ECE 451 Advanced Microwave Measurements. Circular and Coaxial Waveguides

Transcription

1 ECE 451 Advanced Microwave Measurements Circular and Coaxial Waveguides Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois ECE 451 Jose Schutt Aine 1

2 Circular Waveguide Fields For a waveguide with arbitrary cross section, it is known that TE Modes H x H 2 2 y H (1) TM Modes E x E 2 2 y E (2) We first assume TM modes in cylindrical coordinates: E 1E 1 E r r r r E tr E 0 See Reference [6]. ECE 451 Jose Schutt Aine 2

3 Circular Waveguide TM Modes Solution will be in the form, E r f r g Which after substitution gives 1 f dr dr g d 2 r d df 2 2 d g r h r 2 (3) where h For equality in (3) to hold, both sides must be equal to the same constant say n 2 where n is an integer in view of the aimuthal symmetry since the fields must be periodic in. ECE 451 Jose Schutt Aine 3

4 Circular Waveguide TM Modes 2 d g 2 ng 0 2 d (4) 2 2 d f 1 df 2 n h f dr r dr r Solution of (4) is of the form cos sin g C n C n 1 2 (5) (6) (5) is Bessel s equation and has solution f r C J hr C Y hr 3 n 4 J n and Y n are the n th order Bessel functions of the first and second kinds respectively n (7) ECE 451 Jose Schutt Aine 4

5 Bessel Functions of the First Kind J n x r0 1 x /2 r n r r n2r! 1 n1 n! ECE 451 Jose Schutt Aine 5

6 Circular Waveguide TM Modes Y n has singularity at 0 and must consequently be discarded C 4 = 0. The general solution then becomes, cos sin E r C3Jn hr C1 n C2 n Since the origin for is arbitrary, the expression can be written as: E r C J hr n, cos n n where C n is a constant. The boundary condition E tan = 0 requires that E r, 0 for r a Jn Solution exists for only discrete values of h such that ha 0 ECE 451 Jose Schutt Aine 6

7 n ha must be a root of the n th order Bessel function. If we assume that t nl is the l th root of J n, we can define a set of eigenvalues h nl for the TM modes so that: tnl htm nl a l Circular Waveguide TM Modes l th root of J n (.)= Each choice of n and l specifies a particular solution or mode n is related to the number of circumferential variations and l describes the number of radial variations of the field. ECE 451 Jose Schutt Aine 7

8 Circular Waveguide TM Modes The propagation constant of the nl th propagating TM mode is: TM nl 2 tnl a The propagation occurs for < ctmnl or f > f ctmnl where the cutoff frequency and wavelength can be found from = 0 as: 2 1/2 ctmnl The other field components can be obtained from E 2 a t nl f ctmnl t 2 a nl tnl E CnJn r cosn e a j nl ECE 451 Jose Schutt Aine 8

9 Circular Waveguide TE Modes The solutions for the TE modes can be found in a similar manner except that we solve for H (r,) to get:, cos H r C J hr n n n To apply the boundary condition E tan = 0, we require r H to be 0 at r = a We must have H nˆ trh 0 at r a r For this, we need the eros of J n (u) given by s nl. The propagation constant, cutoff frequency and wavelength have the same expressions as in the TM case with t nl s nl. ECE 451 Jose Schutt Aine 9

10 Circular Waveguide TE Modes The propagation constant of the nl th propagating TE mode is: TE nl 2 s a nl 2 1/2 n l l th root of J n (.)= From the tables, it can be seen that the lowest cutoff frequency is the TE 11 mode. and for TE modes, snl H CnJn r cosn e a j nl ECE 451 Jose Schutt Aine 10

