ECE Spring Prof. David R. Jackson ECE Dept. Notes 9

Transcription

1 ECE 6341 Spring 2016 Prof. David R. Jackson ECE Dept. Notes 9 1

2 Circular Waveguide The waveguide is homogeneously filled, so we have independent TE and TM modes. a ε r A TM mode: ψ ρφ,, ( ) Jυ( kρρ) sin( υφ) ψ e Yυ( kρρ) cos( υφ) jk k k k ρ 2

3 Circular Waveguide (cont.) (1) φ variation φ [0, 2 π] ψ( ρφ, + 2 π, ) ψ( ρφ,, ) υ n (uniqueness of solution) Choose cos( nφ ) Jn ( kρ ρ) ψ Yn ( kρ ρ) cos( nφ) e jk 3

4 Circular Waveguide (cont.) 0, φ, (2) The field should be finite on the axis ( ) ψ Y ( ) n k ρ ρ is not allowed ψ cos( nφ) J ( k ρ) e n ρ jk k k k ρ 4

5 Circular Waveguide (cont.) E a, φ, 0 (3) B.C. s: ( ) E 1 k jωµε 1 jωµε k ρ ψ jωµε ( k k ) ψ ψ so ψ( a, φ, ) 0 Hence J ( ) 0 n ka ρ 5

6 Circular Waveguide (cont.) J ( ) 0 n ka ρ J n (x) Plot shown for n 0 x n1 x n2 x n3 x ka ρ x np k ρ x np a x J x 0 Note: is not included since (trivial soln.) n0 0 n ρ np a 6

7 Circular Waveguide (cont.) TM np mode: ρ A cos( ) jk nφ Jn xnp e n 0,1, 2 a 1/2 2 x 1, 2,3,... a 2 np k k p 7

8 Cutoff Frequency: TM k k k ρ k 0 k k ρ x np a 2π f µε c x np a f TM c c 2πa ε r x np 8

9 Cutoff Frequency: TM (cont.) x np values p \ n TM 01, TM 11, TM 21, TM 02,.. 9

10 TE Modes F ψ ρφ,, ( ) ψ cos( nφ) J ( k ρ) e n ρ jk H 2 kρ ψ jωµε Note: ψ a, φ, 0 ( ) 10

11 TE Modes (cont.) Set E a,, 0 ( ) φ φ E φ 1 ψ ε ρ so ψ ρ a ρ 0 Hence J ( ) 0 n ka ρ 11

12 TE Modes (cont.) J ( ) 0 n ka ρ J n ' (x) Plot shown for n 1 Recall: n 1 Jn( x) ~ x, n 0,1, 2,... n 2 n! x' n1 x' n2 x' n3 x ka k ρ ρ x np x np p 1,2,3,... a Note: p 0 is not included (see next slide). 12

13 TE Modes (cont.) ρ ψ cos( nφ) J jk n x np e p 1, 2, a Note: If p 0 x np 0 n 0 ρ Jn x np Jn ( 0) 0 a (trivial soln.) ρ n 0 J0 x np J0( 0) 1 a ψ jk jk e e (trivial fields) k ρ 0 13

14 Cutoff Frequency: TE k k k ρ k 0 k ρ k x np a 2π f µε c x np a f TE c c x 2πa ε r np 14

15 Cutoff Frequency:TE x np values p \ n TE 11, TE 21, TE 01, TE 31,.. 15

16 TE 11 Mode The dominant mode of circular waveguide is the TE 11 mode. Electric field Magnetic field (from Wikipedia) TE 10 mode of rectangular waveguide TE 11 mode of circular waveguide The TE 11 mode can be thought of as an evolution of the TE 10 mode of rectangular waveguide as the boundary changes shape. 16

17 Attenuation Property of TE 01 Mode Goal: We wish to study the high-frequency dependence of attenuation on frequency for circular waveguide modes, and show the interesting behavior of the TE 01 mode (the loss decreases as frequency increases). α c < Pd > 2 < P > f R s 1 1 σδ 2 σ ωµσ < Pd > C 1 2 R H s t 2 dl ωµ 2σ O ( ω ) TE Mode: H H φ F jωµε ρ φ 1 jωµε 2 kρ F Assume that F is order 1 as the frequency increases. Recall that k ρ is a constant. 17

18 Attenuation Property (cont.) H H φ 1 O ω O (1) n 0 0 n 0 Note: ( ) O( ) k O k ω n 0 1 < Pd > O O + O ω O ( ω ) 2 ( 1) 1/2 ( ω ) n 0 ( ) 1 < Pd > O ω O ω O 3/2 ( ω ) 18

19 Attenuation Property (cont.) From the TE table: E H O (1) e.g. 1 F Eφ ε ρ O (1) e.g. 1 Hρ jωµε 2 F ρ Hence Pf O ( 1) 19

20 Attenuation Property (cont.) Hence If n 0: ( 1/2 α ) c O ω If n 0: ( 3/2 α ) c O ω Usual behavior for rectangular waveguides Decreases with frequency! n 0: E H Note: The mode TE 0p mode can be supported by a series of concentric rings, since there is no longitudinal (-directed) current (H φ 0). 20

21 Attenuation Property (cont.) α c TE 11 TM 01 TE 21 TM 11 αc O ( ω 1/2 ) TE 01 αc O ( ω 3/2 ) f c, TE11 f c, TM01 f c, TE21 f c, TE01 f 21

22 Attenuation Property (cont.) The TE 01 mode was studied extensively as a candidate for long-range communications but was not competitive with antennas. Also, fiber-optic cables eventually became available with lower loss than the TE 01 mode. It is still useful for some applications (e.g., high power). From the beginning, the most obvious application of waveguides had been as a communications medium. It had been determined by both Schelkunoff and Mead, independently, in July 1933, that an axially symmetric electric wave (TE 01 ) in circular waveguide would have an attenuation factor that decreased with increasing frequency [44]. This unique characteristic was believed to offer a great potential for wide-band, multichannel systems, and for many years to come the development of such a system was a major focus of work within the waveguide group at BTL. It is important to note, however, that the use of waveguide as a long transmission line never did prove to be practical, and Southworth eventually began to realie that the role of waveguide would be somewhat different than originally expected. In a memorandum dated October 23, 1939, he concluded that microwave radio with highly directive antennas was to be preferred to long transmission lines. Thus, he wrote, we come to the conclusion that the hollow, cylindrical conductor is to be valued primarily as a new circuit element, but not yet as a new type of toll cable [45]. It was as a circuit element in military radar that waveguide technology was to find its first major application and to receive an enormous stimulus to both practical and theoretical advance. K. S. Packard, The Origins of Waveguide: A Case of Multiple Rediscovery, IEEE Trans. MTT, pp , Sept

23 Attenuation Property (cont.) VertexRSI's Torrance Facility is a leading supplier of antenna feed components for the various commercial and military bands. A patented circular polaried 4-port diplexer meeting all Intelsat specifications leads a full array of products. Products include: 4-Port Diplexers, CP or Linear; 3-Port Diplexers, 2xRx & 1xTx; 2-Port Diplexers, RxTx, X-Pol or Co-Pol, CP or Linear; TE21 Monopulse Tracking Couplers; TE01 Mode Components; Transitions; Filters; Flex Waveguides; Waveguide Bends; Twists; Runs; etc. Many of the items are "off the shelf products". Products can be custom tailored to a customer's application. Many of the products can be supplied with standard feed horns for prime or offset antennas. 23