Explicit Algebraic Approximations for MTPA, MTPV, and Loss-Minimization Optimal Control of PMSMs

Abstract

This paper presents explicit algebraic methods for approximating optimal $dq$-axis current references in permanent magnet synchronous motors (PMSMs) under given torque commands. The proposed approach addresses three key optimal control strategies: maximum torque per ampere (MTPA), maximum torque per voltage (MTPV), and loss-minimization control. For MTPA operation, a closed-form explicit formula is derived to approximate the d-axis current that minimizes copper losses. For MTPV operation, an analytical expression is developed to approximate the optimal current vector, effectively addressing iron losses in the high-speed region. Furthermore, a simplified formulation for loss-minimization control is proposed to enhance overall efficiency by balancing both copper and iron losses. These formulas are computationally efficient and eliminate the need for iterative numerical procedures while maintaining high accuracy. Supplementary expressions are also provided to facilitate practical implementation under current and voltage constraints. The mathematical fidelity and computational efficiency of the proposed formulas are rigorously validated through numerical simulations using representative PMSM models. The results demonstrate that the proposed explicit approximations closely match the true numerical optimal trajectories, offering a practical alternative to complex iterative methods without the need for extensive experimental characterization.

1. Introduction

The rapid advancement of semiconductor and power electronics technology has propelled the widespread adoption of permanent magnet synchronous motors (PMSMs) in numerous industrial applications. PMSMs offer compelling advantages such as low noise and vibration, high power density and torque-to-weight ratio, high efficiency and robustness, and low maintenance requirements, making them ideal for embedded systems and applications demanding high performance and reliability. Energy efficiency is a critical concern in today’s industrial landscape. A substantial portion of global electricity consumption is attributed to electric motors used in pumps, fans, and compressors. Variable speed drives can significantly improve efficiency, especially during steady-state operations. As energy efficiency becomes increasingly important, driven by environmental concerns and regulatory standards, advanced control techniques are essential for optimizing the performance of electric drives, particularly in emerging applications like electric vehicles and autonomous systems. An optimal torque control problem for PMSM speed drives is a well-researched area focused on efficiency maximization, and the problem can be analytically formulated as an optimization problem that minimizes motor losses under hardware constraints. Recent studies have explored various control strategies to achieve this goal [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Key efficiency-maximizing torque control methods include maximum torque per ampere (MTPA) control, which prioritizes minimizing copper losses dominant at low speeds [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], and maximum torque per voltage (MTPV), which focuses on reducing iron losses significant at high speeds [1,2,3,4,5,6,7,21]. Loss-minimization control considers both copper losses and iron losses for a balanced approach [22,23,24,25,26,27].

Most prior analytical optimal torque control methods, designed to maximize efficiency, rely on solving quadratic equations to determine optimal $dq$-axis reference current vector values for a given torque reference. This process is computationally intensive and time-consuming. To mitigate this, approximation methods employing an approximate second-order equation have been considered for MTPA control in [11,12,13,14,15,16,17,18,19,20]. Nonetheless, the precision of these approximations deteriorates considerably as the operating point diverges from the zero point. This decline in accuracy is especially marked in the vicinity of the nominal torque, a consequence of the approximation’s inherent bias towards the zero point. The previous approximate MTPA control methods, which are introduced in [11,12,13,14,15,16,17,18,19,20], involve solving cubic equations to obtain approximate $dq$-axis optimal reference current vector for a given torque reference. Alternatively, numerical algorithm-based methods such as [7,8,9] have been proposed to alleviate the computational burden associated with solving quadratic equations. While effective, these methods often require numerous iterations to achieve accurate solutions. Moreover, their convergence performance can be significantly compromised by a poor initial starting point guess. This is particularly true for methods based on Newton’s method or linearized quadratic programming, which can be highly sensitive to the initial conditions.

To address these limitations, this paper presents straightforward, explicit formulas to calculate approximate $dq$-axis optimal reference current vector values for MTPA, MTPV, loss-minimization control. By leveraging the proposed method, these reference values can be computed using simple first-order explicit functions of the specified torque. Furthermore, the paper derives supplementary formulas to facilitate optimal control while adhering to current and voltage constraints. In contrast to the previous methods in [11,12,13,14,15,16,17,18,19,20], which exhibit significant accuracy degradation near the nominal torque, the proposed approach maintains high precision across the entire operating range. These approximate values also serve as reliable initial guesses for conventional numerical algorithms, leading to faster convergence. The simplicity and efficiency of the proposed formulas make them ideal for real-time implementation. Finally, the effectiveness of these algebraic approximations is verified via numerical simulations. It should be noted that this study focuses on the analytical accuracy of the optimal current trajectory mapping rather than the dynamic response of a specific motor drive system. Therefore, numerical verification using standard PMSM parameters is employed to provide a clear and precise comparison between the proposed formulas and the true numerical optima.

2. Problem Statement

In the synchronously rotating $dq$ reference frame, a three-phase mounted PMSM can be represented by the following continuous-time nonlinear equation:

$$\left\{\begin{array}{ccc}\hfill \dot{\omega }& =& k_1{i}_q-k_2\omega +k_3{i}_d{i}_q-k_4{T}_L\hfill \\ \hfill {\dot{i}}_q& =& -k_5{i}_q-k_6\omega -k_7\omega i_d+k_8{V}_q\hfill \\ \hfill {\dot{i}}_d& =& -k_9{i}_d+k_{10}\omega i_q+k_{11}{V}_d\hfill \end{array}\right.$$

(1)

where $T_L$ represents the load torque, $\omega$ is the electrical rotor angular speed, $i_q$ is the q-axis current, $V_q$ is the q-axis voltage, $i_d$ is the d-axis current, $V_d$ is the d-axis voltage, and $k_i>0,i=1,\dots ,11$ are parameter values given by

$$\begin{array}{c}{\displaystyle k_1=\frac{3}{2}\frac{1}{J}\frac{p^2}{4}{\lambda }_m,k_2=\frac{B}{J},k_3=\frac{3}{2}\frac{1}{J}\frac{p^2}{4}(L_d-L_q),k_4=\frac{p}{2J},k_5=\frac{R}{L_q}},\\ {\displaystyle k_6=\frac{{\lambda }_m}{L_q},k_7=\frac{L_d}{L_q},k_8=\frac{1}{L_q},k_9=\frac{R}{L_d},k_{10}=\frac{L_q}{L_d},k_{11}=\frac{1}{L_d}}\end{array}$$

