Numerical Modeling of the Area of Phase Shifter Operation of the Azimuthally Magnetized

Circular Ferrite Waveguide

Mariana Nikolova Georgieva–Grosse Consulting in Physics and Computer Sciences,

Meterstrasse 4, D–70839 Gerlingen, Germany

Georgi Nikolov Georgiev Faculty of Mathematics and Informatics,

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

gngeorgiev@yahoo.com

Abstract—The borderlines of the area in which the circular wave-guide, entirely filled with azimuthally magnetized ferrite, works as a phase shifter for the normal 01TE mode, are determined. For the purpose the special features of the cut-off regime of the configuration and of its limiting one, observed at higher fre-quencies in which the propagation stops, provided the magnetic bias is negative, are used. For a fixed numerical equivalent of the normalized in an appropriate way guide radius, the off-diagonal ferrite permeability tensor element and the phase constant of the wave are counted by the roots of the structure’s characteristic equation, derived through the Kummer confluent hypergeomet-ric function. This is done repeatedly for a varying negative (posi-tive) value of the imaginary part of its complex first parameter. The process continues, until the computed value of the element coincides with that of the same, corresponding to the cut-off state for the radius chosen (to the limiting one, if the latter is greater than certain number) with a prescribed accuracy. The relevant calculated numerical equivalent of the phase constant is accepted as the one, searched for. The differential phase shift equals the constant mentioned in the first case, whereas in the second one, it is found from it by a simple arithmetic. Changing the normalized radius, the boundaries of the domain studied are traced. The out-comes are presented graphically in a normalized form.

Keywords-Boundary-value problems; circular ferrite-dielectric waveguides; eigenvalues and eigenfunctions; ferrite phase shifters; numerical techniques.

I. INTRODUCTION

The circular transmission lines, containing transversely magnetized ferrite medium are employed in constructing vari-ous devices for the modern microwave equipment [1-17]. In particular, the configurations of azimuthal magnetization of the anisotropic load, under normal 01TE mode excitation, may operate as nonreciprocal digital phase shifters [1,3,5-8,11-14, 16,17] and for this reason are attractive for the development of electronically scanned antenna arrays [18,19]. Different lines of attack have been suggested to examine their properties [1,3, 5-8,11-14,16,17]. Very successful proved to be recently the method, grounded on the boundary-value approach and har-nessing confluent hypergeometric functions [6-8,12-14,16,17].

The object of this investigation is the numerical modeling of the area in which the simplest of the class of geometries discussed – the ferrite one provides differential phase shift. The peculiarities of wave propagation are used and an iterative technique, based on the characteristic equation of the struc-ture, written by the Kummer function is applied. The main results of analysis are the computation and the portrayal of the limits of the area mentioned and the plotting of the phase shifting diagram of the structure.

II. SYNOPSIS OF THE BOUNDARY-VALUE PROBLEM

The circular waveguide of radius 0r , entirely filled with latching ferrite, magnetized in azimuthal direction to rema-nence by a central switching conductor of a negligible thick-ness is considered, propagating normal 01TE mode. It is as-sumed that it is infinitely long, lossless and perfectly conduct-ing and that the ferrite is described by a Polder permeability tensor [ ]ijμμμ 0=

�, i , =j 1, 2, 3, with nonzero components

1=iiμ and αμμ j−=−= 3113 , ωγα /rM= , 11 <<− α , ( γ – gyromagnetic ratio, rM – ferrite remanent magnetization, ω – angular frequency of the wave) and a scalar permittivity

rεεε 0= . The characteristic equation of the geometry [6-8]:

( ) 0;, 0 =Φ xca , (1)

is written in terms of the Kummer confluent hypergeometric function [20] in which ,5.1 jka −= ,3=c ,00 jzx =

( ) ,2/ 2ββα=k ,+∞<<−∞ k ( ) ,1 2/1222 βαβ −−= =0z

,2 02rβ .00 >z The phase constant ,β the radial wave- number ( )[ ] 2/1220022 1 βαεμεωβ −−= r and the guide radius are normalized by the relations ( ),/ 0 rεβββ =

( ),/ 022 rεβββ = rrr εβ 000 = with .000 μεωβ = If ( )cnk ,ζ

stands for the n th ( ,...3,2,1=n ) positive purely imaginary zero of ( )xca ;,Φ , eqn. (1) holds when ( ) ( )0,2 2/ rc

nkζβ = which gives the eigenvalue spectrum of the configuration.

