Speed/Torque Ripple Reduction of High-Speed Permanent Magnet Starters/Generators With Low Inductance for More Electric Aircraft Applications

With the electrification trend of future aircraft, high-speed permanent magnet starters/generators (PMSs/Gs) will potentially be widely used in onboard generation systems due to their high power density and high efficiency. However, the per-unit reactance of such high-speed machines is normally designed to be very low due to limited onboard power supply voltage and large electrical power demand, which can result in large current ripples in the machine and thus large torque ripples especially when the machine is fed with a semiconductor-based inverter of a lower switching frequency. The torque ripples may further lead to speed oscillation and generate severe vibrations and noises that are harmful to the mechanical system and human beings around. In this article, a speed/torque ripple reduction method for high-speed PMS/Gs with a low inductance is proposed to improve their performance within a wide speed range. An active damping technique is applied to the speed loop to increase the anti-disturbance capability and generate a smoother reference for the current loop, whereas an adaptive output voltage saturation limit is utilized for the current loop to limit the peak value of current to prevent overcurrent and torque spikes. The parameter tuning criteria are derived through a thorough analysis. Finally, the proposed method is validated on a high-speed PMS/G with an inductance of 99 $\mu \text{H}$ . The results show that the speed ripples and torque ripples are reduced by over 50% within a speed range of 2–14 krpm.


I. INTRODUCTION
R EDUCTION of greenhouse gas emissions is more mandatory now than before. The aviation sector contributes significantly to these emissions, where the sector's CO 2 emissions resembled 2.2% of the global CO 2 emissions in 2017 [1]. The conventional aircraft system is complex, and the multiple power-conversion steps reduce the entire system's efficiency and reliability [2]. Electrical systems are more and more widely implemented in more electric aircraft (MEA) over the last few decades due to their potential in improving efficiency and reducing emissions [3], [4]. Many subsystems that previously used to be driven by hydraulic, mechanical, and pneumatic power have been fully or partially replaced with electrical systems [1]. As the aerospace industry keeps moving toward greener and more electric solutions [5], [6], electrical systems on aircraft will continue to play an everincreasing role [7].
A key technology of the MEA is the electrical drive system that serves as the interface between the aircraft engine and the onboard electrical power system [8]. An electrical drive system can operate as a starter/generator (S/G) for the MEA with the appropriate power converter, electrical machine, and control scheme. An electrical machine can be used as a starter to drive the aircraft engine and can alternatively function as a generator when the engine drives the electrical machine [9]. The S/G system may simplify the power generation system by reducing the complexity of the mechanical subsystem, especially in aerospace and automotive applications, which would result in increased reliability and reduced overall weight [10]. Among its counterparts, highspeed permanent magnet starter/generators (PMS/Gs) have been a popular candidate due to their high power density and flexible bidirectional operation, which contributes to high instant power/torque output and better efficiency over a wide speed range [11]- [14].
However, the PMS/Gs are expected to deliver hundreds of kilowatts electrical power as a generator due to the increasing electric demand of MEA and operate at a wide speed range as a starter to drive the engine to the ignition speed [15]. To achieve high power density in a 270-V onboard dc power supply system, electrical machines for aerospace applications are preferable of higher speeds (>10 000 rpm), larger current density compared to conventional machines, and multiple pole pairs. The high fundamental frequency would result in the increase of coil resistance and decrease of coil inductance due to proximity and skin effect [16]. Moreover, the increased effective air gap in surface mounted PMS/Gs and low coil turn count due to multiple pole design attempts result in a low inductance of the high-speed PMS/Gs. As a result, the highspeed PMS/Gs are normally with low inductance and resistance. The downside part of low inductance in the machine is that when these machines are fed with semiconductor-based ac/dc converters, torque ripples resulting from distorted ac currents become significant, especially when the modulation ratio (modulation frequency over fundamental frequency of the machine) is low [17]. In severe cases, the torque ripples may lead to speed oscillation and generate severe vibrations and noises that are harmful to the mechanical system and human beings around. In addition, the low inductance and resistance increase the risk of overcurrent in case of current controller saturation, especially at low speed when the back electromotive force (back EMF) is relatively lower.
