INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

The inductance of a transmission line is calculated as flux linkages per ampere. If permeability µ is constant, sinusoidal current produces sinusoidally varying flux in phase with the current. The resulting flux linkages can then be expressed as a phasor λ, and

 If i, the instantaneous value of current, is substituted for the phasor I in above eq., then λ should be the value of the instantaneous flux linkages produced by i. Flux linkages are measured in weber-turns, Wbt.

To obtain an accurate value for the inductance of a transmission line, it is necessary to consider the flux inside each conductor as well as the external flux. Let us consider a long cylindrical conductor whose cross section is shown in Fig. We assume that the return path for the current in this conductor is so far away that it does not appreciably affect the magnetic field of the conductor shown. Then, the lines of flux are concentric with the conductor.

  

By Ampere's law the magnetomotive force (mmf) in ampere-turns around any closed path is equal to the net current in amperes enclosed by the path. The mmf equals the line integral around the closed path of the component of the magnetic field intensity tangent to the path and is given by,

Where, H = magnetic field intensity, At/m

s = distance along path, m

I = current enclosed, A

Note that H and I are shown as phasors to represent sinusoidally alternating quantities.

Let the field intensity at a distance x meters from the center of the conductor be designated H_x. Since the field is symmetrical, H_x is constant at all points equidistant from the center of the conductor. If the integration indicated in Eq. above is performed around a circular path concentric with the conductor at x meters from the center, H_x is constant over the path and tangent to it. We get,

Where, I_x is the current enclosed. Then, assuming uniform current density, 

  Where, I is the total current in the conductor. Then, we obtain,

 The flux density x meters from the center of the conductor is,

Where, µ is the permeability of the conductor.

In the tubular element of thickness dx the flux dφ is B_x times the cross-sectional area of the element normal to the flux lines, the area being dx times the axial length. The flux per meter of length is,

 The flux linkages dλ per meter of length, which are caused by the flux in the tubular element, are the product of the flux per meter of length and the fraction of the current linked. Thus,

Integrating from the center of the conductor to its outside edge to find λ_int, the total flux linkages inside the conductor, we obtain, 

  For a relative permeability of 1, µ = 4π X 10^-7 H/m, and,

Hence, we have computed the inductance per unit length (henrys per meter) of a round conductor attributed only to the flux inside the conductor.