THE FIELD in a cascaded, cylindrical cavity comprising

Transcription

1 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 3, AUGUST Transmission Through Axisymmetric, Cascaded Cylindrical Cavities Coupled by Apertures Part II: Structures With Varying Cross-Sections John C. Young, Member, IEEE, and Chalmers M. Butler, Life Fellow, IEEE Abstract A technique is described for determining the field in a series of cascaded, axisymmetric cylindrical cavities excited by a -independent source. The constituent cavities are axisymmetric but may have a cross-section that varies with axial displacement. A set of coupled integral equations are solved for the unknown electric fields in the apertures that separate the constituent cavities. Separate integral equations are formulated to determine the field in each cavity with a variable cross-section. Index Terms Apertures, cavities, integral equations, transient analysis. Fig. 1. Cascaded cavity with variable cross-section section. I. INTRODUCTION THE FIELD in a cascaded, cylindrical cavity comprising coaxial and circular-cylindrical sections may be computed by expanding the field in each section in terms of the eigenfunctions of that section and enforcing continuity of the tangential field in the apertures joining the sections. When a section s cross-section varies with axial displacement, the eigenfunctions of the section may not be known and, hence, another method must be devised to determine the field in that section. If the cavity is composed of a combination of sections with fixed cross-sections and sections with variable cross-sections, then the eigenfunction expansions may still be used to determine the fields in the fixed cross-section cavities. To determine the field in the sections with variable crosssections, aperture theory is used. First, the apertures are shorted, and an equivalent magnetic current that is related to the electric field in the aperture is placed on the shorted apertures. If the aperture electric field is assumed to be known, then the magnetic current(s) are considered sources in the cavity. Next, the perfect electric conductor (pec) walls of the cavity are removed and an equivalent electric current is placed on the wall surface. If this new wall surface is a closed surface, then either an electric field integral equation (EFIE) or a magnetic field integral equation (MFIE) may be used to solve for the wall current. If the wall surface is not closed, e.g., the wall has a baffle, then only the EFIE may be used to solve the problem. Although the EFIE is Manuscript received December 5, 2002; revised June 22, This work was supported by the National Science Foundation under its Graduate Fellowship Program, by the U.S. Army Research Office under Grant DAAD , and by the U.S. Air Force Office of Scientific Research under F The authors are with the Department of Electrical and Computer Engineering, Clemson University, Clemson, NC USA ( johny@ces. clemson.edu). Digital Object Identifier /TEMC more flexible then the MFIE in that the wall surface may be more general, it requires more computation time when a computer is used to solve the integral equation. Once the wall currents are known, the field in the section may be computed and the continuity of tangential fields in the apertures may be enforced. II. DISCUSSION Consider a set of I, cylindrical cavities connected in cascode as shown in Fig. 1. Each component cavity may be either a coaxial cavity, a circular-cylindrical cavity, or a -symmetric cavity whose cross-section varies with axial displacement. So that measurements may be made to support the theory, the first section is constrained to be a coaxial cavity, and, if the last section is terminated in a load, it is also constrained to be a coaxial cavity. The component cavities are connected at the planes z = z 1,z 2,...,z I 1 so that the total structure is -symmetric. The material in the i th cavity is characterized by (ε i,µ i ).The field in both a coaxial cavity and in a circular-cylindrical cavity, with the terminal conditions incorporated, has been treated previously in [1]. Let the electric and magnetic field in section i be denoted by E ζ i (ρ, z) and Hζ i (ρ, z), respectively, in a coaxial cavity (ζ =co), in a circular-cylindrical