The effect of Sb on the electrical and magnetic properties of Ni-Zn ferrites prepared by sol–gel autocombustion method

Polycrystalline Ni-Zn ferrites with a well-defined composition of Ni_0.4Zn_0.6Fe_2-xSb_xO₄ synthesized using sol–gel method. Morphological characterizations on the prepared samples were performed by high resolution transmission electron and field emission scanning electron microscopy. The powders were densified using microwave sintering method. The room temperature complex permittivity (ε′ and ε″) and permeability (μ′ and μ″) were measured over a wide frequency range from 1 MHz–1.8 GHz. The real part of permittivity varies as ‘x’ concentration increases and the resonance frequency was observed at much higher frequencies and there is a significant decrease in the loss factor (tanδ). The electrical resistivity and permeability of NiZn ferrites increased with an increase of Sb content. As the concentration of ‘x’ increases from 0 to 0.08 the saturation magnetisation decreases. The saturation magnetization (M_s)?≈?52.211 A.m²/Kg for x?=?0 at room temperature. The room temperature electro paramagnetic resonance (EPR) were studied.     

Synthesis and characterization of rare-earth elements substituted Ni-Zn ferrites

ABSTRACT

The spinel Ni_0.7Zn_0.3RE_0.04Fe_1.96O₄ ferrites with RE?Ce and Y are fabricated by the polyacrylamide gel method. The effect of rare earth ions substitution for iron ions on the structure, morphology, electromagnetic and microwave-absorbing properties of Ni-Zn ferrites are investigated by X-ray diffraction (XRD), transmission electron microscope (TEM), thermogravimetric and differential thermal analysis (TG-DTA) and HP8510B network analyzer, respectively. The results indicate that Ni_0.7Zn_0.3RE_0.04Fe_1.96O₄ ferrites form a pure cubic spinel structure, and Ce³⁺ and Y³⁺ substitute into the crystal lattice. Both the real ?′ and imaginary ?″ parts values of complex permittivity for Ni_0.7Zn_0.3RE_0.04Fe_1.96O₄ ferrites are higher than those for Ni-Zn ferrites in X-band (8.2-12.4 GHz) frequency range. When RE ions substitute for iron ions, the real μ′ and imaginary μ″ parts values of complex permeability for Ni_0.7Zn_0.3RE_0.04Fe_1.96O₄ ferrites decrease and increase, respectively. The microwave-absorbing properties are better for RE-substituted Ni-Zn ferrites than Ni-Zn ferrites. For Ni_0.7Zn_0.3Ce_0.04Fe_1.96O₄ ferrites, the minimum loss is ?12.5 dB at 18 GHz for an absorber thickness of 2 mm.     

An example of dB/dt characteristics of the dynamic magnetic loss parameter

Ferrite cores are used for power electronics circuits operating under excitation conditions of frequency ranging from hundreds of kilohertz to several tens of megahertz. Development of a mathematical model enabling the estimation of the power loss of ferrites in high‐frequency regions, contributes to the designing of power circuits. The power loss in high‐frequency regions consists of hysteresis and dynamic magnetic losses. Although hysteresis loss does not depend on the excitation frequency, the dynamic magnetic loss increases with increasing excitation frequency. The dynamic magnetic loss parameter, λ_f, has been introduced for identifying the power loss in high‐frequency regions. In this study, λ_f − dB/dt characteristics are experimentally obtained by a newly introduced hysteresis model that uses a normal distribution function. It is found that the λ_f − dB/dt characteristics do not depend on the size of B − H loops. The instantaneous power loss of a ferrite core in the high‐frequency region can be calculated using the λ_f − dB/dt characteristics. Copyright © 2007 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

