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 is satisfied when the ratio of conductance G (in siemens per meter) to capacitance C (in farads per meter) equals the ratio of resistance R (in ohms per meter) to inductance L (in henries per meter)

\frac{G}{C} = \frac{R}{L}

This 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

Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}}

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,

\frac{G}{C} = \frac{R}{L}
G = \frac{RC}{L}

Substituting in,

Z_0=\sqrt{\frac{R + j\omega L}{RC/L + j\omega C}}
Z_0=\sqrt{\frac{LR + j\omega L^2}{RC + j\omega LC}}
Z_0=\sqrt{\left(\frac{L}{C}\right) \frac{R + j\omega L}{R + j\omega L}}
Z_0=\sqrt{\frac{L}{C}}

This is a real number, and independent of frequency. The line can therefore be matched with just a resistor at either end. This expression for Z_0 = \sqrt{L/C} 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.


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