Azimuthally Propagating Modes in an Axially-Truncated, Circular, Coaxial Waveguide

Transcription

1 Azimuthally Propagating Modes in an Axially-Truncated, Circular, Coaxial Waveguide Clifton C. Courtney Voss Scientific Albuquerque, NM There is considerable interest in transition structures that efficiently transport high power RF fields from transmission lines to antenna apertures. By efficient, we mean ways that minimize the amount of reflected power, and the resulting field enhancement of standing waves, that occurs at the interface of the transmission line and the transition structure. In this note we derive and examine the modal properties of the Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide. This guide is defined by an inner and outer radii, and a width. In this note we derive the governing equations for the electromagnetic fields of the TE z and TM z azimuthally propagating modes. First, the characteristic equations 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 a 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. Acknowledgement - This work was supported in part by the Air Force Research Laboratory, Directed Energy Directorate, under a Small Business Innovation for Research (SBIR) Phase I program, Contract No. F M-OlOl.

2 The double-baffled, coaxial waveguide transmission line, shown in Figure 1, is defined by inner and outer radii, and an arc length. In conventional applications, the propagating modes are assumed in the z-direction. The TEz and TMz modes of this geometry, propagation constants, and guide wavelengths were studied in detail in [Ref. 1]. In this memo, the waveguide is truncated in the z-direction, and propagation is assumed in the azimuthal direction. Shown in Figure 2, the Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide is defined by inner and outer radii, and a depth a in the z-dimension. Weare interested in solving for the azimuthally, or 4>-directed, propagating modes. First, the characteristic equations that define the cut off frequencies 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 frequencies are computed and graphs of the normalized field components are presented. 2. Geometry The geometry of the azimuthally propagating, truncated, coaxial waveguide transmission line is shown in Figure 2. The transmission line is defined by is defined by inner (Pl) and outer radii (P2 ), and a depth a in the z-dimension. Propagation is assumed in the azimuthal direction. 3. Wave Equation The natural coordinate system for the Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide is the cylindrical coordinate system. The scalar wave equation in cylindrical coordinates is l-~( 2 8lf/(P,rp,Z)] 1 8 lf/(p,rp,z) 8 2 lf/(p,rp,z) k2 ( )- 0 P If/ p,rp,z - P 8p 8p P 8cp 8z P:p(p~~}[(kA -q']r(ph d 2-2 <D(cp)+q 2 <D(rp)= 0 (2b) dcp d 2-2 Z(z)+ n 2 Z(z) = 0 (2c) dz where: If/(p,cp,z) = R(p)<D(cp)Z(z), the q are dimensionless propagation constants, and k~ + n 2 = k 2 (2a)

3 Baffle I y X (b) --P2 COS~O P1 COS~O 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 xz-plane.

4 y y P1 -P2 COS~o _ P1 COSt/J O (c) The geometry of the azimuthally propagating, truncated, cylindrical coaxial waveguide: (a) 3-D perspective drawing; (b) plane view of the xy-plane; and (c) plane view of the xz-plane.

5 4. Boundary Conditions E<p = 0 for P = PI, P = P2, Z = 0, and z = a 5. Solution of the Separated Wave Equation The <p(ep) and Z(z) equations are harmonic equations with harmonic functions as solutions; these will be denoted h(qep) and h(nz). The equation in R(p) is a Bessel equation, and has Bessel function solutions: Jq(kpp) = the Bessel function of the first kind of order q N q (kpp) = the Bessel function of the second kind of order q H~I)(kpp) = the Hankel function of the first kind of order q H~2) (k pp) = the Hankel function of the second kind of order q Let the function Bq(kpp) represent the linearly independent combination of two of the above. Then, the general solution to the scalar Helmholtz wave equation is: 6. TEz and TMz Field Components The electric and magnetic field components can be written in terms of fields that are TEz and TMz. See Appendix I. The TMz field components are found by letting A = uz~' where A = the magnetic vector potential, and U z = unit vector in the z-direction. Then

6 1 E=-jwA+-.-V(V A), JWJ1E 1 and H=-VxA. J1 o [Olf/] 1 0 [Olf/] 0 [Olf/] V(V A)=V(V,uzlf/)=u u uz- --, p op oz rp p orp oz oz oz V x A = V x Uzlf/ = up J.. 0 If/ - urp0 If/, then expanded in cylindrical coordinates the components p orp op of the above equations become: H = J..J.. olf/ p J1 p orp H =_J..o~ rp Hz =0 J1 op The TE z field components are found by letting F = uzlf/, where F = the electric vector potential, and U z = unit vector in the z-direction. Then 1 E=--VxF, E H=-iwF+-.-1-V(V.F). JWJ1E Since V(V F)=V(V,uzlf/)=u ~[Olf/l]+u J..~[OIf/]+Uz~[OIf/], and p op oz rp p orp OZ oz OZ 1 0 If/! olf/ VxF=Vxuzlf/=up-_ -urp-' p orp op when the above equations are expanded in cylindrical coordinates, the components become: E = _1.J.. olf/ (8a) H = _ i_1_ 021f/ PEp orp p WJ1EOZop E =1. Olf/ (8b) H = _ i_l_j.. 021f/ rp E op rp WJ1EP orp OZ (8d) (8e)

