Circuit and Electromagnetic Sytem Deign Note Note 6 8 June 9 Raiing Cavity Q for Microwave-Pule Compreion by Reducing Aperture Skin-Effect Loe Carl E. Baum Univerity of New Meico Department of Electrical and Computer Engineering Albuquerque New Meico 873 Abtract Thi paper dicue the loe in a waveguide-cavity ocillator. Particular attention i given to an inductive iri for feeding into (charging) the ocillator. `
. Introduction In microwave pule compreion it i important to minimize loe in the reonant cavity. It i thee loe which limit the power gain G in going from ome ource to the power in the reonant mode which will be witched into a load (e.g., antenna). Thinking of the reonant cavity a a length of horted waveguide, the output pule ha a width of roughly the round trip tranit time in the guide baed on the group velocity, if the energy i witched out at one end. A dicued in [4] one can deign a magic tee which can be ued to take waveguide power in two direction and match it into a ingle output guide. Thi double the power out, but with the magic tee being in the center of the guide, the pule width i halved. Beide the loe in the waveguide wall there are loe at both end and at any perturbation in the waveguide wall (probe, witch, magic tee, etc.). A common technique for feeding power into the waveguide cavity involve an iri a indicated in Fig... The iri ha width in the + direction, and length b (the full guide height in the +y direction). It i cut into what we can call the plate. plate a incident power y iri z reonant microwave cavity a a View perpendicular to broad wall Fig.. Inductive Iri for Feeding Power into Waveguide Cavity.
. Symmetry Decompoition of Magnetic Field To implify the analyi let u note that the incident power i much le than the power in the reonant cavity (Fig..). A illutrated in Fig.. we can then decompoe the reonant cavity field into ymmetric and antiymmetric part [5] with repect to the z = ymmetry plane. Note that it i the magnetic field here illutrated which are at right angle to the urface current denity, i placed). J, on the waveguide-dicontinuity plane (in which the iri waveguide feed plate cavity y large field in cavity z H ymmetric part = -/ of cavity field / of cavity field antiymmetric part + / of cavity field / of cavity field Fig.. Decompoition of Cavity Magnetic Field 3
3. Current Denity and Power Lo Near Knife-Edge Boundary of Iri Thi i dicued in [-3]. In thi cae the urface current denity near the edge behave aymptotically a / d J J for y (3.) uing the coordinate in Fig.. for the edge at. Here we have J A/ m = order of H in cavity d (m) = caling contant (3.) There are of coure, two uch edge which will have the ame effect due to ymmetry. Our analyi will concentrate then on the half for >. To make a rough etimate of the additional power lo due to iri edge, let u firt aume that (3.3) a o that we can approimate the urface current denity on the horted aperture a J J y (3.4) Thi i uniform (not a function of or y) and i on the +z ide of the horting plate. For the ymmetric part (Fig..) J J J y y for (3.5) in the ame direction on both ide of the plate. For the antiymmetric part we have J J y for z, for z (3.6) 4
with thi applying all over the z = plane, ecept in the iri. The boundary condition are eactly matched on both the plate (infiniteimally thick) and the aperture. Figure 3. illutrate the ymmetric part of the magnetic field near the aperture. For a conformal tranformation we have w( ) u( ) jv( ) comple potential u( ) v( ) electric potential magnetic potential jz comple coordinate (3.7) v = on = and on z = in aperture u = on z = with > The ymmetrie of thi problem give u the imple reult: a) The magnetic field at the origin i zero. b) For < on z = the magnetic field i perpendicular to the z = plane (and i an odd function of ). c) For > on z = the magnetic field i parallel to the z = plane (oppoitely directed on the two ide). d) The magnetic field i perpendicular to the = plane for <. z H y Fig. 3. Symmetric Part of Magnetic Field Near Aperture 5
