Numerical Study of the Differential Phase Shift in the Azimuthally Magnetized Coaxial Ferrite Waveguide

Mariana Nikolova Georgieva–Grosse1, Georgi Nikolov Georgiev2 1 Consulting and Researcher in Physics and Computer Sciences, D–70839 Gerlingen, Germany

2 Faculty of Mathematics and Informatics, University of Veliko Tirnovo “St. St. Cyril and Methodius”, BG–5000 Veliko Tirnovo, Bulgaria

Abstract—An iterative method is elaborated for counting the differential phase shift produced by the coaxial waveguide, entirely filled with azimuthally magnetized ferrite, under normal

01TE mode excitation. It consists in a repeated numerical sol-ution of the characteristic equation of configuration, written by means of certain complex Kummer and Tricomi confluent hyper-geometric functions, for varying in accordance with a definite scheme imaginary part of their complex first parameters, follow-ed by finding the normalized in an appropriate way guide radius and phase constant of the wave. In the computations the value of central switching conductor to guide radius ratio is assumed fix-ed and the one of the modulus of off-diagonal ferrite permeability tensor element is changed equidistantly. An end is put to the procedure when the reckoned numerical equivalent of the radius coincides with the preliminary singled out one of the same within the framework of the prescribed accuracy. The corresponding to it figured out value of the phase constant is accepted as that, sought. The calculations are accomplished for both signs of the imaginary part in question (of the ferrite magnetization) which results in the phase shift looked for in normalized form. The borders of the area of phase shifter operation of the geometry are specified, as well. The quantity of principle interest is worked out in the lower section of its existence region. The outcomes of investigation are presented in Tables. The analysis is restricted to the case of a thin central conductor.

Index Terms—boundary-value problems; coaxial ferrite waveguides; ferrite phase shifters; numerical techniques.

I. INTRODUCTION The transversely magnetized circular ferrite waveguides are

applicable in designing various components for the modern microwave systems [1-15]. In particular, the transmission lines of azimuthal magnetization of the load that support normal

01TE mode, exhibit properties of nonreciprocal digital phase shifters [1,3,6-8,10-15] and therefore are eligible for the elabor-ation of electronically scanned antenna arrays [16]. The elec-trodynamic analysis of these set-ups, being a precondition for the construction of the devices mentioned, faces serious obstacles of mathematical character [1-3,6-8,10-15]. One of the methods to overcome them which proved to be especially efficacious recently is the one, suggested by Georgiev and Georgieva-Grosse [7,8,10-15,17]. It is based on the boundary-value approach, takes advantage of complex confluent hyper-geometric (real Coulomb wave) [18] and in some cases of real cylindrical functions, too, depending on the specific geometry treated and widely employs iterative schemes [7,8,10-15].

The aim of this work is twofold: i) the development of a technique, using successive approximations for determining the differential phase shift provided by the coaxial waveguide, loaded with azimuthally magnetized ferrite and propagating normal 01TE mode; ii) the practical calculation of this quantity for selected parameters of configuration considered, harnessing the method in question.

Unlike similar approaches, reported lately [7,8,12-14], the basic equation profited is expressed through complex Kummer and Tricomi confluent hypergeometric functions. Besides, the main parameters altered are the central switching conductor to guide radius ratio, the modulus of off-diagonal ferrite permea-bility tensor element and the suitably normalized guide radius. As earlier [7,8,12-14], the iterative process is performed toward the imaginary part of the complex first parameter of wave func-tions. The procedure is complicated and very time consuming. It however can ensure a very high accuracy of the outcomes.

