Complementary Cooperation of Dual Power Circuits to Drive Active Magnetic Bearings

: A compact-design hybrid power ampliﬁer for AMB (active magnetic bearing) units, named as dual cooperative drive circuit (DC 2 ), is designed and veriﬁed by intensive experiments. DC 2 can operate under dual mode: either digital driving mode (DDM) or analog driving mode (ADM). By taking advantages of the complementary cooperation between DDM and ADM, the proposed DC 2 manifests its superiorities mainly in three aspects: (i) Compared to a traditional 2-level PWM (pulse width modulation) drive circuit, the improvement degree of reducing the amplitudes of current ripples is up to 62%. (ii) By taking advantages of mutually complementary cooperation between DDM and ADM, the steady-state errors of output current, which is exerted to the coils of AMBs, can be much reduced by DC 2 . (iii) DC 2 can provide a high-current slew rate so that the response of AMBs is fast and quick enough to regulate the high-speed rotor back to the neutral position within a very short time period below 0.40 ms. In addition, the proposed DC 2 is particularly suitable to be compliant with digital controllers and DSPs (digital signal processors) since the commands or drive sequences are all in DC (direct current) manner.


Introduction
Active magnetic bearings (AMBs) are able to levitate the interested objects in a fashion of no friction and no mechanical wear. Therefore, they are increasingly applied to the industries for which high speed and high precision are both required, such as high-speed compressors, turbo-machineries, high-precision machine tools, and flywheel energy storage systems. One of the most important components of a superior AMB unit is its drive circuit. Most of the drive circuits applied to AMBs are of analog type [1,2]. Analog drive circuits possess the merits of high linearity and low electromagnetic interference (EMI). However, the energy efficiency of analog drive circuits is inherently poor because a great percentage of the energy power is converted into heat at the corresponding driver integrated circuits (ICs). By contrast, the power loss is relatively limited for the digital drive circuits. As a consequence, a large number of studies on digital drive circuits for AMBs were proposed during the past decades [3][4][5][6][7][8][9]. The main shortcoming of digital drive circuits to AMBs is: severe current ripple induced. The most popular technique for current ripple reduction is the so called three-level pulse width modulation (PWM) scheme [8,9]. Nevertheless, the current ripple problem by digital drive circuits is still more serious than that by analog drive circuits.
Due to analog drive circuits and digital drive circuits having their own irreplaceable advantages, many types of switch-linear hybrid (SLH) power amplifiers were proposed recently [10][11][12][13][14]. The SLH amplifier is mainly composed by a set of linear power amplifier and a set of two-level switching power supply. However, the switching frequency of the power supply has to be much higher than the bandwidth of the linear power amplifier. As well known, high switching frequency results in serious

Digital Drive Mode (DDM)
From the viewpoint of signal processing, the corresponding circuit under DDM is nothing but a switching power amplifier. The operation states of the switching power amplifier are illustrated in Figure 2. Under DDM, solely one PWM signal, PWM#1, is required to synchronously switch on/off the two MOSFETs (metal-oxide-semiconductor field-effect transistors), marked as M#1 and M#2 in Figure 1. When the two MOSFETs are turned on, shown in Figure 2a, the switching power amplifier operates at charge state. On the other hand, as the two MOSFETs are both turned off, shown in

Digital Drive Mode (DDM)
From the viewpoint of signal processing, the corresponding circuit under DDM is nothing but a switching power amplifier. The operation states of the switching power amplifier are illustrated in Figure 2. Under DDM, solely one PWM signal, PWM#1, is required to synchronously switch on/off the two MOSFETs (metal-oxide-semiconductor field-effect transistors), marked as M#1 and M#2 in Figure 1. When the two MOSFETs are turned on, shown in Figure 2a, the switching power amplifier operates at charge state. On the other hand, as the two MOSFETs are both turned off, shown in Figure 2b, the power amplifier is at discharge state. Under DDM, the two MOSFETs are switched on/off synchronously so that the output current is able to track the desired output as fast as possible. Accordingly, AMBs can adjust the levitation force promptly against the external disturbance to the rotor. That is, DDM is suitable to be applied for the transient state of the rotor/AMB dynamics. As the rotor/AMB dynamics is at steady state, instead, ADM takes over to provide a relatively stable levitation force to the rotor, with a relatively mild ripple effect that makes the rotor wobble reciprocally.
Appl. Sci. 2018, 8,1270 3 of 17 Figure 2b, the power amplifier is at discharge state. Under DDM, the two MOSFETs are switched on/off synchronously so that the output current is able to track the desired output as fast as possible. Accordingly, AMBs can adjust the levitation force promptly against the external disturbance to the rotor. That is, DDM is suitable to be applied for the transient state of the rotor/AMB dynamics. As the rotor/AMB dynamics is at steady state, instead, ADM takes over to provide a relatively stable levitation force to the rotor, with a relatively mild ripple effect that makes the rotor wobble reciprocally.

