Transient thermal analysis of power devices based on Fourier-series thermal model

A new thermal model based on Fourier series expansion method has been presented for dynamic thermal analysis on power devices. The thermal model based on the Fourier series method has been programmed in MATLAB SIMULINK and integrated with a physics-based electrical model previously reported. The model was verified for accuracy using a two-dimensional Fourier model and a two-dimensional finite difference model for comparison. To validate this thermal model, experiments using a 600V 50A IGBT module switching an inductive load, has been completed under high frequency operation. The result of the thermal measurement shows an excellent match with the simulated temperature variations and temperature time-response within the power module. ©2008 IEEE.

Fourier series expansion method for gain measurement from amplified spontaneous emission spectra of Fabry-Perot semiconductor lasers

A gain measurement technique, based on Fourier series expansion of periodically extended single fringe of the amplified spontaneous emission spectrum, is proposed for Fabry-Perot semiconductor lasers. The underestimation of gain due to the limited resolution of the measurement system is corrected by a factor related to the system response function. The standard deviations of the gain-reflectivity product under low noise conditions are analyzed for the Fourier series expansion method and compared with those of the Hakki-Paoli method and Cassidy's method. The results show that the Fourier series expansion method is the least sensitive to noise among the three methods. The experiment results obtained by the three methods are also presented and compared.

Developments and extensions of Roth's method of double Fourier series with applications to the solution of electromagnetic field problems

The first part of the thesis compares Roth's method with other methods, in particular the method of separation of variables and the finite cosine transform method, for solving certain elliptic partial differential equations arising in practice. In particular we consider the solution of steady state problems associated with insulated conductors in rectangular slots. Roth's method has two main disadvantages namely the slow rate of con­vergence of the double Fourier series and the restrictive form of the allowable boundary conditions. A combined Roth-separation of variables method is derived to remove the restrictions on the form of the boundary conditions and various Chebyshev approximations are used to try to improve the rate of convergence of the series. All the techniques are then applied to the Neumann problem arising from balanced rectangular windings in a transformer window. Roth's method is then extended to deal with problems other than those resulting from static fields. First we consider a rectangular insulated conductor in a rectangular slot when the current is varying sinusoidally with time. An approximate method is also developed and compared with the exact method.The approximation is then used to consider the problem of an insulated conductor in a slot facing an air gap. We also consider the exact method applied to the determination of the eddy-current loss produced in an isolated rectangular conductor by a transverse magnetic field varying sinusoidally with time. The results obtained using Roth's method are critically compared with those obtained by other authors using different methods. The final part of the thesis investigates further the application of Chebyshdev methods to the solution of elliptic partial differential equations; an area where Chebyshev approximations have rarely been used. A poisson equation with a polynomial term is treated first followed by a slot problem in cylindrical geometry.

Modelling of surface identifying characteristics using Fourier series

Texture and small-scale surface details are widely recognised as playing an important role in the haptic identification of objects. In order to simulate realistic textures in haptic virtual environments, it has become increasingly necessary to identify a robust technique for modelling of surface profiles. This paper describes a method whereby Fourier series spectral analysis is employed in order to describe the measured surface profiles of several characteristic surfaces. The results presented suggest that a bandlimited Fourier series can be used to provide a realistic approximation to surface amplitude profiles.

Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.

Simultaneous ultrafast optical pulse train bursts generation and shaping based on Fourier series developments using superimposed fiber Bragg gratings

We propose an all-fiber method for the generation of ultrafast shaped pulse train bursts from a single pulse based on Fourier Series Developments (FDSs). The implementation of the FSD based filter only requires the use of a very simple non apodized Superimposed Fiber Bragg Grating (S-FBG) for the generation of the Shaped Output Pulse Train Burst (SOPTB). In this approach, the shape, the period and the temporal length of the generated SOPTB have no dependency on the input pulse rate.

