# Harmonic Current Injection Method for Torque Ripple Suppression over Wide Speed Range Considering Magnetic Saturation in PMSMs

Kodai Takeda, Shun Taniguchi, Takafumi Hara, Kentaro Matsuo

► To cite this version:

Kodai Takeda, Shun Taniguchi, Takafumi Hara, Kentaro Matsuo. Harmonic Current Injection Method for Torque Ripple Suppression over Wide Speed Range Considering Magnetic Saturation in PMSMs. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0024 . hal-05074034

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Submitted on 19 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Harmonic Current Injection Method for Torque Ripple Suppression over Wide Speed Range Considering Magnetic Saturation in PMSMs

Kodai Takeda, Shun Taniguchi, Takafumi Hara
Hitachi, Ltd.
Omika-cho 7-1-1, Hitachi-shi
Ibaraki, Japan
+81-(294)52-5111
kodai.takeda.vm@hitachi.com
shun.taniguchi.wu@hitachi.com
takafumi.hara.wq@hitachi.com

**Index Terms**—«PMSM», «Torque control», «Vector control», «Harmonic injection».

**Abstract**—We propose a flux-based harmonic current injection method that accounts for magnetic saturation to suppress torque ripple in permanent magnet synchronous motors feedforwardly. The key feature of the proposed method is converting a harmonic-current command into a flux command by pre-acquired nonlinear flux information and converting the flux command into voltage commands through an approximate equation. Numerical simulations demonstrate that this method reduces 12th-order torque ripple by over 70% within a wide frequency range. Experimental results indicate that the amplitude and phase errors of the injected harmonic flux can be held to as low as 0.2% and 2.9°, respectively, even without additional feedback control for harmonics.

## I. INTRODUCTION

Permanent magnet synchronous motors (PMSMs) have been widely used in mobility due to their superior attributes including elevated efficiency and substantial power density. However, in high-power-density PMSMs, the presence of harmonic components within the spatial distribution of the magnetic flux engenders torque ripple, which in turn causes vibration and noise during motor operation. This compromising the precision of position and speed control. As the challenge of torque ripple exacerbates concomitantly with the increase in PMSM power density, methodologies aimed at mitigating torque ripple have been proposed.

To mitigate the torque ripple caused by spatial harmonics, two principal strategies are typically

Kentaro Matsuo
Hitachi Astemo, Ltd.
Takaba 2520, Hitachinaka-shi
Ibaraki, Japan
+81-(3)4232-5300
kentaro.matsuo.xb@hitachiastemo.com

used: structural optimization of the motor and using compensation-control methods. Structural optimization includes the optimization of design and arrangement of magnets, adjustments to the pole-slot ratio, and implementation of skewing. While these optimizations can effectively reduce torque ripple, they often lead to higher manufacturing complexity and costs. Conversely, compensation-control methods involve injecting harmonic currents to suppress torque ripple. This strategy is generally more cost-effective since it does not require physical changes to the motor, thus avoiding additional expenses.

Previous research on compensation-control methods has proposed feedforward (FF) and feedback (FB) strategies for injecting harmonic currents. The FF method calculates harmonic voltage commands from current commands to cancel torque ripple using motor-inverse models. Although this approach is simple and offers a wide bandwidth, it requires a precise motor model to accurately calculate the voltage commands. In contrast, the FB method involves detecting speed fluctuations and calculating harmonic-current commands to reduce these fluctuations using FB-based control [1]–[4]. Using internal motor models or observers to calculate harmonic-current commands is effective and widely studied [5]–[13]. FB-loop or observer methods are robust to disturbance or variations in motor parameters, such as deadtime or magnetic-flux distribution, compared with the FF method. However, the effective bandwidth and stability are limited by the gain of FB-loop control for fundamental
<figure><img src="image_2.png" /><figcaption>Fig. 1: Definition of static and dynamic inductance</figcaption></figure>

current or observers, making it difficult to suppress torque ripple in high-speed regions. To address this limitation, proportional-resonant controllers or multiple-reference-frame methods have been proposed to suppress specific harmonics, which is higher frequency than the bandwidth [2], [3], [10]–[13]. Some of these methods also include an optimization step to compute harmonic commands, helping to suppress or minimize torque ripple and other factors such as losses at different speeds and torques.

