Progress In Electromagnetics Research, PIER 57, 33–54, 2006

FIELD STRUCTURES OF WAVEGUIDE JUNCTIONCIRCULATORS WITH IRREGULAR SHAPED FERRITESIMULATED BASED ON EXACT TREATMENT

H. C. Wu and W. B. Dou

State Key Laboratory of Millimeter WavesSoutheast UniversityNanjing 210096, China

Abstract—Field structures of waveguide Y -junction circulatorscontaining irregular shaped ferrite material, such as Y -junction withpartial-height cylinder/triangle ferrite posts and metal step impedancetransformer, Y -junction with a ferrite sphere, and dual higher ordermodes circulator, which are classified as different junction resonancemodes, are depicted to illustrate the circulating process. The analysismethod used is a combination of the mode expansion method andfinite element method, which is based on the weak form of Helmholtzequation. Calculated results (return loss, insertion loss and isolation)of these structures are shown and compared respectively to those inliteratures. Consistencies have been observed.

1. INTRODUCTION

It is well known that waveguide junction circulators are thefundamental elements in microwave and millimeter wave systems. Inorder to get good performance and eliminate try-and-error methodin circulator design, many methods, including MMM [1–5], MoM,FEM [6], BEM, FDTD [7] and etc., have been used to analyzethe circulators. However, so far the physical pictures about thecirculating process of waveguide junction circulator with differentirregular shaped ferrite have not been described in detail based onthe exact analysis. The physical picture of the junction circulatorcirculating is important for practical engineers to understand thecirculating mechanism of junction circulator and to design a suitablecirculator for meeting requirement of a system. For treatment ofvarious junction circulator structures and drawing the pictures of fieldstructures, a flexible approach that combines finite element method,

34 Wu and Dou

which is based on Helmholtz equation weak form, and mode expansion,is used. The Helmholtz equation weak form was introduced byPeterson and Castillo to analyze some two dimensional problems ofelectromagnetic scattering from inhomogeneous cylinder [8]. Basedon the weak form, a final set of linear equations can be acquired byusing interpolation to present the electric field and magnetic field in thecomplicated structure containing anisotropic media, and combining themode expansion in the regular region to truncate the computing region.Various circulators such as Y -junction circulator with partial-heightferrite cylinder/triangle post and metal step impedance transformer,Y -junction circulator with a ferrite sphere and dual higher order modescirculator are studied. Then E-field distributions in the junction of thecirculators are presented to illustrate the circulating process.

2. APPROACH

For the waveguide junction circulator with regular shaped ferrite suchas circular post, field in ferrite can be expressed in eigenmodes, thefields in other zones also can be expressed in eigenmodes of the zones,then field matching is done on the interfaces between them so as to get alinear equations. Solving the equations the scattering parameters (Sij)of the junction circulator can be obtained, the treatment procedurehas been described in ref. [2] in details. The field in junction is mainlycontrolled by the field in ferrite material, so knowing the field propertiesin ferrite is important to know the performance of the circulator. Here,some field expressions in ferrite post are given as follows for describingthe circulation of circulator. Field in ferrite post can be expressed bythe superposition of eigenmodes. The eigenmodes are classified intotwo categories: volume modes and surface modes. They are given asfollowings (only Ez and Hϕ are shown here, detail seen ref. [2]):

Volume mode (0 < k2c < k2

fµz):

Ez =∑n

[A1n cos β1z + A2nchβ2z] Jn(kcr) exp(jnϕ) (1)

Hϕ =∑n

(−j)YFkc

kf

{[p1J

′n(kcr) − r1

nJn(kcr)kcr

]× A1n cos β1z

+[p2J

′n(kcr) − r2

nJn(kcr)kcr

]× A2nchβ2z

}exp(jnϕ) (2)

Surface mode (−∞ < k2c < 0):

Ez =∑n

2∑i=1

Ain cos βizIn(kcr) exp(jnϕ) (3)

Progress In Electromagnetics Research, PIER 57, 2006 35

Hϕ =∑

i

2∑n=1

jYFkc

kfAin

[piI

′n(kcr) − ri

nIn(kcr)kcr

]cos βiz exp(jnϕ)

(4)Here,

[µ] =

µ −jκ 0jκ µ 00 0 µz

(5)

k2f = ω2ε0µ0εf (6)

β21,2 = k2

fµ − k2c

µ + µz

2µz±

[k4

c

(µ − µz

2µz

)+

k2fκ

µz

(k2

fµz − k2c

)]1/2

(7)

pi = k2f/k2

c (8)

ri =

(k2

fµe − k2c − β2

i

)κk2

c

(9)

YF =(

ε0εf

µ0

)1/2

(10)

kc is obtained by solving transcend equation, which results from thefield matching at interface between ferrite post and dielectric post asshown in Fig. 1. It is detailed in ref. [2]. Jn(x) is Bessel functionand In(x) is modified Bessel function. And we have β2

vm < β2sm, βvm

denote the propagation constant of volume mode in z direction andβsm denote that of surface mode. For each structure analyzed in thispaper, elements of tensor permeability (5) are specified in appendix.

