Torque Ripple Reduction in Surface-Mounted Permanent Magnet Machine with Model-Based Current Reference Control

Abstract

Permanent magnet synchronous machines (PMSMs) are widely used in high-performance drive systems. However, parasitic torque ripple remains a critical limitation, causing acoustic noise, mechanical vibration, and speed fluctuations. This study presents a compact, model-based torque control strategy for surface-mounted PMSMs (SPMSMs) that suppresses torque ripple by generating a structured current reference. Grounded in the magnetic co-energy principle, the proposed method utilizes a deterministic analytical model to compensate for cogging torque and inductance harmonics, avoiding computationally intensive iterative estimators. A primary contribution involves adapting the harmonic injection profile to varying loads and magnetic saturation levels. Comprehensive finite element analysis (FEA) co-simulations demonstrate that the proposed method reduces torque ripple by approximately 87.5% and speed ripple by over 90% at 1500 RPM compared to conventional maximum torque per ampere (MTPA) strategies. Furthermore, extended dynamic analysis confirms superior robustness during start-up, transients, and low-speed operation (100 RPM), maintaining high control authority even under deep magnetic saturation (2.0 p.u.). Performance evaluations verify that this significant enhancement in torque quality is achieved with a negligible increase in total power losses (~2.1%), presenting a computationally feasible solution for industrial embedded platforms.

1. Introduction

Permanent magnet synchronous motors (PMSMs) have become increasingly popular in industrial and traction applications such as electric vehicles (EVs), robotic systems, machine tools, and household appliances, owing to their remarkable features, including high efficiency, compact size, high power density, reliability, and torque–speed characteristics [1,2,3,4,5,6].

Specifically, surface-mounted PMSMs (SPMSMs) are widely preferred for their simple rotor construction and cost-effective design. However, these machines inherently suffer from torque ripple, which leads to undesirable acoustic noise, mechanical vibration, and even shaft fatigue or failure, thereby degrading system performance and reliability [7,8,9,10,11].

The primary sources of torque ripple in SPMSMs are typically associated with distortions in the air-gap magnetic flux density and harmonic components in the back-electromotive force (EMF) [12,13]. In addition, current measurement inaccuracies, unbalanced phase currents, and nonideal switching behaviors further exacerbate the ripple phenomenon [7].

Over the years, a variety of torque ripple minimization strategies have been proposed, which can generally be divided into design-based and control-based methods [2,3,14].

From a design perspective, techniques such as fractional-slot configurations, stator or magnet skewing, and magnet reshaping have been explored to reduce torque pulsations [2,3,7,15]. Although effective, these techniques increase the manufacturing complexity and production cost, and are often inapplicable to existing motor geometries [6,8,16,17].

On the other hand, control-based approaches have gained prominence since they can reduce torque ripple without altering the machine’s physical structure.

Typical methods include maximum torque per ampere (MTPA) operation under zero d-axis current (i_ds = 0) using proportional–integral (PI) control [2,18], deadbeat control [19,20], iterative learning control (ILC) [21,22,23], repetitive control [24], sliding mode control (SMC) [25], and model predictive control (MPC) [16,26,27,28,29].

While PI control is simple and easy to implement, it suffers from limited disturbance rejection and poor robustness to parameter variations [13,18].

Deadbeat control offers faster transient performance, but requires feed forward compensators that must be specifically tuned for each machine, which increases system complexity [19,20].

Similarly, ILC and repetitive controllers are capable of compensating periodic torque disturbances, but they exhibit slow dynamic responses and require careful gain selection [21,22,24].

SMC provides strong robustness against both periodic and nonperiodic disturbances, though chattering and parameter dependency remain key challenges [25].

Meanwhile, MPC has been recognized for its high dynamic performance and constraint-handling capability, but its computational complexity and sensitivity to model uncertainties limit its use in embedded motor drive systems [16,26,29].

Beyond classical model-based strategies, data-driven approaches—such as those utilizing artificial intelligence, adaptive learning, or iterative optimization—have recently shown promising ripple suppression performance [30,31].

However, these methods often require large datasets, high computational resources, and offline training, making them less practical for real-time embedded applications [32,33]. Their performance also tends to be sensitive to data quality and operating condition variations, which restricts their general applicability.

In contrast, model-based control strategies rely on analytical or finite element analysis (FEA)-derived models to accurately represent the electromagnetic behavior of the machine and compensate for the underlying causes of torque pulsations [34,35].

Among them, current reference optimization methods based on magnetic co-energy and flux linkage variations have gained attention because they enable direct compensation of harmonic torque components without relying on learning structures or auxiliary filters [36].

Nevertheless, many of the existing model-based methods still depend on iterative tuning or additional filtering processes, increasing their implementation burden [37].

To address the limitations of conventional torque control strategies, this paper proposes a compact model-based current reference generation method for effectively minimizing torque ripple in SPMSMs. The proposed model leverages an analytical torque estimation model derived from the magnetic co-energy principle [38], enabling precise identification of the harmonic components responsible for torque pulsations. Thus, structured and predictable current references are synthesized to suppress ripple-inducing harmonics without causing unpredictable harmonic interactions.

In the current literature, various advanced control strategies have been proposed to mitigate torque ripple and enhance the robustness of PMSM drives. Notable approaches include Disturbance Observer-Based Control (DOBC) [24], SMC [39], and ILC [13]. While these methods provide effective disturbance rejection and precision, they often require complex gain tuning, auxiliary filters, or extended learning periods to converge. Furthermore, advanced strategies like MPC [40,41,42] and Artificial Neural Networks (ANNs) offer notable performance advantages for nonlinear drive applications. However, their practical implementation is often hindered by severe hardware limitations [8]. These methods typically require computationally intensive iterative optimizations or massive matrix multiplications, necessitating expensive high-performance processors or FPGAs. In contrast, the model presented in this study provides a highly precise yet computationally light standalone solution that is economically feasible for standard industrial DSPs.

To rigorously validate this computationally efficient model, this study is specifically aligned with the Model-Based Design (MBD) methodology. In the V-cycle of industrial drive development, the analytical modeling and virtual validation phase is crucial to eliminate design flaws before costly hardware iterations. Moreover, the experimental analysis of phenomena requiring high dynamic response and wide bandwidth sensor hardware, such as torque ripple, is often constrained by the low-pass filtering effects of physical test setups. Consequently, this paper focuses strategically on high-fidelity coupled simulations (co-simulations) as a virtual environment. This approach establishes a distinctive foundation that minimizes hardware implementation expenses and enhances design precision.

The main contributions of this paper can be summarized as follows:

A compact, model-based current reference generation strategy is developed based on the magnetic co-energy principle. This model precisely compensates for both structural cogging torque and spatial inductance harmonics without relying on computationally intensive iterative estimators, auxiliary filters, or AI-assisted learning algorithms. The analyses conducted show that the proposed model is economically feasible and has a low computational burden for standard industrial DSPs.

The study introduces an offset-scaling approach that seamlessly adapts the required harmonic current injection profile to varying load levels. This allows the control algorithm to maintain high torque precision under magnetic saturation without requiring complex online parameter re-identification.

Through an advanced 2D-FEA/MATLAB R2024b co-simulation framework, it is demonstrated that the proposed method achieves an 87.5% reduction in torque ripple and an over 90% reduction in speed ripple. The loss analysis verifies that this high precision is attained with a negligible increase (~2.1%) in total power losses.

Extended dynamic analysis confirms the superior robustness of the proposed method during startup, transient load steps, and under deep magnetic saturation (up to 2.0 p.u. load), demonstrating its high control authority across diverse operating conditions.

2. Analytical Modeling of PMSM and Parameter Characterization

2.1. PMSM Mathematical Model

Figure 1 shows the structural model of the SPMSM employed in the analysis. In this model, the three-phase stator winding voltage equations are expressed by (1)–(3). Here, u_a, u_b, and u_c represent the stator phase voltages, λ_a, λ_b, and λ_c denote the magnetic flux linkages of the phase windings, and R_a, R_b, and R_c are the stator resistances. The terms L_aa, L_bb, and L_cc correspond to the self-inductances of each phase, while L_ab = L_ba, L_bc = L_cb, and L_ca = L_ac represent the mutual inductances between phases. Additionally, Ψ_Ma, Ψ_Mb, and Ψ_Mc are the magnet-induced flux components, and i_a,b,c denote the stator currents.

$${\mathrm{u}}_{\mathrm{a}}={ \mathrm{R}}_{\mathrm{a}}{\mathrm{i}}_{\mathrm{a}}+{\displaystyle \frac{\partial {\mathsf{\lambda }}_{\mathrm{a}}}{\partial \mathrm{t}}}$$

(1)

$${\mathrm{u}}_{\mathrm{b}}={\mathrm{R}}_{\mathrm{b}}{\mathrm{i}}_{\mathrm{b}}+{\displaystyle \frac{\partial {\mathsf{\lambda }}_{\mathrm{b}}}{\partial \mathrm{t}}}$$

(2)

$${\mathrm{u}}_{\mathrm{c}}={\mathrm{R}}_{\mathrm{c}}{\mathrm{i}}_{\mathrm{c}}+{\displaystyle \frac{\partial {\mathsf{\lambda }}_{\mathrm{c}}}{\partial \mathrm{t}}}$$

(3)

The equations between the magnetic flux linkages, phase currents, and permanent magnet (PM) flux are given by (4)–(6).

$${\mathsf{\lambda }}_{\mathrm{a}}={ \mathrm{L}}_{\mathrm{aa}}{\mathrm{i}}_{\mathrm{a}}+{ \mathrm{L}}_{\mathrm{ab}}{\mathrm{i}}_{\mathrm{b}}+{\mathrm{L}}_{\mathrm{ac}}{\mathrm{i}}_{\mathrm{c}}+{\mathsf{\Psi }}_{\mathrm{Ma}}$$

(4)

$${\mathsf{\lambda }}_{\mathrm{b}}={\mathrm{L}}_{\mathrm{ba}}{\mathrm{i}}_{\mathrm{a}}+{\mathrm{L}}_{\mathrm{bb}}{\mathrm{i}}_{\mathrm{b}}+{\mathrm{L}}_{\mathrm{bc}}{\mathrm{i}}_{\mathrm{c}}+{\mathsf{\Psi }}_{\mathrm{Mb}}$$