Crystal band-stop filters with improved spurious resonance behaviour

It is a well known fact that piezoelectric band-stop filters manifest not only the desired stopband at the fundamental resonant frequency of the crystal, but also additional stopbands at its spurious resonant frequencies. The greater the degree to which such parasitic attenuation peaks are to be suppressed, the more complex is the required circuitry. The most common crystal band-stop filter circuits can be considered as all-pass sections in which a capacitance is ‘disturbed’ over a narrow bandwidth by a resonant circuit. If this ‘disturbance’, possibly after conversion by means of known one-port equivalents, consists of a high impedance series resonant circuit with the inductance L_s in parallel to the entire lattice arm R/jx₁, the 3 dB bandwidth of the stopband becomes Δω₀= R/L_s(1+x) in the loss-free case. Sometimes the dual representation of the ‘disturbance’ is more appropriate. With a finite crystal Q-factor of Q_q, the insertion loss of a desired or parasitic stopband assumes the finite maximum value ǎ ～ In (1+Q_qΔω0/ω_∞). The width and height of the possible attenuation peaks are given in formulas, normalized curves and worked examples for seven band-stop filter circuits of increasing complexity, each with a single crystal.     

添加Co_2O_3对NiCuZn铁氧体磁性能的影响

采用传统氧化物陶瓷工艺制备NiCuZn铁氧体。利用扫描电子显微镜、阻抗分析仪、磁滞回线分析仪,分别对样品的微观形貌、复数磁导率频谱、静磁性能和高频功耗进行了观察和测试。结果表明,在0~0.12wt%的范围内,随Co2O3添加量的增大,样品的平均晶粒尺寸略有减小,起始磁导率逐渐下降,截止频率逐渐升高。在3MHz、10m T、25~140℃条件下,随着Co2O3添加量的增加,由于截止频率逐渐升高,磁导率虚部在高频下得到抑制,剩余损耗降低,导致磁芯功率损耗单调减小。     

Complex permittivity and permeability of Co-substituted NiCuZn ferrite at rf and microwave frequencies

NiCuZn ferrite has recently attracted a lot of attention for its application in high frequency (up to a few GHz) multilayer chip inductors (MLCIs) and for other microwave devices owing to their favorable electromagnetic properties and low densification temperature. In order to study the effect of substitution of cations by cobalt in small concentration on the dielectric and magnetic properties at low and high frequencies, bulk polycrystalline ferrite samples of starting composition (Ni_0.2Cu_0.2Zn_0.6)_{1 − x} Co _x Fe₂O₄, having x = 0, 0.01, 0.03 and 0.05, were prepared by citrate precursor method. Pure spinel (cubic) ferrite formation was confirmed by powder X-ray diffraction technique. Complex permittivity and permeability were measured at microwave frequencies (X-band) using the cavity perturbation method, which is a non-contact method. The values of real part of permittivity (ε ′) vary in the range 7–9.6 and of the imaginary part (ε ″) vary from 0.020–0.120, whereas real part of the permeability (μ′) lies in the range 2.6–14.0 and the imaginary part of permeability (μ″) varies from 0.5–6.0. It is observed that there is an increase in μ′ and decrease in the magnetic loss (tan δ _μ) on increasing the cobalt concentration from x = 0 to x = 0.05. The variation of these parameters, both with frequency in X-band and with the cobalt concentration, is discussed in this paper.     

Impedance and elektrodynamic force acting on a cylindrical conductor partly filling a semi-closed slot

Contents In this paper the inner impedance and the electrodynamic force acting on a cylindrical conductor partly filling a semi-closed slot are calculated. The investigations are made by using the separation of variables method.

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Übersicht Es werden die innere Impedanz und die elektrodynamische Kraft, die auf einen kreisförmigen Leiter wirkt, der eine halbgeschlossene Nut partiell ausfüllt, berechnet. Zur Berechnung wird die Methode der Trennung der Variablen angewendet. Die Ergebnisse sind in Kurvenform dargestellt.