II. SYNOPSIS OF BOUNDARY-VALUE PROBLEM The structure under study is an infinitely long lossless

perfectly conducting coaxial waveguide of radius 0r , uniformly filled with latching ferrite, magnetized in azimuthal direction to remanence by a central switching wire of radius

1r . The anisotropic load is characterized by a Polder permeability tensor [ ]ijμμμ 0= , i , =j 1, 2, 3, with nonzero components 1=iiμ and =13μ =− 31μ αj− , ωγα /rM= ,

11 <<− α , (γ – gyromagnetic ratio, rM – ferrite remanent magnetization, ω – angular frequency of the wave) and a scalar permittivity rεεε 0= . The case of normal nTE0 modes is regarded, described by the equation [10,12]:

( ) ( ) ( ) ( )0000 ;,/;,;,/;, xcaxcaxcaxca ρρ ΨΦ=ΨΦ (1)

in which ( )xca ;,Φ and ( )xca ;,Ψ are the Kummer and Tricomi confluent hypergeometric functions [18]. Here

jkca −= 2/ , 3=c , 00 jzx = , =k ( )22/ ββα , 020 2 rz β= , 01 / rr=ρ ( k , 0z , ρ – real, +∞<<−∞ k , 00 >z , 10 << ρ ),

=β ( )rεββ 0/ , =2β ( )rεββ 02 / , rrr εβ 000 = , =1r rr εβ 10 , 000 μεωβ = , ( β – phase constant of the wave,

( )[ ] 2/12200

22 1 βαεμεωβ −−= r – radial wavenumber, ρ –

central conductor to guide radius ratio) [10,12]. Provided

The 8th European Conference on Antennas and Propagation (EuCAP 2014)

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TABLE I. NUMERICAL VALUES OF THE QUANTITIES crr0 and −enr0 FOR NORMAL 01TE MODE ( 3=c , 1=n ) AS A FUNCTION OF α IN CASE 1.0=ρ .

α 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

crr0 3.9411 3867 3.9417 3004 3.9427 1623 3.9440 9815 3.9458 7704 3.9480 5452 3.9506 3255 3.9536 1348 3.9570 0004 3.9607 9535

−enr0 765.0504 7006 382.5826 3247 255.1189 0154 191.4062 4044 153.1940 5572 127.7321 6129 112.0676 0018 95.9340 0866 85.3477 1846 76.8866 2068

α 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20

crr0 3.9650 0288 3.9696 2656 3.9746 7069 3.9801 4000 3.9860 3968 3.9923 7532 3.9991 5303 4.0063 7937 4.0140 6139 4.0222 0669

−enr0 69.9711 7914 64.2150 4290 59.3507 4436 55.1872 4110 51.5844 4097 48.4372 8032 45.6654 2165 43.2063 8543 41.0108 5083 39.0393 6620

α 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

crr0 4.0308 2339 4.0399 2017 4.0495 0633 4.0595 9175 4.0701 8699 4.0813 0328 4.0929 5255 4.1051 4751 4.1179 0166 4.1312 2933

−enr0 37.2599 9952 35.6466 2956 34.1776 8291 32.8351 8670 31.6040 4879 30.4715 0397 29.4266 8334 28.4602 7591 27.5642 6034 26.7316 9022

α 031 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40

crr0 4.1451 4576 4.1596 6713 4.1748 1061 4.1905 9445 4.2070 3803 4.2241 6194 4.2419 8806 4.2605 3966 4.2798 4147 4.2999 1982

−enr0 25.95652 104 25.2334 6961 24.5578 9899 23.9257 2389 23.3333 3217 22.7775 1957 22.2554 3504 21.7645 3466 21.3025 4277 20.8674 1899

α 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50

crr0 4.3208 0273 4.3425 2004 4.3651 0358 4.3885 8729 4.4130 0739 4.4384 0259 4.4648 1428 4.4922 8676 4.5208 6749 4.5506 0740

−enr0 20.4573 3004 20.0706 2567 19.7058 1804 19.3615 6393 19.0366 4937 18.7299 7644 18.4405 5172 18.1674 7635 17.9099 3741 17.6672 0037

α 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60

crr0 4.5815 6114 4.6137 8750 4.6473 4973 4.6823 1604 4.7187 6005 4.7567 6135 4.7964 0610 4.8377 8777 4.881 0079 4.9261 7701

