This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Magnetic flux can penetrate a type-II superconductor in the form of Abrikosov vortices (also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux phi 0=h/2e. These tiny vortices of supercurrent tend to arrange themselves in a triangular flux-line lattice (FLL), which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-Tc superconductors (HTSCs) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSCs the FLL is very soft mainly because of the large magnetic penetration depth lambda . The shear modulus of the FLL is c66~1/ lambda 2, and the tilt modulus c44(k)~(1+k2 lambda 2)-1 is dispersive and becomes very small for short distortion wavelengths 2 pi /k<< lambda . This softness is enhanced further by the pronounced anisotropy and layered structure of HTSCs, which strongly increases the penetration depth for currents along the c axis of these (nearly uniaxial) crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may `melt` the FLL and cause thermally activated depinning of the flux lines or ofthe two-dimensional `pancake vortices` in the layers. Various phase transitions are predicted for the FLL in layered HTSCs. Although large pinning forces and high critical currents have been achieved, the small depinning energy so far prevents the application of HTSCs as conductors at high temperatures except in cases when the applied current and the surrounding magnetic field are small.

https://iopscience.iop.org/article/10.1088/0034-4885/58/11/003/pdf

The flux-line lattice in superconductors

Magnetic flux can penetrate a type-II superconductor in form of Abrikosov vortices. These tend to arrange in a triangular flux-line lattice (FLL) which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-$T_c$ supercon- ductors (HTSC's) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSC's the FLL is very soft mainly because of the large magnetic penetration depth: The shear modulus of the FLL is thus small and the tilt modulus is dispersive and becomes very small for short distortion wavelength. This softness of the FLL is enhanced further by the pronounced anisotropy and layered structure of HTSC's, which strongly increases the penetration depth for currents along the c-axis of these uniaxial crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may melt the FLL and cause thermally activated depinning of the flux lines or of the 2D pancake vortices in the layers. Various phase transitions are predicted for the FLL in layered HTSC's. The linear and nonlinear magnetic response of HTSC's gives rise to interesting effects which strongly depend on the geometry of the experiment.

The Flux-Line Lattice in Superconductors

High-Performance Piezoelectric Energy Harvesters and Their Applications

As the field of robotics is expanding from the fixed environment of a production line to complex human environments, robots are required to perform increasingly human-like manipulation tasks, moving the state-of-the-art in robotics from grasping to advanced in-hand manipulation tasks such as regrasping, rotation and translation. To achieve advanced in-hand manipulation tasks, robotic hands are required to be equipped with distributed tactile sensing that can continuously provide information about the magnitude and direction of forces at all contact points between them and the objects they are interacting with. This paper reviews the state-of-the-art in force and tactile sensing technologies that can be suitable within the specific context of dexterous in-hand manipulation. In previous reviews of tactile sensing for robotic manipulation, the specific functional and technical requirements of dexterous in-hand manipulation, as compared to grasping, are in general not taken into account. This paper provides a review of models describing human hand activity and movements, and a set of functional and technical specifications for in-hand manipulation is defined. The paper proceeds to review the current state-of-the-art tactile sensor solutions that fulfil or can fulfil these criteria. An analytical comparison of the reviewed solutions is presented, and the advantages and disadvantages of different sensing technologies are compared.

Tactile sensing for dexterous in-hand manipulation in robotics-A review

Levitation and rotation of cylindrical rare‐earth magnets on yttrium‐barium‐copper‐oxide superconducting bearings has been sustained at speeds of up to 120 000 rpm with a surface speed of 40 m/s. The decay of the free rotation rate has been measured at both atmospheric pressure and a partial vacuum to 2.6 μm Hg. The decay measurements in vacuum indicate that the flux drag torques are constant and independent of speed. The magnetic shear stress on the rotor magnet is estimated to be 150 dyn/cm2. It is believed that drag torques on rotating magnets are related to asymmetry in the flux density pattern of the magnet.

