### I. Introduction

### II. Modular Motor Principle

*Z*and the number of pole pairs

*p*satisfies

*Z*= 2

*p*+ 1. Therefore, the pole–slot ratio of the unit motor in this paper is 10:9.

*F*

_{end}_{1}and

*F*

_{end}_{2}are the end forces generated by unit 1 and unit 2;

*n*is the order of Fourier expansion,

*x*is the displacement of the mover,

*F*

*is the Fourier decomposition factor,*

_{n}*L*is the distance between the unit motors,

*L*

_{1}and

*L*

_{3}are the lengths of the primary core and flux barrier, respectively, and

*τ*is the pole pitch.

*L*between the unit motors satisfies the following equation to minimize the end force:

*L*

_{1}of the unit motor is an integer multiple of the pole pitch, the width of the flux barrier can be described as follows:

*a*and

*b*are non-negative integers and

*a*>

*b*.

### III. Electromagnetic Characteristic Analysis

### 1. Analytical Model of the Slotless M-PMLSM

The materials used in the motor are all isotropic.

The permeability of the primary and secondary cores is infinite.

The permeability of PMs is the same as that of air.

*ρ*

*and*

_{s}*ρ*

*denote the primary and secondary coordinate systems, respectively;*

_{r}*R*

*is the outer radius of the air gap;*

_{g}*R*

*and*

_{m}*R*

*are the inner and outer radii of the PMs, respectively;*

_{p}*R*

*is the outer radius of the primary core; and*

_{s}*ρ*

*,*

_{m}*ρ*

*,*

_{p}*ρ*

_{3}, and

*ρ*

_{4}are the span angles of PMs, pole pitch, domain 3, and domain 4, respectively. To accurately solve the distribution of the internal magnetic field of the motor, the motor parameters need to be converted as follows:

*v*

*and the two-dimensional approximate angular velocity is*

_{s}*ω*

*, the relationship between the two satisfies the following equation:*

_{r}### 2. Analytical Model of the Slotless M-PMLSM

*M*

*and*

_{r}*M*

*are the normal and tangential components of the magnetization M*

_{ρ}_{0}, respectively, whose expressions can be expanded in the Fourier series as follows:

*M*

*=*

_{ρcn}*M*

*= 0.*

_{ρsn}#### Domain 1

##### (20)

#### Domain 3

*ρ*

_{3}; thus, its general solution form differs from that of domains 2 and 5. The general solution of the magnetic vector potential in this domain is obtained by the separation of variables method as follows:

*G*

_{41}

*= (*

_{k}*r*/

*R*

*)*

_{s}

^{kπ}^{/}

^{ρ}^{4}and

*G*

_{42}

*= (*

_{k}*r*/

*R*

*)*

_{g}^{−}

^{kπ}^{/}

^{ρ}^{4}.

### 3. Boundary Conditions

### 4. Slot Effect

*ϕ*

_{0}, the relative permeability function of the two-dimensional air gap of the M-PMLSM is expressed as follows [24, 25]:

*ρ*

*is the span angle of the slot,*

_{w}*h*

*is the height of the PMs,*

_{m}*β*(

*r*) is a nonlinear function, and Λ is the permeance.

### IV. Back EMF and End Force

### 1. Back EMF

*ψ*

*produced by a single-turn coil is as follows:*

_{c}*ρ*

_{a}_{y}is the span angle of the coil pitch.

*N*is the number of turns of the coil,

*s*is the number of coils in the series, and

*ρ*

*is the starting position of the*

_{i}*i*th coil.