Advanced Iterative Methods for Exact Computation of the Differential Phase Shift in the Coaxial Waveguide

Entirely Filled with Azimuthally Magnetized Ferrite: Review of Recent Results

Mariana Nikolova Georgieva-Grosse1 Georgi Nikolov Georgiev 2

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, e-mail: gngeorgiev@yahoo.com.

Abstract − A survey of the modern techniques for exact counting of the differential phase shift, afforded by the azimuthally mag-netized coaxial ferrite waveguide which sustains normal nTE0 modes, is presented. The discussion ranges over two groups of methods, each of that includes three routines, linked with cer-tain physical working conditions of the configuration. The back-bone of all of them is the done again putting into practice of an iterative scheme, resulting in the positive purely imaginary roots of the structure’s characteristic equation, derived through complex Kummer and Tricomi confluent hypergeometric func-tions ( )xca ;,Φ and ( )xca ;,Ψ , resp. of expressly chosen par-ameters and variable, for a varying imaginary part of their complex parameter a . The first set of procedures enables to figure up the phase shifting diagram (the combination of the domain of phase shifter operation and of the differential phase shift curves) of the geometry for the normal 01TE mode in the

βΔ-0r – plane, while the second one – the same in the βα Δ- – plane. ( 0r and βΔ are the normalized in an apposite way guide radius and phase shift of the wave and α is the off-diagonal ferrite permeability tensor element.) The basic outcome of the study is that for every allowable 0r , βΔ depends almost lin-early on α in the whole region in which it might be provided.

1 INTRODUCTION

The design of digital phase shifters, built on the coaxial waveguide with azimuthally magnetized ferrite, operating in the normal 01TE mode, requires the knowledge of the differential phase shift, the structure might bring forth [1-8]. It could be found numerically solely, applying ingenious iterative schemes [1-8].

This is a comparative study of the avant-garde methods for precise calculation of the quantity told, using certain zeros of the characteristic equation of the geometry, put down by definite complex conflu-ent hypergeometric functions [1-8]. Six approaches, classified in two groups, are regarded. Changing spe-cific parameters, the phase shift is figured up in the interior and at the two borders of its existence area. The results of reckonings are pictured graphically.

2 COAXIAL WAVEGUIDE ENTIRELY FILLED WITH AZIMUTHALLY MAGNETIZED FERRITE

An infinitely long, uniform, lossless, perfectly conducting coaxial waveguide of radius 0r , totally

filled with latching ferrite, magnetized in azimuthal direction to remanence by a central switching wire of radius 1r that supports normal nTE0 modes with phase constant β , is considered. The ferrite has a Polder permeability tensor [ ],0 ijμμμ = ,i =j 1, 2, 3, with nonzero components 1=iiμ and == 3113 μμ

,αj− ωγα /rM= ( 11 <<− α ), γ – gyromagnetic ratio, rM – ferrite remanent magnetization, ω – angular frequency of the wave and a scalar permittivity .0 rεεε = ( 0ε and 0μ are the free space permittivity and permeability, resp.)

If ( )xca ;,Φ and ( )xca ;,Ψ are the Kummer and Tri-comi confluent hypergeometric functions [9], the char-acteristic equation of configuration is written as [1-8]: ( ) ( ) ( ) ( )0000 ;,/;,;,/;, xcaxcaxcaxca ρρ ΨΦ=ΨΦ . (1)

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 ). The normalized quantities

=β ( )rεββ 0/ , =2β ( )rεββ 02 / , rrr εβ 000 = , and =1r rr εβ 10 , ( =2β ( )[ ] 2/122

002 1 βαεμεω −−r –

radial wavenumber, 000 μεωβ = – free space phase constant, ρ – central conductor to guide radius ratio) permit to obtain generalized results, independent of the operating frequency. In case ( ) ( )ρχ c

nk , denotes the n th ( ,...3,2,1=n ) positive purely imaginary root of equation (1) in 0x (in 0z ), it holds, when [1-3,5-8]: ( ) ( ) ( )0,2 2/ rc

nk ρχβ = (2) which yields the eigenvalue spectrum of transmission line for the fields inspected. (Below the quantities, concurring to the positive ( 0>+α , 0>+k ) and negative ( 0<−α , 0<−k ) magnetization of the load, are marked off by the subscripts “+” and “–”, resp.)