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III. PHYSICAL BACKGROUND OF THE METHODS FOR NUMERICAL MODELING OF THE AREA OF

PHASE SHIFTER OPERATION

A. Phase Shifter Operation The structure may act as a phase shifter (may produce

differential phase shift +− −=Δ βββ when switching rM between its two stable states) for each pair of parameters { }0, rα , satisfying the condition [6,8]:

( ) ( ) ααζ /,12/ 120,0 ncLrcn <−< (2)

in which ( )ncL ,1 (written before also as ( )ncL , [6,7]) denotes certain positive real numbers, linked with ( )c

nk ,ζ , corresponding to −∞→−k [16]. (Provided 3=c , 1=n , ( ) =c

n,0ζ 7.66341 19404 and ( ) =ncL ,1 6.59365 41068 [6-8].) The area of phase shifter operation is finite. The first of its limits is connected with the cut-off state of the guiding line (with the left-hand part of the sequence of inequalities (2)) and the second one – with a peculiar −1En – envelope characteristic in its phase portrait (with the right-hand part of the criterion (2)) [6,8].

B. Magnetically Controlled Cut-Off At the cut-off point, if ,0>+α no transmission may take

place and in this case it is fulfilled: =+crβ 0 ( 0=+crk ). Provided 0<−α , however, in addition to the mode, being in a cut-off regime and for which =−crβ 0 ( 0=−crk ), a second one may get excited, propagating with phase constant 0≠−cβ ( 0≠−ck , −− < crc kk ) [8]. Latching the ferrite magnetization may bring the structure from a transmission to a cut-off state. Thus, differential phase shift may be afforded, determined by the expression: −=Δ ccr ββ [8].

C. Envelope Curve There exists an −1En – envelope line in the phase picture of

configuration [6] of equation ( )−−− = enenen r0ββ where [6,8,16]:

=−enr0 ( ) ( )[ ]2/121 1/, −− − enenncL αα , (3)

=−enβ ( ) 2/121 −− enα , (4)

( 3=c and −enα is a parameter), marking off the end of the domain of propagation of the normal 01TE mode ( 1=n ) in the waveguide examined in case of negative magnetization from the side of higher frequencies, at which the relevant phase curves terminate. For 0>+α there is no such boundary. Thus, in the β−0r – plane [6] the normalized radii −enr0 are co-ordinates also of the upper limit of the area in which phase shift is produced. Its value here is given by the expression [8] :

+−− −=Δ eenen βββ . (5)

(The subscripts “+” (“–”) are attached to the quantities,

relating to the positive (negative) magnetization; the ones “ cr ” (“ −c ”) – to those, corresponding to cut-off (to transmission state for 0<−α , bounded with the cut-off) and the subscripts “ −en ” (“ +e ”) – to the quantities, charac-terizing the envelope (the point from the ( )0r+β – curve for certain −= enαα of abscissa equal to that of the end of the

( )0r−β – one for the same α at the −1En – line).

IV. METHOD FOR DIFFERENTIAL PHASE SHIFT COMPUTATION

A value of normalized guide radius ( ) 2/,00cn

chr ζ> for which propagation is possible [6,8] is chosen. Next, a numerical equivalent of the modulus of off-diagonal ferrite tensor element chα is picked out, such that the relevant { }chch r0,α pair satisfies the condition for phase shifter operation (2). For an arbitrarily selected value chk of the parameter k , subject to the relation Lch kk ≤ , the positive purely imaginary root ( )c

nk ,ζ of eqn. (1) is found, using a special numerical root finding technique ( 1=n ). The numbers, corresponding to the quantities chr0 , chk and ( )compc

nk ch ,ζ are introduced in the formula:

( ) ( )( )[ ]{( ) ( )( )[ ] ( )( )

2/12

0,

220,

20,

22,1

/42/1

2/15.0

������

� −−

±−=

rkr

r

cnk

cnk

cnk

ζζ

ζα, (6)

giving the roots of biquadratic equation:

( ) ( )[ ]{ } ( )( ) 0/2/12

0,2

20,

4 =+−− rkr cnk

cnk ζαζα . (7)

Finally, comp2,1α are put in the expression:

( ) ( )( )[ ]{ } 2/122,1

22,12,1 2/1/1 kααβ +−= . (8)

The computed value of the normalized phase constant comp2,1β is accepted as the one looked for, if the determined in

the same way value of the off-diagonal element comp2,1α

coincides with that of chα within the boundaries of the set in advance accuracy ( ksgnsgn =α ). Otherwise, the parameter k is changed and the procedure is repeated. The calculations are performed twice for 0>+k and 0<−k , yielding the numerical equivalents of phase constants +β and −β , resp. the normalized differential phase shift +− −=Δ βββ . (The subscripts “ ch ” and “ comp ” mark the symbols, standing for the parameters chosen, resp. the quantities computed.)

