US 3818383 A
An elliptical-to-rectangular waveguide transition has a passage of the cross-sectional shape formed by concave top and bottom walls of generally elliptical form and side walls of no concavity. The curvature of the top and bottom walls varies continuously along the length to produce matching to the shapes at the respective ends. The side walls are those of the rectangular end at that point and diminish to zero height at the elliptical end. Non-linear tapering of cross-sectional dimensions is employed to minimize reflections at the ends.
Description (OCR text may contain errors)
United States Patent [191 Willis June 18, 1974 ELLIPTICAL-TO-RECTANGULAR WAVEGUIDE TRANSITION  Inventor: Frank R. Willis, South Holland, Ill.
 Assignee: Andrew Corporation, Orland Park,
 Filed: Feb. 27, 1973  Appl. No: 336,195
 US. Cl. 333/21 R, 333/98 R  Int. Cl. I-I0lp U115  Field of Search 333/21 R, 21 A, 34, 98 R  References Cited UNITED STATES PATENTS 3,336,543 8/1967 Johnson et al. 333/21 R X 3,388,352 6/1968 Ramona! 333/98 R OTHER PUBLICATIONS Wide-l3and Rectangular to Circular Waveguide Mode and Impedance Transformer, IEEE Trans. On MTT, May 1965.
Primary E.ranzinerP-aul L. Gensler Attorney, Agent, or FirmWolfe, Hubbard. Leydig, Voit & Osann, Ltd.
 ABSTRACT An elliptical-to-rectzuigular waveguide transition has a passage of the cross-sectional shape formed by concave top and bottom walls of generally elliptical form and side walls of no concavity. The curvature of the top and bottom walls varies continuously along the length to produce matching to the shapes at the respective ends. The side walls are those of the rectangular end at that point and diminish to zero height at the elliptical end. Non-linear tapering of crosssectional dimensions is employed to minimize reflections at the ends.
9 Claims, I] Drawing Figures PATENTEDJUNI 8l974 5.818.383
SHEEN or 2 v PATENTED JUN 1 8 I974 SHEET 2 BF 2 ELLIPTICAL-TO-RECTANGULAR WAVEGUIDE TRANSITION This invention relates to waveguide transitions and more particularly to the internal shaping of transitions for coupling between elliptical and rectangular waveguide for dominant-mode transmission.
A variety of shapes have heretofore been used for the internal waveguide passage of connectors or transitions for coupling between elliptical and rectangular waveguide, as is commonly required in systems employing elliptical guide. Prior to the present invention, transitions for coupling waveguides of different shapes together appear to have been devised without consideration of the dominant-mode electric field geometry in the various cross-sections of the transition. The present invention flows from analysis of the alteration of the electromagnetic field pattern which must be produced in transition from dominant-mode propagation in one type of guide to dominant-mode propagation in the other type of guide, and lies in providing internal shaping of the transition which produces the most simple and direct desired alteration of the field pattern along the length of the transition, thus reducing the vswr of a transition of any given length or shortening the required length for any given vswr.
By comparison of the well-known dominant-mode field patterns of elliptical and rectangular guide it is seen that the primary function performed by the transition passage is to bend the electric field vector in regions laterally remote from the axis of the guide. In the rectangular guide the dominant-mode pattern in this region has electric field vectors parallel with that at the center, while in an elliptical guide the electric field direction is bent to a degree varying with lateral distance from the axis, the direction of any electric field vector at its termination on a waveguide wall being perpendicular to the wall. Converting to an elliptical waveguide pattern from a rectangular waveguide pattern (or vice versa) may thus be thought of as a process for bending the electric field vectors in the lateral regions.
Because of the boundary conditions for propagation (perpendicular termination of the electric vector and absence of field at the wall in the direction parallel to the wall), the most direct and simple deformation of the field from one shape .to the other is gradual increase of the curvature of the top and bottom walls from their straightness at the rectangular end to the ultimate semielliptical shape of each at the elliptical end. Ideally, the side walls should be shaped to intersect perpendicularly with the top and bottom walls thus formed, thus producing concave top and bottom walls and convex side walls in an ideal transition. However it has been found that the ideal performance is acceptably approximated without the necessity for the complexity of convex side walls, with the side walls at all cross-sectional points straight and parallel with each other. Stated otherwise, the important restriction in this respect is that the side walls be free of concavity. As will be apparent from the foregoing upon study, the height of the side walls inherently diminishes to zero at the elliptical end, with the described shaping of the top and bottom walls.