−enr0 17.4386 0260 17.2235 4778 17.0215 0107 16.8319 8496 16.6545 7578 16.4889 0083 16.3346 3599 16.1915 0389 16.0592 7253 15.9377 5451

α 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70

crr0 4.9734 1567 5.0228 5562 5.0746 4114 5.1289 3064 5.1858 9842 5.2457 3682 5.3086 5858 5.3748 9977 5.4447 2314 5.5184 2205

−enr0 15.8268 0655 15.7263 2967 15.6362 6985 15.5566 1924 15.4874 1799 15.4287 5677 15.3807 8007 15.3436 9043 15.3177 5365 15.3033 0530

α 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80

crr0 5.5963 2524 5.6788 0245 5.7662 7123 5.8592 0513 5.9581 4368 6.0637 0473 6.1765 9960 6.2976 5200 6.4278 2194 6.5682 3602

−enr0 15.3007 5875 15.3106 1497 15.3334 7455 15.3700 5257 15.4211 9676 15.4879 1026 15.5713 7967 15.6730 1056 15.7944 7215 15.9377 5451

α 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90

crr0 6.7202 2652 6.8853 8222 7.0656 1541 7.2632 5175 7.4811 5248 7.7228 8398 7.9929 5782 8.2971 7887 8.6431 6404 9.0411 4012

−enr0 16.1052 4232 16.2998 1112 16.5249 5451 16.7849 5501 17.0851 1676 17.4320 8879 17.8343 2343 18.3027 4170 18.8517 2601 19.5006 4879

α 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

crr0 9.5052 1757 10.0555 1672 10.7219 1416 11.5511 0443 12.6211 1409 14.0747 9146 16.2108 6764 19.8039 7673 27.9365 9438

−enr0 20.2763 1549 21.2170 4685 22.3798 7969 23.8541 5444 25.7894 7240 28.4602 7591 32.4416 4711 39.2278 6769 54.7780 5878

( ) ( )ρχ c

nk , denotes the n th ( ,...3,2,1=n ) positive purely imaginary root of (1), it is true when ( ) ( ) ( )0,2 2/ rc

nk ρχβ = which yields the eigenvalue spectrum of the configuration.

III. PHASE SHIFTER OPERATION For any triad of parameters { }0,, rαρ , satisfying the

criterion: ( ) ( ) ( ) αραρχ /,,12/ 220,0 ncLrc

n <−< , (2)

the geometry inspected affords differential phase shift, determined in normalized form by the expression: =Δβ +− − ββ (3) [10,12]. The symbol ( )ncL ,,2 ρ designates certain real positive numbers, connected with the ones ( ) ( )ρχ c

nk , , concuring to −∞→−k [17]. (For example, if 3=c , 1=n and

1.0=ρ , ( ) ( ) =ρχ cn,0 7.88188 32204 and ( ) =ncL ,,2 ρ 7.65012

21658 [17]. Moreover, obviously for any α it holds

( )( ) 2/,00 ρχ cnr > .) The sequence of inequalities (2) is called

condition for phase shifter operation of the coaxial ferrite waveguide with azimuthal magnetization for normal nTE0 modes [10,12]. It is equivalent to the relation [12]: −<< encr rrr 000 (4) in which ( ) ( ) ( )2

,00 12/ αρχ −= cncrr , (5)

( ) ( )2

20 1/,, −−− −= enenen ncLr ααρ . (6) Note that crr0 ( −enr0 ) attains its minimum =crr0min 3.94094 16102 at 0min, =crα ( ( )ncLr en ,,2min 20 ρ=− at

2/1min, =−enα ) and tends to infinity for 1max, →crα (both for 0→−enα and 1→−enα .) (The subscripts “+”, “–” label the quantities, conforming to counterclockwise, assumed as positive ( 0>+α ), resp. to clockwise, taken as negative ( 0<−α ) ferrite magnetization and the ones “ cr ” and “ −en ” – to those, corresponding to the conventional and to the specific for the anisotropic transmission line examined cut-

The 8th European Conference on Antennas and Propagation (EuCAP 2014)

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TABLE II. NUMERICAL VALUES OF THE QUANTITIES chk− , chk+ , ( )( )compcnk

chch ,−

ρχ , ( )( )compcnk

chch ,+

ρχ , comp−β AND comp

+β FOR NORMAL 01TE MODE

( 3=c , 1=n ) IN CASE chα = 0.1, =chr0 4 AND =chm 10 AS A FUNCTION OF THE NUMBER OF ITERATION N.

