According to Kirchhoff’s voltage law, the voltage balance equation of the kth phase winding of a switched reluctance motor can be expressed as: U k = R k i k + d ψ k d t (1) where U _k is the voltage of the kth phase winding, R _k is the resistance of the kth phase winding, i _k is the current of the kth phase winding, and ψ _k is the flux linkage of the kth phase winding.

Considering that SRM mainly operates in the magnetic saturation region and the highly nonlinear electromagnetic characteristics in this region, it is challenging to establish an accurate nonlinear model of an SR motor based on common electromagnetic characteristics and mathematical formulas. Based on the nonlinear model analyzed in (?), the flux linkage profile of SRM can be expressed: Ψ i , θ = L dsat i + A 1 − e − B i − L q i f θ + L q i (2) where L − q is the inductance of stator and rotor salient pole misalignment (q axis), and L _dsat is the saturation inductance for stator and rotor salient pole alignment (d axis). A, B, f(θ) are defined as presented in 3, 4, and 5, respectively. A = Ψ m − L dsat I m (3) B = L d − L dsat Ψ m − L dsat I m (4) f θ = 2 N r 3 π 3 θ 3 − 3 N r 3 π 3 θ 3 + 1 (5) where L _d is the unsaturated inductance of the d axis, I _m is the rated current corresponding to flux linkage Ψ _m , and N _r is the number of rotor poles.

The electromagnetic torque of the motor is equal to the sum of the torque generated in each interval. Due to the nonlinearity of the magnetization profile, the generated torque is a nonlinear function of phase current and rotor position. According to the principle of virtual displacement, the instantaneous electromagnetic torque at any operating point can be expressed as: T e i , θ = ∂ W ′ θ , i ∂ θ i = c o n s t (6) where W′ is the magnetic common energy of the winding, its expression is: W ′ θ , i = ∫ 0 i Ψ θ , i d i (7)

Substituting Equation (1) into Equation (7), the expression of torque T _e is: T e i , θ = f ′ θ L dsat − L q i 2 2 + A i − A 1 − e − B i B (8) where f′(θ) can be expressed as: f ′ θ = 6 N r 3 π 3 θ 2 − 6 N r 3 π 2 θ (9)

According to the laws of mechanics, the rotor mechanical motion equation of SRM under the action of electromagnetic torque and load torque is expressed as follows: T e = J d 2 θ d t + D d θ d t + T L (10) where T _e is electromagnetic torque, T _L is load torque, J is the rotary inertia, and D is the friction factor, among d θ d t .

The prototype model’s static finite element analysis uses ANSYS software. Considering the A-phase winding as an example, to analyze the torque generation capacity, the step is set as 0.5, the angle value is set from 0 to 45, and the Maxwell software calculates the torque and inductance values under the fixed phase current. The characteristic profiles of the inductor-rotor position and torque-inductor derivative are plotted using Matlab, as illustrated in Figures 1A,B.

Under the linear model, the magnetic saturation characteristic is ignored, and the phase current is fixed. Thus, the inductance derivative is linearly related to the torque. In order to analyze the influence of the inductance value on torque in practical applications, the step of the phase current is set to 1A. Figures 1A,B highlight that the coincidence degree of the inductance curves is high when the phase current is 1 6A. When the current exceeds 6 A, the motor is operated at the magnetic circuit saturation state, and the inductance value in the central area gradually decreases, showing a concave state. When the center lines of the stator’s and rotor’s salient poles are completely aligned, the motor enters a saturation state of the magnetic circuit. Then, the saturation effect increases as the phase current increases, the inductance value decreases, and the torque generation ability becomes weaker. When the front edge of the salient poles of the stator and rotor approach, the sensitivity of inductance increases, and the torque generation strengthens. Therefore, when the phase current is constant, the torque gradually increases as the inductance derivative increases and the relation between the inductance derivative and torque tends to be linear.

In order to analyze the influence of phase current on the torque generation capacity, the phase A winding is given a phase current 12A with a step of 1A, and the torque value varying with the phase current at a fixed angle is acquired. The torque-current and torque derivative-current characteristics profiles are plotted in Matlab, as depicted in Figures 1C,D.