The improvement in intermediate cross-sectional passage shape of the invention is found to reduce the contributions to vswr of these cross-sectional shapes, so that the only relatively substantial contribution to vswr is that caused by the angular relation between the side walls and the connecting elliptical and/or rectangular waveguides, as found in previously known transitions. Such angular relations are eliminated as a further feature of the invention. The side walls are parallel to the axis at the extreme rectangular end and gradually curve to vary the width therebetween which is the H-plane dimension. The Eplane maximum dimension follows a curve having a change of sign of its second derivative, being parallel to the axis at both ends.
The above and further aspects of the invention will be better understood from further description in connection with the annexed drawing in which:
FIG. 1 is a plan view of a dominant-mode waveguide transition embodying the invention, partially in H- plane cross-section;
FIG. 2 is an E-plane section of the transition taken along the line 22 of FIG. 1;
FIG. 3 is an end elevational view of the transition at the rectangular end;
FIG. 4 is an end elevational view of the transition at the elliptical end;
FIG. 5 is a schematic x-y coordinate plot of an idealized transition cross-section shape;
FIG. 6 is a similar plot of a modified shape;
FIG. 7 is a graph or plot illustrative of E-plane shaping of the transition;
FIG. 8 is a generally similar illustration relating to the H-plane shape; and
FIG. 9, l0 and 11 are successive cross-sectional views taken along the correspondingly numbered sec-. tion lines of FIG. 2.
The embodiment of the invention illustrated in FIGS. 1 through 4 and 9 through 11 is of the type made by casting and machine-finishing, having a circular flange 20 with an elliptical aperture 22 at one end and a rectangular flange 24 with a rectangular aperture 26 at the other end, connected by a body portion 28. The flanges are suitably apertured and devised for conventional bolted coupling, with gasket seals (omitted from the drawing) provided for gas-tight connections to elliptical and rectangular guides.
As will be recognized, these features of the illustrated embodiment are no part of the invention, being shown and described solely for completeness. The novelty of the invention lies in the shape of the internal passage which forms the transition from the elliptical end 22 to the rectangular end 26. Prior to description of the shape of this passage, it will aid understanding to first discuss its underlying theoretical basis, in connection with FIGS. 5 through 8.
FIG. 5 is a generalized representation or diagram of a cross-sectional waveguide shape which is neither rectangular not elliptical but has electromagnetic field patterns for dominant-mode (or fundamental-mode) transmission which provide an ideal or optimum smooth transition from the. dominant-mode field pattern of an elliptical guide to the dominant-mode field pattern of a rectangular guide, or vice versa. Such a shape, as shortly discussed, is found to have a minimum of reflection (normally measured and referred to as vswr) in the transition or conversion of the field pattern from one form to the other. As shown in solid lines in FIG. 5, this shape is formed by concave top and bottom walls 30 and 32 and convex side walls 34 and 36. (The terms concave and convex" as used herein and in the annexed claims refer to shapes as viewed from the interior or axis, not as viewed externally.)
The ideal nature of this general configuration as an intermediate between an elliptical shape and a rectangular shape from the standpoint of electromagnetic theory in dominant-mode transmission derives from the configuration of dominant-mode electric field vectors shown as arrows in FIG. 5. (Relative dimensions and curvature magnitudes, etc., of the shapes shown in FIGS. 5 and 6 will be later discussed along with other portions of the showings of these FIGS.) The electric field pattern of the dominant mode in the laterally central region of the guide is the same in elliptical and rectangular guide and this portion of the field accordingly remains constant in direction throughout the transition length. It is the electric field pattern at the sides (the ends of the larger dimension) of the guide which most greatly distinguishes the elliptical guide pattern from the rectangular guide pattern. With the insertion of side walls 34 and 36 perpendicular at their ends to the top and bottom walls 30 and 32 of a guide originally elliptical (as represented by the dotted extensions in FIG. 5), the electric field pattern in the lateral regions remote from the center provides a very smooth transition at the lateral side walls between the extremely curved electrical field which exists in this region in the elliptical guide and the straight electric field vector, parallel to the direction at the center, which characterizes the rectangular guide.