Analog Drive Mode (ADM)
The corresponding operation states under ADM are depicted in Figure 3. PWM#1 is applied to adjust the drain-source voltage of M#2, while PWM#2 is utilized to regulate the AMB coil current. Under ADM, M#2 always operates within saturation region so that its characteristics can be analogous to a voltage-controlled power transducer. As long as the MOSFET operates within its saturation region, the output current is strongly dependent of the gate-source voltage, GS V .
Accordingly, the digital control sequence to adjust the output current of DC 2 , i.e., PWM#2, is converted into analog fashion via a digital-to-analog converter (DAC), shown in Figure 1, prior to feeding to M#2. It is noted that the higher the duty cycle of PWM#1, the more energy consumed by M#2. That is, a trade-off for real-time adjustment strategy is required to be included. As a result, the duty cycle of M#1 is regulated by PWM#1 to minimize, to some extent, the energy consumption by M#2.

Analog Drive Mode (ADM)
The corresponding operation states under ADM are depicted in Figure 3. PWM#1 is applied to adjust the drain-source voltage of M#2, while PWM#2 is utilized to regulate the AMB coil current. Under ADM, M#2 always operates within saturation region so that its characteristics can be analogous to a voltage-controlled power transducer. As long as the MOSFET operates within its saturation region, the output current is strongly dependent of the gate-source voltage, V GS . Accordingly, the digital control sequence to adjust the output current of DC 2 , i.e., PWM#2, is converted into analog fashion via a digital-to-analog converter (DAC), shown in Figure 1, prior to feeding to M#2. It is noted that the higher the duty cycle of PWM#1, the more energy consumed by M#2. That is, a trade-off for real-time adjustment strategy is required to be included. As a result, the duty cycle of M#1 is regulated by PWM#1 to minimize, to some extent, the energy consumption by M#2. Appl. Sci. 2018, 8, 1270 3 of 17 Figure 2b, the power amplifier is at discharge state. Under DDM, the two MOSFETs are switched on/off synchronously so that the output current is able to track the desired output as fast as possible. Accordingly, AMBs can adjust the levitation force promptly against the external disturbance to the rotor. That is, DDM is suitable to be applied for the transient state of the rotor/AMB dynamics. As the rotor/AMB dynamics is at steady state, instead, ADM takes over to provide a relatively stable levitation force to the rotor, with a relatively mild ripple effect that makes the rotor wobble reciprocally.

Analog Drive Mode (ADM)
The corresponding operation states under ADM are depicted in Figure 3. PWM#1 is applied to adjust the drain-source voltage of M#2, while PWM#2 is utilized to regulate the AMB coil current. Under ADM, M#2 always operates within saturation region so that its characteristics can be analogous to a voltage-controlled power transducer. As long as the MOSFET operates within its saturation region, the output current is strongly dependent of the gate-source voltage, GS V .
Accordingly, the digital control sequence to adjust the output current of DC 2 , i.e., PWM#2, is converted into analog fashion via a digital-to-analog converter (DAC), shown in Figure 1, prior to feeding to M#2. It is noted that the higher the duty cycle of PWM#1, the more energy consumed by M#2. That is, a trade-off for real-time adjustment strategy is required to be included. As a result, the duty cycle of M#1 is regulated by PWM#1 to minimize, to some extent, the energy consumption by M#2.

