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Detailed numerical calculations are presented in this note. 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.
This basic structure can be considered as the proto-type of simulators considered in other notes 3 4. 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 key modified to a greater or lesser extent, depending on the degree of the ground proximity. 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 field distribution of a semi-infinite parallel plate transmission line placed in the proximity 211477 a perfectly conducting plane ground is solved rigorously by confurJllli l transformation. The derivation and the essential formulas are presented in the Appendix. The theoretical results so obtained are discussed in Section 2.
In Section 3, results of numerical computations are presented for field-line distribution and electric field intensity for several ground proximities. 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 the distribution of the field lines and electric field intensities.
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. The geometric configuration oey such a system is shown in Fig.
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The plate separation distance is designated by 2b and the distance between the near edge of the plates to the ground by d. If the ground and the plates are assumed to be perfectly conducting, the lley can support 1.
The “proxinmity factor” to appearing in eq. Thus E E 1 ly b Yuniform It seems appiropriate, therefore, to normalize the field intensity of the tr-ansmission line with respect to the uniform field.
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.
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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 relative electric field intensity on the ground is found from eq.
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. 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 in this section in graphical form with a view to showing the parametric effects on the field distributions.
The dimension of the length of the transmission line system i. 2147 mapping of the flux functions 2117 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.
Since the method requires an 8. Because of a large number of graphs to be presented, an individual detailed caption will not be shownoneach graph.
Instead we present a table showing the parameters involved and the Figure numbers related to those parameters.
Lines In this section we present the electric and magnetic field lines of a semiinfinite parallel plate transmission line near a perfectly conducting 21147 ground, i. The dotted lines superimposed on each figure represent the field lines of the similar transmission line without the ground in the Figs. It is also seen 9. Henceforth, by the field intensity, we mean the relative field intensity.
It is not readily apparent from this equation how Eyre and Exrel would behave there. In order leh examine the behavior of the x- and y-component of Erel in a close 2147 of the singular point, one needs to combine the eqs. This was done solely for the sake of graphical clarity. Similarly, the x-components of the electric field intensities of the transmission line are shown in Figs.
ExrorI db – 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 21417 separation of the ground is less than one-half the separation distance of the parallel plates i.
I Exreil without ground.
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. 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. The ground effect is negligible in the region Aa E YrcI for dib 0.
Ground PI,-ne I 0 I Upper Plate y b 0. Center Platie – 1. C0 CC 4’4 -0 0 co co U ppe, —r 1. 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. 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 the results shown in Sections’ 2 and 3, the following observations are made for field lines and electric field intensities. Acknowledgment The authors wish to thank Captain Carl E. 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.
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 configuration of the line segments so transformed in the t-plane is shown in Fig.
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The coordinates of the corresponding points in each plane effected by the successive transformations are tabulated below. The point to in the t-plane corresponds to the point C in the z-plane Fig. A-1the coordinates of which represent the separation distance between the transmission line system and the plane ground. A-1 is represented by the differential equation 21417 semi-infinite parallel plate transmission line system in the z-plane in the presence of the perfectly conducting ground.
The use of these in A. Ltinr the real and imaginary parts 21174 both sides, one finds 2b C1 7r A. In this case, A.