Active resistance and internal inductance of a symmetric cable circuit

Lection 7

In order to determine the active resistance R and internal inductanceof a symmetric cable circuit according to expression (2.35) it is necessary to know components

and .

Longitudinal intensity component of the electric field can be defined from the Helmholtz equation.

In symmetric cable conductors are located close to each other, therefore in this case it is necessary to consider proximity effect. Then the Helmholtz equation will become:

,

In view of that conductors in the middle of a cable are covered by isolation (by dielectric) the equation (2.112) is necessary to solve for conductors and dielectrics separately.

Thus it will be as follows:

а) for metal

в) for dielectrics.

Considering a fact, that , since the factor of eddy currents (losses) , and of dielectric equals to zero, the equation (2.113) will become:

.

The decision of the above mentioned differential equation for metal will be following:

,

where and the modified Bessel functions of the first and second types of

n-th order; А, В, С, D — integration constants;

As the field in a conductor will increase from the centre to periphery, and function is decreasing with argument increase, it is necessary to accept that .As the result of symmetric conductors arrangement relatively to the horizontal axis, from which angleis counted, odd function is absent, therefore .Then, taking into account presence of п field components, we will receive expression for conductors:

. (2.116)

The magnetic field component according to a ratio (2.65) is equal:

. (2.117)

The received equations are similar to the equation for a single conductor of aerial line circuit. Difference is following: as the result of axial symmetry for an internal conductor, field change was not considered on and . If we consider proximity effect , as there appear components of a field (except the basic components of the first conductor field) at the result of fields interaction of the near located conductors.

For definition of integration constants we will write down expressions of the electric and magnetic fields in dielectric, that surrounding conductors. For dielectric the decision of the equation (2.114) is

. (2.118)

Magnetic field component:

, (2.119)

where , ,- integration constants for finding of which the following conditions are used:

ñ continuity of longitudinal components of electric field on the interface conductor – dielectric if ;

ñ continuity of tangential components of a magnetic field: if

ñ the total current law: ;

ñ correspondence of decrease and increase laws of magnetic fields for conductors а and б.

At figure 2.15, magnetic fields for identical conductors on a straight line connecting the centres of conductors, are equal among themselves: (if r) = (if (a-r)).

Knowing the integration constants, it is possible to define values of and on the conductors’ surface at ().

Figure 2.15 – Azimuthal magnetic fields of a symmetric circuit

To define resistance R, Ohm/km, and internal inductance L, G/km, in (2.35) we will substitute values of and , and after corresponding transformations we will receive:

, (2.120)

where - diameter of a conductor, mm;- distance between conductors, mm.

The equation for resistance of a circuit calculation consists of three components: resistance on a direct current , resistance at the result of skin effect and resistance at the result of proximity effect . It is true for resistance of a circuit calculation if we have pair twisting. If it is necessary to define resistance at other kind of twisting (the star or double pair) it is necessary to consider additional losses on eddy currents in other conductors of group for account of which the parameter р is introduced. Taking into account the twisting effect of conductors we introduce parameter, that can be changed in a range 1,02 – 1, depending on diameter of a cable.

The final equation for resistance of a symmetric cable calculation looks like following, Ohm/km,

(2.121)

where - resistance of losses in surrounding metal mass.

For pair twisting р =1, for star - р =5, for double pair- р=2. Values of ; ; ; ; are resulted in tab. 2.1

There available several quads in communication cables, as a rule. Conductors of the adjacent quads, adding additional losses on eddy currents, increase resistance of a circuit. Besides, resistance will increase at the result of losses in a metal cover. For definition of the additional resistance , equivalent to these losses, used data at kHz, resulted in tab. 2.4.

Table 2.4

Recalculation of losses in metal () for another frequency is made by the formula:

, (2.122)

where - the tabular data; - frequency, kHz.

Value of internal inductance of conductors was defined above. Inductance of a circuit as a total is defined by the sum of external and internal inductances: . Value of external inductance is defined under the formula (2.83). As , we will receive external inductance per 1 km:

. (2.123)

Then the total inductance of a symmetric cable circuit, G/km,

.

For low-frequency symmetric cables, at which it is possible not to consider proximity effect, resistance R, Ohm/km, and inductance L, G/km, are defined by the simplified formulas:

;

. (2.125)

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