Pulse Width Modulation-Tuning Processors (PWM-TPs)
Since DC 2 conducts under two distinct operation modes, i.e., DDM and ADM, two pulse width modulation-tuning processors (PWM-TPs) are required to appropriately tune the output current to the AMB unit. In addition, a switch is applied to determine the timing of change between these two modes, i.e., DDM and ADM. The schematic diagram for the PWM-TPs of DC 2 is shown in Figure 4. i re f AMB is the desired current applied to the AMB coils while u re f AMB is the corresponding voltage command. On the other hand, i AMB is the actual current at the AMB coils, andû AMB is the converted voltage with respect to the measured current by the equipped current sensor (CS) upon i AMB . If the error between u re f AMB and u AMB is significant, DC 2 operates under DDM so that the output current by DC 2 ,û AMB , can quickly catch up the desired quantity, u re f AMB . On the contrary, once the output current gets close to the desired quantity, DC 2 operates under ADM to generate a relatively stable current. The design procedure of PWM-TPs is shown in Figure 5.

Pulse Width Modulation-Tuning Processors (PWM-TPs)
Since DC 2 conducts under two distinct operation modes, i.e., DDM and ADM, two pulse width modulation-tuning processors (PWM-TPs) are required to appropriately tune the output current to the AMB unit. In addition, a switch is applied to determine the timing of change between these two modes, i.e., DDM and ADM. The schematic diagram for the PWM-TPs of DC 2 is shown in Figure 4.  The steps to determine the PI gains for DDM and ADM are described by Remark#1 and Remark#2 respectively as follows:

Remark 1. The steps to tune the PI gains for DDM
Step 1 Based on the step-wise reference shown in Section 4, the gain, P K , with respect to proportional action at PI controller is tuned from zero to infinity.
Step 2 As P K is tuned from zero to 300 = * P K , the steady-state errors are all decreased gradually.
By contrast, as P K is tuned from * P K to infinity, the corresponding steady-state errors are all increased gradually.
Step 3 It is concluded that 300 = * P K is the gain finally chosen.
Step 4 Secondly, the gain, I K , with respect to the integral action at PI controller is tuned from zero to infinity.
Step 5 It is observed that the steady-state errors are altered from negative values to positive values as I K from zero to infinity while 300 = * P K is fixed. Therefore, 1000 = * I K is chosen since the corresponding steady-state errors are approximated to zero.

Remark 2. The steps to tune the PI gains for ADM
Step 1 Based on the step-wise reference shown in Section 4, the gain, P K , with respect to proportional action at PI controller is tuned from zero to infinity.
Step 2 As P K is tuned from zero to 100 = P K , the steady-state errors are all decreased gradually. By contrast, as P K is tuned from P K to infinity, the corresponding steady-state errors are all increased gradually. The steps to determine the PI gains for DDM and ADM are described by Remark#1 and Remark#2 respectively as follows:

Remark 1. The steps to tune the PI gains for DDM
Step 1 Based on the step-wise reference shown in Section 4, the gain, K P , with respect to proportional action at PI controller is tuned from zero to infinity.
Step 2 As K P is tuned from zero to K * P = 300, the steady-state errors are all decreased gradually. By contrast, as K P is tuned from K * P to infinity, the corresponding steady-state errors are all increased gradually.
Step 3 It is concluded that K * P = 300 is the gain finally chosen.
Step 4 Secondly, the gain, K I , with respect to the integral action at PI controller is tuned from zero to infinity.
Step 5 It is observed that the steady-state errors are altered from negative values to positive values as K I from zero to infinity while K * P = 300 is fixed. Therefore, K * I = 1000 is chosen since the corresponding steady-state errors are approximated to zero.

