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Zucco",slug:"massimo-e.-zucco"}]},{id:"15163",title:"Application of Fast Fourier Transform for Accuracy Evaluation of Thermal-Hydraulic Code Calculations",slug:"application-of-fast-fourier-transform-for-accuracy-evaluation-of-thermal-hydraulic-code-calculations",signatures:"Andrej Prošek and Matjaž Leskovar",authors:[{id:"21775",title:"Dr.",name:"Andrej",middleName:null,surname:"Prošek",fullName:"Andrej Prošek",slug:"andrej-prosek"},{id:"22838",title:"Dr.",name:"Matjaž",middleName:null,surname:"Leskovar",fullName:"Matjaž Leskovar",slug:"matjaz-leskovar"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"66142",title:"Ferrite Materials and Applications",doi:"10.5772/intechopen.84623",slug:"ferrite-materials-and-applications",body:'\nFerrimagnetism is similar to ferromagnetism in many ways [1, 2, 3, 4]. They all have hysteresis curves as the applied magnetic field changes, resulting in the saturation magnetization (\n
The hysteresis (B-H) curves for (a) hard ferrimagnetism and (b) soft ferrimagnetism. The remnant polarization (\n\n\nB\nr\n\n\n) and the coercive field (\n\n\nH\nc\n\n\n) for the hard ferrites should be as large as possible. On the other hand, for the soft ferrites, the remnant polarization (\n\n\nB\nr\n\n\n) and the coercive field (\n\n\nH\nc\n\n\n) are very small or even close to zero.
Soft magnetic materials include electrical steels and soft ferrites [3, 4]. Unlike the ferromagnetic metals which are conductors, soft ferrites have low electric conductivity, i.e., they are dielectric materials. The electrical steels have extensive applications in low-frequency systems, such as generators, motors, and transformers, while the soft ferrites are suitable for the high-frequency applications, such as circulators, isolators, phase shifters, and high-speed switches.
\nThis chapter will focus on the properties of the soft ferrites and their applications at high-frequency systems. The ferrites are crystals having small electric conductivity compared to ferromagnetic materials. Thus they are useful in high-frequency devices because of the absence of significant eddy current losses. Ferrites are ceramic-like materials with specific resistivities that may be as much as 1014 greater than that of metals and with dielectric constants around 10 to 16 or greater. Ferrites are made by sintering a mixture of metal oxides and have the general chemical composition MO·Fe2O3, where M is a divalent metal such as Mn, Mg, Fe, Zn, Ni, Cd, etc. Relative permeabilities of several thousands are common [5, 6]. The magnetic properties of ferrites arise mainly from the magnetic dipole moment associated with the electron spin [2].
\nThe magnetic dipole moment precesses around the applied DC magnetic field by treating the spinning electron as a gyroscopic top, which is a classical picture of the magnetization process. This picture also explains the anisotropic magnetic properties of ferrites, where the permeability of the ferrite is not a single scalar quantity, but instead is a generally a second-rank tensor or can be represented as a matrix. The left and right circularly polarized waves have different propagation constant along the direction of the external magnetic field, resulting in the nonreciprocity of a propagating wave. Since the permeability should be treated as a tensor (matrix), not a scalar permeability, it is generally much difficult to understand and to have intuition, even for the researchers.
\nThe properties of ferrites are very intriguing. Without a DC bias magnetic field \n
Figure 2 shows the Larmor precession with the circularly polarized fields [7]. Circular polarization may be referred to as right handed or left handed, depending on the direction in which the electric (magnetic) field vector rotates. For a right-hand circularly polarized (RHCP) wave, the fields rotate clockwise at a given position from the source looking in the direction of propagation. The magnetic dipole moment m processes around the \n
Larmor precession of a magnetic moment m around the applied DC bias field H0 (\n\n=\n\nH\n0\n\n\nz\n̂\n\n\n) with (a) a right-hand circularly polarized (RHCP) wave and (b) a left-hand circularly polarized (LHCP) wave. The frequency of the Larmor precession in both cases are the same, i.e., the Larmor frequency \n\n\nω\n0\n\n\n\n\n=\n\nμ\n0\n\nγ\n\nH\n0\n\n\n\n\n. \n\n\nH\nt\n+\n\n\n and \n\n\nH\nt\n−\n\n\n are the transverse components of the incident waves which rotate clockwise (RHCP) and counter-clockwise (LHCP) from the source viewpoint looking in the direction of propagation. The thumb points the direction of the wave propagation, and the fingers give the rotation of the transverse components [7].
