AZIMUTHALLY PROPAGATING MODES IN AN AXIALLY-TRUNCATED, CIRCULAR, COAXIAL WAVEGUIDE

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1 AFRL-DE-TR AFRL-DE-TR AZIMUTHALLY PROPAGATING MODES IN AN AXIALLY-TRUNCATED, CIRCULAR, COAXIAL WAVEGUIDE Clifton C. Courtney Voss Scientific 48 Washington St SE Albuuerue, NM 8708 September 003 Interim Report APPROVED FOR PUBLIC RELEASE; DISTRIBUTION IS UNLIMITED. AIR FORCE RESEARCH LABORATORY Directed Energy Directorate 3550 Aberdeen Ave SE AIR FORCE MATERIEL COMMAND KIRTLAND AIR FORCE BASE, NM

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3 REPORT DOCUMENTATION PAGE Form Approved OMB No Public reporting burden for this collection of information is estimated to average hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headuarters Services, Directorate for Information Operations and Reports ( ), 5 Jefferson Davis Highway, Suite 04, Arlington, VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.. REPORT DATE (DD-MM-YYYY). REPORT TYPE Interim 4. TITLE AND SUBTITLE Aimuthally Propagating Modes in an Axially-Truncated, Circular, Coaxial Waveguide 6. AUTHOR(S) Clifton C. Courtney 3. DATES COVERED (From - To) to a. CONTRACT NUMBER F M-00 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6550F 5d. PROJECT NUMBER e. TASK NUMBER DP 5f. WORK UNIT NUMBER CE 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Voss Scientific 48 Washington St SE Albuuerue, NM SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 0. SPONSOR/MONITOR S ACRONYM(S) AFRL/DEHE 3550 Aberdeen Ave SE Kirtland AFB, NM DISTRIBUTION / AVAILABILITY STATEMENT Approved for public release; distribution is unlimited.. SPONSOR/MONITOR S REPORT NUMBER(S) AFRL-DE-TR SUPPLEMENTARY NOTES 4. ABSTRACT In this note we derive and examine the modal properties of the Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide. This guide is defined by an inner and outer radii, and a width. We derive the governing euations for the electromagnetic fields of the TE and TM aimuthally propagating modes. First, the characteristic euations that define the propagation constants of each mode are derived; then, the electric and magnetic fields are explicitly expressed. With these results the mode impedances are formulated, and the impedance of the propagating mode in the transition section can be calculated. This value of the transition section s impedance can then be used to impedance match the transition section to the transmission line. Finally, as an example, a transmission line transition geometry is defined for which the electric and magnetic fields of the lowest order TE and TM modes are computed and graphed. An Appendix to this note briefly discusses the selection of the form of the vector potentials used to formulate the TE and TM modes. 5. SUBJECT TERMS Electromagnetics; waveguide modes 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT a. REPORT Unclassified b. ABSTRACT Unclassified i 8. NUMBER OF PAGES c. THIS PAGE Unclassified Unlimited 6 9a. NAME OF RESPONSIBLE PERSON Andrew Greenwood 9b. TELEPHONE NUMBER (include area code) Standard Form 98 (Rev. 8-98) Prescribed by ANSI Std. 39.8

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5 Contents. Introduction.... Geometry Wave Euation Boundary Conditions Solution of the Separated Wave Euation TE and TM Field Components TM Field Components TE Field Components Solution of the Separated Wave Euation Subject to the Boundary Conditions of the Generalied Geometry TM Field Components TE Field Components Waveguide Impedance Example Conclusion...7 APPENDIX A- About the Wave Euation in Cylindrical Coordinates...9 iii

6 List of Figures Figure. The geometry of the double baffled cylindrical coaxial waveguide.... Figure. The geometry of the aimuthally propagating, truncated, cylindrical coaxial waveguide....3 Figure 3. The values of for the TE mode as a function of freuency for an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide....3 Figure 4. The characteristic euation of the TE mode for an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide...4 Figure 5. The electromagnetic field distribution of the various field components in an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide...5 Figure 6. The value of the E component of the electric field of the TE mode as a function of aimuthal position...6 Figure 7. The electric field distribution of the various field components in an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide....6 Figure 8. The electromagnetic field distribution of the various field components for the TM mode...7 iv