In order to analyze the influence of phase current on torque in practical application, the step is set to 2.5, and the angle value of 0 22.5 is selected. When the angle is 0–7.5, the trailing edge of the stator pole is gradually approaching the leading edge of the rotor pole. When the angle is 7.5–22.5, the salient poles of the stator and rotor gradually coincide. Figures 1C,D highlight that the A-phase winding of the motor is conducting when the rotor position is 0–15, and the torque gradually increases as the current increases. At this time, the torque generation ability is strong. When the rotor position is 15–22.5, the A-phase winding of the motor is operated at the turn-off freewheeling state, the torque derivative-current value decreases obviously as the phase current exceeds 6A, and the torque generation ability weakens. Therefore, when the angle is fixed, the torque value gradually increases with the increase of phase current, and the relationship between current and torque tends to be square.

The torque profile is obtained by a two-dimensional static field analysis, as illustrated in Figure 2. By analyzing the influence of the current and inductance on the torque generation capacity, it is demonstrated that the torque generation capacity is stronger at points A and B.

To realize the control goal of the TSF method based on constant synthetic instantaneous torque, the following equation must be satisfied: T k θ = T ref f k θ , k = 1,2 , ⋯ ⋯ , m , ∑ k = 1 m f k θ = 1 , 0 ≤ f k θ ≤ 1 (11) where T _k (θ) is the instantaneous torque of the kth phase winding, T _ref is the synthesized instantaneous reference torque, f _k (θ) is the torque share function of the kth phase winding, and m is the number of phases of SRM.

The torque share function directly affects the torque ripple and the reference peak current of each phase. Hence, the TSF function can effectively improve the steady-state performance of SRM. The common TSF functions include linear, cosine, cubic, and exponential. This paper analyzes the linear and cosine types, with their profiles and current diagrams depicted in Figure 4.

In a single rotor angular period, the reference torque of the kth phase can be expressed as: T k _ r e f θ = 0 , 0 ≤ θ ≤ θ o n T ref f rise , θ o n ≤ θ ≤ θ o n + θ o v T ref , θ o n + θ o v ≤ θ ≤ θ off T ref f fall θ off ≤ θ ≤ θ off + θ o v 0 , θ off + θ o v ≤ θ ≤ τ r (12) where T _ref is the total reference torque, f _rise is the slope of the rising phase of the input phase, and f _fall is the slope of the falling phase of the output phase. θ _on , θ _off , and θ _ov represent the turn-on, turn-off, and overlapping angles, respectively, and τ _r is the rotor angle period. The relationship between f _rise and f _fall is defined as: f fall θ = 1 − f rise θ − ε (13) θ _ov should satisfy the following relational equation: θ o v ≤ τ r 2 − θ off (14)

Generally, the tracking ability of the current is constrained by the power converter, and the traditional torque share function ignores the influence of current tracking characteristics. Therefore, the output torque of each phase cannot track the reference torque, resulting in a large torque ripple. This paper considers this limitation in the proposed torque distribution curve design. Figure 5A illustrates the schematic diagram of the TSF control strategy based on cosine, where the solid line presents the reference torque distributed by the TSF function, and the dotted line is the actual torque generated in the motor running process. The diagram reveals that the torque at the position with a small inductance cannot track the given torque in time, resulting in a large torque error. Aiming to solve the torque ripple problem, this paper proposes a method based on the partition correction of the torque share function. According to Figure 5B, the commutation period is divided into three sub-intervals (intervals I, II, and III) with θ _c1and θ _c2 denoting the midpoints. The torque generation capacity of each phase at a different rotor position is analyzed, and the phase with a strong torque generation capacity is selected as the torque compensation phase. The performance of the torque share function is optimized according to the real-time torque tracking error, and the control ability of the actual current is improved. Thus, the optimization goal of torque ripple suppression is achieved. θ c 1 = θ o n + 1 3 θ o v (15) θ c 2 = θ o n + 2 3 θ o v (16)

The torque characteristic at different rotor positions is presented in Figure 6, and the difference in torque generation capacity between the adjacent phases during commutation is presented in Figure 7. By analyzing the rate of change for the torque at the commutation, the torque share function is constructed by partition optimization for the torque ripple minimization.