cavity (ζ =cy),orina variable cross-sectioncavity (ζ = vc). Quantities vary harmonically in time at angular frequency ω according to e jωt, which is suppressed, and the wave number in cavity i is k i = ω ε i µ i. A set of coupled integral equations for the unknown aperture electric fields Eρ(ρ) i is developed in [1] for the case that each component cavity has a fixed cross-section. To incorporate sections with variable cross-section into this integral equation, it is necessary to determine T ζ,s i (ρ) and U ζ,s i (ρ) (Fig. 2) of [1]. T ζ,s i (ρ) represents a term proportional to the contribution from the ρ-directed electric field in aperture s to the -directed magnetic field in that aperture which joins cavity i and is adjacent to aperture s. U ζ,s i (ρ), on the other hand, represents a term /$ IEEE

2 418 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 3, AUGUST 2005 Fig. 5. Equivalent model for section with variable cross-section. Fig. 2. Coupling between apertures. Fig. 6. Arc for determining cavity shape. Fig. 3. Cavity with variable cross-section. and E = E z ẑ + E ρ ˆρ. The open-space term is readily determined from the magnetic current to be H os = jωf and E os = (1/ε i ) (F ˆ) where F (r) =ε i M (r )G(r, r ) ds. (1) Fig. 4. Cavity with apertures shorted. proportional to the contribution to the -directed magnetic field in aperture s which joins cavity i due to the ρ-directed electric field in the same aperture s. Let section i (Fig. 3), 1 <i<i, be a section that has a variable cross-section and that is partially bounded by SA i 1 at z = z i 1 and SA i at z = z i. The field in section i can be determined in terms of the aperture electric fields Eρ i 1 and Eρ. i An equivalent problem, shown in Fig. 4, is constructed by shorting the two apertures and placing magnetic surface currents M i 1 = Eρ i 1 on SA i 1 and M i = Ei ρ on SA i. Then, the field in the equivalent problem is the same as the field in the original cavity section. If the Green s function for a magnetic ring current inside the cavity with shorted apertures were known, then the field in the cavity could be determined. For a cavity of arbitrary cross-section, this Green s function is usually not known, but it may determined numerically [2]. [To avoid cumbersome notation, let E vc i (ρ, z) =E(ρ, z) and H vc i (ρ, z) =H(ρ, z).] The field in the equivalent cavity may be written as the sum of a term due to the magnetic surface current radiating in open space (superscript os) and a term due to the scattering from the cavity walls (superscript sc): E = E os + E sc and H = H os + H sc. Due to -symmetry of both the source and the cavity, H = H ˆ S i 1 A +S i A The open-space Green s function is G(r, r )= exp( jk i r r ) 4π r r, (2) where r and r denote an observation point and a source point, respectively. The scattered field can be determined by removing the pec walls and placing an equivalent electric current J on the wall surface S c as depicted in Fig. 5. These equivalent currents serve to force the tangential electric field on the surfaces vacated by the walls to be zero. Because the surface is axisymmetric, it may be generated by rotating an arc C c (Fig. 6) about the z axis. The arc is parameterized by t, and the unit vector tangent to the arc at t in the direction of increasing t is ˆt. Since the magnetic field has only a component, the electric current only has a -symmetric, t-directed component J(t, ) =J t (t)ˆt. Next, the electric field tangential to the surface S c is forced to be zero. Because magnetic current resides on the apertures, the point of observation r must be approached from a path in the exterior region, which is denoted by a + superscript [ ] E t r + ; M i 1 [ + E t r + ; M] i + Et [r; J t ]=0, r S c. (3) There is no ambiguity in which region is the exterior because, even though the surface is not required to be closed, the outer surface of the wall must be closed. Alternately, if the wall surface is a closed surface, then the component of the magnetic field tangential to the wall surface is forced to be zero in the

