INVESTIGATIONS OF CYLINDRICAL FERRITE COU- PLED LINE ... · on structures realized in planar line technology [19,21]. Such structures allow one to obtain fully integrated FCL devices

Progress In Electromagnetics Research, Vol. 120, 143–164, 2011

INVESTIGATIONS OF CYLINDRICAL FERRITE COU-PLED LINE JUNCTION USING HYBRID TECHNIQUE

A. Kusiek*, W. Marynowski, and J. Mazur

Faculty of Electronics, Telecommunications and Informatics, GdanskUniversity of Technology, Gdansk, Poland

Abstract—In this paper, a novel longitudinally magnetized cylindricalferrite coupled line (CFCL) junction is proposed. In comparisonto planar ferrite coupled line (FCL) configurations, which are wellknown in literature, in such structure the higher gyromagnetic couplingoccurs. This allows to obtain the required in FCL devices Faradayrotation angle π/4 for the ferrite with shorter length and lower valueof magnetization. As a result the total insertion losses in the ferritesection can be reduced using the proposed topology. In the analysisof the proposed CFCL junction a hybrid technique combining methodof moments and coupled mode method (MoM/CMM) is applied. Theresults are compared with the ones obtained from commercial softwareHFSS and a good agreement is obtained.

1. INTRODUCTION

Nonreciprocal devices have been extensively used in modern microwaveand millimeter systems [1–16]. In order to obtain the nonreciprocaleffects one needs to utilize the ferrite materials [1–15] or active elementssuch as amplifiers [16]. Recently, the longitudinally magnetized ferritecoupled strip-[2, 4–10, 15, 17, 18] or slotlines [14, 19] are being developedand employed to realize integrated nonreciprocal devices. Significantinterest in these devices results from their advantages which are weakbiasing magnetic field and wide operation bandwidth [6, 10].

The basic part of ferrite coupled line (FCL) devices islongitudinally magnetized FCL section obtained as a section of twocoupled lines placed on ferrite substrate [1–3]. According to the

Received 27 July 2011, Accepted 22 August 2011, Scheduled 3 September 2011* Corresponding author: Adam Kusiek ([email protected]).

144 Kusiek, Marynowski, and Mazur

coupled-mode model [1] in such section the gyromagnetic couplingoccurs resulting in Faraday rotation effect. The wide operationbandwidth and high isolation is obtained, when the Faraday rotationphenomenon is optimal. This optimal effect is achieved when the ferritematerial is placed in the region where the wave is linearly polarizedand occurs in cylindrical waveguide with coaxially located ferrite rodor suspended stripline junction. In order to construct devices such ascirculators [7, 19], gyrators [20] or isolators [17] the FCL section has tobe cascaded with reciprocal section providing input signals to the FCLwhich are either in phase or out of phase.

So far, studies concerning FCL devices have been focused mainlyon structures realized in planar line technology [19, 21]. Such structuresallow one to obtain fully integrated FCL devices. However, due tothe significant length of ferrite section, the main drawbacks are highinsertion losses occurring in ferrite material and large dimensions ofthe structure. Also due to the geometry of line in planar junction it isdifficult to obtain a linear polarization of the wave and to eliminate theisotropic coupling which results in significant deterioration of isolation.

There were several attempts to improve performance and toreduce total dimensions of planar FCL devices. Promising resultsconcerning low insertion losses and high isolation were obtainedfor the nonreciprocal devices employing ferrite coupled slotline [19]and stripline junctions [2, 7]. For the fabricated devices obtainedinsertion losses were not lower than 3dB and isolation was better than12 dB [7, 15, 19]. Moreover in order to reduce the dimensions of theplanar FCL devices in [8] the circulator with appropriate matchingnetworks at the ports ensuring multiple reflections was proposed. Forpresented device the FCL junction length reduction by a factor oftwo was obtained. The drawback of this structure was high value ofinsertion losses caused by multiple transmission of signal through thelossy ferrite junction. Also similar length reduction of FCL junctionwas achieved with the use of periodic left-handed/ferrite coupled line(LH-FCL) structures [22, 23]. However, for the simulated circulatorutilizing LH-FCL section the insertion losses were not better than 4 dB.

Due to the high insertion losses and low level of isolationthere is still need for developing novel FCL junctions offering betterperformance than currently proposed planar configurations. In thispaper, we propose for the first time a cylindrical ferrite coupledline junction (CFCL). Due to the similar geometry to the circularwaveguide with coaxially located ferrite rod such structure allowsto obtain close to optimal Faraday rotation effect. Moreover, insuch configuration stronger gyromagnetic coupling occurs which isa result of high magnetic field concentration in the ferrite medium.

