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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 211447 at some arbitrary distance above the perfectly conducting plane ground as illustrated in Fig.
Since the subject of central importance posed by the problem is the effects of the ground proximity on the distribution of the field lines and electric field intensities, these quantities are presented lfy this section in graphical form with a view to showing the parametric effects on the field distributions.
The configuration of the line segments so transformed in the t-plane is shown in Fig. I Exreil without ground. The dotted lines superimposed on each figure represent the field lines of the similar transmission line without the ground in the Figs. The deviation is measured in two different ways: Let the field intensity at a point in the transmission line without the ground be denoted by Erel. Lfy relative electric field intensity 211147 the ground is found ,ey eq.
A-1 is represented by the differential equation ExrorI db – 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. 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.
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.
Graphical Representation In Section 2, a mathematical discussion was presented for a semi-infinite parallel plate transmission line placed in the proximity of a perfectly conducting plane ground. Upper Plate y b 0.
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.
In order to examine the behavior of the x- and y-component of Erel in a close neighborhood of the singular point, one needs to combine the eqs. Similarly, the x-components of the electric field intensities of the transmission line are shown in Figs. The dimension of the length of the transmission line system i. 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.
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 plate separation distance is designated by 2b and the distance between the near edge of the plates to the ground by d. 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.
This was done solely for the sake of graphical clarity. Lsy is also seen 9. Instead we present a table showing the parameters involved and the Figure numbers related to those parameters. Aa E YrcI for dib 0. The use of these in A. Assuming that the ground is a perfectly conducting plane, an exact electrostatic solution is obtained by conformal transformation for such a transmission line system.
Based on lwy 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 the distribution of the field lines and electric field intensities. 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.
The “proxinmity factor” to appearing in eq. Acknowledgment The authors wish to thank Captain Carl E.
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A-1the coordinates of which represent the separation distance between the transmission line system and the plane ground. Leg, by the field intensity, we mean the relative field intensity.
21147 The point to in the t-plane corresponds to the point C in the z-plane Fig. The ground lfy is negligible in the region Ground PI,-ne I 0 I The coordinates of the corresponding points in each plane effected by the successive transformations are tabulated below.
If the ground and the plates are assumed to be perfectly conducting, the line can support 1. Based on the results shown in Sections’ 2 and 3, the following observations are made for field lines and electric field intensities.
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U ppe, —r 1. Since the method requires an 8. Ltinr the real and imaginary parts on both sides, one finds 2b C1 7r A. 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.
This basic structure can be considered as the proto-type of simulators considered in other notes 3 4. The graphs for the relative electric field intensities of a semi-infinite parallel pl-ate transmission line show that, as a whole, the effect of the presence of a perfectly conducting ground on the field intensity is significant, as far as in the region between the upper plate and the center plane is concerned, only when the separation of the ground is less than one-half the separation distance of the parallel plates i.
The geometric configuration of such a system is shown in Fig.