7 . Introduction The double-baffled, coaxial waveguide transmission line, shown in Figure, is defined by inner and outer radii, and an arc length. In conventional applications, the propagating modes are assumed in the -direction. The TE and TM modes of this geometry, propagation constants, and guide wavelengths were studied in detail in [Ref. ]. In this memo, the waveguide is truncated in the -direction, and propagation is assumed in the aimuthal direction. Shown in Figure, the Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide is defined by inner and outer radii, and a depth a in the -dimension. We are interested in solving for the aimuthally, or φ-directed, propagating modes. First, the characteristic euations that define the cut off freuencies of each mode are derived, then the electric fields are explicitly expressed. Finally, an example geometry is defined for which the lowest TE and TM mode cutoff freuencies are computed and graphs of the normalied field components are presented.. Geometry The geometry of the aimuthally propagating, truncated, coaxial waveguide transmission line is shown in Figure. The transmission line is defined by is defined by inner ( ) and outer radii ( ), and a depth a in the -dimension. Propagation is assumed in the aimuthal direction. 3. Wave Euation The natural coordinate system for the Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide is the cylindrical coordinate system. The scalar wave euation in cylindrical coordinates is ψϕ (,, ) ψϕ (,, ) ψϕ (,, ) k ψϕ = ϕ (,, ) 0 Using the standard separation of variables techniue the wave euation can be written as ()

8 (a) (b) (c) Figure. The geometry of the double baffled cylindrical coaxial waveguide: (a) 3-D perspective drawing; (b) plane view of the xy-plane; and (c) plane view of the x-plane.

9 x φ Direction of propagation b a y (a) x f r b r y (b) x r b r r cos f 0 r cos f 0 (c) Figure. The geometry of the aimuthally propagating, truncated, cylindrical coaxial waveguide: (a) 3-D perspective drawing; (b) plane view of the xy-plane; and (c) plane view of the x-plane. 3

10 d dr( ) ( ) k + R( ) = 0 d d (a) d ( ϕ) ( ϕ ) 0 dϕ Φ + Φ = (b) d Z n Z ( ) + ( ) = 0 (c) d where: ψϕ (,, ) = R( ) Φ ( ϕ) Z( ), the are dimensionless propagation constants, and k + n = k. 4. Boundary Conditions The boundary conditions on the electric field for the geometry are: E = 0 for = 0, and = a (3a) E ϕ = 0 for =, =, = 0, and = a (3b) E = 0 for = and =. (3c) 5. Solution of the Separated Wave Euation The Φ ( ϕ) and Z( ) euations are harmonic euations with harmonic functions as solutions; these will be denoted hϕ ( ) and hn ( ). The euation in R( ) is a Bessel euation, and has Bessel function solutions: J ( k ) = the Bessel function of the first kind of order N( k ) = the Bessel function of the second kind of order H () ( k ) = the Hankel function of the first kind of order H () ( k ) = the Hankel function of the second kind of order Let the function B ( k ) represent the linearly independent combination of two of the above. Then, the general solution to the scalar Helmholt wave euation is: ψ = B( k ) h ( ϕ) hn ( ) (4) 4

11 6. TE and TM Field Components The electric and magnetic field components can be written in terms of fields that are TE and TM. See Appendix I. 6. TM Field Components The TM field components are found by letting potential, and u = unit vector in the -direction. Then Since and A = u ψ, where A = the magnetic vector E= jωa+ ( A ), (5a) jωµε and H= A. (5b) µ ψ ψ ψ ( A ) = ( u ψ ) = u + ϕ + ϕ u u, ψ ψ A = u ψ = u ϕ ϕ u, then expanded in cylindrical coordinates the components of the above euations become: E ψ = jωµε (6a) H ψ = µ ϕ (6d) E ϕ = jωµε ψ ϕ (6b) H ϕ ψ = µ (6e) E = jωψ + jωµε ψ (6c) H = 0 (6f) 6. TE Field Components The TE field components are found by letting potential, and u = unit vector in the -direction. Then F = u ψ, where F = the electric vector 5