Surface modes are determined by In(x) in the radial direction, sothey are non-propagation modes in the radial direction and their fieldenergy is concentrated nearby the cylindrical surface. In z direction thefield is determined by triangular function, so it is always oscillation.Because the energy of surface mode always concentrates nearby theferrite post surface, so the field distribution is not sensitive to thefrequency. However, volume modes are determined by Jn(x), so volumemodes represent the propagating modes in the radial direction andtheir energy is concentrated inside the body of ferrite. This fielddistribution will be changed with frequency changing. However, somevolume modes may not be oscillation in z direction as shown inEquation (1). The field expressions out of the ferrite post can beobtained similarly. Details are given in ref. [2].

The circulation of circulator was thought of a resonance of thejunction containing ferrite, which is considered a resonator. So the

36 Wu and Dou

resonance frequency of the junction is corresponding to the circulatingfrequency that depends on the ferrite parameters. The resonance modemay be thought of that of a cylinder cavity such as TM11 or otherresonance modes. So the junction resonance mode is a superposition ofmany eigenmodes. Different contribution of each eigenmode results indifferent resonance mode. Changing the ferrite parameters such as size,permittivity and saturated magnetization will results in the changingof contribution of each eigenmode, so the circulating performance isalso changed.

In ref. [2] several junction resonance modes of waveguide junctioncirculator containing ferrite cylinder have been denominated. Theyare low order modes (CV -mode, CV −S-mode, CS−V -mode) and higherorder modes that are dual higher order modes. All they result fromdifferent ferrite size under the condition of same permittivity andsaturated magnetization. Both simulation and experimental resultsof the circulators with ferrite post for different junction modes arealso presented. However, the analysis is limited to the circulator withregular ferrite post. And the field structures in junction have not beengiven there, so we do not know the difference of field structures betweenthem. Here we wish to depict the field structures of waveguide junctioncirculators with irregular shaped ferrite; it will be useful for practicalengineers to understand the circulating process of the circulators andto chose a suitable junction resonance mode to meet the performancerequirement of a system. However, the field matching method based onthe eigenmode expression is difficult for solving the junction circulatorcontaining irregular shaped ferrite such as triangular ferrite post andferrite sphere exactly. So another method that will be described in thefollowing section is used. In spite of this, the eigenmode expression ishelpful for us to understand the field in junction.

3. HYBRID MODE MATCHING AND FINITEELEMENT METHOD

Waveguide Y -junction circulator, as show in Figure 1, is taken as anexample to show the treatment procedure. Parameter a is the widthof the waveguide. The fields are expressed with different functions atdifferent regions.

In the junction with irregular ferrite material (Region I):

�E =N∑

n=1

An�Nn(x, y, z) (11)

Progress In Electromagnetics Research, PIER 57, 2006 37

In the waveguide 1 (Region II):

�E = ein11�e

in11 (x, y, z) +

L∑l=1

es1l�e

s1l(x, y, z) (12)

In the waveguide 2 (Region III):

�E =L∑

l=1

es2l�e

s2l(x, y, z) (13)

In the waveguide 3 (Region IV):

�E =L∑

l=1

es3l�e

s3l(x, y, z) (14)

Where �Nn(x, y, z): the first-order edge elements in vector finiteelement; �eil(x, y, z): electric vector eigenfunction of the lth mode inthe ith waveguide; eil: the coefficient of electric vector eigenfunctionof the lth mode in the ith waveguide; An: the interpolated value of Eat that vector.

Substitute test function �T (x, y, z) with �Ni(x, y, z), i = 1, 2, · · · , Nin Helmholtz weak form [9] results in

∫∫I

∫ [∇× �Ni ·

(µ−1r ∇×

N∑n=1

An�Nn

)−k2 �Ni ·

(εr

N∑n=1

An�Nn

)]dV

= −3∑

j=1

∫∫Sj

�Ni ·[n × µ

−1r

(∇× �E

)]dS (15)

Thus N linear equations are obtained. The properties of anisotropicmedia have been characterized by tensor µ

−1r and εr in (15). For the

situation considered here, µr is equal to [µ] in Equation (5), and εr

reduced to a scalar quality εr. For a lossy anisotropic media [µ] isgiven in Appendix, which is easy to implement in (15).

For the fields matching on the interface areas Sj , there exist thefollowing equations:

ein11�e

in11 (x, y, z) +

L∑l=1

e s1l�e

s1l(x, y, z) =

N∑n=1

An�Nn

(x,−

√3

6a, z

)(16)

38 Wu and Dou

L∑l=1

e s2l�e

s2l(x, y, z) =

N∑n=1

An�Nn

(x,

√3x + −

√3

3a, z

)(17)

L∑l=1

e s3l�e

s3l(x, y, z) =

N∑n=1

An�Nn

(x,−

√3x + −

√3

3a, z

)(18)

By multiplication crossing �hjl(x, y, z), l = 1, 2, · · · , L, on both sides of(16), (17) and (18), and integrating over Sj , j = 1, 2, 3, respectively,3 ∗ L linear equations are gained. Where, �hil(x, y, z) eigenfunction ofthe lth mode in the ith waveguide.

