—The Linear Synchronous Motor with DC Excitation is presented based on its steady -state equations. Until now, many methods based on solving the magnetic equations of the motor or using numerical methods have been presented in various types of research , and complex mathematical equations and time -consuming design are their main pr oblems . In the proposed method, this type of motor has been designed using the electrical relations of the circuit of the Linear Synchronous Motor. Due to the simplicity of relationships and faster design, this method can be easily used in some topics , such as optimization that require repetition algorithms. For validation, the ©MATLAB has been used . Keywords — Linear Motors (LMs), Linear Synchronous Motors (LSMs) , DC Excitation, Finite Element Metho d (FEM), Optimization Algorithm I. INTRODUCTION In the contemporary era, the achievement of linear motion is being considered due to its wide application in various industries, such as industrial automation systems, transportation, and military industries . Linear Motors (LMs) can be used for the linear motion because they have many advantages over Rotary Motors (RMs) and rotary -to-linear motion converters [1]. An LM is an Electric Motor (EM) whose stator and rotor are out of circular mode so that instead of producing rotational torque, it makes linear force longitudinally. The most common mode of operation is the Lorentz force operator ; whose applied force is linearly proportional to the electric current and magnetic field. Many designs have been proposed for LMs , and they are divided into two main categories ; high-speed and high -acceleration LMs [2]. A Linear Synchronous Motor (LSM) is a Linear motor in which the mechanical movement's speed equals the speed of the moving magnetic field. Typically , multi -phase AC synchronous motors are motors with DC electromagnetic excitation in which the moving force has two components ; the first component is caused by the interaction of the traveling magnetic field with the DC magnetic field or the magnetic field generated by permanent magnets. The second component is also caused by the interaction of the traveling magnetic field with variable reluctance in the (d) and (q) axes [3-5]. The Linear Synchronous Motor is one of the types of LM. LSM is the linear counterpart of the R otary Synchronous Motor in which the electrical power input to the machine is converted into mechanical work at their output by linear movement. High efficiency , high -power factor, and an internal magnetic source make LSM suitable for various applications [6]. In the previous resear ch about LSM, various aspects such as modeling, design, and control have been investigated. Attempts to model the LSM led to models with complex mathematical relationships. Therefore, using these design models has limitations [7-8]. In this article, a method for modeling and designing an LSM with DC excitation will be presented. The main purpose of this article is to reduce the complexity of the design to achieve a suitable method for optimizing LSM . After introducing the structure of the LSM in section II, the model used for its design and calculations will be presented in section III. Then the design process will be presented in section IV. In section V, the validity of the obtained results will be evaluated using ©MATLAB, and finally, the s ummary and conclusions will be presented in section VI. II. STRUCTURE OF THE LSM Figure 1 shows the general schematic of a one -way LSM with DC excitation. The primary motor consists of three -phase windings with an iron core along the movement path. Copper wind ings are placed in grooves embedded in the iron core. The movement path consists of two parts, one part is electrified , and the other is not electrified. The electrified