# Shortening factor

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 .