(2)

depending on the number of poles p, the stator resistance R, the d-axis stator inductance $L_d$, the q-axis stator inductance $L_q$, the rotor inertia J, the viscous friction coefficient B, and the magnetic flux ${\lambda }_m$. The output torque $T_e$ can be expressed as

$$T_e={\displaystyle \frac{(k_1+k_3{i}_d)}{k_4}}{i}_q$$

(3)

In PMSM control system design, the following hardware inequality constraints should be considered

$$i_d^2+i_q^2\le I_M^2,$$

(4)

$$V_d^2+V_q^2\le V_M^2$$

(5)

where $I_M$ and $V_M$ are the maximum allowable stator current and voltage, respectively. The results of [2,3,4,5] imply that the voltage drops on the stator resistance is negligible in comparison to $V_M$ and the following equality conditions hold in steady state:

$$V_q={\displaystyle \frac{\omega (k_6+k_7{i}_d)}{k_8}},V_d=-{\displaystyle \frac{k_{10}\omega i_q}{k_{11}}}=-{\displaystyle \frac{\omega i_q}{k_8}}$$

(6)

Following the approach in [2,3,4,5,7,12,13,14,15,16,17,21,24], the voltage limit constraint (5) can be recast as the following inequality through the application of (6)

$${\displaystyle \frac{i_q^2}{k_8^2}}+{\displaystyle \frac{{(k_6+k_7{i}_d)}^2}{k_8^2}}\le {\displaystyle \frac{V_M^2}{{\omega }^2}}$$

(7)

An optimal torque control problem is then formulated to determine the optimal $dq$-axis reference current vector that minimizes a cost function J. This optimization problem is subject to the constraints (3), (4), and (7), and is solved for a given torque reference $T_r$ [1]. The resulting problem can be analytically represented as follows:

$$\arg \underset{(i_d,i_q)}{min}J\mathrm{s}.\mathrm{t}.\left(3\right),\left(4\right),\left(7\right),T_e=T_r$$

(8)

It is important to note that the MTPA control problem is equivalent to the optimization problem (8) with the specific cost function $J=J_I=i_d^2+i_q^2$. Similarly, The MTPV control problem corresponds to (8) with the cost function $J=J_V=V_d^2+V_q^2={\omega }^2[i_q^2+{(k_6+k_7{i}_d)}^2]/k_8^2$. The loss-minimization problem is equivalent to (8) with a cost function that can be approximated as a weighted combination of $J_I$ and $J_V$.

3. Operation Regions of PMSM

The prior optimal torque method of [5] introduced three PMSM optimal control operation regions: MTPA, field-weakening region 1 (FW 1), and field-weakening region 2 (FW 2). The operation region FW 1 is equivalent to maximum current (MC) region of [1,7]. The operation region FW 2 corresponds to the MTPV region of [1,7]. Figure 1 shows typical characteristic curves of a PMSM with ${\lambda }_m<L_d{I}_M$. When ${\lambda }_m\ge L_d{I}_M$, the MTPV region does not exist. Given this limitation, we will initially consider the case where ${\lambda }_m<L_d{I}_M$. The scenario of ${\lambda }_m\ge L_d{I}_M$ will be addressed separately.

The previous results imply that the followings holds:

$$i_{da}^{*}=-{\displaystyle \frac{{\lambda }_m}{4(L_d-L_q)}}-\sqrt{{\displaystyle \frac{{\lambda }_m^2}{16{(L_d-L_q)}^2}}+{\displaystyle \frac{I_M^2}{2}}}$$

(9)

$$i_{dv}^{*}=-{\displaystyle \frac{\left(a+b+\sqrt{b^2-c^2(a^2+2ab-(1+c^2)I_M^2)}\right)}{(1+c^2)}}$$

(10)

$$i_{qa}^{*}=\sqrt{I_M^2-i_{da}^{*2}},i_{qv}^{*}=\sqrt{I_M^2-i_{dv}^{*2}}$$

(11)

$$T_a={\displaystyle \frac{i_{qa}^{*}(k_1+k_3{i}_{da}^{*})}{k_4}},T_v={\displaystyle \frac{i_{qv}^{*}(k_1+k_3{i}_{dv}^{*})}{k_4}}$$

(12)

$${\omega }_a={\displaystyle \frac{V_M}{\sqrt{L_q^2{i}_{qa}^{*2}+{({\lambda }_m+L_d{i}_{da}^{*})}^2}}}$$

(13)

$${\omega }_v={\displaystyle \frac{V_M}{\sqrt{L_q^2{i}_{qv}^{*2}+{({\lambda }_m+L_d{i}_{dv}^{*})}^2}}}$$

(14)

where $a={\lambda }_m/L_d$, $b=L_q{\lambda }_m/2L_d(L_d-L_q)$, and $c=L_q/L_d$. The MTPA trajectory is represented by

$$i_d={\displaystyle \frac{\sqrt{k_1^2+4k_3^2{i}_q^2}-k_1}{2k_3}}=-{\displaystyle \frac{{\lambda }_m}{2(L_d-L_q)}}+\sqrt{{\displaystyle \frac{{\lambda }_m^2}{4{(L_d-L_q)}^2}}+i_q^2},0\le i_q\le i_{qa}^{*}$$

(15)

The MTPV trajectory is given in $(i_d,i_q)$ space as follows:

$$i_d=-a-b-\sqrt{b^2+c^2{i}_q^2},0\le i_q\le i_{qv}^{*}$$

(16)

and the MC trajectory exists on the boundary of the current limit constraint (4) and it is described by

$$i_d=-\sqrt{I_M^2-i_q^2},i_{qv}^{*}\le i_q\le i_{qa}^{*}$$

(17)

The base speed of [5] corresponds to ${\omega }_a$. The torque value $T_a$ is the nominal torque of the previous optimal torque methods.

Remark 1.

In the specific case of a surface-mounted PMSM, the inductance parameters become equal, i.e., $L_d=L_q=L$. Additionally, $k_3=0$, $k_7=k_{10}=1$, $k_5=k_9$, $k_8=k_{11}$. The output torque expression reduces to $T_e=k_1{i}_q/k_4$. With these simplifications, the Equations (9)–(14) are reduced to the following:

$$\begin{array}{c}{\displaystyle i_{da}^{*}=0,i_{dv}^{*}=-\frac{{\lambda }_m}{L},i_{qa}^{*}=I_M,i_{qv}^{*}=\sqrt{I_M^2-{\lambda }_m^2/L^2},}\hfill \\ {\displaystyle T_a=\frac{I_M{k}_1}{k_4},T_v=\frac{k_1\sqrt{I_M^2-{\lambda }_m^2/L^2}}{k_4},}\hfill \\ {\displaystyle {\omega }_a=\frac{V_M}{\sqrt{L^2{I}_M^2+{\lambda }_m^2}},{\omega }_v=\frac{V_M}{\sqrt{L^2{I}_M^2-{\lambda }_m^2}}}\hfill \end{array}$$