Finally, a matrix equation is obtained:

AX = B (19)

The coefficient matrixes can be expressed as

A =

[[Γ]N×N [C]N×3L

[D]3L×Ndiag[T ]3L

](N+3L)×(N+3L)

,

B : (N + 3L) × 1, X : (N + 3L) × 1 (20a)

[Γ]ij =∫∫

I

∫ [∇× �Ni ·

(µ−1r ∇× �Nj

)− k2 �Ni ·

(εr

�Nj

)]dV,

i = 1, 2, . . . , N, j = 1, 2, . . . , N (20b)

[C]i[(j−1)∗L+k)] =∫∫Sj

�Ni ·[n × µ

−1r

(∇× �e s

jk(x, y, z))]

dS,

i = 1, 2, . . . , N, j = 1, 2, 3, k = 1, 2, . . . , L (20c)

[D][(i−1)∗L+k)]j =∫∫Si

�Nj × �hsik(x, y, z)dS,

i = 1, 2, 3, j = 1, 2, . . . , N, k = 1, 2, . . . , L (20d)

diag [T ](i∗L+k) =∫∫Si

�e sik(x, y, z) × �hs

ik(x, y, z)dS,

i = 1, 2, 3, k = 1, 2, . . . , L (20e)

[B]i =

∫∫Sj

�Ni ·[n × µ

−1r

(∇× �e in

11 (x, y, z))]

dS i = 1, 2, . . . , N

∫∫Si

�e in11 (x, y, z) × �h in

11 (x, y, z)dS i = N + 1

0 i = N + 1, N + 2, . . . , N + 3∗L(20f)

Progress In Electromagnetics Research, PIER 57, 2006 39

X are unknowns to be solved, as shown in followings:

X =(A1, A2, . . . , AN , es11, e

s12, . . . , e

s1L, es

21, es22, . . . , e

s2L, es

31, es32, . . . , e

s3L)′

(20g)[Γ]N×N is acquired by applying N sets of basis functions astesting functions to Eq. (15). diag[T ]3L is acquired because of theorthogonality of expansion modes in interfaces. [C]N×3L and [D]3L×N

are the results of the field matching on the ports boundary.Because the plane z = b/2 (Parameter b is the height of the

waveguide) is a symmetry plane and incident wave is H10 mode, itcan be considered an electric wall, so only half volume of the junctionspace need to be considered in calculation. Then for different time t,E-field at plane z = b/2 in the junction of the circulator is expressedas:

E = Re[Eze−jωt

] (t =

i

N

2π

ω, i = 1, 2, · · ·N

)(21)

Usually numerical computation is used to get the scattering parametersof circulators, but the circulating process of the circulators is seldomrevealed. Now the field structure is depicted to discover the circulatingprocess or physical mechanism.

4. NUMERICAL RESULTS AND DISCUSSIONS

4.1. Numerical Results for Y -Junction with Partial-HeightFerrite Posts (Lower Junction Resonance Mode: CV −S

Mode)

A calculation example of Y -junction circulator with partial-heightferrite post is given in Fig. 2. The field structures from Equation (21)in circulating frequency 36 GHz are depicted in Fig. 3. The ports arecorresponding to Fig. 1. Circulation direction is from port 1 to port 2to port 3. The red circle in the junction denotes the ferrite post. Redregion in junction denotes the field in positive value and blue region innegative value.

When t = 0 the field in the region near interface S1 is positiveand in the region near S2 is negative, in the region near the mid ofport 3 the field is zero, so no power coupled into port 3. Power istransmitted from port 1 to 2, and port 3 is isolated. From the fielddistribution in different time, we can see the field is rotated becauseof the action of ferrite post. When t = 11T/30 and 12T/30, althoughthe field in the region near port 3 is positive, the field has been shiftedclosed to ferrite post, so that the field in S3 is still zero and no poweris coupled into port 3. When t = T/2, the field is the same as that

40 Wu and Dou

a

1port

3port2port AA

R

AA −

hb

dielectric

ferrite0 x

z

0 x

y

S1

S2 S3I

II

III IV r

φ

Figure 1. Y -junction circulator loaded with a partial-height ferritepost.

26 28 30 32 34 36 38 400

-10

-20

-30

-40

S-P

aram

eter

s(dB

)

Frequency(GHz)

S11 S21 S31

Figure 2. Performance of Y -junction circulator with a partial-heightferrite post a = 7.12 mm, b = 3.65 mm, R/h = 1.1, εf = 13.85,εd = 2.08, Hi = 200 Oe, p = ωm/ω = 0.4.