Remark 2. The steps to tune the PI gains for ADM
Step 1 Based on the step-wise reference shown in Section 4, the gain, K P , with respect to proportional action at PI controller is tuned from zero to infinity.
Step 2 As K P is tuned from zero toK P = 100, the steady-state errors are all decreased gradually.
By contrast, as K P is tuned fromK P to infinity, the corresponding steady-state errors are all increased gradually. Step 3 Different from the DDM, asK P ≥ 95, the tracking errors of output current would be more than the threshold such that ADM is cut off and then switched to DDM. In this case, we finally choose K * P = 90 to prevent any potential switch to DDM.
Step 4 Secondly, the gain, K I , with respect to the integral action at PI controller is tuned from zero to infinity.
Step 5 It is observed that the steady-state errors are altered from negative values to positive values as K I from zero to infinity while K * P = 90 is fixed. Therefore, K * I = 300 is chosen since the corresponding steady-state errors are approximated to zero. Step 3 Different from the DDM, as 95  P K , the tracking errors of output current would be more than the threshold such that ADM is cut off and then switched to DDM. In this case, we finally choose 90 = * P K to prevent any potential switch to DDM.
Step 4 Secondly, the gain, I K , with respect to the integral action at PI controller is tuned from zero to infinity.
Step 5 It is observed that the steady-state errors are altered from negative values to positive values as I K from zero to infinity while 90 = * P K is fixed. Therefore, 300 = * I K is chosen since the corresponding steady-state errors are approximated to zero.

PWM-TP for DDM
One of the major merits of DC 2 under DDM is: relatively high current slew rate of output current. In other words, DDM is suitable to be employed to the transient state of rotor/AMB dynamics. For there always exists high-frequency current ripples under DDM, the derivative action of a PID (Proportional-Integral-Derivative) controller would certainly downgrade the quality of output current. Therefore, instead a PI (Proportional-Integral) controller, shown in Figure 6, is employed to tune the duty ratio of PWM#1.

PWM-TP for DDM
One of the major merits of DC 2 under DDM is: relatively high current slew rate of output current. In other words, DDM is suitable to be employed to the transient state of rotor/AMB dynamics. For there always exists high-frequency current ripples under DDM, the derivative action of a PID (Proportional-Integral-Derivative) controller would certainly downgrade the quality of output current. Therefore, instead a PI (Proportional-Integral) controller, shown in Figure 6, is employed to tune the duty ratio of PWM#1.

PWM-TP for ADM
Owing to one of the main shortcomings of DDM: inevitable fluctuation of the supplied voltage downgrades the quality of the output current to AMB coils, another operation mode, i.e., ADM, is joined to provide a relatively stable output current to regulate the steady state of rotor position. Under ADM, M#2 operates, within saturation region, to work like a voltage-controlled power transistor. The schematic diagram for PWM-TP under ADM is depicted in Figure 7. Within the PWM-TP under ADM, a low pass filter (LPF) is employed to eliminate the current ripples lurked in the feedback signal. As a consequence, the degree of fluctuation of the feedback signal is much decreased from "intermediate level" down to "mild or low level". Afterwards, the error between AMB u and AMB û is fed to a PI controller to adjust the desired output current precisely. In addition, this error is also applied for fine-tuning the duty ratio of PWM#1 such that the energy consumption can be reduced. Furthermore, there are two compensators, i.e.,

Simulations
In this section, intensive simulations, via MATLAB/Simulink, for PWM-TPs to track the desired output are undertaken to verify the feasibility and efficacy of DC 2 . The Simulink model of

PWM-TP for ADM
Owing to one of the main shortcomings of DDM: inevitable fluctuation of the supplied voltage downgrades the quality of the output current to AMB coils, another operation mode, i.e., ADM, is joined to provide a relatively stable output current to regulate the steady state of rotor position. Under ADM, M#2 operates, within saturation region, to work like a voltage-controlled power transistor. The schematic diagram for PWM-TP under ADM is depicted in Figure 7. Within the PWM-TP under ADM, a low pass filter (LPF) is employed to eliminate the current ripples lurked in the feedback signal. As a consequence, the degree of fluctuation of the feedback signal is much decreased from "intermediate level" down to "mild or low level". Afterwards, the error between u AMB andû AMB is fed to a PI controller to adjust the desired output current precisely. In addition, this error is also applied for fine-tuning the duty ratio of PWM#1 such that the energy consumption can be reduced. Furthermore, there are two compensators, i.e., K V b and K I b shown in Figure 6, are included to adjust the duty ratios of PWM#1 and PWM#2 respectively.

