A transmission line which meets the Heaviside condition, named for Oliver Heaviside (1850–1925), and certain other conditions can transmit signals without dispersion and without distortion. The importance of the Heaviside condition is that it showed the possibility of dispersionless transmission of telegraph signals.In some cases, the performance of a transmission line can be improved by adding inductive loading to the cable.
The condition
A transmission line can be represented as a distributed-element model of its primary line constants as shown in the figure. The primary constants are the electrical properties of the cable per unit length and are: capacitance C (in farads per meter), inductance L (in henries per meter), series resistance R (in ohms per meter), and shunt conductance G (in siemens per meter). The series resistance and shunt conductivity cause losses in the line; for an ideal transmission line, .
The Heaviside condition is satisfied when :
An ideal line trivially meets the Heaviside condition.
Background
A signal on a transmission line can become distorted even if the line constants, and the resulting transmission function, are all perfectly linear. There are two mechanisms: firstly, the attenuation of the line can vary with frequency which results in a change to the shape of a pulse transmitted down the line. Secondly, and usually more problematically, distortion is caused by a frequency dependence on phase velocity of the transmitted signal frequency components. If different frequency components of the signal are transmitted at different velocities the signal becomes "smeared out" in space and time, a form of distortion called dispersion.
A transmission line is dispersionless, if the velocity of signals is independent of frequency. Mathematically .
A transmission line is distortionless if it is dispersionless and the attenuation coefficient is independent of frequency. Mathematically .
This was a major problem on the first transatlantic telegraph cable and led to the theory of the causes of dispersion being investigated first by Lord Kelvin and then by Heaviside who discovered in 1876 how it could be countered. Dispersion of telegraph pulses, if severe enough, will cause them to overlap with adjacent pulses, causing what is now called intersymbol interference. To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of baud. This is an exceptionally slow data transmission rate, even for human operators who had great difficulty operating a morse key that slowly.
For voice circuits (telephone) the frequency response distortion is usually more important than dispersion whereas digital signals are highly susceptible to dispersion distortion. For any kind of analogue image transmission such as video or facsimile both kinds of distortion need to be mitigated.
Derivation
The transmission function of a transmission line is defined in terms of its input and output voltages when correctly terminated (that is, with no reflections) as
where represents distance from the transmitter in meters and
.
are the secondary line constants, α being the attenuation constant in nepers per metre and β being the phase constant in radians per metre. For no distortion, α is required to be independent of the angular frequency ω, while β must be proportional to ω. This requirement for proportionality to frequency is due to the relationship between the velocity, v, and phase constant, β being given by,
and the requirement that phase velocity, v, be constant at all frequencies.
The relationship between the primary and secondary line constants is given by
If the Heaviside condition holds, then the square root function can be carried out explicitly as:
where
.
Hence
. . . Velocity is independent of frequency if the product is independent of frequency. Attenuation is independent of frequency if the product is independent of frequency.
Characteristic impedance
The characteristic impedance of a lossy transmission line is given by
In general, it is not possible to impedance match this transmission line at all frequencies with any finite network of discrete elements because such networks are rational functions of jω, but in general the expression for characteristic impedance is complex due to the square root term. However, for a line which meets the Heaviside condition, there is a common factor in the fraction which cancels out the frequency dependent terms leaving,