3 PHYSICAL BACKGROUND

A close look at the phase chart of geometry [1] reveals that there are { }0,, rαρ – triads, satisfying the sequence of inequalities [3,5-8]:

978-1-4673-5710-4/13/$31.00 ©2013 IEEE

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( ) ( ) ( ) αραρχ /,,12/ 220,0 ncLrcn <−< (3)

for which it may propagate waves with different phase constants +β and −β (may produce differential phase shift +− −=Δ βββ , if rM is switched) [1-3,5-8]. The relation (3) is named condition for phase shifter opera-tion of the waveguide. In it ( )ncL ,,2 ρ [denoted also as ( )ncL ,, ρ ] are definite real positive numbers [1-3,5-8]. (For example, provided 3=c , 1=n and 1.0=ρ ,

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

21658 [1-3,5-8].) The instance in that the inequality in the left- (right-) hand side of the criterion in question is replaced by an identity, answers to the limiting cut-off regime at lower (higher) frequencies in which βΔ breaks off to be afforded. The graphical images of the first (second) one are the bifurcation points at the horizontal axis (the envelope) at that the phase curves for positive and negative (for negative) magnetization terminate (cf. Figs. 1,3 in Refs. [1,3,5]).

4 GENERAL CHARACTERISTIC OF THE METHODS FOR EXACT COMPUTATION OF THE DIFFERENTIAL PHASE SHIFT

The existence of a variety of physical states in that the structure may work as a phase shifter and the availabi-lity of a lot of parameters in its characteristic equation preconditions the possibility to invent different schemes for calculation of βΔ (of +β and −β ). The idea in all of them is to fluctuate the parameter ±k and to figure up the relevant to it root of equation men- tioned in 0z by an earlier elaborated technique. They, however, possess their own specific features [2-8].

By and large, the approaches considered could be categorized in two sets. Each of them embraces three methods: general, yielding the differential phase shift for any { }0,, rαρ – triad which meets the criterion (3) and two modifications of it, permitting to obtain the quantity in question provided the left-, resp. right-hand side of the same converts into an equality [2-8].

At both general techniques the outset is the selection of triad. Then, they branch forth. In the first the { }αρ, – couple is assumed fixed and the iterative procedure is done, about 0r , while in the second, the { }0, rρ – one is unchanged and α is varied. The concept is { }αρ, ({ }0, rρ , ( ) ( ) 2/,00 ρχ c

nr > ) accept-ed, ±k chosen and the figured up roots ( ) ( )ρχ c

nk ,±, tied

up to it, to be used for counting the corresponding to them numerical equivalents of 0r ( α ) and ±β by formulae, linking these quantities. The computations go on, until the reckoned 0r ( α ) becomes identical with the firstly singled out one within the framework of the foreseen accuracy. The concurring to it calculated value of ±β is taken as that, searched. This is done two

times for 0>+k and 0<−k , resulting in the numerical equivalents of constants +β and −β , resp. of βΔ . Then another { }0,, rαρ – triad is selected and every-thing starts again. The practice indicates that usually ρ

remains fixed and only the { }0, rα – pair alters [2-8]. In the modifications of the first (second) main

scheme not three, but two primary parameters { }αρ, ({ }0, rρ ) only are chosen. The third one in the triad is summed up from a certain formula, giving the relation between α and 0r at the lower (upper) cut-off point.

Further, two ways for fluctuating ±k are proposed. In the first it is varied arbitrarily, until the sized up numerical equivalent of 0r ( α ) gets indistinguishable from its initially determined one. In the second, an arbitrary interval for ±k is selected at the beginning and it is moved, unless an interval for the cast up values of 0r ( α ) is found, incorporating the set one. Then, the interval for ±k is narrowed, til the length of the totted up one for 0r ( α ) becomes less than the prescribed error [2-8].

5 MATHEMATICAL FUNDAMENTALS OF THE FIRST GROUP OF METHODS

5.1 ABC of the First general method

The numbers, relevant to the constants +β and −β , (to their difference βΔ ) for any { }0,, rαρ – triad, subject to the requirement (3) are (is) obtained from the expressions [2,3,5,6,8]:

( ) ( )( ) ( )( )[ ] ( ){ } 2/1

22,0 1/2/1/ αααρχ −+= kkr cnk , (4)

( ) ( )( )[ ]{ } 2/122 2/1/1 kααβ +−= . (5)

5.2 Principles of the First modified method at cut-off at lower frequencies

For chosen ρ and α the value of 0r at cut-off ( crr0 ) is counted up from the formula [2,3,5,6,8]:

( ) ( ) [ ]2

,00 12/ crcncrr αρχ −= . (6)

The calculations are performed like in the preceding

instant exclusively for negative 0<−k , yielding [3,5]: −=Δ ccr ββ (7)

at the point regarded. (The subscripts “ cr ” (“ −c ”) mark off the quantities, concurring to the cut-off (to the propagation state for 0<−α , connected with it) [3,5].)

