Zero-Power PEMS System with Full-Bridge PWM Inverter: from Mechanism to Algorithm

The permanent-electro magnetic suspension (PEMS) technology takes advantage of the attractive magnetic force between the magnet and the iron core and reduces the power consumption eventually to zero. However, the current of the zero-power PEMS system fluctuates around zero due to distu rbances and suffers from the electronic nonlinearity. This work presents that the 2 μs turn-off delay ( one electronic defect ) of the integrated circuit (IC) L298N ( one commercial full-bridge pulse-width-modulation (PWM) inverter produced by STMicroelectronics ) leads to the nonlinear current-duty cycle characteristic, which undermines the control stability and limits the PWM frequency of the zero-power PEMS system. Moreover, the nonlinear mechanism is experimentally and theoretically analyzed for the critical PWM frequency and the sensitivity transition. Furthermore, this work proposes the compensation algorithm to overcome the electronic nonlinearity. It is demonstrated that the three-piece linearization approach stabilizes the PEMS system with only a few milliampere current and outperforms the two-piece counterpart with stronger robustness and smoother dynamics under the current-step-change test, especially for the PWM frequency higher than the critical value. Besides, the breakthrough of the critical PWM frequency by the compensation algorithm is of great significance for the dynamic performance of the high-speed PEMS transportation system.


Introduction
The electromagnetic suspension (EMS) technology has played an essential role in the maglev transportation industry in Germany, Japan, and China since 1960s 1 .Besides, the active magnetic bearing 2 is another successful application of the EMS technology and is widely found in flywheel systems, high-speed drives, and turbomolecular pumps.However, the EMS technology consumes considerable large power to generate sufficient attractive force 3 and requires high-performance electronics, such as the isolated gate bipolar transistor (IGBT) 4 and the LLC resonant converter 5 .
In 1980s, the Nd-Fe-B permanent magnet was invented as a special functional material with the largest magnetic energy accumulation.Since then, the permanent-electro magnetic suspension (PEMS) technology 3 is economically effective by using the Nd-Fe-B permanent magnet to produce the attractive force and reduce the power consumption eventually to zero 6 , whereas the electromagnet only plays a dynamic regulatory role 4 .Mizuno and et al. 7 developed an active vibration isolation system with an infinite stiffness against disturbances based on the zero-power PEMS technology.Zhang and et al. 8 numerically optimized the geometry for the zero-power PEMS system as undergraduate project kits in terms of the zero-power force and the controller-gain requirement.
Due to its outstanding energy-saving and robust features, the zero-power PEMS technology also attracted great attention from the maglev transportation industry.Tzeng and Wang 9 presented a rigorous dynamic analysis for the high-speed maglev transportation system with the robust zero-power PEMS strategy.Zhang and et al. 4 focused on the optimal structural design of the PEMS magnet and proposed optimized parameters with better carrying capability and lower suspension power loss.Cho and et al. 10 reported a successful quadruple PEMS system with the improved zero-power control algorithm.
Importantly, unlike the non-zero-mean current () in the EMS technology 11,12 , the current for the zeropower PEMS technology fluctuates about zero due to external disturbances 10,13 .Though the full-bridge pulsewidth-modulation (PWM) inverter 2 (also known as the power converter 14 or the chopper 3 ) can drive the zero-mean current in the electromagnet according to the PWM duty cycle (), there are few works analyzing its electronic nonlinearity that could be one of the most critical factors for the zero-power PEMS technology.
Moreover, the sampling rate (  ) and the PWM frequency (  ) are two crucial parameters for the dynamic performance of the high-speed PEMS transportation system, e.g., the disturbance rejection and the gap tracking 15 .In the literature,   is usually set from 75 Hz to 2 kHz 12,[16][17][18] for the digital controller, whereas   is usually set from 10 kHz to 20 kHz 3,[11][12][13] .Generally,   is at least 5-time smaller than   .Hence, in order to enhance the dynamic performance of the high-speed PEMS transportation system, higher   is under huge demand but may amplify the electronic nonlinearity of the full-bridge PWM inverter.
This work aims to analyze the electronic nonlinearity of the full-bridge PWM inverter for the zero-power PEMS system.The paper starts with the preliminaries of the single-axis PEMS system.Then, the nonlinear mechanism of the  −  characteristic is analyzed.Moreover, two piecewise linearization approaches are compared under the current-step-change test for two   with respect to (w.r.t.) the critical PWM frequency (  ).Nevertheless, discussion is addressed.