3 YOUNG AND BUTLER: TRANSMISSION THROUGH AXISYMMETRIC, CASCADED CYLINDRICAL CAVITIES COUPLED 419 limit as an observation point on the wall is approached from the exterior region [ ] H r; M i 1 [ ] + H r; M i + H [r + ; J t ]=0, r S c. (4) Because all quantities are -symmetric, (3) and (4) need to be enforced for only one value of, e.g., =0. Or, body-of-revolution (BOR) theory may be used, in which case only the zeroth order equation need be solved. The scattered field is given by E sc = jωa Φ and H sc =(1/µ i ) A, where A(r) =µ i J t (r )ˆt (t )G(r, r ) ds (5) S c and Φ(r) = 1 q(r)g(r, r ) ds. (6) ε i S c Here, the current and charge are related by s J(r)+ jωq(r) =0, where the subscript s indicates surface divergence, and the Green s function G is again the open-space Green s function in (2). For known or assumed aperture fields Eρ i 1 and Eρ, i which determine M i 1 and M i, (3) or (4) may be solved for the current J t. Once J t is known, the magnetic field in the apertures is computed directly from H (t) =J t (t). ThetermsT ζ,s i (ρ) and U ζ,s i (ρ) are then given by T vc,s [ i (ρ) =H,i ρ, z2i s 1 ; Eρ s ] (7) and U vc,s [ i (ρ) = H,i ρ, zs ; Eρ] s, (8) where s denotes the source aperture i or i 1 and the appropriate transformation is used to determine (ρ, z) from t at a point on the apertures. If the magnetic current (electric field) in each aperture s = i, i 1 is approximated with pulse basis functions by Q s ρm(ρ) s MnΠ s s n(ρ), (9) n=1 where Q s is the number of basis functions in aperture s, then, for each basis function Π s n, (3) or (4) may be solved for the unknown current J t, i.e., E t [r + ; 1 ] ρ Πs ˆ n + E t [r; J tˆt] =0, r S c (10) or H [r; 1 ] ρ Πs ˆ n + H [r + ; J tˆt] =0, r S c. (11) In (10) or (11), Π s n(ρ) represents the nth pulse basis function in the s th aperture, and the source is considered to be a magnetic surface current M (ρ) =(1/ρ)Π s n(ρ) on aperture s. In effect, the solution of (10) or (11) numerically determines a Green s function for the field in cavity i due to a pulse of magnetic surface current on aperture s since the field anywhere in cavity i due to this pulse may be calculated once the current J t is known. In a computational solution of (10) or (11), the matrix computed from the current J t must be filled and inverted only once and then multiplied by the source vector determined from each pulse basis function. Instead of solving a separate integral equation for the equivalent electric current in region i, one may also incorporate the integral equation (3) or (4) into the set of coupled integral equations for the aperture fields. Either method requires approximately the same computation time, but the first method fits well with the derivation presented in [1]. Triangle expansion and pulse testing are used to solve the EFIE equation, and pulse expansion and point matching are used to solve the MFIE. For these two solution schemes, the MFIE requires much less computation time than the EFIE. III. RESULTS To validate the technique, measurements were performed. A three-section cavity (I =3) was constructed. It comprises a source section, a center section, and a load section. The center section has a constant outer radius of cm and a length of cm. The center cavity is connected to the source through a coaxial section of length cm, of outer radius cm, and of inner radius cm. The center cavity is connected to the load through a coaxial section of length l 3, of outer radius cm, and of inner radius cm. The generator impedance is Z g =50Ω. The input admittance of the various cavity configurations looking into the second cavity section at z = z 1 was measured in the frequency range 50 MHz to 4 GHz. Data computed by both the EFIE and MFIE methods are presented. All frequency domain measurements are normalized by the input admittance of the first cavity section (0.02 S). In addition, the time-domain behavior of the source voltage v 0 (t) at the plane z = z 0 and the load voltage v L (t) at the plane z = z 3, due to a step input, were measured with a time-domain reflectometer (TDR) and computed with a fast Fourier transform (FFT). Several different axisymmetric pieces with nonuniform crosssections were machined for use as an inner conductor in the center cavity. For the first two models, the inner conductor pieces were constructed from two cylinders (S1 and S2), two conical pieces (S3 and S4), and two connectors (S5 and S6), which could be joined