Progress In Electromagnetics Research, Vol. 120, 2011 145

These make possible to design shorter ferrite junctions ensuring lowerinsertion losses and better performance, e.g., better isolation and wideroperation bandwidth in comparison to planar ones.

In the analysis of the proposed structure the hybrid techniquecombining method-of-moment (MoM) with coupled-mode method(CMM) [24, 25] is applied. The analysis using MoM/CMM involvesintroducing an isotropic basic guide. This guide is complementary tothe ferrite one, however instead of ferrite the dielectric characterizedby scalar value of µr = 1 is used. In this approach the basic guidemodes obtained from MoM are used to determine the ferrite modesof the investigated structure. As a result of analysis the dispersioncharacteristics of ferrite line, the gyromagnetic coupling coefficientsand the scattering parameters of ferrite junction are obtained. Thepresented results are compared with the commercial software HFSS.A good agreement between presented results and the reference ones isobtained.

2. FORMULATION OF THE PROBLEM

The proposed four-port CFCL junction is presented in Fig. 1. It con-sists of a cascade connection of two dielectric and ferrite loaded cou-pled transmission lines, where the ferrite is magnetized longitudinally.In order to determine wave properties (e.g., dispersion characteristics,gyromagnetic coupling factor) and scattering parameters of the con-sidered structure we utilize the hybrid MoM/CMM technique. In thisapproach at first the propagation coefficients and field distributionsof modes in a basic cylindrical dielectric coupled line (CDCL) are de-termined with the use of MoM (see Section 2.1). This guide has thesame cross section as CFCL, where instead of ferrite the dielectric rodwith the same permittivity and relative permeability µr = 1 is applied.In our analysis we take into considerations only two propagated fun-damental basic modes whereas higher ones such as evanescent modesare neglected. Next, utilizing CMM the dispersion characteristics oftwo basic modes in ferrite guide, the gyromagnetic coupling coeffi-cient and scattering parameters of CFCL junction are calculated (seeSection 2.2). It should be noted that proposed model allows one forfast and accurate determination of an optimal length of CFCL sec-tion. Moreover, having calculated scattering matrix of such junction,it can be easily used to predict the behavior of nonreciprocal devices,e.g., isolators and circulators employing proposed ferrite coupled linejunction [19, 25].


ferrite rod

4

dielectricsection

dielectric

section

ferritesection

12

3

(a)

I

II

III

IV

r1

r2

r3

ϕ s1

ϕ s2

∆ϕs1

∆ϕs2

(b)

Figure 1. Cylindrical ferrite coupled line junction (CFCL): (a) 3D-view and (b) cross-section.

2.1. Cylindrical Dielectric Coupled Line

The cross-section of considered line is presented in Fig. 1(b). Thelongitudinal components of electric and magnetic fields in each regiontake the following form:

Elz =

M∑

m=−M

(AE, l

1m R(1)m (kρlρ) + AE, l

2m R(2)m (kρlρ)

)ejmϕe−jβz, (1)

H lz =

j

ηl

M∑

m=−M

(AH, l

1m R(1)m (kρlρ) + AH, l

2m R(2)m (kρlρ)

)ejmϕe−jβz. (2)

where ω = 2πfc , kρl =

√k2

l − β2, kl = ω√

µlεl

c , ηl =√

µlεl

, µl and εl

denote permeability and permittivity in lth region, respectively. Inabove equations R

(1)m (·) and R

(2)m (·) denote two linearly independent

radial cylindrical functions and depending on the region they aredefined as follows:

region 1 R(1)m (·) = Jm(·), R

(2)m (·) — neglected

region 2, 3 R(1)m (·) = Jm(·), R

(2)m (·) = Ym(·)

region 4 R(1)m (·) = Jm(·), R

(2)m (·) = Ym(·) — shielded guide

R(1)m (·) — neglected, R

(2)m (·) = H

(2)m (·) — open guide

where Jm, Ym and H(2)m are mth order Bessel, Neumann and Hankel

function of second kind, respectively. Utilizing (1) and (2) thetransverse electric and magnetic fields components can be found


directly from Maxwell’s equations [26]. Now, in order to determinethe propagation coefficient β and field distributions in the consideredguide we utilize the continuity conditions of z and ϕ field componentsin the boundaries between neighboring regions. At first we use theboundary continuity conditions between regions 1-2 (ρ = r1) and 3-4(ρ = r3):

Enz,ϕ(rn, ϕ) = En+1

z,ϕ (rn, ϕ), (3)