(5)

$${\mathsf{\lambda }}_{\mathrm{c}}={\mathrm{L}}_{\mathrm{ca}}{\mathrm{i}}_{\mathrm{a}}+{\mathrm{L}}_{\mathrm{cb}}{\mathrm{i}}_{\mathrm{b}}+{ \mathrm{L}}_{\mathrm{cc}}{\mathrm{i}}_{\mathrm{c}}+{\mathsf{\Psi }}_{\mathrm{Mc}}$$

(6)

In these expressions, both the magnet flux and the inductance components vary periodically with the electrical rotor angle θ_e reflecting the inherent spatial periodicity of the SPMSM magnetic field distribution. The flux components generated by the permanent magnets in each stator phase are defined in (7)–(9).

$${\mathsf{\Psi }}_{\mathrm{Ma}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)={\mathsf{\Psi }}_{\mathrm{M}}\sin {\mathsf{\theta }}_{\mathrm{e}}$$

(7)

$${\mathsf{\Psi }}_{\mathrm{Mb}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)={\mathsf{\Psi }}_{\mathrm{M}}\sin {(\mathsf{\theta }}_{\mathrm{e}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})$$

(8)

$${\mathsf{\Psi }}_{\mathrm{Mc}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)={\mathsf{\Psi }}_{\mathrm{M}}\sin {(\mathsf{\theta }}_{\mathrm{e}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})$$

(9)

Furthermore, the self and mutual inductances are functions of the rotor position, as indicated by (10)–(15). In these equations, L_la, L_lb, and L_lc denote the leakage inductances, L_avr represents the average phase inductance in the a-b-c reference frame, and L_θe indicates the inductance component dependent on the rotor electrical position.

$${\mathrm{L}}_{\mathrm{aa}}{(\mathsf{\theta }}_{\mathrm{e}})={\mathrm{L}}_{\mathrm{la}}+{\mathrm{L}}_{\mathrm{avr}}-{\mathrm{L}}_{{\mathsf{\theta }}_{\mathrm{e}}}\cos {(2\mathsf{\theta }}_{\mathrm{e}})$$

(10)

$${\mathrm{L}}_{\mathrm{bb}}{(\mathsf{\theta }}_{\mathrm{e}})={\mathrm{L}}_{\mathrm{lb}}+{\mathrm{L}}_{\mathrm{avr}}-{ \mathrm{L}}_{{\mathsf{\theta }}_{\mathrm{e}}}\cos {(2\mathsf{\theta }}_{\mathrm{e}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})$$

(11)

$${\mathrm{L}}_{\mathrm{cc}}{(\mathsf{\theta }}_{\mathrm{e}})={\mathrm{L}}_{\mathrm{lc}}+{\mathrm{L}}_{\mathrm{avr}}-{\mathrm{L}}_{{\mathsf{\theta }}_{\mathrm{e}}}\cos {(2\mathsf{\theta }}_{\mathrm{e}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})$$

(12)

$${\mathrm{L}}_{\mathrm{ab}}{(\mathsf{\theta }}_{\mathrm{e}})=-{\displaystyle \frac{{\mathrm{L}}_{\mathrm{avr}}}{2}}-{ \mathrm{L}}_{{\mathsf{\theta }}_{\mathrm{e}}}\cos {(2\mathsf{\theta }}_{\mathrm{e}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})$$

(13)

$${\mathrm{L}}_{\mathrm{bc}}{(\mathsf{\theta }}_{\mathrm{e}})=-{\displaystyle \frac{{\mathrm{L}}_{\mathrm{avr}}}{2}}-{ \mathrm{L}}_{{\mathsf{\theta }}_{\mathrm{e}}}\cos {(2\mathsf{\theta }}_{\mathrm{e}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})$$

(14)

$${\mathrm{L}}_{\mathrm{ca}}{(\mathsf{\theta }}_{\mathrm{e}})=-{\displaystyle \frac{{\mathrm{L}}_{\mathrm{avr}}}{2}}-{\mathrm{L}}_{{\mathsf{\theta }}_{\mathrm{e}}}\cos {(2\mathsf{\theta }}_{\mathrm{e}})$$

(15)

The third and triplen harmonics (3k) do not contribute to torque production under isolated winding conditions. Therefore, only the dominant non-triplen harmonics (5th, 7th, 11th, etc.) are considered in the analytical model. This assumption allows accurate representation of torque pulsations while maintaining computational efficiency in the electromagnetic analysis.

2.2. Magneto-Static Analysis and Parameter Extraction

Accurate estimation of electromagnetic torque provides significant advantages in suppressing torque fluctuations. Therefore, a detailed analysis of machine parameter variations is essential for precise torque prediction. The most effective approach for this purpose is the use of a finite element (FE)–based analysis tool capable of advanced magnetostatic and transient electromagnetic calculations. In this study, a SPMSM model was analyzed using ANSYS Electronics 2025R2. The geometric configuration of the machine is shown in Figure 2, and the main parameters are listed in

Table 1. SPMSM parameters.

Symbol	Description	Value	Unit
ωn	Rated Speed	1500	RPM
Tn	Rated Torque	3.80	Nm
In	Rated Current	3.05	Arms
Ld	d-Axis Inductance	10.90	mH
Lq	q-Axis Inductance	11.20	mH
P	Number of Pole Pairs	2

. The stator and rotor cores were modeled using M700-50A electrical steel, which exhibits a nonlinear B–H characteristic. Consequently, magnetic saturation effects and the nonlinear behavior of the core are inherently included in the FE simulations.

Magnetic saturation occurs when the magnetic permeability decreases with increasing flux density, influencing the spatial distribution of inductances. However, in SPMSMs, the majority of the magnetic flux closes through the air gap due to the surface-mounted magnet configuration. Hence, the saturation effect is relatively limited, and it can be assumed that the effect of magnetic saturation on torque estimation accuracy is negligible [38]. Nevertheless, to enhance modeling fidelity, the FE model developed in this work explicitly incorporates saturation characteristics.

The simulations were performed at 100 RPM and 1500 RPM to evaluate operating conditions at low and high speeds. Similar trends were observed at both speeds. The analyses were conducted with the motor operating at a constant speed of 1500 RPM. Changes in parameters under steady-state operation were observed, and the results at 1500 RPM are presented in Figure 3.

Balanced and pure sinusoidal excitation was employed to distinctly observe the influence of machine parameter variations on electromagnetic torque production.

Figure 3a presents the variation in the d- and q-axis inductances (L_d, L_q). As can be seen, both inductances vary periodically with the rotor position due to spatial magnetic nonuniformity. Figure 3b illustrates the T_cog waveform, which originates from the interaction between stator slots and rotor poles, producing a periodic torque component. Figure 3c shows the d- and q-axis flux linkages (ψ_d, ψ_q), which exhibit smooth waveforms, clearly reflecting the interaction between the magnet-induced flux and inductive components. Figure 3d displays the three-phase sinusoidal current waveforms (I_a, I_b, I_c) applied to the SPMSM at 1500 RPM.

Finally, Figure 3e shows the electromagnetic torque waveform. The average torque remains constant, while small-amplitude periodic oscillations are observed, primarily due to T_cog and the variation in inductances with rotor position. The results confirm that the machine parameters (L_d, L_q, ψ_d, ψ_q) vary periodically, with a frequency related to the slot–pole combination.

3. Precise Torque Estimation and Proposed Torque Control

3.1. Analysis and Comparison of Conventional and Magnetic Co-Energy Torque Models

Accurate estimation of electromagnetic torque is highly effective in suppressing torque ripple in PMSMs. Therefore, developing an analytical model that can precisely represent the torque behavior under various operating conditions is essential. Conventional torque models typically rely on simplified assumptions, which may neglect spatial variations in magnetic parameters and cogging effects. To address these limitations, this subsection presents a comparative analysis between the conventional torque expression and the magnetic co-energy-based torque model, highlighting their underlying assumptions, accuracy, and applicability to high-performance torque control.

3.1.1. The Conventional Torque Model

In SPMSMs, the magnets placed on the rotor surface generate a constant magnetic flux. Due to this structure, the d-axis and q-axis inductances are generally assumed to be nearly equal (L_d ≈ L_q). Under this condition, the reluctance torque component originating from the rotor geometry is neglected, and the total torque of the motor is determined solely by the interaction between the PM flux and the stator current.

In the stationary rotor reference frame, the voltage equations along the d–q axes can be expressed as follows:

$${\mathrm{u}}_{\mathrm{d}} = {\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{d}} + {\displaystyle \frac{\partial {\mathsf{\lambda }}_{\mathrm{d}}}{\partial \mathrm{t}}}-{\mathsf{\omega }}_{\mathrm{e}}{\mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}$$

(16)

$${\mathrm{u}}_{\mathrm{q}}={\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{q}}+{\displaystyle \frac{\partial {\mathsf{\lambda }}_{\mathrm{q}}}{\partial \mathrm{t}}}+{\mathsf{\omega }}_{\mathrm{e}}{\mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}$$

(17)

$${\mathsf{\lambda }}_{\mathrm{d}}={ \mathrm{L}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{d}}+{\mathsf{\Psi }}_{\mathrm{M}}$$

(18)

$${\mathsf{\lambda }}_{\mathrm{q}}={ \mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}$$

(19)

where u_d,q denotes the stator voltages, λ_d,q the flux linkages, R_s the stator resistance, Ψ_M the flux generated by the permanent magnets, i_d,q the stator currents, and ω_e the electrical angular speed. Since the PM flux is aligned with the d-axis, Ψ_Mq = 0.