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Symbole A z-component of the vector potential (complex r.m.s. value) - B _x, B_y components of the magnetic induction (complex r.m.s. value) - F electrodynamic forece - I current (r.m.s. value) - I _n(z) modified Bessel function of first kind - imaginary unit - J z-component of the current density (complex r.m.s. value) - L(o) inductance for direct current - R resistance - X reactance - Z impedance - z ^* conjugate complex number ofz - Rez, Imz, |z| real part, imaginary part and modulus of complex numberz - ₀ magnetic permeability - pulsation - Kronecker symbol - 1 atn=m o otherwise     

Measurement of dynamic magnetic loss parameter in ferrites

The measurement of the iron loss in ferrites is important for developing high‐efficiency switching power supplies. The authors have proposed the dynamic magnetic loss parameter, λ_f, for evaluating the iron loss in ferrites. In previous studies, the parameter was assumed to be a constant value for an individual ferrite material and defined for one period of a small B–H loop. In this paper, assuming that λ_f is a function of the time derivative of the magnetic flux density, dB/dt, a novel measurement method of λ_f of a Ni‐Zn ferrite is proposed using rectangular wave voltage excitation and the Fourier expansion of the exciting current. In order to realize an iron loss measurement system with the rectangular wave voltage excitation, a high‐frequency FET inverter has been developed. The results of measuring λ_f show that it is uniquely determined by dB/dt regardless of the B–H loop size. The measured dB/dt characteristics of λ_f are applied to practical examples for switching power supplies and sinusoidal wave voltage excitations. Their experimental and computational results coincide and it is clarified that the measured dB/dt characteristics are effective and useful. © 2006 Wiley Periodicals, Inc. Electr Eng Jpn, 156(1): 1–6, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20101     

A design guideline of parasitic inductance for preventing oscillatory false triggering of fast switching GaN‐FET

Gallium nitride field‐effect transistors (GaN‐FETs) are attractive devices because of its low on‐state resistance and fast switching capability. However, they can suffer from false triggering caused by fast switching. Particularly, a disastrous oscillation of repetitive false triggering can occur after a turn‐off, which may deteriorate the reliability of power converters. To address this issue, we give a design guideline to prevent this phenomenon. We analyze a simple circuit model to derive the condition of occurrence of this phenomenon, which is then verified experimentally. Results show that the parasitic inductance of the gating circuit, L_g, and that of the decoupling circuit, L_d, should be designed so that the LC resonance frequency of L_g and the gate–source capacitance of the GaN‐FET does not coincide with that of L_d and the drain–source capacitance, respectively. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

Magnetic properties of Ni-Zn ferrites prepared by microwave sintering method

The effect of zinc ion substitution for nickel on structural and magnetic properties of NiZn ferrites is reported. The spinel ferrite system Ni_1-xZn_xFe₂O₄ with x = 0.2, 0.3, 0.4 and 0.5 was prepared by microwave sintering method. The uniaxially pressed samples were sintered at various temperatures such as 900°C, 1000°C and 1100°C for 30 min. X-ray diffraction patterns of the samples indicate the formation of single-phase cubic spinel structure. SEM micrographs show that grain size increases with increasing zinc content and sintering temperature. The elemental composition of these ferrites was analyzed by EDS. Lattice constant increases with increase in zinc content, obeying Vegard’s law. The effect of composition and sintering temperature on initial permeability as the function of frequency and temperature was studied. The initial permeability of NiZn ferrite increases greatly with increasing Zn content and sintering temperature. The dependence of initial permeability with respect to temperature shows the decrease in the Curie point with increase in zinc content, is the normal behavior of ferrites. The relative loss factor (tand

Preparation, thermal stability and permeability behavior of substituted Z-type hexagonal ferrites for multilayer inductors