Despite these advancements, the optimization phase or the use of complex harmonic-current/speed-FB controllers increases computational costs, limiting their use in industrial and other settings where computational resources are constrained. Longer control periods limit the range of ripple frequencies that can be suppressed with control methods. Therefore, we propose a flux-based harmonic current injection method using the FF method to simultaneously reduce computational costs and error. Although this method does not include a FB loop for harmonic currents, it achieves accurate injection by taking into account the effects of magnetic saturation in the motor.

This paper is organized as follows. Section II introduces the proposed method and details the derivation of the voltage command required for this injection. Section III discusses the numerical validation of the proposed method, and Section IV explains the experimental validation. Finally, the conclusions of this research are presented in Section V.

## II. FLUX-BASED HARMONIC INJECTION

### A. Derivation of voltage equation

A voltage equation that accounts for magnetic saturation is shown in (1), where $i_d$ and $i_q$ are the

direct and quadrature axes (dq-axis) currents, $v_d$ and $v_q$ are the dq-axis voltages, $R$ is the motor resistance, $L_d$ and $L_q$ are the dq-axis inductances, $L_{dq}$ and $L_{qd}$ are the cross-coupling inductances between the d- and q-axes, $K_{e0}$ is the back electromotive force constant, $\omega_1$ is the fundamental electrical angular frequency, and $s$ is the Laplace operator. The subscript $t$ on the inductance denotes the dynamic inductance, defined by differentiating the flux with respect to the current. The inductance obtained by dividing the flux by the current value is referred to as static inductance. A graphical definition of dynamic inductance and static inductance is shown in Fig. 1.

$$ 
\begin{aligned}
v_{dq} &= Z i_{dq} + \omega_1 \begin{pmatrix} 0 \\ K_{e0} \end{pmatrix} \\
Z &= \begin{pmatrix} R + \frac{d}{dt} L_{dt} - \omega_1 L_{dq} & \frac{d}{dt} L_{dqt} - \omega_1 L_q \\ \frac{d}{dt} L_{qdt} + \omega_1 L_d & R + \frac{d}{dt} L_{qt} + \omega_1 L_{qd} \end{pmatrix} \\
v_{dq} &= \begin{pmatrix} v_d \\ v_q \end{pmatrix}, \quad i_{dq} = \begin{pmatrix} i_d \\ i_q \end{pmatrix}
\end{aligned} 
\quad (1) $$

In PMSMs, the dq-axis flux linkages $\phi_d$ and $\phi_q$ are defined as

$$ 
\begin{aligned}
\phi_{dq} &= \mathbf{L} i_{dq} + \begin{pmatrix} K_{e0} \\ 0 \end{pmatrix}, \\
\mathbf{L} &= \begin{pmatrix} L_d & L_{dq} \\ L_{qd} & L_q \end{pmatrix}, \quad \phi_{dq} = \begin{pmatrix} \phi_d \\ \phi_q \end{pmatrix}.
\end{aligned} 
\quad (2) $$

By using the definition of dynamic inductance $L_t = \frac{d\phi}{di}$, the following equation can be derived.

$$ \dot{\phi}_{dq} = \begin{pmatrix} L_{dt} & L_{dqt} \\ L_{qdt} & L_{qt} \end{pmatrix} \dot{i}_{dq}. \quad (3) $$

Equation (1) can be transformed into

$$ 
\begin{aligned}
v_{dq} &= R i_{dq} + \begin{pmatrix} L_{dt} & L_{dqt} \\ L_{qdt} & L_{qt} \end{pmatrix} \dot{i}_{dq} \\
&\quad + \omega_1 \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \left( \mathbf{L} i_{dq} + \begin{pmatrix} K_{e0} \\ 0 \end{pmatrix} \right). \quad (4)
\end{aligned} $$

By substituting Eqs. (2) and (3) into Eq. (4), the relation between the voltage and flux linkage of PMSMs can be derived as

$$ 
\begin{aligned}
v_{dq} &= R \mathbf{L}^{-1} \begin{pmatrix} \phi_d - K_{e0} \\ \phi_q \end{pmatrix} \\
&\quad + \dot{\phi}_{dq} + \omega_1 \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \phi_{dq}.
\end{aligned} 
\quad (5) $$
<figure><img src="image_3.png" /><figcaption>Fig. 2: Harmonic-current command derivation</figcaption></figure>

