Shortening factor

The title of this article is ambiguous. This is about signals on lines, not antennas .

The shortening  factor VKF (also NVP from English Nominal Velocity of Propagation ) is a dimensionless number of lines . It is defined as the ratio of the signal speed on a line to the speed of light . In the limit case of high frequencies, it corresponds to the reciprocal of the refractive index for homogeneous optical propagation media , but depends not only on the material, but also on the geometry of the line cross-section.

Determination and typical values

The VKF can be determined experimentally by time domain reflectometry . For this purpose, the signal propagation time is determined which a very short square pulse needs to run through the cable.

High propagation velocities (and low losses) can u. a. in coaxial cables . An inner conductor is held in place by a foamed dielectric . The low permittivity of the dielectric reduces according to u. G. Formulas determine the capacitance per unit length of the line and thus increase the shortening factor . Some values ​​for high frequency cables:

Cable type	VKF
Open band line	95-99%
Belden  9085 (ribbon cable)	80%
RG-8X Belden 9258 (coaxial cable)	82%
RG-213 CXP213 (coaxial cable)	66%

Conversely, highly permittive material is used for delay lines with a particularly low shortening factor.

calculation

The shortening factor is calculated as:

${\displaystyle \mathrm {VKF} ={\frac {v_{\mathrm {p} }}{c}}}$

With

• the speed of light ${\displaystyle c}$
• the phase velocity of the electromagnetic wave . It is the quotient of the vacuum speed of light and the effective refractive index of the medium: ${\displaystyle v_{\mathrm {p} }}$ ${\displaystyle n}$

${\displaystyle v_{\mathrm {p} }={\frac {c}{n}}={\frac {c}{\sqrt {\varepsilon _{\mathrm {r} }\,\mu _{\mathrm {r} }}}}}$

with the two sizes

• Effective permittivity number ${\displaystyle \varepsilon _{\mathrm {r} }(f)}$
• Permeability number of the medium ${\displaystyle \mu _{\mathrm {r} }(f)}$

Both permittivity and permeability depend on the frequency of the signal under consideration.

Insertion into the formula of the shortening factor results in:

${\displaystyle \Rightarrow \mathrm {VKF} ={\frac {1}{n}}={\frac {1}{\sqrt {\varepsilon _{\mathrm {r} }\,\mu _{\mathrm {r} }}}}}$

To calculate the shortening factor, short square-wave pulses are considered, which correspond to high frequencies at which a limit value is approaching. ${\displaystyle \varepsilon _{\mathrm {r} }}$

For most cables (e.g. copper , aluminum ) the following applies:

${\displaystyle \mu _{\mathrm {r} }\approx 1\Rightarrow \mathrm {VKF} \approx {\frac {1}{\sqrt {\varepsilon _{\mathrm {r} }}}}}$

The following applies to a lossless line :

${\displaystyle v_{p}={\frac {1}{\sqrt {L'\,C'}}}\Rightarrow \mathrm {VKF} ={\frac {1}{c{\sqrt {L'\,C'}}}}}$

With

• the capacity coverage ${\displaystyle C'}$
• the inductance of the line ${\displaystyle L'}$

literature

• Klaus W. Kark: Antennas and radiation fields . Electromagnetic waves on lines in free space and their radiation, Springer-Verlag, Berlin / Heidelberg 2016, ISBN 978-3-658-13965-0 .
• Andres Keller: Broadband cables and access networks. Technical principles and standards. Springer-Verlag, Berlin / Heidelberg 2011, ISBN 978-3-642-17631-9 .
• Frieder Strauss: Basic course in high frequency technology. An introduction. 2nd Edition. Springer Verlag, Berlin / Heidelberg 2016, ISBN 978-3-658-11899-0 .

Individual evidence

1. ↑ Chapter 22: Component Data and References . In: H. Ward Silver, N0AX (Ed.): The ARRL Handbook For Radio Communications , 88th. Edition, ARRL , 2011, ISBN 978-0-87259-096-0 , p. 22.48.
2. ↑ Television technology without ballast, Otto Limann, Franzis-Verlag 1973, ISBN 3-7723-5270-7 , p. 179 / transit time cable
3. ↑ Nominal Propagation Velocity. Retrieved July 23, 2015 . 

Web links

• List of impedance and shortening factor of the most common cables (amateur radio info: Afug-Info.de)
• Reduction factor for antennas and coax cables (accessed on November 16, 2017)