(18)

The MTPA, MTPV, and MC trajectories, originally defined in (15), (16), and (17), are reduced to the simplified forms presented in the following Equations (19), (20) and (21), respectively.

$$i_d=0,0\le i_q\le I_M$$

(19)

$$i_d=-{\displaystyle \frac{{\lambda }_m}{L}},0\le i_q\le \sqrt{I_M^2-{\lambda }_m^2/L^2}$$

(20)

$$i_d=-\sqrt{I_M^2-i_q^2},\sqrt{I_M^2-{\lambda }_m^2/L^2}\le i_q\le I_M$$

(21)

4. Explicit Algebraic Approximations for MTPA, MTPV, Loss-Minimization Control

4.1. Explicit MTPA Approximation

The previous approximate MTPA control methods, which are considered in [11,12,13,14,15,16,17,18,19,20], have used the following quadratic equation:

$$i_d={\displaystyle \frac{k_3}{k_1}}{i}_q^2={\displaystyle \frac{(L_d-L_q)}{{\lambda }_m}}{i}_q^2$$

(22)

The above Equation (22) is derived via a Taylor series expansion around zero, based on the nonlinear exact MTPA trajectory Formula (15). While this Taylor-series-based Equation (22) remains highly accurate near the zero point, it exhibits a significant loss of precision as it approaches the nominal torque due to the expansion’s local nature. Moreover, determining an approximate optimal d-axis current for a given torque reference $T_r$ using (22) necessitates solving a complex cubic equation.

Figure 1 suggests that a linear approximation of the MTPA trajectory can be obtained by drawing a straight line connecting points $(i_{da}^{*},T_a)$ and $(0,0)$ in $(i_d,T_r)$ space. This results in the following linear first-order function from $T_r$ to $i_d$

$$i_d={\displaystyle \frac{i_{da}^{*}}{T_a}}{T}_r,0\le T_r\le T_a$$

(23)

It should be noted that using (3) the above linear function from $T_r$ to $i_d$ can be rewritten as the following rational function from $i_q$ to $i_d$:

$$i_d={\displaystyle \frac{k_1{i}_{da}^{*}}{(k_4{T}_a-k_3{i}_{da}^{*}{i}_q)}}{i}_q$$

(24)

In contrast to the previous approximate MTPA control methods, an approximate optimal $i_d$ can be straightforwardly computed by (23) for a given $T_r$.

Figure 1 also suggests that the nonlinear MTPA trajectory can be approximated as the following linear first-order function represented by a straight line connecting points $(i_{da}^{*},i_{qa}^{*})$ and $(0,0)$ in $(i_d,i_q)$ space:

$$i_d={\displaystyle \frac{i_{da}^{*}}{i_{qa}^{*}}}{i}_q,0\le i_q\le i_{qa}^{*}$$

(25)

By combining (3) and (25), the following can be obtained for a given $T_e=T_r$:

$$i_d={\displaystyle \frac{i_{da}^{*}{k}_4}{(k_1+k_3{i}_d)i_{qa}^{*}}}{T}_r$$

(26)

leading to the following quadratic equation

$$k_3{i}_{qa}^{*}{i}_d^2+i_{qa}^{*}{k}_1{i}_d-i_{da}^{*}{k}_4{T}_r=0$$

(27)

After all, the following explicit solution can be derived in the form of a nonlinear function of $T_r$:

$$i_d={\displaystyle \frac{-i_{qa}^{*}{k}_1+\sqrt{i_{qa}^{*2}{k}_1^2+4k_3{k}_4{i}_{da}^{*}{i}_{qa}^{*}{T}_r}}{2k_3{i}_{qa}^{*}}}$$

(28)

It should be noted that all the approximate Equations (23)–(25) and (28) are primarily accurate near the nominal torque $T_a$. For a given torque reference $T_r$, an approximate optimal $i_d$ can be readily computed using either (23) or (28). Due to its simpler form, the linear Equation (23) is computationally more efficient than (28). Therefore, (23) is preferable to (28) in terms of computational burden. When designing the optimal $i_d$ for a given $i_q$, as in [11,12,13,14,15], both (24) and (25) can be used to directly determine an approximate optimal $i_d$. While both approximations offer reasonable accuracy near the nominal torque $T_a$, unlike the quadratic approximation (22) used in [11,12,13,14,15], the linear approximation (25) is computationally more efficient than the nonlinear approximation (24). All the Formulas (23), (24), (25), and (28) can serve as supplementary tools to estimate an approximate optimal d-axis current for MTPA control. Depending on the given conditions, the most suitable formula can be selected, allowing for a tailored approach to MTPA control.

In terms of computational complexity, the proposed first-order function (23) requires only one multiplication to determine the optimal reference current for the given torque reference $T_r$. The refined Formula (28) is an explicit solution to the quadratic Equation (27). In contrast, the prior analytical method [16] involves solving a quartic equation, while the methods in [11,12,13,14,15,16,17,18,19,20] require solving a cubic equation. Based on the relative computational cost analysis in [28], where a single multiplication is the unit cost, the complexity of solving quadratic, cubic, and quartic equations increases more than twofold at each successive degree. This confirms that the proposed formulas are significantly more efficient for real-time implementation compared to conventional higher-order analytical methods.

4.2. Explicit MTPV Approximation

Figure 1 implies that an approximate MTPV trajectory can be represented by the following first-order function in $(i_d,T_r)$ space

$$i_d={\displaystyle \frac{k_6+k_7{i}_{dv}^{*}}{k_7{T}_v}}{T}_r-{\displaystyle \frac{k_6}{k_7}},0\le T_r\le T_v$$

(29)

Using (3) the above function from $T_r$ to $i_d$ can be rewritten as the following rational function from $i_q$ to $i_d$:

$$i_d={\displaystyle \frac{k_4{k}_6{T}_v-k_1(k_6+k_7{i}_{dv}^{*})i_q}{k_3(k_6+k_7{i}_{dv}^{*})i_q-k_4{k}_7{T}_v}}$$

(30)

Figure 1 also suggests that the MTPV trajectory can be approximated as the following first-order function represented by a straight line connecting points $(i_{dv}^{*},i_{qv}^{*})$ and $(-k_6/k_7,0)$ in $(i_d,i_q)$ space:

$$i_d={\displaystyle \frac{k_6+k_7{i}_{dv}^{*}}{k_7{i}_{qv}^{*}}}{i}_q-{\displaystyle \frac{k_6}{k_7}},0\le i_q\le i_{qv}^{*}$$

(31)

Along the similar line to obtain (28), an alternative approximate MTPV trajectory can be obtained in $(i_d,T_r)$ space as follows:

$$i_d={\displaystyle \frac{-b_1+\sqrt{b_1^2-4a_1{c}_1}}{2a_1}}$$

(32)

where $a_1=i_{qv}^{*}{k}_3{k}_7$, $b_1=i_{qv}^{*}(k_3{k}_7+k_1{k}_7)$, $c_1=i_{qv}^{*}{k}_1{k}_6-k_4{T}_r(k_7{i}_{dv}^{*}+k_6)$, $0\le T_r\le T_v$.

