Open AccessFeature PaperArticle Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter 1 School of Intelligent Engineering, Shaoxing University, Shaoxing 312000, China 2 School of Electrical Engineering, Southeast University, Nanjing 210096, China 3 Taizhou Jinyu Electromechanical Co., Ltd., Taizhou 318000, China * Author to whom correspondence should be addressed. Electronics 2026, 15(12), 2506; https://doi.org/10.3390/electronics15122506 (registering DOI) Submission received: 27 May 2026 / Revised: 3 June 2026 / Accepted: 5 June 2026 / Published: 7 June 2026 Abstract This article proposes a double-pre-warped Tustin bi-quad filter (DPT-BQF) to suppress mechanical resonance for non-contact integrated permanent magnet vernier motors (NIPMVMs). First, this article analyzes how conventional notch filters suffer from distortion in notch frequency, bandwidth, and depth when discretized with low-precision methods, which degrades filtering performance and weakens resonance suppression. To address this issue, the inherent limitations of the pre-warped Tustin discretization for BQFs are discussed. On this basis, a double pre-warped Tustin method is proposed by compensating for the bandwidth distortion, and its discretization performance is comprehensively evaluated. Furthermore, the principle, parameter design, and robustness range of the filter are deeply analyzed in the discrete domain. Finally, experimental validation on an NIPMVM test platform demonstrates the effectiveness and feasibility of the proposed method. 1. Introduction The most common passive suppression method is to use filters for resonance attenuation. However, the suppression performance is degraded by digital signal processing and relies on the accuracy of the identified resonance frequency [ 11]. To address this issue, adaptive notch filters capable of rapidly identifying the resonance center frequency can provide an effective solution [ 12]. Similarly, ref. [ 13] adopts an online adaptive notch filter for resonance mitigation, wherein the resonance frequency is identified with respect to the rotational speed via a fast FFT algorithm. Consequently, the notch filter parameters are updated in real time to enhance the suppression performance. However, when the notch filter is enabled, digital delay may cause a shift in the resonance frequency. In this regard, ref. [ 14] analyzes the resonance mechanism of a two-inertia system. To accurately identify the resonance frequency, the SNR for online FFT is improved by increasing the sampling period of the speed loop and reducing the output saturation of the speed loop. However, this method requires intentionally inducing a high-SNR resonance segment prior to each identification process, imposing significant limitations on its practical applicability. To address this issue, a DPT-BQF is proposed, and the operating principle, parameter design, and robustness range of the filter are thoroughly analyzed in the discrete domain within a closed-loop system, achieving effective resonance suppression for NIPMVMs and other two-inertia systems. The specific contributions of this paper are as follows: (1) Compared with the conventional BQF, the proposed method mathematically establishes the mapping relationship between the actual bandwidth and the theoretical bandwidth and provides a clear quantitative stability range for the notch filter. (2) From the implementation perspective, the proposed DPT-BQF has a simple structure. It only requires modification of the discretization coefficients during digital implementation and is, therefore, easy to implement. (3) The proposed DPT-BQF is general and can be applied to NIPMVMs as well as other resonance-prone applications, such as inflatable Savonius wind turbines [ 17]. The robust notch-filter design process in this paper is closely based on the NIPMVM model and is therefore system-specific; however, the underlying design concept can also be extended to other similar applications. 