Figure 3. give a equence of tranformation. The aforementioned ymmetrie allow u to concentrate on the firt quadrant of the plane ( >, y > ). In Fig. 3.A the jz ai i rotated by / to the negative real ( ) ai. Thi i then hifted (Fig. 3.B) to the left o that the conductor edge i moved to the origin. The negative ai i then rotated (Fig. 3.C) by -/ to the jv ai. The magnetic field i now uniform and directed to the right. Summarizing, the final tranformation give / / w, w dw / /, w w d (3.8) From (3.8) we have on the ai / u, v for dw / a d (3.9) From (3.5) the urface current denity for the ymmetric part goe to directed urface current denity a J / a. Hence we have the y- / J J, y (3.) Thi applie equally to both ide of the z = plane. Adding the antiymmetric part we have (for ) J / J for z / J for z (3.) The power lot in the plate per unit width with no aperture i proportional to P J R (3.) 6
jz jz y A. Firt tranformation (firt quadrant to firt two) jz B. Second tranformation (hift to left) jv w / u C. Third tranformation (two quadrant to firt) Fig. 3. Conformal Tranformation 7
8 taken out to ome ditance (here conidering the > contribution). The power with the aperture i proportional to P J R d (3.3) With thi applying to both ide of the z = plane. We then have / / lim lim 4 4 P P P J R d lim lim J R d d J R d J R d (3.4) where the integral i taken from becaue of the ingularity. Changing variable we have, coth d d arc n n (3.5)
Defining (3.6) then we have P J R n a 4 (3.7) So P logarithmically, a weak ingularity. 4. Comparion to Power Lo in Plate with Cloed Aperture With the aperture cloed ( ) the current on the plate i co J J a (4.) for the H,, mode. For the portion of the plate from to a the power lo i a a P J R co d J R (4.) a For a (4.3) then we have P n a P a 4 (4.4) 9
a the fractional increae in the power lo in the plate. A a practical matter, i not taken to zero due to the finite thickne of the plate. A an eample let = mm (for mm thick plate) a = 6.5 cm (L band guide WR-65) (4.5) a = 4. cm Thi give P n8 P a 4. (4.6) which i not a large effect. There i ome uncertainty in thi reult ince there i ome ditribution of J around the aperture edge at. There i le enhancement of J at the plate edge due to the finite thickne of the plate. For further reduction of thi effect one can round the edge of the aperture and increae the plate thickne a in Fig. 4.. The reulting plate thickne would be omething on the order of (more or le). incident power y z reonant microwave cavity Fig. 4. Inductive Iri with Rounded Edge
5. Output-Port Skin-Effect-Lo Reduction While the cavity i being rung up toward full field trength there are alo loe in other dicontinuitie. One of thee might be the H-plane bend output of a magic tee a in Fig. 5.. A indicated, the junction can have rounded edge at the connection to the output waveguide. Thi applie not only to the narrow wall, but alo to the broad wall. A dicued in [4] the height of the output guide might be le than that of the ocillator guide for impedance-matching purpoe. In thi cae the broad-wall connection of the output guide to the ocillator guide can alo be rounded. Figure 5. alo how the connection of a poible input guide for charging the ocillator. The connection edge here can alo be rounded. input a a. output a E Before witching Fig. 5. Rounding Edge in magic Tee
6. Concluding Remark The general leon i that harp edge are to be avoided to minimize loe in the waveguide-cavity ocillator. Thi i not a large effect, but it help. Reference. D. V. Giri and C. E. Baum, Equivalent Diplacement for a High-Voltge Rollup on the Edge of a Conducting Sheet, Senor and Simulation Note 94, October 986; INCEMIC 987, Bangalore, India, September 987, pp. 9-3.. C. E. Baum and J. S. Tyo, Tranient Skin Effect in Cable, Meaurement Note 47, July 996. 3. J. S. Tyo and C. E. Baum, Reduced Skin Lo Diipation at the Edge of a Conducting Plate Uing High- Voltage Rollup, Meaurement Note 5, April 997. 4. C. E. Baum, Impedance-Matched Magic Tee, Circuit and Electromagnetic Sytem Deign Note 5, March 6. 5. C. E. Baum and H. N. Kritiko, Symmetry in Electromagnetic, Ch., pp. -9, in C. E. Baum and H. N. Kritiko (ed.), Electromagnetic Symmetry, Taylor & Franci, 995.