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In this paper, a complete sensitivity analysis of the optimal parameters for the axial flux permanent magnet synchronous machines working in the field weakening region is implemented. Thanks to the presence of a parameterized accurate analytical model, it is possible to obtain all the required parameters of the machine. The two goals of the ideal design are to maximize the power density: $P_{\text{density}}$ and the ratio of maximal to rated speed: $n_{\max}/n_r$ , which is an inductance related parameter keeping the efficiency at the target speed above 90 $\%$ . Different slots/poles/phases combinations are studied to reveal the optimum combination for each phase. This paper has studied the effect of the ratio of number of stator slots to number of rotor poles on the $P_{\text{density}}$ and $n_{\max}/n_r$ . It is shown that a low value of this parameter results in a better $P_{\text{density}}$ and $n_{\max}/n_r$ . The effect of the outer diameter, and the inner to outer diameter ratio are studied with respect to the two design goals. In addition, a comparison between the finite and the theoretical infinite speed designs is implemented. A complete 3D finite element validation has proven the robustness of the analytical model.

The Superconducting (SC) machines have the merits of higher power and torque density due to the fact of higher electrical or magnetic loading. Especially, the field modulation SC machine (FMSCM) recently attracts more and more attention from scholars, since this kind of SC machines successfully overcome the drawbacks of the traditional SC synchronous machines, such as the rotating seal of the coolant and the electrical brush-slip-ring for current transfer. However, the research method of the FMSCM mainly adopts the finite element analysis (FEA). Although the FEA can provide accurate results, the time consumption is huge, particularly at the initial design of the large capacity FMSCM, the time cost will be unacceptable. Aiming at solving this problem, this paper attempts to establish the mathematical analysis model of a double-stator field modulation HTS machine (DSFM-SCM) using a new theory of electrical machine, i.e. general airgap field modulation theory for the first time, to meet the requirement at the initial design, and sufficiently exhibit the modulation effect of geometric structure. Both the FEA and experiments are carried out on the DSFM-SCM, to verify the effectiveness of the proposed mathematical analysis model.

This paper proposes a monolayer Halbach array based permanent magnet spherical motor (PMSpM) using cubic permanent magnets (PM). The research aims to provide a fast computing model to calculate the magnetic field for this PMSpM, which helps for future PMSpM optimization and control study. Three thickness equivalence principles are provided and compared for cubic PMs. The magnetic scalar potential is introduced, and the spherical harmonics expansion is implemented for rotor magnetic field analysis. Engineering revision is used to reduce the Legendre function deviation in polar angle direction. Torques and forces are deduced using the Lorentz force law, and relatively simple models of this PMSpM for both magnetic fields and torques are acquired. In addition, a prototype motor has been developed, and the analytic results of air-gap flux density and torque are compared with their finite element simulation and experimental results, which are proved consistent. It is concluded that the fundamental component of the spherical harmonics can be used to present the magnetic field model, which uses fewer integrals than the Biot-Savart law based solution, and the modeling results of the magnetic field flux density, forces and torques are in line with the expectations. The analytic models are proved correct.

In this paper, a coupled finite-element/state space modeling technique is applied in the determination of the steady-state parameters of a 733 MVA turbogenerator in the abc frame of reference. In this modeling environment, the forward rotor stepping-finite element procedure described in a companion paper is used to obtain the various machine self and mutual inductances under short-circuit and load conditions. A fourth-order state-space model of the armature and field winding flux linkages in the abc frame of reference is then used to obtain the next set of flux linkages and forcing function currents for the finite-element model. In this process, one iterates between the finite-element and state-space techniques until the terminal conditions converge to specified values. This method is applied to the determination of the short-circuit, and reduced- and rated-voltage load characteristics, and the corresponding machine inductances. The spatial harmonics of these inductances are analyzed via Fourier analysis to reveal the impact of machine geometry and stator-to-rotor relative motion, winding layout, magnetic saturation, and other effects. In the full-load infinite-bus case, it is found that, while the three-phase terminal voltages are pure sinusoidal waveforms, the steady-state armature phase currents are nonsinusoidal and contain a substantial amount of odd harmonics which cannot be obtained using the traditional two-axis analysis. >