Although the application of widegap devices with higher switching frequency to an inverter can significantly reduce torque ripples and speed ripples, it is still worthwhile if a modified control method can effectively improve the performance of an inverter based on IGBTs to a satisfactory level. The price of widegap devices is 5-10 times or even more than that of conventional devices with the same voltage and current. The requirements for the drive circuit and the accessory circuit also become higher. The multilevel converter can be adopted with a reduced voltage drop of each power device, but the number of power devices increases significantly resulting in increased system volume and cost and more complex control algorithms [18]. Apart from the higher device and drive circuit cost, higher switching loss, and more significant EMI issues of the widegap devices, the increase of switching frequency would result in the reduction of allowed computation time for the microcontroller between sampling of current feedbacks and updating of voltage references from the control perspective. As a result, more advanced microcontrollers with higher computation speed need to be used, which in turn increases the cost of the overall system. Many torque ripple reduction methods for high-speed PMSMs have been published in the literature in recent years. In [19], the cogging torque of a high-speed PMSM is mitigated through a closed slot design to reduce the 11th and 13th harmonics of air-gap flux and weaken the slotting effect. In [20], the reluctance torque ripple and cogging torque are reduced through the magnet shifting technique. In [21] and [22], torque ripples are identified or measured off-line to be compensated through a tailored lookup table. Similarly, periodic torque ripples are observed by a repetitive observer [23]- [25] or isolated by iterative learning control [26], [27] in real time and compensated accordingly. In [28] and [29], torque ripples caused by distorted back electromotive and nonsinusoidal flux distribution are settled by tailoring the current reference according to off-line measured the back EMF and inductances of the motor. Torque ripples due to current harmonics and parameter mismatches in direct torque control (DTC) [30]- [32] and model predictive control (MPC) [33], [34] are reduced through optimized modulation methods and improved torque or current control accuracy. All these methods aim to solve some torque ripples from a specific source through a change of the motor structure, premeasurement, or online observation of torque ripples or more accurate current control. However, few articles are particularly focused on speed and torque ripples resulting from the low inductance of high-speed PMSMs and relatively lower switching frequency of the inverter within the proportional-integral (PI)-based field oriented control (FOC) frame, which is most widely used in the industry due to its simple structure and parameter independence.
The contribution of this article is to fill this gap and develop a modified control technique that can reduce torque ripples and speed ripples of high-speed PMS/Gs with low inductance and relatively lower switching frequency within the PI-based FOC frame to improve their performance within a wide speed range. As the torque ripple is a disturbance of the speed loop, an active damping scheme is applied to the speed loop to increase its anti-disturbance capability and generate a smoother reference for the current loop. Different forms of active damping are compared, based on which parameter tuning criteria are concluded. Overcurrent issues at d-and q-axes current controller saturation are considered, from which an adaptive reference voltage saturation limit is proposed and a back-calculation anti-windup strategy is utilized for the current loop to limit the peak value of current to prevent torque spikes and speed oscillations. The proposed adaptive saturation limit in combination with the active damping scheme can reduce the torque and speed ripples by over 50% and thus effectively reduce the noises and vibrations.
This article is organized as follows. The deciding factors of torque ripples are analyzed in Section II. The proposed torque ripple reduction method is elaborated in Section III, whereas simulation results on a tradeoff study in comparison with widegap devices are presented in Section IV. In Section V, the performance of the proposed strategy is validated on a high-speed PMS/G with the inductance of 99 μH within a speed range from 2 to 14 krpm. At last, a conclusion is drawn in Section VI.

