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This paper aims to study the relationship between leakage flux coefficient and the coreless axial magnetic field permanent magnet synchronous generator (AFPMSG) size and obtain the expressions of leakage flux coefficient.

In this paper, a magnetic circuit model of coreless AFPMSG is proposed. Four kinds of leakage permeances of permanent magnet (PM) are considered, and the expression of no-load leakage flux coefficient is obtained. Solving the integral region of leakage permeances by generator size, which improves the accuracy of the solution.

Finite element method and magnetic circuit method are used to obtain the no-load leakage flux coefficient and its variation trend charts with the change of pole arc coefficient, air gap length and PM thickness. The average errors of the two methods are 2.835%, 0.84% and 1.347%, respectively. At the same time, the results of single-phase electromotive force obtained by magnetic circuit method, three dimensional finite element method and prototype experiments are 19.36 V, 18.82 V and 19.09 V, respectively. The results show that the magnetic circuit method is correct in calculating the no-load leakage flux coefficient.

The special structure of the coreless AFPMSG is considered in the presented equivalent magnetic circuit and equations, and the equations in this paper can be applied for leakage flux evaluating purposes and initial parameter selection of the coreless AFPMSG.

The purpose of this paper is to propose an improved method which can shorten the calculation time and improve the calculation efficiency under the premise of ensuring the calculation accuracy for calculating the response of dynamic systems with periodic time-varying characteristics.

An improved method is proposed based on Runge–Kutta method according to the composition characteristics of the state space matrix and the external load vector formed by the reduction of the dynamic equation of the periodic time-varying system. The recursive scheme of the holistic matrix of the system using the Runge–Kutta method is improved to be the sub-block matrix that is divided into the upper and lower parts to reduce the calculation steps and the occupied computer memory.

The calculation time consumption is reduced to a certain extent about 10–35% by changing the synthesis method of the time-varying matrix of the dynamics system, and the method proposed of paper consumes 43–75% less calculation time in total than the original Runge–Kutta method without affecting the calculation accuracy. When the ode45 command that implements the Runge–Kutta method in the MATLAB software used to solve the system dynamics equation include the time variable which cannot provide its specific analytic function form, so the time variable value corresponding to the solution time needs to be determined by the interpolation method, which causes the calculation efficiency of the ode45 command to be substantially reduced.

The proposed method can be applied to solve dynamic systems with periodic time-varying characteristics, and can consume less calculation time than the original Runge–Kutta method without affecting the calculation accuracy, especially the superiority of the improved method of this paper can be better demonstrated when the degree of freedom of the periodic time-varying dynamics system is greater.

This study aims to analyze the moderating effect of managerial discretion on the relationship between cross-border knowledge search and the high-tech firms’ innovation quality in a global health emergency and addresses the following issues: the influence mechanism of different types of cross-border knowledge search on the high-tech firms’ innovation quality in a global health emergency; and the moderating role of different dimensions of manager discretion on the above relationship.

Based on the firms’ strategy selection methods, the authors divided cross-border knowledge search into three aspects, namely, breadth, depth and balance, and analyzed the impact of cross-border knowledge search on the innovation quality of high-tech firms in a global health emergency, taking managerial discretion as the moderating variable, and divided it into position rights, pay rights and operation rights according to the key rights of firms’ strategic management. Furthermore, the authors constructed a theoretical model, and for an analysis sample, the authors collected data from Chinese high-tech firms from 2013 to 2021.

The empirical results show that there is an inverted U-shaped relationship between cross-border knowledge search breadth, depth and balance and the innovation quality of high-tech firms. The position rights, pay rights and operation rights of managerial discretion have partially significant moderating effects on the relationship between cross-border knowledge search breadth, depth and balance and the high-tech firms’ innovation quality.

Considerable literature has grown around the theme of the impact of knowledge search on the firms’ innovation quality. Nevertheless, only a few studies draw on the combination of global health emergency, cross-border knowledge search and the innovation quality; in particular, no literature has analyzed the relationship between the three from the managerial discretion perspective. Exploring the above relationships has great theoretical value for enriching and improving knowledge management and innovation management theories and provides a theoretical basis and practical support for high-tech firms to face challenges of a global health emergency and to break through the innovation dilemma.