Heaviside condition

The Heaviside condition , named after Oliver Heaviside is, a condition of an electric line under the line theory must satisfy, so that no distortions of the occurring signal to be transmitted. In the case of transmission lines, this condition can be met by increasing the inductance of the line, as was historically achieved, for example, by the so-called coiled line .

definition

Equivalent circuit diagram for a line element of a two-wire line of infinitesimal length dx.

A transmission line can be represented in the equivalent circuit diagram as a sum of line sections of infinitesimal length , as shown in the adjacent figure. The electrical properties of this conductor are based on : The inductance coating L ' , the capacitance coating C' , the resistance coating R ' and the dissipation coating G' . ${\displaystyle dx}$ ${\displaystyle dx}$

For an ideal lossless line, and . ${\displaystyle R'=0}$ ${\displaystyle G'=0}$

In the case of a real line, on the other hand, the resistance coating and the discharge coating cause losses and distortions on the line. Practically always applies ${\displaystyle R'\neq 0}$ ${\displaystyle G'\neq 0}$

${\displaystyle {\frac {G'}{C'}}\ll {\frac {R'}{L'}}}$

However, it is the Heaviside condition

${\displaystyle {\frac {G'}{C'}}={\frac {R'}{L'}}}$

is fulfilled, then the transmission takes place without distortion. In addition, it can be seen that in this case (with the same resistance and discharge coating) the losses on the line are minimal. The increase in L ′ required for this was previously achieved using Pupin coils .

background

The signal to be transmitted can also be distorted on a linear transmission line. The phase velocity of the frequency components of the signal is itself frequency-dependent due to its phase constant, which is non-linearly dependent on the frequency. If different frequency components are transmitted at different speeds, the signal “smears” ( dispersion ). In addition, the attenuation of the line can vary with the frequency (e.g. due to the skin effect ), so that the signal shape is changed.

This was a major problem with the first transatlantic telecommunications cables, which led to the problem of dispersion through research by Lord Kelvin , and which was eventually resolved by Heaviside, who considered measures to prevent it. If the dispersion is very large, successive pulses can overlap and lead to symbol cross-talk . To prevent this, the walking speed had to be reduced to 1/15  baud . This is very slow even for Morse transmission.

Derivation

In line theory , with the help of the complex alternating current calculation, it is shown that the relationship between the complex amplitude of a sinusoidal voltage wave between two points of a transmission line and the distance , excluding reflections, applies ${\displaystyle U(x)}$ ${\displaystyle \Delta x}$

${\displaystyle {\frac {U(x+\Delta x)}{U(x)}}=e^{-\gamma \Delta x}}$

Therefore the properties of the wave propagation are exclusively determined by the propagation constant

${\displaystyle \gamma =\alpha +j\beta ={\sqrt {(R'+j\omega L')(G'+j\omega C')}}}$

where the real part α is called the damping constant and the imaginary part β is called the phase constant.

If the wave is to be transmitted without distortion, then α must not be dependent on the angular frequency ω , while β must be proportional to ω . The latter means that the phase velocity

${\displaystyle v_{ph}={\frac {\omega }{\beta }}}$

is constant over all frequencies.

The square of the propagation constant

${\displaystyle \gamma ^{2}=(\alpha +j\beta )^{2}=(R'+j\omega L')(G'+j\omega C')\,}$

must give the shape if it is distortion-free . This is only the case if and do not differ by more than a constant factor. Since both have a real and an imaginary part, they must differ by the same factor so that applies ${\displaystyle (A+j\omega B)^{2}}$ ${\displaystyle (R'+j\omega L')}$ ${\displaystyle (G'+j\omega C')}$

${\displaystyle {\frac {R'}{G'}}={\frac {j\omega L'}{j\omega C'}}={\frac {L'}{C'}}}$

what the Heaviside condition is.

Properties of the distortion-free line

A transmission line that fulfills the Heaviside condition has the following characteristic features:

damping

The damping has the frequency-independent value of the direct current damping:

${\displaystyle \alpha ={\sqrt {R'G'}}}$

In particular, it can be shown that if the Heaviside condition is fulfilled, this becomes minimal with regard to the variation of the capacitance or inductance per unit length, which is just as practical as freedom from distortion.

Phase constant

The phase constant increases linearly with the frequency and corresponds to that of the lossless line:

${\displaystyle \beta =\omega {\sqrt {L'C'}}}$

Phase velocity

The phase velocity is constant and corresponds to that of the lossless line:

${\displaystyle v_{ph}={\frac {\omega }{\beta }}={\frac {1}{\sqrt {L'C'}}}}$

Therefore it does not differ from the group speed :

${\displaystyle v_{g}={\frac {d\omega }{d\beta }}={\frac {1}{\sqrt {L'C'}}}}$

Line impedance

The line impedance of a lossy transmission line is given by

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

It is generally not possible to adapt the transmission line precisely across all frequencies, since the function of the characteristic impedance is irrationally dependent on the frequency due to the square root , so that it cannot be represented as a network of discrete components. But if a line fulfills the Heaviside condition, then the wave resistance becomes frequency-independent and purely real. It corresponds to that of the lossless line as well as that of direct current:

${\displaystyle Z_{l}={\sqrt {\frac {L'}{C'}}}={\sqrt {\frac {R'}{G'}}}}$

Such a line can then be adjusted without reflection by only terminating it with ohmic resistors at the ends.

literature

• K. Küpfmüller and G. Kohn: Theoretical electrical engineering and electronics, an introduction . 16th edition. Springer, 2005, ISBN 3-540-20792-9 .
• Eugen Philippow : Fundamentals of electrical engineering . Academic publishing company Geest & Portig K.-G., Leipzig 1967.

See also

• Line equation
• Krarup cable