PWM-TP for DDM
One of the major merits of DC 2 under DDM is: relatively high current slew rate of output current. In other words, DDM is suitable to be employed to the transient state of rotor/AMB dynamics. For there always exists high-frequency current ripples under DDM, the derivative action of a PID (Proportional-Integral-Derivative) controller would certainly downgrade the quality of output current. Therefore, instead a PI (Proportional-Integral) controller, shown in Figure 6, is employed to tune the duty ratio of PWM#1.

PWM-TP for ADM
Owing to one of the main shortcomings of DDM: inevitable fluctuation of the supplied voltage downgrades the quality of the output current to AMB coils, another operation mode, i.e., ADM, is joined to provide a relatively stable output current to regulate the steady state of rotor position. Under ADM, M#2 operates, within saturation region, to work like a voltage-controlled power transistor. The schematic diagram for PWM-TP under ADM is depicted in Figure 7. Within the PWM-TP under ADM, a low pass filter (LPF) is employed to eliminate the current ripples lurked in the feedback signal. As a consequence, the degree of fluctuation of the feedback signal is much decreased from "intermediate level" down to "mild or low level". Afterwards, the error between AMB u and AMB û is fed to a PI controller to adjust the desired output current precisely. In addition, this error is also applied for fine-tuning the duty ratio of PWM#1 such that the energy consumption can be reduced. Furthermore, there are two compensators, i.e.,

Simulations
In this section, intensive simulations, via MATLAB/Simulink, for PWM-TPs to track the desired output are undertaken to verify the feasibility and efficacy of DC 2 . The Simulink model of