The shape of the side walls 34 and 36, wherein they are perpendicular to the top and bottom walls 30 and 32 at the points of intersection of corners, is somewhat difficult to fabricate, but it is found that the ideal shape of FIG. 5 is closely approximated in performance as a practical matter by the shape shown in FIG. 6,. wherein the same elliptically concave top and bottom walls 30 and 32 are employed with straight side walls 38 and 40, i.e., that the smoothness of transition is practically attained by avoiding sidewall concavity.
The discussion thus far of the generalized half-way shape shown in FIG. 6 (more ideally in FIG. 5) is without reference to any specific relation to the shape and dimensions of the elliptical and rectangular terminations at the opposite ends of the transition, particularly the manner in which the underlying principle is employed for determining the cross-section of the transition in regions closely adjacent to the ends.
As is well-known in the art, the term elliptical as commonly applied to waveguide is merely an approximation, and does not necessarily imply a shape meeting the mathematical criteria of a true ellipse. The invention is applicable to any of the more or less oval-shaped configurations commonly called "elliptical in the waveguide art. However it is convenient for explanation of the invention, and for facilitating commercial application in production design, to consider the case where the end of the transition which couples to the elliptical waveguide is a mathematical ellipse, particularly in view of the fact that a truly elliptical shape can usually be selected to provide a satisfactory impedance match to most commercial elliptical wave-guide.
As shown by dotted lines in FIGS. 5 and 6, the concave top and bottom walls 30 and 32 are segments of a true ellipse having a major axis of length 2C and a minor axis of length 20, i.e., having major and minor semi-axes of length C and D, respectively. On the shown x-y coordinates with the center of the passage as origin, the semi-minor axis D is the y coordinate at the center (x 0). The half-width of the passage is X,
the value of x at the sidewall, so that the guide width or sidewall spacing is 2X. The parameters of any specific shape of the type thus described are of course completely specified by these quantities, an ellipse being fully specified by the major and minor axes, and the distance between the straight and parallel side walls completing the detailed specification without necessity for reference to the vertical height of the walls. It will be observed that both a rectangle and a full ellipse may be considered as limiting cases of the geometry thus generalized. The rectangular end of the transition. of long dimension A and short dimension B. constitutes the generalized shape of FIG. 5 with D equal to onehalfB and Cinfinite, and with x equal to one-halfA. At the elliptical end, the parameters C and D are the semiaxes of the ellipse at this end, and X is equal to C. At the latter end, the sides 38 and 40 are of course of zero height or non-existent.
Thus for transition from any given rectangle of dimensions A and B to an ellipse of major and minor axes 2a and 21), intermediate cross-sections of the passage along the transition desirably employ successive intermediate values of C, D and X between these limiting values. The top and bottom walls at any point along the length conform to the ellipse equation:
with C infinite at the rectangular end and equal to a at the elliptical end, and with D equal to one-half B at the rectangular end and equal to b at the elliptical end. The side walls are in the form of parallel lines spaced by 2X. equal to the rectangle width at the rectangular end and to the major axis at the elliptical end.
The exact manner of varying the quantities C, D and X between the terminal values is not highly critical as regards interdependence. For best performance of a transition of any given length, monotonic variation of these quantities throughout the length is of course desirable. However it is undesirable to employ a gradation or taper which produces an angular relation such as that which results at one or both ends from linear dimensional tapering as employed in prior art transitions. For example, if values of D vary linearly from one end to the other, an angle is formed at each end in the laterally central region of the guide (the region of the center E-vector in FIG. 5), and a source of reflections is produced.
