13th International Conference on Mathematical Methods in Electromagnetic Theory September 6 – 8, 2010, Kyiv, Ukraine

NUMERICAL RESULTS FOR THE DIFFERENTIAL PHASE SHIFT IN THE CIRCULAR WAVEGUIDE WITH AZIMUTHALLY MAGNETIZED FERRITE

Mariana Nikolova Georgieva–Grosse 1, Georgi Nikolov Georgiev 2

1 2, Tcherny vrikh Str., BG–5138 Polikraishte, Bulgaria

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

e-mail: gngeorgiev@yahoo.com

Abstract – Employing an iterative approach, an exact computation of the differential phase shift, provided by the cir-cular waveguide, entirely filled with azimuthally magnetized ferrite, is performed in normalized form for the normal

01TE mode. The results obtained correspond to the upper part (consisting of two separate areas) relevant to small and large values of the off-diagonal ferrite permeability tensor element, of the frequency band in which the structure ex- hibits phase shifting properties. They complement previous ones for the lower part of the same. The joint consideration of the new and earlier outcomes shows that the phase shift increases almost linearly with the tensor element (is almost proportional to the reciprocal of normalized guide radius) in the whole band, if the radius (the tensor element) is fixed.

I. INTRODUCTION

Under certain conditions the azimuthally magnetized circular ferrite and ferrite-dielectric waveguides,

operating in the normal 01TE mode, might produce differential phase shift [1-3]. The computation of this quantity, of use in the development of digital phase shifters for electronically scanned antenna arrays, based on the geometries mentioned, is a complicated mathematical task [1-11]. For its exact solution sophisticated iterative techniques have been suggested [4-8]. Results for several configurations have been obtained and presented in a tabular and graphical form [4-7]. The complexity of the methods in question necessitated the elaboration of approximate ones in parallel to them. They allowed to reckon the phase shift with a negligible error directly from the structure parameters by simple formulae, written through suitably defined new real numbers [9,10].

It was believed that the phase and phase shifting behaviour of the simplest of transmission lines regarded (the circular ferrite one) is known to a large extent [1,5]. Quite recently, however, it has been established that this waveguide affords phase shift in a restricted frequency band and its boundaries have been traced [11]. It has been found out, as well that above a certain limit the domain referred to splits in two [11]. This has shown that the values of the phase shift, calculated earlier iteratively [5], answer to the lower part of the latter. Accordingly, the problem for counting the numerical equivalents of quantity examined in the upper part of the band (composed actually of two detached regions [11]), remains open. Its working-out namely is the object of the present study.

II. AZIMUTHALLY MAGNETIZED CIRCULAR FERRITE WAVEGUIDE

The propagation of normal nTE0 modes in the circular waveguide, uniformly filled with ferrite, magnetized in

azimuthal direction to remanence by an infinitely thin central switching wire, is subject to the equation [1,5,9-11]: ( ) 0;, 0 =Φ xca , (1) expressed by the Kummer confluent hypergeometric function [12] in which jkca −= 2/ , 3=c , 00 jzx = ,

=k ( )22/ ββα , 020 2 rz β= ( ωγα /rM= – off-diagonal ferrite permeability tensor element, 11 <<− α , γ –gyromagnetic ratio, rM – ferrite remanent magnetization, ω – angular frequency of the wave, =β

( )rεββ 0/ , =2β ( )rεββ 02 / , rrr εβ 000 = , β – phase constant, =2β ( )[ ] 2/12200

2 1 βαεμεω −−r – radial wavenumber, 0r – guide radius, =0β 00μεω , rε – ferrite relative permittivity) [1,5,9-11]. The eigenvalue spectrum of geometry is determined by the relation ( ) ( )0,2 2/ rc

nkζβ = , ( ( )cnk ,ζ – n th positive purely imaginary root

of eqn. (1), ,...3,2,1=n ). The structure brings forth differential phase shift +− −=Δ βββ , if it holds [1,10]: ( ) ( ) ααν /,1 2

0,2/1 ncLrnc <−<− , (2)

978-1-4244-8861-2/10/$26.00 ©2010 IEEE

13th International Conference on Mathematical Methods in Electromagnetic Theory September 6 – 8, 2010, Kyiv, Ukraine

with ( ) nc ,2/1−ν – n th zero of Bessel function ( ) ( )2/02/1 zJ c− and ( )ncL , – a certain positive real number ( 3=c , ,...3,2,1=n ) [1]. Especially, for 01TE mode ( 1=n ) ( ) =1,3L 6.59365 41068, =1,1ν 3.83170 59702 [1,10,11].