2. NIPMVM and Notch Structure The NIPMVM studied is structurally composed of a PMVM rigidly connected to a magnetic gear. The magnetic gear can be simplified and modeled as a two-inertia system, and its equation can be expressed as follows: ( J i s + B i ) · ω m i = T e − g m − 1 T c − d s · ( p i ω m i − p o ω m o ) ( J o s + B o ) · ω m o = T c − T L + d s g m · ( p i ω m i − p o ω m o ) , (1) where Te, Tc, and TL are the electromagnetic torque, magnetic gear transmission torque, and load torque. pi,o are the pole pair numbers of the inner and outer rotors of the NIPMVM, which are 4 and 24, respectively. Ji,o, ωmi,o, and Bi,o ≈ 0 represent the rotational inertia, mechanical angular velocity, and friction coefficient of the inner and outer rotors, respectively. ds ≈ 0 is the damping constant, and s denotes the differential operator. Furthermore, a simplification is performed, and the simplified variables are shown as Tε = gmTe, Jε = g m 2 Ji, ωmε = g m − 1 ωmi, θeε = s−1 ( ωmε − ωmo), and keff = poTmaxcos( θerr). In addition, the motion and simplified model of the NIPMVM are illustrated in Figure 1. It is noteworthy that the equivalent anti-resonant frequency and natural resonant frequency of the NIPMVM can be derived from the simplified two-inertia model, as ω a ε 2 = keff/ Jo, ω n ε 2 = ω a ε 2 ( γε + 1), and γε = Jo/ Jε, respectively. In this regard, a structure where the BQF is connected in series between the speed loop and current loop controllers is adopted in this article to suppress the resonance. 3. Resonance Suppression Based on DPT-BQF The discretization methods traditionally used for BQF mainly include the back Euler (BE-BQF), the Tustin (T-BQF), and the pre-warped Tustin (PT-BQF) methods. However, the notch filters still suffer from low center-frequency accuracy. To solve this, this article proposes a double-pre-warped discretization that significantly reduces the distortion of the filter. 3.1. Pre-Warped Tustin Bi-Quad Filter When a BQF is discretized using the back Euler or Tustin method, significant deviations often occur. To mitigate such discretization distortion, a commonly used approach, the pre-warped Tustin method, is first introduced. After applying this method, the resulting discretized model GPT-BQF(z) is given by G PT - BQF z = k 2 + ω T 2 + k T 2 ω T k z 2 − 2 k 2 − ω T 2 z + k 2 + ω T 2 − k T 2 ω T k k 2 + ω T 2 + k T 1 ω T k z 2 − 2 k 2 − ω T 2 z + k 2 + ω T 2 − k T 1 ω T k , (2) where k = ωn/tan( ωnTs/2). It is worth noting that the structure of the PT-BQF is identical to that of the T-BQF except for the difference in the coefficient k. Then, substituting z = ejωTs to analyze its frequency characteristics yields G PT - BQF e j ω T s = k 2 + ω T 2 cos ω T s − k 2 − ω T 2 + j k T 2 ω T k sin ω T s k 2 + ω T 2 cos ω T s − k 2 − ω T 2 + j k T 1 ω T k sin ω T s , (3) According to (3), the Bode plots for ωn = 2 π × 350 rad/s, bt = 350 rad/s, and xt = –30 dB are obtained when the sampling period is stepped from 0.0002 s to 0.001 s, as shown in Figure 2. As shown, the pre-warped Tustin, due to its high accuracy at the notch frequency, effectively prevents the notch frequency from shifting after discretization. As the sampling period increases, the PT-BQF maintains both its notch frequency and depth without deviation. However, the accuracy of the PT-BQF deteriorates away from the notch frequency, leading to a narrowing of the bandwidth as the sampling period increases, which, likewise, affects the performance of the filter. 