A linearly polarized incident wave can be decomposed into RHCP and LHCP waves of equal amplitude. The orientation of the linearly polarized wave changes after the wave propagates a certain distance because of the distinct propagation constants. The phenomenon is the famous Faraday’s rotation [5, 6]. This unique property has various applications, such as phase shifters, isolators, and circulators. However, it is difficult to follow for students and even researchers in that the permeability is a tensor, not just a simple proportional constant.
\nHere we consider the simplest case for the pedagogic purpose—a circularly polarized plane wave is normally incident upon a semi-infinite medium. The wave characteristics such as the propagation constant k and the wave impedance Z are associated with the permeability μ, which is a tensor for the ferrite medium [5]. By finding the preferred eigenvalues, it will be shown that the properties of μ depend on the DC bias field \n
The permeability \n
Since the torque is equal to the time change rate of the angular momentum, we have
\nBy comparing Eqs. (1) and (2), we obtain
\nA large number of the magnetic dipole moment \n
M and H in Eq. (4) differ slightly from m and \n
Since the AC terms have an exp(−iωt) dependence, by substituting Eq. (5) and (6) into Eq. (4) the transverse component terms read
\n\n\n
where \n
The eigenvectors corresponding to these two eigenvalues are the right-hand circularly polarized wave (RHCP, denoted as +) and the left-hand circularly polarized wave (LHCP, denoted as -), respectively. The symbols, + and -, represent positive helicity and negative helicity, respectively. The LHCP wave has a relatively mild response over the entire frequency range. On the contrary, the RHCP wave has a much more dramatic response.
\nThe permeabilities of the RHCP and LHCP waves are
\nEq. (12) has a singularity when the wave frequency \n
For a resonant cavity with a quality factor (\n
the permeability for the RHCP wave now reads
\nTo conduct a complete simulation of a ferrite device, we need to know its complex permittivity, the saturation magnetization, and the resonance linewidth. We will discuss how to characterize the ferrite’s properties in the next section.
\nHere we will discuss the measurement of the most important properties of ferrites, including the dielectric properties (\n
Ferrites are ceramic-like materials with relative dielectric constants around 10 to 16 or greater. The resistivities of ferrites may be as high as 1014 greater than that of metals. Since ferrites are dielectric materials. The dielectric properties (\n
Figure 3 shows the ideal of the field enhancement method. Figure 3(a) shows the resonant frequency as functions of the dielectric constant (\n
(a) Resonant frequency versus dielectric constant based on full-wave simulations. The solid curve can be divided into three regions: low, transition, and ultrahigh. The dashed line is simulated with a much thinner sample of 1.00 mm in thickness, which exhibits the properties similar to those of perturbation. (b) Schematic diagram of the field enhancement method. It consists of a cylindrical resonant cavity and a metal rod. The sample is placed on the top of the metal rod. The metal rod focuses and enhances the electric field significantly. An SubMiniature version A (SMA) 3.5-mm adapter couples the wave from the top of the cavity [11].
Ferrites have a strong response to the applied magnetic field. The magnetic properties of ferrites arise mainly from the magnetic dipole moment associated with the electron spin. Relative permeabilities of several thousands are common. The saturation magnetization (\n
The hysteresis curve regarding the magnetization \n\nM\n\n and the internal bias \n\n\nH\n0\n\n\n. When the applied internal magnetic field \n\n\nH\n0\n\n\n is large enough, the magnetization will be saturated, denoted as (\n\n\nM\ns\n\n\nor\n\n4\nπ\n\nM\ns\n\n\n). When \n\n\nH\n0\n\n\n decreases to zero, the remnant polarization is denoted as \n\n\nM\nr\n\n\n. The polarization will change sign (from positive to negative) when \n\n\nH\n0\n\n\n is greater than \n\n−\n\nH\nc\n\n\n which is called the coercive field.