Progress In Electromagnetics Research, PIER 57, 2006 41

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.405

-1.051

-0.6963

-0.3419

0.0125 0

0.366 9

0.721 2

1.07 6

1.43 0

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.835

-1.384

-0.9337

-0.4831

-0.03250

0.4181

0.8688

1.319

1.770

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.405

-1.051

-0.6963

-0.3419

0.01 250

0.36 69

0.72 12

1.07 6

1.43 0

(a) 0t (b)6T

t (c)36

2 TTt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.405

-1.051

-0.6963

-0.3419

0.0125 0

0.366 9

0.721 2

1.07 6

1.43 0

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y-1.405

-1.051

-0.696 3

-0.341 9

0.01250

0.3669

0.7212

1.076

1.430

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.405

-1.051

-0.6963

-0.3419

0.0125 0

0.366 9

0.721 2

1.076

1.430

(d)30

11Tt (e)

3012T

t (f) 30

13Tt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.405

-1.051

-0.696 3

-0.341 9

0.01250

0.3669

0.7212

1.076

1.430

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.405

-1.051

-0.6963

-0.3419

0.0125 0

0.366 9

0.721 2

1.07 6

1.43 0

(g)30

14Tt (h)

230

15 TTt =

=

=

= =

= =

= = =

Figure 3. E-field distribution in the junction on plane z = b/2(Frequency: 36 GHz).

in t = 0 except the positive and negative value is exchanged. WhereT = 2π/ω is time cycle. Therefore, the circulation of circulator is acomplex process, which contains field rotation and field shift. In ϕ.direction field can be distinguished as n = 1 mode (here n is equal tothe n in eigenmode expressions in Equations (1)–(4)). This junctionresonance mode has been denominated as CV −S mode in ref. [2].

4.2. Numerical Results for Y -Junction with Partial-HeightFerrite Cylinder Posts and Metal Step ImpedanceTransformer (Lower Mode: CS−V Mode)

The configuration of circulator is given in Fig. 4. Fig. 5 showsits performance. Because there are some inhomogeneity of the

42 Wu and Dou

a

1port

3port2port

AA

R

l

AA

hb

dielectric

ferrite

ï ïmetal step

0 x

y

0 x

z

__

a R hl M Tf d s

=7.12 mm, =1.2 mm, =0.78 mm, =0.8 mm,=13.85, =2.08, =4.05 mm, =0.485

0 δεε

Figure 4. Schematic diagram of Y -junction circulator with a partial-height cylinder ferrite post and metal step impedance transformer.

26 28 30 32 34 36 38 400

-5

-10

-15

-20

-25

-30

-35

)Bd(ssoL nrute

R

Frequency(GHz)

Calculated Results Measured Results

26 28 30 32 34 36 38 400

-5

-10

-15

-20

-25

-30

)Bd(ssoL tresnI

Frequency(GHz)

Calculated Results Measured Results

26 28 30 32 34 36 38 400

-5

-10

-15

-20

-25

-30

)Bd(noita losI

Frequency(GHz)

Calculated Results Measured Results

Figure 5. Performance of Y -junction circulator with a partial-heightcylinder ferrite post and metal step impedance transformer.

magnetization in ferrite used in the experiment and the error of themanufacture such as the diameter of dielectric piece may be largera little than that of ferrite post, there is a little difference betweenthe calculation results and measured results. This resonance modeis denominated as CS−V mode in ref. [2]. Because of its broadbandperformance, the field structures are depicted at two frequencies pointand shown in Fig. 6 and Fig. 7, respectively. The circulation process

Progress In Electromagnetics Research, PIER 57, 2006 43

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.340

-1.049

-0.7575

-0.4663

-0.1750

0.1162

0.4075

0.6987

0.9900

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.065

-0.7506

-0.4363

-0.1219

0.1925

0.5069

0.8212

1.136

1.450

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-0.9750

-0.7344

-0.4938

-0.2531

-0.01250

0.2281

0.4688

0.7094

0.9500

(a) t = 0 (b) 306T

t (c)309T

t= =

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-0.9650

-0.7575

-0.5500

-0.3425

-0.1350

0.0725 0

0.280 0

0.487 5

0.695 0

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y-0.927 5

-0.739 1

-0.550 6

-0.362 2

-0.173 7

0.0146 9

0.203 1

0.391 6

0.580 0

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-0.850 0

-0.640 6

-0.431 3

-0.221 9

-0.0125 0

0.196 9

0.406 3

0.615 6

0.825 0

(d)30

10Tt (e)

3011T

t (f) 30

12Tt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-0.900 0

-0.635 0

-0.370 0

-0.105 0

0.160 0

0.425 0

0.690 0

0.955 0

1.220

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-0.990 0

-0.698 7

-0.407 5

-0.116 2

0.175 0

0.466 3

0.757 5

1.049

1.340

(g)30

14Tt (h)

23015 TT

t =

= =

=

=

=

Figure 6. E-field distribution in the junction on plane z = b/2(Frequency: 28.5 GHz).

of mode is different from that of CV −S mode. The field rotation is notobvious. It can be seen from Fig. 6 that the negative value part of fieldmoves from port 2 through mid of junction to port 1 with the time tchanging from 6T/30 through 9T/30, 10T/30 and 11T/30, 12T/30 to14T/30. The positive value part of field moves along the left region ofjunction around negative part to port 2; that is, negative part seemsstanding in hall as if a post, positive part circulates around the postfrom port 1 to port 2 as if stream. From Fig. 7 similar circulationprocess also can be observed. In ϕ direction n = 1 mode does notclearly displaced as CV −S mode does. It is considered the result of thedifferent contribution of each eigenmode.