Hnz,ϕ(rn, ϕ) = Hn+1

z,ϕ (rn, ϕ), (4)

where n = 1, 3 and ϕ ∈ [0, 2π]. Applying the orthogonality propertiesof ejmϕ to Equations (3) and (4) we obtain relation between coefficientsA

(·),21m , A

(·),22m and A

(·),31m , A

(·),32m , respectively in a form:

A22 = T2A2

1 and A31 = T3A3

2,

where Alp = [AE, l

p(−M), . . . , AE, lpM , AH, l

p(−M), . . . , AH, lpM ]T and

T2 =(Z1M

H,2A2 −ME,2

A2

)−1 (ME,2

A1 − Z1MH,2A1

), (5)

T3 =(Z4M

H,3A1 −ME,3

A1

)−1 (ME,3

A2 − Z1MH,3A2

), (6)

Zl = ME,lA1

(MH,l

A1

)−1for l = 1, 4. (7)

For the sake of brevity matrices ME,·A1 , MH,·

A1 , ME,·A2 and MH,·

A2 are definedin Appendix A.

Next, we consider the boundary between regions 2 and 3. In orderto include in the analysis the metallization layer at first we define zand ϕ components of electric field at ρ = r2. These electric field isnon-zero in each ith slot area of the line and is expressed as follows:

Esz =

I∑

i=1

K∑

k=1

Ask sin

(ϕ− ϕsi

∆ϕsikπ

), (8)

Esϕ =

I∑

i=1

K∑

k=0

Ask cos

(ϕ− ϕsi

∆ϕsikπ

), (9)

where ϕi ∈ [ϕsi, ϕsi + ∆ϕsi] and I denotes a number of slots in theconsidered guide. By applying the boundary continuity conditionsbetween electric fields in regions 2, 3 and in the slots area:

Elz(rl, ϕ) = Es

z(rl, ϕ), (10)

Elϕ(rl, ϕ) = Es

ϕ(rl, ϕ) for l = 2, 3 (11)


and utilizing orthogonality of ejmϕ we obtain the following set ofequations:

A21 =

[(ME,2

A1 + ME,2A2 T12

)]−1IosAs, (12)

A32 =

[(ME,3

A1 T34 + ME,3A2

)]−1IosAs, (13)

where Ios is a matrix of integrals defined in Appendix A. Finally, weenforce the continuity of z and ϕ components of magnetic field fromregions 2 and 3 in the slot areas of line:

H2z (r2, ϕi) = H3

z (r2, ϕi), (14)H2

ϕ(r2, ϕi) = H3ϕ(r2, ϕi), (15)

where ϕi ∈ [ϕsi, ϕsi+∆ϕsi]. Using basis functions (8) and (9) as testingfunctions to solve (15) and (14), respectively and applying relations(12) and (13) to resulting set of equations we obtain:

DAs = 0, (16)where

D = Iso

[MH,3

(ME,3

)−1 −MH,2(ME,2

)−1]Ios (17)

and ME,(·), MH,(·), Iso are defined in Appendix A. SolvingEquation (16) the propagation coefficients β and distributions ofelectric E and magnetic H modal fields in the considered cylindricaldielectric guide are obtained.

2.2. Cylindrical Ferrite Coupled Line

Applying solution of (16) to coupled-mode method (CMM) a gyro-magnetic coupling coefficients, propagation coefficients of ferrite modesand scattering matrix of CFCL junction can be determined [3, 11, 27].In this approach the transversal electric and magnetic fields in theinvestigated ferrite guide are expressed in terms of basic guide fieldeigenfunctions et,e(o) and ht,e(o) as follows:

Eft = Ue(z)et,e + Uo(z)et,o, (18)

Hft = Ie(z)ht,e + Io(z)ht,o. (19)

In above equations subscripts e and o denote even and odd mode,Ue(o)(z) = Ue(o)e−jkz and Ie(o)(z) = Ie(o)e−jkz are modal voltage andcurrent, k is the unknown propagation coefficient in ferrite guide andeigenfunctions et,e(o) and ht,e(o) =−iz×et,e(o) are normalized as follows:∫∫

S

et,e(o) × ht,e(o)~ds = 1. (20)


The Maxwell’s equations defined for CDCL and CFCL are combinedtogether and after some mathematical manipulations we obtainsimilar to [3, 27] set of coupled mode equations defining equivalenttransmission line problem of CFCL:

∂

∂zUe(z) + jβeZeIe(z) = CeoIo(z),

∂

∂zUo(z) + jβoZoIo(z) = CoeIe(z),

∂

∂zIe(z) + jβeYeUe(z) = 0,

∂

∂zIo(z) + jβoYoUo(z) = 0,

(21)

where:

Ceo = k0η0µa

∫

Ωf

(ht,e × h∗t,o

) · iz dΩf ,

Coe = −C∗eo

(22)

define the coupling between two fundamental modes in basic guide,Ωf is a ferrite area in the cross section. Taking into account thatin such type of structures the field distributions of the basic modesEt,e(o) and Ht,e(o) in cross-section of the dielectric guide are mucheasier determined than eigenfunctions ht,e(o) and et,e(o), we introducethe following relation between fields and eigenfunctions in the dielectricCDCL guide:

Et,e(o) = U e(o)(z)et,e(o), (23)

Ht,e(o) = Ie(o)(z)ht,e(o), (24)

where U e(o)(z), Ie(o)(z) are modal voltage and current in the dielectricbasic guide. Then, assuming that the power of even and odd mode isdefined as:

Pe(o) =∫∫

S

Et,e(o) ×H∗t,e(o)ds (25)

and the impedance of even and odd mode is:

Ze(o) =U e(o)

Ie(o)

=k0η0

βe(o), (26)

the coupling coefficient (22) can be formulated as:

Ceo = k0η0µa

√ZeZo√PePo

∫

Ωf

(Ht,e ×H∗

t,o

) · iz dΩf , (27)


where η0 =√

µ0/ε0 and βe(o) are propagation coefficients of twofundamental modes in dielectric guide. The components Ht,e(o) =Hρ,e(o)iρ + Hϕ,e(o)iϕ and Et,e(o) = Eρ,e(o)iρ + Eϕ,e(o)iϕ are transverseeven and odd electric and magnetic modal fields obtained from (16).Equation (27) indicates that the gyromagnetic coupling occurs whenthe ferrite is located in the area of line where the magnetic field vectorsof two basic isotropic modes are orthogonal.

The defined above transmission line model of the ferrite guide canbe used to determine scattering matrix of CFCL junction. At firstutilizing symmetry properties of the investigated structure the modaleven/odd voltage and current can be related to voltage and currentdefined for each of two coupled lines (see Fig. 2) as follows:

Ue(z) = U1(z) + U2(z), Ie(z) = I1(z) + I2(z),

Uo(z) = U1(z)− U2(z), Io(z) = I1(z)− I2(z).

Next, applying the above relations to (21) we obtain the eigenvalueproblem of the following form:

QK = Kk, (28)

which solution are the eigenvalues k1 and k2 defining propagationcoefficients of two fundamental modes in ferrite guide. For the sakeof brevity matrix Q, diagonal matrix of eigenvalues k and matrix ofeigenvectors K are defined in Appendix B.

Using the solution of (28) the voltage and current in each of two

i1v1

i2v2

i3v3

i4 v4

z=0 z=Lz

line 2

line 11

2

3

4

(a)

b1

a1

b3

a3

b2

a2

b4

a4S

1

2

3

4

(b)

Figure 2. Four-port FCL junction: (a) circuit model, (b) networkrepresentation.


coupled lines can be defined at considered z cross-section as follows:

U1(z)

U2(z)

I1(z)

I2(z)

= K

e−jk1z 0 0 00 ejk1z 0 00 0 e−jk2z 00 0 0 ejk2z

A+1

A−1A+

2

A−2

, (29)

where A+(−)1 and A

+(−)2 are unknown amplitudes of the forward and

backward partial waves in the equivalent transmission line. Assumingnotation from Fig. 2(a) and utilizing (29) we can write the relationsbetween voltage and current in the ports of the considered junctiondefined at interfaces z = 0 and z = L:

U1(z)

U2(z)

I1(z)

I2(z)

z=0

=

v1

v2

i1i2

and

U1(z)

U2(z)

I1(z)

I2(z)

z=L

=

v3

v4

−i3−i4

. (30)

Combining Equations (29) and (30) we obtain the following relation:

v3

v4

−i3−i4

= K

e−jk1L 0 0 00 ejk1L 0 00 0 e−jk2L 00 0 0 ejk2L

K−1

v1

v2

i1i2

. (31)

Finally, we define incident and reflected waves in each ith port of thestructure:

ai =vi√Z0

+ ii√

Z0 and bi =vi√Z0

− ii√

Z0,

where Z0 is a wave impedance of the port. Applying above relationsto (31) we obtain the scattering matrix of four-port CFCL junctionwhich is defined as follows:

b1

b2

b3

b4

=

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

a1

a2

a3

a4

. (32)

Assuming instead of the voltage and current waves the real voltageand current distribution in the proposed transmission line model ofCFCL junction, the wave impedances Ze(o) and Z0 can be treated ascharacteristic impedances.