3.2. Double-Pre-Warped Tustin Bi-Quad Filter This article introduces improvements to mitigate the distortion of the notch bandwidth. The DPT-BQF is proposed. First, the correction method for the bandwidth of the DPT-BQF is introduced. Let the actual bandwidth of the BQF be Ω, and the magnitude gain at frequencies ωn ± Ω/2 be xΩ/2 dB. According to (3), its amplitude frequency characteristic can be shown as γ = k 2 + ω T 2 cos ω T s − k 2 − ω T 2 η = b T k sin ω T s x Ω / 2 = 20 lg γ 2 + 10 x T / 20 η 2 / γ 2 + η 2 , (4) It can be further simplified as k 2 + ω T 2 cos ω T s − k 2 − ω T 2 2 1 − 10 x Ω / 2 / 10 10 x Ω / 2 / 10 − 10 x T / 10 = b T k sin ω T s 2 , (5) Then, by substituting ω = ωn ± Ω/2 into the above equation, it can be further simplified as cos Ω T s = λ Ω / 2 2 k 2 + ω T 2 2 − b T 2 k 2 / λ Ω / 2 2 k 2 + ω T 2 2 + b T 2 k 2 λ Ω / 2 2 = 1 − 10 x Ω / 2 / 10 / 10 x Ω / 2 / 10 − 10 x T / 10 , (6) Finally, using trigonometric identities, the mapping relationship between the ideal notch bandwidth bT and the actual notch bandwidth Ω is provided in Appendix A, and the final result can be expressed as b T = λ Ω / 2 k 2 + ω t 2 k tan Ω T s 2 , (7) Then, the discretized DPT-BQF model GPT-BQF( z) is deduced as G DPT - BQF z = k 2 + ω T 2 + k T 2 ω T k z 2 − 2 k 2 − ω T 2 z + k 2 + ω T 2 − k T 2 ω T k k 2 + ω T 2 + k T 1 ω T k z 2 − 2 k 2 − ω T 2 z + k 2 + ω T 2 − k T 1 ω T k , (8) where the parameter bT in the kT1 is tuned according to (7). The tuning procedure of the DPT-BQF is as follows: (a) Obtain the resonance frequency ωT using the online resonance identification method [ 18]. (b) Select the Ω, and map it to the tuning parameter bT according to (7). (c) Choose an appropriate notch depth parameter xT and, together with bT, determine the parameters kT1 and kT2. (d) Finally, using k = ωT/tan( ωTTs/2) and according to (8), complete the notch filter design, where the filter output y can be expressed as y k = a 1 + a 3 u k − 2 a 2 u k − 1 + a 1 − a 3 u k − 2 + 2 a 2 y k − 1 − a 1 − a 4 y k − 2 a 1 + a 4 a 1 = k 2 + ω T 2 , a 2 = k 2 − ω T 2 , a 3 = k T 2 ω T k , a 4 = k T 1 ω T k , (9) At this point, the design and tuning of the DPT-BQF has been completed, and the digital implementation of the DPT-BQF can be achieved according to (9). 3.3. Performance Analysis of the DPT-BQF (1) Notch frequency: By differentiating the amplitude–frequency characteristic of the DPT-BQF to determine its notch frequency according to (8), the following is obtained: d G DPT - BQF d ω = φ γ 2 + η 2 γ = k 2 + ω T 2 cos ω T s − k 2 − ω T 2 η = k T 1 ω T k T s sin ω T s φ = 2 γ η k T 2 2 k T 1 2 − 1 k 2 + ω T 2 T s sin ω T s η + cos ω T s γ , (10) It can further be solved to obtain: cos ω T s = k 2 − ω T 2 k 2 + ω T 2 ⇒ ω T k = tan ω T s 2 ⇒ ω = ω T , (11) It can be seen that the DPT-BQF, benefiting from the pre-warped tuning of k, achieves an accurate mapping between the actual notch frequency ω and the tuned notch frequency ωT, effectively preventing notch frequency deviation. (2) Notch depth: As analyzed above, the DPT-BQF achieves accurate mapping between the actual notch frequency ω and the tuned notch frequency ωT by pre-correcting the discretization coefficients. Furthermore, analyzing the notch depth at the frequency ωT yields: G DPT - BQF ω = ω T = γ 2 + k T 2 η 2 γ 2 + k T 1 η 2 = k T 2 k T 1 = 10 x T 20 γ = k 2 + ω T 2 cos ω T T s − k 2 − ω T 2 η = ω T k sin ω T s , (12) As shown, the notch depth of the DPT-BQF at ωT is xT = 20lg| GPT-BQF| |ω = ωT, i.e., the magnitude gain at the notch frequency. Therefore, when using the pre-warped Tustin method for discretization, the actual notch depth is consistent with the tuned value, and no distortion occurs. Hence, the DPT-BQF, benefiting from the pre-warping of the coefficient k and the proposed bandwidth pre-warping, improves the discretization accuracy of the notch frequency, notch depth, and notch bandwidth in a comprehensive manner. This enables the controller to achieve precise suppression of resonance. Finally, to illustrate the discretization accuracy of the DPT-BQF more intuitively, the Bode plots are also presented in Figure 3. As shown, the notch frequency, bandwidth, and depth of the DPT-BQF exhibit no significant deviation. 