Preliminaries
This subsection addresses the preliminaries of the single-axis PEMS system.Firstly, the experimental setup is detailed with the hardware and the three nonlinear relationships.Secondly, the current in the electromagnet is theoretically modelled.Thirdly, the nonlinear  −  characteristic of the full-bridge PWM inverter with the electromagnet is briefly presented and to be further analyzed and compensated in the present work.

Experimental setup
Figure 1 visualizes the close-loop architecture of the single-axis PEMS system, Figure 2 visualizes the associated hardware, and Table 1 lists out physical properties.The six elements are: 1) Floater includes the payload and the Nd-Fe-B permanent magnet with the grade of N35.The magnet holder keeps the minimum gap between the magnet and the iron core to protect the ITR8307 sensor, while the three vertical rods and the soft connection maintains the upwards orientation of the magnet regardless of the gravity center of the payload; 2) ITR8307 distance sensor converts the floating height (ℎ) between the floater and the electromagnet into the height signal ( ℎ ) by the reflection intensity of the infra light.Besides, a light-proof housing is installed to entirely cover the single-axis PEMS system; 3) Analog signal processor converts  ℎ into the proportional and differential signals (  and   ); 4) Digital STM32 controller outputs the PWM signal with   and  (based on   ,   and the doubleloop control algorithm).Besides, the real-time data are upload to the computer by the serial port; 5) L298N full-bridge PWM inverter converts the PWM signal into  through the electromagnet. is measured by the ammeter, while the voltage across the electromagnet is measured by the oscilloscope; and 6) Electromagnet generates the attractive magnetic force () on the magnet to balance the gravity () of the floater.Moreover, there are three nonlinear relationships for the single-axis PEMS system in Fig. 1: 1)  is a nonlinear function with ℎ and  as, where  has two components as, (i) Permanent magnetic force (  ):   is the attractive magnetic force between the permanent magnet and the iron core of the electromagnet.And,   is a nonlinear decreasing function of ℎ 19 ; and (ii) Electromagnetic force (   ):   is the magnetic force between the permanent magnet and the electromagnet and is linear with  8 .As a sign conventional,  > 0 leads to attractive   , and vice versa.
2)  ℎ is a nonlinear function with ℎ as, Figure 3 visualizes the nonlinear characteristic of the ITR8307 distance sensor.It is observed that  ℎ is an increasing function of ℎ with a decreasing slope.3)  is a nonlinear function with  ∈ (0,1) at a given   as,  =   ().
The present work aims to propose the suitable compensation algorithm by analyzing the nonlinear mechanism associated with the L298N full-bridge PWM inverter.Besides,   and  ℎ have been well acknowledged and compensated by various approaches, such as the gain scheduling method 19 .

Theoretical modelling
Figure 4 shows the electric circuit of the L298N full-bridge PWM inverter produced by STMicroelectronics.1 and 2 are two TTL logic-level input ports and control the voltages at Nodes M and N, denoted as   and   , respectively.For example,   ≈ + when 1 = HIGH, and   ≈ 0 V when 1 = LOW.Besides, the electromagnet can be modelled as a serial combination of an inductor () and a resistor ().Denote the current in the electromagnet as   and the voltage across as   =   −   .
Hence, the relationship between   and   can be expressed as, where   is driven by the full-bridge PWM inverter and   varies continuously due to the inductor.Consequently, Table 2 lists out three modes of the full-bridge PWM inverter with the electromagnet.When 1 ≠ 2, the electromagnet is either positively charged (+C) or negatively charged (-C) depending on the sign of   ; when 1 = 2 , the electromagnet is effectively short-circuited with   ≈ 0  and its magnetic energy is dissipated during the discharging (DC) mode.Meanwhile, in Table 3, four quadrants 2 are defined according to   and   .Generally, the energy flows from the power supply into the electromagnet where the PWM signal can be explicitly expressed as, where  (= 1   ⁄ ) denotes the period of the PWM signal and  is any arbitrary integer.
Assuming the ideal full-bridge PWM inverter,   can be expressed by referring to Moreover, denote the equilibrium current at  =  as  1 and that at  = ( + ) as  2 .Solving Eq. ( 4) together with Eq. (7) gives the two equilibrium currents as, which indicates that   fluctuates between  1 and  2 .Since the time constant of the electromagnet (  ⁄ ≈ 5 to 9 ms to be fully charged or discharged in Fig. 9) is much larger than the PWM period (e.g.,  = 0.1 ms for 10 kHz), i.e.,   ⁄ ≪ 1, Eq. ( 8) can be approximated as, which indicates that ideal   in the electromagnet is linear with the duty cycle under a high   .
Nonlinear   −  characteristic Furthermore, since  is linear with   8 , the large sensitivity change about the zero current (around 20 times for 100 kHz) will undermine the control stability of the zero-power PEMS system.