12 and E= F, ε (7a) H= jωf + ( F ). jωµε (7b) Since ψ ψ ψ ( F ) = ( u ψ ) = u + ϕ + ϕ u u, and ψ ψ F = u ψ = u ϕ ϕ u, when the above euations are expanded in cylindrical coordinates, the components become: E ψ = (8a) ε ϕ H ψ = j (8d) ωµε E ϕ ψ = ε (8b) H ϕ ψ = j ωµε ϕ (8e) E = 0 (8c) ψ H = jωψ j ωµε (8f) 7. Solution of the Separated Wave Euation Subje ct to the Boundary Conditions of the Generalied Geometry Propagating waves in the φ-direction in the truncated coaxial waveguide give rise to harmonic functions jϕ h ( ϕ) = e (9) and, for hn ( ) The scalar wave function is then hn ( ) = a sin( n) + b cos( n) (0) n jϕ ψ = B ( k ) e hn ( ) subject to the boundary conditions. The solutions for the TE and TM modes in the guide are as follows. n () 6

13 7. TM Field Components The TM electric and magnetic field components in terms of the general wave function are nk jϕ jϕ E = { B( k) e hn ( )} = B ( k) e h ( n) jωµε jωµε (a) ( j) jϕ { } jϕ E = B ( k ) e hn ( ) = B ( k ) e hn ( ) ϕ jωµε ϕ jωµε (b) jϕ n jϕ E = jω+ B ( ) ( ) ( ) ( ) k e hn = jω + B k e h n jωµε k0 (c) jϕ H = j B( k) e hn ( ) µ (d) ψ k jϕ H = = B ( k ) e hn ( ) ϕ µ µ (e) d Note that B ( k) = B( k). Since d( k ) H = 0 (f) E ϕ = 0 for =, =, = 0, and = a then hn ( ) = ( a sin( n) + b cos( n)) = 0 = 0, a n n = 0, a is satisfied if: mπ a n =, b n = 0, n =, and m = 0,,, K. (3) a Note, then, that the cutoff freuencies are defined in the same way as the cutoff freuencies of standard rectangular guide, i.e., the cutoff freuencies of the TM modes of the aimuthally propagating, truncated, cylindrical, coaxial waveguide. The boundary conditions are also satisfied if B( k ) = = 0 Let B( k) = aj ( k) + bn ( k), then aj ( k) + bn ( k ) =, = 0., 7

14 And, aj ( k ) + bn ( k ) = 0 aj ( k ) + bn ( k ) = 0 Solving the first euation for a : N( k ) a = b (4) J ( k ) Substitution into the second euation yields: Or, N ( k ) aj ( k ) + bn ( k ) = b J ( k ) + an ( k ) = 0 J( k ) N ( k) b N( k) J( k) = 0 J( k ) For specific values of k, and, the values of that solve N( k) N( k) = (5) J ( k ) J ( k ) are the sought after mode numbers that are true for any non-ero value of b. Hence, b = and a N ( k) =. J ( k ) Finally, the scalar wave function for the TM modes is: N( k ) jϕ mπ ψ = N( k) J( k) e sin( n), for n =, m =,,3, K, and k + n = k. J( k ) a Note that just a single solution for is possible [Ref. 4]. The TM field components are then found explicitly as: 8

15 nk N ( k) E = N ( k) J ( k) cos( n) e jωµε J ( k ) jϕ N ( k ) Eϕ = N ( ) ( ) sin( ) k J k n e jωµε J( k ) n N( k ) = ω + k0 J ( k) E j N ( k ) J ( k ) sin( n) e jϕ jϕ (6a) (6b) (6c) N( k ) H = j N( k) J( k) sin( n) e µ J ( k ) k N ( k) Hϕ = N ( k) J ( k) sin( n) e µ J( k ) jϕ jϕ (6d) (6e) H = 0 (6f) 7. TE Field Components The TE electric and magnetic field components in terms of the general wave function are: jϕ E = j B ( k) e hn ( ) ε (7a) k j E B ( k ϕ = ) e hn ( ) ϕ ε (7b) E = 0 (7c) nk j H j B ( k ϕ = ) e h ( n) ωµε (7d) n jϕ Hϕ = B ( k) e h ( n) ωµε (7e) Since then n jϕ H = jω + B ( ) ( ) k e h n k0 E = 0 for = 0, and = a. hn ( ) = a sin( n) + b cos( n) = 0, a n n = 0, a (7f) 9