This first of two companion papers centers on applying a coupled finite-element/state-space technique to the determination of self and mutual winding inductances of a 733 MVA turbogenerator and computing its open-circuit characteristic, in the natural abc frame of reference. In this method, the apparent self and mutual inductance profiles of the armature and field windings, expressed as functions of rotor position angle, are computed from a series of magnetic field solutions performed at uniformly-distributed samples of rotor positions covering the entire 360/spl deg/ electrical cycle, using the energy perturbation method. These inductances, which are obtained at no-load for three different excitation levels, include the full effect of space harmonics introduced by the magnetic circuit geometry, winding layouts and magnetic circuit saturation. The abc-frame/finite-element computed open-circuit characteristic is in excellent agreement with the test results. This computed no-load set of parameters forms the initial data for simulation of the full-load performance given in the companion paper, including the full impact of space harmonics and saturation on the flux linkage, current and voltage waveforms, and other performance parameters. >

Fractional-slot permanent magnet (PM) machines with an odd number of stator slots are attractive candidates for industrial applications due to their high torque density and low torque ripple. However, the unbalanced magnetic force is detrimental to the bearing and yields noise and vibration. This paper determines the existence criterion of UMF on both open-circuit and on-load conditions by analysis of air gap field and magnetic force waves. Only when the machine periodicity is one, will the open-circuit UMF be generated due to the mutual interaction between the modulated PM harmonics and the original PM harmonics. In addition, the existence of on-load UMF in machines with a periodicity of one is confirmed even when the slotting effect is ignored. Furthermore, among the five investigated machines with a given odd slot number of 9 but different pole numbers, the UMF in a machine with two poles is the smallest since the magnitude of the 2nd harmonic in the armature reaction is the smallest compared with other contributing harmonics. Meanwhile, the UMF in a machine with pole number 2 p = N s−1 is the largest due to the relatively larger magnitude of the ( p + 1)th armature reaction harmonic and the additive effect between the first order radial and tangential stresses.

This paper describes an optimal method of evaluating the synchronous reactance of the d - and q -axes simultaneously with rotor angle measurement. First, interactions among the variables such as the synchronous reactance of the d - and q -axes, the field current and the rotor angle through magnetic saturation are theoretically reviewed with a standard machine model. Next, with the steady-state test data, optimal values of Xd and Xq are evaluated simultaneously using an optimization technique to minimize the sum of 2-norm errors between the calculated and measured values in the field current and the rotor angle. The proposed method has been verified against both the 282 MVA salient-pole and 612 MVA round-rotor generators. The optimal value of the Xq / Xd ratio was found to be much smaller than that of manufacturer's data, and mismatches between the measured and calculated values in both the rotor angle and field current were significantly reduced. This systematic method is so simple that field engineers would find it easy to use. Thus, the rotor angle measurement makes it possible to derive the optimal values of Xd and Xq simultaneously, and the use of high-precision speed pickup sensor from all the generators is essential for improving machine model accuracy.

The magnetic potential drop in nonlinear iron can be replaced by equivalent current sheets on linear iron edges. Based on this principle, this paper proposes a nonlinear subdomain and magnet circuit hybrid model (SMCHM) for open-circuit field prediction in surface-mounted permanent-magnet machines. The values of current sheets are calculated from a magnetic circuit and used as boundary conditions in the subdomain model. The finite element (FE) validation is performed, which shows the SMCHM can accurately predict the magnetic field distribution and electromagnetic performance as nonlinear FE method does. The accuracy of SMCHM is significantly higher than the conventional subdomain model neglecting a nonlinearity effect, and the computation speed of SMCHM is dramatically faster than the FE model.

This paper presents a two-dimensional (2-D) analytical model for plannar linear permanent-magnet (PM) synchronous motors. The general and particular solutions are presented for a slotless single-sided linear PM synchronous motor as a case study. The governing partial differential equations of the motor are formulated for seven regions: mover side exterior, mover back iron, PMs, air-gap, winding, stator back-iron, and stator-side exterior. Four different magnetization patterns, i.e., parallel, ideal Halbach, 2-segment Halbach, and bar magnets in shifting directions magnetization patterns are considered to calculate the tangential and normal components of the open-circuit magnetic flux density and armature reaction of the motor under the study. The winding-induced voltage, flux linkage, self- and mutual inductances, and normal and tangential components of forces are computed. The validity of the proposed technique is confirmed by comparing the analytical results with the results obtained from a 2-D finite-element analysis.

