Microwave behavior of ferrites
The small signal approximation of the precession of the magnetic moments (Eq. I.8), sub-jected to a static magnetic field Hdc and a microwave magnetic field h(t), leads to the tensor relationship between the magnetic flux density b and the field h. b = µ0µhˆ , (I.13).
where µ0 is the permeability of vacuum (4π ∗ 10−7 H/m ), and µˆ is the tensor permeability. If the magnetic field is applied along the z axis of the Cartesian coordinate system, the tensor permeability is given by  : µ jκ 0 0 0 µz µˆ = jκ −µ 0 (I.14).
When a ferrite material is placed in a static magnetic field Hdc, its EM properties vary according to the direction of wave propagation. Under the action of an external magnetic field, the extra diagonal term κ becomes non-zero and material becomes anisotropic in nature. This induced anisotropy is the reason for the nonreciprocal behavior of ferrites. On the other hand, for a demagnetized ferrite κ = 0, and the permeability tensor becomes diagonal. In this case the ferrite is considered as an isotropic material . Ferrites are widely used in microwave devices due to this nonreciprocal behavior towards EM wave propagation. In the field of microwave electronics, ferrites are widely used in many signal processing and telecommunication applications. Devices like circulators/isolators are based on the non-reciprocal behavior of EM wave propagation in magnetized ferrites. Devices like tunable filters, delay lines, phase shifters and variable attenuators etc. exploit the non linearity of the EM behavior of ferrites with respect to a static magnetic field. Ferrites are also used for miniaturization of antennas and realization of absorbers in UHF band. These devices exploit the high permeability exhibited by the ferrite materials in the demagnetized state. The flexibility of the magnetic properties of the ferrite is obtained by controlling its magne-tization state using the static magnetic field, Hdc. This static field determines the location of the magnetization of the ferrite in the hysteresis loop. In practice, anisotropy of the ferrites and the frequency tunability of the devices are achieved by the application of a static magnetic field . Finally, ferrite in the state of remanence oﬀers a fast and stable switching, which is very useful for phase shifters and makes it pos-sible to avoid the use of permanent magnets in microwave devices
For majority of applications of reciprocal devices, there are one or more semiconductor-based devices that meet the same specifications. This is not the case for nonreciprocal de-vices. Semiconductor-based alternatives often have limitations in power, mechanical stress, and non-linearity. So, undoubtedly microwave devices based on magnetic materials play a very important role in microwave technology, especially in nonreciprocal devices.
Precise control of the performance of these devices requires prior knowledge of the dynamic behavior of ferrite materials. Dynamic behavior of the ferrites is first modeled by the sus-ceptibility tensor, which defines the relation between the microwave magnetizationm and the microwave excitation h in the form: m = µ0χh (I.15) However, this behavior is usually represented by the permeability tensor, which connects the microwave magnetic induction b and the microwave excitation h, (Eq. I.13).
EM characterization of ferrite materials
Characterization of materials is an important and necessary step, even before the design and realization of microwave devices. There are diﬀerent methods for characterizing the properties of materials. These methods vary according to their specifications such as the range of operating frequencies, the isotropic or anisotropic nature of the material, the shape of the sample and its dielectric or magnetic character.
This section gives a brief overview on the state of the art of the microwave theory and tech-niques for the characterization of magnetic materials. Figure I.9 illustrates qualitatively the techniques used to characterize the EM properties of materials as a function of frequency. The resonant cavity methods make it possible to precisely determine the permittivity or the permeability of the magnetic material for a fixed frequency value. These methods are usually very accurate but they are mono-frequency techniques. There are constraints re-garding the size and shape of the cavity and the sample to be measured. In order to obtain parameters at diﬀerent frequencies, we have to use diﬀerent cavities of diﬀerent sizes. The frequency dependent properties of the material cannot be obtained with these methods. In addition, they are only suitable for low-loss materials. High magnetic losses will reduce the quality factor of the cavity, as well as the sensitivity of the measurement. On the other hand with transmission/reflection techniques, it is possible to determine con-stituent parameters of the materials over a broad frequency band from the transmission and reflection coeﬃcients. For the characterization of the constituent parameters of the material, it is necessary that the number of independent parameters measured, is greater than or equal to the number of constituent parameters to be determined. That is to say, to determine ǫ and µ of an isotropic material such as a demagnetized ferrite, it is necessary to measure at least two distinct parameters (S11 or S22 and S12 or S21).
In the case of an anisotropic material, such as a magnetized ferrite where the permeability is a tensor quantity, it is necessary to measure three distinct S-parameters (S11, S12 and S21) in order to find µ, κ and ǫ in their complex form. Thus we have to make sure that the mea-surement cell is nonreciprocal in nature (S21 = S12). Although the transmission/reflection techniques are very practical, S-parameter measurement in a wide frequency band leads to a reduction in the accuracy with respect to single-frequency resonant methods, in particular for low loss materials. Experimentally, EM characterization of ferrites is commonly carried out in demagnetized or saturated states. For example the transmission/reflection technique in a coaxial line [18, 19] is used to extract the scalar permittivity and scalar permeability of isotropic ferrites in the demagnetized state. In saturation, a resonant cavity is used for the linewidth measurements (∆H and ∆Hef f ). This quantity, which represents the magnetic losses of the material, is an input parameter of the Polder model. The characterization of ferrites in partial mag-netization states is less easy. For partially magnetized states which are found in practice in self-biased circulators/ isolators, phase shifters and tunable antennas or filters, diﬀerent characterization techniques have been developed in the laboratory, Lab-STICC, using micro strip line , rectangular waveguides  and strip lines [22, 23]. The main advantage of these techniques is directly related to the fact that they provide access to the constituent material parameters, scalar permittivity and permeability tensor components.