Co₂Z-type hexagonal ferrites with iron excess Ba₃Co_2???x Fe_24?+?x O₄₁ (0?≤?x?≤?0.8) and deficiency Ba₃Co_2?+?y Fe_24???y O₄₁ (0?≤?y?≤?0.6) were prepared by an oxalate coprecipitation technique. This synthesis route leads to almost single phase Z-type ferrites for x?=?0 after calcination and sintering at 1330 °C. The Z-type formation is enhanced for x?>?0 and single phase ferrites are obtained for 0.4?≤?x?≤?0.8. The permeability of Z-type ferrites varies with composition x: Maximum permeability of μ′?=?28 is observed for 0.4?≤?x?≤?0.6 for samples sintered at 1330 °C. The frequency dispersion shows broad peaks of μ″ stretching from 200 MHz to >1 GHz. For iron deficient samples 0?≤?y?≤?0.6 multi-phase mixtures were obtained. For Ag-based multilayer inductor applications sintering at 950 °C is required. Co₂Z ferrites with Fe excess are not stable at this temperature as demonstrated by XRD. The permeability of samples sintered at 950 °C is drastically reduced to μ′?=?3. This demonstrates that these materials are not able to provide sufficient permeability for multilayer inductors for high-frequency operations since they are not compatible with the low temperature ceramic cofiring technology.     

铁基软磁合金薄带高频磁导率测量方法研究

给出一种铁基软磁合金薄带高频磁导率的测量方法,详细介绍了该方法的测量原理,并利用Agilent E4991A射频阻抗/材料分析仪研究了退火温度对20μm厚铁基软磁合金薄带高频磁导率的影响。结果表明,随退火温度的提高,磁导率实部单调提高,磁导率虚部则是先升高后降低。尤其是在550℃下退火,在1MHz和10MHz下样品磁导率实部分别为2210和330;比较了不同温度退火样品的磁导率的测量结果,得知550℃是一个比较理想的退火温度。研究结果对以铁基非晶、纳米晶软磁合金薄带为磁心的高频微电感、微变压器等磁性器件的设计具有重要的指导意义。     

On a mathematical model of squirrel-cage induction motors

Contents This paper presents a mathematical model of 3-phase squirrel-cage induction motor accounting for a very high number of space harmonics in which, due to application of a special transformation of voltages and currents, a differential equation system with constant coefficients is obtained. The number of space harmonics is so high that it is possible to perform a direct computation of the alternating component of the electromagnetic torque which decides on the parasitic synchronous torque.An application example of this model for the analysis of steady and dynamic states of a concrete squirrel-cage motor is given.

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Über ein mathematisches Modell des Käfigläufermotors

Übersicht Im Artikel wird ein mathematisches Modell dreiphasiger Käfigläufermotoren mit Berücksichtigung einer großen Zahl räumlicher Harmonischer vorgestellt. Unter Anwendung einer speziellen Transformation der elektrischen Spannungen und Ströme erhält man ein Gleichungssystem mit konstanten Koeffizienten Die Zahl der räumlichen Harmonischen ist so groß, daß die Berechnung der Wechselkomponenten des elektromagnetischen Moments und damit des parasitären synchronen Moments möglich ist. Die Anwendung des mathematischen Modells wird anhand eines Beispiels für dynamischen und statischen Betrieb vorgestellt.

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List of symbols and abbreviations N number of bars of the rotor cage - L ₈ total self inductance of stator phase winding - M ₈ mutual inductance of stator windings - R _rg,L _rg resistance and leakage inductance of a rotor end ring segment-respectively - R _b,L _b resistance and leakage inductance of a rotor cage bar-respectively - k=0,1, ... N—1 number of the rotor currents symmetric component and theM _sr matrix column corresponding to it - l equivalent axial length of stator core - equivalent width of air-gap - p pole pair numer - z number of turns per phase - v order of harmonic - k _sv stator winding coefficient for thev-th harmonic - k _rv rotor winding coefficient for thev-th harmonic - k _skv skewness coefficient for thev-th harmonic - U _s stator voltage vector in symmetrical components - U _r rotor voltage vector in symmetrical components - i _s stator current vector in symmetrical components - O zero matrix - R _s stator resistance matrix - R _r rotor resistance matrix - LL _s stator inductance matrix - L _rr rotor inductance matrix - M _sr matrix of stator-rotor mutual inductances - T _e electromagnetic torque - T _m motor load torque - rotor position angle - ₀ initial rotor position angle - rotor angular velocity - (*) conjugation index of a complex number - Re {y} real part of a complex number - Im {y} imaginary part of a complex number - T transposition index - [X] ^x highest integer not greater thanX and of the same sign asX - U r.m.s. value of the phase voltage - _u pulsation of supply voltage     