While both approximate Equations (29) and (32) provide reasonable accuracy in computing an approximate MTPV optimal current vector for a given $T_r$ near $T_v$, the first-order approximation (29) is computationally more efficient due to its simpler form. Similarly, when designing the optimal $i_d$ for a given $i_q$, the first-order approximation (31) is computationally more efficient than the rational function (30).

If $T_v\le T_r\le T_a$ and $\omega$ satisfies the voltage limit constraint (7), then the optimal current vector that minimizes iron losses exists within the operation region MC. This implies that operating a PMSM along the MC trajectory constitutes a form of MTPV control, enabling efficient operation for given $T_v\le T_r\le T_a$ while adhering to the current limit constraint (4). Thus, approximating the MC trajectory will be considered in this subsection.

Figure 1 implies that the MC trajectory can be approximated by the following linear segment between points $(i_{da}^{*},T_a)$ and $(i_{dv}^{*},T_v)$ in $(i_d,T_r)$ space:

$$i_d={\displaystyle \frac{(i_{da}^{*}-i_{dv}^{*})(T_r-T_v)}{(T_a-T_v)}}+i_{dv}^{*}$$

(33)

Using (3) the above function from $T_r$ to $i_d$ can be rewritten as the following rational function from $i_q$ to $i_d$:

$$i_d={\displaystyle \frac{k_1{i}_q(i_{da}^{*}-i_{dv}^{*})-k_4{T}_v{i}_{da}^{*}+k_4{T}_a{i}_{dv}^{*}}{k_4(T_a-T_v)-k_3(i_{ds}^{*}-i_{dv}^{*})i_q}}$$

(34)

Figure 1 also suggests that the MC trajectory can be linearized by approximating it with the following straight line passing through points $(i_{dv}^{*},i_{qv}^{*})$ and $(i_{da}^{*},i_{qa}^{*})$ in $(i_d,i_q)$ space:

$$i_d={\displaystyle \frac{(i_{dv}^{*}-i_{da}^{*})(i_q-i_{qa}^{*})}{(i_{qv}^{*}-i_{qa}^{*})}}+i_{da}^{*}$$

(35)

By adopting a similar approach to the derivation of (28), an alternative approximate MC trajectory equation can be derived for a given $T_r$. This equation is given by a nonlinear function of $T_r$ as follows:

$$i_d={\displaystyle \frac{-b_2+\sqrt{b_2^2-4a_2{c}_2}}{2a_2}}$$

(36)

where $a_2=(i_{qv}^{*}-i_{qa}^{*})k_3$, $b_2=k_3(i_{qa}^{*}{i}_{dv}^{*}-i_{qv}^{*}{i}_{da}^{*})+k_1(i_{qv}^{*}-i_{qa}^{*}$, $c_2=k_1(i_{qa}^{*}{i}_{dv}^{*}-i_{qv}^{*}{i}_{da}^{*})-k_4{T}_r(i_{dv}^{*}-i_{da}^{*})$, $T_v\le T_r\le T_a$.

When designing an approximate MC optimal $i_d$ for a given $T_r$, the first-order approximation (33) is computationally more efficient than the nonlinear approximation (36). Similarly, when designing an approximate MC optimal $i_d$ for a given $i_q$, the first-order approximation (35) is computationally more efficient than the rational function (34).

While the maximum speed limit ${\omega }_M\left(T_r\right)$ for the MTPA region is readily available from

$${\omega }_M\left(T_r\right)={\omega }_a,0\le T_r\le T_a$$

(37)

determining the maximum speed limit ${\omega }_M\left(T_r\right)$ for the MTPV or MC regions is computationally intensive. However, by employing the first-order approximations (29) and (33), the maximum speed limit ${\omega }_M\left(T_r\right)$ can be easily estimated for a specific $T_r$. Combining (3) with the voltage limit constraint (7) yields the following equation:

$$\omega \le {\displaystyle \frac{k_8{V}_M}{\sqrt{k_4^2{T}_r^2/{(k_1+k_3{i}_d)}^2+{(k_6+k_7{i}_d)}^2}}}$$

(38)

This, combined with the first-order approximation (29), implies that an approximation of the maximum speed limit ${\omega }_M\left(T_r\right)$ in the MTPV region can be given by

$${\omega }_M\left(T_r\right)={\displaystyle \frac{k_8{V}_M}{\sqrt{k_4^2{T}_r^2/{({\gamma }_1+{\gamma }_2{T}_r)}^2+{\gamma }_3^2{T}_r^2}}}$$

(39)

where ${\gamma }_1=k_1-k_3{k}_6/k_7$, ${\gamma }_2=k_3(k_6+k_7{i}_{dv}^{*})/k_7{T}_v$, ${\gamma }_3=(k_6+k_7{i}_{dv}^{*})/T_v$, $0\le T_r\le T_v$. Similarly, the maximum speed limit ${\omega }_M\left(T_r\right)$ in the MC region can be approximated by

$${\omega }_M\left(T_r\right)={\displaystyle \frac{k_8{V}_M}{\sqrt{k_4^2{T}_r^2/{({\beta }_1+{\beta }_2{T}_r)}^2+{({\beta }_3+{\beta }_4{T}_r)}^2}}}$$

(40)

where ${\beta }_1=k_1+k_3{i}_{dv}^{*}-k_3(i_{da}^{*}-i_{dv}^{*})T_v/(T_a-T_v)$, ${\beta }_2=k_3(i_{da}^{*}-i_{dv}^{*})/(T_a-T_v)$, ${\beta }_3=k_6+k_7{i}_{dv}^{*}-k_7(i_{da}^{*}-i_{dv}^{*})T_v/(T_a-T_v)$, ${\beta }_4=k_7(i_{da}^{*}-i_{dv}^{*})/(T_a-T_v)$, $T_v\le T_r\le T_a$.

By employing (39) and (40) for a given $T_r$, a maximum speed limit can be estimated, which is beneficial for MTPV control. Likewise, the proposed approximation formulas allow for the effective operation of a PMSM within the operation regions for a specified $T_r$ while ensuring that the current and voltage limit constraints, as outlined in (4) and (7), are not exceeded.