in any arbitrary order. These parts as well as their dimensions are shown in Fig. 7. A typical cavity is shown in Fig. 8. The arrangement of the inner conductor parts in the center section is given from left to right in the format S5-S4L- S2-S3R-S1-S6, where the L or R on S3 and S4 indicates that the large end of the conical section is situated on the left or on the right, respectively. For model 1, the agreement between measured and computed data, shown in Fig. 9, is good at most frequencies but does exhibit some differences near 1 GHz and 2 GHz for a 50 Ω load. The two computational solutions agree very well. When the load is a short, the agreement in the susceptances is very good as seen in Fig. 10. Because in the computational model lossless walls are assumed, there is no mechanism for loss when the load is a short. The presence of a nonzero conductance in the computed data for a cavity loaded with a short is due to numerical noise

4 420 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 3, AUGUST 2005 Fig. 9. Normalized input admittance for model 1 terminated in a 50 Ω load (S5-S4L-S2-S3R-S1-S6, l 3 =10.21 cm). Fig. 7. Parts comprising variable cross-section center conductor (all dimensions in cm). Fig. 8. Typical cavity shape for model 1 and model 2. in the solution. The measured data for a cavity terminated in a short does, of course, have a nonzero conductance due to wall losses. Time-domain data are measured and computed at the source and load for a 0.4 V pulse input v g (t) with a 50 ps rise and fall time and a duration of 40 ps. In the experiment, the cavity was connected to the TDR through 1 m coaxial cables with 50 Ω characteristic impedances, and the actual voltages measured were the voltages at the input and output of these cables. The measured voltages are time shifted to account for the transit time of the wave in the coaxial cables. This is valid because the TDR impedances, the coaxial cable characteristic impedances, and the characteristic impedances of the first and last cavity sections are all 50 Ω. For model 1, measured and computed time-domain waveforms at the source and the load are shown in Fig. 11 for a 50 Ω load and Fig. 12 for a 100 Ω load. The agreement is very good, but, if the input waveform had contained significant spectral con- Fig. 10. Normalized input admittance for model 1 terminated in a short (S5- S4L-S2-S3R-S1-S6, l 3 =12.44 cm). tent at frequencies where the measured and computed data differ significantly, then the agreement may not have been so good. The early time discrepancy in the measured and computed waveforms in Fig. 11(a) is due to the fact that the coaxial cable that connects the TDR and the cavity is not modeled in the computations. For model 2, the agreement with measurements is not as good as it is in the data of model 1 as is observed in Fig. 13 for a 50 Ω load. The agreement between the EFIE and MFIE solutions is excellent. Model 3 (Fig. 14) has an inner conductor with a smoothly varying cross-section whose cross-sectional radius is defined by ρ(z) =[ cos(196.85z)exp( z)] cm, where z is the axial displacement measured from the left side of the smoothly varying section. The normalized input admittance

5 YOUNG AND BUTLER: TRANSMISSION THROUGH AXISYMMETRIC, CASCADED CYLINDRICAL CAVITIES COUPLED 421 Fig. 14. Cavity shape for model 3. Fig. 11. Measured and calculated time-domain (a) source voltage and (b) load voltage for model 1 terminated in a 50 Ω load (S5-S4L-S2-S3R-S1-S6, l 3 =10.21 cm). Fig. 12. Measured and calculated time-domain source voltage for model 1 terminated in a 100 Ω load (S5-S4L-S2-S3R-S1-S6, l 3 =15.31 cm). Fig. 15. Normalized input admittance for model 3 terminated in a 50 Ω load (l 3 =10.21 cm). Fig. 13. Normalized input admittance for model 2 terminated in a 50 Ω load (S5-S3R-S4L-S2-S1-S6, l 3 =10.21 cm). for this cavity is shown in Fig. 15 for a 50 Ω load. In addition, the time-domain behavior of the waveform at the source and load is plotted in Fig. 16. Both the time-domain behavior and frequency-domain behavior exhibit excellent agreement in measurements and computations. Fig. 16. Measured and calculated time domain. (a) Source voltage and (b) load voltage for model 3 terminated in a 50 Ω load (l 3 =10.21 cm).