Formulation and mixing conditions of magnetic composite materials used for magnetic wedges in an interior permanent magnet synchronous motor

An interior permanent magnet synchronous motor (IPMSM) is characterized by its high efficiency. However, torque ripple and loss occur because of the spatial harmonics generated by air-gap permeance fluctuations. This study clarified that slot harmonic components can be suppressed by inserting magnetic wedges made of magnetic composite material in the slot opening of an IPMSM. First, the authors examined the torque and loss characteristics by varying relative permeability and saturation magnetic flux density of magnetic wedges using finite element analysis (FEA). Results indicated that the torque ripple and loss could be suppressed when the relative permeability of the magnetic wedge was μ_r = 10 to 22 and the saturation magnetic flux density was B_s = 0.5 to 0.75 T. Furthermore, the authors produced magnetic composite materials made of various soft magnetic material powders and examined their magnetic properties and viscosities. Results showed that a magnetic composite material with optimal magnetic properties and injectable viscosity could be manufactured using Fe-Si-Al with a low volume fraction.     

An induction motor model with deep-bar effect and leakage inductance saturation

Contents A two-phase dynamic induction motor model is derived which includes the effects of rotor bar eddy currents and leakage inductance saturation. The model is based on approximate analysis of the air-gap and slot electromagnetic fields which leads directly to lumped circuit equations. New expressions are given for the variation of the zig-zag and slot leakage inductance with saturation. — The model is applied to the calculation of both steady state and dynamic characteristics for a large induction motor.

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Ein Modell des Asynchronmotors mit Hochstabläufer-Effekt und Sättigung der Streuinduktivität

Übersicht Es wird ein zweisträngiges Modell eines Asynchronmotors abgeleitet, das Wirbelströme in den Läuferstäben und die Sättigung der Streuinduktivität berücksichtigt. Grundlage des Modells ist eine angenäherte Analyse der Felder im Luftspalt und in den Nuten, die direkt auf Gleichungen mit konzentrierten Elementen führt. Es werden neue Ausdrücke für die Abhängigkeit der doppeltverketteten und der Nutstreuung von der Sättigung angegeben. Das Modell wird für die Berechnung stationärer und dynamischer Vorgänge bei einem großen Asynchronmotor angewendet.

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List of Symbols A magnetic vector potential - B flux density - G describing function - g air-gap length - H magnetic field intensity - I _Om rotor saturation current - I _r rotor current matrix - l _r ^j current in thejth rotor slot - l _rm amplitude of the fundamental ofI _bir ^j distribution - I _rr rotor harmonic conceptual current matrix - I _s stator current matrix - I _ss stator harmonic conceptual current matrix - i _mne m, nth modal eddy current in thejth rotor slot - i _nb nth harmonic conceptual current at the interface betweenjth rotor slot and slot neck regions - i _n ^rj nth harmonic conceptual current at the opening of thejth rotor slot - i _n ^sj nth harmonic conceptual current at the opening of thejth stator slot - i _Ob rotor zero order conceptual current matrix - i _Oi ^j conceptual current at thejth rotor slot due to daturation of the tooth tips - K _nw nth harmonic winding distribution factor effective length of the motor - L _nc self inductance of the eddy current modei _One ^j - L _r rotor inductance matrix - L _s stator inductance matrix - P _p pole pitch - M _ne mutual inductance betweeni _One ^j andI _r ^j and rotor modal eddy currents - M _ri mutual inductance matrix between rotor winding and rotor conceptual winding due to saturation of the tooth tips - M _rr mutual inductance matrix between rotor winding and rotor conceptual winding - M _rss mutual inductance matrix between rotor winding and stator conceptual winding - M _si mutual inductance matrix between stator winding and rotor conceptual winding due to saturation of the rotor tooth tips - M _sr mutual inductance matrix between stator and rotor windings - M _srr mutual inductance matrix between stator winding and rotor conceptual winding - M _ss mutual inductance matrix between stator winding and stator conceptual winding - N ₁,N ₂ number of stator and rotor slots - R _ne resistance of eddy current modei _One ^j - R _r rotor resistance matrix - R _s stator resistance matrix - s ₁,d ₁ dimensions of a stator slot - s ₂,d ₂ rotor slot opening and rotor slot neck height - s _b ,d _b dimensions of a rotor bar - s _p slot pitch - N _c number of turns per slot - U _s applied voltage matrix - q number of slots per pole phase group - rotor flux linkage matrix - y_s stator flux linkage matrix - , inductance coefficients - O angular position - conductivity of rotor bars - eigenvalues - resistivity of rotor bars     