Remark 2.

When ${\lambda }_m\ge L_d{I}_M$, the MTPV region disappears. Consequently, the proposed formulas for the MTPV region are not applicable. Nevertheless, the proposed formulas derived for the MC region remain effective, as long as $i_{dv}^{*}$, $i_{qv}^{*}$, and $T_v$ are set as $i_{dv}^{*}=-I_M$, $i_{qv}^{*}=0$, and $T_v=0$. Because operating a PMSM along the MC trajectory is a particular instance of MTPV control, the proposed formulas for the MC region can be utilized for MTPV control in this scenario. The simplified voltage limit constraint (7) is derived under the assumption that the stator resistance voltage drop is negligible. However, as highlighted in [6], the MTPV trajectories for PMSMs—particularly those operating with a low voltage limit—can deviate from conventional trajectories because the resistive voltage drop becomes significant relative to $V_M$ as the speed increases. Consequently, the MTPV approximation Formulas (29) and (32) may exhibit increased estimation errors in applications with low-battery voltages or high-stator resistance, where the applicable boundary of the simplified constraint is exceeded.

4.3. Explicit Loss-Minimization Approximation

Following the previous methods [26,27], the loss-minimization control problem can be approximated as

$$\arg \underset{(i_d,i_q)}{min}{J}_L\mathrm{s}.\mathrm{t}.\left(\right),\left(\right),\left(\right),T_e=T_r$$

(41)

where $J_L=R{J}_I+J_V/R_i$, $R_i$ is the iron loss resistance, and $R_i\gg R$. Let $i_{dI}$ and $i_{dV}$ denote the optimal solutions to the optimization problem (8) with cost functions $J=J_I$ and $J=J_V$, respectively. Then, inspired by [27], an approximate solution $i_d$ to the loss-minimization control problem (41) is represented by a weighted combination of $i_{dI}$ and $i_{dV}$ as follows:

$$i_d={\displaystyle \frac{i_{dI}R{R}_i+i_{dV}{L}_d^2{\omega }^2}{R{R}_i+L_d{\omega }^2}}$$

(42)

It is important to note that $i_{dI}$ and $i_{dV}$ are the solutions to the MTPA and MTPV control problems, respectively. By leveraging the proposed approximations (23) and (29) for $i_{dI}$ and $i_{dV}$, an approximate solution $i_d$ to the loss-minimization control problem can be derived for a given $T_r$. This solution takes the form of a first-order function of $T_r$ as follows:

$$i_d=\left({\displaystyle \frac{i_{da}^{*}R{R}_i}{T_a(R{R}_i+L_d{\omega }^2)}}+{\displaystyle \frac{(k_7{i}_{dv}^{*}+k_6)L_d^2{\omega }^2}{k_7{T}_v(R{R}_i+L_d{\omega }^2)}}\right)T_r-{\displaystyle \frac{k_6{L}_d^2{\omega }^2}{k_7(R{R}_i+L_d{\omega }^2)}}$$

(43)

Remark 3.

The approximate solution (43) is limited by the applicability of the Formulas (10) and (12). For the specific case where ${\lambda }_m\ge L_d{I}_M$, the MTPV region vanishes, rendering the Formulas (10) and (12) inapplicable for determining valid $i_{dv}^{*}$ and $T_v$. However, the approximate solution (43) remains viable for loss-minimization control. In this scenario, $i_{dv}^{*}$ and $T_v$ can be computed using the following alternative formulas:

$$i_{dv}^{*}=-{\displaystyle \frac{a+b+\sqrt{b^2-c^2(a^2+2ab-(1+c^2)I_T^2)}}{1+c^2}}$$

(44)

$$T_v={\displaystyle \frac{(k_1+k_3{i}_{dv}^{*})\sqrt{I_T^2-i_{dv}^{*2}}}{k_4}}$$

(45)

where $I_T$ is a sufficiently large constant guaranteeing $I_T>{\lambda }_m/L_d$, and $a={\lambda }_m/L_d$, $b=L_q{\lambda }_m/2L_d(L_d-L_q)$, and $c=L_q/L_d$.

Remark 4.

For a surface-mounted PMSM, $L_d=L_q=L$, and the above approximate solution (43) is reduced to

$$i_d=-{\displaystyle \frac{{\lambda }_m{L}^2{\omega }^2}{R{R}_i+L{\omega }^2}}$$

(46)

The exact solution derived in [22] is given by

$$i_d=-{\displaystyle \frac{{\lambda }_m{L}^2{\omega }^2(R+R_i)}{R{R}_i^2+L{\omega }^2(R+R_i)}}$$

(47)

This exact solution can be approximated to (46) when $R_i\gg R$, a common condition in PMSMs. This in turn implies that the proposed approximate solution (43) is reasonably accurate.

5. Accuracy Refinement and Design Procedure

To further enhance the accuracy of the proposed approximate optimal solutions, the previous numerical algorithm-based methods like those presented in [7,8,9] can be applied. However, it’s crucial to recognize that the convergence performance of these methods, especially those rooted in Newton’s method or linearized quadratic programming, can be significantly impacted by the quality of the initial starting point. By utilizing the proposed approximate values as robust initial estimates, we can significantly improve the performance of these numerical algorithm-based methods and accelerate their convergence rates.

Alternatively, we can use the following analytic methods inspired by Newton’s method, which involve solving quadratic equations, to improve the accuracy of the proposed approximate optimal solutions computed using the Formulas (23), (29), (33), and (43).

5.1. Refined MTPA Approximation

Initially, let’s focus on refining the accuracy of the approximate optimal solution of (23) within the MTPA region. For a given $T_r$ satisfying $0\le T_r\le T_a$, the proposed formula (23) yields an approximate MTPA optimal solution vector $(d_a,q_a)$ where $d_a=i_{da}^{*}{T}_r/T_a$ and $q_a=(k_1+k_3{d}_a)/k_4{T}_r$. We can approximate the output torque curve of $T_r=i_q(k_1+k_3{i}_d)/k_4$ as a tangent line at the point $(d_a,q_a)$, which can be expressed as:

$$i_q=s_{a1}{i}_d+s_{a0}$$

(48)

where $s_{a1}=-k_3{k}_4{T}_r/{(k_1+k_3{d}_a)}^2$ and $s_{a0}=q_a-s_{a1}{d}_a$. By referring to Figure 2a, we observe that a refined solution $i_{dr}$ can be determined from the intersection point of the MTPA curve (15) and the straight line (48). Combining (15) and (48), we obtain the following refined approximate MTPA optimal solution vector $(i_{dr},i_{qr}=(k_1+k_3{i}_{dr})/k_4{T}_r)$:

$$i_{dr}={\displaystyle \frac{-b_a+\sqrt{b_a^2-4a_a{c}_a}}{2a_a}}$$

(49)

where $a_a=k_3(1-s_{a1}^2)$, $b_a=k_1-2s_{a0}{s}_{a1}{k}_3$, and $c_a=-k_3{s}_{a0}^2$.