3. NUMERICAL RESULTS

In the numerical calculation CFCL junction presented in Fig. 1(a)is taken into account. In Section 3.1, parametric studies of theinvestigated structure are presented. In Section 3.2 the CFCL junctionwith Faraday rotation angle π/4 is designed and applied to realizecirculator.

3.1. Wave Properties of CFCL Junction

The fabrication process of the proposed CFCL junction from Fig. 1(a)based on metallization etching on cylinder is quite difficult. Howeversuch structure can be easily fabricated by etching the circuit on thindielectric planar substrate and then folding on a dielectric/ferritecircular cylinder.

According to above fabrication constraints in the analysis weassume the CFCL junction with following parameters of ferrite rod:r1 = 2.2mm, εrf = 15.04, saturation magnetization Ms = 1800Gs,internal bias Hi = 0, resonance line width ∆H = 0 and dielectriccoating: hd = r2 − r1 = 0.127 mm, εrd = 2.2 (Taconic TLY-5), angular slots spacing: ϕs1 = −30, ∆ϕs1 = 60, ϕs2 = 85,∆ϕs2 = 70. For the simulation of the presented examples the followingnumbers of eigenfunctions M = 30 and K = 5 were assumed in ourapproach. Utilizing MoM the normalized dispersion characteristics(βe(o) = βe(o)/k0) and magnetic field distributions for two basic modes

6 7 8 9 10 11 12 13 142.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

Norm

aliz

ed P

ropagation C

oeffic

ients

β

Frequency (GHz)

e

o

(a)

(3 (2 (1 0 1 2 3-3

-2

-1

0

1

2

3

x (mm)

y (

mm

)

Ho

He

(b)

~

β∼

β∼

Figure 3. Wave parameters of fundamental even and odd mode inCDCL: (a) normalized dispersion characteristics and (b) transversemagnetic field distributions at f0 = 10 GHz.


(even and odd) in CDCL are calculated and presented in Figs. 3(a)and 3(b), respectively. From Fig. 3(b), it can be noticed that inthe proposed guide there exists region where magnetic fields of twobasic modes are perpendicular to each other. Since, the ferritematerial is located in such region, the gyromagnetic coupling occurs(see Equation (27)). Moreover, one can see from Fig. 4, that withthe increase of frequency the field is more concentrated in region 1,resulting in increasing of gyromagnetic coupling coefficient.

The parametric studies of gyromagnetic coupling versus param-eters of the line are presented in Fig. 5. In Fig. 5(a), the frequencydependent characteristics of normalized gyromagnetic coupling coeffi-cient Ceo = Ceo/(k0η0µa

√ZeZo) determined for different angular slot

∆ϕs1 and strip ∆ϕm widths are presented. It can be found that withincreasing value of ∆ϕs1 and ∆ϕm the normalized gyromagnetic cou-pling coefficient Ceo is also increasing. In Fig. 5(b), the parametriccharacteristics of Ceo, determined in a function of radius of ferrite rodand for different height hd and permittivity εrd of dielectric coating,are presented. It can be noticed that Ceo increases with the increaseof ferrite rod radius and coating height which results from higher field

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x (mm)

y (

mm

)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x (mm)

y (

mm

)

(a)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x (mm)

y (

mm

)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x (mm)

y (

mm

)

(b)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x (mm)

y (

mm

)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x (mm)

y (

mm

)

(c)

Figure 4. Poynting vector distribution of fundamental even and oddmode in CDCL at frequency: (a) f0 = 7 GHz, (b) f0 = 10GHz and (c)f0 = 13GHz.


8 8.5 9 9.5 10 10.5 11 11.5 120.6

1

1.4

1.8

2.2

2.6

3

3.4

3.8

4.2 x 10 -3

Frequency (GHz)

Ceo (

1/Ω

)

m

s1 = 10 o

s1 = 20 o

s1 = 60 o

(a)

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.50

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2 x 10 -3

r1 (mm)

rd

hd = 0.127

hd = 0.381

hd = 0.635

(b)

Ceo (

1/Ω

)

∆ϕ

∆ϕ

∆ϕ ∆ϕε

~~

Figure 5. Characteristics of normalized gyromagnetic couplingcoefficient Ceo versus: (a) frequency for different slot ∆ϕs1 and strip∆ϕm angular widths, (b) radius of ferrite rod for different heighthd = r2 − r1 and permittivity εrd of the coating at f0 = 8 GHz.Parameters: ∆ϕm = ϕs2 − ϕs1 − ∆ϕs1 = 10, 30, 50deg and =εrd = 2.2, 4.5, 6.

concentration in the ferrite rod. On the other hand with the increasingvalue of dielectric permittivity εrd, the field becomes to concentrate inthe coat, hence the gyromagnetic coupling decreases.

