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College of Automation & College of Artificial Intelligence，Nanjing University of Posts and Telecommunications，Nanjing 210023，P.R.China

（Received 15 May 2020；revised 20 July 2020；accepted 25 July 2020）

Abstract:For the advantages of easy realization and rapidly intelligent response，the one-cycle control was applied in five-phase six-leg switching power amplifier for magnetic bearing.This paper improves the one-cycle control considering resistance voltage drop and derives its mathematical models.The improved algorithm is compared with the former one.The simulation and experimental results show that the improved algorithm can effectively reduce the output current ripple，achieve good tracking of the given current，improve the control accuracy，and verify the effectiveness and superiority of the method.

Key words：one-cycle control；magnetic bearing；switching power amplifier；voltage drop

0 Introduction

Owing to the advantages of no friction-loss，low noise and long life，the magnetic bearing has wide application prospects in flywheel energy storage，high-speed machine tools and other fields［1-4］.As a key part of magnetic suspension system，the control accuracy of power amplifier directly affects the levitation performance of magnetic bearing.At present，the permanent magnet biased magnetic bearing［5］requires power amplifier to provide bi-directional current，which usually uses the H-bridge structure as its power topology.In the magnetic bearing systems of multiple degrees-of-freedom（DOFs），the multiple H-bridges structure［6-7］is commonly used，which takes the larger bulk and increases the complicacy and cost of the system.And the introduction of multi-leg switching power amplifier（SPA）topology［8-10］reduces the bulk and the cost of SPA.Li［11］proposed the topology of fivephase six-leg SPA（FPSL-SPA），which reduced the power switch amount by 40% compared to five-H-bridge power amplifiers.

At present，there are many kinds of control strategies in the research of multi-leg SPA of magnetic bearing.Some research used the space vector pulse width modulation（SVPWM）［12-14］to control the topology.However，the control algorithm was complex and had high demand for the controller.Li et al.［15］proposed proportion-integral-derivative（PID）regulator to control the topology of multi-leg SPA.Due to the existence of neutral leg，the control of 5-DOF was easy to be coupled，which was difficult to implement the decoupled control.In addition，the existing maximum current error method，sampling and holding method，node potential method，hysteresis control method and other control strategies［16-17］also have the problem of output coupling.Some scholars［18-19］studied the control of neutral legs in multi-leg structure，which reduced the coupling between each leg to some extent，but the coupled was still strong.Liu et al.［20］proposed the application of one-cycle control（OCC）in the topology of FPSL-SPA.The algorithm had the advantages of simple control，small computation，fast response，high control precision and versatility，and obtained control effects.At the same time，the coupling between 5-DOF was eliminated by fixing duty cycle of the neural leg for enabling separate control of each phase.However，there is little research on OCC of FPSL-SPA，and it is still in the basic research stage.In the process of deriving the mathematical model of OCC，the voltage drop caused by the coil resistance of magnetic bearing is ignored［20］.Therefore，the established model is not accurate enough to track the current accurately.

This paper aims at the above problems and improves the unipolar and bipolar OCC algorithm［20］considering the coil resistance voltage drop.The mathematic model of this algorithm is constructed and the implementation process is designed.The algorithms before and after improvement are compared，and emulational and experimental results are given out to verify the effectiveness of the algorithm.

1 Principle of FPSL-SPA

The topology of FPSL-SPA is shown in Fig.1，constituted by twelve switches and one DC voltage source，andMwith different subscripts is signals of different switches.In SPA，legs of phasesA—Eare loaded，the leg of phaseNis neutral，andLa—Leare five inductive loads，which are connected to the loaded legs and the neutral leg.In order to achieve independent control of each phase current，the duty cycle of neutral leg is fixed as 0.5［20］.

Fig.1 Five-phase six-leg switching power amplifier

The control schematic diagram of single DOF SPA system for magnetic bearing is shown in Fig.2.It is composed of controller，switching power amplifier and current sensor.

The controller is made up of digital signal processing（DSP），which is the main control chip.According to the principle of OCC，the duty cycle of each switch is calculated and the corresponding switching signal is generated.SPA is the main circuit，which realizes the dynamic control of coil current of magnetic bearing，and the current sensor is responsible for detecting the real-time current of coil.

Fig.2 Control diagram of single degree of freedom SPA

When the duty cycle of neutral leg is fixed as a constant，the operation process of each phase is similar，and here we discuss the operation process of loaded phaseA.

