This paper proposes a determination method of the cascaded number for lumped parameter models (LPMs) of the transmission lines. The LPM is used to simulate long-distance transmission lines, and the cascaded number significantly impacts the simulation results. Currently, there is a lack of a system-level determination method of the cascaded number for LPMs. Based on the theoretical analysis and eigenvalue decomposition of network matrix, this paper discusses the error in resonance characteristics between distributed parameter model and LPMs. Moreover, it is deduced that optimal cascaded numbers of the cascaded π-type and T-type LPMs are the same, and the Γ-type LPM has a lowest analog accuracy. The principle that the maximum simulation frequency is less than the first resonance frequency of each segment is presented. According to the principle, optimal cascaded numbers of cascaded π-type, T-type, and Γ-type LPMs are obtained. The effectiveness of the proposed determination method is verified by simulation.

The reliability of electromagnetic relay (EMR) which contains a permanent magnet (PM) can be improved by a robust design method. In this parameter design process, the calculation of electromagnetic system is very important. In analytical calculation, PM is often equivalent to a lumped parameter model of one magnetic resistance and one magnetic potential, but significant error is often caused; in order to increase the accuracy, a distributed parameter calculation model (DPM) of PM bar is established; solution procedure as well as verification condition of this model is given; by a case study of the single PM bar, magnetic field lines division method is adopted to build the DPM, the starting point and section magnetic flux of each segment are solved, a comparison is made with finite element method (FEM) and measured data; the accuracy of this magnetic field line based distributed parameter model (MFDPM) in PM bar is verified; this model is applied to the electromagnetic system of a certain type EMR, electromagnetic system calculation model is established based on MFDPM, and the static force is calculated under different rotation angles; compared with traditional lumped parameter model and FEM, it proves to be of acceptable calculation accuracy and high calculation speed which fit the requirement of robust design.