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This paper describes a control framework that enables distributed battery energy storage systems (BESS) connected to distribution networks (DNs) to track voltage setpoints requested by the transmission system operator (TSO) at specific interconnection points in an optimal and coordinated manner. The control design is based on an optimisation problem whose objective is to minimise the real-time voltage-tracking mismatch while satisfying local physical network constraints. A novel agent-based control scheme adopting an online convex optimisation (OCO) framework is developed and solved in a distributed fashion to guarantee the solution’s scalability and the service provision within the required time. The BESS agents under the proposed control framework automatically adapt to the time-varying network conditions so as to track the required voltage setpoints whilst fulfilling the technical operating requirements of the local network. The designed OCO-based framework addresses the existing conflict between the accuracy and optimality of the solution and the communication and computational efficiency. The convergence to the optimal solution is demonstrated. Several case studies are performed to corroborate the analytical results and assess the performance of the proposed approach.

This paper develops and validates an equivalent model of a wind farm for probabilistic harmonic propagation and mitigation studies in power electronics rich transmission or distribution networks. The methodology is based on Monte Carlo simulations and probabilistic distributions. Field measurements from a single wind turbine and PCC bus are used for the validation of the equivalent model. It is found out that the simulated results using developed model are in good agreement with the available field measurements. This equivalent model is therefore deemed to be suitable for the first level approximation of harmonic propagation and the identification of potential harmonic issues in the power electronics rich networks.

Hybrid renewable energy source (HRES) plants have been recognized as a promising option for obtaining stable power production from renewable energy sources. The research on HRES plants has been mostly focused on the economic benefits of this concept. Much less attention has been paid to the influence of economic dispatch of HRES plants on system dynamic performance. This paper illustrates the necessity of defining the optimal technology mix in the HRES plant for certain system conditions from the perspective of both economy and system dynamic behaviour. In this study, the focus is on transient system stability. Furthermore, the methodology for developing dynamic equivalent of the whole HRES plant for transient stability studies in the form of a low-order transfer function model is presented. Illustrative results are obtained using an HRES plant consisting of several non-dispatchable renewable generation technologies.

We review our recent theoretical studies on multimode instabilities in Fabry–Perot cavity mid-infrared quantum cascade lasers (QCLs) caused by parametric excitation of Rabi flopping oscillations. Numerical simulations are based on the semiclassical traveling wave Maxwell–Bloch equations. QCLs with a few mm cavity without an absorber exhibit intermittent RNGH self-pulsations, while regular self-pulsations are possible in short-cavity QCLs, with the cavity length of 100 μm or smaller. However, the second threshold in short-cavity QCLs is significantly increased compared to the values for a few mm long devices. We provide here a new insight on RNGH instability via bifurcation analysis of the output waveform and studies of the recurrence period density entropy. We propose an interpretation of the broadening/narrowing of the optical spectrum of a QCL i.e. switching the RNGH instability on and off observed in experiment.

We consider the influence of additional carrier confinement, achieved by application of strong perpendicular magnetic field, on inter Landau levels electron relaxation rates and the optical gain, of two different GaAs quantum cascade laser structures operating in the terahertz spectral range. Breaking of the in-plane energy dispersion and the formation of discrete energy levels is an efficient mechanism for eventual quenching of optical phonon emission and obtaining very long electronic lifetime in the relevant laser state. We employ our detailed model for calculating the electron relaxation rates (due to interface roughness and electron-longitudinal optical phonon scattering), and solve a full set of rate equations to evaluate the carrier distribution over Landau levels. The numerical simulations are performed for three- and four-well (per period) based structures that operate at 3.9 THz and 1.9 THz, respectively, both implemented in GaAs/Al0.15Ga0.85As. Numerical results are presented for magnetic field values from 1.5 T up to 20 T, while the band nonparabolicity is accounted for. (C) 2016 Elsevier B.V. All rights reserved.

This paper presents a computational technique to identify the region of attraction (ROA) for transient stability analysis in power systems with uncertainty and high-dimensionality. This is subsequently translated to a number of established transient stability indices in order to establish the multidimensional index-specific stability boundary. The uncertainty in this paper is represented in up to 4-dimensions in form of different collections of loads of the system spanning a full range of feasible loading levels. The advantage of this technique is its simplicity to implement and good accuracy. To verify performance of this tool, NETS-NYPS 68-Bus test system is used.