The paper describes a general model of a hybrid-excited synchronous machine with multiple voltage levels each generated by a 3 -phase stator system in a voltage-behind-reactance formulation. The different voltage level of each 3 -phase stator system is realized in the same machine design. The rotor is excited through an excitation winding and a permanent magnet. The model incorporates magnetic saturation of the main inductance, magnetic and electric coupling between all phases, and the field winding and a mechanical displacement angle. The cross coupling between q and d axes of the main inductances is included as a constant saliency factor $\alpha$ for all saturation levels. The turn ratio of each 3 -phase stator winding system to another 3 -phase system is also modeled as a constant value. Each 3 -phase system saturates the iron core with a different weighting represented by the number of turns relative to the reference stator system. The proposed model is included in a field-oriented control scheme. Simulations are carried out with requested $\boldsymbol{I_{q,d}}$ and field winding currents to demonstrate model's dynamics, and the results are verified with measurements on a test bench.

The partitioned stator switched-flux permanent magnet (PM) (SFPM) (PS-SFPM) machine is a novel stator-PM machine in which PMs and armature windings are separately placed in two different stators. Compared with the conventional SFPM machine with a single stator in which both PMs and armature windings are located, the PS-SFPM machine can generate higher torque density due to the utilization of the machine's inner space. In this paper, the magnetic gearing effect in the PS-SFPM machine is investigated in terms of air-gap field harmonics based on a simple magnetomotive force (MMF)-permeance model. It is found that the PS-SFPM machine is one type of magnetically-geared machine, in which the modulations of rotor iron pieces to the open-circuit PM and armature reaction fields are similar to those found in the magnetic gear and the conventional magnetically geared machine. The pole-pair numbers and rotating speeds of the air-gap field harmonics obtained by the MMF-permeance model are verified by finite element (FE) analysis. More than 93% of the average electromagnetic torque is contributed by the main air-gap field harmonics having pole-pair numbers $(2i - 1){\boldsymbol p}$ PM and $\rm{| N}_{\rm{r}} \,\pm \, $ (2 i -1) p PM| ( i = 1,2,3) in a PS-SFPM machine with p PM-pole-pair PMs and Nr -rotor-pole. A prototype machine is fabricated and tested to verify the FE analysis.

This paper investigates the terminal voltage distortion in the interior permanent magnet machines with fractional slot concentrated windings, with particular emphasis on the influence of stator slot Ns and rotor pole 2 p combinations. The 12-slot/10-pole machine is first employed to show the voltage distortion phenomenon. Using the frozen permeability method, the mechanism is then investigated, which reveals that the variation of armature flux paths due to rotor saliency is the origin of such phenomenon, especially when the current advancing angle $\beta $ approaches 90°. Further, the terminal voltage distortion of the six machines are compared, i.e., ${N_s} \pm 1 = 2p$ (9-slot/8-pole, 9-slot/10-pole), ${N_s} \pm 2 = 2p$ (12-slot/10-pole, 12-slot/14-pole), ${N_s}/2p = 3/2$ (12-slot/8-pole), 3/4 (12-slot/16-pole), which shows that the machines with ${N_s} suffer more severe voltage distortion than their counterparts with ${N_s} > 2p$ . In addition, considering the inevitable machine saturation, a design trade-off is proposed by selecting proper ${N_s}/2p$ combinations to minimize the influence. Finally, prototypes are fabricated and tested to validate the analyses.

De Mello and Hannett [1] have theoretically proposed a method for determining the operational inductances Ld(j2w), Lq(j2w) of a synchronous machine at low values of generator speeds. Eitelberg and Harley [2] extended these ideas to obtain the inductances and time constants from the Ld(j2w), Lq(j2w) curves by a numerical procedure which allows one to attach different levels of importance to either the magnitude or phase fitting of the operational inductance expressions to the computed curves. The first part of this two-part paper extends these ideas [1-2] by proposing an improved MODIFIED data-processing method which uses a Fourier analysis of the open-phase voltage and short-circuit current to compute the operational inductances Ld(j2w), Lq(j2w) and avoids integration of the d,q axes flux linkages; providing that the fundamental and the third harmonic present in the recorded waveforms are considered. The second part [3] of this paper applies the improved MODIFIED data-processing method to a 3 kVA, 220 V microalternator.

