A new analytical expression of a previously proposed Klein-Gordon dipole matrix elements in the quasiclassical approach (including quantum defects) is presented. The intermediate state method in the semiclassical Coulomb approximation is used to derive the Klein-Gordon dipole radial integrals corresponding to single-electron nlj to n'l'j' transitions with arbitrary quantum numbers in non-hydrogenic ions. This last approach is extended to the second-order Dirac-Coulomb equation. Similar expressions are obtained in the two electromagnetic field gauges which in the non-relativistic limit give the length and velocity forms of the transition operator. A computational procedure for the evaluation of the Dirac formulae by the use of recursion relations, expressed in terms of Anger's functions, is also described. Numerical applications of the above-mentioned WKB methods, starting from the well known Schrodinger dipole matrix elements, are carried out for the calculation of the lowest 2s1/2-2p1/2,3/2 transitions in the lithium isoelectronic sequence for atomic number Z=3-92. Oscillator strengths for Rydberg 2s1/2-np1/2(Ca17+, n=8-16 and Zr37+, n=3-20) and ns1/2-20p1/2,3/2(Ca17+, Zr37+, W71+, U89+, n=19,20) transitions are also reported. The values obtained are compared and discussed with available experimental and other theoretical treatments.
We show a technique for external direct current (DC) control of the amplitudes of limit cycles both in the Phase-shift and Twin-T oscillators. We have found that amplitudes of the oscillator output voltage depend on the DC control voltage. By varying the total impedance of each oscillator oscillatory network, frequencies of oscillations are controlled using potentiometers. The main advantage of the proposed circuits is that both the amplitude and frequency of the waveforms generated can be independently controlled. Analytical, numerical, and experimental methods are used to determine the boundaries of the states of the oscillators. Equilibrium points, stable limit cycles, and divergent states are found. Analytical results are compared with the numerical and experimental solutions, and a good agreement is obtained.
This paper studies the dynamics of a self-excited oscillator with two external periodic forces. Both the nonresonant and resonant states of the oscillator are considered. The hysteresis boundaries are derived and the hysteresis domains are defined in terms of the system parameters. Making use of the properties of Hill's equation, we derive the stability conditions of oscillation in the resonant case. Phase portraits are obtained numerically and experimentally. One of the most important contributions of this study is to validate a set of reliable analytical expressions (formulae) describing the system behaviour. These are of great importance for design engineers. The reliability of the analytical formulae is demonstrated by the very good agreement between the results obtained by numerical and experimental analyses.
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