N chk− ( )( )compcnk

chch ,−

ρχ compr −0 comp−β N chk+ ( )( )compc

nkch

ch ,+ρχ compr +0 comp

+β

1 –0.009 7.84716 83615 4.00672 35208 0.17626 50078

1 0.005 7.90122 30981 3.99031 71854 0.09900 49504

–0.008 7.85101 93986 3.99546 62229 0.15719 85638 0.006 7.90509 56752 4.00095 95243 0.11854 79977

2 –0.0085 7.84909 36869 4.00090 74336 0.16675 54110

2 0.0059 7.90470 83486 3.99982 47139 0.11659 95577

–0.0084 7.84947 87983 3.99978 91327 0.16484 77450 0.0060 7.90509 56752 4.00095 95243 0.11854 79977

3 –0.00842 7.84940 17748 4.00001 15924 0.16522 94285

3 0.00591 7.90474 70805 3.99993 74903 0.11679 44633

–0.00841 7.84944 02865 3.99990 02875 0.16503 85961 0.00592 7.90478 58127 4.00005 04233 0.11698 93553

4 –0.00841 9 7.84940 56260 4.00000 04552 0.16521 03461

4 0.00591 5 7.90476 64466 3.99999 39372 0.11689 19110

–0.00841 8 7.84940 94771 3.99998 93194 0.16519 12635 0.00591 6 7.90477 03198 4.00000 52313 0.11691 14001

5 –0.00841 9 7.84940 56260 4.00000 04552 0.16521 03461

5 0.00591 55 7.90476 83832 3.99999 95841 0.11690 16556

–0.00841 89 7.84940 60111 3.99999 93415 0.16520 84379 0.00591 56 7.90476 87705 4.00000 07135 0.11690 36045

6 –0.00841 896 7.84940 57800 4.00000 00097 0.16520 95828

6 0.00591 553 7.90476 84994 3.99999 99229 0.11690 22403

–0.00841 895 7.84940 58185 3.99999 98984 0.16520 93920 0.00591 554 7.90476 85381 4.00000 00358 0.11690 24351

7 –0.00841 896 7.84940 57800 4.00000 00097 0.16520 95828

7 0.00591 5536 7.90476 85226 3.99999 99906 0.11690 23572

–0.00841 8959 7.84940 57839 3.99999 99986 0.16520 95637 0.00591 5537 7.90476 85265 4.00000 00019 0.11690 23767

8 –0.00841 89592 7.84940 57831 4.00000 00008 0.16520 95675

8 0.00591 55368 7.90476 85257 3.99999 99997 0.11690 23728

–0.00841 89591 7.84940 57835 3.99999 99997 0.16520 95656 0.00591 55369 7.90476 85261 4.00000 00008 0.11690 23747

9 –0.00841 89591 3 7.84940 57834 4.00000 00000 0.16520 95662

9 0.00591 55368 2 7.90476 85258 3.99999 99999 0.11690 23732

–0.00841 89591 2 7.84940 57834 3.99999 99999 0.16520 95660 0.00591 55368 3 7.90476 85258 4.00000 00000 0.11690 23734

TABLE III. QUANTITIES compk± , ( )( )compc

nkch

comp ,±ρχ , comp

±β FOR NORMAL 01TE MODE ( 3=c , 1=n ) AS A FUNCTION OF 0r AND α IN CASE 1.0=ρ .