Simulations
In this section, intensive simulations, via MATLAB/Simulink, for PWM-TPs to track the desired output are undertaken to verify the feasibility and efficacy of DC 2 . The Simulink model of DC 2 is depicted in Figure 8. The corresponding environmental setup for the simulations undertaken later is listed in Table 1. The reference signal to track, i.e., the desired output, is a step-wise function while the applied voltage to drive the AMB coil, V P , is set as 40 VDC. i re f AMB is the desired current applied to the AMB coils while u re f AMB is the corresponding voltage command. The constant resistance with respect to u re f AMB over i re f AMB , i.e., R iu in Figure 5, is set as 1 Ω. On the other hand, i AMB is the actual current at the AMB coils, andû AMB is the converted voltage with respect to the measured current by the equipped current sensor (CS) upon i AMB . The constant resistance for i AMB converted toû AMB is set as 1 Ω. In addition, a switch is applied to determine the timing of change between these two modes, i.e., DDM and ADM. As long as the error between i AMB and i re f AMB is over 0.03 A, then DC 2 is controlled by PWM-TP under DDM. On the contrary, as the error between i AMB and i re f AMB is less than or equal to 0.03 A, then DC 2 is controlled by PWM-TP under ADM.
Appl. Sci. 2018, 8, 1270 7 of 17 DC 2 is depicted in Figure 8. The corresponding environmental setup for the simulations undertaken later is listed in      The desired output, i.e., i re f AMB shown in Figure 8, to be tracked is nothing but a sequence of step-wise signal whose quantity in ampare is limited by [0, 1.5 A]. The time duration of each step is set as 5 ms. The output currents and the tracking errors with respect to output current by two-state Power Amplifier (PA), i.e., DDM, three-state PA, DC 2 and ADM, are shown in Figure 9. Unfortunately, the peak-to-peak amplitudes of current ripples under DDM are up to 30 mA. Since the current ripples are too severe so that the regulation performance of AMB upon the position deviation of rotor is much downgraded, the other operation mode, namely ADM, has to be joined to reduce the aforesaid current ripples. It is noted that, under ADM, M#2 is ensured to operate in saturation region. As M#2 operates in its saturation region, the higher the gate-source voltage, V GS , the steeper the slope of the curve of i D versus V DS . Therefore, under ADM, once V DS of M#2 is altered during a very short time interval which is taken by the on-off action of M#1, the lower i D is induced, the minor current ripple would be carried by the resulted output current. From the simulation results under ADM, when i AMB is 1.5 A, the amplitude of current ripples is about 7.2 mA. Moreover, the amplitude of current ripples is down to 2.1 mA as i AMB is reduced to 0.01 A. On the other hand, it can be observed that steady-state errors are within [−10 mA, +5 mA] under DDM. In comparison, the steady-state errors are, all the time, within the range [0, +8 mA] under ADM. This range is about half of that under DDM. Besides, it is obvious that there are a few spikes induced at three-state PA while no any spikes at DC 2 .
Appl. Sci. 2018, 8, 1270 8 of 17 The desired output, i.e., ref AMB i shown in Figure 8, to be tracked is nothing but a sequence of step-wise signal whose quantity in ampare is limited by [ 1.5A 0, ]. The time duration of each step is set as ms 5 . ms 5 . The output currents and the tracking errors with respect to output current by two-state Power Amplifier (PA), i.e., DDM, three-state PA, DC 2 and ADM, are shown in Figure 9. Unfortunately, the peak-to-peak amplitudes of current ripples under DDM are up to mA 30 . Since the current ripples are too severe so that the regulation performance of AMB upon the position deviation of rotor is much downgraded, the other operation mode, namely ADM, has to be joined to reduce the aforesaid current ripples. It is noted that, under ADM, M#2 is ensured to operate in saturation region. As M#2 operates in its saturation region, the higher the gate-source voltage, GS  Since DDM and ADM possess their own particular merits, DDM and ADM are combined together to constitute the proposed DC 2 . As a matter of fact, taking advantage of joint DDM and ADM, the tracking of DC 2 can be further reduced to be within [ ]. To be more explicitly