4. Parameter Design and Analysis of the DPT-BQF 4.1. Parameter Design in a Closed-Loop System The system performance after inserting a BQF is analyzed from the perspective of a closed-loop system. First, the closed-loop transfer function of the NIPMVM after inserting the DPT-BQF can be derived as follows: ω m i ω r e f = d 3 z 3 + d 2 z 2 + d 1 z b 2 z 2 + b 1 z + b 0 k p + k i T s z − k p k t z − 1 c 3 z 3 + c 2 z 2 + c 1 z + c 0 a 2 z 2 + a 1 z + a 0 + k p + k i T s z − k p d 3 z 3 + d 2 z 2 + d 1 z b 2 z 2 + b 1 z + b 0 ω o ω r e f = g m k e f f T s 3 z 3 b 2 z 2 + b 1 z + b 0 k p + k i T s z − k p k t z − 1 c 3 z 3 + c 2 z 2 + c 1 z + c 0 a 2 z 2 + a 1 z + a 0 + k p + k i T s z − k p d 3 z 3 + d 2 z 2 + d 1 z b 2 z 2 + b 1 z + b 0 a 2 = k 2 + ω T 2 + k T 1 ω T k b 2 = k 2 + ω T 2 + k T 2 ω T k c 3 = J ε J o + ( J ε + J o ) k e f f T s 2 d 3 = g m 2 J o T s + k e f f T s 3 a 1 = − 2 k 2 − ω T 2 b 1 = − 2 k 2 − ω T 2 c 2 = − 3 J ε J o − ( J ε + J o ) k e f f T s 2 d 2 = − 2 g m 2 J o T s a 0 = k 2 + ω T 2 − k T 1 ω T k b 0 = k 2 + ω T 2 − k T 2 ω T k c 1 = 3 J ε J o , c 0 = − J ε J o d 1 = g m 2 J o T s , (13) Furthermore, keeping the controller parameters unchanged, the notch bandwidth is tuned to bt = ୧୨ ୍ଠ ωt rad/s, and the notch depth to xt = −20 dB. To compare the performance differences when tuning the notch frequency based on the previously mentioned vibration frequency and natural resonance frequency, the performance comparison results for notch frequencies ωt of 31.9 Hz and 34.9 Hz are shown in Figure 4. As shown in Figure 4a, the performance at vibration and natural resonance frequency is evaluated. Without the notch filter, the dominant poles yield a damping ratio of only 0.11. With the filter applied, the damping ratio increases to 0.503 at the vibration frequency and 0.405 at the natural resonance frequency. Moreover, the Bode plot results indicate comparable suppression performance under both conditions, with slightly improved attenuation at the vibration frequency. 4.2. Robustness Analysis of the System with the DPT-BQF For convenient tuning of DPT-BQF parameters, it is meaningful to analyze the tuning range of notch parameters under the premise of system stability. Then, the robust parameter range of the closed-loop system in (13) is shown in Figure 5. It should be added that, when plotting Figure 5, the controller parameter tuning method from [ 19] is adopted. In this case, z1 = 0.9, p1 = p2 = 0.5, and the sampling period is 0.0001 s. First, Figure 5a shows the poles for ωt = 5 to 42 Hz. When the frequency is in the low range, the system remains stable; however, when the frequency increases to 42 Hz, a pair of poles moves outside the unit circle. Figure 5b shows the poles for xt = 0 to −32 dB. Notably, from (6), xt should not be set at −3 dB. When xt < −3 dB, one pair of poles lies on the boundary and moves inside the unit circle as the notch depth increases. However, another pole moves outside the stability circle when the notch depth reaches −32 dB. Finally, Figure 5c shows the poles for bt = 0 to 15.7 times the notch frequency. When the bandwidth reaches 15.7 times, the system becomes unstable. In summary, the tuning of filter parameters significantly affects system stability. An excessively small or large notch frequency, depth, or bandwidth may cause instability. Therefore, the robust parameter range should be analyzed, and