On the efficacy of Fourier series approximations for pricing European options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the halfrange cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together,these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

Biorthogonal series method for Oseen type flows

A biorthogonal series method is developed to solve Oseen type flow problems. The theory leads to a new set of eigenfunctions for a specific class of linear non-selfadjoint operators containing the biharmonic one. These eigenfunctions differ from those given earlier in the literature for the biharmonic operator. The method is applied to the problem of thermocapillary flow in a cylindrical liquid bridge of finite length with axial through flow. Flow and temperature distributions are obtained at leading order of an expansion for small surface tension Reynolds number and Prandtl number. Another related problem considered is that of cylindrical cavity flow. Solutions for both cases are presented in terms of biorthogonal series. The effect of axial through flow on velocity and temperature fields is discussed by numerical evaluation of the truncated analytical series. The presence of axial through flow not only convectively shifts the vortices induced by surface forces in the direction of the through flow, but also moves their centers toward the outer cylindrical boundary. This process can lead to significantly asymmetric flow structures.

Lacunary Fourier series and a qualitative uncertainty principle for compact Lie groups

We define lacunary Fourier series on a compact connected semisimple Lie group G. If f is an element of L-1 (G) has lacunary Fourier series and f vanishes on a non empty open subset of G, then we prove that f vanishes identically. This result can be viewed as a qualitative uncertainty principle.

Fourier Transform Method for Measuring the Micro Amplitu of Water Surface Waves

采用一种非接触的光学方法傅立叶变换莫尔法(Fourier transform method),结合数字图像处理技术,对微幅振荡的水表面波的振幅进行测量.它是对全场中每一个像素点进行测量,比接触测量法具有更高的灵敏度.它为微幅水表面波振幅的测量提供了一种手段.通过将计算机生成的周期性光栅图像经投影机直接投影到被测物体的参考平面,经CCD摄像头、图像板捕捉存储形成数字化的光栅图像,利用傅立叶变换莫尔法处理光栅图像,从而获得包含有水表面波的振幅的相位信息,再经适当的几何变换获得振幅信息.我们在垂直振荡装置上进行了不同激励频率和不同振幅的表面波的振幅测量.

Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

Lq estimates for rearrangements of Fourier series

This work is concerned with estimating the upper envelopes S* of the absolute values of the partial sums of rearranged trigonometric sums. A.M. Garsia [Annals of Math. 79 (1964), 634-9] gave an estimate for the L₂ norms of the S*, averaged over all rearrangements of the original (finite) sum. This estimate enabled him to prove that the Fourier series of any function in L₂ can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the L_q norms of the S*, for all even integers q. This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the L₂ case. This estimate for finite sums is extended to Fourier series of L_q functions; it is shown that there are functions to which the Men’shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series is in L_q.

Measurement of gain spectrum for Fabry-Perot semiconductor lasers by the Fourier transform method with a deconvolution process

To improve the accuracy of measured gain spectra, which is usually limited by the resolution of the optical spectrum analyzer (OSA), a deconvolution process based on the measured spectrum of a narrow linewidth semiconductor laser is applied in the Fourier transform method. The numerical simulation shows that practical gain spectra can be resumed by the Fourier transform method with the deconvolution process. Taking the OSA resolution to be 0.06, 0.1, and 0.2 nm, the gain-reflectivity product spectra with the difference of about 2% are obtained for a 1550-nm semiconductor laser with the cavity length of 720 pm. The spectra obtained by the Fourier transform method without the deconvolution process and the Hakki-Paoli method are presented and compared. The simulation also shows that the Fourier transform method has less sensitivity to noise than the Hakki-Paoli method.

Study on tapered crossed subwavelength gratings by Fourier modal method

Fourier modal method incorporating staircase approximation is used to study tapered crossed subwavelength gratings in this paper. Three intuitive formulations of eigenvalue functions originating from the prototype are presented, and their convergences are compared through numerical calculation. One of them is found to be suitable in modeling the diffraction efficiency of the circular tapered crossed subwavelength gratings without high absorption, and staircase approximation is further proven valid for non-highly-absorption tapered gratings. This approach is used to simulate the "moth-eye" antireflection surface on silicon, and the numerical result agrees well with the experimental one.