available as 300 Ohm or 450 Ohm impedance

cheers
Dave

 

[gnurf]

RF ladderline is readily available as 300 Ohm or 450 Ohm impedance

For anyone who's interested, the impedance of a twin-lead line (distance D between conductors, and wire diameter d) is:

 

[skeptic2]

So we are told in our class that the characteristic impedance of a coaxial cable is impractical to be at a value like 500Ohms. Why is that?

The assumption is that the transmission line is lossless and that the dielectic constant (E_r) is 1

Didn't anyone ask the instructor why they are impractical?

 

[Bob S]

If you calculate the attenuation due to copper skin-effect losses as a function of characteristic impedance, the minimum impedance will be below 100 ohms. The minimum impedance is also slightly dependent on the dielectric.
Thw characteristic impedance of a coaxial line is (see Eq(10) in http://kom.aau.dk/~okj/te7/coaxnote.pdf)
$$Z_o=\frac{1}{\surd\varepsilon}\frac{377}{2\pi}Ln \frac{D}{d}$$where ε is the relative permittivity of the dielectric, and d and D are the inner and outer conductor diameters. There is no fundamental upper limit to Z_o, but impedances over 377 ohms are very impractical, except for high impedance delay lines.

I have used special coaxial delay lines (HH1600, 1600 ohms, 1 microsec per foot) which have a helical center conductor to add inductance, but they are very dispersive and lossy.

 

[yungman]

ummm that goes against practice, unless your whole statement and formula relate soley to coax cable ??
but your second sentence suggests not ?

RF ladderline is readily available as 300 Ohm or 450 Ohm impedance

cheers
Dave

I could be wrong, but the formula is for any transmission lines in TEM mode. OP did talked about coax cable in the first post which is TEM mode.

 

[yungman]

Looks like I am wrong about the upper limit of the impedance. Sorry.

 

[KasraMohammad]

Didn't anyone ask the instructor why they are impractical?

I should have mentioned in the original post, but the instructor asked it expecting us to go home and ponder on it and see if we can figure out the answer.