# Electrical Circuits Simulator in Null-Flux Electrodynamic Suspension Analysis
## 2 Unveiling the Inner Workings: Exploring the Principles and Model of an EDS System
### The Research Gap
The Fourier transform analysis approach is utilized to decompose complex waveforms into their constituent sinusoidal components, enabling a thorough exploration of the spatial and harmonic properties of magnetic force and field distributions. This technique operates in the frequency domain, representing a time function of a signal as a combination of different frequencies. In the domain of EDS systems, the application ofthis analysis is particularly notable in developing a comprehensive theory of moments for EDS magnetic levitation systems. This theory, incorporating double Fourier series and dynamic circuit principles, facilitates the calculation of angular derivatives of moments and forces, as extensively discussed by [PERSON] (1982) [PERSON] (1982). Fourier analysis achieves a deeper understanding and analysis of EDS systems, providing valuable insights into their behaviors and performances. Experimental testing and measurements play a crucial role in studying the performance of EDS systems and validating theoretical models. These experiments use physical setups, apparatus, and data-gathering techniques to record and analyze forces, vibrations, and other relevant factors. While measuring devices inherently introduce some level of error and uncontrollable random variables exist in natural systems, it is through experimentation that theories are validated. In this work's case study, the experimental data obtained from the test track substantiated the feasibility of the proposed method. Integrating experimental data ensured more comprehensive understanding of, and validation of, the proposed approach, and strengthened its credibility and applicability in real-world scenarios. Comparative Analysis of Related Studies  
The intuitive model for eddy currents in thin sheets was presented by [PERSON] (1972) in 1972, predating the understanding of electron properties and serving as a foundational work in EDS. [PERSON] (1973) described the fields within metallic plates using diffusion equations. However, these analyses were conducted solely for point magnetic charges. [PERSON] (1973) utilized the method of image currents to describe magnetic potential in perfect conductors. [PERSON] (1974) applied the image method and [PERSON]'s formula to calculate the resultant force on a coil moving over an infinite conducting plane. [PERSON] (1975) derived the image force formula acting on a real coil due to its image, employing the conservation of energy theorem. Subsequent studies began to employ empirically obtained constants. [PERSON] (1976) established the relationship between lift and drag forces for various magnetic sources. [PERSON] (1977) demonstrated the need for different calculations of maximum drag force, based on the thinness/thickness, or penetration depth, of a metallic plate. [PERSON] (1977) proposed a similar correction to [PERSON]'s recoil velocity. The FEM is a numerical tool for solving [PERSON]'s equations, considering electromagnetic fields with specified boundary conditions and design geometries [PERSON] (1977); [PERSON] (1977). However, the method becomes computationally intensive when dealing with systems involving relative movements and temporal and spatial dependencies, due to the dynamic meshing required at each simulation step, particularly in 3D analyses. As shown by [PERSON] et al. [PERSON] (2018), efforts to circumvent the need for meshing at each step aim to streamline the computational process. Utilizing the vector magnetic potential boundary condition, they avoided the moving mesh of onboard superconducting (SC) coils. However, as the number of coils in a model increases, so does the computation time required to calculate the forces. The Argonne Laboratory introduced the DCT for EDS systems in the 1990s. Considering the many components involved, its elegance and efficiency stem from representing an EDS system as a set of differential equations in matrix form. The focus is on directly calculating forces without additional transformations [PERSON] (1977).