11 Circular Waveguide TE & TM Modes See Reference [6]. ECE 451 Jose Schutt Aine 11

12 TE 11 Mode in Circular Waveguide See Reference [1]. E H ECE 451 Jose Schutt Aine 12

13 Modes in Circular Waveguide TE 11 E H TM 11 See Reference [1]. ECE 451 Jose Schutt Aine 13

14 Example: Circular Waveguide Design Design an air filled circular waveguide such that only the dominant mode will propagate over a bandwidth of 10 GH. Solution: the cutoff frequency of the TE 11 mode is the lower bound of the bandwidth. f cte c 2 a The next mode is the TM 01 with cutoff frequency: f ctm c 2 a ECE 451 Jose Schutt Aine 14

15 Example: Circular Waveguide Design The BW is the difference between these two frequencies c BW fctm f cte GH 11 2 a From which we find a = cm So that f 32.7 GH and f GH cte ctm ECE 451 Jose Schutt Aine 15

16 Coaxial Waveguide Most common two conductor transmission system Dielectric filling in most microwave applications is polyethylene or Teflon ECE 451 Jose Schutt Aine 16

17 Coaxial Waveguide TEM Mode Two conductor system Dominant mode is TEM Tangential E field and normal H field must be 0 in conductor surfaces E 0andH r 0at r a, b ECE 451 Jose Schutt Aine 17

18 Coaxial Waveguide TEM Mode TEM solution can exist only with ˆ E re ˆ r, and H H r, r with no dependence because of aimuthal symmetry we get H o o jer jh r jer r o 1 H 1 H o H 0 H r 0 r r r r Where propagation in direction is assumed. ECE 451 Jose Schutt Aine 18

19 We get Coaxial Waveguide TEM Mode H H e r ˆ o j E rˆ H o j where H o is a constant. No cutoff condition for TEM mode. The voltage between the two conductors is given by ln / j V Ho b a e The current in the inner conductor is given by 2 I j Hoe The characteristic impedance Z o is thus given by ln( b/ a) Zo 2 r ECE 451 Jose Schutt Aine 19 e

20 Coaxial Waveguide TE and TM Modes TE and TM modes may also exist in addition to TEM. In a coaxial line, they are generally undesirable. For TM modes, we have:, cos o E r C3J n hr C4Yn hr n For TE modes, we have: ' ', cos o H r C3Jn hr C4Yn hr n With boundary conditions at r =a, b of E r H r, 0 for TM modes 0 for TE modes ECE 451 Jose Schutt Aine 20

21 Coaxial Waveguide TE and TM Modes These conditions lead to for TM modes J ha Y hb J hb Y ha n n n n J ha Y hb J hb Y ha ' ' ' ' n n n n for TE modes Solutions of these transcendental equations determine the eigenvalues of h for given a, b. As in the circular waveguide case, the modes for coaxial waveguide are denoted TE nl and TM nl. ECE 451 Jose Schutt Aine 21

22 Coaxial Waveguide TE and TM Modes The mode with the lowest cutoff frequency is the TE 11 mode for which the eigenvalue h is approximated as: h a 2 b The cutoff frequency and cutoff wavelength are given by 2 ab and f 1 h a b c11 c11 ECE 451 Jose Schutt Aine 22

23 Coaxial Waveguide TE and TM Modes TM 01 See Reference [3]. ECE 451 Jose Schutt Aine 23

24 References [1]. C. S. Lee, S. W. Lee, and S. L. Chuang, "Plot of modal field distribution in rectangular and circular waveguides", IEEE Trans. Microwave Theory and Techniques, 33(3), pp , March [2]. J. H. Bryant, "Coaxial transmission lines, related two-conductor transmission lines, connectors, and components: A U.S. historical perspective", IEEE Trans. Microwave Theory and Techniques, 32(9), pp , September [3]. H. A. Atwater, "Introduction to Microwave Theory", p. 76, McGraw-Hill, New York, [4]. N. Marcuvit, "Waveguide Handbook", IEEE Press, Piscataway, New Jersey, [5]. S. Ramo, J. R. Whinnery, and T. Van Duer, "Fields and Waves in Communication Electronics", John Wiley & Sons, New York, [6]. U. S. Inan and A. S. Inan, "Electromagnetice Waves", Prentice Hall, ECE 451 Jose Schutt Aine 24