US 2767380 A Abstract available in Claims available in Description (OCR text may contain errors) Oct. 16, 1956 o. J. zoBEl. IMPEDANCE TRANSFORMER Filed Sept. 30, 1952 37 /N VE N TOR O. J. ZOB EL M2M A 7' TORNE V United States Patent O IMPEDANCE TnANsFoRMER Otto J. Zobel, Morristown, N. J., assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application September 30, 1952, Serial No. 312,190 Claims. (Cl. S33-35) This invention relates to wave transmission networks and more particularly to broad-band microwave impedance transformers. An object of the invention is to connect without mismatch two resistive impedances which differ in the magnitude and slope of their impedance-frequency characteristics. A more specific object is to match, over a band of frequencies, two wave guides of rectangular cross section which differs in one or both transverse dimensions. In microwave systems, it is sometimes necessary to connect two wave guides having real characteristic impedances which differ from each other over the frequency band it is desired to transmit. The difference may be in magnitude only, in slope only, or in both magnitude and slope. For eicient transmission of energy from one wave guide to the other over a broad band of frequencies, they must be joined through a transducer, or .impedance transformer, having image impedances which substantially match the characteristic impedances, respectively, of the guides throughout the band. The present invention is directed to a broad-band microwave impedance transformer for connecting two resistive impedances whose impedance-frequency characteristics differ from each other in magnitude, slope, or both. The transformer comprises one or more pairs of tandemconnected sections of rectangular wave guide which, in general, differ from each other in length and in both transverse dimensions. The design method employed is based on the assumption that, since each section alone can be treated mathematically like a length of non-dissipative smooth transmission line, the physically connected sections can be similarly treated, neglecting any small errors arising from physical changes of cross section at the junction points. Actual results obtained have shown that this assumption is entirely justified. Explicit design formulas are presented for determining the required length, width, and height of each section of a twosection transformer. These formulas are based on the magnitude `and slope of each of the desired image impedances of the transformer at the selected mid-band frequency. Also, a simple procedure is outlined for applying the formulas to a transformer comprising two or more pairs of sections connected in tandem. Each of the sections has a phase shift approximately equal to 1r/2 radians at the mid-band frequency. The transformer is especially useful in connecting two dissimilar wave guides of rectangular cross section. In this case, the transformer can be designed to provide a very close impedance match over a comparatively broad band of frequencies. The nature of the invention and its various objects, features, and advantages will appear more fully in the following detailed description of preferred embodiments illustrated in the accompanying drawing, of which Fig. l is a perspective view, with one corner cut away, of a two-section microwave impedance transformer in accordance with the invention connecting two rectangular wave guides; Figs. 2 and 3 are, respectively, a sectional top view and a sectional side view of the transformer shown in Fig. l; Figs. 4 and 5 present typical impedancefrequency characteristics showing the matches obtainable with a transformer of the type shown in Figs. l, 2, and 3; and Fig. 6 is a side View of a transformer in accordance with the invention comprising three pairs of sections connected in tandem. In the embodiment shown in Figs. l, 2, and 3, two hollow-pipe wave guides 10 and 13 of oblong cross section are connected by an impedance transformer 14 comprising two sections l1 and l2 of rectangular wave guide connected in tandem. 'The guide 10 has an inside width as and an inside height ba. For the guide 13 the correspending dimensions are ab and bb, in the section 11 they are ai and b1, and in the section 12 they are a2 and b2. The sections 11 and 12 have respective lengths l1 and l2, each equal to a quarter wavelength within the section. The side walls are made continuous between the different sections of guide to prevent Athe escape of energy. In general, this results in small steps in the width, at the junction points 16, 17, and 18 in Fig. 2, and small steps in the height, at the points 19, 2l), and 21 in Fig. 3. It will be assumed that microwave energy is to be transmitted from the guide l0 through the transformer 14 to the guide 13, as indicated by the arrow 16 in Fig. l. The tlow of energy