Nonlinear Mechanism
This subsection analyzes the nonlinear   −  characteristic for the electromagnet driven by the L298N full-bridge PWM inverter.Firstly, into the voltage dynamics is investigated for  = 0.5 and two   , one ultra-high and one ultra-low.Secondly, DC, FC and BC modes are respectively characterized.Thirdly, the critical PWM frequency (  ) is modeled from the energy perspective.Fourthly, the sensitivity transition is analyzed w.r.t.  .Nevertheless, several remarks are addressed.

Voltage dynamics
   = 100 kHz (ultra-high) Figure 6 visualizes the whole-period variations of 1, 2,   ,   , and   for   = 100 kHz and  = 0.5 w.r.t. the time ().It is observed that the dynamic responses of 1 and 2 are so fast that the step change is completed in less than 0.1 μs, the rising and falling step changes of   and   are completed in 0.2 μs and 1 μs, respectively.However, by comparing Figs.6(a-b), there is a significant delay for around 2 μs between the falling step changes of   (or   ) and 1 (or 2), which is referred to as the turn-off delay in the datasheet of IC L298N produced by STMicroelectronics.Therefore, the 2 μs turn-off delay results in the DC mode as defined in Table 2 and highlighted in Figure 6(c).   = 100 Hz (ultra-low) Figure 7 visualizes the whole-period variation of   for   = 100 Hz and  = 0.5 w.r.t..Since the 2 μs turn-off delay is negligible compared with the 10 ms PWM period, the variations of 1, 2,   , and   are trivial and not shown.As defined in Table 3, the BC and FC modes are separated by the boundaries at   = ±12  in Fig. 7.

 Theoretical analysis
For  = 0.5, The magnetic energy () in the electromagnet is charged by the FC mode and dissipated by the DC and BC modes.And, the energy balance equation can be expressed as, where   ,   , and   denote the charged energy by the FC mode and the dissipated energies by the DC and BC modes, respectively.
Moreover, assuming  ≪   ⁄ ,   ,   , and   can be linearized and approximated as, where   ,   , and   denote the energy-charging rate by the FC mode, and the energy-dissipating rates by the DC and BC modes, respectively.
which indicates that the energy-dissipating rate by the DC mode is around twice as much as that by the BC mode.Therefore, the DC mode gradually replaces the BC mode with the increase of   .Table 5 lists out the estimated relative portions of the three modes by Eq. ( 14) and the associated fitting errors for   = 1, 2, 5, and 10 kHz and  = 0.5.For example, the fitting error (=0.18%) for   = 1 kHz is the difference between    ⁄ (=11.23%) and estimated    ⁄ (=11.05%).Hence, the first-order estimation for the magnetic energy is generally acceptable with the low fitting error.
For   = 1 kHz, the boundaries between the FC and BC regimes are almost continuous as a result of the negligible DC regime; on the contrary, for   = 100 kHz, the BC regime is heavily squeezed by the DC regime.Besides, the cyan dotted lines indicate the termination points of the FC regimes.
By referring to Tables 6-7, the variations of  can be characterized into three zones from the energy perspective: (i) Low-sensitivity zone (Stages I, II, and III, between the cyan dotted lines): Both +FC and -FC modes co-exist, which indicates that the magnetic energy stored during the FC mode (both positive and negative) is completely dissipated by the DC or BC mode.Therefore, the low-sensitivity zone is dominated by the energy cancellation between the FC and BC modes for   <   and between the FC and DC modes for   >   .
(ii) Sensitivity-transition zone (Stage IV, between the purple and cyan dotted lines): The 2 μs DC mode has a finite energy-dissipation capability and buffers the sensitivity transition until the capability is entirely consumed, as outlined by the purple and cyan dotted lines.Therefore, the sensitivity transition expends with the increase of   .
(iii) High-sensitivity zone (Stage V, outside the purple dotted lines): The expending of the FC regime and the shrinking of the BC regime occur simultaneously with the increase of the duty cycle.Therefore, the magnetic energy accumulates more effectively in the high-sensitivity zone than that in the lowsensitivity zone.