## B. Derivation of harmonic-flux command

The method for generating the harmonic-flux-command values is explained. The output torque $\tau$ can be represented as

$$ \tau = p(\phi_d i_q - \phi_q i_d) \\ = p(\phi_{d0} i_{q0} - \phi_{q0} i_{d0} + \phi_{dh} i_{q0} - \phi_{qh} i_{d0} \\ + \phi_{d0} i_{qh} - \phi_{q0} i_{dh} + \phi_{dh} i_{qh} - \phi_{qh} i_{dh}). \quad (6) $$

The variables in Eq. (6) are defined in Eq. (7), and $p$ denotes pole pairs. Here, $x$ represents $\phi$ or $i$, and $a$ represents $d$ or $q$. The subscript 0 denotes the direct current (DC) component, and $h$ denotes the harmonic component.

$$ x_a = x_{a0} + x_{ah} \quad (7) $$

Assuming that the harmonic components are sufficiently small compared with the DC components $x_{a0} \gg x_{ah}$, the last two terms in Eq. (6) are negligible. The torque-ripple component caused by spacial flux distortion can be canceled by applying a harmonic current satisfying $\phi_{d0} i_{qh} - \phi_{q0} i_{dh} = \phi_{dh} i_{q0} - \phi_{qh} i_{d0}$. Since there are numerous combinations of harmonic currents that satisfy this condition, we explored the appropriate combination on the maximum torque per ampere (MTPA) trajectory from the perspective of efficiency, as shown in Fig. 2.

Once the calculated harmonic currents are set as the current commands $i_{dh}^*$ and $i_{qh}^*$, the predetermined relationship between current and flux linkage, represented as $\hat{\phi}_d$ and $\hat{\phi}_q$, is used. The reference harmonic-flux commands $\phi_{dh}^*$ and $\phi_{qh}^*$ are then determined by the difference from the DC component at each current operating point, as respectively expressed by the following equations.

$$ \phi_{dh}^* = \hat{\phi}_d(i_{dc}^* + i_{dh}^* + i_{qc}^* + i_{qh}^*) - \hat{\phi}_d(i_{dc}^*, i_{qc}^*) \quad (8) $$

$$ \phi_{qh}^* = \hat{\phi}_q(i_{dc}^* + i_{dh}^* + i_{qc}^* + i_{qh}^*) - \hat{\phi}_q(i_{dc}^*, i_{qc}^*) \quad (9) $$

## C. Harmonic-voltage command

The harmonics of the voltage command can be derived by considering small fluctuation in the flux command as follows:

$$ \begin{pmatrix} v_{dc}^* + v_{dh}^* \\ v_{qc}^* + v_{qh}^* \end{pmatrix} = R \mathbf{L}^{-1} \begin{pmatrix} \phi_{dc}^* + \phi_{dh}^* - K_{e0} \\ \phi_{qc}^* + \phi_{qh}^* \end{pmatrix} \\ + \frac{d}{dt} \begin{pmatrix} \phi_{dc}^* + \phi_{dh}^* \\ \phi_{qc}^* + \phi_{qh}^* \end{pmatrix} \\ + \omega_1 \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \phi_{dc}^* + \phi_{dh}^* \\ \phi_{qc}^* + \phi_{qh}^* \end{pmatrix}. \quad (10) $$

Though $\mathbf{L}$ depends on the dq-axis current, (10) can be assumed as a linear function to the small fluctuations around the DC components of $\phi_{dc}^*$ and $\phi_{qc}^*$. By removing the DC components and extracting only the harmonic components of the voltage command, $v_{dh}^*$ and $v_{qh}^*$ can be calculated as

$$ \mathbf{v}_{dqh}^* = \frac{R}{L_d L_q - L_{dq} L_{qd}} \begin{pmatrix} L_q & -L_{qd} \\ -L_{dq} & L_d \end{pmatrix} \phi_{dqh}^* \\ + \dot{\phi}_{dqh}^* + \omega_1 \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \phi_{dqh}^*, \quad (11) $$

$$ \mathbf{v}_{dqh}^* = \begin{pmatrix} v_{dh}^* \\ v_{qh}^* \end{pmatrix}, \quad \phi_{dqh}^* = \begin{pmatrix} \phi_{dh}^* \\ \phi_{qh}^* \end{pmatrix}. $$