Demagnetized state – Coaxial line method
The coaxial line characterization method was developed by Weir  using the work of Nicolson and Ross  and is called Nicholson-Ross-Weir (NRW) method. This method be-came the reference method for the permittivity and permeability measurements of isotropic materials in the demagnetized state. Main advantage of this method is the simplicity in the analysis and calculations using the classical transmission line theory. It is possible to calculate the magnetic permeability and the electrical permittivity from the measured S-parameters simultaneously with this approach. The sample is toroidal in shape and inserted in between the inner and outer conductors of the coaxial line. The fundamental mode of propagation in a coaxial line is the TEM mode.
The validity of the method is related to the frequency (fc) of the occurrence of the first higher order mode. The appearance of higher order modes in a coaxial line depends on the dimensions of the line and the EM properties of the propagation medium (permeability and permittivity).
Saturated state – Resonant cavity methods
Resonant cavity methods are widely used for measuring the dielectric or magnetic properties of the materials due to their sensitivity and high accuracy [4, 16]. The general principle of resonant cavity perturbation methods consists of measuring the shift in the resonance frequency (Fr ) and quality factor (Q) of the loaded cavity with respect to that of an empty cavity. The resonance frequency and quality factor of the cavity is determined with and without the sample. From these values, the permittivity and permeability are then extracted using theoretical relations.
The sample must be very small compared with the size of the cavity so that there is only a small shift in frequency when the sample is inserted. The sample length must be less than λ/4 of the cavity to avoid dimensional resonances. These are mono-frequency methods based on the perturbation theory. To measure the dielectric properties of the material, sample should be placed in a cavity where the electric field is at maximum and magnetic field is at minimum. When the sample is placed at a position where magnetic field is at maximum, magnetic properties of the material can be characterized.
Table of contents :
I EM Characterization of Anisotropic Ferrites – State Of The Art
I.2.1 Static properties
I.2.2 Dynamic properties
I.2.3 Microwave behavior of ferrites
I.3 EM characterization of ferrite materials
I.3.1 Demagnetized state – Coaxial line method
I.3.2 Saturated state – Resonant cavity methods
I.3.2.1 Permeability measurement
I.3.2.2 Resonance linewidth measurements
I.3.3 Partially magnetized state
I.3.3.1 Nonreciprocal microstrip line – Quasi-TEM method
I.3.3.2 Partially filled waveguide method
I.3.3.3 Nonreciprocal strip line method
I.4 Motivation and Objectives
II A Coaxial Line Method For Damping Factor Measurement
II.2 Permeability tensor models
II.2.1 Saturated state – Polder model
II.2.2 Demagnetized state – Schloemann model
II.2.3 Partially magnetized state
II.2.3.1 Permeability model by Rado
II.2.3.2 Green and Sandy model
II.2.3.3 Igarashi and Naito
II.2.4 Any magnetization state: Generalized permeability tensor model (GPT)
II.3 Measurement cell
II.4 General description of the method
II.5 Direct problem
II.5.1 Analytical functions for constituent parameters
II.5.2 EM analysis of the measurement cell
II.5.2.1 Theoretical solutions for propagation constant
II.5.2.2 Theoretical solutions for scattering parameters
II.5.3 Direct problem – Results
II.6 Validation of the direct problem
II.6.1 Dielectric material
II.6.2 Magnetic material
II.6.2.1 Demagnetized ferrites
II.6.2.2 Saturated ferrites
III Inverse Problem- Experimental Results
III.2 Inverse problem
III.2.1 Optimization procedure
III.2.2 Choice of the permittivity and permeability models
III.2.3 Optimization algorithm
III.3 Measurement setup
III.4 Measurement results
III.4.1 Composite ferrites
III.4.2 Bulk ferrites
III.4.3 Validation of results
IV EM Modeling Of Anisotropic Ferrites – Application To Y-Junction Microstrip Circulators
IV.2 Anisotropic ferrites – Non-homogeneous internal fields
IV.2.1 Demagnetizing field effects
IV.2.2 Effect of non-uniform internal fields in the power absorption spectrum
IV.2.3 Commercial software solutions: Magneto-static simulations using Ansys Maxwell 3D
IV.3 Electromagnetic modeling tool
IV.3.1 Magneto-static solver (Lab-STICC)
IV.3.2 EM modeling of anisotropic ferrites – Ansys HFSS
IV.3.2.1 Integration of theoretical permeability tensor models with HFSS
IV.3.2.2 Ansys HFSS – Macro programming
IV.3.2.3 Validation of results
IV.3.2.4 Comparison with classical HFSS simulations
IV.3.2.5 Non-Uniform biasing – Comparison with Maxwell 3D simulations
IV.4 Application – Microstrip Y-junction circulator
IV.4.1 Circulator Design
IV.4.2 Experimental results
IV.4.3 Non-uniform biasing fields- Electromagnetic analysis
Conclusion and Perspectives