0r compk± ( )( )compc

nkch

comp ,±ρχ comp

±β compk± ( )( )compc

nkch

comp ,±ρχ comp

±β compk± ( )( )compc

nkch

comp ,±ρχ comp

±β

1.0=α 2.0=α 3.0=α

4 –0.00841 89591 7.84940 57834 0.16520 95661 0.00591 55368 7.90476 85258 0.11690 23733

5 –0.04057 00614 7.72638 99892 0.62692 02320 –0.08218 48193 7.56958 19612 0.62210 47254 –0.12095 02670 7.42600 98600 0.59878 525000.03665 46647 8.02454 68702 0.58827 41498 0.06689 58481 8.14377 02497 0.54478 44172 0.08800 68903 8.22779 82378 0.48273 52914

6 –0.05987 22829 7.65331 53744 0.76370 24389 –0.12295 49755 7.41865 15700 0.76013 34353 –0.18576 99547 7.19144 01006 0.74219 638990.05422 74069 8.0936 60506 0.73149 70362 0.10083 57133 8.27917 68090 0.69569 72490 0.13784 58629 8.42871 55757 0.64547 97621

7 –0.07695 86712 7.58912 30337 0.83435 54630 –0.15952 63550 7.28557 36578 0.83017 21498 –0.24443 32927 6.98519 09485 0.81305 391610.06926 45477 8.15316 58642 0.80675 04937 0.12925 15390 8.39381 80773 0.77493 85034 0.17839 99785 8.59476 32581 0.73014 55144

8 –0.09315 73559 7.52869 73748 0.87669 19264 –0.19480 20609 7.15930 82582 0.87165 50020 –0.30207 71304 6.78820 74133 0.85440 092340.08309 15203 8.20817 55970 0.85253 72366 0.15500 23692 8.49868 71987 0.82332 29069 0.21457 80365 8.74476 33072 0.78184 75582

9 –0.10894 94481 7.47019 59031 0.90430 41339 –0.22985 00651 7.03592 10007 0.89844 82777 –0.36066 38841 6.59383 20184 0.88079 891360.09618 62950 8.26052 87859 0.88283 29543 0.17910 34034 8.59766 31694 0.85548 37416 0.24805 66908 8.88508 92653 0.81629 84596

10 –0.12456 64187 7.41274 15216 0.92337 86644 –0.26521 75223 6.91351 31835 0.91679 24185 –0.42134 23577 6.39874 86936 0.89868 795360.10877 67450 8.31109 86320 0.90405 42566 0.20203 93565 8.69258 01217 0.87812 16471 0.27962 81964 9.01871 41501 0.84062 89239

11 –0.14013 86706 7.35584 86185 0.93712 62242 –0.30124 91398 6.79099 66978 0.92990 08699 –0.48500 13944 6.20090 70625 0.91134 805200.12098 92023 8.36036 63577 0.91955 82329 0.22408 16516 8.78445 16227 0.89474 29218 0.30973 62596 9.14727 55324 0.85855 84573

12 –0.15574 80377 7.29922 03087 0.94736 60332 –0.33819 91890 6.66766 64571 0.93958 30784 –0.55247 65256 5.99880 70288 0.92061 112910.13290 09403 8.40862 39533 0.93126 16920 0.24539 89851 8.87389 73124 0.90735 22478 0.33865 81804 9.27177 69844 0.87221 19784

13 –0.17145 03278 7.24266 14193 0.95519 74420 –0.37628 03812 6.54301 76598 0.94692 66074 –0.62466 07497 5.79118 60181 0.92757 092310.14456 26970 8.45606 11775 0.94033 15459 0.26610 44256 8.96132 45216 0.91717 23516 0.36658 13709 9.39288 23126 0.88288 60705

14 –0.18728 63243 7.18603 60689 0.96131 87727 –0.41568 89230 6.41665 63727 0.95261 89203 –0.70259 02076 5.57685 26892 0.93291 478310.15600 96536 8.50280 79627 0.94751 43750 0.28627 82796 9.04701 63491 0.92498 72415 0.39364 02139 9.51105 69277 0.89141 29721