section is selected to be five times the moving length on which the DC excitation is l ocated (not specified in Figure 1) and is powered by a voltage source. τshpthree -phase windings (a) l -Vdc Lsτ hs (b) Fig. 1. Structure of the LSM: a) Primary (Part of Fix), b) Secondary (Part of moving) The moving part of the motor consists of an iron core and DC excitation electromagnetic poles, which are placed on top of the primary. There is no physical contact between the secondary and the primary, and an air gap separates these two parts . Secondary (moving) windings are made of copper. III. PERFORMANCE AND MODELING OF THE LSM To obtain a method for designing the LSM with DC excitation, the steady -state model of the motor is used. Therefore, in this section, the steady -state equations of the motor will be discussed . A. Equations of the Steady -State Based on the phasor diagram of the LSM excited by the salient pole , the input voltage will be calculated according to the following equation [9]: 𝑉𝑠⃗⃗⃗ =𝑅𝑝ℎ𝐼𝑠⃗⃗ +𝑋𝑠𝑑𝐼𝑠𝑑⃗⃗⃗⃗⃗ +𝑗𝑋𝑠𝑞𝐼𝑠𝑞⃗⃗⃗⃗ +𝐸𝑓⃗⃗⃗⃗ (1) The equations of the d and q axes are as follows: 𝑉𝑠sin𝛿=−𝑅𝑝ℎ𝐼𝑠𝑑+𝑋𝑠𝑞𝐼𝑠𝑞 (2) 𝑉𝑠cos𝛿=𝑅𝑝ℎ𝐼𝑠𝑞+𝑋𝑠𝑑𝐼𝑠𝑑+𝐸𝑓 (3) In the above relationship , Vs is the value of the stator 's voltage , and Is is the value of the current of the stator . Isd and Isq are the images of the Is on the d and q axes . Also, Ef is the effective value of the no -load induced voltage on one phase of the stator windings due to DC excitation and this voltage will be expressed in the following . δ is the load factor between V s and E f. Also, Rph is the resistance of each primary phase, Xsd and X sq are the reactance of the d and q axes, respectively, and are calculated from the following relationships [9]: 𝑅𝑝ℎ=𝜌𝑙𝑤 𝐴𝑤 (4) 𝑋𝑠𝑑=𝑋𝑝ℎ+𝑋𝑎+𝑋𝑠𝑑 (5) 𝑋𝑠𝑞=𝑋𝑝ℎ+𝑋𝑎+𝑋𝑠𝑞 (6) In the above equations, ρ is the specific resistance of the conductors used in the primary winding, lw is the length of the primary winding, and A w is the cross -section of the conductors. Xph is the scattering reactance of each preliminary phase, X a is the leakage reactance of the air gap, X sd and X sq are the magnetization reactance of the d -axis and q-axis, respectively. They can be calculated from the following relations [1]. 𝑋𝑝ℎ=2𝜇0𝜔[(𝜆𝑠+(𝑁1 𝑁2)𝜆𝑑)𝑙+𝜆𝑒𝑙𝑒𝑐]𝑁1𝑁2 𝑝𝑞 (7) 𝑋𝑎=6𝜇0𝜔𝑙(𝑁2−𝑁1) 𝜋 (8) 𝑋𝑠𝑑=6𝜇0𝜔𝜏𝑙𝑁12 𝜋2𝑘𝑐𝑔0𝑘𝑑 (9) 𝑋𝑠𝑞=𝑘𝑞 𝑘𝑑𝑋𝑠𝑑 (10) In which, μ0 is the magnetic permeability coefficient of air , and ω is the initial angular frequency . Also, λs, λd and λe are the specific magnetic conductivity of slot, differential , and end connections , respectively. N1 and N 2 are the number of turns in the primary phase for the overlapping part with the moving part and the electrified part empty of moving, respectively. Also, l is the motor's width, l ec is the length of end connections, p is the number of pole pairs, q is the number of slots in each phase, τ is the pole pitch, g 0 is the distance of air gap , kc is the Carter coefficient, k d and k q are coefficients of the effect of the armature on the d -axis and q -axis, respectively, and can be calculated from the following equation s [10]: 𝑘𝑑=(𝜋𝛼𝑝+sin𝜋𝛼𝑝)/𝜋 (11) 𝑘𝑞=(𝜋𝛼𝑝−sin𝜋𝛼𝑝)/𝜋 (12) That, α p is the ratio of pole arc to pole pitch. Also, Ef can be calculated from the following equation [10]: 𝐸𝑓=√2𝐵𝑓𝑙𝑁1𝜔𝑠 (13) In the above relation Bf is the density of the flux of the air gap which caused by the excitation of DC . ωs is the synchronous speed. These parameters can be calculated by the following relations [10]: 𝜔𝑠=2𝜏𝑓 (14) 𝐵𝑓=𝜇0𝑁𝑓𝐼𝑓 𝑘𝑐𝑔0(1+𝑘𝑠𝑎𝑡) (15) Where, f is frequency, Nf is the number of turns of the DC excitati on winding, I f is the DC excitation current, k sat is the coefficient of iron saturation. Also, we have: 𝐼𝑓=𝑉𝑑𝑐 𝑅𝑓 (16) That, V dc is the voltage of the DC excitation, and R f is the excitation winding resistance. B. Calculations