Eqns. (7),(8) are derived from the expressions [6,8]:

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

,0 1/2/1/ αααζ −+= kkr cnk , (9)

( ) ( )( )[ ]{ } 2/122 2/1/1 kααβ +−= . (10)

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Figure 1. Normalized differential phase shift βΔ , produced by the

azimuthally magnetized circular ferrite waveguide for the normal 01TE mode vs. α with 0r as parameter, in case <0r 13.18730 82136.

Figure 2. Normalized differential phase shift βΔ , produced by the

azimuthally magnetized circular ferrite waveguide for the normal 01TE mode vs. α , in case =0r 13.18730 82136.

The condition Lkk ≤ , (11)

equivalent to

( ) ( )( ) ( )( )0,2

0, /22/1 rkr cnk

cnk ζζ ≥− (12)

(the discriminant Δ of eqn. (7) to be non-negative) specifies the set of real solutions of the same. Lk is the limiting value

Figure 3. Normalized differential phase shift βΔ , produced by the

azimuthally magnetized circular ferrite waveguide for the normal 01TE mode vs. α with 0r as parameter, in case >0r 13.18730 82136.

at which relation (12) turns to an equality. In case Lkk = , it holds 0=Δ and .21 Lααα == Provided ,Lkk < then

0>Δ (the left-hand side of the expression (12) is larger than its right-hand side one) and .21 αα > If Lch αα < ( Lch αα > ), the root 2α ( 1α ) is taken (resp. the quantity

2β ( 1β ) is computed). When Lkk = the common value of the latter Lα (i.e. Lβ ) is used.

Numerical Example: In case 50 =chr , 6.0)1.0(1.0=chα , the calculations yield: =Δ compβ 0.04109 02548, 0.08223 90544, 0.12350 19898, 0.16492 84822, 0.20655 80368, 0.24841 56969, resp. If ( ) == ncLr ch ,2 10 13.18730 82136 ( ,3=c 1=n ), 9.0)1.0(1.0=chα , then =Δ compβ 0.01558 70562, 0.03124 05095, 0.04701 87217, 0.06296 24010, 0.07908 21161, 0.09534 22859, 0.11164 19761, 0.12779 44118, 0.14351 03073. Provided ,150 =chr ,5.0)1.0(1.0=chα

=Δ compβ 0.01370 57130, 0.02748 32325, 0.04139 55342, 0.05548 57877, 0.06976 25808 and ,9.0=chα =Δ compβ 0.12654 72340 (for 8.0)1.0(6.0=chα , βΔ does not exist).

The dependence of βΔ on α with 0r as a parameter is shown with solid lines in Fig. 1-3. In Fig. 1 0r varies from 4 to 12, in Fig. 2 =0r 13.18730 82136 and in Fig. 3 =0r 15, 20, 30 and 50. The analysis indicates that the ( )αβΔ – curve for

== − min,00 enrr 13.18730 82136 (the minimum of −1En – line [6,8]) is a limiting one. The characteristics, corresponding for smaller (larger) values of 0r are represented by continuous lines, consisting of one (two pieces) in Fig. 1 (Fig. 3). In the second case for each 0r there is an interval of numerical equivalents of α in which phase shift is not provided (the area between the two pieces for given 0r in Fig. 3).

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Figure 4. Area of phase shifter operation of the azimuthally magnetized

circular ferrite waveguide for the normal 01TE mode.

V. METHOD FOR MODELING OF THE FIRST LIMIT OF THE AREA OF PHASE SHIFTER OPERATION

As above, a value of normalized radius ( ) 2/,00cn

chr ζ> is chosen. Then, the modulus of off-diagonal element ±crα , linked with the cutoff frequencies is counted from the formula:

( ) ( )( )[ ]{ } 2/120,0 2/1 rc

ncr ζα −=± , (13)

obtained from expression (6) in case 0=±crk . Next, an arbit-rary negative numerical equivalent chk− ( Lch kk ≤− ) of the par-ameter k is taken and the relevant root

( ) ,,cnkζ ( )1=n of eqn. (1)

is reckoned. Afterwards, the values of chr0 , chk− and ( )compcnk ch ,−

ζ are substituted in the expression (8). Then k is changed, until

comp1α coincides with ±crα with the beforehand determined

accuracy. The relevant to the last step numerical equivalent of the constant comp

1β is accepted as one of −cβ , resp. of crβΔ .

Numerical Example: For 50 =chr , 13.18730 82136 and 15 (the same like before) the calculations yield: =±crα 0.64243 37898, 0.95685 67352 and 0.96682 30433, and =Δ crβ 0.26624 84977, 0.15210 50315 and 0.13517 88659, resp.

