[theThinker (OP)]

What is AC phase velocity?

It seems when we talk about AC current circuit theory, we only deal with the frequency and voltage, but seldom the phase velocity. 

I am not sure if there is any such  "phase velocity" where velocity = wave_length x frequency. We know electrical signals along conducting lines travel almost near light speed. So what is the general  relation of voltage = function(x, t) for x along a power line?     

[evan_au]

Phase velocity is an important factor in designing microwave transmission lines and optical fibers, where:
- the length of the transmission line is longer than a wavelength
- transmission occurs over a broad range of frequencies, so dispersion is a problem
- dispersion is where different frequencies travel at different velocities, distorting the waveform
https://en.wikipedia.org/wiki/Phase_velocity

However, in an AC power distribution network (eg 50 or 60Hz):
- The wavelength is around 3,500km
- There are very few transmission lines this long
- The frequency carried is almost purely the base frequency (50 or 60Hz)
- With a few harmonics from electronic circuits (eg 150 or 180Hz), which should be controlled at the point of generation (eg variable-speed lift motors).

So the AC power network does not need to worry too much about phase velocity, but they do need to match the phase of generated electrical power in different parts of the network, otherwise power will be wasted in the distribution network, instead of being sold to the end-user.

To manage geographically-widespread power grids, they are often linked by High-Voltage DC links, which don't have these phase problems.

[theThinker (OP)]

Phase velocity is an important factor in designing microwave transmission lines and optical fibers, where:
- the length of the transmission line is longer than a wavelength
- transmission occurs over a broad range of frequencies, so dispersion is a problem
- dispersion is where different frequencies travel at different velocities, distorting the waveform
https://en.wikipedia.org/wiki/Phase_velocity

However, in an AC power distribution network (eg 50 or 60Hz):
- The wavelength is around 3,500km
- There are very few transmission lines this long
- The frequency carried is almost purely the base frequency (50 or 60Hz)
- With a few harmonics from electronic circuits (eg 150 or 180Hz), which should be controlled at the point of generation (eg variable-speed lift motors).

So the AC power network does not need to worry too much about phase velocity, but they do need to match the phase of generated electrical power in different parts of the network, otherwise power will be wasted in the distribution network, instead of being sold to the end-user.

To manage geographically-widespread power grids, they are often linked by High-Voltage DC links, which don't have these phase problems.

Holy cow! "The wavelength is around 3,500km!"

But how - in the first place - do we know what is phase velocity in a power cable. Can you give a link to the circuit theory involved. 

[Colin2B]

But how - in the first place - do we know what is phase velocity in a power cable. Can you give a link to the circuit theory involved. 

As @evan_au points out, phase velocity isn’t appropriate here as we are talking single frequency and hence no dispersion. Propagation speed can be measured and is an undergrad lab experiment - compare input signal into a long cable loop with output signal, you will get approx 90% speed of light depending on quality of copper and cable construction. Most often the measurements are done on coax and other transmission cables. You can work out the velocity factor as % = 100/√ε₀ε_r

[Bored chemist]

100/√ε0εr

Sure about that?

[Colin2B]

Yes, but it depends on definitions and I wasn’t specific, and should have been.
Usually quoted as a fraction 1/√ε_eff ie effective dielectric constant, but some use ε_r for effective or just ε, others use a relative dielectric constant ε_r where ε_eff = ε_r x ε₀.

[evan_au]

Sure about that?

The classic calculation for the speed of light in a vacuum:
c=1/SQRT(ε₀μ₀)

Electrical transmission lines normally have a capacitance constant (ε) very different from ε₀ and a inductive constant (μ) slightly different from μ₀, so the propagation velocity in this medium is calculated as:
v=1/SQRT(εμ)

See: https://en.wikipedia.org/wiki/Speed_of_light#Propagation_of_light
...and the subsequent section has a nice diagram showing the difference between group velocity and front velocity.