Mg_(0.6)Cu_(0.1)Zn_(0.3)Fe_(2+x)O_(4+3/2x)铁氧体的结构和磁性能

分别采用过铁、正铁和缺铁配方通过固相反应法制备MgCuZn铁氧体,分析了Fe3+对铁氧体的磁性能和烧结特性的影响。微量缺铁有助于促进烧结并改善磁性能,过铁情况下,饱和磁化强度随x值增大迅速下降,在x=0.06处下降至38.84 A·m2/kg,相应的磁导率下降,截止频率向高频移动。并研究了微量V2O5掺杂对改善磁性能的作用,在掺杂量为0.4wt%处获得虚部损耗的有效提升(截止频率处提升近30%)。在此基础上探讨了MgCuZn铁氧体用作抗EMI磁珠的可行性,其低廉的价格相较于传统的Ni Zn/Ni Cu Zn铁氧体具有明显的优势。     

Inner Intrinsically Safe Criterion and Design Considerations of Buck Converter Based on Equivalent Inductance

To obtain the intrinsic safety criterion of the Inner-intrinsic-Safety-Buck-Converter (ISBC) based on equivalent-inductance to guide its optimal design, the Inductor Disconnected Discharge (IDD) behavior of ISBC is researched, and the Equivalent-Simple-Inductive-Circuit (ESIC) based on the equivalent inductance is defined. According to ESIC based on the equivalent-current and the energy equivalence, the expression of the equivalent inductance L_ei is derived. The effect of the inductance L, the capacitance C, the input voltage V_i and the load resistance R_L on L_ei is deeply studied, and it’s pointed out that L_ei increases with the increase of V_i, L, C, and R_L, respectively. According to the minimal ignition current curve of the Simple-Inductive-Circuit (SIC), the nonexplosive intrinsically safe criterion based on L_ei is proposed. Combining the proposed criterion, the output ripple voltage and the expression of L_ei, the design region of the capacitance and inductance of the Buck converter meeting the demands of both inner intrinsic safety and electric performance is obtained. The feasibility of the design method and the proposed criterion is proved by the simulation and experimental results.     

Analysis of transients in electrical circuits containing solenoids with initially magnetized cores for impulse excitation

Contents The paper is concerned with the solution of transients in electrical circuits containing elements with the pronounced magnetic permeability. In the proposed method, the circuit and field equation are solved simultaneously without using the equivalent circuit idea, adjustments are made for circuit current after each time step. The elaborated method is developed to devices with D.C. initial magnetization, and an approximate method of solving transients in electrical circuits containing elements with D.C. initial magnetization has been proposed.

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Die Analyse nichtstationärer Vorgänge bei der Impulserregung von elektrischen Schaltungen, die vormagnetisierte Eisendrosselspulen enthalten

Übersicht In diesem Beitrag wird eine Methode zur Berechnung nichtstationärer Vorgänge in elektrischen Schaltungen mit nichtlinearen, vor magnetisierten Eisendrosselspulen beschrieben. Vorgestellt wird das Iterationsverhalten zur Bestimmung nichtstationärer Vorgänge des Eingangsstromes im Zuge der gleichzeitigen Auflösung der Maschen- und Feldgleichungen. Die Berechnungen werden unter Annahme der Ersatzkennlinie fürB=f(H) durchgeführt.