compared, the tracking errors (TEs) by two-state power amplifier (PA), three-state PA and DC 2 are summarized in Table 2. As the applied current to AMB coils, AMB i , is 0.01A, the tracking Since DDM and ADM possess their own particular merits, DDM and ADM are combined together to constitute the proposed DC 2 . As a matter of fact, taking advantage of joint DDM and ADM, the tracking of DC 2 can be further reduced to be within [−1.5 mA, +2 mA]. To be more explicitly compared, the tracking errors (TEs) by two-state power amplifier (PA), three-state PA and DC 2 are summarized in Table 2. As the applied current to AMB coils, i AMB , is 0.01 A, the tracking error by DC 2 is below 15% while the tracking error by two-state PA is up to 133.6%. As i AMB is much greater than 0.01 A, the tracking by DC 2 is reduced to 0.3% or below. On the other hand, the comparison, in terms of the peak-to-peak amplitude of current ripple (PPACR), by a two-state PA, three-state PA and DC 2 is concluded by Table 3. During the k th time interval, PPACR is defined as the difference between max{i AMB (t), (k − 1)T S < t ≤ kT S } and min{i AMB (t), (k − 1)T S < t ≤ kT S } where k ∈ N, T S is the time period and i AMB (t) the actual current applied to coils at AMB. The current ripples by two-state PA are over 27 mA. Compared with two-state PA, the improvement degree of reducing peak-to-peak amplitude of current ripples by DC 2 is up to 94% as i AMB is by 0.01 A, i.e., low drive current, under steady-state circumstance. On the contrary, as i AMB is by 1.5 A, i.e., high drive current, under transient-state circumstance, the improvement degree of reducing the peak-to-peak amplitude of current ripples by DC 2 is about 74%. Besides, the power dissipations at MOSFETs by two-state PA, three-state PA and DC 2 are summarized in Table 4. The power dissipation at MOSFETs is mainly determined by the applied current, on-resistance of MOSFET and the time duration at on-state, i.e., duty cycle. It is noted that the duty cycle under ADM are normally below 20% while it is usually above 75% under DDM. Therefore, though the on-resistance of M#2 under ADM is about twice of that under DDM, the power dissipation at M#2 under ADM is still lower than that under DDM. To sum up, owing to the mutually complementary cooperation between DDM and ADM, the proposed DC 2 at least has three outstanding advantages such as: (i) the slew rate of output current is high, (ii) the tracking errors can be greatly reduced, and (iii) the current ripples are significantly suppressed.   Note: TE is the abbreviation of "tracking error". TE is the error between the desired current and the actual output current in RMS sense. Table 3. Peak-to-peak amplitudes of current ripples.  Note: 1. "PPACR" is the abbreviation of "peak-to-peak amplitude of current ripple"; 2. "Imp." is the abbreviation of "improvement".   Note: 1. "PD" is the abbreviation of "power dissipation". 2. "Imp." is the abbreviation of "improvement". .
The simulation results to track the sinusoidal wave by two-state power amplifier (PA), three-state PA and DC 2 are shown in Figures 10 and 11. The reference signal in Figure 10 is a sinusoidal wave with 50 Hz and amplitude 0.7 A while the reference signal in Figure 11 is a sinusoidal wave with 500 Hz amplitude 0.1 A. It is noted that the current ripples are almost eliminated by DC 2 (see Figures 10i and  11ii). On the other hand, the tracking errors and the current ripples by DC 2 are very close to those by the three-state PA either for low-speed case or for high-speed case. To sum up, the performance to track a sinusoidal wave by DC 2 is superior to that by the three-state PA.
The simulation results to track the sinusoidal wave by two-state power amplifier (PA), three-state PA and DC 2 are shown in Figures 10 and 11. The reference signal in Figure 10 is a sinusoidal wave with 50 Hz and amplitude 0.7 A while the reference signal in Figure 11 is a sinusoidal wave with 500 Hz amplitude 0.1 A. It is noted that the current ripples are almost eliminated by DC 2 (see Figures 10i and 11ii). On the other hand, the tracking errors and the current ripples by DC 2 are very close to those by the three-state PA either for low-speed case or for high-speed case. To sum up, the performance to track a sinusoidal wave by DC 2 is superior to that by the three-state PA.