may, of course, as well be in the reverse direction. For analysis, it will be assumed further that the mode of transmission is the TE1,0, in which the elec-tric field E is perpendicular to the longer transverse dimension of the guide, as indicated in Figs. 1 and 3. The section 11 will transmit waves above the critical or cutoif frequency for, given by the expression (1) where c is the velocity of light and is equal to 3 101 centimeters per second. The phase constant, ,31, of the section at any frequency, f, is 1,=t,.,/\/1- f.,/f 2 where km is the characteristic quency and ,is given by ka, 1=601r2b1/a1 (4) The design formulas are greatly simplied by introducing a parameter D1, defined as where fq is the quarter-wave frequency at which the phase shift B1 of the section is 1r/2 radians. The frequency fq is ordinarily chosen as the geometric mean of the limits of the desired transmission band. lt will be noted that D1 will have a value greater than unity, since fq will always be greater than fet. For the section 12, the cut-ofi frequency, constant, z, the characteristic impedance, k2, its value, km2, at infinite frequency, and the value of a second parameter, D2, may be found from expressions similar, respectively, to (l), (2), (3), (4), and (5) except that the subscripts l are all changed to 2. The composite transformer 14, comprising the sections 11 and 12 connected in tandem, has a real image impedance Wa at the left end and a real image impedance Wb at the right end. The slops of the image impedancefrequency characteristics are, respectively, Wa and Wb. In accordance with the invention, the impedances Ws and Wb may be chosen arbitrarily at the frequency fq. and their slopes We. and Wb may be given any desired negaimpedance at innite frefcz, the phase tive values at fq. This imposes four conditions. There are also two more conditions, namely, that the phase shift Bi in the section 11 and the phase shift B2 in the section 12 shail each be 1r/2 radians at fq', that is, Since the transformer 14 has six variables, the dimensions ai, az, b1, b2, l1, and l2, these six conditions can all be satisfied at the frequency fq. A recommended design procedure and explicit design formulas will now be presented, using the known quantities Wa, Wb, Ws', and Wb'. As the derivation of the formulas is lengthy and somewhat involved, it will be omitted. First, the parameter D1 is obtained as the root, greater than unity, Vof the cubic expression Then, the dimensions of the section 11 are found from the formulas Also, convenient expressions for the cut-off frequency fel of the section and its characteristic impedance km, at infinite frequency may be written in tenms of the parameter D1 as follows: mwa/9571 (17) and k.,.=k./\l. us) For the section 12, the characteristic impedance ka is given by kz=W1/4Wb3/4U (19) The dimensions as, bz, and Iz, the cut-off jez, and the impedance ke? may now be found from the following formulas: C- ik. V21, 1),-1 (20) @menacent/Dwi (21) MFM/JE (24) The transformer 14 will give a particularly close im pedance match over a comparatively wide band if the image impedances Wn and Wb are to match the characteristic impedances of rectangular wave guides such as 10 and 13. It will be assumed that the guide 10 has a cut off frequency fea and, at the frequency j'q, a characteristie `impedance ks with a slope ka. For the guide 13, the corresponding quantities are frs, kb, and kb'. The dimensions of the sections 11 and 12 are found from the formulas (14), (15), (16), (20), (21), and (22), but in using formula (7) to find D1, the expression for p becomes ,teva/k.. (z5) and the expressions for s and t can be simplified to This simplification can be made since, at fq, the following relationships hold: In this case, the characteristic impedances k1 and k2 are given by the formulas k1=ka3f4kbU4U (32) and kz==ka1/4kb3/4U (33) The impedance-frequency characteristics in Fig. 4 show the match obtained with a two-section transformer 14 in accordance with the invention designed to connect a rectangular wave guide 10 in which as is 0.90() inch and ba is 0.400 inch to a rectangular wave guide 13 in which ab is 1.122 inches and bb is 0.497 inch. The band to be transmitted extends between 8.5 and 9.6 ltiiomegacycles. The quarter-wave frequency iq is taken as 9.033 kilomegacycles, the geometric mean of these frequencies. The required dimensions in inches of the sections 11 and 12, found from the design formulas given above, are as follows: en 0.929 b1=0.410 l1=0.460 a2=1.027 122:0.452 12:0.424 The solid-line curves 24 and 27 show the characteristic impedances ks and kb, respectively, of the guides 10 and 13 to be matched. The curves 25 and 26 show the characteristic impedances k1 and k2, respectively, of the transformer sections 11 and 12. The broken-line curves 23 and 28 show the image