parameters should be configured within this range. For convenient parameter selection, the detailed parameter configuration procedure is given as follows: First, after the load and controller parameters are determined, the resonance frequency is obtained using the parameter identification method in [ 18], and this frequency is used as the notch frequency of the filter. According to (13), and following the form shown in Figure 5, the system pole-zero trajectories under different notch frequencies, notch depths, and bandwidths are plotted. The parameters are then selected according to the actual trajectories under different system parameters. Specifically, the notch frequency is selected based on the identification result, and it only needs to be ensured that it lies within the stable range. The notch bandwidth and notch depth can be repeatedly tuned within the stable range to obtain better performance. It is worth noting that, since the stable range is generally limited, the tuning process is convenient. 5. Experimental Verification To evaluate the effectiveness of the proposed DPT-BQF for the NIPMVM, experiments were conducted on the test platform shown in Figure 6. The NIPMVM is driven by the inverter based on IPM IM513-L6A. Then, the TMS320F28335 is used to implement the proposed method. The notch frequency is tuned according to the identified actual frequency [ 18], set to ωt = 34.9 Hz, while the other notch parameters are set as follows: notch bandwidth bt = ୧୫ ୍ଠ ωt rad/s and notch depth xt = −22 dB. It is worth noting that, to verify the notch distortion issue, the initial parameters of all notch filters are identical. To verify that the DPT method achieves the highest accuracy for the BQF, a frequency-sweep signal with an amplitude of 1 A and a frequency range of 0–500 Hz is used. The notch frequency is set to ωt = 200 Hz, the bandwidth to bt = 200 Hz, and the depth to xt = −30 dB. The results are shown in Figure 7. As seen, only the DPT-BQF shows no distortion, with both frequency and bandwidth at the preset 200 Hz. In contrast, the BE-BQF shifts the notch frequency to 180 Hz with a depth reduction to −6 dB; the T-BQF shifts the frequency to 165 Hz and narrows the bandwidth to 120 Hz; and the PT-BQF maintains the frequency but also narrows the bandwidth to 120 Hz. To verify the resonance suppression capability of the DPT-BQF at the rated speed, the comparison results of the systems with the BQF inserted, under an inner-rotor speed variation from 0 r/min−500 r/min−0 r/min, are shown in Figure 8. The DPT-BQF, benefiting from higher accuracy, provides optimal resonance suppression across a wide range. Systems without a notch filter or with conventional notch filters exhibit significant resonance, which becomes more pronounced at high speeds. Simultaneously, severe resonance is also observed in the current. These speed variation experiments confirm the excellent mechanical resonance suppression performance of the DPT-BQF in the NIPMVM. Furthermore, to quantitatively analyze the performance of the DPT-BQF, a comparison table is provided, as shown in Table 1. The BE-BQF shows obvious notch frequency and notch depth errors, while the T-BQF suffers from both notch frequency and bandwidth errors. The PT-BQF eliminates the notch frequency error, but its bandwidth error remains 40%. In contrast, the proposed DPT-BQF achieves zero error in notch frequency, bandwidth, and depth. Moreover, it reduces the current ripple to 0.8 A and the speed ripple to 7.9 r/min, demonstrating better discretization accuracy and resonance suppression performance. Furthermore, to compare the computational cost of different BQFs, the computation time of each method is also provided in Table 1. It can be seen that even the DPT-BQF only requires 4.26 