16 is satisfied if: mπ a n =, b n = 0, n =, and m =,,3, K. (8) a From the boundary condition on the E ϕ component, the general Bessel function, B ( k ), also satisfies the boundary conditions if d B( k ) = 0, ( = ) d k Let then B ( k ) = aj ( k ) + bn ( k ), d ( ) { aj } k bn k d k ( ) + ( ) =, = 0, or aj ( k ) + bn ( k ) 0 = =, Substituting for the boundaries = and = give the euations: aj ( k ) + bn ( k ) = 0 and aj ( k ) + bn ( k ) = 0. Solving the first euation above for a : N ( k ) a = b J ( k ) (9) Substitution into the second euation yields: Or, N ( k ) b J ( k ) + bn ( k ) = 0 J ( k ) N ( k ) b J ( k) N ( k ) = 0 J ( k ) For specific values of k, and, the values of that solve N ( k ) J ( k ) = N ( k ) J ( k ) (0) 0

17 are the sought after mode numbers that are true for any non-ero value of b. Hence, b = and N ( k ) a = b J ( k ) solves the boundary condition for =, and the value of k that satisfies En. 0 solves the boundary condition for =. Finally, the scalar wave function for the TE modes is: N ( k ) ψ = N( k) J( k) sin( n) e J ( k ) N ( k ) B( k) = N ( k ) J( k) J ( k ) jϕ mπ, for n =, m = 0,,, K, and a k + n = k. Note that again, just a single solution for is possible [Ref. 4]. The TE field components are then found explicitly as: N ( k ) E = j N( k) J( k) sin( n) e ε J ( k) k N ( k ) Eϕ = N ( k) J ( k) sin( n) e ε J ( k ) jϕ jϕ (a) (b) E = 0 (c) nk N ( k ) H = j N ( k) J ( k) cos( n) e ωµε J ( k ) n N ( k) Hϕ = N( k) J( k) cos( n) e ωµε J ( k ) jϕ jϕ n N ( k) H = jω + N ( ) ( ) sin( ) k J k n e k0 J ( k ) jϕ (d) (e) (f)

18 8. Waveguide Impedance The characteristic wave impedances for the Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide are defined in terms of the dominant field components for the TE and TM modes. The wave impedance for the TE mode is given by while wave impedance for the TM mode is given by 9. Example TE Z = E / H ; (a) TM Z = E / H. (b) Determine the fundamental mode and that mode s cutoff freuency of the Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: = 5in= 0.7 m, = 6in= 0.54 m and a = 6.5 inches. Plot the distributions of all field components at f =.3 GH, determine the guide wavelength in the propagating direction. The fundamental mode is the TE mode ( m = and = ). The cutoff freuency is determined by the standard rectangular guide cutoff freuency f c a 0 =, c where c 0 = speed of light and a = broad wall dimension. For this case: 8 f c = /( 0.65) = GH. Shown in Figure 3 is a plot of the numerically determined values of as a function of freuency for the TE mode. To compute the field distributions of the TE mode at f =.3 GH, the value of must first be determined. Shown in Figure 4 is a graph of the characteristic euation of the TE mode for an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters : = 5in= 0.7 m, = 6in= 0.54 m and a = 6.5 inches. The point at which the curve crosses ero defines the reuired value of, in this case =.7.