Remarks
This subsection investigates into the nonlinear mechanism of the   −  characteristic.The experimental results indicate the following remarks: 1) In micro level, it is observed that the IC L298N possesses a 2 μs turn-off delay that leads to the passive DC mode; in macro level, such a tiny delay results in the nonlinear   −  characteristic.
2) With the increase of   , the relative portion of the 2 μs DC mode is significantly amplified and replaces the BC mode entirely at  = 0.5 and   = 15.28 kHz, which is theoretically modelled from the energy perspective.
3) The low-sensitivity zone is dominated by the energy cancellation between the FC and BC modes for   <   and by the energy cancellation between the FC mode and the 2 μs DC mode for   >   .
4) The sensitivity transition results from the finite energy-dissipation capability of the 2 μs DC mode and expends with the increase of   .5) Though higher   can enhance the dynamic performance of the high-speed PEMS transportation system 15 , suitable compensation algorithm is necessary to take care of the nonlinear   −  characteristic especially for   >   .

Compensation algorithm
Since  is linear with   8 , this subsection proposes two piecewise linearization approaches to approximate the nonlinear   −  characteristic with the full-bridge PWM inverter.The compensation algorithm is verified by the current-step-change test over the sensitivity transition under two   w.r.t.  .
For   =  kHz <    Piecewise Linearization () = 1072.0− 590.6 Note that the sensitivity ratio between   (1) and   (2) is more than 8 times, which will undermine the control stability of the single-axis PEMS system over the medium sensitivity transition of the nonlinear   −  characteristic.Besides, the three intersection points among the three asymptotic lines are highlighted in for the two piecewise linearization approaches.
Practically, the control algorithm generates the target current (  * ), the compensation algorithm solves for the suitable duty cycle according to the piecewise linearization approach, and the real current (  ) is obtained in the electromagnet driven by the full-bridge PWM inverter.Hence, the two-piece linearization approach consists of Asymptotic Lines 1 and 2 as, And, the three-piece linearization approach consists of the three asymptotic lines as,  Moreover, Figures 15(d-f) compare the two piecewise linearization approaches by the averaged results (  ℎ ,   * , and  ).In Figs.15(d-e),  ℎ and   * with the two-piece linearization approach significantly overshoot around  = 1, 7, and 11 s , when bypassing Intersection Point C, (0.4025, −26.18 mA) and (0.5975, 26.18 mA).Meanwhile, in Fig. 15(f),  with the two-piece linearization approach significantly deviates from that with the three-piece counterpart around  = 13.0 s when bypassing Intersection Point B (0.5650, 6.18 mA) .Therefore, the three-piece linearization approach greatly outperforms the two-piece linearization approach with stronger robustness and smoother dynamics under the current-step-change test over the harsh sensitivity transition for   = 50 kHz >   .

Remarks
This subsection verifies the proposed compensation algorithm for the L298N full-bridge PWM inverter in the zero-power PEMS system.The experimental results indicate the following remarks: 1) The piecewise linearization approach with more pieces can better approximate the sensitivity transition of the nonlinear   −  characteristic.Besides, other fitting approaches, e.g., polynomial fitting, can serve the same purpose as well.
2) Under the current-step-change test over the sensitivity transition, the three-piece linearization approach successfully stabilizes the single-axis PEMS system with only a few milliampere current and demonstrates stronger robustness and smoother dynamics than the two-piece counterpart especially for   = 50 kHz >   .