II. TORQUE RIPPLE ANALYSIS
The mathematical model of PMS/Gs in the synchronous rotating frame is listed as follows: where i d and i q denote the d-and q-axes currents, whereas v d and v q are the d-and q-axes voltages, respectively. L d and L q are the d-and q-axes inductance, respectively .ψ m is the flux linkage. ω r and ω m are the electrical and mechanical angular velocities, respectively. T e and T L are the electrical magnetic torque and mechanical torque, respectively. J and K f are the inertia and frictional damping ratio, respectively. Considering a surface-mounted PMS/G, L d = L q , the derivative of (3) can be given as Substituting (2) into (5) and assuming the flux linkage is constant, (6) is derived as Considering the machine is fed by a dc/ac inverter, with the switching behavior of the inverter, using (6), the torque ripple in a certain switching period can be expressed as where the number k denotes the average value of the state variable in the kth switching period, whereas f s denotes the switching frequency. It can be concluded that the torque ripple is proportional to Pψ m /( f s L q ). For comparison studies, the parameters of two high-speed PMS/Gs are listed in Tables I and II, respectively. PMS/G-I is driven by a SiC MOSFET-based inverter with a switching frequency of 40 kHz and a phase inductance of 874 μH, whereas PMS/G-II [33] is driven by a silicon IGBT-based inverter with a switching frequency of 16 kHz and a phase inductance of 99 μH. The Pψ m /( f s L q ) value of PMS/G-II is 12.2 times of PMS/G-I, which means that the torque ripple of PMS/G-II can be 12.2 times of PMS/G-I if the same voltage is applied to the q-axis inductance. For a high-speed PMS/G with low inductance and limited switching frequency like PMS/G-II, a possible way is to reduce the torque ripple through control strategies to limit the q-axis voltage and d-and q-axes currents at each switching interval. Fig. 1 shows the control diagram of a PMS/G at the starter mode, which is the commonly used field-oriented cascaded PI control with flux weakening control. The internal control loop is for the d-and q-axes current control and the outer loops are for speed and flux weakening control. For generator operation of the PMS/G, the speed loop will be replaced by the dc-link voltage control or droop control [35].
The speed and current loop controllers in Fig. 1 are expressed as As the q-axis current reference is generated by the speed loop controller, any speed fluctuation would result in current reference ripples [from (8)] and voltage reference ripples (10). This in turn leads to torque ripples and further increases the speed ripples.
Therefore, a properly designed speed controller, reducing the ripples in i * q , can potentially reduce the torque ripples. Also, the q-axis voltage needs to be dynamically adjusted with the variation of speed and cross-coupling voltages to prevent overcurrent, which is normally realized by the integral action of the PI controller. However, a larger integral gain may easily lead to the current controller saturation since the d-and q-axes current cannot track the reference in a transient response due to the cross-coupling voltage especially when the error between current reference and feedback is large, which happens at sudden load changes. At saturation conditions, the voltage reference is clamped to the maximum modulated voltage of the inverter, which is 1/ √ 3 of dc-link voltage for space vector pulsewidth modulation (SVPWM) or 1/2 of dc-link voltage for sinusoidal pulsewidth modulation (SPWM). The saturation can cause severe overcurrent of PMSM/Gs at a lower speed with lower back EMF due to low inductance and resistance and thus results in instant torque spikes and speed overshoot. To settle these issues, a control algorithm with active damping and adaptive saturation limit is proposed in Section III.

A. Active Damping for the Speed Loop
The concept of virtual resistor or active damping was proposed by Dahono [36] and [37], for the application of input LC filter of PWM converter and then for LCL filter of dc/dc converter to dampen the LC resonance. A resistor can be connected in various ways to dampen the oscillation in the LC/LCL circuit, but the power loss is thereby increased. The idea of active damping or virtual resistor can be generalized as the effect of a real resistor can be replaced by an equivalent control strategy. In this way, the damping effect can be realized, but the power loss with a real resistor can be avoided.
In this article, this idea of active damping is transplanted to the speed loop of high-speed PMS/Gs. Unlike most literature dealing with the torque ripples by direct harmonic rejection, an indirect way through improving the anti-disturbance capability of speed loop by active damping is used to reduce speed ripples and generate a smoother current reference and thus reduce torque ripples. The control loop diagram of the PMS/G with the active damping scheme is presented in Fig. 2, where the left-side block is the controller and the right-side represents the PMS/G plant. The damping ratio K f of the plant is very small for low mechanical loss, which results in a very small integral gain and weak anti-disturbance capability for the PI controller designed based on pole-zero cancellation criteria. However, a lot of disturbance in the speed loop (lumped as T L in Fig. 2), such as torque ripples, load torque, and other unmodeled uncertainties, would deteriorate the stability and dynamic performance of the speed controller. Similar to the virtual resistance effect in current circuits, the virtual damping ratio of the mechanical plant can be increased by adding an active damping term to the output of the speed controller. In Fig. 2, an active damping term −H ad (s)ω m is added to the q-axis current reference.