of the Power, Force and Efficiency By ignoring the core losses, the electromagnetic power can be calculated as follows [11]: 𝑃𝑒=𝑚𝑉𝑠𝐼𝑠cos𝜑−𝑚𝑅𝑠𝐼𝑠2 (17) In which, m is the number of primary power supply phases and cosφ is the input power factor. So, the electromagnetic force of the motor is obtained from the following equation: 𝐹𝑒=𝑃𝑒 𝜔𝑠 (18) Using the relations (2), (3), and (18), the electromagnetic force can be expressed as a function of the δ (load factor): 𝐹𝑒=𝑚[𝑉𝑠(𝑅𝑠cos𝛿+𝑋𝑠𝑑sin𝛿)−𝐸𝑓𝑅𝑠 𝜔𝑠(𝑋𝑠𝑑𝑋𝑠𝑞+𝑅𝑠2)2 (19) The figure (3) shows the force -load factor characteristic (Rs=0). Also, the efficiency can be calculated as follows [11]: 𝜂=𝑃𝑒 𝑃𝑖𝑛=𝐹𝑒𝜔𝑠 𝑚𝑉𝑠𝐼𝑎cos𝜑 (20) Fig. 2. Force -load factor characteristic for a motor with R s=0 IV. PROPOSED ALGORITHM FOR DESIGN ING MOTOR In this section, a new method for designing a Linear Synchronous Motor with DC excitation at the desired speed and thrust force will be presented. This method uses the steady -state model of the motor which is presented in the previous section. A. Design of the Stator Knowing the motor's speed, the pol e pitch can be calculated from equation ( 14). Now the length of the motor is calculated from the following equation [11]: 𝐿𝑠=2𝑝𝜏 (21) To have a proper rail, we should choose the proper length for the electrified area. The length of the electrified area can be considered as an integral coefficient of the length of the motor using the following relationship [12]: 𝐿𝑒𝑛=𝑘𝑒𝑛𝐿𝑠 (22) In the above equation , ken is an integral coefficient . By increasing this coefficient, the active area will be increase , but the primary resistance will also increase, and this will cause more losses, and reduce efficiency. On the other hand, by choosing a small coefficient, the length of the active area will be reduced . Now, knowing the pole pitch and choosing the appropriate q (number of grooves in each phase in each pole), the groove pitch can be calculated from the following equation [11]: 𝜏𝑠=𝜏 𝑚𝑞 (23) To calculate the dimensions of the groove, tooth width is considered as a coefficient of the groove pitch, and we have: 𝑤𝑡=𝐶𝑡𝜏𝑠 (24) Where, Ct is a coefficient between 0 and 1 because the width of the tooth is smaller than the pitch of the groove or is actually a part of the pitch of the groove. Now the width of the groove can also be calculated. To calculate the number of turns N1 and N2, it is necessary to specify the number of turns in each coil ( nc) and the number of pairs of poles. After choosing the type of winding, we first consider nc equal to 1 and proceed wi th the calculations; If the thrust force obtained is at the desired level, we stop the calculations, otherwise , we increase n c, and we repeat the calculations until the output thrust force is within the acceptable range. In this way, having nc and choosing the type of winding for the motor, we will have: 𝑁1=𝑛𝑠𝑝𝑞 (25) That, ns is the number of conductors in each slot and it is obtained according to the type of winding . So: • Single layer winding : 𝑛𝑠=𝑛𝑐 (26) • Double layer winding : 𝑛𝑠=2𝑛𝑐 (27) And for N 2: 𝑁2=𝑘𝑒𝑛𝑁1 (28) To calculate the estimated current of the stator input (Ia), the relation (20) is used: 𝐼𝑎=𝐹𝑥𝑑𝜔𝑠 𝑚𝑉𝑠𝜂cos𝜑 (29) Where F xd is the desired output thrust force . In this regard, to estimate the required input current to produce the desired thrust force , we must have the values of the input power factor and efficiency. For this purpose, we first assume that the product of two values is an arbitrary number between 0 and 1, after calculating the parameters of the circuit and obtaining the actual current of the motor, these values will be corrected. Now, by estimating the initial current and selecting the allowed current density for the primary winding, the cross section of the groove can be calculated. With t he groove's width and