Discussion
The present work investigates into the nonlinear mechanism of the L298N full-bridge PWM inverter, proposes the compensation algorithm, and realizes for the zero-power PEMS system with record-breaking   = 50 kHz.Based on the experimental observation and the theoretical analysis, we can draw the following conclusions: 1) Nonlinear mechanism: the 2 μs turn-off delay of the IC L298N leads to the DC mode that accounts for   and the sensitivity transition.
2) Compensation algorithm: the piecewise linearization approach overcomes the sensitivity transition and stabilizes the zero-power PEMS system especially for   >   .The lower fitting error, the stronger robustness and the smoother dynamics under the current-step-change test.
Therefore, in order to enhance the dynamic performance of the high-speed PEMS transportation, higher   can be realized by reducing (i) the turn-off delay of the full-bridge PWM inverter and (ii) the fitting error of the compensation algorithm.

Analog Signal Processor
where   and   are the two inputs to the 12-bit analog-digital converters of the STM32 controller.
where   * is the target current and    is the setpoint of   .Then,  is calculated by the compensation algorithm, and the L298N full-bridge PWM inverter drives the electromagnet.
where    is the setpoint of the target current.For example,    = 0 results in the zero-power PEMS system.
Nevertheless, Table 8                  Table 1.Physical properties of the single-axis PEMS system.
Table 2. Three modes of full-bridge PWM inverter with electromagnet.
Table 3.Four quadrants of full-bridge PWM inverter with electromagnet.
Table 4. Relative portions of three modes for various   and  = 0.5.
Table 5.Estimated relative portions of three modes and associated fitting errors for various   and  = 0.5.
Electric circuit of the analog signal processor.

Figure 1 .
Figure 1.Close-loop architecture of the single-axis PEMS system.

Figure 2 .
Figure 2. Hardware of the single-axis PEMS system, (a) stereoscopic view and (b) cross-sectional view for the electromagnet and the permanent magnet, (c) electronic devices, and (d) experiment rig.

Figure 3 .
Figure 3. Nonlinear characteristic of the ITR8307 distance sensor.

Figure 5 (( 11 )Figure 5
Figure 5(a) visualizes the variations of   w.r.t. for   = 0.1,1,10,100 kHz.It is observed that the characteristic   −  function (  ) is an odd function about Point (0.5, 0) as,   (1 − ) = −  ().(10) Also, the characteristic   −  curve deviates from the ideal line in Eq. (9), and the deviation increases with the increase of   .Moreover, define the sensitivity of   w.r.t. as,  =    ⁄ .(11) Figure 5(b) visualizes the variations of  w.r.t..It is observed that  at  = 0.5 decreases with the increase of   (as low as 70 mA for 100 kHz), while the four sensitivity curves converge to around 1700 mA for  < 0.3 and  > 0.7.

Figure 6 .
Figure 6.Whole-period variations of (a) 1 and 2, (b)   and   , and (c)   for   = 100 kHz and  = 0.5 w.r.t..The 2 μs turn-off delay due to the IC L298N can be clearly observed for both   and   in (b).Three modes are highlighted in (c) according to   .

Figure 7 .
Figure 7. Whole-period variation of   for   = 100 Hz and  = 0.5 w.r.t..The red dashed lines (  = ±12 V) indicate the boundaries between the FC and BC modes.

Figure 10 .
Figure 10.Contour of   w.r.t.log 10 (  ) and   ⁄ for  = 0.5.The green dotted curves (  = ±9.5 V) indicate the boundaries between the DC mode and the two charging modes.
) visualize the associated variations of .The purple dotted lines indicate the transition points with  = 1000 mA.It is observed that the three sensitivity curves converge to the high sensitivity outside the purple dotted lines; the three curves stay near the respective minimum values between the cyan dotted lines; meanwhile, the sensitivity transitions take place between the purple and cyan dotted lines.

Figure 11 .
Figure 11.(a-c) Phase diagrams of   w.r.t. and   ⁄ , and (d-f) variations of  w.r.t. for   = 1, 10, and 100 kHz.The cyan dotted lines indicate the termination points of the FC regimes, whereas the purple dotted lines indicate the transition points with  = 1000 mA.The dashed curve in (d&f) is the benchmark curve with   = 10 kHz.

