*3.1. Analytical Solutions Applied for the LIM Evaluation*

LIMs are made of several major components such as the magnetic steel primary core, distributed three-phase copper winding with a three-phase excitation terminal, aluminum top cap, and magnetic BI. The primary is finely laminated and during LIM electromagnetic calculations, for simplicity, the conductivity of the primary can be set to zero. However, the BI conductivity cannot be assumed as zero because the BI is usually only coarsely laminated—mainly for economic reasons. Instead, for the case of the slab-shaped BI, the BI conductivity must be set to its true material value and a respectively lower value for the case of coarse laminations. This value of conductivity is usually determined based on matching the test LIM's longitudinal performance with the simulation model results. Theoretical methods of assessing the BI conductivity have been proposed but, so far, they only serve as a guidance for the empirical derivation. Figure 4 shows the three-dimensional model of the LIM [29].

**Figure 4.** Three-dimensional model of the LIM [29].

The development of analytical methods aimed at solving the LIM problems (separation of variables method, integral equations method) took place in the 1970s. Currently, the main method of analysis is Finite Element Analysis (FEA). Many recently published papers contain both analytical and numerical solutions, where the analytical calculations have mostly been made for validating the results of the numerical models.

Despite significant progress in numerical algorithm efficiency and modern computer speeds, the 3D simulation models of large LIMs are still prohibitively time consuming to solve. The following simplifying assumptions are usually made to the LIM to simplify the calculation process:


The equation that describes the electromagnetic field distribution within the 2D LIM model has the following form:

$$\frac{\partial^2 A}{\partial \mathbf{x}^2} + \frac{\partial^2 A}{\partial y^2} = \mu(-J + j\omega \sigma A + \sigma v\_x \frac{\partial A}{\partial \mathbf{x}}) \tag{1}$$

where *A* is the *z*-component of the magnetic vector potential, *J* is the *z*-component of current density, ω is the angular frequency, *vx* is the velocity in the *x*-direction, and σ and μ are the conductivity and permeability, respectively.

Analytical solutions for the evaluation of the LIM properties have been discussed in many papers. In most of them, the standard mathematical approach to LIM modeling is to define the currents of the primary windings as sinusoidally distributed current sheets [17,30–35].

Mathematical and experimental research of coils or filaments moving above a conducting plate (limited to DC excitation) were performed in the majority of the Maglev application studies [36–40]. The system of a stationary filament or coil with AC excitation above a conducting plate has also been studied [41]. In [42], a method of LIM winding optimization has been discussed.

The finite length of a LIM results in a number of well-recognized effects such as the previously mentioned highly speed dependent end effect responsible for the demagnetization of the front end of the machine [17,35]. The end effect has been described by many authors. One description in particular [26], based on an average model of the non-saturated LIM and later expanded to include saturation [2], has been adopted for practical, real-life LIM control algorithms [29,43].

The end effect causes a drop in the effective magnetizing inductance [44,45] and a total motor impedance, which results in a more complex LIM control scheme [43]. Analytical equations defining the end-effect induced magnetizing inductance correction factor are derived in [46]. Extensive studies on the compensation of the magnetizing inductance, reported in [29], propose the adaptation of traction LIM control methods to the depth of demagnetization of the machine in real time. Correction of the end effect for vector controls was also proposed by others, as in [43] and [46], with the equivalent circuit of the LIM based on the formula proposed by Duncan [46]. The equivalent circuit concept developed by Duncan inspired many subsequent investigations [47–52]. In [43] and [53], the control circuit is based on *d*-axis and *q*-axis equivalent circuits with parameter correction based on Duncan's method. A fuzzy logic controller is suggested in [54] based on the flux linkages of the primary and secondary. LIM performance calculations of both the motoring and braking forces, impacted by the end effect, based on the space-vector equivalent circuit of the LIM, are calculated in [55–57]. Applications of the field-oriented controls and a model reference adaptive speed observer for LIMs are also reported in [58,59], respectively. The transverse edge effect and the saturation effect are added to Duncan's model in [60]. The last examples prove that any improvement in Duncan's approach have a high potential to enhance subsequent control schemes.

Experimental measurements carried out on a traction LIM demonstrate that the leakage inductance can be, depending on the design of the reaction rail, as high as ten percent of the secondary inductance. The effects of the secondary leakage become significant as the LIM speed increases. The non-immediate current response of the RR due to its non-zero leakage inductance causes the demagnetization to be less pronounced. The assumption, in [26], that only the constant magnetizing current of the primary elicits the RR current response has been adapted by many researchers. In [28], a more accurate assumption is made. It is observed that the induced current in the RR is excited by the sum of the averaged primary magnetizing current and the RR induced current. The resultant steady-state current response is obtained by an application of a recursive algorithm converging to a finite sum of the infinite series. The final form of the sum of the infinite series represents the magnetizing current correction for the equivalent LIM circuit. As has been demonstrated by comparing experimental results with measurement, a derived magnetizing inductance correction factor predicts the LIM performance much better.

The Fourier series method was applied in [61,62] for the LIM evaluations. Instead of simplifying the primary excitation to a form of current sheets, the authors modeled the primary excitations as discrete coils. The discrete coils approach leads to a more intuitive and realistic model of the LIM and allows for the representation of spatial harmonics arising from a discrete current distribution. Frequency domain solver FEA simulations are also used here to validate and cross check the analytical model results. This is important as it establishes the practicality of the FEA frequency domain computation as a preferred replacement for the time-consuming transient computations. First, the evaluation is done by analytically solving a simple pair of filaments moving relative to a RR constructed of aluminum and iron plates and carrying a harmonic current. This solution becomes a building block for modeling a complete LIM. Because of speed and convenience, the analytical model is a practical and efficient way of rapidly assessing the impact of design changes on the performance of the LIM and helps to qualify the adapted FEA solution method.

Figure 5a shows a steady-state 2D calculation model of a discretely distributed coil with AC current excitation at an arbitrary frequency, moving over and in parallel to a conducting plate with an arbitrary speed [61], and Figure 5b shows the extension of this model.

The approach presented in [61] is similar to the algorithm given in [63,64], where Fourier transform and a mixture of magnetic scalar and vector potential formalism were used. However, the solution presented in [61] makes use of the Fourier series instead of transform and of vector potential formalism without the need of scalar potential to achieve the desired results. The analytical treatment of a 2D LIM model with a primary source moving relative to a conducting plate has many applications, including Maglev and traditional linear propulsion machines.

The analytical approach presented in [62] is similar to the work first presented in [32] and [61], where models of the LIM with current sheet finite primary excitations were presented using Fourier transforms and series methods, respectively. Two papers, [61,62], satisfy the analytical solution for the entire LIM by referring to a vector potential formalism only. The validation of results obtained in [61] and [62] was performed by means of FEA, described in the next chapter.