cross section values, the height of the groove can be calculated. By considering the proper coefficients (such as the Carter 's coefficient) for the electrical relations , that presented in the previous section, the electrical parameter s of the motor can be calculated. By calculating these parameters and applying the relations (1) and (2) and using the phasor diagram of the real input current can be calculated. It should be noted that the suitable angle of the load can be calculated according to the characteristic force -load angle according to equation (19). To calculate Ef in the mentioned equations , first we choose a value for the flux density of the stator tooth according to the magnetization curve of the iron , which used in the stator core. First, by ignoring the effect of the armature on the air gap's flux density, the upper limit for the air gap 's flux density caused by DC excitation can be obtained from equation (15). With the appropriate choice for this flux density, the induced voltage Ef can be calculated. It should be noted that this is a suitable choice because the main flux of the motor is the same flux caused by DC excitation , and the changes in the electrical characteristics of the stator will not chan ge much in this flux density. Of course, it should be noted that at the end, the actual value of the air gap's flux density will be obtained by considering the effect of the armature , and accordingly, the actual flux density of the stator teeth. Now, with the determination of the actual current of the motor, the power factor can be calculated using the relations (1), (2) , and using the phasor diagram , and the efficiency from the relation (20). By calculating the actual values of the power factor and effic iency, the actual value of the product of these two values can be calculated. If this real value is within the acceptable range of the initially selected value, we stop the calculations for this section, and if not, a new selected value is determined acc ording to the following relationship and we repeat the calculations to reach convergence for this value: (𝜂𝑐𝑜𝑠𝜑)𝑛𝑒𝑤=(𝜂𝑐𝑜𝑠𝜑)𝑐𝑎𝑙+(𝜂𝑐𝑜𝑠𝜑)𝑒𝑠 2 (30) Now the output thrust force is calculated; If the difference between the obtained value and the desired thrust force is acceptable, we stop the calculations, and if not, put n c+1 instead of n c and repeat the calculations until the desired value for the thrust force is obtained. At t he end, by determining the number of turns in each coil, the number of turns in each phase, the corresponding initial current and the cross -sectional area obtained for the conductors, the desired wire is selected from the standard table. According to the d iameter of the selected conductor, the width of the groove and tooth should be recalculated. By calculating the width of the groove and tooth, the flux density in the teeth is calculated from relations (15) and (16) and compared with the maximum value. If the value of the obtained flux density is higher than its default value, the tooth width should be increased. At the end, the output parameters of the machine and the weight of its different parts are recalculated. The block diagram of this method is shown in Figure 3. start Selection of the parameters like synchronous speed, thrust force, etc. Calculation of the polar pitch, groove pitch, length of overlapping part and active area, width of groove, and tooth Select the number of turns )cn per coil ( Calculation of the number of turns in each phase (N 1 and N 2) Select ηcosφ in the range of [0-1] Estimating the initial current and selecting the conductor cross -section Calculation of the output parameters ηcosφ = [η cosφ]cal[ηcosφ]New =([ηcosφ]Cal+[ηcosφ]es )/2 No Fx = F xdYes1+c= ncn NoChoosing a conductor from among the available conductors YesRecalculating