The conventional electromagnetic torque expression of the SPMSM is given by:

$${\mathrm{T}}_{\mathrm{e}} = {\displaystyle \frac{3}{2}}\mathrm{P}\left[{\mathsf{\Psi }}_{\mathrm{M}}{\mathrm{i}}_{\mathrm{q}}+\left({\mathrm{L}}_{\mathrm{d}}-{\mathrm{L}}_{\mathrm{q}}\right){\mathrm{i}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{q}}\right]$$

(20)

where P represents the number of pole pairs. For a surface-mounted configuration, if L_d = L_q, the second term becomes zero, and the torque depends solely on the q-axis current component. Under this condition, the maximum torque is achieved when i_d = 0, satisfying the MTPA criterion naturally, which ensures high torque output with minimal copper losses. Equation (21) represents the average torque, forming the basis of conventional control methods such as MTPA and Field-Oriented Control (FOC).

$${\mathrm{T}}_{\mathrm{e}}={\displaystyle \frac{3}{2}}\mathrm{P}{\mathsf{\Psi }}_{\mathrm{M}}{\mathrm{i}}_{\mathrm{q}}$$

(21)

This expression indicates that the electromagnetic torque depends only on the q-axis current. In other words, the torque production of the motor is determined only by the interaction between the PM flux and the q-axis current.

Although the inductances L_d and L_q are assumed to be equal and constant in this approach, in practice, their values slightly vary with the rotor position (L_dθe, L_qθe) and may not be identical. These variations arise from slotting effects, magnetic saturation, and spatial harmonics. Therefore, even in surface-mounted PMSMs, a small reluctance torque component exists, which contributes to torque ripple [43].

However, this expression represents only the average torque. Conventional models neglect the torque components caused by the energy stored in the stator windings and the T_cog generated by slot–magnet interaction. Hence, for precise torque control, these components should also be taken into consideration. Therefore, to effectively suppress torque ripple, the influence of variations in the stator magnetic energy and T_cog must be incorporated into the analysis.

3.1.2. The Magnetic Co-Energy Torque Models

The main sources of torque ripple in surface-mounted PMSMs are T_cog, magnetic flux harmonics, parameter variations, and current measurement errors. T_cog arises from the periodic magnetic interaction between the permanent magnets and the stator slots [44]. This component can be modeled as a periodic function of the rotor position, with its amplitude determined by the number of stator slots and rotor poles.

Magnetic flux harmonics are another major cause of torque pulsations. Since the air-gap flux density is not perfectly sinusoidal, the linkage flux between the stator current and the magnet flux in the three-phase (a-b-c) frame contains higher-order harmonics such as the 3rd, 5th, 7th, and 11th components. When transformed into the d–q reference frame, these harmonics appear as multiples of the sixth order. Accordingly, the d-axis flux can be expressed as:

$${\mathsf{\lambda }}_{\mathrm{d} }({\mathsf{\theta }}_{\mathrm{e}}) = {\mathsf{\lambda }}_0 + {\mathsf{\lambda }}_{\mathrm{d}6}\cos ({6\mathsf{\theta }}_{\mathrm{e}}) + {\mathsf{\lambda }}_{\mathrm{d}12}\cos ({12\mathsf{\theta }}_{\mathrm{e}}) + \dots$$

(22)

where λ₀ = Ψ_M denotes the flux generated by the permanent magnets, while λ_d6 and λ_d12 represent the amplitudes of the 6th- and 12th-order harmonics. Based on (20) and (22), the torque expression can then be expanded as in (23), consisting of the average torque (T₀) and harmonic components (T₆, T₁₂).

$${\mathrm{T}}_{\mathrm{e} }={\mathrm{T}}_0+{ \mathrm{T}}_6\cos \left({6\mathsf{\theta }}_{\mathrm{e}}\right)+{\mathrm{T}}_{12}\cos \left({12\mathsf{\theta }}_{\mathrm{e}}\right)+\dots$$

(23)

Measurement offset and gain errors in the current sensors also contribute to torque ripple. A DC offset in the measured current produces torque oscillations at the fundamental frequency, whereas gain errors introduce oscillations at twice the fundamental frequency. Consequently, the electromagnetic torque includes not only a DC component but also 1st, 2nd, 6th, and 12th harmonic components [45].

The electromagnetic torque can be defined as the derivative of the magnetic co-energy of the machine with respect to the electrical angle [38]. In the air gap, both magnetic energy and co-energy vary periodically, and the flux density depends on these variations. The electromagnetic torque is therefore expressed as a function of the magnetic co-energy (W_f′).

$${\mathrm{T}}_{\mathrm{e}}=\mathrm{P}{\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{f}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}$$

(24)

The total magnetic co-energy consists of three main components:

$${\mathrm{W}}_{\mathrm{f}}^{\prime } = {\mathrm{W}}_{\mathrm{fI}}^{\prime } +{\mathrm{W}}_{\mathrm{fIM}}^{\prime } +{\mathrm{W}}_{\mathrm{fM}}^{\prime }$$

(25)

In these terms, W_fI represents the energy stored in the stator windings, W_fIM denotes the co-energy resulting from the interaction between the stator flux and the magnet flux, and W_fM signifies the self-energy of the magnets. Equation (26) is obtained by (24) and (25).

$${\mathrm{T}}_{\mathrm{e}}=\mathrm{P}\left[{\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fI}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fIM}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fM}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}\right]$$

(26)

${\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fI}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}$ gives the torque term obtained from the energy stored in the stator windings and W_fI′ is defined as

$${\mathrm{W}}_{\mathrm{fI}}^{\prime }={\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}{\mathrm{i}}_{\mathrm{a}}& {\mathrm{i}}_{\mathrm{b}}& {\mathrm{i}}_{\mathrm{c}}\end{array}\right]\left[\begin{array}{ccc}{\mathrm{l}}_{\mathrm{aa}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)& {\mathrm{l}}_{\mathrm{ab}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)& {\mathrm{l}}_{\mathrm{ac}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)\\ {\mathrm{l}}_{\mathrm{ab}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)& {\mathrm{l}}_{\mathrm{bb}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)& {\mathrm{l}}_{\mathrm{bc}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)\\ {\mathrm{l}}_{\mathrm{ca}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)& {\mathrm{l}}_{\mathrm{bc}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)& {\mathrm{l}}_{\mathrm{cc}}\left({\mathsf{\theta }}_{\mathrm{e}}\right)\end{array}\right]\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{a}}\\ {\mathrm{i}}_{\mathrm{b}}\\ {\mathrm{i}}_{\mathrm{c}}\end{array}\right]$$

(27)

${\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fIM}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}$ gives the torque term obtained from the co-energy stored in the mutual inductances of stator windings and permanent magnets on the rotor. W_fIM′ is defined as

$${\mathrm{W}}_{\mathrm{fIM}}^{\prime }=\left[\begin{array}{ccc}{\mathrm{i}}_{\mathrm{a}}& {\mathrm{i}}_{\mathrm{b}}& {\mathrm{i}}_{\mathrm{c}}\end{array}\right]\left[\begin{array}{c}{\mathsf{\Psi }}_{\mathrm{Ma}}\\ {\mathsf{\Psi }}_{\mathrm{Mb}}\\ {\mathsf{\Psi }}_{\mathrm{Mc}}\end{array}\right]$$

(28)

${\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fM}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}$ is the torque value obtained from the co-energy of the magnets, and it is equal to T_cog.

$${\displaystyle \frac{\partial {\mathrm{W}}_{\mathrm{fM}}^{\prime }}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}={\mathrm{T}}_{\mathrm{cog}}$$

(29)

Equations (27)–(29) are replaced in (26) to obtain (30).

$$\begin{gathered} {\mathrm{T}}_{\mathrm{e}}=\mathrm{P}{\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}{\mathrm{i}}_{\mathrm{a}}& {\mathrm{i}}_{\mathrm{b}}& {\mathrm{i}}_{\mathrm{c}}\end{array}\right]{\displaystyle \frac{\partial \left[{\mathrm{l}}_{\mathrm{Habc}}\right]}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{a}}\\ {\mathrm{i}}_{\mathrm{b}}\\ {\mathrm{i}}_{\mathrm{c}}\end{array}\right]+\mathrm{P}{\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}{\mathrm{i}}_{\mathrm{a}}& {\mathrm{i}}_{\mathrm{b}}& {\mathrm{i}}_{\mathrm{c}}\end{array}\right]{\displaystyle \frac{\partial \left[{\mathrm{l}}_{\mathrm{Fabc}}\right]}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{a}}\\ {\mathrm{i}}_{\mathrm{b}}\\ {\mathrm{i}}_{\mathrm{c}}\end{array}\right] \\ +\mathrm{P} \left[\begin{array}{ccc}{\mathrm{i}}_{\mathrm{a}}& {\mathrm{i}}_{\mathrm{b}}& {\mathrm{i}}_{\mathrm{c}}\end{array}\right]{\displaystyle \frac{\partial \left[{\mathsf{\phi }}_{\mathrm{MHabc}}\right]}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+\mathrm{P} \left[\begin{array}{ccc}{\mathrm{i}}_{\mathrm{a}}& {\mathrm{i}}_{\mathrm{b}}& {\mathrm{i}}_{\mathrm{c}}\end{array}\right]{\displaystyle \frac{\partial \left[{\mathsf{\phi }}_{\mathrm{MFabc}}\right]}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{ \mathrm{T}}_{\mathrm{cog}} \end{gathered}$$

(30)

l_Ha, l_Hb, l_Hc, l_Fa, l_Fb, and l_Fc are the harmonic and fundamental inductance in the a-b-c frame and Ψ_MHa, Ψ_MHb, Ψ_MHc, Ψ_MFa, Ψ_MFb, and Ψ_MFc are the harmonic and fundamental flux of permanent magnets in the a-b-c frame, respectively. In the proposed magnetic co-energy-based model, magnetic saturation effects are also incorporated [43]. This approach allows for a more accurate estimation of both the average and harmonic components of torque, improving the precision of torque control and ripple analysis.

3.1.3. The Comparison of Conventional and Magnetic Co-Energy Torque Models

The parameter variations in the surface-mounted PMSM were derived from advanced FEA. These variations, which periodically depend on the rotor position, are the primary cause of torque ripple. Consequently, the electromagnetic torque also oscillates periodically. When the machine parameters d- and q-axis inductances (L_d, L_q), flux linkages (ψ_d, ψ_q), and T_cog are known, the analytical model expressed in (30) provides highly accurate torque estimation.

The validity of the proposed model was verified through simulations in MATLAB/Simulink. The position-dependent parameter data obtained from finite element method (FEM) analyses were imported into MATLAB and utilized within the analytical model. In this way, the influence of magnetic and geometric variations on the torque was accurately reflected.