The idea is to place the sample at the maximum of the H-field. It exhibits resonant absorption when the internal bias field is changed to \n\n\nH\nr\n\n\n. By changing the magnetic field, we will obtain the absorption. \n\n\nH\n1\n\n\n and \n\n\nH\n2\n\n\n are associated with the 3-dB absorption. The difference between these two values is the resonance linewidth \n\nΔ\nH\n\n.
Note that the saturation magnetization is denoted as \n
The loss of ferrite material is related to the linewidth, \n
The idea is normally implemented using a TE10n (n even) cavity in the X-band region [9, 12]. The test sample is placed at the H-field maximum. The sample is spherical with a diameter of approximately 0.040 inches, which is much easier to estimate than the internal bias field \n
The three key parameters are obtained in three experimental setups under different sizes and geometries of the samples. If the samples’ properties are slightly different or the machining error is not negligible, the error will be large or even unacceptable. The ultimate goal is to integrate the measurements and to extract the parameters using one experimental setup. The three key parameters will be used in the design of the microwave ferrite devices in the next session.
\nThe ferrites are crystals having small electric conductivity compared to ferromagnetic materials. Thus they are useful in high-frequency situations because of the absence of significant eddy current losses. Three commonly used ferrite devices are discussed below. These are phase shifters, circulators, and isolators [13, 14, 15, 16].
\nThe phase shifters are important applications of ferrite materials, which are two-port components that provide variable phase shift by changing the bias magnetic field. Phase shifters find application in test and measurement systems, but the most significant use is in phase array antenna where the antenna beam can be steered in space by electronically controlled phase shifters. Because of the demand, many different types of phase shifters have been provided. One of the most useful designs is the latching nonreciprocal phase shifter using a ferrite toroid in the rectangular waveguide. We can analyze this geometry with a reasonable degree of approximation using the double ferrite slab geometry.
\nFigure 6 shows the full-wave simulation for a two-port phase shifter using high-frequency structure simulator (HFSS, ANSYS). A standard waveguide WR-90 is employed with a width of 22.86 mm and height of 10.16 mm. The field patterns are displayed in Figure 6(a) for an empty waveguide and a waveguide with two ferrite slabs. The phase difference \n
Simulation results. (a) The field pattern to the left is for the empty waveguide. The right figure shows field strength with ferrites. (b) The phase difference \n\nΔ\nϕ\n\n as a function of the internal bias field \n\n\nH\n0\n\n\n.
Circulator, a nonreciprocal device, has been widely used in various microwave systems. Figure 7 schematically shows the function of a stripline circulator. The circulator is, in general, a three-port device. If the incident wave is injected from Port 1, then the wave will ideally go to Port 2, while Port 3 will be isolated as shown in Figure 7(a). On the other hand, if the wave is injected from Port 2, it will go to Port 3 and isolate from Port 1 as shown in Figure 7(b). There are three figures of merit for a circulator: transmission, reflection, and isolation. The transmission from Port 1 to Port 2 should be as high as possible, i.e., the insertion loss should be as small as possible. The reflection received at Port 1 due to the incident wave of Port 1 (S11) and the isolation from Port 1 to Port 3 (S13) should be as small as possible. The nonreciprocity of the circulator can be used to protect the oscillators from the damage of the reflected power in plasma or material processing systems. It can also be used to separate the transmitted and the received waves in radar or communication systems [13, 14, 15, 16, 17].
\nSchematic diagrams of the operation of a stripline circulator. (a) The incident wave is injected from Port 1 and the transmitted wave ideally goes to Port 2. (b) The incident wave from Port 2 will go to Port 3. The color spectrum is the electric field pattern inside the ferrite disks. It is a three-port, nonreciprocal device. A full-wave solver, high-frequency structure simulator (HFSS), is used [18].
In addition to the stripline circulator, there are other types such as the microstrip circulator and the waveguide circulator. The microstrip circulator is similar to the stripline circulator in many ways. Here we show a waveguide circulator which is capable of high-power operation. Figure 8(a) shows the structure of the nonreciprocal device and the simulated electric field strength. The simulation parameters are described in the caption. The circulator is, in general, a three-port device. If the incident wave is injected from Port 1, then the wave will ideally go to Port 2, while Port 3 will be isolated as shown in Figure 8(b). On the other hand, if the wave is injected from Port 2, it will go to Port 3 and be isolated from Port 1 as shown in Figure 8(b).