We assumed $L_q, L_d \gg L_{dq}, L_{qd}$, and a previous study [6] suggested that cross-coupling inductance $L_{dq}$ and $L_{qd}$ can be ignored and approximated into (12) for the first term of (11).

$$ \mathbf{v}_{dqh}^* = \begin{pmatrix} R/L_d & -\omega_1 \\ \omega_1 & R/L_q \end{pmatrix} \phi_{dqh}^* + \dot{\phi}_{dqh}^* \quad (12) $$

Since Eq. (12) contains differential terms and cannot be directly computed, the harmonic-flux command is expressed as a sum of frequency components through Fourier transformation (FT) with respect to the electrical angle $\theta_e$ as (13). The amplitude of the n-th order harmonic-flux command is denoted as $\phi_{ndh}^*$ and $\phi_{nqh}^*$, and the phase as $\theta_{ndh}^*$ and $\theta_{nqh}^*$. The derivative values can then be obtained analytically and delays due to pseudo-differentiation can be suppressed. Suppression of
<figure><img src="image_4.png" /><figcaption>Fig. 3: Block diagram of proposed method.</figcaption></figure>

the delay is especially important for accurately injecting high-frequency harmonics.

$$ \begin{pmatrix} \phi_{dh}^* \\ \phi_{qh}^* \end{pmatrix} = \begin{pmatrix} \sum_n \phi_{ndh}^* \cos(n\theta_e + \theta_{ndh}^*) \\ \sum_n \phi_{nqh}^* \cos(n\theta_e + \theta_{nqh}^*) \end{pmatrix} \quad (13) $$

Differentiating Eq. (13) yields the following equation:

$$ \begin{pmatrix} \dot{\phi}_{dh}^* \\ \dot{\phi}_{qh}^* \end{pmatrix} = \begin{pmatrix} -\sum_n n \phi_{ndh}^* \sin(n\theta_e + \theta_{ndh}^*) \\ -\sum_n n \phi_{nqh}^* \sin(n\theta_e + \theta_{nqh}^*) \end{pmatrix}. \quad (14) $$

### D. Simplified harmonic-voltage command

Equation (12) can be further simplified by comparing the order of each term. For calculating the FF-voltage command, as shown in (12), inductance and resistance of the motor are still included. Therefore, multiple tables should be referred, and motor constants depending on the current should be explored to accurately calculate the voltage command.

The off-diagonal elements of the matrix in the first and second terms of Eq. (12) are proportional to $\omega_1$. However, the diagonal elements of the matrix in the first term are constant. Therefore, as the rotational speed increases, Eq. (12) can be approximated using the following equation.

$$ \begin{pmatrix} v_{dh}^* \\ v_{qh}^* \end{pmatrix} = \begin{pmatrix} \dot{\phi}_{dh}^* \\ \dot{\phi}_{qh}^* \end{pmatrix} + \omega_1 \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \phi_{dh}^* \\ \phi_{qh}^* \end{pmatrix} \quad (15) $$

This equation has no constants depending on current except for flux linkages. Therefore, the proposed method can save computational resources and is feasible.

## III. NUMERICAL VERIFICATION

To verify the effectiveness of the proposed control method, we conducted an analysis by using MATLAB/Simulink.

### A. Flux-based cascaded vector control

We introduced the flux-based cascade-vector control system for this simulation. For more detailed information, please refer to a previous study [14]. In this control system, as shown in Fig. 3, the dq-axis current commands $i_{dc}^*$ and $i_{qc}^*$ are calculated on the basis of the given torque command $\tau^*$, and transformed into flux commands $\phi_{dc}^*$ and $\phi_{qc}^*$ by the flux converter. Harmonic commands $\phi_{dh}^*$ and $\phi_{qh}^*$ are then added to ensure accuracy at low rotational speed and prevent the cancelling of harmonic current injection. An auto current regulator generates voltage commands $v_{dc}^*$ and $v_{qc}^*$ by using (16). Here, $\phi_d$ and $\phi_q$ are the measured flux linkages, $i_d$ and $i_q$ are dq-axis currents transformed from the measured three-phase current $I_u$, $I_v$, and
$I_w$, $s$ is the Laplace operator, and $\omega_c$ is the control gain.

