# Electrical Circuits Simulator in Null-Flux Electrodynamic Suspension Analysis
## 1 Introduction  
The contactless nature of magnetic levitation (Maglev) trains offers increased speed and efficiency in transportation systems [1]. Currently, three magnetic levitation techniques are employed for Maglev transportation: high-temperature superconducting (HTS) flux pinning suspension, electromagnetic suspension (EMS), and electrodynamic suspension (EDS) [2]. HTS relies on the non-ideal diamagnetism and flux-pinning properties of the type-II superconductors to achieve stable electromagnetic forces between permanent magnet guideways and superconductors [3, 4]. EMS utilizes the interaction between ferromagnetic materials and electromagnets, requiring a closed control loop, due to the unstable nature of the attractive force [5, 6]. EDS, the focus of this work, is based on the induced currents forming in conductive materials when exposed to a variable magnetic field, generating repulsive forces [7, 8]. Superconducting EDS is suitable for high-speed trains that lack commercial operation vehicles, with a Japanese prototype which is expected to launch in 2027 [9]. High-speed trains have proven to be a viable option for distances up to 1000 km, offering advantages, such as time and cost savings, travel comfort, and reliability, compared to other modes of transport [10]. Due to their contactless nature, Maglev trains distribute the vehicle's load along its entire length, reducing track stress and allowing higher speeds [11]. This results in a lighter, less expensive infrastructure with reduced maintenance requirements [12]. The U-shaped guideway structure enhances safety, as derailments are highly unlikely [13]. Additionally, the absence of traditional rotary engine components leads to a lighter vehicle with a reduced cross-sectional area, contributing to a more compact design and increased energy efficiency [14, 15]. The calculation of magnetic forces in a Maglev system has peculiarities from a linear machine [16]. Radial forces between the stator and rotor provide lift for the vehicle, with spatial harmonics affecting the performance of the EDS system [17]. Transient effects of the temporal-spatial dependence of magnetic forces impact component vibration, noise, vehicle motion, and ride quality [18]. Various approaches are used to calculate these forces, including numerical models based on harmonic analysis via Fourier transform [19] or on finite element methods [20; 21]. However, these methods often require significant computation time. Dynamic circuit theory (DCT) offers a more efficient alternative, considering the EDS system in terms of space and time-dependent circuit parameters governed by differential equations [22; 23]. This method supports edge and end effects and is well-suited for dynamic and transient analysis and computer simulation. This research was motivated by the need to develop a time-efficient method for calculating the forces generated by a null-flux EDS system, eliminating the lengthy computational processes required by finite element methods. This approach, which has not been explored in previous studies, aimed to simplify the research process involved in investigating electrodynamic levitation phenomena. By using this approach, researchers would no longer be burdened with writing extensive algorithms or relying on computationally demanding methods. Another motivation was to provide a straightforward way to study the non-cross-connected null-flux system. Implementing the system in the simulator could solve induced currents and generate forces quickly. Additionally, the research aimed to facilitate the transition to analyzing cross-connected systems by connecting conductive wires. This flexibility would facilitate the introduction of other electrical components and modifications to the electrical circuit, enabling further exploration and experimentation. This paper presents a numerical calculation model using an electrical circuit simulator to analyze the interaction of superconducting (SC) coils and 8-shaped coils, based on DCT and Yamanashi test line data [24; 25].