We present a systematic method for the construction of a discrete state embedded in the continuum part of the spectrum of the differential equation - y ? + f ( x ) y = λ y . Starting from an arbitrary preselected eigenvalue λ 0 , we generate a family of functions yielding identical eigenspectrum as f(x). The nature of the corresponding eigenfunctions remains unaltered, except at λ 0 , for which we obtain a discrete eigenfunction. The procedure is exemplified using the simplest case of f(x)=0.

Solar cells are unable to absorb photons with energies lower than the constituent material's band-gap, which eliminates a significant part of the spectrum. The efficiency of the process may be increased by converting low-energy photon pairs, via nonlinear optical effects, into photons optimal for the solar cell. The converter structures consist of GaN/AlGaN super lattice series perturbed by asymmetric quantum wells which form the Bragg confined structures (BSCs). BCSs support additional bound electron states in the super lattice minigaps, including states above the barrier. Parameters of the structure are optimized for each photon pair, by using the genetic algorithm, to obtain a continual converter.

Optical and structural properties of materials can be characterized by spectroscopic ellipsometry which represents an experimental technique that mea sures polarized light reflected from a material surface. Since the most complicated part of this me thod is data analysis and modeling, we present the calculation of the ellipsometric parameters for bi- isotropic materials. We consider the ellipsometric model for these materials which require not just pe rmittivity like in conventional model, but also magnetic permeability and magnetoelectric coupling.

The power system research community and industry acknowledge the importance of accurate load modelling for power system studies, however, many still use typical representation of static loads by the constant impedance/current/power load models, while dynamic loads, if represented, are usually modelled with an induction motor (IM) model. The last systematic update of load models was performed in the mid 1990s, since when significant changes have occurred in the structure, type and composition of loads at all network buses. General inadequacy of currently used load models was highlighted in several unsuccessful attempts to reproduce the behaviour observed in recent blackouts during the corresponding “post-mortem” simulations and analysis. Over the last several years, there has been a renewed interest in both industry and academia for load modelling due to appearance of new types of loads, offering increased efficiency and controllability. Different types of modern non-linear power electronic loads are now responsible for a significant part of the total demand in almost all load sectors. Furthermore, there are currently no appropriate load models available for the correct representation of various directly connected and inverter-interfaced micro and small-scale distributed generation technologies, which, in some of the future network scenarios, may strongly impact real and reactive power demands and behaviour in future network scenarios, as they would be installed in large numbers. In a response to this renewed interest in load modeling, CIGRÉ Study Committee C4 established, in late 2009, the Working Group (WG) C4.605: “Modelling and Aggregation of Loads in Flexible Power Networks”. The WG started work in February 2010 with the aim to: i) provide a critical and updated overview of existing load models and their parameters for power system studies at all voltage levels, and identify types of loads and load classes for which adequate load models are presently missing; ii) provide a comprehensive overview of existing methodologies for load modeling, with a critical overview of component based and measurement based approaches, clearly identifying their advantages and disadvantages; iii) develop a set of recommendations and step-by-step procedures for load model development and validation, using either component based or measurement based approaches, or their combination; iv) develop load models for all typical devices and classes of customers for which there are no existing models and recommend their typical parameter values and ranges; v) provide recommendations on developing equivalent static and dynamic models of networks with significant amount of distributed generation, including equivalent models of micro-grids and active distribution network cells.

In order to avoid losses in metamaterial unit cells at frequencies of interest, caused by metallic inclusions, an active medium design has been proposed. As candidate structures for this active medium, we have chosen quantum cascade lasers because of their high output gain. Here we analyze and compare two quantum cascade structures that emit at 4.6 THz and 3.9 THz, respectively, placed under the influence of a strong magnetic field. We first solve the full system of rate equations for all relevant Landau levels, and obtain the necessary information about carrier distribution among the levels, after which we are able to evaluate the permittivity component along the growth direction of the structure. With these data one can determine the conditions under which negative refraction occurs, and calculate the values of the refractive index of the structure, as well as the range of frequencies at which the structure exhibits negative refraction for a predefined total electron sheet density.