44 Wu and Dou

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.515

-1.11 9

-0.723 7

-0.328 1

0.0675 0

0.463 1

0.858 8

1.254

1.650

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.335

-0.985 6

-0.636 2

-0.286 9

0.0625 0

0.411 9

0.761 2

1.11 1

1.460

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.030

-0.772 5

-0.515 0

-0.257 5

1.11 0E-16

0.2575

0.5150

0.7725

1.030

(a) 0t (b)3T

t (c) 30

5Tt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-0.845 0

-0.610 6

-0.376 2

-0.141 9

0.0925 0

0.326 9

0.561 2

0.795 6

1.030

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y-0.797 5

-0.569 7

-0.341 9

-0.114 1

0.113 7

0.341 6

0.569 4

0.797 2

1.025

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.180

-0.9100

-0.6400

-0.3700

-0.1000

0.170 0

0.440 0

0.710 0

0.980 0

(d)30

7Tt (e)

30

9Tt (f)

30

11Tt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.475

-1.123

-0.7712

-0.4194

-0.06750

0.284 4

0.636 2

0.988 1

1.340

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.645

-1.249

-0.853 8

-0.458 1

-0.0625 0

0.333 1

0.728 7

1.124

1.520

==

= ==

=

(g) t =3013T____ (h) t =

30____15T = 2

___T

30

Figure 7. E-field distribution in the junction on plane z = b/2(Frequency: 34 GHz).

4.3. Numerical Results for Y -Junction with Partial-HeightTriangle Ferrite Posts and Metal Step ImpedanceTransformer (Lower Mode: CS−V Mode)

Fig. 8 depicts the configuration of the circulator and Fig. 9 shows theperformance. The field structures at circulating frequency are givenin Fig. 10. The red triangle line in junction denotes the ferrite. Thisjunction resonance mode may also be denominated as CS−V mode. Itcan be seen that the negative part of field moves from port 2 to mid ofjunction then to port 1. The positive part of field moves from port 1 toport 2 along the left of junction around negative part. The circulationprocess given in Fig. 10 is similar to that given in Fig. 6 and Fig. 7.

Progress In Electromagnetics Research, PIER 57, 2006 45

a

1port

3port2port

AA

l

AA

hb

dielectric

ferrite

ïmetal

d

stepδ

a b d hl H pOe,f d i m ωωεε

=7.12 mm,=13.85,

=3 mm, =0.93 mm,=3.56 mm, =0.7 mm,=4.15 mm, =2.08, =0.4

δ=200 = /

−

Figure 8. Y -junction circulator with a partial-height triangle ferritepost and metal step impedance transformer.

26 28 30 32 34 36 38 400

-5

-10

-15

-20

-25

-30

)Bd(srete

m araP-

S

Frequency(GHz)

S11_Calculated S21_Calculated S31_Calculated

Figure 9. Performance of Y -junction circulator with a partial-heighttriangle ferrite post and metal step impedance transformer.

46 Wu and Dou

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y-1.665

-1.237

-0.8087

-0.3806

0.0475 0

0.475 6

0.903 7

1.33 2

1.76 0

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.520

-1.103

-0.685 0

-0.267 5

0.150 0

0.567 5

0.985 0

1.402

1.820

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.195

-0.810 6

-0.426 2

-0.0418 7

0.342 5

0.726 9

1.111

1.496

1.880

(a) 0t (b) 30

2Tt (c)

30

4Tt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.990

-1.674

-1.358

-1.043

-0.7267

-0.4108

-0.0 9500

0.220 8

0.536 7

0.852 5

1.168

1.484

1.800

1.850

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-2.000

-1.655

-1.309

-0.9636

-0.6182

-0.2727

0.07273

0.4182

0.7636

1.109

1.455

1.800

1.850

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.940

-1.552

-1.164

-0.7760

-0.3880

0

0.388 0

0.776 0

1.164

1.552

1.900

(d)30

6Tt (e)

30

8Tt (f)

30

10Tt

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.630

-1.252

-0.875 0

-0.497 5

-0.120 0

0.257 5

0.635 0

1.013

1.390

-3 -2 -1 0 1 2 3

2

1

0

-1

-2

-3

-4

X

Y

-1.760

-1.331

-0.902 5

-0.473 8

-0.0450 0

0.383 7

0.812 5

1.241

1.670

(g)30

12Tt (h)

230

15 TTt =

=

=

=

= =

=

=

=

Figure 10. E-field distribution in the junction on plane z = b/2(Frequency: 32.5 GHz).