$$ \begin{pmatrix} v_{dc}^* \\ v_{qc}^* \end{pmatrix} = \omega_c \begin{pmatrix} \phi_{dc}^{**} - \phi_d \\ \phi_{qc}^{**} - \phi_q \end{pmatrix} + \frac{\omega_1 \omega_c}{s} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \phi_{dc}^{**} - \phi_d \\ \phi_{qc}^{**} - \phi_q \end{pmatrix} + \begin{pmatrix} \frac{R}{L_d} & 0 \\ 0 & \frac{R}{L_q} \end{pmatrix} \begin{pmatrix} \frac{\omega_c}{s} \begin{pmatrix} \phi_{dc}^{**} - \phi_d \\ \phi_{qc}^{**} - \phi_q \end{pmatrix} - \begin{pmatrix} K_{e0} \\ 0 \end{pmatrix} \end{pmatrix} \quad (16) $$

The proposed method is used to calculate harmonic-voltage commands $v_{dh}^*$ and $v_{qh}^*$ by using Eq. (12) or (15) from $\phi_{dh}^*$ and $\phi_{qh}^*$, which are determined from the lookup table calculated from Eqs. (8) and (9), as described in Section II-B. The final voltage commands, which are calculated by adding $v_{dh}^*$ and $v_{qh}^*$ to $v_{dc}^*$ and $v_{qc}^*$, respectively, are transformed into the three-phase voltage commands $V_u^*, V_v^*$, and $V_w^*$, which are used to control the inverter through gate-drive signals generated using a pulse width modulation (PWM) method appropriate for the rotational speed and torque.

## B. Simulation conditions

We focused on the 12th-order torque ripple with an amplitude of 5% of the DC torque. Figure 4 illustrates a typical flux-linkage curve exhibiting magnetic saturation, with the operating point marked as a red circle. The current and flux-linkage values in Fig. 4 are normalized by their respective maximum values. Given that the d-axis current is predominantly negative, the horizontal axis for the d-axis flux-linkage ranges from -1 to 0. Analysis of these graphs revealed that magnetic saturation is particularly pronounced in the q-axis direction for this motor.

The harmonic commands for flux, denoted as $\phi_{dh}^*$ and $\phi_{qh}^*$, are computed using the method outlined in Section II-B. The target for torque-ripple suppression in this simulation is the 12th harmonic. However, due to the non-linear nature of the conversion from torque to current and from current to flux linkage, as illustrated in Fig. 4, $\phi_{dh}^*$ and $\phi_{qh}^*$ contain harmonic components other than the 12th order. Despite this, numerical calculations indicate that these additional harmonics constitute less than 0.1% of the amplitude of the

<figure><img src="image_5.png" /><figcaption>Fig. 4: Typical flux-linkage pattern with magnetic saturation of motor for simulation</figcaption></figure>

12th harmonic. Therefore, these minor harmonics were disregarded in this verification.

## C. Simulation results

Figure 5 illustrates the remaining torque ripple after implementing the proposed method at various rotational speeds. The black and red bars were calculated using Eqs. (12) and (15) for the harmonic-voltage-command values, respectively. To verify the accuracy of harmonic-voltage calculation, the harmonic flux injected into the FB loop was set to zero, that is, $\phi_{dh}^*$ and $\phi_{qh}^*$ in (16) were 0.

From Fig. 5, accuracy sharply decreased lower frequencies, particularly at 6.67 Hz. This indicates that the assumptions of the approximation method described in Section II-D are no longer valid. When calculating the $R/L_d$ and $R/L_q$ of the simulation model, the values were approximately 50 to 100, respectively. Conversely, at 6.67 Hz, the 12th-order electrical angular velocity was about 280
<figure><img src="image_6.png" /><figcaption>Fig. 5: Remaining 12th order torque ripple</figcaption></figure>

(rad/sec), which is too close for the approximation
to be valid.

One factor contributing to the decrease in ac-
curacy at lower frequencies is the cancellation
of injected harmonics by the flux FB. Harmonic
commands were not superimposed on the FB com-
mands in the black and red cases shown in Fig. 5.
Therefore, if the frequency of the harmonic com-
mand is within the bandwidth of the FB control,
the harmonics applied by $v_{dqh}^*$ are suppressed by
the FB control.

