Unstable Roots Research Articles

The influence of vaporization upon the roots of a high current arc. II

By the use of graphite cathodes in argon the normally unstable arc root exhibiting strong vaporization of the cathode can be made stationary with a well defined, axially symmetrical vapour jet. Spectroscopic measurements of the C2 molecular bands give a vapour temperature between 4000 K and 5000 K. On strongly heated electrodes the hot plasma is to some extent hindered by the relatively cold vapour: this results in a displacement, supported by magnetic forces, of the arc root onto colder areas and thus in instability.

Van der Pol’s equation and nonlinear oscillations in a beam plasma system

A nonlinear dispersion relation for beam plasma oscillations is obtained by expanding the usual dispersion relation about both the most unstable root and the perturbed beam electron orbits. This procedure leads to a nonlinear rate equation similar to the van der Pol equation.

Stability analysis of nonlinear control systems with characteristics equations having complex coefficients

In this paper, the stability equation method is applied to the analysis of nonlinear systems with characteristics equations having complex coefficients. Three types of systems are studied: those with unstable open-loop poles, unstable characteristics roots or an equal number of open-loop poles and zeros.

Theory and Simulation of Turbulent Heating by the Modified Two-Stream Instability

Results of an analytical and numerical study of the nonresonant, modified plasma two-stream instability, which is driven by relative streaming of electrons and ions across a magnetic field B0 are presented. The instability has characteristic frequency and growth rate comparable to the lower-hybrid frequency. The linear theory is discussed both in the electrostatic and fully electromagnetic cases, and a detailed numerical study of the dependence of the unstable roots of the dispersion relation for a wide range of plasma parameters is presented. The nonlinear theory includes discussions of (1) quasilinear theory, (2) trapping, which is responsible for nonlinear stabilization, (3) a derivation of a fully nonlinear scaling law which shows how results scale with electron-ion mass ratio, and (4) the effect of cross-field vortex-like motion caused by turbulence induced E × B drifts. One-and two-dimensional computer simulations with dense k-space spectra are presented in support of this theory. The simulations show that the instability can be a very important turbulent heating mechanism that heats the ions (perpendicular to B0) and the electrons (parallel to B0) comparably. The final state has (Ti⊥/mi)1/2 ≈ (Te‖/mi)1/2∼12U, where U is the initial relative drift speed. Applications to experimental situations are discussed.

Self-Oscillation in a Control System with Dead Time and Saturation

In this paper, the author analytically investigates a second order control system with dead time and saturation and he elucidates selective occurrence of self-oscillation in the system. The conclusions obtained are as follows: The system has several pairs of unstable characteristic roots in some intervals of dead time. Among a number of self-excited modes of oscillations which correspond to unstable roots, only one mode of oscillation develops and settles down to a steady state periodic oscillation under the effect of saturation (the oscillation is referred to as self-oscillation). Although occurrence of two or more modes of self-oscillations may be possible in some intervals of dead time, these modes of oscillations can never occur simultaneously, but only one mode of them can occur depending upon the initial condition. The condition of occurrence of self-oscillation is analytically expressed and also amplitudes and frequencies of self-oscillations are theoretically obtained. The theoretical results are in good agreement with the experimental results by analog computer.

Generalisation of the Routh-Hurwitz criterion and its applications

A generalisation of the Routh-Hurwitz stability criterion for testing the presence of unstable roots of a polynomial with complex coefficients is presented. Numerical examples are given to illustrate the application of the generalisation.

Electrostatic Instabilities in a Plasma with Anisotropic Velocity Distribution

An anisotropic magnetized plasma, in which the distribution of particle velocities is a bi-Maxwellian characterized by the temperatures ${T}_{\ensuremath{\perp}}$ and ${T}_{\mathrm{II}}$, is considered in order to delimit as much as possible the location of possibly unstable roots to the dispersion equation. We find that marginally unstable roots of the dispersion equation can occur only if both $[l+\frac{1}{2}]l\ensuremath{\omega}l[l+1\ensuremath{-}(\frac{{T}_{\mathrm{II}}}{{T}_{\ensuremath{\perp}}})]$ and $\ensuremath{\omega}l\frac{({T}_{\ensuremath{\perp}}\ensuremath{-}{T}_{\mathrm{II}})}{{T}_{\mathrm{II}}}$, where $l$ is an integer and $\ensuremath{\omega}$ is the real part of the frequency in units of the cyclotron frequency. Thus, in particular, the system is stable if ${T}_{\ensuremath{\perp}}l2{T}_{\mathrm{II}}$.

A new application of the Hurwitz-Routh stability criteria

WHEN THE HURWITZ-ROUTH stability criteria are applied to an algebraic polynomial with real coefficients, the existence, or nonexistence, of unstable roots of the polynomial is revealed. However, if the polynomial contains all stable roots, the criteria do not give an indication of the relative stability of the roots. The latter information is frequently desired.