5.2. Refined MTPV Approximation

Now, consider the problem of enhancing the precision of the approximate optimal solution of (29) within the MTPV region. For a given $T_r$ satisfying $0\le T_r\le T_v$, we can obtain an approximate MTPV optimal solution vector $(d_v,q_v)$ where $d_v=(k_7{i}_{dv}^{*}+k_6)T_r/k_7{T}_v-k_6/k_7$ and $q_v=(k_1+k_3{d}_v)/k_4{T}_r$. We can also define the following tangent line at the point $(d_v,q_v)$:

$$i_q=s_{v1}{i}_d+s_{v0}$$

(50)

where $s_{v1}=-k_3{k}_4{T}_r/{(k_1+k_3{d}_v)}^2$ and $s_{v0}=q_v-s_{v1}{d}_v$. By referring to Figure 2b and combining (16) and (50), we can obtain the following refined approximate MTPV optimal solution vector $(i_{dr},i_{qr}=(k_1+k_3{i}_{dr})/k_4{T}_r)$:

$$i_{dr}={\displaystyle \frac{k_8(-b_v+\sqrt{b_v^2-4a_v{c}_v})}{2k_7{a}_v}}-{\displaystyle \frac{k_6}{k_7}}$$

(51)

where $a_v={\alpha }_2(1-e_1^2)$, $b_v={\alpha }_1-2e_0{e}_1{\alpha }_2$, and $c_v=-e_0^2{\alpha }_2$, ${\alpha }_2=-k_3{k}_8{k}_{11}$, ${\alpha }_1=k_3{k}_6{k}_{11}-k_1{k}_8$, $e_1=s_{v1}/k_7$, and $e_0=s_{v0}/k_8-s_{v1}{k}_6/k_7{k}_8$.

Similarly, in order to reduce the approximation error of the approximate optimal solution for the MC region, we can define a point $(d_c,q_c)$ and a tangent line at the point $(d_c,q_c)$:

$$i_q=s_{c1}{i}_d+s_{c0}$$

(52)

where $d_c=(i_{da}^{*}-i_{dv}^{*})(T_r-T_v)/(T_a-T_v)+i_{dv}^{*}$, $q_c=(k_1+k_3{d}_v)/k_4{T}_r$, $T_v\le T_r\le T_a$, $s_{c1}=-k_3{k}_4{T}_r/{(k_1+k_3{d}_c)}^2$ and $s_{c0}=q_c-s_{c1}{d}_c$. By referring to Figure 2c and combining (17) and (52), we can obtain the following refined approximate MC optimal solution vector $(i_{dr},i_{qr}=(k_1+k_3{i}_{dr})/k_4{T}_r)$:

$$i_{dr}={\displaystyle \frac{-b_c+\sqrt{b_c^2-4a_c{c}_c}}{2a_c}}$$

(53)

where $a_c=s_{c1}^2+1$, $b_r=2s_{c1}{s}_{c0}$, and $c_r=s_{c0}^2-I_M^2$.

5.3. Refined Loss-Minimization Approximation

The approximate solution (42) to the loss-minimization control problem (41) is derived from the solutions to the MTPA and MTPV optimal control problems. This interdependence suggests that employing more precise approximate MTPA and MTPV optimal solution vectors can directly improve the accuracy of the overall approximate solution. To enhance the precision of the approximate solution of (43) for a given $T_r$, we simply replace (43) with

$$i_d={\displaystyle \frac{i_{drI}R{R}_i+i_{drV}{L}_d^2{\omega }^2}{R{R}_i+L_d{\omega }^2}}$$

(54)

where $i_{drI}$ and $i_{drV}$ are refined approximate MTPA and MTPV optimal current values given by

$$i_{drI}={\displaystyle \frac{-b_a+\sqrt{b_a^2-4a_a{c}_a}}{2a_a}}$$

(55)

$$i_{drV}={\displaystyle \frac{k_8(-b_v+\sqrt{b_v^2-4a_v{c}_v})}{2k_7{a}_v}}-{\displaystyle \frac{k_6}{k_7}}$$

(56)

$a_a=k_3(1-s_{a1}^2)$, $b_a=k_1-2s_{a0}{s}_{a1}{k}_3$, $c_a=-k_3{s}_{a0}^2$, $s_{a1}=-k_3{k}_4{T}_r/{(k_1+k_3{d}_a)}^2$, $s_{a0}=q_a-s_{a1}{d}_a$, $d_a=i_{da}^{*}{T}_r/T_a$, $q_a=(k_1+k_3{d}_a)/k_4{T}_r$, $a_v={\alpha }_2(1-e_1^2)$, $b_v={\alpha }_1-2e_0{e}_1{\alpha }_2$, $c_v=-e_0^2{\alpha }_2$, $s_{v1}=-k_3{k}_4{T}_r/{(k_1+k_3{d}_v)}^2$, $s_{v0}=q_v-s_{v1}{d}_v$. ${\alpha }_2=-k_3{k}_8{k}_{11}$, ${\alpha }_1=k_3{k}_6{k}_{11}-k_1{k}_8$, $e_1=s_{v1}/k_7$, $e_0=s_{v0}/k_8-s_{v1}{k}_6/k_7{k}_8$, $d_v=(k_7{i}_{dv}^{*}+k_6)T_r/k_7{T}_v-k_6/k_7$, and $q_v=(k_1+k_3{d}_v)/k_4{T}_r$. As noted in Section 4, if the Formulas (10) and (12) are inapplicable for determining valid $i_{dv}^{*}$ and $T_v$, we should resort to (44) and (45).

5.4. Overall Design Procedure

Our results to compute approximate optimal current vector values for a given torque reference can be summarized as the following design procedure:

Step 1:

Parameter Calculation

• Compute the important values $i_{da}^{*}$, $i_{dv}^{*}$, $i_{qa}^{*}$, $i_{qv}^{*}$, $T_a$, $T_v$, ${\omega }_a$, and ${\omega }_v$ using (9)–(14).
• If the reference torque satisfies operating conditions (e.g., $T_r\le T_{mtpa}$), proceed to Step 2. Otherwise, reselect $T_r$ as a feasible value.