15 –0.20328 76746 7.12924 50110 0.96619 17601 –0.45661 94111 6.28825 10495 0.95711 24970 –0.78753 49327 5.35457 80353 0.93709 272280.16726 72989 8.54895 70551 0.95330 73034 0.30598 02829 9.13117 85348 0.93132 01971 0.41993 56335 9.62664 26331 0.89834 89495

The 8th European Conference on Antennas and Propagation (EuCAP 2014)

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TABLE IV. NORMALIZED DIFFERENTIAL PHASE SHIFT βΔ FOR NORMAL 01TE MODE ( 3=c , 1=n ) AS A FUNCTION OF 0r AND α IN CASE 1.0=ρ .

0rα 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

4.0 0.04830 71928 4.5 0.04293 98991 0.08590 99587 0.12893 96021 0.17205 67580 5.0 0.03864 60823 0.07732 03082 0.11604 99586 0.15486 05659 0.19377 49823 0.23281 23733 5.5 0.03513 29760 0.07029 25410 0.10550 42992 0.14079 18601 0.17617 57771 0.21167 24035 6.0 0.03220 54027 0.06443 61863 0.09671 66278 0.12906 87685 0.16151 12127 0.19405 79177 0.22671 66476 6.5 0.02972 82392 0.05948 09181 0.08928 12548 0.11914 99951 0.14910 41009 0.17915 53405 0.20930 87353 7.0 0.02760 49693 0.05523 36464 0.08290 84017 0.11064 88998 0.13847 06655 0.16638 35463 0.19439 00175 0.22248 32583 7.5 0.02576 48142 0.05155 27719 0.07738 55646 0.10328 19464 0.12925 61810 0.15531 63764 0.18146 26519 0.20768 50540 8.0 0.02415 46861 0.04833 20951 0.07255 33652 0.09683 64830 0.12119 45929 0.14563 42113 0.17015 32567 0.19473 88070 8.5 0.02273 40026 0.04549 04046 0.06828 99262 0.09114 98931 0.11408 24477 0.13709 26362 0.16017 63045 0.18331 76980 9.0 0.02147 11796 0.04296 45360 0.06450 04540 0.08609 56974 0.10776 15036 0.12950 15503 0.15130 97181 0.17316 75338 9.5 0.02034 12949 0.04070 46228 0.06111 01117 0.08157 40435 0.10210 68328 0.12271 08739 0.14337 81730 0.16408 76356 0.18480 18017

10.0 0.01932 44078 0.03867 07714 0.05805 90297 0.07750 50498 0.09701 84931 0.11660 05434 0.13624 14097 0.15591 74600 0.17558 7748010.5 0.01840 43758 0.03683 06875 0.05529 87420 0.07382 40500 0.09241 55788 0.11107 33638 0.12978 58844 0.14852 70431 0.16725 2482511.0 0.01756 79913 0.03515 79481 0.05278 95947 0.07047 81353 0.08823 18942 0.10604 98123 0.12391 86975 0.14181 00506 0.15967 6168611.5 0.01680 43438 0.03363 07257 0.05049 88303 0.06742 35985 0.08441 27558 0.10146 41956 0.11856 31056 0.13567 86409 0.15275 9810812.0 0.01610 43412 0.03223 08306 0.04839 91507 0.06462 40167 0.08091 25940 0.09726 17694 0.11365 51556 0.13005 96126 0.14642 0912812.5 0.01546 03461 0.03094 29835 0.04646 76257 0.06204 87949 0.07769 31303 0.09339 65482 0.10914 11269 0.12489 14763 0.14059 0177513.0 0.01486 58961 0.02975 42558 0.04468 48526 0.05967 20449 0.07472 19751 0.08982 96212 0.10497 55633 0.12012 22021 0.13520 8968913.5 0.01431 54862 0.02865 36343 0.04303 43033 0.05747 17136 0.07197 15371 0.08652 78419 0.10111 97427 0.11570 74683 0.13022 7337614.0 0.01380 43977 0.02763 16789 0.04150 18109 0.05542 88986 0.06941 81668 0.08346 27990 0.09754 04746 0.11160 92847 0.12560 2470014.5 0.01332 85630 0.02668 02517 0.04007 51624 0.05352 73045 0.06704 14763 0.08060 99999 0.09420 91461 0.10779 49008 0.12129 7457415.0 0.01288 44567 0.02579 22999 0.03874 37733 0.05175 28087 0.06482 37963 0.07794 82174 0.09110 09594 0.10423 59328 0.11728 03125