FIGS. 7 and 8 show plots of D and X, respectively, as a function (later discussed more specifically) of z, the distance along the guide from the rectangular end to the elliptical end. The total length of the guide or transition is shown as L. These plots of course correspond to longitudinal configuration of the transition in the E- plane and l-I-plane respectively. As illustrated, to correspond with the embodiment of FIGS. 1 through 4, both D and X are greater at the elliptical end than at the rectangular end. However the principles are equally applicable where one or both dimensions of the rectangular end are larger than the corresponding dimension at the elliptical end. As may be seen from FIG. 7, the lengthwise transition in E-plane dimension (the direction of the electric field vector in the central region in dominant-mode transmission) is made without angular discontinuity by employing a dimension which is a function of length having a change of sign of second derivative at some place along the length, with a zero first derivative at each end. The flare or taper of the H- plane dimension (perpendicular to the central electric field vector of the dominant mode) is likewise nonlinear, with a zero first derivative at the rectangular end. At the elliptical end, however, although this may likewise terminate in a manner smoothly matching the major axis of the elliptical guide (i.e., with a zero first derivative at this end) if so desired, such shaping is unnecessary at this point. The apparent angular relation of the side wall of the transition to the corresponding portion of the elliptical guide at the point of juncture seen in FIG. 8 cannot cause reflection by reason of the fact, not observed in FIG. 8, that the height of the side wall tapers to zero at this point so that the angle discontinuity is merely apparent rather than existent. Accordingly, although it is important that the shaping of the taper of X at the rectangular end correspond to a zero derivative, the function defining the manner of variation of X at the other end has no similar requirement.
The end-to-end shaping thus generally described may obviously be accomplished with a large variety of detailed implementations, particularly when it is considered that there is no necessity that the variation of curvature of the top and bottom walls along the length conform to segments of true mathematical ellipses. Accordingly the invention may be practiced in its broader aspects without mathematical expression of intermediate cross-sectional shapes reached by application of the above principles. However for commercial manufacture, wherein transitions have to be made from time to time for matching a variety of elliptical guides to a variety of rectangular guides, it is desirable to implement the generalized shapes described above by reasonably simple mathematical expressions from which a transition may be constructed without substantial experimentation or subjective evaluation of the relative merit of passage-tapering geometries devised by artistic drawing or instinctive fabrication. The transition shape at any given point may be calculated from fairly simple formulae which have been devised to meet the above criteria. Formulae experimentally verified to be highly suitable for this purpose are as follows:
D (8/2) cos (1rz/2L) [2 sin (1rz/2L) X a (a A/2) cos (1rz/2L) The calculated values of C and D of the above formulae are used in the general ellipse equation earlier given to obtain an x-y plot for a sufficient number of spaced cross-sections to serve as an adequate guide in fabrication.
The selection of the total length L of the transition is made in conventional fashion, except that shorter length may be used for a low vswr value. It will be observed that for any given elliptical and rectangular ends the equations above are universal as regards the length of the transition to which they are applicable, the 2 variable being in essence in terms of fractional portions of the total length L, whatever that may be.
FIGS. 9, l and 11 show the resulting cross-sectional shapes of the passage defined by top and bottom walls 44 and 46 and side walls 48 and 50 in the specific construction whose general features were earlier described. The three cross-sections shown were calculated from the above expressions for a particular inch and a short dimension of approximately inch, the elliptical end having a major axis of approximately 1.6 inch and a minor axis of approximately 0.85 inch. The total length of the transition illustrated is approximately 3.2 inches. Although all the parameters used in determining the shape at each cross-section vary monotonically along the length, this does not result, in the present instance, in a monotonic change in the height of the side walls 48. Both from FIG. 2 and from comparison of FIGS. 9, l0, and 11, it will be seen that in this embodiment the maximum height of the side walls 48 and 50 occurs in an intermediate region rather than at either end.
As in the case of prior art transitions, one or more tuning screws (not shown) may be used to further reduce vswr, either at particular frequencies or over an entire band. The present transition shape, where tuning screws are employed, greatly reduces the tuning required for making the vswr wholly negligible over a wide band of frequencies.
As in the case of the broader aspects of the invention, utility of the above formulae is not confined to the type of embodiment specifically illustrated, wherein the elliptical guide at one end is larger in both dimensions than the rectangular guide at the other end, being equally usable with any other dimensional relationship.