High‐speed rotation of magnets on high Tc superconducting bearings

In this paper, we present the development of two piezoelectric MEMS generators, {3–1} mode and {3–3} mode, which have the ability to scavenge mechanical energy of ambient vibrations and transform it into useful electrical power. These two piezoelectric MEMS generators are of cantilever type made by a silicon process and which can transform mechanical energy into electrical energy through its piezoelectric PZT layers. We developed a PZT deposition machine which uses an aerosol deposition method to fabricate the high-quality PZT thin film efficiently. Our experimental results show that our {3–1} mode device possesses a maximum open circuit output voltage of 2.675 VP-P and a maximum output power of 2.765 µW with 1.792 VP-P output voltage excited at a resonant frequency of 255.9 Hz under a 2.5 g acceleration level. The {3–3} mode device possessed a maximum open circuit output voltage of 4.127 VP-P and a maximum output power of 1.288 µW with 2.292 VP-P output voltage at its resonant frequency of 214 Hz at a 2g acceleration. We also compared the output characteristics of both the {3–1} mode and the {3–3} mode piezoelectric MEMS generators which were both excited at a 2g acceleration level.

/pdf/piezoelectric-mems-generators-fabricated-with-an-aerosol-1jg5gjxjop.pdf

Piezoelectric MEMS generators fabricated with an aerosol deposition PZT thin film

This work fabricates a novel type of micromachined microwave switch on a semi-insulating GaAs substrate using a microactuator and a coplanar waveguide (CPW) using electrostatic actuation as the switching mechanism. The microactuator uses several continuously-bent cantilevers connected in series. A cantilever with two sections, a straight-beam section and a curved-beam section, forms the basic unit of the microactuator. The straight-beam section is made of an aluminum (Al) layer, 0.5 μm thick. The curved-beam section is made of an Al layer of the same thickness, combined with a 0.1-μm layer of chromium (Cr) film. This section is initially curled due to the different residual stress of Al and Cr. The low temperature (250°C) process ensures that the switch is capable of monolithic integration with microwave and millimeter wave integrated circuits (MMIC). When no dc potential is applied, the actuator is curled far from the signal line of the CPW, and therefore the insertion loss at this `on' state is only 0.2 dB at 10 GHz. Because the metal microactuator is far from the signal line of the CPW at this `on' state, the microwave propagation is hardly disturbed by the microactuator. When an applied electrostatic force pulls the actuator tip down into contact with the signal line of the CPW, it creates a large capacitance between the actuator and the CPW. The isolation at this `off' state is −17 dB at 10 GHz. Maintaining the actuator in the `off' state requires only a very low actuation voltage of 26 V. Once the dc potential is removed, the residual stress of the actuator structure pulls it to the up position. The microactuator moving back and forth between these two switching states, acts like the movement of a frog's tongue. This switch has excellent performance at the wide-band RF frequencies used in transmit/receive modules of wireless communication. This study measured the critical corrupt (activating) voltage and recovery voltage of the microactuator. The 10-ms switching time of this switch is slower than the switching time of solid-state switches.

Innovative micromachined microwave switch with very low insertion loss

Levitation forces between a small permanent magnet and a disk of bulk high‐temperature superconductor at 77 K were measured as a function of vertical separation for disks of composition Y‐Ba‐Cu‐O, Ag/Y‐Ba‐Cu‐O, (Pb,Bi)‐Sr‐Ca‐Cu‐O, and Tl‐Ba‐Ca‐Cu‐O. The forces were highly hysteretic; however, for all samples, on the initial descent of the magnet toward the disk, the force was unique, independent of magnet speed, and varied approximately as the negative exponential of the separation distance. Magnetic stiffness, associated with minor hysteresis loops, was found to be approximately proportional to the levitation force, and nearly independent of magnet configuration and superconductor composition.

Levitation force and magnetic stiffness in bulk high-temperature superconductors

Dynamic forces between small permanent magnets and bulk high‐Tc superconducting ceramic materials have been measured, including magnetic stiffness and damping effects. The vibration frequencies and the related magnetic stiffness show a large dependence on the magnet‐superconductor distance. The dynamic magnetic stiffness is shown to be related to small static reversible magnetic forces, but is not correlated with the static hysteretic forces. The magnetic damping is inversely related to the magnet‐superconductor distance and can critically damp the oscillations at small gaps. These data also show that the flux pinning forces depend on the prior flux history in the superconductor.

Pei-Zen Chang

Papers

High‐speed rotation of magnets on high Tc superconducting bearings

Piezoelectric MEMS generators fabricated with an aerosol deposition PZT thin film

Innovative micromachined microwave switch with very low insertion loss

Levitation force and magnetic stiffness in bulk high-temperature superconductors

Dynamic magnetic forces in superconducting ceramics