Fig.3 shows the circuit of the loaded phaseAand the neutral leg.To facilitate description，switching functionSaandSnare introduced as

Fig.3 Equivalent circuit of the loaded phase A and neutral leg

According to the condition of switches，there are four working states of phaseAas follows：

（1）WhenSa=0 andSn=0，the coil of phaseAis in the continuous current state；

（2）WhenSa=0 andSn=1，the coil of phaseAis in the adverse charging state；

（3）WhenSa=1 andSn=0，the coil of phaseAis in the forward charging state；

（4）WhenSa=1 andSn=1，the coil of phaseAis in the continuous current state.

From the above analysis，it can be seen that when the OCC of FPSL-SPA works，the phaseAconverts among four working states and enables the coil current of magnetic bearing to follow the change of the given signal.

2 Improvement of One-Cycle Control Method

According to the different switching status，OCC can be divided into unipolar OCC and bipolar OCC［20］，and the control process of the two OCCs is quite different.Using the unipolar OCC，the charging and discharging status of the one DOF magnetic bearing coil do not exist simultaneously in one switching cycle，but the phenomenon exists in the bipolar OCC.This paper optimizes the mathematical model of unipolar and bipolar OCC algorithm considering the influence of coil resistanceRof magnetic bearing，and comparatively analyzes the OCC algorithm before and after improvement.Because the measured conduction voltage drop of the switch is small，it is neglected in the analysis process.

2.1 Improvement of bipolar OCC algorithm

The upper and lower switches of the same leg are complementarily conducted by the bipolar OCC.The duty cycle of the upper switch of the loaded leg is calculated by the OCC algorithm.Pulse width modulation（PWM）waveform generation mode and the corresponding current variation are shown in Fig.4，whereMuais the signal of the upper switch of loaded leg of phaseA，Munthe signal of the upper switch of the neutral leg，PRD the value of the time reference period register in DSP，Dathe duty cycle ofMua，andDnthe duty cycle ofMun.From Fig.4，we can see that：

（1）During the periods fromt0tot1and fromt4tot5，the voltages at both ends of the coil areUdc，and the current is in charging state and keeps rising.It can be expressed as

whereLis the coil inductance，Rthe coil resistance，andithe coil current.

（2）During the periods fromt1tot2and fromt3tot4，the voltages at both ends of the coil are zero，and the current is in the continuous state and keeps decreasing slowly.It can be expressed as

Fig.4 Waveforms of bipolar OCC

（3）During the period fromt2tot3，the voltages at both ends of the coil are-Udc，which is in the state of discharging，and the current decreases.In this case，it can be expressed as

According to Eqs.（3）—（5），we can derive that

Thus，the value of current at the end of one cycle can be obtained that

Also it can be seen from Fig.4 that

Therefore，we have

whereTsis the switching cycle andDthe duty cycle of loaded leg.So，in order to make the current keep up with the given value in one cycle，it should be

Then there is

whereiREFis the given value of current andΔi（n）the deviation between the given current and the actual current att0moment.

Therefore，in order to ensure that the current follows the given value within one cycle under the bipolar OCC，the duty cycle of switchMuacan be expressed as

According to Eq.（12），the improved bipolar OCC algorithm is proposed.

Because the influence of coil resistance is neglected when deriving the mathematical model of bipolar OCC［20］，the model is not accurate enough.The duty cycle of switchMuabefore improvement of bipolar OCC can be expressed as

By substituting Eq.（13）into Eq.（9）and using the control algorithm before improvement，the current at the end of the cycle can be obtained as

It can be seen that when the influence of coil resistance is taken into account，there is still a deviation between the current obtained by the original OCC algorithm and the given one at the end of the switching cycle，and then the deviation of the current can be expressed as

2.2 Improvement of unipolar OCC algorithm

Different from the bipolar OCC algorithm，the unipolar OCC algorithm changes the switching sequence of the upper and lower switches of the neutral leg，and the control result is also different from that of the bipolar OCC algorithm.The PWM generation mode and current waveforms of unipolar OCC are shown in Fig.5.

Fig.5 Waveforms of unipolar OCC

（1）During the periods fromt1tot2and fromt3tot4，the voltages at both ends of the coil areUdc，and the current is in charging state and keeps rising.It can be expressed as

（2）During the periods fromt0tot1，fromt2tot3and fromt4tot5，the voltages at both ends of the coil are zero，and the current is in the continuous state and keeps decreasing slowly.It can be expressed as

According to Eqs.（16）and（17），we can derive that

Thus，the current at the end of one cycle can be obtained as

Also it can be seen from Fig.5 that

By substituting Eq.（20）into Eq.（19），we can obtain that

In order to make the current keep up with the given value，it should be

Then there is

Therefore，in order to ensure that the current follows the given value in one cycle using unipolar OCC，the duty cycle of switchMuacan be obtained as

According to Eq.（24），the improved unipolar OCC algorithm is proposed.