Step 2:

Maximum Speed Limit Calculation

• Compute the maximum speed limit values ${\omega }_M\left(T_r\right)$ using the proposed Formulas (37), (39), and (40) based on the operation regions and the given torque reference value $T_r$.
• If a reference speed ${\omega }_r$ is required, design ${\omega }_r$ adhering to the speed limit Formulas (37), (39), and (40), that is, find a feasible pair $(T_r,{\omega }_r)$.

Step 3:

Explicit Optimal Current Vector Calculation

• Compute the initial approximate optimal current vector values using the proposed first-order explicit Formulas (23), (29), (33), and (43).
• If higher precision is required, proceed to Step 4 for further refinement. Otherwise, the current values can be directly used for the torque control system.

Step 4: (Optional):

Accuracy Refinement

• For enhanced precision, compute refined approximate optimal current vector values using the proposed Formulas (49), (51), (53), and (54), taking the initial values obtained in Step 3 as the starting points.
• (Optional Iteration) If even higher accuracy is required, a conventional numerical algorithm can be employed, using the refined values from the previous step as a robust initial estimate to ensure rapid convergence. Otherwise, the current values may be used directly for control.

Remark 5.

To further enhance the accuracy of the approximate optimal solutions obtained via (49), (51), (53), and (54), the corresponding formulas can be applied iteratively. For instance, the accuracy of the refined solution from (49): (1) set the refined vector $(i_{dr},i_{qr}=(k_1+k_3{i}_{dr})/k_4{T}_r)$ calculated from (49) as a new initial point $(d_a,q_a)$; (2) determine a new tangent line (48) at this updated point; and (3) compute the next refined vector using (49). By repeating this procedure, the approximation error can be reduced to the desired level. Note that the convergence of this iterative process generally depends on the initial starting point. In this study, the explicit formulas in Section 4 provide sufficiently accurate initial estimates to ensure reliable and rapid convergence in practical PMSM operating ranges, as validated through the numerical examples in Section 6.

Remark 6.

The robustness of the proposed method is inherently linked to the accuracy of the motor parameters. Like conventional numerical and analytical approaches, the proposed formulas are developed under the assumption that motor parameters are precisely known. Consequently, variations in these parameters due to temperature changes or magnetic saturation may lead to increased approximation errors. However, because the proposed methods provide explicit algebraic solutions, they can be more easily integrated with real-time parameter estimation techniques to maintain control accuracy in practical applications, unlike computationally intensive iterative algorithms.

6. Numerical Verification

The primary objective of this study is to verify the accuracy of the proposed explicit algebraic formulas in approximating the theoretical optimal trajectories for MTPA, MTPV, and loss-minimization control. Since the focus is on the precision of reference current generation rather than the dynamic performance of the motor drive system, numerical validation, which allows for direct comparison with exact theoretical values, is sufficient to demonstrate the effectiveness of the proposed method. To validate the proposed approach, we consider a PMSM with the following parameters [15]: $p=4$, $R=2.48\left[\mathrm{\Omega }\right]$, $L_d=74.98\left[\mathrm{mH}\right]$, $L_q=113.91\left[\mathrm{mH}\right]$, ${\lambda }_m=0.193[\mathrm{V}·\sec /\mathrm{rad}]$, $J=0.0042[\mathrm{kg}·{\mathrm{m}}^2]$, $B=0.0001[\mathrm{N}·\mathrm{m}·\sec /\mathrm{rad}]$, and $T_L=1.5[\mathrm{N}·\mathrm{m}]$. These parameters result in the following dynamic model [15]:

$$\left\{\begin{array}{ccc}\hfill \dot{\omega }& =& 2757i_q-0.2381\omega -556.1i_d{i}_q-4762T_L\hfill \\ \hfill {\dot{i}}_q& =& -21.77i_q-1.694\omega -0.6582\omega i_d+8.779V_q\hfill \\ \hfill {\dot{i}}_d& =& -33.08i_d+1.519\omega i_q+13.34V_d\hfill \end{array}\right.$$

(57)

Assume that $I_M=10$ and $V_M=196.7$. Using (9)–(14) for the given motor parameters, we can obtain $i_{da}^{*}=-5.9395$, $i_{dv}^{*}=-8.2279$, $i_{qa}^{*}=8.045$, $i_{qv}^{*}=5.6834$, $T_a=10.2387$, $T_v=8.7521$, ${\omega }_a=206.90$, and ${\omega }_v=254.14$.

Figure 3a,b illustrate the approximated MTPA (dotted green), MC (dotted red), and MTPV (dotted blue) curves. Figure 3a shows these curves derived from the proposed first-order Formulas (23), (29), and (33), respectively. The solid black curves in this figure represent the output torque curves for two specific conditions: $T_e=T_a$ and $T_e=T_v$. Figure 3b displays the curves derived from the refined nonlinear Formulas (49), (51), and (53). For comparison, each figure also includes the true MTPA, MC, and MTPV curves as solid lines. To quantitatively evaluate the approximation accuracy, the mean and maximum Euclidean distance errors are calculated for the proposed formulas. The first-order Formulas (23), (29), and (33) exhibit mean/maximum errors below 0.23/0.46, 0.13/0.30, and 0.51/0.78, respectively. The refined nonlinear Formulas (49), (51), and (53) demonstrate exceptional precision, reducing the mean/maximum errors to less than 0.05/0.1.

The quadratic approximation Formula (22), which is used for MTPA control in [11,12,13,14,15,16,17,18,19,20], can also be used to derive an approximated MTPA curve. For a more in-depth comparison, Figure 4a illustrates the approximated MTPA curves generated by the conventional quadratic Formula (22) and the proposed Formulas (23), (25), and (49). The lines are color-coded as follows: green for the true MTPA (15), cyan for (22), blue for (23), black for (25), and magenta for (49). Figure 4b presents a comparative analysis of Euclidean distance errors. The cyan line represents the Euclidean distance error between the true optimal current vector and the approximated optimal current vector obtained using the conventional Formula (22) for a given torque reference $T_r$. Similarly, the blue, black, and magenta lines depict the Euclidean errors corresponding to the approximated optimal current vector calculated by the proposed Formulas (23), (25), and (49), respectively. Figure 4c depicts Euclidean distance errors versus $i_q$. The mean/maximum Euclidean distance errors are calculated for both the conventional and the proposed algebraic formulas in order to quantitatively evaluate the approximation accuracy. The conventional Formula (22) exhibits mean/maximum errors of 1.9/7.1. In contrast, the proposed Formulas (23) and (25) achieve much higher accuracy, with mean/maximum errors below 0.23/0.46 and 0.51/0.80, respectively. Furthermore, the refined nonlinear Formula (49) demonstrate exceptional precision, reducing the mean/maximum errors to less than 0.003/0.008, which are practically negligible. These quantitative data, along with the trajectories in Figure 4, confirm that the proposed methods offer significant improvements in precision compared to the conventional formula.