This paper presents a closed-form solution of Maxwell's equations in squirrel-cage induction motors. The solution is obtained for a cylindrical multilayer geometry. The squirrel cage is represented by an equivalent anisotropic homogeneous medium. The effect of stator slots and teeth is included by a second anisotropic homogeneous medium. The induction motor is modeled as six concentric cylindrical layers representing the different construction components of the motor. The governing partial differential equations are solved for the excitation source, conducting, and nonconducting regions. The obtained formulas allow for efficient calculations of machine performance which may help the motor designer to select the proper parameters of the machine that fit the design requirements. Accuracy of the method is verified against finite element simulations, a commercial motor design program, and experimental results.

Magnetic-gearing effect has become increasingly attractive when designing direct-drive low-speed permanent-magnet machines. The machines derived from the magnetic-gearing effect can be termed as harmonic machines. Unlike the conventional types, harmonic machines rely on the field harmonics to achieve energy conversion and transmission. The detailed knowledge of the field distributions in the air gap is vitally important for predicting and optimizing its performance. In this paper, we present an analytical approach to calculate the magnetic field distribution in a low-speed permanent-magnet harmonic machine. A series-slot model which is composed of a group of partial differential equations concerning the scalar magnetic potential is built up. Then, the field solutions are obtained by using the method of separating variables and analyzing the field boundary conditions. Finally, the flux densities are derived from the scalar magnetic potentials. All the results agree well with those obtained from the finite element method.

It has been recognized that the cross-magnetizing phenomenon of saturated synchronous machines plays an important role in their analysis using the two-axis frame models. In this paper, the finite-element method (FEM) is applied to determine from 2-D static electromagnetic field solutions the parameters representing this phenomenon. A specially designed synchronous machine that can be excited from both its direct and quadrature axes is used in this study in order that the parameters representing the cross-magnetizing effect could be measured experimentally and, thus, the accuracy of the numerical solution can be verified. The effects of some of the factors that may affect the accuracy of the FEM analysis have also been investigated. The results of these studies have shown that the calculated parameters representing the cross-magnetizing phenomenon using the FEM are consistent with the measured values.

This paper presents a general mathematical model of a five-phase induction machine including the effects of higher space and time harmonics in the air gap field. These harmonic waves play a decisive role in the behavior of machines with more than three phases. Mathematical expressions for the calculation of the self and mutual inductances are presented, and results are compared with values obtained by finite element analysis and measurements. Based on the air gap field distribution produced by the stator and rotor, all the field harmonics are included in a direct and simple way in the self-inductances. The mutual inductances are obtained from a Fourier series description of the air gap field, resulting in a different inductance for each harmonic field. The machine equations are then simplified using coordinate transformations, which result in equivalent d-q models and equivalent circuits for given harmonic groups. In their final form, the equations are appropriated for the simulation of the machine behavior and developing new control strategies including higher space and time harmonics. Finally, practical results of a prototype machine are compared with simulations demonstrating the accuracy of the model

In this paper, the distribution pattern of harmonic waves in AC electric motors is investigated. On the basis of this, a novel exact and universal approach for calculating differential leakage related to the harmonic waves was developed. This approach applies to any AC electric motor, regardless of winding type, single-phase or three-phase, regular or nonregular, integral-slot or fractional-slot, symmetrical or asymmetrical. Since this novel approach substitutes the approximation method adopted by conventional approaches with an exact one, it distinguishes itself by offering precise results, general applicability, and fast calculation.

In this paper, the nonlinear phenomenon known as Hopf bifurcation, chaos and asynchronous operation of a simple power system are explored. Firstly, taking into account the nonlinearity of the generator shaft and the interaction of mechanics and electrics in the generator sets, the authors obtain a transient model by combining Park equations and mechanics equations. Then the Hopf bifurcation, period-doubling bifurcation and chaos caused by too large a line resistance are investigated with nonlinear mode and Floquet theory. The bifurcation figure of the system is also given. Further study shows that the chaos attractor breaks up into asynchronous operation when the resistance becomes larger. This way of loss-of-stability is different from that caused by loss-of-excitation.

