Phase velocity of electromagnetic waves and medium constants

In most of the literature, propagation velocity of electromagnetic waves along a transmission cable is shown as

  Vp = c / sqrt(us*es)                                        (1)
  here,
	Vp = phase velocity (m/s)
	c = speed of light in vacuum (m/s)
	  = 2.99792458e8 (m/s) (Footnote 1)
	us = relative permeability of medium
	es = relative dielectric constant of medium

(Footnote 2).

For example, in a vacuum us = es = 1, and it is natural that the propagation velocity is equal to the speed of light in a vacuum (Footnote 3).

In polyethylene-insulated coaxial standards meeting JIS C 3501 or MIL standards, both the conductor and insulator is non-magnetic, and the relative dielectric constant of polyethylene is 2.3, and therefore us = 1, es = 2.3 to yield

  Vp = c / sqrt(2.3) = 0.66 * c .

This speed is about 66% the speed of light in a vacuum and agrees with the JIS standard "shortening coefficient of wavelength 66+-2%".

The shortening coefficient of a wavelength is defined as

  shortening_coefficient_of_a_wavelength
	= wavelength_of_an_electromagnetic_wave_along_a_transmission_cable
		/ wavelength_of_an_electromagnetic_wave_in_a_vacuum

but from the standard relation for waves

  Vp = l*f                                                 (2)
  here,
	Vp = phase velocity (m/s)
	l = wavelength (m)
	f = frequency (Hz)

holds, the shortening coefficient of a wavelength is equivalent to the decrease in the ratio of velocity, and we confirm equation (1) is correct.

In the IEC standards, the velocity ratio

  velocity_ratio = velocity_of_electromagnetic_wave_along_a_transmission cable
	/ velocity_of_electromagnetic_wave_in_vacuum

that takes relativity into account is used instead of the shortening coefficient of a wavelength. I prefer to use the velocity ratio. However, in the old days, measuring the wavelength of electromagnetic waves was much easier than measuring velocity, and therefore, the concept of a shortening coefficient of a wavelength was natural.

In 1VHF (feeder cord for television reception), the conductor is 7/0.23 (seven bare 0.23mm-copper wires twisted) and the distance between conductors is about 7.5mm. The JIS C 3330 standard gives a representative value of capacitance 13pF/m and the shortening coefficient of a wavelength, 85%.

Since glasses-shaped PE (polyethylene) insulators are used to hold the conductors, precise calculations must be done numerically, for example, by using finite element methods. However, here we use the approximate capacitance for a twin lead

  C = 27.8*e/log(d/a)
  here,
	C = capacitance (pF/m)
	e = (apparent) relative dielectric constant of insulator
	d = distance between conductors (m)
	a = radius of conductor (m)

From the capacitance C = 8.8 pF/m when the dielectric is a vacuum (e = 1) and the representative value in the standard 13 pF/m, the apparent relative dielectric constant can be obtained as e = 1.3. Therefore, if equation (1) is correct the shortening coefficient of the wavelength would be about 88%. The error compared to the representative value in the standard is about 3%, and hence this is not a bad approximation.

The question is that equation (1) seems to have no problem when applied to real cables, but is this equation really correct?

Footnote 1 - Speed of light in a vacuum

The speed of light in a vacuum is not an observed value, but is a defined value. If you are not sure, this is a good chance to study the modern system of units.

Footnote 2

For example, out of the abundant literature, I have referred to the following:

1) Moriya Yoshiaki, Kazuo Seki, - High frequency wave measurement
(Tokyo Denki University Press) ISBN4-501-31950-X
page 3 

2) Lamont V.Blake,- Transmission Lines and Waveguides
(John Wiley 7 Sons, Inc.)
page 48

3) Transistor Gijutsu SPECIAL No.42,- Measurement of high speed digital circuits and trouble analysis
(CQ press)
page 20

In reference 1), the well-known approximation

  beta = w*sqrt(L*C)
  here,
	beta = phase constant (rad/m)
	w = 2*PI*f (rad/s)
	PI = 3.14159265..
	f = frequency (Hz)
	L = inductance of line (H/m)
	C = capacitance of of line (F/m)

applicable to normal high-frequency circuit conditions w*L >> R and w*C >> G and equation (2) is used to obtain

  Vp = f*l = 2*PI*f/beta = 1/sqrt(L*C)

Furthermore, "L and C of the transmission line is also given by

  L*C = u*e = us*es/c^2
  here
	c: speed of light
	u, e : permeability and dielectric constant of medium
	us, es: relative permeability and relative dielectric constant of medium ,

thus

  Vp = 2*PI*f/beta = 1/sqrt(L*C) = c/sqrt(us*es)

Therefore, both the transmission wavelength wl and the phase constant Vp is inversely proportional to the product of relative permeability us and relative dielectric constant ��s of the transmission line."

Reference 2) gives some more detail. "The rigorous formula, valid for all frequencies and for lines with a loss, is quite complicated. However, for either a lossless line or for a line with moderate loss operated at high (radio) frequencies, the formula can be simplified. It becomes

  Vp = 1/sqrt(L*C)

This formula is exact for a lossless line, but may be used without serious inaccuracy for the majority of lines at radio frequencies.

It was mentioned in Sec.1-2 that the velocity of a voltage wave on a transmission line is actually the velocity of the accompanying electromagnetic wave in the medium between the line wires. For a nondissipative medium, this velocity is given by the formula

  v = 1/sqrt(u*e)

where u is the magnetic permeability of the medium and e is the electric permittivity. It can be shown that these two different formulas for the wave velocity lead to the same result - that is, no matter what the configuration of the line, the product LC will always be equal to the product u*e."

This is basically the same thing. The bottom line is that, the relation LC = u*e holds for lines that are lossless or have some loss at high frequency, and equation (1) can be used to calculate the phase velocity.

Footnote 3

I was surprised to see the advertisement "Light is faster than electromagnetic waves" by a wire company in the first page of the Nikkei newspaper some time ago. Physically, light is a type of electromagnetic wave. Light seems like a special electromagnetic wave because of the functional limitations of human visual perception.

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