The aspects of light propagation through a semiconductor metamaterial (SMM) which consists of two alternating differently doped In0.53Ga0.47As layers are studied analytically and numerically. A general analytical expression for the effective index of refraction of the Poynting vector is derived for periodic structure composed of two alternating layers of absorptive and dispersive semiconductors and light propagation is simulated by finite-difference-time-domain method. Negative values of the effective refractive index with low absorption are observed in the mid-infrared frequency range. The main advantage of this design is the possibility to control the frequency region with negative effective refractive index by changing the doping densities of the layers.

We present a detailed analysis of GaAs/AlGaAs terahertz quantum cascade laser in the presence of an intense external magnetic field. One of the objectives in further development of THz quantum cascade laser is the realization of structures operating at higher temperatures. This is difficult to obtain as the operating photon emission energy is smaller than the longitudinal-optical phonon energy in the semiconductor material. With increased temperature, electrons in the upper radiative state gain sufficient in-plane energy to emit an longitudinal-optical phonon, which represents a non-radiative scattering and reduces the optical gain. By applying strong magnetic field, two-dimensional continuous energy subbands become split into series of discrete Landau levels, and at particular values of B it is possible to quench these non-radiative channels. Numerical simulations are performed on two-well design quantum cascade laser operating at 4.6 THz, implemented in CaAs/Al(0.15)Ga(0.85)As, and the magnetic field is perpendicular to the epitaxial layers. Strong oscillations of carrier lifetimes for the upper state of the laser transition, as a function of magnetic field are observed, which can be attributed to interface roughness scattering and longitudinal-optical phonon scattering between Landau levels. 12th Annual YUCOMAT Conference, Sep 06-10, 2010, Herceg Novi, Montenegro

We present an analysis of the influence of the Goos-Hanchen effect on tunneling times, group delay and dwell time, of electromagnetic waves propagating through an obstacle made of left-handed metamaterial embedded in a dielectric which exhibits saturable type of nonlinearity. The derived equations show that only the group delay, is affected by the Goos-Hanchen shift without any impact on the dwell time. Besides the reduction of the group delay, the most remarkable result is the possibility for total reduction of the Goos-Hanchen shift for finite incident angles. These phenomena are observable in the frequency region for which metamaterial exhibits negative index of refraction. (C) 2011 Elsevier B.V. All rights reserved.

in generating optimal structures which emit radiation at specified wavelengths corresponding to absorption fingerprints of particular harmful pollutants found in the atmosphere. We also illustrate another interesting prospective application of quantum cascade laser-type structures: the design of metamaterials with tunable complex permittivity, based on amplification via intersubband transitions. In this case, the role of the magnetic field is to assist the attainment of sucient optical gain (population inversion), necessary to eectively manipulate the permittivity and fulfill the conditions for negative refraction (left-handedness).

The Goos-Minchen effect is an optical phenomenon defined as a displacement of the reflected beam of linearly polarized light after the total internal reflection, which occurs perpendicularly to the direction of propagation. Due to this effect, when an optical wave propagates through a barrier, tunneling times may change accordingly. In this paper, we consider the impact of the GoosHanchen effect on group delay and dwell time for electromagnetic wave propagating through a nonlinear dispersive slab placed inside linear dispersive surroundings. Numerical calculations are performed for the special case, namely a double negative index metamaterial embedded into a material with a saturable nonlinearity, when a background medium is vacuum. The numerical results for tunneling times are calculated when the Goos-Hanchen effect is observed. It is shown that this approach gives more accurate expressions for tunneling times when the angle of incidence has a non-zero value.

In this Letter we present precise derivation of the second boundary condition for the wavefunctions (the first boundary condition is the continuity of the wavefunctions) at the interface of two III–V compound semiconductors, starting from an accurate expression for the bulk conduction-band structure expanded up to fourth order in wavevector. The obtained boundary condition is valid for all states (both bound and continuous) of the quantum well, and follows directly from constantness of the probability current along the quantum well and does not conflict with the double integrating of the Schrodinger equation around the interface.