6 422 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 47, NO. 3, AUGUST 2005 Fig. 17. Cavity shape for model 4. Fig. 19. Measured and calculated time-domain source voltage for model 4 terminated in (a) a short (l 3 =12.44 cm)and(b)a50ω load (l 3 =10.21 cm). Fig. 18. Normalized input admittance for model 4 terminated in a short (l 3 = cm). The last cavity constructed, illustrated in Fig. 17, has a disjoint center conductor. In addition, the cross-section contains a junction. Because the surface is not a closed surface, the EFIE analysis is used in the computations. The normalized input admittance for this cavity is shown in Fig. 18 for a short-circuit termination. The agreement is excellent at lower frequencies and is good at higher frequencies. In addition, the time-domain behavior of the waveform at the source is depicted in Fig. 19 for a short circuit and a 50 Ω load. As is expected, the late time response approaches 0.4 V due to an open-circuited center conductor at dc. section varies with axial displacement, an additional integral equation for the current on the walls of the section with the apertures shorted is solved. The solution of this integral equation effectively provides a numerical Green s function from which the section s field may be determined. The technique applies equally well to sections whose cross-section is fixed. However, the numerical solution of the additional integral equation is usually much slower than that of the eigenfunction expansion method, unless the number of required basis functions is small. An experimental apparatus comprising three cavities was constructed. The middle cavity has a fixed-radius outer conductor and an inner conductor with a variable cross-section. The input admittance of this structure was measured for various inner conductor configurations in the frequency range 50 MHz to 4 GHz. Both an EFIE and MFIE are used to analyze the middle section, and the computed admittances compare well with the measured admittances. In addition, the voltage at the source and load terminals are calculated and compared to measured time-domain data. The analysis applies equally well to a cavity whose outer wall cross-section varies with axial displacement, but no data is presented for this case due to the high cost of machining a cavity with a variable cross-section outer conductor. ACKNOWLEDGMENT The assistance of C. Bopp with some measurements is appreciated. IV. CONCLUSION A technique is presented to compute the field in cascaded cavities comprising axisymmetric sections whose cross-section may or may not vary with axial displacement. When a section s cross-section is fixed, the field in the section is expanded in terms of the eigenfunctions of that section. When a section s cross- REFERENCES [1] J. C. Young, C. M. Butler, and M. G. Harrison, Transmission through axisymmetric, cascaded cylindrical cavities coupled by apertures Part I: Structures with coaxial and circular-cylindrical cross-sections, IEEE Trans. Electromagn. Compat., to be published. [2] A. W. Glisson and C. M. Butler, Analysis of a wire antenna in the presence of a body of revolution, IEEE Trans. Antennas Propag., vol. AP- 28, pp , Sep

7 YOUNG AND BUTLER: TRANSMISSION THROUGH AXISYMMETRIC, CASCADED CYLINDRICAL CAVITIES COUPLED 423 John C. Young (S 97 M 03) received the B.E.E. degree in electrical engineering from Auburn University, Auburn, AL, in 1997 and the M.S. and Ph.D. degrees in electrical engineering from Clemson University, Clemson, SC, in 2000 and 2002, respectively. He received a National Science Foundation Graduate Fellowship in 1998 and served as a Graduate Research Assistant at Clemson University from 1997 to From January 2003 to April 2003, he served as a Postdoctoral Researcher at Clemson University. In July 2003, he joined the Ando and Hirokawa Lab at the Tokyo Institute of Technology, Tokyo, Japan, as a Postdoctoral Researcher sponsored by the Japan Society for the Promotion of Science. His research interests