Heaviside condition
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The Heaviside condition, stated by Oliver Heaviside, is used in the construction of telegraph cables, etc. to balance the effects of the cable’s capacitance and inductance. It is a requirement for distortionless transmission of pulses through an electrical transmission line.
The condition depends on a number of electrical properties of the wire pair that forms the cable – see figure. These are the capacitance C (in farads per meter), the inductance L (in henries per meter), the series resistance R (in ohms per meter), and the shunt conductance G (in siemens per meter). The series resistance and shunt conductivity cause losses in the line; for an ideal transmission line, R = G = 0.
The Heaviside condition is satisfied if
which is achieved by adding capacitors, inductors, and resistors to the circuit as appropriate. In the past, in order to satisfy the Heaviside condition in long distance telephone lines, Pupin coils were added.
The telegraph line will give distortionless transmission if and only if it is matched at both ends; otherwise reflections will occur at the ends of the line. The characteristic impedance of a lossy transmission line is given by
This is complex, and a function of frequency, ω. In general, there is no way to match this transmission line at all frequencies. The line will therefore have a frequency-selective response, and tend to "smear" the telegrapher's rectangular pulses, which contain a broad range of frequencies, into wider non-rectangular shapes. Under the Heaviside condition, however, the above equation can be simplified to
which is a real number, and independent of frequency. The line can therefore be matched with just a resistor at either end. This expression for 
is the same as for a lossless line (R = 0,G = 0) with the same L and C, although the attenuation (due to R and G) is of course still present.
[edit] Example
An unshielded twisted pair cable could have C = 50 pF/m, R = 0.1 Ω/m, L = 0.5 μH/m, and G = 0 S/m. Nominally, Z0 = 100 Ω, but due to the resistance this is only a good approximation for 
= 32 kHz. At lower frequencies, this would become a problem for cable runs that are comparable to or longer than the wavelength, i.e. above 50 km at 4 kHz. Adding a shunt conductivity G = 10 μS/m would make the impedance independent of frequency. However, if the shunt is implemented as a discrete resistors at fixed spacings along the cable, it would affect the response at high frequencies where the wavelength is comparable to or smaller than the distance between the shunt resistors. At a 1 km spacing, this effect would become important in the 100 kHz range.
Alternatively, the inductance could be increased to about 5 mH/m by placement of loading coils at regular intervals, thus allowing a flat frequency response for voice frequencies from about 50 Hz. A side effect is that this would increase the cable impedance to 10 kΩ.