off at lower and upper frequencies, resp. [12].) In what follows the discussion is confined to 01TE mode ( 1=n ) solely.

IV. ITERATIVE METHOD FOR DIFFERENTIAL PHASE SHIFT COMPUTATION

To calculate βΔ produced first a set of parameters { }chchch r0,, αρ , subject to the condition (2), is chosen. Next, two arbitrary negative values of the parameter chk− are picked out and the relevant roots of (1) are reckoned for 1=n . Afterwards, the numbers, squaring with chα , chk− and

( ) ( )chcompcnkch ρχ

,− are substituted in the formulae:

( ) ( )( ) ( )( )[ ] ( ){ } 2/122,0 1/2/1/ αααρχ −+= kkr cnk , (2)

( ) ( )( )[ ]{ } 2/122 2/1/1 kααβ +−= , (3) obtained from the expressions for k , 2β and 0r in Section II. The numerical equivalents of −k are changed, until chr0 winds up between the pertinent to them counted values of the nor-malized radius compr −0 . Then, the found in this way interval for

−k is shrunk, dividing it in parts and repeating the above scheme for that of them, resulting in an interval for compr −0 , in-corporating chr0 . The calculations stop when it is fulfilled:

chcompch rr ε<− 00 ( chε – positive real number, less than unity, specifying the prescribed accuracy). Any of the figured up numerical equivalents ( )compN

−β , corresponding to the ends

of (or to a point from) the interval for compr −0 , could be accepted as a value of the normalized phase constant −β for 0<−α , searched for ( N – number of iteration). Then the procedure is done again for +k , yielding the quantity +β ( >−β +β [12]) for 0>+α , resp. βΔ ( 0>Δβ ), provided by the structure for the set { }chchch r0,, αρ , singled out. Next, the latter is turned round and everything starts anew. The superscripts “ ch ” and “ comp ” are attached to the symbols, standing for the parameters chosen, resp. the quantities computed which are used in the numerical analysis. In the final outcomes and where no ambiguity might arise, they are dropped.

V. NUMERICAL STUDY All data, presented here, relate to a waveguide with a thin

central conductor (with characteristic parameter 1.0=ρ , see Ref. [10]). Specifically, Table I lists the normalized radii crr0 and −enr0 (the left- and right-hand limits of the area of phase shifter operation of geometry), depending on the modulus of off-diagonal ferrite tensor element α . The iterative process (the basis of investigation), is illustrated in Table II for =chα

0.1 and =chr0 4, assuming the step of subsequent approxima-tions (the ratio of the length of successive intervals of variation of parameter −k , resp. +k ) =chm 10. The preliminary calcula-tions for locating the first of intervals, including chr0 , are omitted. Besides, in this Table the digits in a given result which

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are identical with these in the final one, are distinguished by bold face type. The values of parameters compk± , of the relevant to them roots of characteristic equation ( )( )compc

nkch

comp ,±ρχ and of the

phase constants comp±β for both directions of magnetization,

pertinent to the normal 01TE mode, determined, applying the iterative approach developed, are written in Table III, on condi-tion that =α 0.1, 0.2 and 0.3. The first (second) of each couple of rows for certain 0r comes up to compk− ( compk+ ). The upshots for the normalized differential phase shift are summar-ized in Table IV for =α 0.1 (0.1) 0.9. In the last two Tables the numerical equivalents of radius 0r , changing from 4 to 15 with a step 1, resp. 0.5 answer to the lower part of the region in which βΔ is afforded ( <crr0min <0r −enr0min or 3.94094 16102 << 0r 15.30024 43316) above whose upper limit it splits [12]. The blank boxes in both Tables concur to combi-nations of parameters for which the phase shift is not provided.

