Line coverings

In addition to the line impedance, the electrical parameters of an electrical line include the so-called line coverings . Cable coverings describe the capacitance , inductance , series resistance in the direction of the line and the transverse conductance across the direction of the line of a line based on the line length. The cable coverings are length-related, summarized electrical properties depending on the material and geometry. As a first approximation, they are constants of a cable type - if this is in free space. For this reason they are also referred to as "primary line constants" (in contrast to the "secondary line constants", characteristic impedance and propagation constant ).

Equivalent circuit diagram

Equivalent circuit diagram for a line element of a two-wire line of length . For any length of the label is simplistic to , , and limited. ${\displaystyle \mathrm {d} x}$ ${\displaystyle L'}$ ${\displaystyle R'}$ ${\displaystyle G'}$ ${\displaystyle C'}$

The figure shows the equivalent circuit diagram of a line section that is repeated according to the infinitesimal length , see line theory . The resistance coating , the dissipation coating , the capacitance coating and the inductance coating represent the values evenly distributed over a homogeneous line with the length . To distinguish between the sizes of a discrete component (including a specific cable), the length-related cable coverings are marked with a dash. ${\displaystyle \mathrm {d} x}$ ${\displaystyle R'}$ ${\displaystyle G'}$ ${\displaystyle C'}$ ${\displaystyle L'}$ ${\displaystyle a}$

Toppings

The length of the line under consideration can be freely selected to determine the coverings of a homogeneous line. Information for a capacity of, for example

• 0.067 μF per km or 67 pF per m or 1 μF per 14.9 km

are permissible and equivalent to one another. A line is called homogeneous if its coverings are constant over its length.

Resistance coating R '

The resistance coating has the unit ohm per meter  . ${\displaystyle R'={\frac {R}{a}}={\frac {\mathrm {d} R}{\mathrm {d} x}}}$ ${\displaystyle [R']=\mathrm {\frac {\Omega }{m}} }$

It describes the ohmic resistance of an electrical line in relation to its length . With the specific resistance and the cross-sectional area applies to a single conductor . In fact, with a two-wire line, the forward and return conductors, which may have different parameters, must be taken into account. This is why, for example, for a symmetrical double line ${\displaystyle R}$ ${\displaystyle a}$ ${\displaystyle \varrho }$ ${\displaystyle A}$ ${\displaystyle R=\varrho \;{\frac {a}{A}}}$

${\displaystyle R'=2{\frac {\varrho }{A}}}$ .

This “direct current” calculation only applies to relatively low frequencies, because the current displacement by the skin effect reduces the penetration depth and the resistance layer increases with the frequency.

Discharge coating G '

The discharge coating has the unit Siemens per meter; . ${\displaystyle G'={\frac {G}{a}}={\frac {\mathrm {d} G}{\mathrm {d} x}}}$ ${\displaystyle [G']=\mathrm {\frac {S}{m}} }$

It describes the losses due to incomplete insulation per length. With the voltages and currents that typically occur , the relative current losses due to the discharge coating are significantly lower than the relative voltage losses due to the resistance coating.

Capacitance per unit length C '

The capacitance layer has the unit farad per meter; . ${\displaystyle C'={\frac {C}{a}}={\frac {\mathrm {d} C}{\mathrm {d} x}}}$ ${\displaystyle [C']=\mathrm {\frac {F}{m}} }$

It is the capacity of one line per length of this line. The capacitance per unit length can be calculated from the permittivity (formerly dielectric constant) and the geometry of the line arrangement. For example, a two-wire line with a wire diameter and a wire spacing has the capacitance per unit length ${\displaystyle \varepsilon }$ ${\displaystyle d}$ ${\displaystyle D}$

${\displaystyle C'={\frac {\varepsilon \cdot \pi }{\operatorname {arcosh} {\frac {D}{d}}}}\quad {\text{mit}}\quad \operatorname {arcosh} x=\ln(x+{\sqrt {x^{2}-1}})}$, or.

${\displaystyle C'={\frac {\varepsilon \cdot \pi }{\ln {\frac {2D}{d}}}}\quad {\text{wenn}}\quad d\ll D\ll a}$.

Often a low line capacitance is desired in order to e.g. B. to keep crosstalk from signal lines or the energy ( reactive power ) stored in supply networks during each network period low. This can be achieved through a low permittivity and / or a large wire spacing compared to the wire diameter . For significant problems due to the capacitance, see, for example, the 380 kV Transversale Berlin . ${\displaystyle \varepsilon }$ ${\displaystyle D}$ ${\displaystyle d}$

Inductance coating L '

The inductance coating has the unit henry per meter; . ${\displaystyle L'={\frac {L}{a}}={\frac {\mathrm {d} L}{\mathrm {d} x}}}$ ${\displaystyle [L']=\mathrm {\frac {H}{m}} }$

It represents the inductance value per length. For example, a two-wire line with a wire diameter and a wire spacing and in the gap with a permeability (formerly induction constant) has the inductance coating ${\displaystyle d}$ ${\displaystyle D}$ ${\displaystyle \mu }$

${\displaystyle L'={\frac {\mu }{\pi }}\operatorname {arcosh} {\frac {D}{d}}}$ , or.

${\displaystyle L'={\frac {\mu }{\pi }}\ln {\frac {2D}{d}}\quad {\text{wenn}}\quad d\ll D\ll a}$ .

While the internal inductance of the conductor must generally also be taken into account when calculating the inductance per unit length, this is not necessary at higher frequencies due to the current displacement caused by the skin effect.

Depending on the application and the required impedance, a high or low line inductance can be desirable. An example of increasing the line inductance to achieve a high impedance is the Krarup cable . Should the inductance coating - z. B. for the transmission of high current pulses - be as low as possible, this can be achieved by a low permeability or small distances between the forward and return conductors. Particularly low inductance layers can be achieved with strip conductors lying close together . However, as the wire spacing decreases, the capacitance per unit length increases to the same extent as the inductance per unit length decreases. A reduction in inductance can also be achieved by connecting several lines in parallel. ${\displaystyle \mu }$ ${\displaystyle D}$

application

The above-mentioned line impedance is determined by the line coverings when operating with sinusoidal alternating voltage and using the complex alternating current calculation (j is the imaginary unit here ):

${\displaystyle Z_{l}={\sqrt {\frac {R'+\mathrm {j} \omega L'}{G'+\mathrm {j} \omega C'}}}}$

literature

• K. Küpfmüller and G. Kohn: Theoretical electrical engineering and electronics . 16th edition. Springer, 2005, ISBN 3-540-20792-9 .

Individual evidence

1. ↑ ^{a b} Hans-Georg Unger : Electromagnetic waves on lines . Dr. Alfred Hüthig Verlag, Heidelberg 1980, ISBN 3-7785-0601-3 .