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h (x, x, y, y) is the adjunct Greens function [6] defined by:

Figure 1: Rectangular structure on study.

h

x, x, y, y

= g

x, x, y, y

(2)

The volume integral (1) is applied to the ferrite slab volume with a possible source s(x; y).However, using inhomogeneous boundary conditions, the final solution of the Ez field is obtainedwithout an internal source and the relation (1) becomes as follows:

Ez

x, y

=

hEz Ezh

d (3)

The initial equation to find Greens function in cartesian coordinates is given by:

2x2 +

2

y2 + k2

g =

x x

y y

(4)

where k =

00rfeff and eff =22

( and are constitutive parameters of the ferrite

permeability tensor).The derivation of Greens function (when a magnetic wall is fixed as a boundary condition along

the ferrite slab) [6] leads to:

g

x, y, x, y

=n

ejn(xx)

2a.

n cos(n(y

b)) n sin(n(y b))

n sin(2nb).

n cos(n(y + b)) n sin(n(y + b))

2n +

2n2

2

|yy

(5)

where n =na

and 2n = k2

2n.

From the integral relation (3) the Ez field can be found on the access lines and the scatteringparameters can be determined [7]. Figure 2 shows some results from this analytical model which

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was sligthly modified to take into account non perfect magnetic wall at the ferrite slab edge. Nonreciprocal effects are observed since S21 and S31 are different from 40 to 45 GHz. These results arein agreement with 3D HFSS simulations which show a large effect in a wide frequency band (seethe following paragraph) although some discrepancies are observed. However the model allows usto determine the length of the rectangle giving priority of the first mode:

a = e

keff

(6)

Then, the width of the rectangle b is determined by empirical means.

3. 3D SIMULATIONS

Now, the same structure is simulated by the HFSS 3D electromagnetic software. Many simulationswere made with several parameters having varied. Figure 3 shows the results (S11, S21 and S31)obtained from the same structure as on Figure 2(a). As we have already indicated: similar nonreciprocal effects are observed that the ones found by the analytical model, on a wide frequencyband around to 40 GHz. However, there are some discrepancies between the results obtained fromthe HFSS and the anaytical model simulations. For example, the bandwidth is larger than the onegiven by the analytical model. This phenomenon is all greater as the boundary conditions on the

ferrite edge depart from a perfect magnetic wall.

(a) (b)

Figure 2: S11, S21 and S31 of two different rectangular structures obtained from the analytical model withferrite magnetization Ms = 382: kA/m; internal field Hi = 1400 kA/m and dimensions: (a) a =740 m andb =300 m; (b) a =800 m and b =500 m.

(a) (b)

Figure 3: S11, S21 and S31 of two different rectangular structures obtained from HFSS with ferrite magneti-

zation Ms = 382: kA/m; internal field Hi = 1400 kA/m and dimensions: (a) a =740 m and b =300 m; (b)a =800 m and b =500 m.

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The three ports are not located in a symmetrical way. Thus, non recirpocal effects are notsimilar between two successive ports. For example, Figure 4 shows S23, S32 parameters part (a)and S13, S31 parameters part (b). This drawback could be overcome by a rigorous impedancematching.

(a) (b)

Figure 4: (a) S23, S32 parameters, (b) S13, S31 parameters for the same structure as on Figure 2 part (a).

The dimensions and the D.C. internal field are determined by using the analytical model, thenthe optimization is performed in making them vary separately. For example, the results obtainedwhen the internal D.C. field value is 1000 kA/m are shown on Figure 5. Only small effects areobserved and this D.C. value is not usable.

The best results are pointed out when the rectangle size is 740 m by 300 m and the D.C. field

value is 1400 kA/m. The bandwidth value can reach 5 GHz and the relative bandwidth 10.5 percent. Other parameter influence (access line width. . .) has been studied but not presented in thispaper.

Therefore, important comments could be done: even if the impedance matching is quite wrong(the S11 parameter is almost close to 1 everywhere), high non reciprocal effects are shown in a largefrequency band. This structure could be used as rectangular circulator.

(a) (b)

Figure 5: S11, S21 and S31 obtained for ferrite magnetization Ms = 382: kA/m and internal fieldHi = 1000 kA/m, (a) from HFSS, (b) from the analytical model.

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4. CONCLUSION

A rectangular junction ferrite component has been studied. It is made from a stripline structureincluding two ferrite slabs which are magnetized along the z-axis direction. Using a rectangularstructure avoids the use of bent access lines that could disturb the TEM line propagation. Accordingto the first results, the component must be improved before it is a rectangular circulator. Theimpedance matching must be performed. The analytical model could be better adjusted if we took

into account non perfect boundary conditions used for Greens function derivation.Barium or strontium ferrites seem to be suitable to operate in millimeter waves. Thin film depo-sition requires high temperature annealing with a mganetic moment direction. Composite magneticmaterials made from directed nano particles scattered into a host dielectric matrix constitute an-other interesting solution owing to their low temperature processes. This will true provided thatthe magnetic moment direction be maintained.

On the same way, other structures as coplanar and microstrip could be investigated.Research domain of self-biased magnetic material still remains a challenge for integrating non

reciprocal components.

REFERENCES

1. Olivier, A., P. Shi, and C. Vittoria, Integrated self-biased hexaferrite microstrip circulators formillimeter-wavelength applications, IEEE Trans. Microwave Theory Tech., Vol. 49, 385387,

February 2001.2. Mincho, A. and L. G. M. Tsankov, Design of self-biased waveguide circulators, J. Appl.

Phys., Vol. 73, No. 10, 70187020, May 1993.3. Bosma, H., On stripline Y-circulation at UHF, IEEE Trans. Microwave Theory Tech.,

Vol. 12, 6172, January 1964.4. Helszajn, J., Fabrication of very weakly and weakly magnetized microstrip circulators, IEEE

Trans. Microwave Theory Tech., Vol. 46, 439449, May 1998.5. Ogasawaram, N., Coplanar-guide and slot-guide junction circulators, Electronics Letters,

Vol. 7, No. 7, 220221, May 1998.6. Dudley, D. G., Mathematical Foundations for Electromagnetic Theory, IEEE Press Oxford

University Press, 1994.7. How, H., T. M. Fang, C. Vittoria, and R. Schmidt, Design of six-port stripline ferrite junction

circulators, IEEE Trans. Microwave Theory Tech., Vol. 42, 12721275, July 1994.