It will also be observed that a circular guide is a limiting case of an elliptical guide, in this case with an eccentricity of zero as opposed to the unity eccentricity of the opposite limiting case of the top and bottom of the rectangle. Accordingly, the same general shaping of bottom and top walls and side walls (whether by use of the above formulae or otherwise) is desirably used in a transition from circular guide to rectangular guide. All intermediate cross-sections are generally similar to the type of progression of cross-section shown in FIGS. 9 through 11.
What is claimed is:
1. In a waveguide transition for dominant-mode transmission of the type having a passage rectangular at one end, generally elliptical at the other end, and of continuously varying cross-section between said ends, the improvement wherein the cross-section at any point between the ends has concave top and bottom walls and straight side walls the concave curvature of said top and bottom walls extending continuously from one of said side walls to the other, the top and bottom walls each having concave curvature continuously increasing from straightness at the rectangular end to the form of semi-ellipses forming the elliptical end with the longitudinal configuration of the transition formed by said top and bottom walls in the E-plane forming a smoothly continuous curve represented by an equation which has a second derivative that changes sign along the length of the transition, and which has a first derivative that is zero at both ends of the transition so that there are no angular discontinuities at the ends of said top and bottom walls, the side walls being tapered in height and having their top and bottom edges meeting at the elliptical end.
2. The transition of claim 1 wherein the rectangular and elliptical ends of said passage have different transverse dimensions, and the longitudinal configuration of the transition formed by said side walls in the H-plane forms a smoothly continuous curve represented by an equation which has a first derivative that is zero at the rectangular end of the transition so that there are no angular discontinuities at the rectangular ends of said side walls.
3. The transition of claim 1 wherein the distance X of each sidewall from the centerline at any longitudinal point is substantially equal to a (a A/2) cos [(1r/2) (z/L)] where a is half the major axis of the elliptical end, A is the width of the rectangular end, is the distance of the point from the rectangular end. and L is the total length.
4. The transition of claim 1 wherein the top and bottom walls are of the cross-sectional general form of segments of ellipses tapering in eccentricity along the length of the transition.
5. The transition of claim 4 wherein the passage at any distance 2 from the rectangular end has an E-plane dimension substantially in accordance with D (B12) cos (1rz/2L) b sin (1rz/2L) where L is the total length of the transition, B is the E- plane dimension of the rectangular end and b is half the E-plane dimension of the elliptical end.
6. The transition of claim 5 wherein the passage at any distance 2 from the rectangular end has top and bottom walls which are substantially segments of the ellipse (x /C (y /D l where C a/sin (rr 1/2 L) a being half the H-plane dimension of the elliptical end, and
D= (8/2) cos (1r 2/2 L) b sin 1/2 7. The transition of claim 6 wherein the spacing 2X between the side walls and any distance 2 from the rectangular end is substantially equal to twice the quantity a (a A/2) cos (tr/2) (z/L) A being the l-l-plane dimension of the rectangular end.
8. In a waveguide transition for dominant-mode transmission of the type having a passage rectangular at one end, generally elliptical at the other end. and of continuously varying cross-section between said ends the improvement wherein the cross-section at any point between the ends has concave top and bottom walls and convex side walls, the concave curvature of said top and bottom walls extending continuously from one of said side walls to the other. the top and bottom walls each having concave curvature continuously increasing from straightness at the rectangular end to the form of semi-ellipses forming the elliptical end with the longitudinal configuration of the transition formed by said top and bottom walls in the E-plane forming a smoothly continuous curve represented by an equation which has a second derivative that changes sign along the length of the transition, and which has a first derivative that is zero at both ends of the transition so that there are no angular discontinuities at the ends of said top and bottom walls, the side walls being tapered in height and having their top and bottom edges meeting at the elliptical end.
9. The transition of claim 8 wherein the rectangular and elliptical ends of said passage have different transverse dimensions, and the longitudinal configuration of the transition formed by said side walls in the H-plane forms a smoothly continuous curve represented by an equation which has a first derivative that is zero at the rectangular end of the transition so that there are no angular discontinuities at the rectangular ends of said side walls.