of groove s dimensions and output parametersCalculating the flux density of the toothBt > B tmaxIncrease the tooth s widthNoCalculation of the ultimate parameters and weight of the different parts of the motor YesEnd Fig. 3. Block diagram of the proposed method B. Design of the Rotor Now, by designing the stator and obtaining the required air gap flux density, it is possible to calculate the WfIf using equation (12). By using equations (13) and (14), other parameters of the DC excitation circuit can be calculated. C. Optimization In order to provide proper optimization, the machine's sensitivity to changes in parameters such as frequency, machine width, and air distance is first examined. These studies are shown in Figures 4 to 6. As the frequency increases, the polar pitch decreas es. The primary resistance will decrease due to the direct relationship with the polar pitch, and on the other hand, the primary reactance will increase with the increase in frequency. With these explanations and precision in Figure 4, we will see that the input current increases initially with the decrease of the resistance and then decreases with the increase of the reactance. Accordingly, the power also increases at first and then decreases. Also, the downward trend of the power factor can be seen. Effic iency is also improved by reducing initial resistance and reducing losses. Increasing the width of the machine to specific points for power, power factor, and efficiency brings an upward trend, which can be seen in Figure 5, which is caused by changes in t he induced voltage and the angle of the input current with this voltage. Figure 6 shows that choosing a small air gap improves the machine's output parameters on the other hand, in applications such as magnetic trains, reducing the air gap is limited due t o high speeds and the possibility of the train hitting the rail. The power to reduce the air gap too much. Of course, parameters such as load angle and tooth width were also investigated, but due to the limitation in the number of pages, only their optimiz ation results are given. The Particle Swarm Optimization (PSO) algorithm has optimized the design [1]. This article uses the normalized multiplication of individual cost functions to convert multi - objective optimization into single -objective optimization. Once the weight cost function is considered, first, the inverse normalized cost functions of rail weight per unit length and moving weight are added together and considered a cost function. This cost function can be expressed as follows: 𝑀=𝑓𝑢1×𝑓𝑢2×𝑓𝑢3×𝑓𝑢4 (31) Where 𝑓𝑢1, 𝑓𝑢2, and 𝑓𝑢3 are the desired objective functions for propulsion force, power factor, and efficiency, respectively. 𝑓𝑢4 is also the objective function for choosing the minimum rail weight per unit length and moving weight, which is presented below how to calculate it: 𝑓𝑢4=(1 𝑊𝑚𝑒𝑡𝑟+1 𝑊𝑡𝑝2) 2 (32) In the above relationship, 𝑊𝑡𝑝2 and 𝑊𝑚𝑒𝑡𝑟 are the weight of moving and the weight per unit of rail length, respectively. The optimized values of the desired parameters and each range are shown in Tables I and II. 1.5 1 0.5 0 10 20 30 40 50 60 Input Frequency [Hz] 15 10 5 0 10 20 30 40 50 60 Input Frequency [Hz] 1 0 10 20 30 40 50 60 Input Frequency [Hz] 100 80 60 10 20 30 40 50 60 Input Frequency [Hz] Fig. 4. The effect of input frequency on the motor's output parameters 1.5 1 0.5 0 50 100 150 Machine Width [mm] 50 0 50 100 150 Machine Width [mm] 1 0 50 100 150 Machine Width[mm] 100 0 50 100 150 Machine Width[mm] Fig. 5. The effect of machine width on the motor's output parameters 1.5 1 0.5 0 10 20 30 40 50 Airgap [mm] 40 0 10 20 30 40 50 Airgap [mm] 1 0 10 20 30 40 50 Airgap [mm] 100 0 10 20 30 40 50 Airgap [mm] Fig. 6. The effect of airgap on the motor's output parameters V. VERIFICATION