Figure 4 illustrates the torque waveforms obtained at 1500 RPM and 100 RPM. The torque obtained from the FEA is denoted as “Te-FEA,” while the estimated torque values obtained by different methods are indicated as “Te-Est1,” “Te-Est2,” and “Te-Est3.”

In Figure 4a, the estimation result using the conventional torque Equation (21) (Te-Est1) is presented. The results show that the conventional model approximately predicts the mean torque value, but it is inadequate to describe the torque oscillations. This limitation arises from the model’s assumption that parameter changes are constant.

Figure 4b shows the torque estimation result when considering only electrical parameter variations (Te-Est2). This model neglects the T_cog term. Although the electromagnetic torque is predicted approximately using this method, certain ripple components still remain.

Figure 4c illustrates the magnetic co-energy-based torque estimation method (Te-Est3). The obtained torque prediction shows excellent agreement with the FEM-based torque waveform. Furthermore, the proposed model achieves high levels of accuracy in torque estimation, even at low speeds. The estimation results at a low speed of 100 RPM are presented in Figure 4d for the magnetic co-energy-based torque estimation method (Te-Est3). As can be seen from the figures, the model outperforms traditional models in its ability to predict the amplitude and harmonic components of torque with greater accuracy.

3.2. Proposed Precise Torque Control Method

In order to simplify the complex, position-dependent spatial matrices in the a-b-c frame in Equation (30) algebraically, these terms are transformed into the synchronized d-q reference frame with Clarke and Park transformations. The resulting torque equation is shown in (31).

$${\mathrm{T}}_{\mathrm{e}}=\mathrm{P}{\displaystyle \frac{1}{2}}\left[\begin{array}{cc}{\mathrm{i}}_{\mathrm{d}}& {\mathrm{i}}_{\mathrm{q}}\end{array}\right]{\displaystyle \frac{\partial \left[\begin{array}{cc}{\mathrm{l}}_{\mathrm{q}}& 0\\ 0& {\mathrm{l}}_{\mathrm{q}}\end{array}\right]}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{d}}\\ {\mathrm{i}}_{\mathrm{q}}\end{array}\right]+\mathrm{P}\left[\begin{array}{cc}{\mathrm{i}}_{\mathrm{d}}& {\mathrm{i}}_{\mathrm{q}}\end{array}\right]{\displaystyle \frac{\partial \left[\begin{array}{c}{\mathsf{\phi }}_{\mathrm{Md}}\\ {\mathsf{\phi }}_{\mathrm{Mq}}\end{array}\right]}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+\mathrm{P}\left[\begin{array}{c}{\mathsf{\Psi }}_{\mathrm{Md}}\\ {\mathsf{\Psi }}_{\mathrm{Mq}}\end{array}\right]\mathrm{x} \left[\begin{array}{cc}{\mathrm{i}}_{\mathrm{d}}& {\mathrm{i}}_{\mathrm{q}}\end{array}\right]+{\mathrm{T}}_{\mathrm{cog}}$$

(31)

In SPMSMs, the effects of T_cog, magnetic flux harmonics, and parameter variations causing torque fluctuations reduce the accuracy of torque control. Conventional current control techniques, which maintain a constant q-axis current, can preserve the average torque but fail to suppress instantaneous torque oscillations. To overcome this limitation, a precise current control approach that accounts for rotor position-dependent parameter variations is proposed. The proposed method is derived from the torque Equation (31) expressed in the d–q reference frame.

In SPMSMs, the MTPA trajectory naturally aligns with setting the d-axis current to zero (i_d = 0) since the reluctance torque is minimal. Applying this control constraint (i_d = 0) simplifies Equation (31) significantly, as all terms multiplied by i_d are eliminated. Furthermore, since the fundamental permanent magnet flux is aligned with the d-axis, it is assumed that Ψ_Md ≈ Ψ_M, while $\partial$Ψ_Mq represents the position-dependent harmonic flux variations on the q-axis. Incorporating these physical justifications yields the simplified torque expression in Equation (32).

$${\mathrm{T}}_{\mathrm{e}}=\mathrm{P}{\mathrm{i}}_{\mathrm{q}}\left[{\displaystyle \frac{{\mathrm{i}}_{\mathrm{q}}}{2}}{\displaystyle \frac{{\partial \mathrm{l}}_{\mathrm{q}}}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{\displaystyle \frac{\partial {\mathsf{\Psi }}_{\mathrm{Mq}}}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{\mathsf{\Psi }}_{\mathrm{M}}\right]+{\mathrm{T}}_{\mathrm{cog}}$$

(32)

To extract the required q-axis reference current (i_qref), which actively cancels the torque ripple, Equation (32) can be rearranged into a standard quadratic polynomial form with respect to i_q (A(i_q)² + B(i_q) + C = 0). Applying the standard quadratic formula yields the discriminant (Δ = B² − 4AC) in Equation (33). The physically meaningful root responsible for generating positive torque provides the exact analytical solution for i_qref, as formulated in Equation (34). The roots of the resulting quadratic expression provide the current references that minimize torque ripples as a function of rotor position. The physically meaningful root corresponding to positive torque generation is selected from Equations (33) and (34).

$$∆ ={\left[\mathrm{P}\left({\displaystyle \frac{\partial {\mathsf{\Psi }}_{\mathrm{Mq}}}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{\mathsf{\Psi }}_{\mathrm{M}}\right)\right]}^2-\left[2\mathrm{P}{\displaystyle \frac{\partial {\mathrm{l}}_{\mathrm{q}}}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}\right]\left[{\mathrm{T}}_{\mathrm{cog}}-{\mathrm{T}}_{\mathrm{e}}\right]$$

(33)

$${\mathrm{i}}_{\mathrm{qref} 1,2}=\left[-\mathrm{P}\left({\displaystyle \frac{\partial {\mathsf{\Psi }}_{\mathrm{Mq}}}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}+{\mathsf{\Psi }}_{\mathrm{M}}\right) \pm \sqrt{∆}\right]/\left[\mathrm{P}{\displaystyle \frac{\partial {\mathrm{l}}_{\mathrm{q}}}{\partial {\mathsf{\theta }}_{\mathrm{e}}}}\right]$$

(34)

The parameters (L_q, Ψ_M, T_cog) obtained from the FEA in Equation (34) utilize the nonlinear B-H curve of the M700-50A core material. These parameters therefore naturally incorporate the complex effects of magnetic saturation, hysteresis and eddy current dynamics within the position-dependent spatial data. This ensures that the proposed analytical model remains highly reliable, even under deep saturation conditions. Thus, the magnetic saturation model ceases to be hypothetical and is inherently integrated into Equation (34) through the obtained parameter information.

To evaluate the moment control performance, the proposed precise torque control method was compared with the traditional MTPA method through a preliminary analysis conducted in MATLAB/Simulink. The parameters required for the proposed control method can be easily obtained from the FE analysis. Since these parameters exhibit periodic variation depending on the number of poles of the machine, only preliminary information for a single pole range was extracted. Therefore, this approach significantly reduces memory requirements and computational complexity. These data were transferred to the MATLAB environment in tabular form and used in the proposed control method.

Figure 5 shows the torque waveform obtained with a constant i_qref current in the traditional MTPA method. When a constant i_qref current is applied, as illustrated in Figure 5, significant torque ripples are observed.

In contrast, the proposed control method calculates the i_q reference using Equation (34). The corresponding electromagnetic torque waveform is shown in Figure 6. The results clearly demonstrate that torque oscillations are effectively suppressed, and the torque is maintained steadily around the reference value of 3.5 Nm.

Equation (34) offers a simple analytical method for calculating the necessary current references to suppress torque ripples. While conventional analytical models often neglect magnetic saturation and iron losses for simplicity, the proposed model and the high-fidelity FEA simulations used in this study account for these nonlinearities to ensure high precision. Moreover, all parameters required in this equation can be easily derived from FEA data.

The inductance profiles (L_dqθe) extracted by FEA inherently contain the necessary spatial harmonic and saturation information. Therefore, the proposed method does not require a separate fast Fourier transform (FFT) analysis or specific harmonic selection. By solving Equation (34) using these position-based inductance maps, the generated i_qref automatically includes the necessary compensation components. These inductance variations are utilized via interpolation for any current or speed value, enabling precise i_qref generation under varying operating conditions, including saturation. As the proposed method operates as a reference generator within a standard cascaded PI control structure (FOC), closed-loop stability is ensured through classical PI bandwidth tuning, eliminating the requirement for the intricate Lyapunov-based stability proofs typically demanded of nonlinear controllers.

On the other hand, many methods, such as observer-based or differential-based methods [13,22], are sensitive to noise because they calculate the current derivative. However, the proposed method is not directly affected by current sensor noise because it uses rotor position-dependent data, and it has high noise immunity because it does not involve a derivative operation.

In conclusion, the proposed precise torque control method effectively mitigates torque ripples and achieves accurate torque control across both low- and high-speed ranges. This approach offers a computationally efficient and adaptable torque control strategy for SPMSMs operating under precise applications, in particular.

4. Test and Results

4.1. Advanced Co-Simulation Structure and the Purposed Torque Control Method

To evaluate the accuracy and practical feasibility of the proposed precise torque control method, an advanced co-simulation framework has been established. The development of a generalized and automated evaluation framework is essential for ensuring the safe operation of advanced electric drive systems, particularly in applications where high torque accuracy and system reliability are critical. In many industrial domains—such as automation, robotics, and transportation—validating safety and performance typically requires extensive experimental testing, which significantly increases development cost and time. A systematic design process that integrates formal modeling techniques and high-fidelity co-simulations in the early stages enables rigorous verification of control performance before hardware prototyping. This approach reduces implementation risks, minimizes the need for repeated field testing, and ensures that the proposed torque control strategy can be implemented reliably in real-time drive applications.

Detecting high-order torque harmonics (6th, 12th, etc.) experimentally requires expensive torque transducers with extremely high bandwidth (>5 kHz) and precision, which are often unavailable in standard setups. Standard sensors act as low-pass filters, masking the very ripple phenomena this study aims to minimize. Therefore, high-fidelity 2D-FEA co-simulation serves as a virtual sensor with infinite bandwidth, providing a more reliable environment for validating the harmonic cancelation logic of the proposed algorithm than a bandwidth-limited physical prototype.