This paper describes a method to calculate saturated reactances of synchronous machines at various loads under steady-state operation by considering influence of cross-magnetizing phenomenon between d- and q-axis circuits. In the method, mutual reactances x/sub dq/ due to the cross-magnetizing phenomenon can be calculated on the basis of two-dimensional magnetic field analysis. The validity of analysis is shown by good agreement between measured and calculated field current values of a 592 MVA turbine generator at various loads. It is clear that quantitative accuracy requires the magnetic coupling between d- and q-axis circuits be considered. Subsequently, steady-state reactances at various loads are analyzed by considering mutual reactances x/sub dq/ due to cross-magnetizing phenomenon. The relation between the load angle /spl delta/ and mutual reactance x/sub dq/ is clarified, and the difference between equivalent synchronous reactances that are obtained by considering x/sub dq/ and conventional defined synchronous reactances that do not consider x/sub dq/ is shown numerically.

The paper presents the use of the finite element method for determining the parameters of a two-axis model of a three-phase cage induction machine. The model parameters are obtained from the finite element field solutions. The saturation of the motor is taken into account in calculation of the flux distribution by using a static nonlinear vector potential solution. The linear time harmonic vector potential field solution is used for the inductances determination. All the inductances are determined as functions of the magnetizing current. Rotor skew is also taken into account. The accuracy of the described identification procedure is verified through laboratory measurements of the parameters.

Cross-saturation in smooth air-gap AC machines has been a subject of controversy in the 1980s. The discussions were based on the model of a saturated AC machine with d-q axis winding currents acting as state-space variables. The recently introduced concept of generalised flux and generalised inductance has enabled the derivation of a series of models of saturated smooth air-gap AC machines for various selections of state-space variables. All the models, except for the winding flux model, contain explicit terms that describe cross-saturation. The aim of this paper is to compare behaviour of various models when cross-saturation is neglected. Two transients of a single-cage induction machine are analysed: motor starting with increased voltage; and induction generator self-excitation. Some experimental results are included. It is shown that the impact of cross-saturation on accuracy depends on the type of the model. All the models that ask for a time derivative of the generalised inductance yield poor results if cross-saturation is neglected. However, the models that require a time derivative of the inverse of the generalised inductance are basically unaffected. Thus these models can be simplified without any loss of accuracy.

In this paper, a possible reason for oscillation phenomena in a diesel-generator set is investigated through digital and analog network simulations. The nonlinear simulations of a swing equation show sudden deviations in the amplitude of rotor angle oscillations, provided that there exist periodic fluctuations of mechanical inputs to the generator unit. This kind of oscillation, different from the one that is identified in linear resonance, is thought to be caused by system nonlinearities and ill-combination of system parameters. The results of the analog simulations show the occurrence of bifurcation phenomenon in a electrical power mode. An approximate nonlinear model of the swing equation is found to hold the form of Duffing's equation. From the viewpoint of Duffing's equation, an instability assessment method is proposed using a catastrophe theory analogy.

Main flux saturation is most frequently modelled by selecting either stator and rotor d-q axis currents or stator and rotor d-q axis flux linkages as state-space variables. This paper attempts to unify main flux saturation modelling in d-q axis models of induction machines by presenting a general method of saturation modelling. Selection of state-space variables in the saturated machine model is arbitrary and appropriate models in terms of different state-space variables result by application of the method. A couple of models, obtainable with different selection of state-space variables, are presented. The cross-saturation effect is explicitly present in all the models, except for the one with stator and rotor flux linkage d-q axis components as state-space variables. The models are verified by simulation and experimental investigation of induction generator self-excitation. >

This paper presents two finite element based methods for the calculation of forces on conductor segments and magnetized ferrous cores in three dimensional magnetic fields. The formulation is based on the three dimensional magnetic vector potential and the curl curl equation. Results of use of the two methods in the calculation of forces in two examples are reported. The first example is that of two currentcarrying conductors (bus bars), while the second example is that of an iron cored armature (coil) and keeper. These two relatively simple examples were chosen for purposes of comparing the three dimensional finite element results with existing two dimensional methods and closed form approximations. The results of comparison of these forces are presented here. This comparison confirms the validity and desirability of the present 3D formulation of forces by highlighting some of the limitations of the known analytical and two dimensional methods in certain applications. The two methods presented here are applicable to a variety of problems of force calculations in various electrical devices and machinery.