include integral equation solution techniques, electromagnetic theory, waveguides, and array antennas. Dr. Young is a member Tau Beta Pi and Eta Kappa Nu. Chalmers M. Butler (S 55 S 65 SM 75 F 83 LF 01) received the B.S. and M.S. degrees from Clemson University, Clemson, SC, and the Ph.D. degree from the University of Wisconsin, Madison. He has been a member of the faculty at Louisiana State University, the University of Houston, and the University of Mississippi, where he was Chairman of Electrical Engineering ( ) and University Distinguished Professor ( ). Since 1985, he has been at Clemson University, where he is presently Alumni Distinguished Professor and Warren H. Owen Professor of Electrical and Computer Engineering. He has served as International Commission B Editor of the Review of Radio Science ( and ), as a member of the Editorial Boards of ELECTROMAGNETICS AND COMPUTER APPLICATIONS IN ENGINEERING EDUCATION, and as guest editor of two special issues of Radio Science. His research has been focused primarily upon integral equation techniques in electromagnetics and numerical methods for solving integral equations. His principal interests are in antennas and aperture penetration. Prof. Butler is a member of Sigma Xi, Tau Beta Pi, Phi Kappa Phi, Eta Kappa Nu, and Commissions B, D, and F of the International Union of Radio Science (URSI). He received the Western Electric Fund Award (1974), the School of Engineering Outstanding Teacher Award at the University of Mississippi, and the 1990, 1991, 1993, and 1994 NCR/AT&T Excellence in Teaching Award at Clemson. He has received two best paper awards: the Best Basic EMP Non-Source Region Paper during from the SUMMA Foundation and the 1986 Oliver Lodge Premium Award from the Institute of Electrical Engineers of London. He received the 1990 and the 2003 Editor s Citation for Excellence in Refereeing in RADIO SCIENCE, the McQueen Quattlebaum Faculty Achievement Award, the Provost s Award for Scholarly Achievement, the Class of 1939 Faculty Achievement Award, and the 2000 Alumni Award for Excellence in Research at Clemson University. From the University of Wisconsin, he received the Centennial Medal for Contributions to Electrical and Computer Engineering. He is a recipient of the IEEE Millennium Medal and the 2003 recipient of the IEEE AP-S Chen-To Tai Distinguished Educator Award. He has served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION ( and ) and of the IEEE TRANSACTIONS ON EDUCATION, as a member of the IEEE Antennas and Propagation Society Administrative Committee ( and ), as an IEEE Antennas and Propagation Society National Distinguished Lecturer ( ), as Chairman of U.S. Commission B of URSI ( ), and as National President of Eta Kappa Nu. He served as Secretary ( ), Vice Chairman ( ), and Chairman ( ) of the U.S. National Committee for URSI. He has been Vice Chairman ( ) and Chairman ( ) of the International Commission B of URSI. Presently, he is a Vice President of URSI. He was a U.S. delegate to the 18th 26th General Assemblies of URSI and chaired the 120-person U.S. delegation for the 24th (Kyoto, 1993). He has served on a number of committees and panels including the IEEE Hertz Medal Committee ( ), the IEEE Awards Board, the National Academy of Sciences Panel for Evaluation of the Center for Electronics and Engineering of the National Bureau of Standards (now NIST) ( ), and the National Research Council s Panel for the Evaluation of the U.S. Army s Mine Detection Program ( ), which he chaired. He has been a member of numerous technical program committees of National Radio Science Meetings and IEEE AP-S International Symposia and served as General Chairman of the NIST Workshop on EMI/EMC Metrology Challenges for Industry. He was the Vice Chairman of the Technical Program Committee for the 1995 URSI Electromagnetic Theory Symposium in St. Petersburg, Russia, and chaired the same committee for the 1998 URSI Electromagnetic Theory Symposium held in Thessaloniki, Greece.