To verify the effectiveness of the active damping scheme, the closed-loop tracking and anti-disturbance transfer functions need to be derived and analyzed in the following. With the active damping feedback added to (8), the q-axis current reference i * q is given as Substituting (11) to (10), the q-axis voltage reference is derived as Assuming the current controller is not saturated, i.e., v q = v * q and solving the equation by substituting (11) into (2), i q can be obtained as Substituting (13) into (3), the mechanic equation of the machine is derived as As ω m is determined by two factors, the reference ω * m and the disturbance T L , the corresponding transfer functions are analyzed separately according to the superposition principle. The closed-loop tracking transfer function ω m /ω * m is derived as, (15), shown at the bottom of the next page, whereas the closed-loop anti-disturbance transfer function ω m /T L is derived as, (16), shown at the bottom of the next page.
The closed-loop tracking transfer function ω m /ω * m is fourthorder with four poles and two zeros, whereas the closedloop anti-disturbance transfer function is fourth-order with four poles and three zeros. Both can be reduced to secondorder through pole-zero cancellation by properly choosing the form of H ad (s) and tuning the PI parameters to improve the controller stability. In this article, two different forms of active damping will be discussed. As proportional feedback is the most used active damping form in LCL applications, it is analyzed first as Case 1. Then, a proportional and differential feedback active damping is analyzed as Case 2. These two cases will be compared with the original speed controller without active damping (Case3) as follows. T ω = (Ĵ /K fa ), and k pi = 2π f cL q , T i = (L q /R s ), for pole-zero cancellation, where parameters with "ˆ" represent corresponding parameters used in the controller, which can potentially be different from the real values in the plant, and f ω and f c are the bandwidths of the speed loop and current loop, respectively. Then, (15) is transformed to the following equation, (17), as shown at the bottom of the next page.
Assuming parameters used in the controller are close to the real values and thus (L q /R s ) = (L q /R s ) and (K f /Ĵ ) = (K f /J ), the pole and zero s = −(R s /L q ) can be canceled and (17) is transformed to, (18), as shown at the bottom of the next page.
Similarly, the anti-disturbance transfer function can be derived as, (19), shown at the bottom of the next page.
Both the closed-loop tracking transfer and closed-loop, antidisturbance transfer function are reduced to third order with pole-zero s = −(R s /L q ) canceled, whereas there is still a zero s = −(K fa /Ĵ ) in (18) and s = −2π f c (L q /L q ) in (19), respectively, without a paired pole to be canceled. The reason is that active damping is used in the outer loop in this article, which is different from applying active damping in the inner loop for LCL filters. Therefore, pure proportional feedback is not the best choice, and some modifications can be made to H ad (s) to allow cancellation of one more pole and reduce (18) and (19) to second order. (18) is replaced by (20). AssumingL q = L q , which means the accuracy of measured or observed L q is high enough, (20), as shown at the bottom of the next page, can be simplified to a second-order system

2) Case 2 [H ad
with the paired pole-zero s = −(K fa /Ĵ ) canceled. Similarly, the closed-loop anti-disturbance transfer function ω m /T L is also reduced to second order with the paired pole-zero s = −2π f c (L q /L q ) canceled. (s) = 0): For comparison studies, the conventional PI-based strategy without active damping can be regarded as a special case H ad (s) = 0, and then the closedloop tracking and anti-disturbance transfer functions can be obtained as
Comparing (21) with (23), it can be found that the closedloop tracking transfer functions of the modified method with active damping and the conventional method without active damping are the same. However, the dominant closed-loop pole of the anti-disturbance transfer function is moved from s = −(K f /J ) in (24) to s = −(K fa /J ) in (22) after the active damping is applied. Therefore, with properly tuned PI parameters k pω = ((2π f ωĴ )/((3Pψ m /2))), T ω = (Ĵ /K fa ), k pi = 2π f cL q , and T i = (L q /R s ), the active damping scheme enhances the stability and anti-disturbance capability of the speed controller while keeping the reference tracking capability unchanged. The corresponding bode diagrams of (21) and (22) are presented in Figs. 3 and 4, respectively.