\n(a) Schematic diagrams of the operation of a waveguide circulator. A full-wave solver, HFSS, is used with the saturation magnetization (\n\n4\nπ\n\nM\ns\n\n\n) is 1600 G, the dielectric constant\n\n\nε\n′\n\n\n is 13.0, and the resonance linewidth \n\nΔ\nH\n\n is 10 Oe. The radius and thickness of the ferrite disks in rust red are R = 21.0 mm and t = 5 mm, respectively. The waveguide is a standard WR 340 with 86.36 × 43.18 mm2. The electric field pattern is displayed in color. (b) Simulation results of the waveguide circulator like the one in Part (a). The solid red curve is the transmission or insertion loss; the blue curve represents the reflection loss, and the black is the isolation.
The isolator is one of the useful microwave ferrite components. As shown in Figure 9, the isolator is generally a two-port device having unidirectional transmission characteristics (nonreciprocity). From Port 1 to Port 2 (S21), the forward transmission is high (i.e., low insertion loss in Figure 9(a)). However, from Port 2 to Port 1 (S12), the reverse transmission is low (i.e., high isolation in Figure 9(b)). Besides, the reflection (S11 and S22) should be as low as possible. The simulation parameters in Figure 9 are the same as Ex. 9.2 of Pozar’s textbook [5]. The simulation parameters and the sample’s geometry are described in the caption.
\nThe simulated field strength for a two-port isolator using the full-wave solver. (a) The high forward transmission (S21) and (b) the low reverse transmission (S12). The saturation magnetization (\n\n4\nπ\n\nM\ns\n\n\n) is 1700 G, the dielectric constant \n\n\nε\n′\n\n\n is 13.0, and the resonance linewidth \n\nΔ\nH\n\n is 200 Oe. The length and thickness of the ferrite are L = 24.0 mm and t = 0.5 mm.
An isolator is commonly used to prevent the high reflected power from damaging the precious and expansive microwave source. For example, the impedance of a plasma system changes a lot when the plasma is ignited. The radical change of the impedance will result in impedance mismatch and cause serious reflection which might kill the source instantly. An isolator can be used in place of a matching or tuning network. However, it should be realized that the reflected power will be absorbed by the ferrite of the isolator, as shown in Figure 9(b). When the ferrite absorbs the reflected energy, the temperature will rise and the performance will be compromised. Therefore, a simple isolator can be implemented by using a circulator with one port well terminated [19]. For example, if Port 3 in Figure 8(a) is matched with water load, the power injected from Port 2 will go to Port 3 and will be isolated from Port 1. The power handling capability can be improved.
\nCirculator and isolator can be implemented using the self-bias [20, 21, 22], just like the latch phase shifter (\n
Ferrimagnetism and ferromagnetism share many magnetic properties in common, such as hard and soft magnets, but the conductivity differentiates these two materials. Ferrites are ceramic materials and suitable for the high-frequency operation. The electromagnetic properties of ferrite materials are difficult to understand in that the magnetic susceptibility is a tensor and depends on the saturated magnetization \n
The complex permittivity \n
At high-power operation, the ferrite devices will be heated. The spin wave linewidth may be taken into account. Besides, the ferrites will become paramagnetism when the temperature exceeded the Curie temperature. These two factors are important for high-power operation, which are not considered in this chapter.
\nThis chapter was supported in part by the Ministry of Science and Technology of Taiwan and in part by China Steel Company/HIMAG Magnetic Corporation, Taiwan. The author is grateful to the Taiwan Branch of ANSYS Inc. for technical assistance and to Dr. Hsein-Wen Chao and Mr. Wei-Chien Kao for their assistance in the full-wave simulation. Dr. Hsin-Yu Yao and Mr. Shih-Chieh Su are appreciated for the discussion of the ferrites’ characterization.
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\n\n5.1 IntechOpen has a right to terminate this Publication Agreement for quality, program, technical or other reasons with immediate effect, including without limitation (i) if the Corresponding Author or any Co-Author commits a material breach of this Publication Agreement; (ii) if the Corresponding Author or any Co-Author (being an individual) is the subject of a bankruptcy petition, application or order; or (iii) if the Corresponding Author or any Co-Author (being a company) commences negotiations with all or any class of its creditors with a view to rescheduling any of its debts, or makes a proposal for or enters into any compromise or arrangement with any of its creditors.