This framework integrates a detailed FEA of the machine with the analytical control models within a unified environment, enabling a simulation behavior that closely reflects the real system. In this way, physical phenomena affecting torque ripple—such as magnetic saturation, harmonic flux components, and T_cog—are assessed simultaneously with the control algorithms.

The electromagnetic model of the machine is analyzed in ANSYS using a 2D-FEA approach, which is suitable due to the radial symmetry of the PMSM. The control system and the proposed torque estimation scheme are implemented in MATLAB/Simulink. As illustrated in Figure 7, both subsystems are interconnected through ANSYS to perform the co-simulation. Compared with simulations based solely on analytical models, this integrated environment provides significantly higher accuracy and yields results that are much closer to the actual behavior of a PMSM operating under torque control [46].

For surface-mounted permanent magnet synchronous machines (SPMSMs), the effective air gap is large, and the end leakage flux is relatively small compared to induction or IPMSM machines. Therefore, 2D-FEA provides sufficient accuracy for torque ripple analysis. The 3D end effects primarily act as constant leakage inductance, which the controller naturally compensates for without affecting the AC ripple cancelation capability of the proposed model.

Within the co-simulation structure, the rotor position and stator currents obtained from ANSYS are transferred to the MATLAB/Simulink domain. Torque estimation and reference current generation are performed in MATLAB, and the resulting reference currents are applied back to the 2D-FEA model. Thus, the control algorithm operates directly against the instantaneous electromagnetic response of the machine, which inherently includes magnetic saturation, harmonic flux interactions, and variations in inductances.

The electromagnetic parameters of the machine—L_d, L_q, flux linkage, and T_cog—are extracted from nonlinear FEAs in ANSYS. Obtaining these quantities for only one pole pitch is sufficient, which significantly reduces memory requirements and computational complexity. The extracted data are subsequently imported into MATLAB to be used in the control loop.

The co-simulation approach offers a major advantage by allowing the electromagnetic and control models to be developed by experts from different domains and then integrated into a unified platform. Compared with conventional monolithic modeling, the method provides superior accuracy and enhances control system design flexibility [46,47,48].

The 2D-FEA results obtained for the proposed torque control method under operating conditions of 3.5 Nm and 1500 RPM are presented in Figure 8. Figure 8a illustrates the injected d- and q-axis reference currents and the corresponding torque waveforms. Figure 8b shows the three-phase reference currents applied to reduce torque ripple. Figure 8c presents the inductance variations caused by the injected harmonic currents, while Figure 8d,e depict the resulting changes in d-q axis fluxes and the a-b-c phase flux characteristics.

4.2. Sampling Time and Delay Effect

Mechatronic systems are constantly growing in complexity and expanding their range of applications. They require increasingly high levels of dependability and performance. Accordingly, system validation should be complemented by rigorous analysis and verification methods [46].

In real-time digital control systems, the sampling time and the inherent computation–communication delay play a critical role in determining the achievable torque control performance. To demonstrate the applicability of the proposed control method in real time, a series of detailed co-simulation analyses was conducted to evaluate the effect of sampling time and associated delays on the performance of the proposed torque control method. By investigating sampling times of 100 µs, 10 µs, and 1 µs, the limits of controller performance and the theoretical potential of the proposed model were demonstrated.

In the co-simulation structure, the rotor position and stator currents obtained from the 2D-FEA model in ANSYS are transferred to MATLAB/Simulink, where torque estimation and current reference generation are performed. After the reference currents are computed, they are sent back to ANSYS. Because this data exchange occurs sequentially, one full control cycle requires two sampling intervals. Consequently, for a sampling time of 100 µs, the current reference suffers from an effective 200 µs delay, as illustrated in Figure 9a. The impact of inverter dead-time and loop delays is lumped into the 200 μs delay analysis (Figure 9).

Figure 9b shows that this delay introduces a noticeable deterioration in torque control performance, even when the analytically correct reference currents are used. The delayed actuation results in a dynamic mismatch between the machine’s instantaneous electromagnetic response and the applied reference currents, producing additional torque ripple. However, as can be seen from the graph, the torque ripples caused by time delay in the proposed torque control method are still much smaller than those in the traditional MTPA method. The results confirm stability even under these aggregated delays. Figure 9c presents the resulting torque behavior when the same reference currents are used without applying any delay compensation, confirming the adverse impact of sampling-induced delays on torque quality.

To fully characterize this situation, the same analysis was conducted for 10 µs and 1 µs sampling times, and the results are summarized in Figure 10. With a 10 µs sampling time (Figure 10a,b), the torque waveform exhibits improved smoothness compared to the 100 µs case. Although a small amount of jitter remains due to the effective delay of 20 µs, this is negligible under normal conditions. Nevertheless, this shows that even a moderate delay in the sampling period can be a limiting factor in precision applications requiring high-precision torque control. When the sampling period is reduced to 1 µs (Figure 10c), the effect of the resulting 2 µs total delay on the torque is quite low. Under these conditions, the controller output aligns closely with the instantaneous electromagnetic behavior of the SPMSM, revealing the theoretical upper bound of the proposed method’s performance in systems equipped with sufficiently fast hardware.

To quantify the effect of sampling time and delay, the torque ripple and RMS torque error were evaluated for each case. Figure 11a–d present the torque waveforms obtained under constant i_q excitation and sampling times of 100 µs, 10 µs, and 1 µs. The results clearly indicate that the RMS torque error decreases significantly as the sampling time is reduced. At 1 µs, the electromagnetic torque follows the reference almost perfectly, and the torque ripple reaches its minimum level. The corresponding RMS error values are summarized in

Table 2. The RMS errors of torque ripples at 1500 RPM.

With Constant Stator Current	With Reference Stator Current
	100 µs Sampling Time	10 µs Sampling Time	1 µs Sampling Time
18.38%	10.21%	3.36%	1.31%

.

These results highlight an important conclusion. Even if the proposed torque control method and approach are analytically correct, time delay is still one of the causes of torque ripple in practical digital control applications. Therefore, the performance of the controller is fundamentally limited by the available computation and communication speed. In systems where advanced hardware enables low-level sampling, the proposed analytical torque control can deliver highly accurate, low-ripple operation. Conversely, although the proposed control method performs better than traditional MTPA despite the delay caused by slower sampling times, the situation must be carefully evaluated in terms of control design and performance assessment.

In summary, the analyses presented in this section demonstrate that:

• The proposed torque control method is inherently capable of achieving high-precision torque regulation,
• Sampling time and delay constitute the primary limitation on practical performance,
• When sampling time is sufficiently small, the control system reaches its theoretical performance limits, providing near-ideal torque behavior even with a time delay.

4.3. Investigation of Torque Ripple Reduction Performance

To rigorously evaluate the effectiveness of the proposed precise torque control method, a comparative study is conducted against three distinct control strategies. This comprehensive comparative framework is designed to clearly demonstrate the specific contributions and performance benefits of the proposed model. These strategies are designed as an ablation study of the proposed analytical model Equation (34) to isolate the contribution of each physical phenomenon (electrical parameter variation vs. cogging torque) to the torque ripple reduction. Method 1 (Conventional MTPA) represents the conventional MTPA control strategy, where a constant i_q reference is applied, and id is maintained at zero [49]. Method 2 (Electrical Parameter-Based) represents the physical principle of analytical approaches in the literature that focus on compensating for torque ripple caused by inductance variations and magnetic saturation while neglecting the cogging torque (T_cog) [50]. For a fair comparison, this method is derived by neglecting the T_cog term in the proposed Equation (34). Method 3 (Cogging Torque-Based) represents control strategies that specifically target cogging torque compensation, often assuming linear magnetic conditions [51]. For a fair comparison, Method 3 is derived from the proposed model Equation (34) by neglecting the variation in the inductances and focusing solely on the T_cog term. Ultimately, this comparative research design is systematically evaluated across a wide range of operating conditions—including steady-state, transient load steps, and deep magnetic saturation—to rigorously validate the robustness of the proposed model.

To ensure a fair comparison and confirm that the performance differences stem solely from the quality of the generated reference currents (i_qref) rather than the tuning of the feedback loop, all four methods (Method 1–3 and Proposed Model) utilize the exact same inner current loop settings. All control strategies were implemented within the same MATLAB/Simulink and Ansys co-simulation environment, utilizing identical machine parameters and solver settings. This rigorous setup eliminates the suboptimal tuning factor and isolates the physical accuracy of the reference generation models. In all cases, the d-axis current is kept at zero. The analyses aim to compare the performance of these methods by evaluating their ability to mitigate torque ripple, their behavior with regard to torque and current harmonics, and the distribution of associated losses, etc., all under identical operating conditions.

4.3.1. Torque Ripple Comparison Under Operating Conditions at Different Load Levels

Figure 12 presents the electromagnetic torque and the corresponding i_q current waveforms obtained with all four methods at 1500 RPM and 3.8 Nm. The results validate the theoretical premise of the ablation study. Method 2, which incorporates the variation in electrical parameters (L_dqθe) but ignores T_cog, partially reduces the ripple compared to Method 1 but fails to suppress the high-frequency pulsations caused by the permanent magnet slot interaction. Conversely, Method 3, which compensates for T_cog but assumes constant inductances, fails to address the reluctance torque variations caused by magnetic saturation and inductance variations. The proposed model, by integrating both the electrical parameter variations and the cogging torque into a unified magnetic co-energy framework (34), achieves the lowest torque ripple. This confirms that simultaneously addressing magnetic saturation and mechanical slotting effects is essential for high-performance torque control, rather than treating them in isolation.

As a result, the proposed model generates a substantially smoother torque profile and a lower harmonic torque content compared to Methods 1, 2, and 3.

To validate the robustness of the proposed model against magnetic saturation and parameter variations across a wide operating range, extensive FEA simulations were conducted. A critical aspect of the proposed control strategy is its ability to adapt to varying load levels without requiring complex online parameter re-identification.

Figure 13 illustrates the variations in the q-axis inductance (L_q) and flux linkage (ψ_q) with respect to the electrical angle (θ_e) under a wide range of current levels, from 0 to 2.0 times the nominal i_q current (0, 0.25, 0.5, 1.0, 1.1, 1.25, 1.5, and 2.0 × i_qnom).