4.4. Numerical Results for Y -Junction with a Ferrite Sphere(Lower Mode: CV −S Mode)

The configuration of the circulator is shown in Fig. 11. Theperformance of the circulator is depicted in Fig. 12. Agreementbetween calculation and experiment has been observed. Also, becauseof the error of the manufacture such as the ferrite may not be exactlya sphere and the thick or diameter of the two dielectric piece on andbelow the ferrite sphere may be different, there is a little differencebetween the calculation results and measured results. It is apparentthat the resonance mode of junction containing ferrite sphere may bedifferent from that of junction containing ferrite post. To draw all fieldcomponents structure in three dimensions is not an easy task. The fieldstructure on the symmetrical plane z = b/2 is depicted again as shown

Progress In Electromagnetics Research, PIER 57, 2006 47

a

1port

3port2port

AA

R

l

AA

h b

dielectric

ferrite

metal step

0 x

z

0 x

y

(a) Y-junction circulator (b) ferrite sphere and dielectric posts (meshed)

_

a

b

R

h

f

H

i H

Oe

Ms

a

b

R

h

f

d

Hi Oe

H Oe T

Ms Tε

εε

εδ

∆

∆=2 =2.54 mm, =0.42 mm, =0.635 mm, =13.5, =2.25,

=1700 , =80 , tan =2.5 X 10 ,-4 =0.5 this page=2 =2.54 mm, =0.4 mm 5%, =0.635 mm, =13.5,=2.25, >1700 , <120 , =0.5 in Ref. [10].d Oe

+_

Figure 11. Schematic diagram of Y -junction circulator with a ferritesphere.

89.0 89.5 90.0 90.5 91.0 91.5 92.0 92.50

-5

-10

-15

-20

-25

-30

-35

)Bd(srete

m araP-

S

Frequency(GHz)

S21 Calculated S11 Calculated S21 Measured in Refence

89.0 89.5 90.0 90.5 91.0 91.5 92.0 92.50

-5

-10

-15

-20

-25

-30

)Bd(srete

m ar aP-

S

Frequency(GHz)

S31 Calculated S31 Measured in Ref.[10]

Figure 12. Performance of Y -junction circulator with a ferrite sphere.

in Fig. 13. Take a survey carefully it may be found that the fieldstructure in Fig. 13 is similar at some extent to that in Fig. 3. So itmay be called CV −S mode. The field is rotated with time t increasingfrom t = 0 to t = T/2. When t = 11T/30, it seems some power maybe coupled into port 3. However, the field has been shifted close to themid of junction, the field along interface S3 is zero so no power flowsinto port 3. In ϕ direction field can also be distinguished as n = 1mode.

48 Wu and Dou

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.600

-1.067

-0.5333

0

0.5333

1.067

1.600

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-2.000

-1.288

-0.5750

0.1375

0.8500

1.563

2.000

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.790

-1.229

-0.667

-0.106

0.4550

1.016

1.578

1.700

(a) = 0 (b) = 30___3

(c) = 30___6 TT

t t t

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.190

-0.9187

-0.6475

-0.3762

-0.1050

0.1663

0.4375

0.7087

0.9800

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y-0.9725

-0.7591

-0.5456

-0.3322

-0.1188

0.09469

0.3081

0.5216

0.7350

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.000

-0.6900

-0.3800

-0.07000

0.2400

0.5500

0.8600

1.170

1.480

(d)30

9Tt (e)

30

11Tt (f)

30

12Tt

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.360

-0.8475

-0.3350

0.1775

0.6900

1.202

1.360

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.600

-1.070

-0.5400

-0.01000

0.5200

1.050

1.580

1.600

(g)30

14Tt (h)

230

15 TTt

== =

= = =

Figure 13. E-field distribution in the junction on plane z = b/2(Frequency: 90.4 GHz).

4.5. Numerical Results for Dual Higher Order ModesY-Junction Circulators (High Mode: Dual Higher Modes)

Compared to the low order junction resonance mode such as CV −S

mode and CS−V mode described above, the higher order resonancemodes are supported by large ferrite volume. Usually increasingferrite radius can get higher order modes. Higher order mode canwithstand higher power because it has larger area attached to metalwall of circulator and the height of ferrite piece is small compare toits diameter, so the heat within ferrite can be dissipated throughthe metal wall more efficiency. At W band the anisotropic actionof ferrite is reduced because the saturated magnetization of ferritecan only reach 0.5T, which results in a small value κ of tensorelement of ferrite. Therefore, low order mode can only support a

Progress In Electromagnetics Research, PIER 57, 2006 49

a

1port

3port2port

AA

R

AA

2/hb

dielectric

ferrite0 x

z

0 x

y

_

a b R/h f d iH Oe p m=2.54 mm, =1.27 mm, =2.1, =15.5, =2.5, =200 , = / =0.15ε ω ωε

Figure 14. Schematic diagram of dual higher order modes circulator.

narrowband performance, higher order mode can support wider bandperformance as it has larger ferrite volume to provide anisotropicaction. Computation and practices show the higher order modeavailable is dual higher modes for which two higher order modescirculating in same direction and close to each other, as expoundedin ref. [2], an example calculated is shown in Fig. 15. The structure ofthe circulator is similar to that given in Fig. 1 except ferrite post withlarger radius is adopted, which is given in Fig. 14.

First higher order mode circulates in band from 94 GHz to 98 GHz;Second higher order mode circulates from 105 GHz to 107 GHz; from99 GHz to 103 GHz the circulator has bad performance that can beconsidered boundary frequency zone between two modes.