Utilizing calculated gyromagnetic coupling coefficients fromFigs. 5(a) and 5(b) and propagation coefficients of even and oddmode the normalized dispersion characteristics of ferrites modes arecalculated from relation (B2) and presented in Fig. 6. In Figs. 6(a),6(b) and Figs. 6(c), 6(d) the normalized propagation coefficientsof two ferrite modes (k1(2) = k1(2)/k0) are shown, respectively.Moreover, in Figs. 6(e) and 6(f) the difference between calculatedpropagation coefficients is depicted, which can be used to determinethe approximate length of ferrite section ensuring Faraday rotationangle π/4 [28] as follows:

Laopt =

π

2(k1 − k2). (33)

In the case of angular variation of slots and strips one can observe thatdespite the increase of coupling coefficient with the increasing valuesof ∆ϕs1 and ∆ϕm, the difference between propagation coefficients isdecreasing (see Fig. 6(e)). The opposite effect is observed with thevariation of height hd and permittivity εrd of dielectric coating. Here,with the increasing value of gyromagnetic coupling the increase ofdifference between ferrite modes propagation coefficients is observed(see Fig. 6(f)).


8 8.5 9 9.5 10 10.5 11 11.5 121.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

No

rma

lize

d p

rop

ag

atio

n c

oe

ffic

ien

t k

1

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.51.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4hd = 0.127hd = 0.381hd = 0.635

8 8.5 9 9.5 10 10.5 11 11.5 121.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.51.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4h d = 0.127h d = 0.381h d = 0.635

8 8.5 9 9.5 10 10.5 11 11.5 120.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

| k −k

|

12

m

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

εrd

hd = 0.127hd = 0.381hd = 0.635

Frequency (GHz)

(a)

r1 (mm)

(b)

s1 = 10o

s1 = 20o

s1 = 60o

∆ϕ

∆ϕ

∆ϕ

No

rma

lize

d p

rop

ag

atio

n c

oe

ffic

ien

t k

1

s1 = 10o

s1 = 20o

s1 = 60o

∆ϕ

∆ϕ

∆ϕ

s1 = 10o

s1 = 20o

s1 = 60o

∆ϕ

∆ϕ

∆ϕ

No

rma

lize

d p

rop

ag

atio

n c

oe

ffic

ien

t k

2

No

rma

lize

d p

rop

ag

atio

n c

oe

ffic

ien

t k

2

Frequency (GHz)

(c)

r1 (mm)

(d)

Frequency (GHz)

(e)

r1 (mm)

(f)

~~

| k

−k

|

12

~~

∆ϕ

m∆ϕ

εrd

εrd

m∆ϕ

~ ~~~

Figure 6. Characteristics of normalized propagation coefficients oftwo fundamental ferrite modes versus parameters of the guide fromFig. 5.

Finally, in Fig. 7 the parametric studies of ferrite section lengthLopt ensuring Faraday rotation angle π/4 are presented. It can benoticed that according to the results obtained in Figs. 6(e) and 6(f)the length of the section Lopt increases with increasing ∆ϕs1 and ∆ϕm


8 8.5 9 9.5 10 10.5 11 11.5 126

7

8

9

10

11

12

13

14

15

16

17

Lo

pt (m

m)

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.510

20

30

40

50

60

70

80

90

100hd = 0.127hd = 0.381hd = 0.635

s1 = 10o

s1 = 20o

s1 = 60o

∆ϕ

∆ϕ

∆ϕ

Frequency (GHz)

(a)

r1 (mm)

(b)

Lo

pt (m

m)

εrd

m∆ϕ

Figure 7. Characteristics of ferrite section length ensuring Faradayrotation angle π/4 versus parameters of the guide from Fig. 5.

for which the decrease of difference between ferrite modes propagationcoefficients is observed. On the other hand the optimal length of thesection becomes shorter when the height hd and permittivity εr ofdielectric coating are increasing.

3.2. Application of CFCL Junction to Circulator

In this section, the scattering parameters of circulator employingproposed CFCL junction are determined. The circulator is obtainedby cascading T -junction and CFCL junction with Faraday rotationangle π/4. The required Faraday rotation angle is obtained when theCFCL junction satisfies amplitude and phase conditions [28]. For theseconditions, the input signal is equally divided between two output portsof CFCL junction and the difference phase between output signals isequal 0 or 180. In Section 3.2.1, the CFCL junction with the requiredamplitude and phase conditions is designed. Next, the scatteringparameters of circulator are calculated by cascading S-parameters ofideal T -junction and the designed CFCL junction.