As observed in Figure 13a,b, increasing the current load leads to magnetic saturation, which naturally varies the average magnitude of the inductance and flux linkage. However, a crucial phenomenon is observed: while the amplitudes of these parameters change due to saturation, their spatial harmonic waveforms (shapes) remain remarkably consistent and do not undergo significant structural distortion. Therefore, even with variation in L_q or T_cog magnitude due to manufacturing tolerances, the spatial harmonic profile (shape) remains largely consistent. This physical characteristic allows the proposed method to produce i_qref currents in accordance with the desired torque load via offset scaling. Since the harmonic orders are determined by the motor’s physical structure (slot/pole combination), the proposed model effectively suppresses torque ripple by interpolating the parameter magnitudes from electrical angle-dependent data. Concurrently, it maintains the precise harmonic injection profile derived from Equation (34).

The effectiveness of this approach under different load conditions is presented in Figure 14. The motor was tested under different load sequences of 1.0 Nm, 0.5 Nm, 3.5 Nm, and 2.0 Nm at 1500 RPM.

The conventional method (Method 1) exhibits significant torque ripples, which vary in amplitude depending on the load (Figure 14a). A fluctuating torque increases mechanical stress, acoustic noise, and long-term fatigue in the machine load system. In contrast, the proposed model successfully suppresses these ripples at all load levels. Notably, during the transition to the 3.5 Nm and even under lighter loads, the proposed model maintains a smooth torque profile.

As can be seen in Figure 14b, the proposed model adjusts the DC offsets of the generated i_qref currents according to the load, in order to meet the average torque demand. Simultaneously, the AC harmonic components—responsible for ripple cancelation—are scaled according to the instantaneous inductance values obtained via interpolation. This confirms that the proposed algorithm is not limited to a single operating point but is robust against the nonlinearities introduced by load changes and magnetic saturation. This characteristic significantly enhances the practical applicability of the method, reducing computational requirements and simplifying real-time implementation.

4.3.2. Torque Ripple Comparison Under Overload Operating Conditions

In industrial applications, PMSMs can be required to operate under overload conditions where magnetic saturation becomes significant. Therefore, to evaluate the robustness of the proposed control strategy against magnetic saturation and parameter variations, the machine was tested under overload conditions of 1.25, 1.5, and 2.0 times the reference torque at 1500 RPM. The resulting reference currents (i_qref) and electromagnetic torque (T_e) waveforms are presented in Figure 15.

As shown in Figure 15a–c, the proposed model successfully generates the required average torque levels of approximately 4.375 Nm (1.25 p.u.), 5.25 Nm (1.5 p.u.), and 7.0 Nm (2.0 p.u.), respectively. Crucially, despite the machine entering the nonlinear magnetic saturation region at these high current levels, the torque ripple was effectively suppressed.

The generated q-axis reference current (i_qref) waveforms in Figure 15 exhibit the specific harmonic injection profile required for ripple cancelation. It is observed that while the DC offset of the current increases significantly to meet the higher load demand, the AC harmonic shape (the ripple-canceling component) is maintained and scaled appropriately. This confirms the validity of the proposed offset-scaling strategy. As analyzed in Figure 13, even under heavy saturation (2.0 p.u.), the spatial distribution of the machine’s magnetic parameters retains its characteristic harmonic structure. Since the machine is an SPMSM, the cross-coupling effect is minimal compared to IPMSMs. Furthermore, the inductance variations accommodate the magnetic saturation profile. Therefore, the harmonic shape (AC component) largely remains valid across the loads, allowing the DC offset scaling method to operate effectively. Consequently, the proposed method can utilize the interpolated parameters to synthesize the correct i_qref without requiring complex online adaptation algorithms.

These results conclusively demonstrate that the proposed analytical model provides high-performance torque control not only at normal points but also under severe overload and saturation conditions, ensuring reliable operation for high-dynamic drive cycles.

4.3.3. Torque Ripple Comparison: Transient Response and Dynamic Performance at Different Operating Speeds

To evaluate the robustness of the proposed control strategy under dynamic operating conditions, extensive analyses were conducted covering transient load steps, startup acceleration, and low-speed operation. These results demonstrate that even during transients where the machine produces torque above normal limits, the algorithm successfully tracks the electrical angle θ_e and generates the ripple-suppressing i_qref, proving its robustness in high-dynamic regimes. The proposed method maintains stability not only in a steady state but also during severe transient conditions where torque limits are exceeded, proving its robustness against saturation-induced parameter variations.

The dynamic performance of the proposed method is compared with the conventional MTPA method (Method 1) in Figure 16.

Figure 16a illustrates the transient torque response and saturation robustness during a severe load step change. The machine initially operates under a high load condition (approximately 10 Nm), significantly exceeding the reference torque (T_e* = 3.5 Nm). In this region, the proposed model effectively suppresses torque ripples even under deep magnetic saturation. The peak-to-peak torque ripple is reduced from 2.8 Nm (Method 1) to 0.25 Nm (Proposed Model), representing a 91% reduction. It maintains stability and high precision during the transient period, settling at the reference torque with a ripple of only 0.25 Nm, compared to 2 Nm for Method 1. This confirms that the offset-scaling strategy is highly robust against parameter variations caused by saturation during dynamic load changes.

Figure 16b shows the startup performance from standstill to 1500 RPM. While both methods track the speed ramp, a magnified view of the steady-state region reveals a crucial difference in speed stability. Method 1 exhibits a speed ripple of 16 RPM due to the oscillating torque. In contrast, the proposed model significantly smooths the speed trajectory, reducing the ripple to just 1.5 RPM. This 90% reduction in speed ripple is a direct consequence of the suppressed torque pulsations. Since speed fluctuations are the primary source of mechanical vibrations and acoustic noise in electric drives, these results quantitatively demonstrate the proposed method’s capability to enhance mechanical longevity and acoustic performance.

Low-Speed Performance Figure 16c,d evaluate the performance at a low speed of 100 RPM. Low-speed operation is often challenging for model-based controllers due to the low signal-to-noise ratio of back-EMF and nonlinear friction effects. Figure 16c shows that even at low speeds, where conventional methods often struggle due to unmodeled harmonics, the proposed method reduces the torque ripple (by 95.2%) from 4.2 Nm to 0.2 Nm. Consequently, as shown in Figure 16d, the speed ripple at 100 RPM is drastically reduced from 44 RPM (Method 1) to 2.2 RPM (Proposed Model). This validates that the proposed analytical model, which relies on position-dependent inductance maps rather than speed-dependent back-EMF integration, maintains its high accuracy and control authority virtually independent of the motor speed.

Table 3. Summary of torque ripple and speed stability performance under various operating conditions.

Operating Condition	Performance Metric	Conventional Method (Method 1)	Proposed Model	Improvement
High Speed (1500 RPM)	Torque Ripple (Trip)	~2.0 Nm	~0.25 Nm	~87.5%
	Speed Ripple (nrip)	~16 RPM	~1.5 RPM	~90.6%
Low Speed (100 RPM)	Torque Ripple (Trip)	~4.2 Nm	~0.20 Nm	~95.2%
	Speed Ripple (nrip)	~44 RPM	~2.2 RPM	~95.0%
Transient Conditions (~10 Nm)	Torque Ripple (Trip)	~2.8 Nm	~0.25 Nm	~91.0%

summarizes the quantitative performance metrics derived from the dynamic response analysis in Figure 16. The proposed model demonstrates superior robustness across all tested conditions.

In summary, the results in Figure 16 and Table 3 confirm that the proposed torque control strategy offers superior dynamic performance, ensuring smooth torque and speed profiles during startup, transients, overload, and low-speed operations, thereby directly addressing the mechanical and acoustic concerns common in PMSM drives.

4.3.4. Torque Harmonic and THD Analysis

To examine the harmonic nature of torque ripple, an FFT-based analysis is performed. Figure 17a shows the THD percentages of the torque for each method. Figure 17b provides the corresponding harmonic spectra.

The conventional methods exhibit dominant low-order harmonics, especially around the 2nd, 4th, and 6th components, which are directly related to magnetic asymmetries and T_cog.

The proposed model minimizes the harmonic components that cause torque ripple by calculating the current harmonics analytically. Consequently, the torque THD is drastically reduced.

This result confirms the method’s ability to generate torque profiles with significantly lower distortion across a wide frequency band.

4.3.5. Analysis of i_q Reference Currents

In order to elucidate the mechanism behind torque ripple suppression and its impact on system efficiency, a detailed spectral and root-mean-square (RMS) analysis of the q-axis reference currents was conducted. Figure 18a shows the i_q currents waveforms and the THD values of the i_q reference currents obtained from all methods, while Figure 18b presents the RMS value of the fundamental i_q component, and Figure 18c illustrates the harmonic spectra.

The THD analysis in Figure 18a shows that Method 1 achieves a nearly distortion-free i_q waveform (THD = 0.01%) due to its use of a constant, purely sinusoidal i_q reference. Although this approach effectively minimizes current harmonics, the absence of any harmonic injection prevents torque ripple compensation, leaving significant torque pulsations that induce mechanical stress and oscillations in both the machine and the load.

In contrast, Methods 2 and 3 introduce harmonic distortion, yielding THD levels of 6.76% and 5.40%, respectively. These harmonics originate from their inability to maintain balanced torque production, which leads to irregular and unstructured harmonic components in the reference currents.

The proposed model exhibits a THD of 7.80%, which is slightly higher than the other methods. However, this difference must be interpreted within the context of the model’s design philosophy.

Unlike Methods 1–3, the proposed model injects additional harmonic components intentionally and analytically to achieve torque ripple suppression. Therefore, the resulting harmonic content is not irregular or parasitic. Instead, it is precisely shaped and fully predictable. This distinction is critical in practical applications. Controlled harmonics can be reliably tracked by standard current controllers without compromising system stability, whereas uncontrolled harmonics from Methods 2 and 3 may increase controller burden and reduce dynamic robustness.

