The quartic equation: invariants and Euler's solution revealed

Nickalls, R. W. D.

doi:10.1017/s0025557200184190

Cited by 46 publications

(31 citation statements)

References 5 publications

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“…The coefficients of the polynomials can be readily written in terms of the network parameters and loads. Quartic polynomials and their roots have been completely characterized, and can be routinely computed in terms of the polynomial coefficients-see for example [17]. The convergence region can thus be easily computed.…”

Section: 11] the Definition Of Contraction Is As Followsmentioning

confidence: 99%

Convergence of the Z-Bus Method for Three-Phase Distribution Load-Flow with ZIP Loads

Bazrafshan

Gatsis

2018

IEEE Trans. Power Syst.

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This paper derives a set of sufficient conditions guaranteeing that the load-flow problem in unbalanced threephase distribution networks with wye and delta ZIP loads has a unique solution over a region that can be explicitly calculated from the network parameters. It is also proved that the wellknown Z-Bus iterative method is a contraction over the defined region, and hence converges to the unique solution.

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Section: 11] the Definition Of Contraction Is As Followsmentioning

confidence: 99%

Convergence of the Z-Bus Method for Three-Phase Distribution Load-Flow with ZIP Loads

Bazrafshan

Gatsis

2018

IEEE Trans. Power Syst.

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“…Euler's solution will be presented (for details, see Nickalls, 2009) which is based on the depressed (and monic) quartic polynomial given by into (A9). To compute the roots of the depressed quartic, one needs to find the three roots z 1 , z 2 and z 3 of Euler's resolvent cubic polynomial given by…”

Section: By Definingmentioning

confidence: 99%

A unified theory for optimal feedforward torque control of anisotropic synchronous machines

Eldeeb

Hackl

Horlbeck

et al. 2017

International Journal of Control

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KEYWORDSMaximum torque per ampere (MTPA); maximum torque per current (MTPC); maximum torque per voltage (MTPV); maximum torque per flux (MTPF); maximum current (MC); field weakening (FW); analytical solution; optimal feedforward torque control; efficiency; copper losses; anisotropy; synchronous machine; interior permanent-magnet synchronous machine; reluctance synchronous machine; permanent-magnet-assisted or enhanced synchronous machine; quadrics; quartics; Lagrangian optimisation; operation management ABSTRACT A unified theory for optimal feedforward torque control of anisotropic synchronous machines with non-negligible stator resistance and mutual inductance is presented which allows to analytically compute (1) the optimal direct and quadrature reference currents for all operating strategies, such as maximum torque per current (MTPC), maximum current, field weakening, maximum torque per voltage (MTPV) or maximum torque per flux (MTPF), and (2) the transition points indicating when to switch between the operating strategies due to speed, voltage or current constraints. The analytical solutions allow for an (almost) instantaneous selection and computation of actual operation strategy and corresponding reference currents. Numerical methods (approximating these solutions only) are no longer required. The unified theory is based on one simple idea: all optimisation problems, their respective constraints and the computation of the intersection point(s) of voltage ellipse, current circle or torque, MTPC, MTPV, MTPF hyperbolas are reformulated implicitly as quadrics which allows to invoke the Lagrangian formalism and to find the roots of fourth-order polynomials analytically. The proposed theory is suitable for any anisotropic synchronous machine. Implementation and measurement results illustrate effectiveness and applicability of the theoretical findings in real world.

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“…The quartic equation can be solved either numerically or analytically. An explicit solution can be derived in terms of the solutions of a related cubic equation, the resolvent cubic (Clark, 1984;Faucette, 1996;Stahl, 1997;Nickalls, 2009). Note that if Q 4 vanishes, then equation 50 reduces to a cubic equation with three roots.…”

Section: The Longitudinal Modes Of Propagationmentioning

confidence: 99%

On the propagation of a disturbance in a smoothly varying heterogeneous porous medium saturated with three fluid phases

Vasco

2013

GEOPHYSICS

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From the equations governing the deformation of a porous medium containing three fluid phases, I derive expressions for the phase velocity of the various modes of displacement. These expressions are valid for a medium with smoothly varying heterogeneity. There is a single mode of transverse displacement, similar in nature to an elastic shear wave. The four-phase velocities of the longitudinal modes of displacement are derived from the solutions of a quartic equation. The coefficients of the polynomial equation are written as linear sums of the determinants of 4 × 4 matrices. The matrices contain various combinations of the parameters from the governing equations. The three-phase expressions are compared to two-phase estimates for the case in which one of the fluid saturations vanishes. Also, in a numerical illustration, velocity variations of around 10% are associated with the cyclic injection of carbon dioxide and water into an oil-saturated reservoir.

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The quartic equation: invariants and Euler's solution revealed

Cited by 46 publications

References 5 publications

Convergence of the Z-Bus Method for Three-Phase Distribution Load-Flow with ZIP Loads

Convergence of the Z-Bus Method for Three-Phase Distribution Load-Flow with ZIP Loads

A unified theory for optimal feedforward torque control of anisotropic synchronous machines

On the propagation of a disturbance in a smoothly varying heterogeneous porous medium saturated with three fluid phases

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