impedances Wa and Wb, respectively, of the transformer 14 at the left end and at the right end. It is apparent that an extremely close match is obtained, throughout a wide band, between the curves 23 and 24, and also between the curves 27 and V2li, In fact, over a considerable portion of the range, the curves 23 and 24 are indistinguishable, and the same is true of the curves 27V and 28. The characteristics shown in Fig. 5 apply to another example of a two-section transformer in accordance with the invention, designed to connect a wave guide in which as is 1.372 inches and Ba is 0.622 inch to a guide 13 in which ab is 1.590 inches and bb is 0.795 inch. The band to be passed extends between 5.925 and 6.425 kilomegacycles, and fq is 6.170 kilomegacycles. The required dimensions in inches of the sections 11 and 12 are as follows: a1=1.408 b1=0.652 [v -:0.652 a2=1.512 b2=0.735 22:0.618 The curves labelled We, Wb, ka, kb, k1, and ka correspond, respectively, to the similarly designated curves in Fig. 4. In this case, also, the image impedances of the transformer 14 closely match the characteristic irnpedances of the guides 10 and 13, as evidenced by the substantial coincidence of the curves for We and ka, and also of the curves for Wb and kb. This is an interesting example in that all of the curves cross at the frequency fx. At this particular frequency, no impedance transformation is required, but the important point is that the transformer impedances match the guide impedances in slope, In each of the examples disclosed herein, the insertion loss within the desired transmission band is not more than 0.00004 decibel greater than that of an ideal transformer. The corresponding voltage standing wave ratio is less than 0.053. This comparison neglects the effects of dissipation, which ordinarily will be very small in the short lengths of wave guide employed'. It should be pointed out that a two-section transformer in accordance with the present invention has a consider able advantage over a tapered transformer in that the former is easier to construct and is generally shorter in physical length. In accordance with an extension of theA invention, if the impedances kt. and kb to be connected differ considerably, their difference is divided into two or more ranges. The formulas presented above are then used to design a two-section transformer for each of the ranges and the transformers are connected in tandem. Thus, by using two or more pairs of transformer sections, good impedance matches will be obtained over a broad frequency band even in this case. Fig. 6 shows a side view of such a composite transformer 29 connecting a wave guide 30 of characteristic impedance ks to a wave guide 37 of characteristic impedance kb. The transformer 29 comprises three twosection transformers 38, 39, and 40 connected in tandem. Each of these component transformers is of the type shown in Figs. l, 2, and 3, comprising two sections of wave guide connected in tandem. In the transformer 38 the sections are designated 31 and 32, in the transformer 39 they are 33 and 34, and in the transformer 40 they are 35 and 36. If the impedance range between ka and kb is to be covered in two equal step-up or step-down ratios, there will be an auxiliary dividing impedance kan such that tra.. 34) giving kaft/b (35) Assuming that the impedance kan follows the law of wave guides, the design parameter Dd.; is given by Dd,i=(De-l-Db)/2 (36) For this case, the composite transformer 29 will consist of the two two-section component transformers 38 and 39. The required dimensions of the sections 31 and 32 are found from the design formulas given above, except that, in determining p, s, and t, kan is substituted for kb in Equation 25 and Dan is substituted for Db in Equations 26 and 27. The required dimensions of the sections 33 and 34 are found in the same way, except that kd.; is substituted for ka in Equation 25 and Dan for Ds in Equations 26 and 27. lf there are n equal step-up or step-down ratios, the corresponding expressions for the ith dividing impedance ka.; and the jth design parameter Das are from which the dimensions of the various pairs of transformer sections may be found by applying the design formulas, after making the proper substitutions. lt is to be understood that the above-described arrangements are illustrative of the application of the principies of the invention. Numerous other arrangements may be devised by those skilled in the art without departing from the spirit and scope of the invention. What is claimed is: l. In combination, two wave guides and a transducer connecting said guides, said guides having characteristic impedances which differ in magnitude and slope at a selected frequency, said transducer comprising two sections of rectangular wave guide connected in tandem, each of said