III. METHOD FOR DIFFERENTIAL PHASE SHIFT COMPUTATION

i) A rectangular network of { }0, rα pairs is formed whose co-ordinates along the α – axis are the same

( =α 0.1(0.1)0.9) and the ones along the 0r – axis ( =0r 13.0(0.5)66.0) are an extension of the co-ordinates ( =0r 4.0(0.5)13.0) of the network, constructed in the previous investigation [5] on which this research is based. The limits of the entire interval of variation of 0r in both studies are specified by the sequence of inequalities (2) and correspond to the smallest of the values of α enumerated ( =α 0.1).

ii) Anyone of the pairs { }chch r0,α that meets the condition (2) is chosen. Hence, for the numerical equivalents of α pointed out, the chr0 selected should equal or should lie between: 4.0 and 66.0; 4.0 and 33.5; 4.5 and 23.0; 4.5 and 17.5; 4.5 and 15.0; 5.0 and 13.5; 5.5 and 13.0; 6.5 and 13.5; 9.0 and 16.5, resp. For smaller (larger) 0r than the first (second) of the numbers given in each of the cases, differential phase shift is not produced.

iii) An arbitrary interval chchch kkk 1,2, +++ −=Δ ( 01,2, >> ++chch kk ) of positive values of the imaginary part k of the

complex first parameter of the Kummer function is picked out and the roots ( )cnkch ,1,+

ζ and ( )cnkch ,2,+

ζ of eqn. (1) are counted up at its ends. Afterwards the numbers, squaring with chk 1,+ , ( )c

nkch ,1,+ζ and chch αα =+ , resp. to chk 2,+ , ( )c

nkch ,2,+ζ

and chch αα =+ are put in the expression [1,5]:

( )( ) ( )( )[ ] ( ){ } 2/122

,0 1/2/1/ αααζ −+= kkr cnk . (3)

chk 1,+ and chk 2,+ are varied, until for the relevant results for the normalized radius, it is fulfilled: compr 2,0+ > chr0 > compr 1,0+ (until an interval is found, containing chr0 ). The quantities compr 1,0+ and compr 2,0+ , and chk 1,+ and chk 2,+ are

assumed as zeroth approximations to compr +0 and compk+ sought. Then, chk+Δ appointed is divided in ten parts and the procedure described is performed for each of them. This yields the first approximations to chr0 (the ends of that of the new intervals for the computed values of the radius, containing chr0 ). The pertinent subinterval for k is divided once more in ten and everything is repeated. The process is terminated when the deviation of the counted numerical equivalents of the radius from chr0 becomes less than the accuracy accepted. Then, any value

Table 1a. Numerical values of the normalized differential phase shift βΔ for normal 01TE mode as a function of the parameters α and 0r in the intervals: 3.0,2.0,1.0=α , 0.20)5.0(0.130 =r ; 4.0=α , 5.17)5.0(0.130 =r ; 5.0=α ,

0.15)5.0(0.130 =r ; 6.0=α , 5.13,0.130 =r ; 7.0=α , 0.130 =r ; 8.0=α , 5.13,0.130 =r ; 9.0=α , 5.16)5.0(0.130 =r .

α 0r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

13.0 0.0158 1139 0.0316 8864 0.0476 8969 0.0638 5502 0.0801 9547 0.0966 7647 0.1131 9923 0.1295 8058 0.1455 362813.5 0.0152 2644 0.0305 2017 0.0459 4031 0.0615 2792 0.0772 9319 0.0931 9885 0.1249 2491 0.1402 5481