Figure 12 (
Figure 12(a) visualizes the nonlinear   −  characteristic for   = 10 kHz <   together with three asymptotic lines.Asymptotic Line 1 is obtained by the two points at  = 0.50 and 0.51, Asymptotic Line 2 is obtained by the two points at  = 0.65 and 0.70, while Asymptotic Line 3 is obtained by the two points at  = 0.57 and 0.58.The numerical expressions of the three asymptotic lines are,

Figure 12 .
Figure 12.Variation of   for   = 10 kHz w.r.t.(a)  together with three asymptotic lines, and (b)

. ( 19 )Figure 13
Figure 13 visualizes the experimental results ( ℎ ,   * , and ) under the current-step-change test with the two piecewise linearization approaches for   = 10 kHz <   .In order to remove the random noise, the averaged value is obtained from the ten consecutive repeats.From Figs. 13(b&e), it is observed that   * with

Figure 14 .
Figure 14.Variation of   for   = 50 kHz w.r.t.(a)  together with three asymptotic lines, and (b)   * for the two piecewise linearization approaches.
Figure 15 visualizes the experimental results ( ℎ ,   * , and ) under the current-step-change test with the two piecewise linearization approaches for   = 50 kHz >   .From Figs. 15(b&e), it is observed that   * with noise varies within ±6 mA and averaged   * varies within ±2 mA for  ≥ 15 s.

Figure 16 .
Figure 16.Electric circuit of the analog signal processor.

Figure 17 .
Figure 17.Double-loop block diagram for the STM32 controller.

Figure 1 .
Figure 1.Close-loop architecture of the single-axis PEMS system.

Figure 2 .
Figure 2. Hardware of the single-axis PEMS system, (a) stereoscopic view and (b) cross-sectional view for the electromagnet and the permanent magnet, (c) electronic devices, and (d) experiment rig.

Figure 3 .
Figure 3. Nonlinear characteristic of the ITR8307 distance sensor.

Figure 6 .
Figure 6.Whole-period variations of (a) 1 and 2, (b)   and   , and (c)   for   = 100 kHz and  = 0.5 w.r.t..The 2 μs turn-off delay due to the IC L298N can be clearly observed for both   and   in (b).Three modes are highlighted in (c) according to   .

Figure 7 .
Figure 7. Whole-period variation of   for   = 100 Hz and  = 0.5 w.r.t..The red dashed lines (  = ±12 V) indicate the boundaries between the FC and BC modes.

Figure 10 .
Figure 10.Contour of   w.r.t.log 10 (  ) and   ⁄ for  = 0.5.The green dotted curves (  = ±9.5 V) indicate the boundaries between the DC mode and the two charging modes.

Figure 11 .
Figure 11.(a-c) Phase diagrams of   w.r.t. and   ⁄ , and (d-f) variations of  w.r.t. for   = 1, 10, and 100 kHz.The cyan dotted lines indicate the termination points of the FC regimes, whereas the purple dotted lines indicate the transition points with  = 1000 mA.The dashed curve in (d&f) is the benchmark curve with   = 10 kHz.

Figure 12 .
Figure 12.Variation of   for   = 10 kHz w.r.t.(a)  together with three asymptotic lines, and (b)   * for the two piecewise linearization approaches.

Figure 16 .
Figure 16.Electric circuit of the analog signal processor.

Figure 17 .
Figure 17.Double-loop block diagram for the STM32 controller.

Figures Figure 1
Figures

Figure 2 Hardware
Figure 2

Figure 4 Electric
Figure 4

Figure 6 Whole
Figure 6

Figure 7 Whole
Figure 7

Figure 9 Initial
Figure 9

Figure 17 Double
Figure 17

Table 1 .
Physical properties of the single-axis PEMS system.

Table 2 .
Three modes of full-bridge PWM inverter with electromagnet.

Table 3 .
Four quadrants of full-bridge PWM inverter with electromagnet.

Table 4 .
Relative portions of three modes for various   and  = 0.5.

Table 5 .
Estimated relative portions of three modes and associated fitting errors for various   and  = 0.5.

Table 8 .
lists out the four control parameters adopted in the present work.Control parameters for stm32 controller.

Table 8 .
Control parameters for stm32 controller.