Characteristic Impedance

Author: Edmund A. Laport

Any type of feeder with only air dielectric between all the conductors of the system has a characteristic impedance that is determined wholly by the geometry of its cross section, that is, the sizes and shapes of the conductors, their mutual spacings, and their distance from ground or other conducting planes or surfaces. When the entire field of the system is contained within a dielectric material with an inductivity e, Eq. (3) applies. When the conductors have a thin covering of insulating material, the remaining dielectric being air, the presence of the insulation on the wires has the effect of slightly increasing the effective metallic radius of the wires by an amount proportional to the inductivity of the material. For very thin insulating coverings, or for materials of very low inductivity, this effect is usually negligible in practice. In so far as line losses are concerned, any insulation on the wires increases the attenuation. This is because the loss factors for any solid dielectric material are greater than air alone and because the potential gradients are maximum at the surfaces of the conductors. The amount of increase thus caused must usually be determined empirically and in some cases can be negligible. Beyond a point, loss can affect the value of the characteristic impedance in magnitude and phase angle.

The presence of supporting insulators, or insulators used for maintaining constant spacing, has the effect of increasing the capacitance of the wire system and therefore of reducing its characteristic impedance and the velocity of propagation. In many open-wire lines this effect is virtually negligible (though it should be considered quantitatively in all applications where the electrical length of a line is critical). In enclosed lines with air dielectric, the spacing insulators always have a considerable influence on the characteristic impedance and the propagation velocity and therefore must always be considered.

The manner in which the characteristic impedance is derived from the cross section of a line is developed in Chap. 6, which gives several examples of the computation of characteristic impedance for different types of lines.

In typical practice, round or cylindrical conductors in the form of wires or tubes are used for the conductors. A uniform transmission line means that its cross section is identical at every point throughout its length, and if it be an open-wire line, it is assumed to be straight. In practice, the supporting members, insulators, small variations in cross-section dimensions, corners or bends in the conductors, and the close proximity of other metallic or dielectric objects cause variations in the characteristic impedance. The importance of these irregularities depends upon their magnitude and also upon the frequency of the guided energy. The electrical distance between identical irregularities (such as supporting members) also affects Z₀. If there are at least seven such uniformly spaced irregularities per wavelength, their effect is the same as a uniformly distributed shunt capacitive loading, which reduces the characteristic impedance slightly below its theoretical idealized value and reduces the propagation velocity.

When the frequency is so high that there are fewer than about seven uniformly spaced irregularities per wavelength, there is reflection of energy between these successive points, if the irregularities are sufficiently large, which causes the input impedance to oscillate above or below its characteristic impedance at different frequencies. This is a condition to be avoided because it also increases the attenuation per unit length of line and presents a complex impedance to the generator (regardless of the load termination), which varies with frequency. In this condition the line is said to be "lumpy."

It is possible to reduce the lumpiness of line impedance in the critical region of reflection when open-wire lines are used. For instance, the presence of insulators, binding wire, metallic insulator caps, and assembly hardware produces the effect of an increase in shunt capacitance over that due to the wires alone in air between supporting poles. The resulting capacitive loading may be of such a magnitude as to constitute a substantial irregularity, especially if the frequency to be transmitted is high.51 Then, for the short section of line near the pole, Z₀ is reduced. All that is required to maintain constant characteristic impedance is to maintain a constant ratio of inductance to capacitance per unit length. A unit of length near a supporting pole has an increased capacitance. Therefore, if the spacing can be increased, or the conductors reduced in diameter, to increase the unit-length inductance by an identical amount, the irregularity due to the support is neutralized and a truly uniform line results.

In practical construction it is usually necessary to change the direction of a feeder. A corner or angular bend introduces an irregularity by changing the series inductance of the line at and near the bend owing to interlinking of fields. For this reason, good construction requires that the bend be gradual and devoid of sharp corners. When such bends are made, it is necessary to use exactly the same length of wire on the two sides of a balanced line. It is more important to maintain equal wire lengths on the two sides of a circuit in making a bend than to try to maintain a strictly constant spacing between wires at the bend.

When feeders must pass through switching devices, great ingenuity must be used to avoid discontinuities of magnitudes that are disturbing in their effect on the line impedance. Any dead-end portions of transmission line also produce impedance irregularities that can be very troublesome in switching devices.

In solid-dielectric feeders (both flexible and rigid), there is a close approach to the ideally uniform line. Where connections or junctions are made with other sections of feeder, the connectors may introduce local irregularities. For most standard solid-dielectric feeder material it is possible to obtain constant-impedance connectors that have very little effect on the uniformity of the feeder impedance.