14.0 0.0146 8330 0.0294 3536 0.0443 1653 0.0593 6852 0.0746 0094 0.1353 5489

14.5 0.0141 7764 0.0284 2554 0.0428 0535 0.0573 5949 0.0720 9700 0.1307 9203

15.0 0.0137 0571 0.0274 8323 0.0413 9553 0.0554 8579 0.0697 6258 0.1265 4723

15.5 0.0132 6426 0.0266 0189 0.0400 7726 0.0537 3434 0.1225 7555

16.0 0.0128 5041 0.0257 7580 0.0388 4195 0.0520 9369 0.1188 5599

16.5 0.0124 6167 0.0249 9995 0.0376 8207 0.0505 5378 0.1153 6568

17.0 0.0120 9581 0.0242 6989 0.0365 9096 0.0491 0573

17.5 0.0117 5088 0.0235 8171 0.0355 6274 0.0477 4167

18.0 0.0114 2513 0.0229 3191 0.0345 9216

18.5 0.0111 1701 0.0223 1739 0.0336 7455

19.0 0.0108 2512 0.0217 3536 0.0328 0574

19.5 0.0105 4822 0.0211 8331 0.0319 8197

20.0 0.0102 8518 0.0206 5901 0.0311 9988

13th International Conference on Mathematical Methods in Electromagnetic Theory September 6 – 8, 2010, Kyiv, Ukraine

Table 1b. Numerical values of the normalized differential phase shift βΔ for normal 01TE mode as a function of the parameters α and 0r in the intervals: 1.0=α , 0.40)5.0(0.200 =r ; 2.0=α , 5.33)5.0(0.200 =r ; 3.0=α ,

0.23)5.0(0.200 =r .

α 0r 0.1 0.2 0.3 α

0r 0.1 0.2 α0r 0.1 0.2 α

0r 0.1

20.0 0.0102 8518 0.0206 5901 0.0311 9988 25.0 0.0082 3423 0.0165 7568 30.0 0.0068 6808 0.0138 6313 35.0 0.0058 932620.5 0.0100 3499 0.0201 6042 0.0304 5640 25.5 0.0080 7345 0.0162 5604 30.5 0.0067 5616 0.0136 4133 35.5 0.0058 1093

21.0 0.0097 9673 0.0196 8571 0.0297 4879 26.0 0.0079 1886 0.0159 4880 31.0 0.0066 4787 0.0134 2678 36.0 0.0057 3090

21.5 0.0095 6957 0.0192 3321 0.0290 7455 26.5 0.0077 7012 0.0156 5327 31.5 0.0065 4302 0.0132 1914 36.5 0.0056 5308

22.0 0.0093 5274 0.0188 0140 0.0284 3140 27.0 0.0076 2690 0.0153 6879 32.0 0.0064 4146 0.0130 1808 37.0 0.0055 7737

22.5 0.0091 4557 0.0183 889 0.0278 1728 27.5 0.0074 8890 0.0150 9477 32.5 0.0063 4304 0.0128 2330 37.5 0.0055 0368

23.0 0.0089 4743 0.0179 9448 0.0272 3029 28.0 0.0073 5584 0.0148 3063 33.0 0.0062 4761 0.0126 3452 38.0 0.0054 3195

23.5 0.0087 5772 0.0176 1696 28.5 0.0072 2747 0.0145 7587 33.5 0.0061 5504 0.0124 5146 38.5 0.0053 6208

24.0 0.0085 7594 0.0172 5529 29.0 0.0071 0353 0.0143 2999 34.0 0.0060 6520 39.0 0.0052 9402

24.5 0.0084 0159 0.0169 0849 29.5 0.0069 8380 0.0140 9256 34.5 0.0059 7798 39.5 0.0052 2769

25.0 0.0082 3423 0.0165 7568 30.0 0.0068 6808 0.0138 6313 35.0 0.0058 9326 40.0 0.0051 6302

Table 1c. Numerical values of the normalized differential phase shift βΔ for normal 01TE mode as a function of the parameters α and 0r in the intervals: 1.0=α , 0.66)5.0(0.400 =r .

