Heaviside condition

The Heaviside condition, named after Oliver Heaviside, is a condition that needs to meet an electrical line through the governance theory, to avoid distortions of the transmitted signal occur. The fulfillment of this condition can be accomplished by increasing the inductance of the line at transmission lines, as for example, was historically achieved by the so-called coil-loaded line.

Definition

A transmission line can be represented in the equivalent circuit diagram as a sum of line sections of infinitesimal length, as in the illustration. The electrical properties of the conductor are referred to: The inductance L ', the capacitance C' of the resistance per unit length R ' and conductance per unit length G'. The resistance per unit length and the conductance per unit length provide for losses in the line, therefore applies to an ideal transmission.

The Heaviside condition is satisfied if:

In a common line with distortion, however, usually applies:

Background

The signal of a transmission line can also be distorted with a linear transfer function. The frequency components of the signal are frequency dependent by their phase velocity. If different frequencies are transmitted at different speeds, " smeared " the signal ( dispersion). In addition, the attenuation of the line may vary with the frequency (e.g., by the skin effect ), so that the waveform is changed.

This was a big problem in the first transatlantic telecommunications cables, which led to the problem of dispersion through investigations by Lord Kelvin and was finally resolved by Heaviside, who changed his measures against it. If excessive dispersion, subsequent pulses may overlap and cause -symbol interference. To prevent this, the step speed to 1/15 baud had to be reduced. This is very slow even for the Morse transmission.

Derivation

The transfer function of a transmission line, in the absence of reflection is defined as:

With the propagation constant

Where α is the damping constant and β is the phase constant. In a distortion equal to 0 has α as a function of angular frequency ω to be constant, while β is proportional to ω. This is given by the speed

And the fact that the phase velocity is constant over all frequencies.

The ratio of the individual constants is given by

Which should be obtained with low distortion in shape. This applies only if do not differ and by more than a constant factor. Since both have a real and imaginary parts, these must differ by the same factor, so that applies:

And thus the Heaviside condition is satisfied.

Properties of the line

The characteristics of a line, which satisfies the condition Heaviside are:

Damping,

The phase constant,

And the phase velocity,

Characteristic impedance

The line impedance of a lossy transmission is given by

It is generally not possible, the transmission line at all frequencies accurately adapt, because the function is irrational by the root, so that it can not be represented as a network of discrete components. But when she meets the Heaviside condition, by a factor of

Which is frequency independent and purely real, the head can only be displayed with resistors at the ends.

This factor is the same as the lossless line ().