Fig. 16 shows the field structure of first higher order mode at97 GHz where red circular denote the ferrite post. Watching the fieldstructure carefully it can be observed that it displays n = 2 modein ϕ. direction. When t = 0, 3T/30, 6T/30 and 9T/30, it seemsthat the field near port 3 can excites H10 mode in port 3, actually, itexcites H30 mode; despite the field at mid of interface S3 has higher

50 Wu and Dou

80 85 90 95 100 105 1100

-5

-10

-15

-20

-25

-30

-35

)Bd(srete

maraP-

S

Frequency(GHz)

S11 S21 S31

Figure 15. Performance of dual higher order modes circulator.

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.095

-0.789 4

-0.483 7

-0.178 1

0.127 5

0.433 1

0.738 8

1.044

1.350

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.240

-0.934 4

-0.628 7

-0.323 1

-0.0175 0

0.288 1

0.593 8

0.899 4

1.205

1.210

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-0.910 0

-0.648 9

-0.387 8

-0.126 7

0.134 4

0.395 6

0.656 7

0.917 8

1.179

1.280

(a) t=0 (b) t=3/30*T (c) t=6/30*T

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-0.600 0

-0.411 1

-0.222 2

-0.0333 3

0.1556

0.3444

0.5333

0.7222

0.9111

1.100

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.100

-0.886 1

-0.672 2

-0.458 3

-0.244 4

-0.0305 6

0.183 3

0.397 2

0.611 1

0.825 0

(d) =9/30* (e) =12/30* T Tt t

Figure 16. E-field distribution in the junction on plane z = b/2(Frequency: 97 GHz).

Progress In Electromagnetics Research, PIER 57, 2006 51

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.410

-0.963 8

-0.517 5

-0.0712 5

0.375 0

0.821 2

1.267

1.714

2.160

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.410

-0.963 8

-0.517 5

-0.0712 5

0.375 0

0.821 2

1.267

1.714

2.160

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.410

-0.963 8

-0.517 5

-0.0712 5

0.375 0

0.821 2

1.267

1.714

2.160

(a) t=0 (b) t=3/30*T (c) t=6/30*T

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.410

-0.963 8

-0.517 5

-0.0712 5

0.375 0

0.821 2

1.267

1.714

2.160

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.750

-1.399

-1.048

-0.697 3

-0.346 4

0.00454 5

0.3555

0.7064

1.057

1.408

1.759

2.110

2.160

(d) =9/30* (e) =12/30* tt T T

Figure 17. E-field distribution in the junction on plane z = b/2(Frequency: 99 GHz).

negative value, the field at two side of interface S3 has positive value,so it excites H30 mode other than H10 mode. However, H30 modeis cutoff in port 3, thus port 3 is isolated. For this high mode therotation of field structure is not displayed as CV −S mode, and it alsois not similar to CS−V mode. When t = 0, the power (positive region)enter the junction from port 1, then moves into ferrite post with timeincreasing. Because the field is dominated by n = 2 junction resonancemode, the entered field was distributed through the ferrite post alongits diameter as shown in Fig. 16(e) in t = 12T/30. When t = T/2 thefield structure is the same as that in t = 0 except the positive regionand negative region is exchanged. For the next half cycle from t = T/2to t = T , the process is same except the positive region and negativeregion exchanged.

When frequency is 99 GHz, the field structure is given in Fig. 17when circulator is not circulating. It display n = 2 mode clearly.Entered power was scattered into all 3 ports so the circulator has badperformance. When t = 0, the entered power has moved into ferritepost, however, when t = 3T/30 and 6T/30, the power was returnedto port 1 a little so the reflect wave in port 1 was excited. Whent = 9T/30, the transmission wave was excited in port 2. In next halfcycle, the transmission wave in port 3 was excited.

The circulation process of second higher order mode, which isgiven in Fig. 18, is different from first higher order mode. It seemsn = 1 mode in ϕ direction. It seems similar to CV −S mode shown

52 Wu and Dou

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.000

-0.781 8

-0.563 6

-0.345 5

-0.127 3

0.0909 1

0.309 1

0.527 3

0.745 5

0.963 6

1.182

1.400

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-2.300

-1.882

-1.464

-1.045

-0.627 3

-0.209 1

0.209 1

0.627 3

1.045

1.464

1.882

2.300

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-2.200

-1.818

-1.436

-1.055

-0.672 7

-0.290 9

0.0909 1

0.472 7

0.854 5

1.236

1.618

2.000

(a) t=0 (b) t=4T/30 (c) t=8T/30

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.100

-0.936 4

-0.772 7

-0.609 1

-0.445 5

-0.281 8

-0.118 2

0.0454 5

0.209 1

0.372 7

0.536 4

0.700 0

-1.0 -0.5 0.0 0.5 1.0

0.5

0.0

-0.5

-1.0

-1.5

X

Y

-1.200

-1.018

-0.8364

-0.6545

-0.4727

-0.2909

-0.1091

0.07273

0.2545

0.4364

0.6182

0.8000

(d) t=12T/30 (d) t=14T/30

Figure 18. E-field distribution in the junction on plane z = b/2(Frequency: 106 GHz).

in Fig. 3; field structure is rotated with t increasing like CV −S mode,but field was distributed on a larger area like a zoom of CV −S mode,which may results from a larger diameter of ferrite post. When timet changes from t = 12T/30 to 14T/30 the positive region of the fieldwas rotated to port 3, which is not concentrated closed to centre of thejunction so some power may leak to port 3 and isolation is degraded.