In Fig. 3, the bode diagram of the tracking transfer function ω m /ω * m is compared at different speed bandwidths f ω = 25, 50, and 100 Hz and current loop bandwidths f c = 250 and 1000 Hz. In Fig. 4, the bode diagram of the antidisturbance transfer function ω m /T L is compared at different speed loop bandwidths f ω = 25, 50, and 100 Hz and damping ratios K fa = 0.1, 1.0, and 10.0. Fig. 3 reveals that the speed loop cutoff frequency is dominated by the speed loop bandwidth f ω . Fig. 4 shows that active damping contributes to obviously improved suppression effects of low-frequency disturbance for the speed controller, and a larger active damping ratio means better anti-disturbance capability. With K fa2 = 10, the amplitude gains of ω m /T L are all decreased by 25 dB compared to K f = 0.1 at the three different f ω values.
In particular, the low-frequency amplitude gains of f ω = 25 Hz and K fa2 = 10 are even smaller than that of f ω = 100 Hz andK f = 0.1. However, a larger active damping ratio also results in a larger integral gain according to k pω = 2π f ωĴ /(3Pψ m /2) and T ω = /K fa , which increases the chance of speed loop saturation considering the output limit. Therefore, a compromise must be made regarding the active damping and PI parameter tuning. Also, H ad (s) contains a parameter f c regarding the bandwidth of the current loop, so the current loop and speed loop control parameters need to be tuned together.

B. Adaptive Saturation Voltage Limit of Current Loops
The previous analysis assumes that the current loops are not saturated and the d-and q-axes current can well follow the references. However, for machines with low inductance, the large current and limited dc-link voltage easily get the controller saturated since the current cannot track the reference in the transient due to the cross-coupling between d-and q-axes when the error between current reference and feedback is large, which normally happens at sudden load change. At saturation conditions, the voltage output is clamped to the saturation limit value determined by the dc supply voltage and the modulation algorithms. In the conventional PI approach, the d-and q-axes voltage reference amplitude , which easily leads to instant overcurrent at a lower speed with smaller back EMF due to small inductance and resistance and thus cause instant torque spike and speed overshoot.
In this article, the saturation voltage reference limit of current loops adaptive to the instant speed as shown in is proposed to limit the maximum current to prevent overcurrent in transient process and eliminate the torque spike and ensure a stable performance at wide speed range operation. The saturation limit v max of current loops in (25) is the minimum value of ω r ψ m + R s i max adaptive to the back EMF and maximum allowed current i max , and the maximum inverter output voltage m max v dc determined by the dc-link voltage v dc and the maximum modulation rate m max dependent on modulation algorithms. The saturation limit curves at i max = 150 A, 250 A, and 500 A are shown in Fig. 5 with R s = 0.1 , and ψ m = 0.0364 Vs. The intersection between m max v dc and ω r ψ m + R s i max determines the saturation limit region at the certain allowed maximum current.
To evaluate the performance of the proposed adaptive voltage limit scheme, the current and voltage limit circle equations are derived as (26) where i N is the nominal current of the machine, and r is the radius of the voltage limit circle r = (v max /ω r ). Normally, R s is neglected when analyzing the voltage limit circle, but it cannot be neglected here for the case of a machine with low inductance. Denoting (R s /ω r ) = α, then the voltage limit circle equation can be expressed in polar coordinates as Solving (27), the boundary of d-and q-axes current limited by the voltage circle is obtained as As α and r are speed-dependent, the voltage limit circles at different speeds from 2 to 18 krpm are compared. Three cases v max1 = m max v dc , v max2 = ψ m ω r + R s i max , and v max3 = min(v max1 , v max2 ) are considered, and the results are presented in Figs. 6-8, respectively. SVPWM is considered here, so m max = 1/ √ 3 . i max is set as 250 A although i N is around 150 A to leave some margin for instant overload. Each circle determines the d-and q-axes current boundary at a certain speed when the output voltage of the current loop is saturated to the corresponding voltage.