7 H j/ (8t) z =-jojij/- j-- OJj.1 8z 2 7. Solution of the Separated Wave Equation Subject to the Boundary Conditions of the Generalized Geometry Propagating waves in the -direction in the truncated coaxial waveguide give rise to harmonic functions The scalar wave function is then Ij/ = Bq(kpp)e-jqffJh(nz) subject to the boundary conditions. follows. The solutions for the TE z and TM z modes in the guide are as I 8 2 { '} nk,. E =-.--- B (k p)e-jqffjh(nz) =-.-p-b (k p)e-jqffjh'(nz) p jojj.1 8z 8p q p jojj.1 q p (-.)2 1. E =-.----{B (k p)e-;qffjh(nz)} =. jq B (k p)e-;qffjh(nz) (l2b) ffj jojj.1 p2 8r/ q p jojj.1 p2 q p E z = {- joj+-. _1_ 8 2 } Bq(kpp)e- jqffjh(nz)= - joj(l + n: J Bq(kpp)e-jqffJh"(nz) (l2c) 2 jojj.1 8z 11 H =-jq--b (k p)e-jqffjh(nz) (l2d) p j.1p q p H = _! 81j/ = _ k p B' (k p)e-jqffjh(nz) (l2e) ffj j.1 8p j.1 q p Hz = 0 k o (l2t)

8 Erp = 0 for P = PI' P = P2, Z = 0, and z = a is satisfied if: an = 1, b n = 0, n = m1r, and m = 0,1,2,... a Note, then, that the cutoff frequencies are defined in the same way as the cutoff frequencies of standard rectangular guide, i.e., the cutoff frequencies of the TM modes of the azimuthally propagating, truncated, cylindrical, coaxial waveguide. The boundary conditions are also satisfied if Bq(kpp) Ip=Pl,P2 = 0 aqjq(kppj)+ bqnq(kppj) = 0 aqjq(kpp2)+bqnq(kpp2) = 0 Nq(kpP2) Jq(kpP2) Nq(kppJ) Jq(kppJ) Nq(kppJ) b = 1 and a = Finally, the scalar wave function for the TM z modes is: q q Jq(kpPl)

9 Note that just a single solution for q is possible [Ref. 4]. The TM z field components are then found explicitly as: q I. E = j--b (k p)e-;q'l'h(nz) P 8p q P k. E =-E.-B'(k p)e-;q'l'h(nz) 'I' 8 q P H E z =0 nk. = - j-p-b' (k p)e-;q'l'h'(nz) P WJl8 q P (l7b) (l7c) (l7d) = ---!!!L~B(k p)e-jq'l'h'(nz) H 'I' WJl8 P q P

10 is satisfied if: an = 1, b n = 0, n = m1r, and m = 1,2,3,... a From the boundary condition on the E'P satisfies the boundary conditions if component, the general Bessel function, Bq (k pp ), also d [B (k )J I = 0 d( kpp) q pp P;fJl,P2

11 For specific values of k p ' PI and P2' the values of q that solve N' (k PI) bq = 1 and aq = -bq ~ p solves the boundary condition for P = PI' and the value of kp Jq(kpPI) that satisfies Eqn. 20 solves the boundary condition for P = P2. N'(k ) B (k ) N (k ) - J (k ) q ppi q pp - q pp q pp J' (k ) q ppi Note that again, just a single solution for q is possible [Ref. 4]. The TE z field components are then found explicitly as: q 1 [ N' (k PI)]. E = j-- N (k p)-j (k p) q p sin(nz)e- NqJ PEp q p q p J;(kpPI) k [ N' (k PI)]. E =-!!.. N'(k p)-j'(k p) q p sin(nz)e-;qqj qj E q p q p J;(kpPl) nk [ N' (k PI)]. H =-j-p- N'(k p)-j'(k p) q p cos(nz)e-;qqj p WJ1E q p q p J;(kpPI) nq 1 [ N' (k PI)]. H = ---- N (k p)-j (k p) q p cos(nz)e-;qqj qj WJ1E P q p q p J;(kpPI)