The results shown in green include harmonic
commands added to the FB. Accuracy improved
significantly at frequencies of 6.67 and 13.3 Hz
and slightly decreased at higher frequencies, such
as 267 Hz. This is probably because the harmonic-
command frequency was close to the Nyquist fre-
quency of the control system.

# IV. EXPERIMENTAL VERIFICATION

## A. *Experimental setup*

We conducted an experimental test to validate
the proposed method. Figure 6 illustrates the ex-
perimental setup. Due to the limitations of the
torque sensor, high-frequency torque could not be
measured directly. Thus, current was measured
using a power meter, which samples faster than the
motor controller, and was evaluated as a proxy for
torque. The measured current was evaluated using
the flux transformed using Eqs. (8) and (9), where
$\hat{\phi}_d$ and $\hat{\phi}_q$ are calibrated for the test motor.

Three fundamental frequency $f_1$ conditions were
selected, considering the harmonic frequency and

<figure><img src="image_7.png" /><figcaption>Fig. 6: Experimental setup</figcaption></figure>

control period. The frequency of the injected har-
monic command was $12f_1$. The flux-based har-
monics injector calculates $v_{dqh}^*$ by using Eq. (15)
because $R/L_d$ and $R/L_q$ were sufficiently small to
be negligible under these experimental conditions.

## B. *Measured results*

Figure 7 plots the electrical angle on the hor-
izontal axis and normalized flux with the am-
plitude of the harmonic commands of flux on
the vertical axis. Measurements were taken over
multiple periods of $f_1$ and displayed by dividing
and overlaying them within the period of $f_1$. The
blue and red lines represent the flux command and
measured flux linkage, respectively. The measured
flux linkage varied due to the effect of carrier
harmonics from the current generated by PWM.
Notably, the command and output aligned well.

To assess accuracy more precisely, each wave-
form was analyzed using fast FT. The results for
the 12th harmonic of $f_1$'s amplitude and phase
accuracy are shown in Table I. For amplitude ac-
curacy, the d-axis had an error margin of less than
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>(a) $f_1=33.3$ Hz</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>(b) $f_1=133$ Hz</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>(c) $f_1=267$ Hz</figcaption></figure>

<caption>Fig. 7: Measured waveforms. q- and d-axes flux linkages are plotted in upper and lower sides, respectively.</caption>

<table><caption>TABLE I: Accuracy of 12-th hamonic flux injection</caption><thead><tr><th rowspan="2">Fundamental frequency</th><th colspan="2">d-axis error</th><th colspan="2">q-axis error</th></tr><tr><th>Amplitude</th><th>Phase</th><th>Amplitude</th><th>Phase</th></tr></thead><tbody><tr><td>33.3 Hz</td><td>3.8%</td><td>-5.8°</td><td>14.2%</td><td>-2.9°</td></tr><tr><td>133 Hz</td><td>2.9%</td><td>5.1°</td><td>7.4%</td><td>4.7°</td></tr><tr><td>267 Hz</td><td>0.2%</td><td>13.5°</td><td>15.2%</td><td>10.8°</td></tr></tbody></table>

5%, regardless of the motor's rotational speed, indicating good agreement. However, the q-axis had an amplitude error exceeding 10%, which is slightly larger. This difference occurs because the injected flux command for the q-axis is about half that of the d-axis, making the constant error more significant.

Phase accuracy remained within ±5° up to 133 Hz, showing good precision. At 267 Hz, the phase error was around 10°, which is roughly a 10-microsecond discrepancy. Although most delay components in the motor controller and sensors were compensated for during command calculations, variations from nominal delay times and their cumulative effects are believed to be the cause.

## V. CONCLUSION

We proposed a flux-based harmonic current injection method for injecting precise harmonic currents and reducing torque ripple with simple calculations, even when magnetic saturation occurs. Numerical simulations confirmed that torque ripple can be suppressed by more than 70% across a wide range of rotational speeds, demonstrating that the simplified approximate model maintains sufficient accuracy even at high speeds. The experimental results indicate that the proposed method achieved injecting harmonic flux with an accuracy of amplitude errors within 4% for the d-axis and slightly higher for the q-axis. Phase accuracy was within ±6° up to a frequency of 133 Hz. The proposed method is thus a practical and efficient means to reduce torque ripple in PMSMs without increasing computational complexity or cost.

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