In Fig. 6, the maximum d-and q-axes currents can be up to over 1000 A at 2 krpm, even if a current limit of 250 A is applied to the current loop. The maximum d-and q-axes currents at 4 and 8 krpm are also far beyond 250 A, which reveals a high risk of over current. In Fig. 7, the d-and q-axes   current are much reduced at 2-8 krpm with the speed adaptive voltage limit applied, but the circle limited by v max1 is smaller at higher speeds. The transition point around 12 krpm can be seen in Fig. 8, which proves that the current is limited at saturation conditions within wide speed range, and thus, intolerable overcurrent and torque spikes are prevented.
Combining the active damping and adaptive voltage saturation limit, the diagram of the proposed method is presented in Fig. 9, whereas the speed and current controllers are expressed in (29)-(37) as follows: Field weakening is applied for the d-axis current reference in (29), whereas the back-calculation anti-windup method is applied in both the speed loop controller (30) and current loop controller (33) and (34). An improved anti-windup scheme as proposed in [38] is adopted to improve the dc bus voltage usage by saturating the root mean square value of the voltage reference to v max by (35)-(37).

IV. SIMULATION RESULTS
For tradeoff studies, the proposed modified method with active damping and adaptive saturation limit is compared with the conventional PI method through simulation at different switching frequencies. Parameters of the machine used in the simulation are listed in Table II in Section II. Figs. 10-13 present the simulation results of speed step response from 0 to  6 krpm and 6 to 10 krpm at a switching frequency of 16 kHz (SiC IGBT) and 40 kHz (SiC MOSFET), respectively, of both the conventional PI method and the proposed modified method. It can be seen in Fig. 10 that the dynamic response of the proposed method is much faster than that of the conventional method at both 16 and 40 kHz when the speed reference steps from 0 to 6 krpm and from 6 to 10 krpm.
The zoomed-in view in Fig. 11 shows that the response time of the proposed method is only 0.06 s when the speed reference steps from 0 rpm to 6 krpm, whereas the response time of the conventional method is almost 0.9 s. Similarly, the zoomed-in view in Fig. 12 shows that the response time of the proposed method is less than 0.05 s when the speed reference steps from 6 rpm to 10 krpm, whereas the response time of the conventional method is almost 0.9 s. In particular, the dynamic response of the proposed method at 16-kHz switching frequency is much better than that of the conventional method at 40 kHz. The accelerated dynamic response is due to the increased integral gain for pole-zero cancellation when the active damping is applied.
For the steady-state speed ripples, it can be found in Fig. 13 that the speed ripples are effectively reduced at both 16 and 40 kHz with the proposed modifications applied. The performance of 16 kHz can be further improved by increasing the active damping gain. The simulation results prove the effectiveness of the proposed modification in reducing the speed ripples and accelerating the dynamic response of the speed loop. Although the performance at 16 kHz is not as good as that at 40 kHz, the purpose of this article is not to challenge the widegap devices of higher switching frequency with the proposed method but to provide an alternative to reduce the torque ripple and speed ripple to improve the performance of the system through a modified control method based on limitations of the hardware.

A. Test Setup Description
The proposed method with active damping and adaptive saturation limit is verified on a PMS/G prototype with an inductance of 99 μH and a switching frequency of 16 kHz. Parameters of the PMS/G and inverter are listed in Table II in Section II, whereas the hardware structure is presented in Fig. 14.
A 150-kW prime mover and the homebrewed PMS/G machine are placed in an isolated room from the safety viewpoint. The three-level neutral point clamped (NPC) converter along with a DSK6713/Actel a3p400 control platform, dc source, prime mover controller, and host PC is placed in the control room. The selected devices for the NPC converter are IGBT modules from Infineon. The prime mover emulates an aircraft engine shaft, coupled with the PMS/G. The NPC converter interfaces the PMS/G and the 270-V dc link. Realtime variables of the inverter are monitored by the host PC through a data cable.