\n\nIn case of termination, IntechOpen will notify the Corresponding Author, in writing, of the decision.
\n\n6. INTECHOPEN’S DUTIES AND RIGHTS
\n\n6.1 Unless prevented from doing so by events outside its reasonable control, IntechOpen, in its discretion, agrees to publish the Chapter attributing it to the Corresponding Author and any Co-Author.
\n\n6.2 IntechOpen has the right to use the Corresponding Author’s and any Co-Author’s names and likeness in connection with scientific dissemination, retrieval, archiving, web hosting and promotion and marketing of the Chapter and has the right to contact the Corresponding Author and any Co-Author until the Chapter is publicly available on any platform owned and/or operated by IntechOpen.
\n\n6.3 IntechOpen is granted the authority to enforce the rights from this Publication Agreement, on behalf of the Corresponding Author and any Co-Author, against third parties (for example in cases of plagiarism or copyright infringements). In respect of any such infringement or suspected infringement of the copyright in the Chapter, IntechOpen shall have absolute discretion in addressing any such infringement which is likely to affect IntechOpen's rights under this Publication Agreement, including issuing and conducting proceedings against the suspected infringer.
\n\n7. MISCELLANEOUS
\n\n7.1 Further Assurance: The Corresponding Author shall and will ensure that any relevant third party (including any Co-Author) shall, execute and deliver whatever further documents or deeds and perform such acts as IntechOpen reasonably requires from time to time for the purpose of giving IntechOpen the full benefit of the provisions of this Publication Agreement.
\n\n7.2 Third Party Rights: A person who is not a party to this Publication Agreement may not enforce any of its provisions under the Contracts (Rights of Third Parties) Act 1999.
\n\n7.3 Entire Agreement: This Publication Agreement constitutes the entire agreement between the parties in relation to its subject matter. It replaces and extinguishes all prior agreements, draft agreements, arrangements, collateral warranties, collateral contracts, statements, assurances, representations and undertakings of any nature made by or on behalf of the parties, whether oral or written, in relation to that subject matter. Each party acknowledges that in entering into this Publication Agreement it has not relied upon any oral or written statements, collateral or other warranties, assurances, representations or undertakings which were made by or on behalf of the other party in relation to the subject matter of this Publication Agreement at any time before its signature (together "Pre-Contractual Statements"), other than those which are set out in this Publication Agreement. Each party hereby waives all rights and remedies which might otherwise be available to it in relation to such Pre-Contractual Statements. Nothing in this clause shall exclude or restrict the liability of either party arising out of its pre-contract fraudulent misrepresentation or fraudulent concealment.
\n\n7.4 Waiver: No failure or delay by a party to exercise any right or remedy provided under this Publication Agreement or by law shall constitute a waiver of that or any other right or remedy, nor shall it preclude or restrict the further exercise of that or any other right or remedy. No single or partial exercise of such right or remedy shall preclude or restrict the further exercise of that or any other right or remedy.
\n\n7.5 Variation: No variation of this Publication Agreement shall be effective unless it is in writing and signed by the parties (or their duly authorized representatives).
\n\n7.6 Severance: If any provision or part-provision of this Publication Agreement is or becomes invalid, illegal or unenforceable, it shall be deemed modified to the minimum extent necessary to make it valid, legal and enforceable. If such modification is not possible, the relevant provision or part-provision shall be deemed deleted.
\n\nAny modification to or deletion of a provision or part-provision under this clause shall not affect the validity and enforceability of the rest of this Publication Agreement.
\n\n7.7 No partnership: Nothing in this Publication Agreement is intended to, or shall be deemed to, establish or create any partnership or joint venture or the relationship of principal and agent or employer and employee between IntechOpen and the Corresponding Author or any Co-Author, nor authorize any party to make or enter into any commitments for or on behalf of any other party.
\n\n7.8 Governing law: This Publication Agreement and any dispute or claim (including non-contractual disputes or claims) arising out of or in connection with it or its subject matter or formation shall be governed by and construed in accordance with the law of England and Wales. The parties submit to the exclusive jurisdiction of the English courts to settle any dispute or claim arising out of or in connection with this Publication Agreement (including any non-contractual disputes or claims).
\n\nLast updated: 2020-11-27
\n\n\n\n
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