19 freuency GH Figure 3. The values of for the TE mode as a function of freuency for an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: = 5in= 0.7 m, = 6in= 0.54 m and a = 6.5 inches. The field components can now be determined using Euations. Shown in Figure 5 are the normalied electromagnetic field distributions of the various field components for the TE mode. One notes that E and H are the dominate field components (as expected), since they closely resemble the fundamental mode field distributions in standard rectangular guide. The guide wavelength can be determined numerically by plotting the real component of the phasor E component of the electric field of the TE mode as a function of aimuthal position as shown in Figure 6. The guide wavelength is found to be λ = ϕ = π =.67 inches g 80 which is just bigger than the free space wavelength at f =.3 GH of 9.08 inches, but less than rectangular the rectangular guide wavelength of λ =.687 inches. The TE mode impedance can TE TE be calculated as Z = E / H, and in this case is found to be Z = 8 Ω. g 3

20 TE Modes Characteristic Euation = Figure 4. The characteristic euation of the TE mode for an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: = 5in= 0.7 m, = 6in= 0.54 m and a = 6.5 inches. As the freuency increases, additional propagating modes become possible. For example, at f = GH, the TE mode will propagate. The value of =.955 is found numerically, and the electric field components are shown in Figure 7. As the freuency further increases, the first TM mode can propagate. For example, at f =. GH, the TM mode will propagate. Note that the cutoff freuencies for the TM modes are not defined by the standard simple rectangular waveguide relations (as was found for the TE modes). Rather, they are determined through Bessel function relations that are solved numerically. The value of =.998 is found numerically, and the electric and magnetic field components are shown in Figure 8. The TM mode impedance can be calculated as TM TM Z = E / H, and in this case is found to be Z = 40 Ω. 4

21 E-field E Eφ Er (a) /a H-field Hr Hφ H (b) /a Figure 5. The electromagnetic field distribution of the various field components in an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: = 5in= 0.7 m, = 6in= 0.54 m and a = 6.5 inches for the TE mode: (a) normalied electric field; and (b) normalied magnetic field. 5

22 Real[Er] φ degrees Figure 6. The value of the E component of the electric field of the TE mode as a function of aimuthal position. E-field E Eφ Er /a Figure 7. The electric field distribution of the various field components in an Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: = 5in= 0.7 m, = 6in= 0.54 m and a = 6.5 inches for the TE mode. 6

23 E-field Er Eφ E (a) /a H-field H H f H r (b) /a Figure 8. The electromagnetic field distribution of the various field components for the TM mode: (a) normalied electric field; and (b) normalied magnetic field. 0. Conclusion This report derives the cut off freuencies and fields of aimuthally propagating modes in a truncated, cylindrical waveguide. Waveguide impedances are also derived, and example computations are shown. The results are useful for designing antenna and transmission structures conformal to curved and cylindrical surfaces. 7

24 REFERENCES. Modes of a double baffled, cylindrical, coaxial waveguide, C. Courtney, et al, submitted for publication as an AFRL SSN memo, September Time Harmonic Electromagnetic Fields, R. Harrington, pg. 99, McGraw-Hill, NY, Waveguide Handbook, ed. N. Marcuwit, MIT Radiation Laboratory Series, vol. 0, pp Encyclopedia of Physics, ed. S. Flugge, pg. 366, Springer- Verlag, Berlin, Propagation in curved and twisted waveguides of rectangular cross-section, L. Lewin, Proceedings of the Institute of Electrical Engineers, Part B 0, pp 75 80,

25 APPENDIX A- About the Wave Euation in Cylindrical Coordinates The natural coordinate system for the Aimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide is the cylindrical coordinate system. And one might be tempted to propose solutions that TE φ and TM φ. This would not be wise. The VECTOR wave euation is ( ) + k 0? = 0 (A) where the vector operator () = ( ()) (). Now this euation reduces to the scalar wave euation for Cartesian coordinates, since the unit vectors are constants. This is not so in cylindrical coordinates, except for the unit vector in the -direction. When written out, Euation No. becomes Ψ Ψ Ψ ϕ + Ψϕ Ψ ϕ + Ψ ϕ + Ψ ( ) + k0 = 0 ϕ u ϕ u u? Separating this into its components yields three euations Ψ Ψ k Ψ ϕ + 0 Ψ = 0 ϕ Ψϕ Ψ k ϕ + Ψ + 0Ψ ϕ = 0 ϕ k0 0 (A) (A3a) (A3b) Ψ + Ψ = (A3c) Only Euation 3c is a scalar wave euation. In forming the solutions for the potentials, it is important that we propose solutions that are TE and TM, so that solutions to the scalar wave euation can be postulated. From Prof. Chalmers Butler s ECE 83 class notes, Clemson University,

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