B. Experimental Results
The proposed method with active damping and adaptive saturation limit is compared with the conventional method at different speeds 14, 10, 6, and 2 krpm, respectively, with a light load of 1 N·m, which is the condition the torque ripples and speed ripples cause the most serious mechanical vibration and noises.
The speed and torque waveforms at 14 krpm are presented in Figs. 15 and 16, respectively, where the waveforms of the   proposed method are in red, whereas that of the conventional method are in blue. The steady-state speed ripple peak-to-peak value of the conventional method is roughly 50 rpm, whereas that of the proposed method is less than 20 rpm. The torque ripple peak-to-peak value is reduced from 9 N·m to less than 3 N·m. The torque FFT results at 14 krpm in Fig. 17 show that the low-frequency harmonics under 400 Hz are effectively suppressed with the proposed modifications applied, which matches the theoretical analysis in Section III.  Hz is shifted to 400 Hz with reduced magnitude. The reason why performance at 14 krpm is slightly better than 10 krpm is that the PMS/G enters the field weakening region at 14 krpm with nonzero d-axis current reference, and the q-axis current saturation limit is reduced.
The speed and torque waveforms at 6 krpm are presented in Figs. 21 and 22, respectively. The steady-state speed ripples are reduced from 70 to 30 rpm, whereas the torque ripples are reduced from 8 N·m to less than 4 N·m. Torque FFT results in Fig. 23 show that the low-frequency ripples are generally suppressed with the proposed algorithm applied, although there is a spike at 100 Hz.
For the results at 2 krpm, the steady-state speed ripples are reduced from 70 to 30 rpm in Fig. 24. The torque ripple results in Fig. 25 do not show an obvious reduction of peak-to-peak value, as the modulation index is rather low at this speed, resulting in a lot of harmonics in the current, but the torque   ripples are shifted to higher frequency according to FFT results in Fig. 26, so that the peak of torque integral value is reduced, which results in a reduction of speed ripples in Fig. 24.
In conclusion, the speed ripple and torque ripple peak-topeak values at different speeds are summarized in Tables III and IV, respectively. In general, the steady-state speed ripples of the PMS/G are reduced by at least 57.1% with the active damping and adaptive saturation limit applied within a wide speed range from 2 to 14 krpm. For the torque ripples, the   peak-to-peak value is reduced by at least 50.0% from 6 to 14 krpm. The performance at high speed is better than low speed as the low modulation ratio at low-speed results in current distortion and harmonics, which cannot be solved through  pure control methods. An optimized modulation algorithm can be applied to further improve the low-speed performance.

VI. CONCLUSION
In 270-V onboard power systems, the high-speed PMS/Gs are designed to be with low inductance and resistance, which results in severe torque ripples and may further lead to speed oscillation and generate severe vibrations and noises that are harmful to the mechanical system and human beings around. The low inductance and resistance also increase the risk of overcurrent in case of current controller saturation, especially at a lower speed with relatively lower back EMF. A modified speed/torque ripple reduction method for high-speed PMS/Gs with low inductance within the PI-based FOC frame is proposed in this article to improve their performance within the wide speed range. As the torque ripple is a disturbance of the speed loop, an active damping scheme is applied to the speed loop to increase the anti-disturbance capability and generate a smoother reference for the current loop. Different forms of active damping are compared, based on which parameter tuning criteria are concluded. Overcurrent issues at current controller saturation are considered, an adaptive output voltage saturation limit is proposed, and a back-calculation antiwindup strategy is utilized for the current loop to limit the peak value of current to prevent instant overcurrent and torque spikes. Both the active damping and adaptive saturation limit contribute to reduce the speed and torque ripples. Tradeoff study through simulation between an IGBT-based inverter with a switching frequency of 16 kHz and a MOSFET-based inverter with a switching frequency of 40 kHz shows that the proposed modified method not only reduces the speed ripples at steady states but also improves the dynamic response of the speed loop. Finally, the proposed method is validated on a high-speed PMS/G test rig with an inductance of 99 uH and a switching frequency of 16 kHz. The results show that the speed and torque ripples are effectively reduced over 50% within the speed range from 2 to 14 krpm.