Furthermore, Figure 18b indicates that the proposed model produces a slightly lower fundamental i_q component compared to the other methods, exhibiting a phenomenon of RMS balancing via fundamental reduction. Although the proposed method introduces a slightly higher THD (7.80%), it successfully synthesizes an optimal current waveform that delivers the desired average torque with a reduced fundamental current component (i_qFund). This reduction in the fundamental component implies a potential improvement in copper losses, as copper losses scale with the square of the current magnitude. In other words, despite introducing structured harmonics for ripple minimization, the proposed model avoids an unnecessary increase in the fundamental current amplitude, which is advantageous for overall machine efficiency. Moreover, by effectively suppressing the ripple-inducing reluctance and cogging components that oppose rotation, the motor operates more efficiently. This reduction in the fundamental component creates a headroom that accommodates the injected harmonics. Consequently, the total RMS current (i_qRMS) of the proposed model (5.40 A) remains comparable to Method 2 (5.46 A) and significantly lower than Method 1 (6.06 A), despite the increased THD.

Overall, the harmonic analysis confirms that the proposed model introduces a predictable and manageable harmonic pattern while simultaneously reducing the fundamental i_q component. These characteristics collectively demonstrate that the proposed model achieves torque ripple mitigation without imposing excessive current distortion or efficiency penalties, thereby offering a favorable balance between performance and practical implementation. Therefore, the proposed method utilizes optimized harmonics not as noise, but as a control lever to enhance the overall electromechanical performance and longevity of the drive system without compromising thermal limits.

4.3.6. Loss Distribution Analysis and Torque Ripple Ratio

A loss decomposition was performed to quantify the impact of the injected harmonic currents on the overall loss profile of the SPMSM. The losses considered copper losses associated with the fundamental current component (P_cuFund), additional copper losses induced by harmonic currents (P_Harm), iron losses (P_Fe), and mechanical friction losses (P_Fric) in this evaluation. The resulting loss distribution for all methods is summarized in Figure 19, and a detailed breakdown of the proposed model’s losses at 3.5 Nm, 1500 RPM is provided in

Table 4. Proposed model detailed loss distribution at 3.5 Nm load and 1500 RPM.

TRR (%)	ղ(%)	iqRMS (A)	Pout (W)	PFe (W)			PFric (W)	PCuFund (W)	PHarm (W)	PTLoss (W)
				PHys (W)	PEddy (W)	PEx (W)
7.14	91.31	5.40	550	~4.20	~3.10	~0.20	~12.00	~32.83	~1.13	~52.33

.

As detailed in Table 4, the total iron loss (P_Fe) of 7.50 W is further decomposed into hysteresis loss (P_Hys ≈ 4.20 W), eddy current loss (P_Eddy ≈ 3.10 W), and excess loss (P_Ex ≈ 0.20 W). According to the Steinmetz Equation, since P_Hys is proportional to frequency (f) and P_Eddy to the square of frequency (f²), and considering that the fundamental operating speed (1500 RPM) and main flux levels are identical across all compared methods, the base iron losses remain consistent. Crucially, the FEA results indicate that the high-frequency flux variations introduced by the injected harmonics (6th and 12th order) are of low amplitude; thus, they do not cause a statistically significant increase in the total P_Fe compared to the reference cases.

In addition, hysteresis losses (P_Hys) and eddy current losses (P_Eddy) are also heavily dependent on the amplitude of the magnetic flux density variation (ΔBⁿ and ΔB²). In SPMSMs, the effective air gap is inherently large since the relative permeability of the surface magnets is close to that of air. Consequently, the high-frequency harmonic currents injected by the proposed model generate only very small high-frequency flux density ripples (ΔB) in the stator core. The amplitude of these harmonic flux variations (ΔB) is extremely low compared to the fundamental flux density. This means that the mitigating effect of the square of this amplitude (ΔB²) dominantly offsets the increase caused by the higher harmonic frequencies (f²). Therefore, as verified by the high-fidelity FEA results, the injected ripples do not induce deep hysteresis loops or severe eddy currents, resulting in only a negligible fluctuation in P_Fe without imposing any additional thermal stress on the machine.

A primary concern in harmonic injection schemes is the potential for increased copper losses (P_Cu). This condition provides a fair basis for isolating the influence of current harmonics on copper losses. The values of P_Fe and P_Fric were determined to be 7.50 W and 12.00 W, respectively, using Ansys at 3.5 Nm and 1500 RPM.

Figure 19 shows that the fundamental copper loss (P_cuFund) constitutes the dominant portion of total losses for all methods. The proposed model injects carefully structured harmonic components to suppress torque ripple. However, the resulting increase in harmonic copper loss (P_Harm) is minimal (1.13 W), representing only a small fraction of the total copper loss.

Since the proposed method suppresses torque ripples that lead to inefficiencies in torque production, it marginally reduces the fundamental current component (i_qFund) required to produce the same average torque. As observed in Figure 18b, the fundamental current component of the proposed model is lower compared to the other methods. This reduction in the fundamental component compensates for the additional load introduced by the injected harmonic currents. Consequently, the total RMS current (i_qRMS) and total copper losses remain at nearly the same level as the method without harmonic injection (Method 2) (32.83 W vs. 32.09 W). Thus, the proposed method achieves a balance of the RMS currents by reducing the fundamental current.

The conventional MTPA method (Method 1) draws an RMS current of 6.06 A, whereas the proposed model requires only 5.40 A to produce the same average torque. By effectively suppressing the ripple-inducing reluctance and cogging components that oppose rotation, the proposed controller reduces the fundamental current component (P_CuFund ≈ 31.7 W). This reduction creates a thermal headroom that accommodates the injected harmonics. The specific copper loss attributed to the harmonic injection (P_Harm) is calculated as 1.13 W (Table 4). This value corresponds to approximately 2.1% of the total machine losses (52.33 W), rendering it thermally negligible. Furthermore, the high-frequency flux variations introduced by these harmonics are of low amplitude and a structured nature; thus, the FEA-calculated iron losses (P_Fe) show no significant increase compared to the reference cases. Consequently, the total copper loss of the proposed method (32.83 W) is maintained at a level comparable to Method 2 (32.09 W), proving that torque quality is enhanced without compromising energy efficiency.

Figure 20 evaluates the trade-off between torque ripple suppression and total copper loss using the torque ripple ratio (TRR), computed as given in (35) [13]. The effectiveness of the control strategy is best understood through the trade-off analysis presented in Figure 20 and

Table 5. Comparison of the torque control methods at 3.5 Nm load and 1500 RPM.

Control Method	TRR (%)	ղ (%)	iqRMS (A)	Pout (W)	PTCuLoss (W)	PTLoss (W)
Method 1	57.14	90.18	6.06	550	40.40	59.90
Method 2	42.86	91.42	5.46	550	32.09	51.59
Method 3	21.43	91.30	5.37	550	32.93	52.43
Proposed Model	7.14	91.31	5.40	550	32.83	52.33

. The superiority of the proposed model is evident in the given values in Table 5.

$$\mathrm{TRR} (\%)={\displaystyle \frac{{\mathrm{T}}_{\max }-{\mathrm{T}}_{\min }}{{\mathrm{T}}_{\mathrm{average}}}} \times 100$$

(35)

The proposed model achieves a drastic reduction in TRR, lowering it from 57.14% (Method 1) to 7.14%. This represents an approximately 87% reduction in torque ripple. In exchange for this significant improvement, the total copper loss increases by only 0.74 W compared to Method 2. This implies that the slight electrical loss is negligible for a substantial gain in mechanical efficiency and system longevity. From an electromechanical perspective, this trade-off is highly advantageous. The elimination of torque ripple significantly mitigates mechanical vibration energy, reduces acoustic noise (NVH), and alleviates mechanical stress on bearings, thereby extending the operational lifespan of the drive system.

Consequently, the proposed model provides a compromise between torque quality and loss characteristics, rendering it a highly effective torque control strategy for high-performance PMSM applications.

4.4. Comprehensive Comparative Analysis and Implementation Feasibility

A holistic evaluation is conducted to provide rigorous validation of the proposed control strategy beyond steady-state torque ripple suppression. This section benchmarks the proposed model against state-of-the-art methods, including conventional FOC, MPC and ANN, focusing on three critical dimensions: computational complexity relative to real-time hardware constraints; multidimensional performance metrics; and the trade-offs between design cost and control precision. Subsequent subsections detail the algorithmic burden and timing determinism (Section 4.4.1), visualize the balance of conflicting performance indicators such as THD and efficiency (Section 4.4.2), and analyze the economic viability for industrial mass production (Section 4.4.3).

4.4.1. Computational Complexity and Implementation Feasibility

Although the validation presented in this study relies on high-fidelity 2D-FEA simulations and a MATLAB/Simulink co-simulation, the proposed model is inherently compatible with real-time implementation on DSP or FPGA platforms. Successful deployment requires a set of routinely measurable electrical quantities and a set of machine parameters obtained through standard offline characterization procedures. The real-time measurements include the two-phase currents and the electrical rotor position θ_e. The offline characterization includes the PM flux, T_cog, the q-axis flux, and L_q inductance. These quantities can be extracted using conventional experimental techniques. The PM flux linkage is determined from the integrated back-EMF observed during no-load operation. T_cog can be obtained using a high-precision torque transducer while the rotor is driven at low speed by a master motor [52,53]. The dq-axis flux characteristics and inductances can be acquired either through FEA or experimental current-flux mapping.

The feasibility of deploying advanced control algorithms in industrial drives is fundamentally determined by the trade-off between control performance and the required computational resources. Based on the theoretical framework and simulation results presented in this study, three primary implementation challenges—computational burden, memory constraints, and synchronization delays—have been identified and addressed. A detailed comparison of the proposed method against conventional FOC and advanced strategies such as MPC that require iterating through all possible voltage vectors [26,54] and ANNs involving complex weight matrix multiplications [39,55] is presented in

Table 6. Comprehensive comparison of the proposed method against state-of-the-art control strategies in terms of computational complexity, hardware requirements, and control performance.