We start from the fourth-order nonparabolic and anisotropic conduction band bulk dispersion relation to obtain a one-band effective Hamiltonian, which we apply to an AlGaSb symmetric double-barrier structure with resonant energies significantly (more than 200meV) above the well bottom. The spin-splitting is described by the k3 Dresselhaus spin-orbit coupling term modifying only the effective mass of the spin eigenstates in the investigated structure. Apart from the bulk-like resonant energy shift due to the band nonparabolicity, we obtain a substantial shift depending on the choice of boundary conditions for the envelope functions at interfaces between different materials. The shift of resonant energy levels leads to the change of spin-splitting and the magnitude of the dwell times. We attempt to explain the influence of both the nonparabolicity and boundary conditions choice by introducing various effective masses.

We investigate tunneling times of a particle with energy dependent effective mass for a one-dimensional real potential. General relations between phase, group and dwell times are obtained for a desired potential shape. For the textbook case of a real rectangular potential barrier the explicit relations for relevant times are derived, which reveal that the nonparabolicity, depending on the energy of incident particle, may substantially increase the group time in realistic structures. Further, we extend this theory to the case of absorptive media, described by complex potentials, via introduction of a new absorptive tunneling time tau(a). Depending on whether the short wavelength or long wavelength limit is considered, maximization of tau(a) results in a very different shape of a complex rectangular potential.

In this work we report on modeling the electron transport in n-Si/SiGe structures. The electronic structure is calculated within the effective-mass complex-energy framework, separately for perpendicular (Xz) and in-plane (Xxy) valleys, the degeneracy of which is lifted by strain, and additionally by size quantization. The transport is described via scattering between quantized states, using the rate equations approach and tight-binding expansion, taking the coupling with two nearest-neighbour periods. The acoustic phonon, optical phonon, alloy and interface roughness scattering are taken in the model. The calculated U/I dependence and gain profiles are presented for a couple of QC structures.

In this paper a procedure for the global optimization of mid-infrared GaAs/AlGaAs quantum cascade lasers that relies on the method of simulated annealing is presented. We propose a double longitudinal optical phonon resonance design obtained via a ladder of three states, with subsequent pairs separated by optical phonon energy. Addition of an extra level decreases the lower laser level population by enabling an efficient extraction into the injector region. The output characteristics of the optimized structures are calculated using the full self–consistent rate equation model, which includes all of the relevant scattering mechanisms. We also presented the experimentally measured output characteristics of an initial device, which are in agreement with the numerically calculated values, confirming the good design capabilities of the applied procedure.

Properly designed waveguides are of great importance for efficient operation of mid-infrared and tera-hertz quantum cascade semiconductor laser. Such waveguides comprise layers with suitably chosen doping, which determines refractive index, which enables bound modes to exist. It is of great interest to tailor the structure so to get minimal losses and maximal modal overlap with the active (gain) layer. In this paper we use the analogy between the Schrodinger and Helmholtz equations, and employ supersymmetric quantum mechanics, extended into the complex domain, in order to vary refractive index profile in search for better waveguide design. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A method is proposed for the design and optimization of structural parameters of GaN-AlGaN Bragg-confined structures with respect to peak intersubband absorption from the ground to the first excited state,1 /spl rarr/ 2 electronic transition, in the near infrared spectral range. An above-the-barrier bound state was used to extend the range of transition energies above the values available in conventional quantum wells. Intrinsic polarization fields and nonparabolicity effects were taken into account. The selection of optimal parameters, maximizing the absorption at wavelengths of 1.55 and 1.3 /spl mu/m, was performed by using a simulated annealing algorithm, and optimal structures with infinite superlattices as confinement regions were thus designed. These optimal parameters were then used to set realistic, finite structures with a small number of layers, the performance of which was re-evaluated by solving the Schrodinger-Poisson equation self-consistently for a few different levels and profiles of doping.

The design of rectangular GaN/AlGaN quantum wells for maximal intersubband absorption in the near infrared wavelength range 1.3–2 μm, on the 0→1 electronic transition is considered, taking into account the effects of internal polarization fields and nonparabolicity. The nonlinear optimization method based on solving a system of nonlinear equations is employed in finding the structural parameters which give the maximal dipole matrix element while keeping the absorption peak at the desired wavelength.

This paper explores the sensitivity of torsional modes and maximum torsional torques to turbine-generator shaft mechanical parameters. The analysis is performed on a typical single machine infinite bus system. Small-signal stability analysis, as well as large-disturbance analysis, are performed on six rotational mass mechanical system with and without inclusion of saturation of the generator. It is shown that mechanical parameters of the shaft, namely, inertia, shaft spring constant, internal shaft damping, and external damping, can significantly influence damping and frequency of torsional modes, mode shapes, and maximum torsional torques.