μs, indicating strong feasibility for real-time implementation. Furthermore, experiments are conducted to validate the robustness of the filter parameters analyzed in Figure 5 and to provide a reference for parameter tuning. The results of sudden changes in parameters under steady-state conditions with the DPT-BQF at optimal tuning are shown in Figure 9. The figure shows the results when the notch frequency, depth, and bandwidth are increased to ωt = 38 Hz, xt = −42 dB and bt = ୧୬ ୍ଠ ωt rad/s. As seen when the notch frequency and bandwidth exceed the robust range, the system instantly becomes unstable. In contrast, increasing the notch depth still allows the system to operate stably, although the resonance suppression performance is significantly reduced. This is consistent with the analysis in Figure 5 above. As the notch frequency and bandwidth increase, a pair of poles moves along the real axis outside the stability circle, with larger values corresponding to greater distances from the stability boundary. Meanwhile, further decreasing the notch depth places the poles on the boundary; however, they essentially move along the boundary, minimally affecting stability but reducing resonance suppression performance. These results also validate the correctness of the discrete-domain analysis method proposed in this article and provide valuable guidance for BQF tuning. 6. Conclusions This article presents a double-pre-warped Tustin bi-quad filter to suppress NIPMVM mechanical resonance. The notch distortion caused by traditional discretization is analyzed, and the mechanism of improved accuracy from double pre-warping of notch frequency and bandwidth is explained. In the discrete domain, the resonance suppression mechanism is studied from a closed-loop perspective, and a parameter tuning method ensuring system robustness is provided. Finally, experimental results confirm that the proposed DPT-BQF offers superior mechanical resonance suppression. Author Contributions Conceptualization, J.C.; methodology, Q.C.; software, J.W.; validation, B.S.; writing—original draft preparation, J.C.; writing—review and editing, Y.F. (Ying Fan); visualization, M.T.; supervision, Y.F. (Yiming Fang); project administration, J.C.; funding acquisition, J.C. All authors have read and agreed to the published version of this manuscript. Funding This research and the APC were funded by the Zhejiang Provincial Natural Science Foundation of China (grant number LQN25E070003), the Shaoxing Basic Public Welfare Research Project (grant number 2025A11004), the “Pioneer” and “Leading Goose” R&D Program of Zhejiang (grant number 2024C03038), and the National Natural Science Foundation of China (grant number 62573114; 62403324). Data Availability Statement Data is contained within the article. Conflicts of Interest Author Junlei Chen was employed by the company Taizhou Jinyu Electromechanical Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Nomenclature Symbol Description Symbol Description T eElectromagnetic torque z Discrete-domain operator T cMagnetic gear transmission torque T sSampling period T LLoad torque k Pre-warped Tustin coefficient pi, poPole-pair numbers of the inner and outer rotors w tTuned notch frequency Ji, JoMoments of inertia of the inner and outer rotors b tTuned notch bandwidth wmi, wmoMechanical angular velocities of the inner and outer rotors Ω Actual notch bandwidth Bi, BoFriction coefficients of the inner and outer rotors x tTuned notch depth d sDamping coefficient of the magnetic gear xΩ/2Magnitude gain at the bandwidth boundary s Laplace operator λ Ω/2Auxiliary coefficient for bandwidth correction g mSpeed ratio of the magnetic gear kT1, kT2Coefficients of BQF T εEquivalent electromagnetic torque θ errElectrical angle error between