EFFECTS OF EARTH ON THE CAPACITANCE OF THREE PHASE TRANSMISSION LINES
Earth affects the capacitance of a transmission line because its presence alters the electric field of the line. If we assume that the earth is a perfect conductor in the form of a horizontal plane of infinite extent, we realize that the electric field of charged conductors above the earth is not the same as it would be if the equipotential surface of the earth were not present. The electric field of the charged conductors is forced to conform to the presence of the earth's surface. The assumption of a flat, equipotential surface is, of course, limited by the irregularity of terrain and the type of surface of the earth. The assumption enables us, however, to understand the effect of a conducting earth on capacitance calculations.
Consider a circuit consisting of a single overhead conductor with a return path through the earth. In charging the conductor, charges come from the earth to reside on a conductor, and a potential difference exist between the conductor and earth. The earth has a charge equal in magnitude to that on a conductor but of opposite sign. Electric flux from the charges on the conductor to the charges on the earth is perpendicular to the earth's equipotential surface, since the surface is assumed to be a perfect conductor. Let us imagine a fictitious conductor of the same size and shape as the overhead conductor lying directly below the original conductor at a distance equal to twice the distance of the conductor above the plane of the ground. The fictitious conductor is below the surface of the earth by a distance equal to the distance of the overhead conductor above the earth. If the earth is removed and a charge equal and opposite to that on the overhead conductor is assumed on the overhead conductor is assumed on the fictitious conductor, the plane midway between the original conductor and the fictitious conductor is an equipotential surface and occupies the same position as the equipotential surface of the earth. The electric flux between the overhead conductor and this equipotential surface is the same as that which existed between the conductor and the earth. thus, the purposes of calculation of capacitance, the earth may be replaced by a fictitious charged conductor below the surface of the earth by a distance equal to that of the overhead conductor above the earth. Such a conductor has a charge equal in magnitude and opposite in sign to that of the original conductor and is called the image conductor.
The method of calculating the capacitance by replacing the earth by the image of an overhead conductor can be extended to more than one conductor. If we locate an image conductor for each overhead conductor, the flux between the original conductors and their images is perpendicular to the plane which replaces the earth, and that plane is equipotential surface. The flux above the plane is the same as it is when the earth is present instead of the image conductors.
If the conductors is high above ground compared with distances between them, the diagonal distances in the numerator of the correction term are nearly equal to the vertical distances in the denominator and the term is very small. This is the usual case, and the effect of ground is generally neglected for three phase lines except for calculations by symmetrical components when the sum of the three line current is not zero.
Consider a circuit consisting of a single overhead conductor with a return path through the earth. In charging the conductor, charges come from the earth to reside on a conductor, and a potential difference exist between the conductor and earth. The earth has a charge equal in magnitude to that on a conductor but of opposite sign. Electric flux from the charges on the conductor to the charges on the earth is perpendicular to the earth's equipotential surface, since the surface is assumed to be a perfect conductor. Let us imagine a fictitious conductor of the same size and shape as the overhead conductor lying directly below the original conductor at a distance equal to twice the distance of the conductor above the plane of the ground. The fictitious conductor is below the surface of the earth by a distance equal to the distance of the overhead conductor above the earth. If the earth is removed and a charge equal and opposite to that on the overhead conductor is assumed on the overhead conductor is assumed on the fictitious conductor, the plane midway between the original conductor and the fictitious conductor is an equipotential surface and occupies the same position as the equipotential surface of the earth. The electric flux between the overhead conductor and this equipotential surface is the same as that which existed between the conductor and the earth. thus, the purposes of calculation of capacitance, the earth may be replaced by a fictitious charged conductor below the surface of the earth by a distance equal to that of the overhead conductor above the earth. Such a conductor has a charge equal in magnitude and opposite in sign to that of the original conductor and is called the image conductor.
The method of calculating the capacitance by replacing the earth by the image of an overhead conductor can be extended to more than one conductor. If we locate an image conductor for each overhead conductor, the flux between the original conductors and their images is perpendicular to the plane which replaces the earth, and that plane is equipotential surface. The flux above the plane is the same as it is when the earth is present instead of the image conductors.
If the conductors is high above ground compared with distances between them, the diagonal distances in the numerator of the correction term are nearly equal to the vertical distances in the denominator and the term is very small. This is the usual case, and the effect of ground is generally neglected for three phase lines except for calculations by symmetrical components when the sum of the three line current is not zero.
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