The influence of permanent dipole moments of quantized states on intersubband second harmonic generation based on quantum-interference induced transparency in semiconductor quantum wells is explored using the harmonic balance method. The permanent moments are found to be quite important: they affect the transparency condition, especially at larger pump intensities. Hence, both the conversion efficiency and the optimal interaction path length change significantly when accounting for the permanent moments, and the conversion efficiency is reduced.

We use a finite-difference model that is capable of describing the single state spin dynamics in a double-barrier AlGaAs heterostructure. The use of Green’s functions enables a description of the double-barrier structure by a finite matrix while the interaction with contacts is described by appropriate self-energies. To account for interface roughness scattering, a self-energy ΣIR(E, k) is derived within the random phase approximation. The dominant part is due to in-plane momentum relaxation while a smaller part describing spin-flip scattering is neglected. The former only decreases the state lifetime while the latter can also affect the spin precession frequency.

Optical nonlinearities in three-level asymmetric semiconductor quantum wells (QW) are analyzed (optical rectification, second harmonic generation (SHG)). The influence of permanent dipole moments on SHG conversion efficiency for various QW structures is investigated. It is found that as the pump photon wavelength increases so does the relative importance of the permanent dipole moments, provided the QW is resonant.

The possibility is explored of obtaining resonantly enhanced nonlinear optical susceptibilities in semiconductor quantum wells with two bound states and the resonant state in continuum as the third state. This would significantly extend the range of input radiation photon energies that may be frequency-doubled under resonance conditions in realistic structures. Calculation for the AlGaAs alloy based wells designed for the second harmonic generation of 240meV radiation indicate the usefulness of this approache.

The electronic structure of a perturbed superlattice of the Bragg-confined resonant structure type is discussed within the effective mass approximation. Expressions for bound-state energies and wavefunctions in these structures are derived in general form, valid for any symmetric form of the potential and effective mass modulation. The special case of rectangular superlattices with a perturbation layer is considered in more detail. The influence of the structure parameters on properties of Bragg-confined states is analysed, and positions of bound states lying in minigaps of the structure with the resonant double-well pertubation layer are determined. The possibilities of finding optimal structure parameters in respect to the maximal localization of the bound-state wavefunctions, in a broad range of above-the-barrier energies, are also discussed.

Using the inverse spectral theory we devise a systematic procedure to generate quantum well profiles optimized in respect to the optical rectification coefficient. Cases that, in course of design, start with an infinite barrier rectangular or with parabolic quantum well allow for a large part of the procedure to be done analytically, and are considered in more detail. Calculations for GaAs based wells indicate somewhat larger values of the rectification coefficient may be obtained than are reported in the current literature. The design procedure may also be extended to optimization of other optical nonlinearities.

The paper discusses the effects that nonlinear dynamic load model parameters have on electromechanical oscillations in power systems. Based on a modified Heffron-Phillips model of a synchronous machine and on a generic model of dynamic loads, analysis shows the dependence of the modal oscillations of the system on load model parameters. A single-machine infinite-bus case has been analyzed. It was found that under some circumstances damping can be improved, but under other conditions dynamic load may cause a decrease in damping.

We have considered the realization of metamaterials based on semiconductor quantum nanostructures, in particular, with the structural arrangement as in quantum cascade laser (QCL) designed to achieve optical gain in the mid-infrared and terahertz part of the spectrum. The entire structure is placed in a strong external magnetic field, which facilitates the attainment of sufficient population inversion, necessary to manipulate the permittivity, and enable a left-handed regime.

The scattering properties of four-parameter one-dimensional point interactions also define time aspect of these interactions. In this paper we investigate how different families of point interactions affect the relevant tunneling times. Particular emphasis is given to various interpretations of so called δ' potential.