(1) Interval I: At the initial stage of commutation, the torque generation capacity of the K phase is lower than the K-1 phase. The torque of the K-1 phase is used as compensation instead of the K phase. For correcting TSF, the control block diagram of interval I is presented in Figure 8A, suggesting that the torque generation capacity of the k-1 phase is greater than that of the K phase in interval I. At this time, the K phase is at the stage of establishing an excitation current. Thus, the current value is small, and the change rate of winding inductance is low. Therefore, the actual torque generated cannot track the reference torque. At this time, the torque generation capacity of the k-1 phase is strong, and the k-1 phase can compensate for the torque error generated by the k phase.

The torque error of the K phase equation is Δ T I = T k _ r e f − T k (17)

In the overlapping area, the total electromagnetic torque equals the sum of the electromagnetic torques of the input and output phases. The updated total reference torque can be expressed as: T ref = T k − 1 new + T k = T ref f fall new + T ref f rise k = T ref f fall k − 1 + Δ f I + T ref f rise k (18)

The relationship between f _rise and f _fall is defined as f fall θ = 1 − f rise θ + θ o n − θ off (19)

(2) Interval II: In the middle of commutation, the torque generation capacity of the k-1 phase is similar to that of the K phase. The K phase is in the current conduction interval, with strong control ability. The K phase compensates for the torque error caused by commutation.

The control block diagram of interval II correction TSF is depicted in Figure 8B, highlighting that the torque-generating capacity of the k-1 phase is similar to that of the K phase. At this time, the k-1 phase is at the turn-off freewheeling state, the current gradually decreases, the torque is in a state with a fixed profile decline, and the current control ability is weak. The k-phase current in this interval is rising, and the controllability is strong. Therefore, the k-phase is selected for torque compensation in interval II.

The torque error generated in this interval is: Δ T I I = T k _ r e f − T k / k − 1 (20)

The compensation function Δf _II can be expressed as: Δ f I I = a I I θ − θ o n θ o v b I I + a I I Δ T I I c I I (21)

(3) Interval III: In the later stage of commutation, the torque generation capability of the k-1 phase is lower than that of the k-phase. The block diagram of the interval III correction TSF control is illustrated in Figure 8C, revealing that the torque-generating capability of the k-phase is greater than that of the k-1 phase. At this time, the rate of change of the k-1 phase winding inductance is small. Thus, the phase torque is large, which causes the actual torque to exceed the reference torque, resulting in torque ripples. Additionally, the torque generation capability of the k-phase is relatively strong, so the k-phase compensates for the torque error generated by k-1.

The torque error of the k-1 phase can be expressed as: Δ T III = T k − 1 − T k − 1 _ r e f (22)

The modified torque share function expression with adding the compensation function in sections is presented below: f θ = 0 , 0 ≤ θ ≤ θ o n f rise , θ o n ≤ θ ≤ θ o n + 1 3 θ o v f rise + Δ f I I , θ o n + 1 3 θ o v ≤ θ o n + 2 3 θ o v f rise + Δ f III θ o n + 2 3 θ o v ≤ θ ≤ θ o n + θ o v 1 , θ o n + θ o v ≤ θ ≤ θ off f fall + Δ f I , θ off ≤ θ ≤ θ off + 1 3 θ o v f fall , θ off + 1 3 θ o v ≤ θ ≤ θ off + 2 3 θ o v f fall , θ off + 2 3 θ o v ≤ θ ≤ θ off + θ o v 0 , θ off ≤ θ ≤ τ r (23)

In order to ensure the stable operation of the motor in speed and torque output, this paper comprehensively considers the torque share function control strategy indicators.

The torque ripple of the motor T _r is defined as: T r = T max − T min T a v × 100 % (24) where T _max is the maximum torque, T _min is the minimum torque, and T _av is the average torque.

Copper loss is one of the important factors affecting the operating efficiency of the motor, which can be expressed by the RMS value I _rms during the conduction period: I rms = 1 θ p ∫ θ o n θ off i k 2 d θ + ∫ θ o n θ off i k − 1 2 d θ (25)

The copper loss of SRM is proportional to the square of the phase current, and the copper loss generated by the motor operation can be reduced by reducing the motor’s phase current.

The flux linkage change rate of the motor is expressed as: d Ψ k d θ = U k − R k i k − L d i d t ω (26) where the voltage drop R _k I _k of the winding resistance is usually small compared to the dc-link voltage, and thus, the term R _k I _k can be ignored in qualitative analysis. The above formula highlights that the flux linkage change rate is only affected by the DC-link voltage, the current change rate, and rotational speed.