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Based on the theoretical and numerical results presented in Sections 2 and 3, conclusions are drawn in Section 4 on the effects of the perfectly conducting plane on ldy distribution of the field lines and electric field intensities. A third computer program was developed to calculate the field intensities at a set of predetermined points in the transmission line system by use of the Newton-Raphson method for the complex function.
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The dotted lines superimposed on each figure represent the field lines of the similar transmission line without the ground in the Figs. The coordinates of the corresponding points key each plane effected by the successive transformations are tabulated below. Let the field intensity at a point in the transmission line without the ground be denoted by Erel.
Since the subject of central importance posed by the problem is the effects of the ground proximity let the distribution of the field lines and electric field intensities, these quantities are presented in this section in graphical form with a view to showing the parametric effects 221147 the field distributions.
Center Platie – 1. In this case, A. Henceforth, by the field intensity, we mean the relative field intensity. The “proxinmity factor” to appearing in eq. A semi-infinite parallel plate transmission line system in the z-plane in the presence of the perfectly conducting ground.
The point to in the t-plane corresponds to the point C in the z-plane Fig. A-1 is represented by the differential equation It is also seen 9.
Because of a large number of graphs to be presented, an individual detailed caption will not be shownoneach graph. Ltinr the real and imaginary parts on both sides, one finds 2b C1 7r A. Ln this note the effect of the ground proximity on the field distribution will be discussed in order to establish such ground effects in explicit numerical terms.
When a semi-infinite parallel plate transmission line is in the proximity of a perfectly conducting plane ground, one expects the field distribution and impedance of a transmission line to be modified to a greater or lesser extent, depending on the degree of the ground proximity.
If the ground and the plates are assumed to be perfectly conducting, the line can support 1. Upper Plate y b 0. Detailed numerical calculations are presented in this note. Thus E E 1 uniform b Yuniform It seems appiropriate, therefore, to normalize the field intensity of the tr-ansmission line with respect to the uniform field.
Lines In this section we present the electric and magnetic field lines of a semiinfinite parallel plate transmission line near a perfectly conducting plane ground, i.
Conclusions The primary objective of this work is to investigate the effect of the ground proximity on the field distribution of a semi-infinite parallel plate transmission line. Theoretical Results In this section we present a theoretical discussion on field lines and electric field intnsiLies in a semi-infinite parallel plate transmission line system in the proximity of a perfectly conducting plane ground.
APPENDLX – The Conformal Transformation Our aim is to find the distribution of electric field and magnetic field lines of a semi-infinite parallel plate transmission line system which is placed at some arbitrary distance above the perfectly conducting plane ground as illustrated in Fig.
Since the method requires an 8. The plate separation distance is designated by 2b and the distance between the near edge of the plates to the ground by d. Ground PI,-ne I 0 I It is not readily apparent from this equation how Eyre and Exrel would behave there.
From the contour plots, it is observed that, as a whole, the presence of a perfectly conducting plane ground causes the field strength to be enhanced in a neighborhood of the upper edge and to be weakened in a region where the center plane intersects the ground plane. Instead we present a table showing the parameters involved and the Figure numbers related to those parameters.
Acknowledgment The authors wish to thank Captain Carl E. In particular, the field distribution and line impedance of a parallel plate transmission has been examined in detail by Baum and extended by Brown and Granzow. Assuming that the ground is a perfectly conducting plane, an exact electrostatic solution is obtained by conformal transformation for such a transmission line system.
The ground effect is negligible in the region U ppe, —r 1. le
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Similarly, the x-components of the electric field intensities of the transmission line are shown in Figs. The mapping of the flux functions were also computVlr-CLdCl1latud by feeding as an input to a program a set of appropriate values of u and v. The relative electric field intensity on the ground is found from eq. The geometric configuration of such a system is shown in Fig.
This region can be transformed by Schwartz-Christoffel transformation, onto the upper half of the t-plane with the line segments A-P-Q-C-D transformed into the real axis of the t-plane. The theoretical results so obtained are discussed in Section 2. The deviation is measured in two different ways: A systematic numerical computation of the electric field intensity of the transmission line has been carried out and the results are presented in Section 3.
It may be instructive, however, to consider briefly some special cases of interest such as field variations on the upper plate, on the center plane and at the ground.
The field distribution of a semi-infinite parallel plate transmission line placed in the proximity of a perfectly conducting plane ground is solved rigorously by confurJllli l transformation.
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This was key solely for the sake of graphical clarity. The configuration of the line segments so transformed in the t-plane is shown in Fig. ExrorI db –