To demonstrate the effectiveness of the proposed approximate solutions (43) and (54) for the loss-minimization control problem (41), we assume a constant iron loss resistance, $R_i=100\left[\mathrm{\Omega }\right]$. Figure 5a illustrates the approximate solution vectors derived from the first-order Formula (43), depicted by the dotted magenta line. Figure 5b shows the approximated solution vectors obtained from the refined nonlinear Formula (54), also represented by the dotted magenta line. For reference, both figures include the true loss-minimization solution (magenta), MTPA (green), MC (red), and MTPV (blue) curves as solid lines. To simplify the analysis, we assume a constant angular speed of $\omega =200$ while calculating the loss-minimization solution vectors. To assess the general applicability of the proposed loss-minimization formula, the performance is evaluated across a broad range of rotational speeds, extending beyond the initial case of $\omega =200$. Figure 6 and Figure 7 show the resuls for $\omega =100$ and 300, respectively. As shown in the figures, the Euclidean distance errors remain consistently small, confirming that the proposed explicit method effectively approximates the optimal trajectories under various operating conditions without being significantly affected by speed-dependent iron loss variations. It should be noted that the first-order Formula (43) exhibits mean/maximum errors below 0.29/0.47, and the refined nonlinear Formula (54) achieves much higher accuracy, with mean/maximum errors below 0.08/0.15.

Furthermore, to evaluate the generality of the proposed methods, numerical verifications are conducted on an additional motor model with a relatively higher saliency ratio. The results demonstrate that the proposed approach is robust and generally applicable to various PMSM designs with diverse magnetic characteristics. We consider a PMSM with the following parameters: $p=4$, $R=1.1\left[\mathrm{\Omega }\right]$, $L_d=59\left[\mathrm{mH}\right]$, $L_q=101\left[\mathrm{mH}\right]$, ${\lambda }_m=0.193[\mathrm{V}·\sec /\mathrm{rad}]$, $J=0.0042[\mathrm{kg}·{\mathrm{m}}^2]$, $B=0.0001[\mathrm{N}·\mathrm{m}·\sec /\mathrm{rad}]$, and $T_L=1.5[\mathrm{N}·\mathrm{m}]$. These parameters result in the following dynamic model:

$$\left\{\begin{array}{ccc}\hfill \dot{\omega }& =& 2757i_q-0.2381\omega -600i_d{i}_q-4762T_L\hfill \\ \hfill {\dot{i}}_q& =& -10.89i_q-1.911\omega -0.5842\omega i_d+9.901V_q\hfill \\ \hfill {\dot{i}}_d& =& -18.64i_d+1.712\omega i_q+16.95V_d\hfill \end{array}\right.$$

(58)

Assume that $I_M=6$ and $V_M=196.7$. Using (9)–(14) for the given motor parameters, we can obtain $i_{da}^{*}=-3.2466$, $i_{dv}^{*}=-5.3717$, $i_{qa}^{*}=5.0457$, $i_{qv}^{*}=2.6729$, $T_a=4.9856$, $T_v=3.3567$, ${\omega }_a=385.91$, and ${\omega }_v=662.07$. Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 present the results obtained using the above model (58), which correspond directly to the results shown in Figure 3 through Figure 7 derived from the model in (57). Figure 8a,b illustrate the approximated MTPA (dotted green), MC (dotted red), and MTPV (dotted blue) curves for the above model (58). The first-order Formulas (23), (29), and (33) exhibit mean/maximum errors below 0.1/0.2, 0.2/0.3, and 0.5/0.8, respectively. The refined nonlinear Formulas (49), (51), and (53) demonstrate exceptional precision, reducing the mean/maximum errors to less than 0.02/0.08. Figure 9a–c are generated by the conventional quadratic Formula (22) and the proposed Formulas (23), (25), and (49) for the above model (58). The conventional Formula (22) exhibits mean/maximum errors of 0.6/2.3. In contrast, the proposed Formulas (23) and (25) achieve much higher accuracy, with mean/maximum errors below 0.08/0.2 and 0.34/0.53, respectively. Furthermore, the refined nonlinear Formula (49) demonstrate exceptional precision, reducing the mean/maximum errors to less than 0.0005/0.001, which are practically negligible. Figure 10, Figure 11 and Figure 12 show the results of the loss-minimization control problem (41) with (58) under $\omega =200$, $\omega =100$, and $\omega =300$, respectively. The first-order Formula (43) exhibits mean/maximum errors below 0.14/0.26, and the refined nonlinear Formula (54) achieves much higher accuracy, with mean/maximum errors below 0.07/0.14.

As shown in the above results, the proposed algebraic formulas consistently provide precise optimal current references, with the approximation errors remaining within a very small range. The above examples demonstrate that the conventional quadratic approximation formula (22), employed for MTPA control in [11,12,13,14,15,16,17,18,19,20], exhibits significant inaccuracies, particularly in the vicinity of nominal torque. This level of inaccuracy renders Formula (22) unsuitable for practical approximation purposes. It should be noted that this paper proposes two types of algebraic approximation methods: the basic first-order formulas for maximum simplicity and the refined formulas for enhanced accuracy in nonlinear motor models. The proposed first-order Formulas (23), (29), (33), and (43) achieve a compelling balance of simplicity and accuracy, providing reliable approximations of optimal values while maintaining computational efficiency. The high computational efficiency of these explicit formulas offers practical advantages for motor drivers by reducing the execution time in the control loop. This enables the implementation of more sophisticated control algorithms or the use of cost-effective microprocessors in real-time drive systems. Although the refined Formulas (49), (51), (53), and (54) involve solving quadratic equations, effectively doubling the computational load compared to the first-order formulas, they offer the significant advantage of substantially minimizing approximation errors. Moreover, a crucial advantage of these refined formulas lies in the elimination of time-consuming iterative processes, which were previously necessary in numerical algorithm-based methods, such as those presented in [7,8,9], to achieve accurate optimal values.

7. Conclusions

This paper presented explicit algebraic methods for approximating the optimal $dq$-axis current references for MTPA, MTPV, and loss-minimization control in PMSMs. The derived closed-form formulas provide a computationally efficient alternative to iterative numerical procedures. Furthermore, refinement techniques were introduced to enhance the accuracy of the approximated values, ensuring robust performance across various operating conditions. The effectiveness of the proposed methods was validated through numerical simulations, which demonstrated high accuracy and potential for real-world implementation. Consequently, the proposed approach offers a practical and efficient framework for developing real-time optimal torque control systems for high-performance PMSM drives.