sections having a phase shift approximately equal to '1r/2 radians at said frequency, the transverse dimensions of one of said sections differing, respectively, from the transverse dimensions of the other of said sections, and said dimensions being so chosen that the image impedance of said transducer at said frequency substantially match said characteristic impedances, respectively, both in magnitude and in slo-pe. 2. In combination, two wave guides and a transducer connecting said guides, said guides having characteristic impedances which differ in magnitude and slope at a selected frequency, said transducer comprising two sections of rectangular wave guide connected in tandem, the length and transverse dimensions of one of said sections differing, respectively, from the length and transverse dimensions of the other of said sections, and said dimensions being so chosen that the image impedances of said transducer at said frequency substantially match said characteristic impedances, respectively, both in magnitude and in slope. 3. In combination, two wave guides and a transducer connecting said guides, said guides having characteristic impedances which differ in magnitude and slope at a selected frequency, said transducer comprising a plurality of pairs of sections of rectangular wave guide connected in tandem, each of said sections having a phase shift approximately equal to 1r/2 radians at said frequency, said sections differing from section to section in both transverse dimensions, and said dimensions being so chosen that the image impedances of said transducer at said frequency substantially match said characteristic impedances, respectively, both in magnitude and in slope. 4. The comibnation in accordance with claim 3 in which said pairs of sections have substantially equal irnpedance stepup ratios at said frequency. 5. The combination in accordance with claim 3 in which the number of said pairs of sections exceeds two. 6. In combination, two wave guides and a transducer connecting said guides, said guides having characteristic impedances which differ in magnitude and slope at a selected frequency, said transducer comprising a plurality of pairs of sections of rectangular wave guide connected in tandem, said sections differing from section to section in length and in both transverse dimensions, and the lengths and transverse dimensions of said sections being so chosen that the image impedance of said transducer at said frequency substantially match said characteristic impedances, respectively, both in magnitude and in slope. 7. The combination in accordance with clairn 6 in which said pairs of sections have substantially equal impedance step-up ratios at said frequency. 8. The combination in accordance with claim 6 in which the number of said pairs of sections exceeds two. 9. In combination, a broad-band microwave impedance transformer and two wave guides, said transformer com prising two sections of rectangular wave guide connected in tandem, each of said sections having a phase shift equal to 1r/2 radians at a selected frequency, and the transverse dimensions of one of said sections differing, respectively, from the transverse dimensions of the other of said sections, whereby at said frequency the transformer has at one end a real image impedance of preselected magnitude and slope and at its other end a real image impedance of different preselected magnitude and slope, one of said guides being connected to said one end of said transformer and having a characteristic impedance which substantially matches, in magnitude and slope, the image irnpedance of said transformer at said one end, and the other of said guides being connected to said other end of said transformer and having a characteristic impedance which substantially matches, in magnitude and slope, the image impedance of said transformer at said other end. 10. In combination, two dissimilar wave guides and a transformer connected therebetween, one of said guides having a cut-olf frequency fes and a characteristic impedance of magnitude ka at a selected frequency fq, the other of said guides having a cut-off frequency feb and a characteristic impedance of magnitude kb at the frequency fq, said transformer comprising two sections of rectangular wave guide connected in tandem, one of said sections having a length lr and transverse dimensions a1 and b1, the other of said sections having a length l2 and transverse dimensions a2 and b2, and said dimensions having approximately the following values: where c is the velocity of light and Di is the root, greater than unity, of the cubic where Ragan, Microwave Transmission Circuits, vol. 9, M. I. T. Radiation Laboratory Series, published 1948 by Mc- Graw-Hill; pages 52, 53 and 217 relied on. Patent Citations
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