12 The characteristic wave impedances for the Azimuthally 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 ZTE =E 1H p z; ~2~ while wave impedance for the TM mode is given by ZTM =E z 1H p Determine the fundamental mode and that mode's cutoff frequency of the Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: PI = 5 in = m, pz = 6 in = m and a = 6.5 inches. Plot the distributions of all field components at f = 1.3 GHz, determine the guide wavelength in the propagating direction. The fundamental mode is the TEz 11 mode (m = 1 and q = 1). The cutoff frequency is determined by the standard rectangular guide cutoff frequency Ie = s-., where Co = speed of light and 2a a = broad wall dimension. For this case: Ie = xl 0 8 /(2 X ) = GHz. Shown in Figure 3 is a plot of the numerically TEz ll mode. determined values of q as a function of frequency for the

13 1~ 2 2~ frequency - GHz The values of q for the TEz ll mode as a function of frequency for an Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: PI =5in=O.l27m, P2 =6in=0.1524m and a=6.5 inches. To compute the field distributions of the TEz ll mode at f = 1.3 GHz, the value of q must first be determined. Shown in Figure 4 is a graph of the characteristic equation of the TEz11 mode for an Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: PI = 5in = m, P2 = 6in = m and a = 6.5 inches. The point at which the curve crosses zero defines the required value of q, in this case q = c: 0 :;:l III ;:, C" W (,) 0.2 :;:l 0 III '': II).. t) III III.c: q The characteristic equation of the TEz11 mode for an Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: PI =5in=0.127m, P2 =6in=0.1524m and a=6.5 inches.

14 The field components can now be determined using Equations 21. Shown in Figure 5 are the normalized electromagnetic field distributions of the various field components for the TEz ll mode. One notes that E p and Hz 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 p component ofthe electric field of the TEz ll mode as a function of azimuthal position as shown in Figure 6. The guide wavelength is found to be p + P A = 1 2 xrp=--x-xjr=12.67 mches g which is just bigger than the free space wavelength at f = 1.3 GHz of 9.08 inches, but less than the rectangular guide wavelength of A;ectangular = inches. The TE mode impedance can be calculated as ZTE = E p / Hz, and in this case is found to be ZTE = 181 D. As the frequency increases, additional propagating modes become possible. For example, at f = 2 GHz, the TEz 21 mode will propagate. The value of q = is found numerically, and the electric field components are shown in Figure 7. As the frequency further increases, the first TM mode can propagate. For example, at f = 2.2 GHz, the TMz ll mode will propagate. Note that the cutoff frequencies 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 q = is found numerically, and the electric and magnetic field components are shown in Figure 8. The TM mode impedance can be calculated as ZTM = E z / H p ' and in this case is found to be ZTM = 1140 D.

15 -c 0.6 a; l;: I ~ :;- 0.6 a; l;: I = (a) Z/a The electromagnetic field distribution of the various field components in an Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: PI =5in=O.127m, pz =6in=O.1524m and a=6.5 inches for the TEzll mode: (a) normalized electric field; and (b) normalized magnetic field.

16 ~ 0 iii Gl 0:: I{J - degrees The value of the Ep component of the electric field of the TEz ll mode as a function of azimuthal position " 0.6 Q) 'i: I w zja The electric field distribution of the various field components in an Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide defined by the parameters: PI =5in=O.127m, P2 =6in=O.1524m and a=6.5 inches for the TEz 21 mode.

17 (a) "C 0.6 (j) I.t= I w (b) 't:l Q) 0.6 q: I :I: The electromagnetic field distribution of the various field components for the TMz ll mode: (a) normalized electric field; and (b) normalized magnetic field. 1. "Modes of a double baffled, cylindrical, coaxial waveguide," C. Courtney, et ai, submitted for publication as an AFRL SSN memo, September Time Harmonic Electromagnetic Fields, R. Harrington, pg. 199, McGraw-Hill, NY, Waveguide Handbook, ed. N. Marcuwitz, MIT Radiation Laboratory Series, vol. 10, 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 102, pp 75-80, 1055.

18 APPENDIX I - About the Wave Equation in Cylindrical Coordinates 1 The natural coordinate system for the Azimuthally Propagating, Truncated, Cylindrical, Coaxial Waveguide is the cylindrical coordinate system. And one might be tempted to propose solutions that TE and TM. This would not be wise. where the vector operator y-20=v(v O)-VxVO. Now this equation reduces to the scalar wave equation for Cartesian coordinates, since the unit vectors are constants. This is not so in cylindrical coordinates, except for the unit vector in the z -direction. When written out, Equation No. I becomes Only Equation 3c is a scalar wave equation. In forming the solutions for the potentials, it is important that we propose solutions that are TE z and TM z, so that solutions to the scalar wave equation can be postulated.