3.2.1. Scattering Parameters of CFCL Junction

Utilizing the results presented in Section 3.1 the CFCL junctionsatisfying amplitude and phase conditions for ensuring Faradayrotation angle π/4 is designed. In the analysis the center operationfrequency f0 = 10 GHz is assumed. The calculated scatteringparameters of the junction for f0 are shown in Fig. 8. From thepresented results, it can be noticed that for length Lopt = 12.94mmwhen one of the structure port is excited the signals are equally divided


0 5 10 15 20 25 30 35 40 45 50 55 −40

−35

−30

−25

−20

−15

−10

−5

0

Length (mm)

Ma

gn

itud

e S

(d

B)

|S11||S21||S31||S41|

(a)

0 5 10 15 20 25 30 35 40 45 50 55−60

−30

0

30

60

90

120

150

180

210

Length (mm)

Ph

ase

diff

ere

nce

(d

eg

)

(arg(S 31)--arg(S41))

(b)

(arg(S 32 )--arg(S42 ))

Figure 8. Length dependent scattering parameters characteristics ofCFCL junction: (a) amplitude and (b) phase difference.

6 7 8 9 10 11 12 13 14−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency (GHz)

Ma

gn

itu

de

S (

dB

)

|S11||S21||S31||S41|

(a)

6 7 8 9 10 11 12 13 14−5

−4

−3

−2

−1

0

1

2

3

4

5

Frequency (GHz)

Ph

ase

diff

ere

nce

(d

eg

)

0

(b)

(arg(S 31)--arg(S41))+180

(arg(S 32)--arg(S42))

Figure 9. Frequency dependent scattering parameters characteristicsof CFCL junction: (a) amplitude and (b) phase difference (solid line— MoM/CMM, dashed line — HFSS).

in the output ports. According to the phase difference the outputsignals are even and odd mode, respectively. Moreover, obtained Lopt

is close to Laopt = 12.92mm obtained from relation (33). For the

considered section with Lopt = 12.94mm the scattering parametersversus frequency are calculated and presented in Fig. 9. It can befound that 3 ± 0.5 dB power division is obtained in frequency rangefrom 9.5 to 10.5GHz. Moreover, the phase difference between outputsignals is varying from −4 to 0 in the considered frequency range. Thefrequency dependent scattering parameters are compared with HFSS.


A good agreement of presented results with reference ones is observed.

3.2.2. Scattering Parameters of CFCL Circulator

Since, the CFCL junction from Section 3.2.1 satisfies requiredamplitude and phase conditions [28] it can be used in realizationof nonreciprocal devices such as isolators, gyrators or circulators.This can be simply done by ensuring the proper excitation of CFCLjunction. In Fig. 10, frequency response of single circulator obtainedby cascading scattering parameters of ideal T -junction with calculatedS-parameters of the investigated CFCL junction is presented. Theinvestigated circulator with the following circulation direction 1 →3 → 2 → 1 clearly shows nonreciprocal behavior. For this structurethe isolation and reflection losses are better than −15 dB in frequencyrange from 8 to 11.5GHz. In the simulation of the CFCL circulatorparameters the lossless ferrite junction was assumed. Bearing in mindthat the losses in FCL devices comes mainly from ferrite section weestimate their value in proposed junction with the use of commercialsoftware HFSS. In the analysis the following material parameterswere assumed for ferrite rod: resonance line width ∆H = 80 Oe,tan δ = 0.0005, dielectric coating: tan δ = 0.0009 and metallizationlayers: σ = 5.8e7 S/m. For assumed parameters calculated losseswere about 0.2 dB. Obtained results are much lower than insertionlosses of coplanar FCL junction [15] operating in similar frequencyrange, for which the simulated in HFSS insertion losses were about

1

Hi 2

3

θ=π/4

CFCLT - junction

ferrite

(a)

6 7 8 9 10 11 12 13 14−30

−25

−20

−15

−10

−5

0

|S| (d

B)

f (GHz)

S11

S22

S33

S12

S23

S31

S13

S21

S32

(b)

Figure 10. Frequency response of single circulator utilizing proposedCFCL junction from Fig. 1: (a) schematic view of the structure and(b) magnitude scattering parameters.


1.2 dB. Moreover, in comparison to [15] the length of the proposedferrite section is about two times shorter, allowing for reduction ofFCL devices dimensions.