the inner and outer rotors J εEquivalent moment of inertia i qq-axis current ω mεEquivalent mechanical angular velocity n miInner rotor speed θ eεEquivalent angular displacement difference n moOuter rotor speed k effEquivalent stiffness coefficient TmaxMaximum transmitted torque of the magnetic gear ω a ε Equivalent anti-resonant angular frequency γ εInertia ratio ω n ε Equivalent natural resonant angular frequency Appendix A The derivation from (6) to (7) is as follows. Let A = λ Ω / 2 2 k 2 + ω T 2 2 B = b T 2 k 2 . (A1) Then, (6) can be rewritten as cos ( Ω T s ) = A − B A + B . (A2) Using the trigonometric identity, then, tan 2 Ω T s 2 = 1 − cos ( Ω T s ) 1 + cos ( Ω T s ) . (A3) Furthermore, it can be deduced as tan 2 Ω T s 2 = B A = b T 2 k 2 λ Ω / 2 2 ( k 2 + ω t 2 ) 2 . (A4) Taking the square root and rearranging gives b T = λ Ω / 2 ( k 2 + ω t 2 ) k tan Ω T s 2 . (A5) This gives the mapping relationship between the tuned bandwidth bt and the actual bandwidth Ω, as shown in (7). References Motor models. ( a) Motion model. ( b) Simplified model. Motor models. ( a) Motion model. ( b) Simplified model. Comparison of PT-BQF under different periods. Comparison of PT-BQF under different periods. Comparison of DPT-BQF under different periods. Comparison of DPT-BQF under different periods. Comparison of the DPT-BQF at vibration and natural resonance frequency. ( a) Poles and zeros. ( b) Bode plot of the inner rotor. ( c) Bode plot of the outer rotor. Comparison of the DPT-BQF at vibration and natural resonance frequency. ( a) Poles and zeros. ( b) Bode plot of the inner rotor. ( c) Bode plot of the outer rotor. Robust range of the DPT-BQF. ( a) With varying ωt. ( b) With varying xt. ( c) With varying bt. Robust range of the DPT-BQF. ( a) With varying ωt. ( b) With varying xt. ( c) With varying bt. Test bench setup. ( a) Motors. ( b) Controller. Test bench setup. ( a) Motors. ( b) Controller. Comparison with various BQFs. ( a) Response. ( b) Amplitude–frequency characteristics. Comparison with various BQFs. ( a) Response. ( b) Amplitude–frequency characteristics. Comparison of speed performance with different BQFs. Comparison of speed performance with different BQFs. Results of the robust parameter range of the DPT-BQF. Results of the robust parameter range of the DPT-BQF. Table 1. Performance comparison table of different BQFs. Table 1. Performance comparison table of different BQFs. ωt Error (%) bt Error (%) xt Error (%) iq Ripple (A) nmo Ripple (r/min) Computation Time (μs) No filter / / / 6 33 / BE-BQF 10 / 80 2.9 17.3 3.21 T-BQF 17.5 40 0 3.2 24.2 3.31 PT-BQF 0 40 0 3.3 23.7 3.98 DPT-BQF 0 0 0 0.8 7.9 4.26 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Share and Cite MDPI and ACS Style Chen, J.; Shi, B.; Wu, J.; Fan, Y.; Chen, Q.; Fang, Y.; Tang, M. Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter. Electronics 2026, 15, 2506. https://doi.org/10.3390/electronics15122506 AMA Style Chen J, Shi B, Wu J, Fan Y, Chen Q, Fang Y, Tang M. Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter. Electronics. 2026; 15(12):2506. https://doi.org/10.3390/electronics15122506 Chicago/Turabian Style Chen, Junlei, Bocheng Shi, Jingying Wu, Ying Fan, Qiushuo Chen, Yiming Fang, and Min Tang. 2026. "Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter" Electronics 15, no. 12: 2506. https://doi.org/10.3390/electronics15122506 APA Style Chen, J., Shi, B., Wu, J., Fan, Y., Chen, Q., Fang, Y., & Tang, M. (2026). Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter. Electronics, 15(12), 2506. https://doi.org/10.3390/electronics15122506 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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Resonance Suppression for NIPMVM Based on Double-Pre-Warped Tustin Bi-Quad Filter