Control Technique	Computational Burden	Trigonometric Functions	Iteration	Relative Execution Time	Memory Requirement	Hardware Cost	Dynamic Response	Relative Torque Ripple Level
Classical PI-FOC [50]	Very Low (Scalar operations)	2 (Park/Clarke)	No	1.0 (Reference)	Very Low	Low	Medium	1.0 (Reference)
Advanced MPC [26,54]	Very High (Cost function minimization)	2+ (Variable over prediction horizon)	Yes (High-order optimization solver)	5.0x–8.0x+	High	High	High	0.7–0.8
Observer-Based (EKF) [56]	High (Matrix multiplication and inversion)	2+ (Model linearization)	No (Recursive)	3.0x–5.0x	Medium	Medium	High	0.4–0.6
Artificial Intelligence (ANN/RL) [39,55]	Extremely High (Massive neuron weight/matrix multiplications)	No (Generally uses LUTs)	No (Feed forward)	8.0x–10.0x+	High	High	Very High	0.2–0.4
Iterative Learning Control [33,45]	Medium/High (Requires memory management)	Low (Time domain) High (Fourier-Based)	Yes (Loops required to learn error signal)	1.5x–2.0x	Very High	Medium	Low (Requires learning time)	0.1–0.3
Proposed Model	Low (Scalar operations)	2 (Park/Clarke)	No (Direct analytical solution)	1.2x–1.5x	Low	Low	High	0.1–0.3

.

Advanced control strategies, particularly Finite-Control-Set MPC (FCS-MPC), typically require iterative optimization loops to minimize the cost function at each sampling step. As highlighted in Table 6 and the literature, this iterative nature leads to a very high computational load, resulting in relative execution times of 5.0×–8.0× compared to the standard FOC. Furthermore, the variable execution time inherent in iterative solvers poses a risk of timing violations and jitter in real-time systems where deterministic behavior is crucial. Similarly, ANN-based methods involve complex weight matrix multiplications, leading to extreme execution times (8.0×–10.0×) and requiring high-cost hardware.

In contrast, the proposed method utilizes a closed-form analytical solution (34), which eliminates the need for iterative solvers or complex matrix operations. As shown in Table 6, this structural advantage ensures a short and deterministic Worst-Case Execution Time (WCET) with a constant time complexity of O(1). According to performance analyses, the total execution time of the proposed algorithm on a standard DSP (e.g., 150 MIPS) is anticipated to be only 1.2 to 1.5 times longer than that of a traditional FOC loop. This negligible overhead leaves a sufficient timing margin within the targeted sampling period for other critical tasks, allowing implementation on low-cost standard embedded platforms without performance bottlenecks.

The use of high-fidelity LUTs to capture magnetic saturation and cogging torque can strain the limited on-chip Flash/RAM capacity of embedded microcontrollers. As indicated in Table 6, advanced methods often require high memory resources. To address this, the proposed method exploits the magnetic symmetry of the SPMSM. As detailed in Section 4.1, instead of storing data for the entire rotor revolution, only the parameters for a single pole pitch are stored. This optimization radically reduces the memory footprint to a low level (Table 6). Furthermore, by employing linear interpolation for intermediate current and speed values, high control precision is maintained even with low-resolution tables, ensuring efficient memory utilization.

In digital control systems, the inevitable time delay between current measurement (ADC), computation, and PWM update can degrade control performance, especially at high speeds. The comprehensive analysis in Section 4.2 (Figure 9) shows that the proposed model is more robust than traditional methods, even when delays of up to 200 μs are considered. For practical real-time implementation, this robustness is further enhanced by introducing a simple angle prediction term, Δθ = ω_e × T_delay, into the position feedback. This compensates for the one-step delay by shifting the data read index, ensuring precise synchronization between the estimated parameters and the actual rotor position.

Furthermore, as demonstrated in the preceding sections, the proposed model exhibits superior performance in mitigating torque ripple. A traditional PI controller perceives ripple caused by structural (cogging torque) or electrical (harmonics) factors in the motor as a disturbance and, due to its limited bandwidth, is unable to suppress it. However, the proposed method eliminates the ripple at its source by generating effective current references. Although AI methods can achieve very low ripple levels, they do so by performing thousands of matrix operations. In contrast, the proposed analytical method delivers similar performance (0.3–0.5 level) using only a fraction of the computational power required by AI.

In summary, unlike observer-based methods (e.g., Extended Kalman Filters) or adaptive nonlinear controllers that treat structural ripples as disturbances, the proposed method proactively suppresses them using a deterministic analytical model. Due to its optimized memory usage and low computational load, it offers a feasible and cost-effective solution for industrial real-time applications, overcoming the hardware limitations often associated with complex control algorithms. Any embedded platform capable of executing a standard FOC algorithm possesses sufficient resources to implement the proposed control law without compromising real-time performance.

4.4.2. Multidimensional Performance Comparison

In order to facilitate a multidimensional comparison of the torque control methods, a six-metric radar chart has been constructed (Figure 21). This chart includes the torque ripple ratio (%), torque THD (%), I_qTHD (%), I_qRMS (A), total copper loss (W), and total loss (W).

The radar chart results indicate that the proposed model provides a distinct advantage in terms of torque quality. This chart visualizes the trade-offs inherent in each simplified approach compared to the proposed model.

As observed in Figure 21, while Method 2 and Method 3 provide improvements in specific metrics (e.g., TRR) compared to the baseline (Method 1), they suffer performance penalties in other areas due to their incomplete physical modeling. For instance, Method 3 reduces ripple but may exhibit higher THD or copper losses due to uncompensated inductance harmonics. The proposed model, represented by the blue line, encompasses the smallest area near the center of the chart for metrics like TRR and THD, indicating superior performance. The proposed model achieves this without any significant deterioration in other performance measures. This balanced superiority effectively bridges the gap between the electrical parameter-focused (Method 2) and cogging-focused (Method 3) approaches.

4.4.3. Analysis of Design Trade-Offs and Economic Viability

Selecting a control strategy involves making a fundamental trade-off between the performance of the algorithm and the cost of implementation. As shown in Table 6, traditional high-performance methods (e.g., FCS-MPC and neural networks) achieve low torque ripple, but typically require high hardware costs (e.g., FPGAs) and impose very high computational loads due to iterative optimization loops. The proposed method challenges this conventional trade-off by offering a solution that balances high-precision control with economic feasibility.

Unlike optimization-based strategies, the proposed method uses a compact analytical structure. It does not require iterative optimization, learning-based complex algorithms or additional sensing hardware. Leveraging offline parameter characterization minimizes the real-time computational burden and memory requirements on the processor. This low computational complexity enables the method to be easily implemented on low-cost, standard embedded controllers (DSP/MCU), rather than on expensive, high-end processors. This significantly reduces development time and system cost.

A key economic benefit of the proposed strategy is its capacity to offset hardware defects via software. Manufacturing tolerances in mass-produced motors often lead to high cogging torque or geometric asymmetries, which have traditionally required expensive mechanical refinements to correct. However, the proposed method enables these imperfect motors to operate with high performance by suppressing the resulting ripples via the control algorithm. As shown in Figure 21, the method achieves balanced superiority, effectively bridging the gap between low-cost hardware and high-quality torque production.

A quantitative comparison of execution metrics and error performance against the conventional method is summarized in

Table 7. Summary of computational efficiency and ripple reduction performance at 3.5 Nm and 1500 RPM.

Performance Metric	Conventional MTPA (Method 1)	Proposed Model	Improvement with Proposed Model
Relative Execution Time	1.0× (Base)	1.2×–1.5×	Low Overhead (Feasible on DSP)
Speed Ripple (nrip)	~16 RPM	~1.5 RPM	~90.6% Reduction
Torque Ripple (Trip)	~2.0 Nm	~0.20 Nm	~87.5% Reduction
Hardware Cost	Low	Low	No FPGA required
Implementation Complexity	Low	Low-Medium	Deterministic Equation (34)

to succinctly demonstrate the viability of the proposed model.

5. Conclusions

This study presents and validates a compact, analytical, model-based torque control strategy designed to mitigate parasitic torque ripple in surface-mounted permanent magnet synchronous machines (SPMSMs). Unlike conventional methods, which rely on high-bandwidth feedback or computationally expensive iterative optimization, the proposed method uses a deterministic algebraic model based on the magnetic co-energy principle. By synthesizing structured current references that proactively compensate for cogging torque and inductance harmonics, the following key conclusions can be drawn.

Comprehensive finite element analysis (FEA) co-simulations and comparative analyses demonstrate that the proposed model reduces the torque ripple ratio (TRR) by approximately 87.5% (from 57.14% to 7.14%) compared to the conventional maximum torque per ampere (MTPA) method. This improvement translates directly into superior mechanical performance, achieving a ~90.6% reduction in speed ripples at 1500 RPM while maintaining high control authority, even at low speeds of 100 RPM, where conventional methods typically degrade. Furthermore, the analyses verify that this substantial improvement in torque quality results in a negligible increase in total power loss of only ~2.1% (~0.74 W), according to Method 2. Extended dynamic validation also proves that the algorithm can effectively suppress ripples during start-ups, transient load steps, and deep magnetic saturation up to a load of 2.0 p.u.

With a relative execution time of only 1.2–1.5 compared to standard FOC and minimal memory requirements, implementation in standard industrial DSPs is feasible, eliminating the need for the expensive FPGA hardware required by advanced MPC or AI-based methods.

In conclusion, the proposed strategy effectively bridges the gap between low-cost hardware implementation and high-precision torque control. It provides a software-based solution that compensates for hardware imperfections, such as cogging torque and manufacturing asymmetries, thereby enhancing the performance and longevity of electric drive systems in industrial and traction applications.

This study employed high-accuracy FEA co-simulation as a virtual sensor to address the challenges in accurately measuring high-order (6th and 12th) torque harmonics, which are typically masked by the low-pass filtering effects of standard physical torque sensors. In addition to the highly promising results obtained from the high-fidelity, co-simulation framework, this limitation can be viewed as the next step for future research. In order to physically measure high-order torque harmonics, future experimental research on the proposed control strategy will require a high-performance dynamometer equipped with a high-bandwidth torque sensor (>5 kHz) on a physical test bench. Furthermore, to accommodate the minimal sampling times as well and prevent execution delays during harmonic current injection, the proposed analytical algorithm should be used on a rapid prototyping platform featuring a DSP, etc., architecture (e.g., dSPACE or Speedgoat). This physical verification has the potential to bridge the gap between the MBD stage described in this article and its application in the real world.