α 0r 0.1 α

0r 0.1 α 0r 0.1 α

0r 0.1 α0r 0.1 α

0r 0.1

40.0 0.0051 6302 45.0 0.0045 9587 50.0 0.0041 4287 55.0 0.0037 7292 60.0 0.0034 6526 65.0 0.0032 0553 40.5 0.0050 9997 45.5 0.0045 4605 50.5 0.0041 0255 55.5 0.0037 3963 60.5 0.0034 3732 65.5 0.0031 8177

41.0 0.0050 3846 46.0 0.0044 9733 51.0 0.0040 6302 56.0 0.0037 0694 61.0 0.0034 0985 66.0 0.0031 5838

41.5 0.0049 7844 46.5 0.0044 4966 51.5 0.0040 2427 56.5 0.0036 7483 61.5 0.0033 8284 66.5

42.0 0.0049 1985 47.0 0.0044 0301 52.0 0.0039 8627 57.0 0.0036 4330 62.0 0.0033 5626 67.0

42.5 0.0048 6266 47.5 0.0043 5735 52.5 0.0039 4900 57.5 0.0036 1232 62.5 0.0033 3012 67.5

43.0 0.0048 0680 48.0 0.0043 1265 53.0 0.0039 1244 58.0 0.0035 8188 63.0 0.0033 0440 68.0

43.5 0.0047 5224 48.5 0.0042 6888 53.5 0.0038 7657 58.5 0.0035 5197 63.5 0.0032 7909 68.5

44.0 0.0046 9892 49.0 0.0042 2601 54.0 0.0038 4137 59.0 0.0035 2257 64.0 0.0032 5418 69.0

44.5 0.0046 4681 49.5 0.0041 8402 54.5 0.0038 0683 59.5 0.0034 9367 64.5 0.0032 2966 69.5

45.0 0.0045 9587 50.0 0.0041 4287 55.0 0.0037 7292 60.0 0.0034 6526 65.0 0.0032 0553 70.0

of k , belonging to the interval which corresponds to the last iteration, is put in the formula [1,5]:

( ) ( )( )[ ]{ } 2/122 2/1/1 kααβ +−= (4) that enables to find the phase constant comp

+β looked for. The calculations are accomplished also for 0<−k to get the constant comp

−β , resp. the normalized differential phase shift compcompcomp+− −=Δ βββ .

iv) A new pair { }chch r0,α is picked out and the scheme from point iii) is done over again. (The subscripts “+” and “–” answer to positive ( 0>+α , 0>+k ) and negative ( 0<−α , 0<−k ) magnet-

ization, resp. The superscripts “ ch ” and “ comp ” label the symbols, standing for the parameters chosen, resp. the quantities computed which are used in the numerical analysis. In the final outcomes they are omitted.)

IV. NUMERICAL RESULTS

The impact of α and 0r on βΔ is presented in Tables 1a-c. The way of ordering the data is predetermined

by the effect of α on the size of interval for 0r in which the phase shift is afforded, through relation (2).

13th International Conference on Mathematical Methods in Electromagnetic Theory September 6 – 8, 2010, Kyiv, Ukraine

Table 1a illustrates the splitting of the upper part of the domain of existence of βΔ is provided in two areas: a larger, conforming to 7.0<α and a smaller one, pertinent to 7.0>α [11]. The results for the first of them for higher frequencies (for 0.200 >r ), are given in Tables 1b,c. As seen from Table 1c, the interval in which 0r alters, is largest for the smallest α ( =α 0.1). The blank spaces in the Tables are relevant to the combinations of parameters α and 0r for which βΔ is not produced (which do not satisfy the inequalities (2)).

The quantity βΔ preserves its properties, established earlier [5]: i) For a fixed α ( 0r ), it decreases (increases) monotonously with 0r ( α ); ii) In the first (second) case it is almost proportional to the function

0/1 r (is almost a linear function of α ) with a factor which increases (diminishes) when α ( 0r ) grows. It is easy to check also that αβ /Δ , 0rβΔ and ( ) 0/ rαβΔ are practically constants for a specific 0r , α and for any allowable { }0, rα pair, resp.

V. CONCLUSION

The differential phase shift, due to the circular waveguide with azimuthally magnetized ferrite which sustains normal 01TE mode, is calculated as a function of the off-diagonal permeability tensor element of the load and of the guide radius. For the purpose an iterative method is applied. The outcomes are obtained in normalized form and are presented in a tabular shape. They are connected with the upper part of the frequency band in that the quantity scrutinized observed and are an extension of the results of a previous study for the lower part of the same. Some properties of the phase shift are found, as well.

ACKNOWLEDGEMENT

We express our gratitude to our mother Trifonka Romanova Popnikolova for her self-denial and for the tremendous efforts she exerts to support all our undertakings.

REFERENCES

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