Tunneling times in an anisotropic structure composed of alternately placed layers of two linear, dispersive and absorptive materials are analyzed in this paper. For such a structure, a general expression for the relation between the two most common tunneling time definitions, namely the dwell time and the group delay, is obtained. Numerical calculations are performed for a new type of semiconductor metamaterial. The results show a peak position frequency mismatch between the dwell time and the group delay, and between the dwell time and the absorption. 3rd International School and Conference on Photonics, Aug 29-Sep 02, 2011, Belgrade, Serbia

A method is developed for the analysis and extracting the optimal structural parameters of GaN/AlGaN Bragg-confined structures, in order to maximize the intersubband absorption on bound–above-the-barrier transitions at wavelengths in the near infrared spectral range. The “built-in” electrostatic field originating from piezoelectric and spontaneous polarization, and band nonparabolicity are taken into account. The optimal GaN/Al0.65Ga0.35N Bragg-confined structure designed for maximal absorption at λ=1.3 μm (950 meV) provides a fractional absorption of 1.5%, at 2×1012 cm−2 doping per active well.

Investigating the propagation of electromagnetic radiation in linear and nonlinear materials (with third-order nonlinearity), we derived a general expression for the dwell time valid for all linear media, as well as its relation with the group delay. The obtained results indicate that, for a nonmagnetic lossless material, the group delay reduces to the sum of the dwell time and the self-interference time. Calculations confirmed the occurrence of the Hartman effect in negative-refractive-index metamaterials (the dwell time saturates with increasing obstacle length), as well as the negative group delay in a certain spectral regime. Analysis of the Goos-Hanchen shift, in case of an obstacle made of negative-refractive-index metamaterial embedded in a dielectric with saturable nonlinearity, showed that there is no impact on the dwell time, while various effects on the group delay may be achieved, including its reduction or even preservation for nonzero incident angles. (C) 2011 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/1.3586782]

The paper presents detail parameter identification procedure and validation of dynamic equivalent model of Distributed Network Cell (DNC) developed based on grey-box approach. The aim is to demonstrate that the equivalent model based on grey-box approach has the potential to significantly improve the modelling capabilities and model accuracy compared to the black-box modelling. The equivalent model consists of a converter-connected generator and a composite load model in parallel. It is presented in six-order nonlinear state space form. The model is evaluated by comparing the responses of case study DNC obtained using non linear simulations in DigSILENT and those obtained by the model. (6 pages)

The paper presents the modeling of Distributed Network Cell (DNC) based on grey-box approach. The equivalent model of DNC comprises of a converter-connected generator and a composite load model in parallel. The detail development of the equivalent model is described in this paper. The grey-box approach is chosen based on the fact that it offers possibility to incorporate some prior knowledge about DNC structure into model development, make developed model more physically relevant and intuitive compared to black box model and potentially improve the accuracy of the model. The dynamic equivalent model is presented in the form of sixth-order nonlinear state space format and developed from the algebraic and differential equations describing assumed typical components of DNC. (6 pages)

We present a method for systematic optimization of quantum cascade laser active region, based on the use of the genetic algorithm. The method aims at obtaining a gain-maximized structure, designed to emit radiation at specified wavelengths suitable for direct absorption by pollutant gasses present in the ambient air. After the initial optimization stage, we introduce a strong external magnetic field to tune the laser output properties and to slightly modify the emission wavelength to match the absorption lines of additional compounds. The magnetic field is applied perpendicularly to the epitaxial layers, thus causing two dimensional continuous energy subbands to split into series of discrete Landau levels. This affects all the relevant relaxation processes in the structure and consequently the lifetime of carriers in the upper laser level. Furthermore, strong effects of band nonparabolicity result in subtle changes of the lasing wavelength at magnetic fields which maximize the gain, thus providing a path for fine tuning of the output radiation properties. Numerical results are presented for GaAs/Al(x)Ga(1-x)As based quantum cascade laser structures designed to emit at particular wavelengths in the mid infrared part of the spectrum. 11th Annual Conference of the Materials-Research-Society-of-Serbia (YUCOMAT 2009), Aug 31-Sep 04, 2009, Herceg Novi, Montenegro

The Levinson's theorem in a semiconductor quantum dot is analysed. If the well and barrier effective masses differ, the zero-energy wavefunction phaseshift is shown to be eta (0)= pi Mb(Mb being the number of bound states) for even values of azimuthal quantum number l, and pi (Mb+1) for odd l. For constant effective mass, however, eta (0)= pi Mb regardless of the parity of l. Furthermore, while eta to 0 in the large-energy limit for the constant effective mass, the essentially different behaviour of eta takes place if the effective mass is nonuniform.