However，by the unipolar OCC algorithm before improvement，the duty cycle of switchMuacan be obtained as［20］

By substituting Eq.（25）into Eq.（21），the current at the end of a period can be obtained as

Therefore，there is still deviation between the current obtained by unipolar OCC algorithm before improvement and the given one at the end of the switching cycle，and then the deviation of the current can be expressed as

3 Simulation and Experiment

In order to verify the feasibility and superiority of the mathematical models of the two improved OCC algorithms，this paper builds a simulation model with MATLAB and carries out the experimental verification.In the simulation and experiments，the inductance coil is used as a load，which includesL=3.5 mH andR=1.0Ω.Both simulation and experimental conditions are set as：bus voltageUdc=20 V and switching frequencyfs=40 kHz.

3.1 Simulation and experiment of bipolar OCC

Fig.6（a）is the simulation result of the improved bipolar OCC for the input of sinusoidal wave with frequency of 400 Hz and amplitude of 0.8 V，which shows the output current tracking effect is good.In order to comparatively analyze the algorithm before and after improvement，simulation and experiment are carried out under the given currentiREF=1.2 A.Fig.6（b）is the simulation waveform of the algorithm before improvement.It can be seen that there is a deviation between the average tracking current and the given value.Therefore，the tracking is not accurate.Compared with the original algorithm，the improved algorithm considers the influence of the resistance voltage drop of the magnetic bearing coil on the current，so the mathematical model is more accurate.The simulation waveform of the improved bipolar OCC is shown in Fig.6（c）.It can be seen that the average tracking current of the improved algorithm is almost unbiased with the given one，which verifies the effectiveness of the improved algorithm.

Fig.7（a）is the corresponding experimental waveform.From the simulation and experimental results，it can be seen that the improved algorithm can track the current well，and the feasibility of the improved bipolar OCC algorithm is verified.The experimental results of Fig.7（b）and Fig.7（c）also prove the superiority of the improved bipolar OCC algorithm.

Fig.6 Simulation waveforms of bipolar OCC

Fig.7 Experimental waveforms of bipolar OCC

3.2 Simulation and experiment of unipolar OCC

Similar to the bipolar OCC，simulation and experiment of unipolar OCC are carried out，and waveforms are shown in Fig.8 and Fig.9，respectively.From Fig.8（a）and Fig.9（a），it can be seen that the improved algorithm can track the current well，which verifies the feasibility of the improved unipolar OCC algorithm.As can be seen from Fig.8（b），there is a deviation between the average tracking current and the given value when using the algorithm before improvement，and the tracking is not accurate.By comparing simulation results in Figs.8（b）and（c），it can be seen that the tracking waveform of the improved algorithm is more accurate than that before improvement，which verifies the effectiveness of the improved algorithm in this paper.The experimental results in Figs.9（b）and（c）also can prove the superiority of the improved unipolar OCC algorithm.

Fig.8 Simulation waveforms of unipolar OCC

Fig.9 Experimental waveforms of unipolar OCC

4 Conclusions

Considering the coil resistance voltage drop，this paper improves the unipolar OCC algorithm and the bipolar OCC algorithm，and optimizes the mathematical models of the two OCC algorithms.The theory，simulation and experiment show that the mathematical model of the improved algorithm is more accurate and the control effect is better than that before improvement.It is helpful to improve the control accuracy of the one-circle control algorithm in the FPSL-SPA of magnetic suspension bearing.

AcknowledgementThis work was supported by the National Science Foundation of China（No.51607096）.

AuthorDr.LIU Chengzi received the B.S.and Ph.D.degrees in electrical engineering from Nanjing University of Aeronautics and Astronautics（NUAA）in 2006 and 2015，respectively.In 2015，she joined the Department of Electrical Engineering，Nanjing University of Posts and Telecommunications（NJUPT）.Her current research interests include power electronics and magnetic bearing system.

Author contributionsDr.LIU Chengzi designed the study，revised and modified the manuscript.Ms.YAN Ting conducted the experimental analysis and wrote the manuscript.Dr.YANG Yan completed the modeling of the algorithm.Dr.LIU Zeyuan made a simulation analysis.All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.