To validate the results which obtained from the design presented in the previous section, a motor with the specifications in table I was designed with the proposed method , and the circuit parameters obtained from this method were simulated with the Finite Element Method. ©Maxwell was used for this simulation. The results of the simulation are shown in Figures 7 to 10. According to the rail's constraint s, the initial speed and weight are considered 5 m/s (synchronous and 1 kg, respectively. The figure 7 shows the motor's speed. According to figure 7, after 100 ms, the speed of the motor reaches the synchronous speed. Figure 8 shows the motor 's tractive force at nominal load. As can be seen in the figure 8, the average output power is 953.42 N. Figures 9 and 10 show the input current and voltage, respectively. Also, the distribution of the flux density is shown in Figure 1 1. In Figure 11, it can be seen that the maximum flux density occurs in the yoke and it is equal to 2T. TABLE I. MACHINE PARAMETERS FOR OPTIMIZATI ON AND THEIR RANGE OF CHANGES Parameters Unit Range of changes (Max -Min) Optimization value Frequency Hz 100-10 10.8 Number of slots/poles/phases - 2-1 2 Motor's width mm 200-20 90 Airgap mm 30-10 10.6 Groove width/Groove pitch - 0.5-0.3 0.5 Load's angle degree 180-0 12.75 TABLE II. OPTIMIZATION RESULTS FOR THE DESIRED OBJECTIVE FUNCTIONS Parameters Unit Optimization results Thrust force N 953.42 Power factor - 0.932 Efficiency [%] 71.7 Rail weight per unit length Kg/m 86.96 Moving's weight Kg 480.364 For a better comparison between the output results of two methods at the nominal load, it is displayed in Table III. In this table, the error is calculated as follows: Error=FEM results−Calculated results FEM results (33) As it is manifest from Table III, the induced voltage predicted by the proposed method is slightly higher than the actual value, which means that the air gap is smaller than the actual value. With this analysis, it can be expected that the current is slightly lower than the actual value predicted. 20 10 0 -10 -20 0 50 100 150 200 250 300 350 400 450 Time [ms] Fig. 7. The speed obtained from the finite element -1 0 50 100 150 200 250 300 350 400 450 Time [ms] Fig. 8. The output force obtained from the finite element method Thrust [kN] Input Current[A] Efficiency [%] Power Factor Thrust [kN] Input Current[A] Efficiency [%] Power Factor Thrust [kN] Input Current[A] Power Factor Efficiency [%] 20 0.5 50 0.5 50 0.5 40 Speed [m/s] Thrust [kN] 0 1 2 3 4 Ec 0 0 50 100 150 200 250 300 350 400 450 Time [ms] Fig. 9. Input current obtained from finite element method 400 200 0 -200 -400 0 50 100 150 200 250 300 350 400 450 Time [ms] Fig. 10. Induced voltage on the stator phases obtained from the finite element method Fig. 11. Distribution of the flux density in different parts of the motor TABLE III. COMPARISON OF THE OUTPUT RESULTS FOR THE PROPOSED METHOD AND THE FINITE ELEMENT METHOD Parameters Unit Results of the proposed method Results of the FEM Error [%] Thrust force N 953.42 953.42 0 Input Current A 10.82 13.06 17.2 Induced Voltage V 153.01 148.98 -2.7 Power factor - 0.932 0.898 -3.78 Efficiency [%] 71.7 65 -10.3 VI. CONCLOTION In this article, a method was presented for designing a linear synchronous motor based on steady state equations. In the previously presented methods, electromagnetic equations or numerical methods such as finite element methods , are usually used, which in addition to complexity, are time - consuming and not suitable for optimization. This is despite the fact that the presente d method solves the mentioned problems and can be used in sensitivity analysis and optimization. The finite element method was used to validate the design 's results, and its results confirm the results of the presented method.