The upshots of computations of the phase shift at the cut-off frequency points for the values of chr0 , given in Figs. 1-3 are presented in the same by circles. They depict the ends of

( )αβΔ – curves. Obviously the latter are finite. It might be expected that there exists a continuous boundary of the area of phase shifter operation, connected with the cut-off regime. It corresponds to the lower part of the frequency band in which

βΔ is afforded for each chr0 ( ±crα ) and restricts the area mentioned in the βα Δ− – plane from above. The points (circles) in question lie on it. Varying chr0 in the interval

( ) +∞<≤ chcn r0,0 2/ζ , the border referred to is traced (see the

upper dashed line in Figs. 4 and 5 labeled by 1LEvn−Δβ in the last one). The values of ±crα for the numerical equivalents of chr0 , pointed out in Figs. 1-3, are also written in Figs. 4 and 5.

VI. METHOD FOR MODELING OF THE SECOND LIMIT OF THE AREA OF PHASE SHIFTER OPERATION

The scheme repeats the previous one, save for the fact that instead for ,±crα the computations are performed for the quantities −en1α and −en2α , determined from the formula:

=−−2

2,1 enenα ( )[ ][ ]201 /,4115.0 rncL−± , (14)

easily recovered from the one (6) in case −∞→−k in which ( ) 0, →c

nkζ [6,7], or from the equation (3). Besides, the iterative procedure is performed for 0>+

chk , resulting in +1β and +2β for −en1α and −en2α , resp. The differential phase shift is counted from the relation (5) in which −enβ is specified by the expression (4).

Numerical Example: If =chr0 13.18730 82136 and 15 , it is obtained: === −−− min,21 enenen ααα 0.70710 67812 and

=Δ −en1β =Δ −en2β =Δ − min,enβ 0.11279 75530 (for the first value) and =−en1α 0.51159 65267, =−en2α 0.85922 58107,

=Δ −en1β 0.07142 88810 and =Δ −en2β 0.12105 55162 (for the second one), resp.

The existence of two solutions for −enα (resp. for −Δ enβ ) at each ( )ncLr ,2 10 > ( 3=c , 1=n ), visualized by rhombs in Figs. 3-5 for the values of 0r , disposed in Fig. 3, follows from the double-valuedness of the −1En – line [6]. Corresponding-ly, the ( )αβΔ – curves in the case considered consist of two parts (see Figs. 3, 5) which for ( )ncLr ,2 10 = have a common point (the square in Figs. 2, 4, 5). Changing chr0 in the interval

( ) +∞<≤ chrncL 01 ,2 , the second border of the domain examined (shown by the 1REvn−Δβ – dotted line in Figs. 4 and 5) is reckoned. It is bounded with the −1En – envelope (with the limiting regime at higher frequencies) and delimits confines the region studied in the βα Δ− – plane from below.

VII. AREA OF PHASE SHIFTER OPERATION. PHASE SHIFTING DIAGRAM

The results of numerical analysis are presented in Figs. 4 and 5, displaying respectively the domain in which the con-figuration investigated may produce differential phase shift for the normal 01TE mode and its phase shifting diagram (the distribution of the differential phase shift characteristics of the structure with 0r as parameter on the area mentioned). The

( )αβΔ – line for ( )ncLr ,2 10 = separates the latter in two parts: upper and lower, coloured by rose and pink, resp. It has a common point with the 1REvn−Δβ – border, portrayed by a square. It is seen that all characteristics slightly deviate from straight lines and are almost proportional to α . The

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Figure 5. Phase shifting diagram of the azimuthally magnetized circular

ferrite waveguide for the normal 01TE mode. ones, lying in the upper (lower) part of the zone under exam-ination are continuous functions of α (are interrupted by the

1REvn−Δβ – border). The 1LEvn−Δβ – limit possesses a maximum =Δ crβ 0.27094 15794 78340 at =α 0.711053 and

=0r 5.44943 34065 16560 which is the largest value of the differential phase shift which the structure might provide.

VIII. CONCLUSION The boundaries of the area of phase shifter operation of the

azimuthally magnetized circular ferrite waveguide for the normal 01TE mode are modeled through an iterative tech-nique, consisting in a repeated numerical solution of the char-acteristic equation of the structure, followed by a computation of the normalized guide radius 0r and phase constant of the wave. The approach is applied also to study the influence of radius 0r and the off-diagonal ferrite tensor element α on the phase shift ,βΔ produced. The analysis shows that for fixed

0r ( α ) βΔ grows when α increases ( 0r diminishes).

ACKNOWLEDGMENT 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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