Darmois matching and C3 matching

We apply the Darmois and the $C^3$ matching conditions to three different spherically symmetric spacetimes. The exterior spacetime is described by the Schwarzschild vacuum solution whereas for the interior counterpart we choose different perfect fluid solutions with the same symmetry. We show that Darmois matching conditions are satisfied in all the three cases whereas the $C^3$ conditions are not fulfilled. We argue that this difference is due to a non-physical behavior of the pressure on the matching surface.

Wireless Power Transfer between Two Self-Resonant Coils over Medium Distance Supporting Optimal Impedance Matching Using Ferrite Core Transformers

An efficient wireless power transfer (WPT) system is proposed using two self-resonant coils with a high-quality factor (Q-factor) over medium distance via an adaptive impedance matching network using ferrite core transformers. An equivalent circuit of the proposed WPT system is presented, and the system is analyzed based on circuit theory. The design and characterization methods for the transformer are also provided. Using the equivalent circuit, the appropriate relation between turn ratio and optimal impedance matching conditions for maximum power transfer efficiency is derived. The optimal impedance matching conditions for maximum power transfer efficiency according to distance are satisfied simply by changing the turn ratio of the transformers. The proposed WPT system maintains effective power transfer efficiency with little Q-factor degradation because of the ferrite core transformer. The proposed system is verified through experiments at 257 kHz. Two WPT systems with coupling efficiencies higher than 50% at 1 m are made. One uses transformers at both Tx and Rx; the other uses a transformer at Tx only while a low-loss coupling coil is applied at Rx. Using the system with transformers at both Tx and Rx, a wireless power transfer of 100 watts (100-watt light bulb) is achieved.

A solvable contact potential based on a nuclear model

We extend previous works on the study of a particle subject to a three-dimensional spherical singular potential including a $$\delta $$ δ –$$\delta '$$ δ ′ contact interaction. In this case, to have a more realistic model, we add a Coulombic term to a finite well and a radial $$\delta $$ δ –$$\delta '$$ δ ′ contact interaction just at the edge of the well, which is where the surface of the nucleus would be. We first prove that the we are able to define the contact potential by matching conditions for the radial function, fixing a self-adjoint extension of the non-singular Hamiltonian. With these matching conditions, we are able to find analytic solutions of the wave function and focus the analysis on the bound state structure characterizing and computing the number of bound states. For this approximation for a mean-field Woods–Saxon model, the Coulombic term enables us to complete the previous study for neutrons analyzing the proton energy levels in some doubly magic nuclei. In particular, we find the appropriate $$\delta '$$ δ ′ contribution fitting the available data for the neutron- and proton-level schemes of the nuclei $${}^{{208}}$$ 208 Pb, $${}^{{40}}$$ 40 Ca, and $${}^{{16}}$$ 16 O.

Stabilization of a class of underactuated Euler Lagrange system using an approximate model

The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.

Approximate perfect fluid solutions with quadrupole moment

We investigate the interior Einstein’s equations in the case of a static, axially symmetric, perfect fluid source. We present a particular line element that is specially suitable for the investigation of this type of interior gravitational fields. Assuming that the deviation from spherically symmetry is small, we linearize the corresponding line element and field equations and find several classes of vacuum and perfect fluid solutions. We find some particular approximate solutions by imposing appropriate matching conditions.