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List of principal symbols d thickness of core, m - D average diameter of core, m - E(x,t) electric field strength, V/m - h height of core, m - H(x, t) magnetic field strength, A/m - H _i initial value of magnetic field strength, A/m - i(t) current in circuit, A - I _i initial current in circuit, A - I _st steady-state value of current, A - J(x, t) eddy-current density, A/m² - L external inductance, H - L ₃ inductance corresponding to external flux, H - R _c resistance of exciting winding, - R ₁,R ₂ external resistances, - u(t) supply voltage, V - t time, s - t _st time after which transients decay, s - w, w ₁ number of turns of exciting windings - x space axis, m - t time increment, s - x space increment, m - D.C. direct current - core conductivity, S/m - _d differential permeability, H/m     

Additional no-load losses in inverter-fed high-speed cage induction motors

An analysis of additional high-frequency losses in high-speed cage induction motors with closed rotor slots under sinusoidal and inverter supply at no load is presented. Calculated losses are compared with experimental results obtained by measurement on a 270 kW, 16,200/min, prototype motor, at both the delta and star connections of the stator winding. The main additional losses of the prototype motor investigated due to inverter supply are eddy current losses in the stator winding. Both measurement and calculation show that, at the delta connection, the additional losses are higher due to the lower modulation degree of the inverter, causing higher harmonic voltage amplitudes.List of symbols B magnetic flux density, T - b _sheet width of iron sheet, m - b _bar mean width of rotor bar, m - b _sd mean stator tooth width, m - b _Qm mean slot width, m - b _LT square wire side, m - f frequency, Hz - h _bar rotor bar height, m - h _br rotor slot bridge height, m - h _sy stator yoke height, m - I current (rms value), A - k ordinal number of voltage harmonic - k _C Carters factor - k _dHy hysteresis deterioration factor for teeth - k _dFt eddy-current deterioration factor for teeth - k _Fe iron stack fill factor - K _l1k first-order skin effect coefficient of stator inductance for kth voltage harmonic - K _l2k second-order skin effect coefficient of stator inductance for kth voltage harmonic - K _1k first-order skin effect coefficient of stator resistance for kth voltage harmonic - _2k second-order skin effect coefficient of stator resistance for kth voltage harmonic - k _yHy hysteresis deterioration factor for yoke - k _yFt eddy-current deterioration factor for yoke - k _w1 winding factor for fundamental space harmonic - l _b length of winding overhang, m - l _Fe core length, m - L _dc inductance at d.c. current, H - L _r rotor inductance, H - L _s stator inductance, H - M torque, Nm - m _T number of vertically arranged coils in slot - m _TZ number of vertically arranged conductors in slot - n _N rated speed, 1/min - p number of pole pairs - p _Ft eddy current losses per iron mass at 50 Hz, 1 T, W/kg - p _Hy hysteresis losses per iron mass at 50 Hz, 1 T, W/kg - P _Cu copper losses, W - P _Fe iron losses, W - P _Fe,Hy hysteresis losses, W - P _Fe,Ft eddy-current iron losses, W - P _fr friction and windage losses, W - Q _r number of rotor slots - Q _s number of stator slots - R _s stator phase resistance, - R _r rotor phase resistance, - R _dc resistance at d.c. current, - s slip - s _k slip for kth harmonic - U voltage (rms value), V - V volume, m³ - N number of turns per phase - z ₁ average number of conductors in one turn, lying side by side - z ₂ average number of vertically arranged conductors per turn - air-gap length, m - magnetic flux, Wb - electric conductivity, S/m - permeability ( ₀=4×10^–7 H/m: permeability of vacuum), H/m - _r relative permeability - _Fe mass density, kg/m³ - _s leakage coefficient for stator winding - _p pole pitch, m - _sd stator tooth pitch, m G. Joksimovi is on research leave from University of Montenegro, Montenegro, Yugoslavia