VI. CONCLUSION Using the method of successive approximations, a numeri-

cal study of the differential phase shift, due to the azimuthally magnetized ferrite-loaded coaxial waveguide, working in the normal 01TE mode, is performed. For the purpose the struc-ture’s characteristic equation, derived by complex Kummer and Tricomi confluent hypergeometric functions of specially select-ed parameters and variable, is harnessed. Results are given in normalized form for a thin central conductor in the lower part of the area of phase shifter operation of configuration as a func-tion of its parameters. Their analysis indicates that as in case of circular ferrite geometry [15] for a fixed guide radius the phase shift increases almost linearly with the modulus of off-diagonal ferrite tensor element, while if the latter is constant, it decreases when the radius grows. The coaxial guiding line of the same radius and tensor element produces smaller phase shift than the circular one. The technique is too hard and its application re-quires great efforts but the accuracy secured is extremely high.

ACKNOWLEDGMENTS

We express our gratitude to our mother Trifonka Romanova Popnikolova and to our late father Nikola Georgiev Popnikolov for their self-denial and for their tremendous efforts to support all our undertakings.

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[11] M.N. Georgieva-Grosse and G.N. Georgiev, “Advances in the theory of the circular waveguide with an azimuthally magnetized ferrite cylinder and a dielectric toroid”, in Proc. 33th Progr. In Electromagn. Res. Symp. PIERS 2013, Taipei, Taiwan, pp. 322-323 in Abstracts, pp. 1220-1224 in PIERS Proc., March 25-28, 2013, (in the Special Session “Advanced mathematical and computational methods in electromagnetic theory and their applications”, organized by M.N. Georgieva-Grosse and G.N. Georgiev).

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[13] M.N. Georgieva-Grosse and G.N. Georgiev, “Iterative method for analysis of the differential phase shift in an azimuthally magnetized circular ferrite-dielectric waveguide,” in Proc. 2013 Int. Symp. Electro-magn. Theory URSI-EMTS 2013, Hiroshima, Japan, 23PM3E-05, pp. 763-766, May 20-24, 2013.

[14] M.N. Georgieva-Grosse and G.N. Georgiev, “Advanced iterative methods for exact computation of the differential phase shift in the circular waveguide completely filled with azimuthally magnetized fer-rite: Review of recent results,” in Proc. Fifteenth Int. Conf. Electro-magn. Adv. Applicat. ICEAA’13, Torino, Italy, pp. 1290-1293, September 9-13, 2013, (Invited Paper in the Special Session „Advances in challenging problems of mathematical and computational electromagnetics: Focus on the applications,” organized by G. N. Georgiev and M. N. Georgieva-Grosse).

[15] G.N. Georgiev and M.N. Georgieva-Grosse, “Circular waveguide, completely filled with azimuthally magnetized ferrite,” Chapter in Wave Propagation: Academy Publish, Cheyenne, Wyoming, U.S.A., in press.

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[17] G.N. Georgiev and M.N. Georgieva-Grosse, “Hypothesis for the identity of the ( )ncL ,,2 ρ and ( )ncL ˆ,ˆ,ˆˆ

2 ρ numbers and its application in the theory of waveguides”, in Proc. 34th Progr. In Electromagn. Res. Symp. PIERS 2013, Stockholm, Sweden, pp. 805-806 in Abstracts, pp. 940-945 in PIERS Proc., August 12-15, 2013, (in the Special Session “Advanced mathematical and computational methods in electromagnetic theory and their applications”, organized by G. N. Georgiev and M. N. Georgieva-Grosse).

[18] F.G. Tricomi, Fonctions Hypergéométrique Confluentes. Paris, France: Gauthier-Villars, 1960.

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