5. CONCLUSION

Waveguide junction circulators with irregular shaped ferrite areanalyzed by a method that combines FEM and mode expansion. Thefield structures of the circulators are depicted to show the circulationprocess clearly for the first time. The description on the circulationprocess is also presented. The correctness of calculation has beenconfirmed by experiments. These pictures depicted will be helpfulfor microwave engineers to understand the physical mechanism ofwaveguide junction circulator and to choose a suitable mode fordesigning a circulator meeting the requirement of practical system.

Progress In Electromagnetics Research, PIER 57, 2006 53

APPENDIX A.

Permeability of FerriteIn ferrite region,

µr =

µ jκ 0−jκ µ 00 0 1

(A1)

with

ω0 = γH0 (A2)ωm = γMs (A3)

µ = 1 +ωmω0

ω20 − ω2

(A4)

κ =ωmω

ω20 − ω2

(A5)

where,γ: the gyromagnetic ratioMs: saturated magnetizationH0: the applied biased dc field

If the ferrite is lossy,

εr = εf (1 − j tan δ) (A6)µ = µ′ − jµ′′, κ = κ′ − κ′′ (A7)

µ′ = 1 +ω0ωm(ω2

0 − ω2) + ωmω0ω2α2[

ω20 − ω2(1 + α2)

]2 + 4ω20ω2α2

(A8)

µ′′ =ωαωm

[ω2

0 + ω2(1 + α2)]

[ω2

0 − ω2(1 + α2)]2 + 4ω2

0ω2α2(A9)

κ′ =ωωm

[ω2

0 − ω2(1 + α2)]

[ω2

0 − ω2(1 + α2)]2 + 4ω2

0ω2α2(A10)

κ′′ =2ω0ωmω2α[

ω20 − ω2(1 + α2)

]2 + 4ω20ω2α2

(A11)

α = γ∆H/(2ω) (A12)

Where,εf : dielectric constant of the ferritetan δ: loss tangent∆H: ferrite line-width

54 Wu and Dou

ACKNOWLEDGMENT

We express our thanks sincerely to the referees for their comments andhelpful suggestion that will improve the quality of our paper.

REFERENCES

1. EL-Shandwily, M. E., A. A. Kamal, and E. A. F. Abdallah,“General field theory treatment of H-plane waveguide junctioncirculators,” IEEE Trans. Microwave Theory Tech., Vol. 21, 392–403, Jun. 1973.

2. Dou, W. B. and Z. L. Sun, “Millimeter wave ferrite circulator androtator,” Int. J. Infrared & MMW, Vol. 17, 2035–2131, Dec. 1996.

3. Akaiwa, Y., “Mode classification of a triangular ferrite post forY -circulator operation,” IEEE Trans. Microwave Theory Tech.,Vol. 25, 59–61, Jan. 1977.

4. Khilla, A. M. and I. Wolff, “Field theory treatment of H-planewaveguide junction with triangular ferrite post,” IEEE Trans.Microwave Theory Tech., Vol. 26, 279–287, Apr. 1978.

5. Okamoto, N., “Computer-aided design of H-plane waveguidejunction with full-height ferrite of arbitrary shape,” IEEE Trans.Microwave Theory Tech., Vol. 27, 315–321, Apr. 1979.

6. Koshiba, M. and M. Suzuki, “Finite-element analysis of H-planewaveguide junction with arbitrarily shaped ferrite post,” IEEETrans. Microwave Theory Tech., Vol. 34, 103–109, Jan. 1986.

7. Hu, S. X. and W. B. Dou, “FDTD analysis of several waveguidejunction circulators with full-height ferrite of arbitrary shape,”Int. J. RF and Microwave CAE, Vol. 14, No. 4, 153–165, 2004.

8. Peterson, A. F. and S. P. Castillo, “A frequency-domaindifferential equation formulation for electromagnetic scatteringfrom inhomogeneous cylinders,” IEEE Trans. Antennas Propag.,Vol. 37, 601–607, May 1989.

9. Wu, H. and W. Dou, “Analysis of waveguide multi-portsdiscontinuities by the helmholtz weak form and mode expansion,”IEEE Proc. Microw. Antennas Propag., Vol. 151, 530–536, Dec.2004.

10. Yung, E. K. N., R. S. Chen, K. Wu, and D. X. Wang, “Analysisand development of millimeter-wave waveguide-junction circulatorwith a ferrite sphere,” IEEE Trans. Microwave Theory Tech.,Vol. 46, 1721–1734, Nov. 1998.

11. Jin, J. M., The Finite Element Method of Electromagnetism,Xi’dian University Press, Xi’an, 2001 (In Chinese).