5.3 Rudiments of the First modified method at cut-off at higher frequencies

Once more the { }αρ, – couple is selected only. The pertinent numerical equivalent of 0r at the higher cut-

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Figure 1. Phase shifting diagram of the azimuthally magnetized coaxial ferrite waveguide for the normal

01TE mode in the βΔ-0r – plane. off (at the 1En – envelope) ( −enr0 ) is found from [2,3,5,6,8]:

=−enr0 ( ) ( )[ ]2/12

2 1/,, −− − enenncL ααρ . (8)

The computations are fulfilled just for 0>+k , giving the constant +eβ . The phase shift is figured up through the correlations [2,3,5,6,8]:

=Δ −enβ +− − een ββ , (9)

=−enβ ( ) 2/121 −− enα . (10)

(The subscript “ −en ” (“ +e ”) labels the quantities, describing the envelope (the point from the ( )0r+β –line for specific α of abscissa, coinciding with that of the termination point of the ( )0r−β – one for identical α at the 1En – characteristic [1,3,5].)

6 MATHEMATICAL FUNDAMENTALS OF THE SECOND GROUP OF METHODS

6.1 ABC of the Second general method

The iterative procedure employs the formula [7]:

( ) ( )[ ]⎭⎬⎫

⎩⎨⎧ ×−−±−=

2/122

222

22

22

2,1 44115.0 kβββα (11)

in which 2β is expressed by eqn. (2) with ( ) ( ) 2/,00 ρχ c

nr > . The sign + (–) answers to 1α ( 2α ), ( 21 αα > ). The requirement

Lkk ≤ (12)

determines the set of real values of the latter. Lk are definite limiting real positive numbers which are

Figure 2. Phase shifting diagram of the azimuthally magnetized coaxial ferrite waveguide for the normal

01TE mode in the βα Δ- – plane. dependent on 0r and ksgn [7]. The phase constant is ferret out from the relationships [7]:

( ) 21 αβ = , (13)

( ) 12 αβ = , (14)

( ( ) ( )21 ββ < ). (The subscripts “(1)” and “(2)” specify the number of the solution for β [7].)

6.2 Principles of the Second modified method at cut-off at lower frequencies

For a singled out numerical equivalent of the normalized radius ( ) ( ) 2/,00 ρχ c

nr > the modulus of off-diagonal element ±crα , bound up with the cut-off frequencies, is reckoned up from the equality [7]: ( ) 2/12

21 βα −=±cr . (15)

The reiterative process, regarding α is accomplished for a negative value of the parameter k solely. The numerical equivalent of −cβ , determined from (13) or (14) is accepted as the one of crβΔ .

6.3 Rudiments of the Second modified method at cut-off at higher frequencies

The process repeats the foregoing one, excepting the fact that instead of for ±crα the calculations are effectuated for the quantities −en1α and −en2α , found out from the term [7]:

=−−2

2,1 enenα ( )[ ]{ }202 /,,4115.0 rncL ρ−± . (16)

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The scheme is done for ,0>+k resulting in ( ) +e1β and ( ) +e2β for −en1α and −en2α , resp. The differential

phase shift is found from the relations (9), (10).

7 PHASE SHIFTING DIAGRAMS

Figs. 1 and 2 present the phase shifting diagrams (the superposition of the area of phase shifter operation and of the differential phase shift characteristics) of the configuration in the βΔ-0r – and in the βα Δ- – plane, resp., for the normal 01TE mode. The blue

( )0rβΔ – and the red ( )αβΔ – solid lines are plot-ted, using the first, resp. the second general methods. The blue dashed 1LEn – and the red dotted 1REn – , and the crimson dashed 1LEnv – and the brown dotted

1REnv – borders are outlined, harnessing their first and second modifications. ( α is varied from 0 to 1 and 0r – from ( ) ( ) 2/,0 ρχ c

n to infinity.) The characteristics, lying in the upper (lilac) section of the area in Fig. 2 are formed of one and those in the lower (pink) region – of two parts. Their dividing line concurs to ( )ncLr en ,,2 20 ρ=− . The maximum of the

1LEn – and 1LEnv – limits (marked by (1) in both Figures) depicts the largest value of the phase shift

=Δ crβ 0.24655 08703 95035 that the structure might provide, reached at =α 0.709183 and =0r 5.58979 42553 16039. The symbol (2) in Fig. 1 shows the leftmost point of the 1REn – boundary (the minimum of −enr0 ) [2,3,5,7].

8 CONCLUSION

The physical bases and the mathematical fundamentals of two up-to-the-minute groups of iterative methods for precise reckoning of the differential phase shift, produced by the coaxial waveguide, wholly filled with azimuthally magnetized ferrite, are brought up. They use certain zeros of the complex characteristic equation of transmission line and give the quantity of interest in its entire existence area. The outcomes are presented graphically. The approaches described could be ex-tended to investigate more involved layered geometries which could find application in the development of ferrite phase shifters for the microwave frequencies.

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.

References

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