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Linear Electrical Constants of Cables

The linear electrical constants of cables are the series resistance R, the series self-inductance L, the shunt capacitance C, and the shunt conductance G. In discussing these constants, two types of cables must be considered: those used for local or exchange purposes and those used for toll and long-distance purposes. Also, the cables must be considered at audio frequencies and at the higher carrier frequencies.

These so-called constants are of such a nature that they should be called parameters.4 They are determined from measurements on actual short cable sections.4 Much of the following information is from reference 4.

Series Resistance R. Both skin effect (page 215) and proximity effect cause the alternating-current or effective resistance to be greater than the direct-current resistance. Proximity effect is defined1 as "the phenomenon of non-uniform current distribution over the cross-section of a conductor caused by variation of the current in a neighboring conductor."

The effective resistance of a cable pair at various frequencies can be computed by means of the curves of Fig. 1. For instance, a 19-gauge conductor at 20° centigrade has a direct-current resistance of about 8 ohms per 1000 feet. At a carrier frequency of 51,200 cycles, the corresponding value of B is B = sqrt(51200/8) = 80. If skin effect only is to be considered, the effective resistance would be about 1.12 times the direct-current resistance, or about 8.96 ohms per 1000 feet. If the proximity effect of the other wire of the pair is to be considered and if k of Fig. 1 is 0.4 (which is about the ratio actually measured in a cable4), then the effective resistance at this same frequency is about 1.3 times the direct-current resistance, or about 10.4 ohms per 1000 feet. Proximity effects of wires other than the companion wire of the pair have been neglected in Fig. 1.

The variations in the effective resistance of a cable pair with changes in temperature are computed4 as outlined on page 217. Values of the effective resistance for 19-gauge cable pairs, which are widely used in toll and long-distance circuits, are shown in Fig. 2.

series resistance of telephone cable
Figure 1. Curves for determining the series resistance of telephone cable conductors when skin effect only is considered (k = 0), and when both skin effect and proximity effect (k = 0.25, 0.3, 0.4) are considered. The factor k is the ratio of wire radius to distance between centers, expressed in the same units. For a typical telephone cable, k = 0.4. The value of B is the square root of the ratio of the frequency in cycles, divided by the direct-current resistance in ohms, per 1000 feet per wire. The direct-current resistance is multiplied by the resistance ratio to obtain the alternating-current resistance (see text). (Reference 4.)

series resistance per mile of telephone cable
Figure 2. Series resistance per mile for both wires of a 19-gauge telephone cable pair at various frequencies and temperatures. (Reference 4.)

Series Inductance L. The series self-inductance of two parallel wires is given by equation 69, page 217. This equation contains a term μδ that corrects for the skin effect. Another term should be added to

series self-inductance per mile of a telephone cable
Figure 3. Series self-inductance per mile for both wires of a 19-gauge telephone cable pair at various frequencies and temperatures, (Reference 4.).

this equation to correct for the proximity effect. For engineering purposes, however, it is advisable to obtain inductance values for cables from sources such as Table I, page 250, Table III, page 253, or from Fig. 3.

shunt capacitance per mile of a telephone cable
Figure 4. Shunt capacitance per mile for 19-gauge telephone cable pairs at various frequencies and temperatures. (Reference 4)

Shunt Capacitance C. The capacitance between two parallel wires is given by equation 70, page 218. This equation applies to the two wires in free space and does not apply with accuracy to twisted pairs surrounded by other wires and in a conducting lead sheath. Also, this formula is for an air dielectric, but the dielectric constant of the paper insulation used in cable pairs is from 1.7 to 1.9, depending on the amount of air and impurities contained in the paper.4 Frequency and temperature affect the dielectric constant in a complicated way,4 and thus again, for engineering purposes, it is advisable to obtain the values of cable capacitance from tables such as Table I, page 250, Table III, page 253, or from Fig, 4.

Shunt Conductance G. The exact nature of the losses that determine the shunt conductance of a cable pair is quite involved, perhaps even more so than for an open-wire line (page 219), Cables offer the advantage, however, that, with the exception of temperature effects, weather conditions have negligible influence on the shunt conductance. The nature and moisture content of the dielectric, the frequency of the

shunt conductance per mile for a telephone cable
Figure 5. Shunt conductance per mile for 19-gauge telephone cable pairs at various frequencies and temperatures. (Reference 4)

test voltage, the spacing of the wires, and the wire sizes are factors affecting the shunt conductance,4 When manufactured, cables are carefully dried to reduce conductance. The shunt conductance of cables is given in Table I, page 250, Table III, page 253, and in Fig. 5.

The data presented in this section have been for R, L, C, and G of twisted-pair cables to 100000 cycles. Such cables have been studied4 at higher frequencies, although at present they are seldom used at these higher frequencies.



Last Update: 2011-05-30