4. CONCLUSION

In this paper, the novel cylindrical ferrite coupled line junction ispresented. Utilizing the MoM/CMM approach the gyromagneticcoupling, dispersion characteristic and scattering parameters of CFCLjunction are calculated. Obtained results show that in comparisonto planar junctions in such section a higher gyromagnetic couplingoccurs. This allows to use the ferrite materials with lowersaturation magnetization and to reduce the length of the junctionby a half. In result such CFCL junction exhibits much lowerinsertion losses and allows to reduce total dimensions of nonreciprocaldevices in comparison to recently developed in literature planar FCLconfigurations.

ACKNOWLEDGMENT

This work was supported by Polish Ministry of Science and HigherEducation from sources for science in the year 2011 under ContractNo. 0215/T02/2010/70.

APPENDIX A.

In Equations (5)–(7) matrices ME,lA1 , MH,l

A1 , ME,lA2 and MH,l

A2 take theform

MEAp =

MEe

z ,lAp 0

MEe

ϕ,l

Ap MEh

ϕ,l

Ap

, (A1)

where

MEez ,l

Ap = diag

R(p)m (kρlrl)

m=M

m=−M, (A2)

MEe

ϕ,l

Ap = diag

−jmγ

k2ρlrl

R(p)m (kρlrl)

m=M

m=−M

, (A3)

2mm]MEh

ϕ,l

Ap = diag− ωµ

ηkρlR′(p)

m (kρlrl)m=M

m=−M

(A4)


and

MH,lAp =

0 MHh

z ,lAp

MHe

ϕ,l

Ap MHh

ϕ,l

Ap

, (A5)

where

MHez ,l

Ap = diag

j

ηlR(p)

m (kρlrl)m=M

m=−M

, (A6)

MHe

ϕ,l

Ap = diag−jωε

kρlR′(p)

m (kρlrl)m=M

m=−M

, (A7)

MHh

ϕ,l

Ap = diag

mγ

ηlk2ρlrl

R(p)m (kρlrl)

m=M

m=−M

. (A8)

p = 1, 2 and R′m(x) = ∂Rm(x)/∂x denotes first derivative of Bessel

function.In Equations (12) and (17) matrices of integrals Ios and Iso take

form:Ios = diag

Iso,1, . . . , I

so,I , I

co,0, I

co,1, . . . , I

co,I

(A9)

where

[Is,ios

]m,k

=12π

ϕsi+∆ϕsi∫

ϕsi

e−jmϕ sin(

ϕ− ϕsi

∆ϕsikπ

), (A10)

[Ic,ios

]m,k

=12π

ϕsi+∆ϕsi∫

ϕsi

e−jmϕ cos(

ϕ− ϕsi

∆ϕsikπ

)(A11)

andIso = 2π (I∗os)

T . (A12)

In Equation (17) matrices of electric field ME,2, ME,3 andmagnetic field MH,2, MH,3 take the form

ME,2 = ME,2A1 + ME,2

A2 T12, (A13)

MH,2 = MH,2A1 + MH,2

A2 T12, (A14)

ME,3 = ME,3A1 T34 + ME,3

A2 , (A15)

ME,3 = ME,3A1 T34 + ME,3

A2 . (A16)


APPENDIX B.

In Equation (28) matrices Q, k and K take a form:

• matrix Q

Q =

0 0 Q1p Q1m + Q3

0 0 Q1m −Q3 Q1p

Q2p Q2m 0 0Q2m Q2p 0 0

, (B1)

where Q1p = βeZe+βoZo

2 , Q1m = βeZe−βoZo

2 , Q2p = βeYe+βoYo

2 ,Q2m = βeYe−βoYo

2 , Q3 = −jCeo,

• matrix of eigenvalues k

k = diag k1,−k1, k2,−k2 , (B2)

where k1 =√

β20 + Γ2, k2 =

√β2

0 − Γ2, β0 =√

β2e+β2

o2 , ∆β =

β2e−β2

o2 and Γ2 =

√∆β2 + C2

eoZeZo

βeβo,

• matrix of eigenvectors K

K =

Keo Keo Koe Koe

K∗eo K∗

eo −K∗oe −K∗

oe

YeeKoe −YeeKoe YooKeo −YooKeo

YeeK∗oe −YeeK

∗oe YooK

∗eo −YooK

∗eo

, (B3)

where Keo = 1 + j(∆β2−Γ2)Ceoβo

√ZoZe

, Koe = 1 + j(∆β2−Γ2)Ceoβe

√ZeZo

,

Yee = βeYe

k1and Yoo = βoYo

k2.

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Documents

INVESTIGATIONS OF CYLINDRICAL FERRITE COU- PLED LINE ... · on structures realized in planar line technology [19,21]. Such structures allow one to obtain fully integrated FCL devices