Experiments
In this section, a few experiments are undertaken to examine the feasibility and capability of the proposed DC 2 in terms of tracking capability and ripple suppression. The photographs and the corresponding layout of DC 2 are shown in Figure 12. Besides, the experimental test rig, shown in Figure 13, is constructed. The schematic diagram of equipped facilities to conduct the corresponding experiments is depicted in Figure 14. The load to drive, with respect to DC 2 , is any one of the axial AMBs of the 5-DOF rotor/active magnetic bearing (Rotor/AMB) unit. The electrical resistance and the nominal inductance of this AMB coil are 0.9 Ω and 7 mH respectively. Two

Experiments
In this section, a few experiments are undertaken to examine the feasibility and capability of the proposed DC 2 in terms of tracking capability and ripple suppression. The photographs and the corresponding layout of DC 2 are shown in Figure 12. Besides, the experimental test rig, shown in Figure 13, is constructed. The schematic diagram of equipped facilities to conduct the corresponding experiments is depicted in Figure 14. The load to drive, with respect to DC 2 , is any one of the axial AMBs of the 5-DOF rotor/active magnetic bearing (Rotor/AMB) unit. The electrical resistance and the nominal inductance of this AMB coil are 0.9 Ω and 7 mH respectively. Two sets of direct current (DC) power supplies are employed. The first set, ITech IT6332A, is employed to provide +5 VDC/+15 VDC for DC 2 . The other set, Keithley 2268-40-21, is applied to drive the AMB coils by +40 VDC. The DSP chip, Texas Instruments TMS320F28335, in which the PWM-TPs are embedded, receives the signal from the CS (Current Sensor),û AMB , and then generates the two control commands, i.e., PWM#1 and PWM#2, to DC 2 .      For the experiments undertaken, the desired output, a sequence of step-wise reference, is preset at the DSP chip. The amplitude and the duty ratio of the step-wise reference are set as 1A and 50% respectively. The tracking performances under DC 2 , DDM and ADM are shown in Figure  15. From Figure 15, it is observed that at the beginning of tracking the step-wise signal, i.e., starting 2.5 ms to 2.8 ms, the DC 2 operates by DDM. Under DDM, the duty ratio of PWM#1 (with respect to the Observation Point OP#1) is adjusted to its maximum, i.e., 95%, so that the output current is able to track the desired output as fast as possible. This is the major advantage of DDM at transient state of the rotor position regulation dynamics. However, M#1 and M#2 operate both in on-off fashion synchronously under DDM which unfortunately induces severe current ripples embedded in the output current. In comparison, unlike the digital control command to M#2 under DDM (with respect to OP#2′), the control command to M#2 under ADM (with respect to OP#2) is in analog manner. Accordingly, as soon as the output current to the axial AMB coil gets close to the desired quantity, DC 2 is switched to ADM. As DC 2 is switched from DDM to ADM, the mechanism of M#2, which is in on-off fashion under DDM, is changed to the voltage-controlled fashion to refine the output current more precisely. In addition, at steady state of the rotor position regulation dynamics, compared with the duty cycle of PWM#1 under DDM (with respect to OP#3′), the duty cycle of PWM#1 under ADM (with respect to OP#3) is reduced to a relatively low degree. As a result, the fluctuation of output current is greatly limited under ADM (see OP#4) so that the amplitudes of current ripples can be certainly much reduced. For the experiments undertaken, the desired output, a sequence of step-wise reference, is preset at the DSP chip. The amplitude and the duty ratio of the step-wise reference are set as 1 A and 50% respectively. The tracking performances under DC 2 , DDM and ADM are shown in Figure 15. From Figure 15, it is observed that at the beginning of tracking the step-wise signal, i.e., starting 2.5 ms to 2.8 ms, the DC 2 operates by DDM. Under DDM, the duty ratio of PWM#1 (with respect to the Observation Point OP#1) is adjusted to its maximum, i.e., 95%, so that the output current is able to track the desired output as fast as possible. This is the major advantage of DDM at transient state of the rotor position regulation dynamics. However, M#1 and M#2 operate both in on-off fashion synchronously under DDM which unfortunately induces severe current ripples embedded in the output current. In comparison, unlike the digital control command to M#2 under DDM (with respect to OP#2 ), the control command to M#2 under ADM (with respect to OP#2) is in analog manner. Accordingly, as soon as the output current to the axial AMB coil gets close to the desired quantity, DC 2 is switched to ADM. As DC 2 is switched from DDM to ADM, the mechanism of M#2, which is in on-off fashion under DDM, is changed to the voltage-controlled fashion to refine the output current more precisely. In addition, at steady state of the rotor position regulation dynamics, compared with the duty cycle of PWM#1 under DDM (with respect to OP#3 ), the duty cycle of PWM#1 under ADM (with respect to OP#3) is reduced to a relatively low degree. As a result, the fluctuation of output current is greatly limited under ADM (see OP#4) so that the amplitudes of current ripples can be certainly much reduced. To be explicitly compared, the corresponding tracking performances under DDM, ADM and DC 2 are depicted by Figure 16. It is observed that the rise time under ADM, i.e., 1.08 ms from 0 A to A' , is much longer than that under DDM or DC 2 , i.e., 0.40ms from 0 A to A . That is, the response of ADM is relatively slow so that ADM is inappropriate to be applied at transient state.
On the other hand, the amplitude of the peak-to-peak current ripples under DDM is about 50 mA while it is merely 19 mA under either ADM or DC 2 . That is, the improvement degree of reducing the peak-to-peak amplitude of current ripples by DC 2 is about 62%. This also implies that DDM is inappropriate to be applied at steady state. In comparison to DDM, DC 2 additionally possesses the merit of significantly limited current ripples. To sum up, DC 2 can provide a relatively stable current to drive AMB coils with a much shorter rise time and relatively minor ripples. To be explicitly compared, the corresponding tracking performances under DDM, ADM and DC 2 are depicted by Figure 16. It is observed that the rise time under ADM, i.e., 1.08 ms from A 0 to A , is much longer than that under DDM or DC 2 , i.e., 0.40 ms from A 0 to A. That is, the response of ADM is relatively slow so that ADM is inappropriate to be applied at transient state. On the other hand, the amplitude of the peak-to-peak current ripples under DDM is about 50 mA while it is merely 19 mA under either ADM or DC 2 . That is, the improvement degree of reducing the peak-to-peak amplitude of current ripples by DC 2 is about 62%. This also implies that DDM is inappropriate to be applied at steady state. In comparison to DDM, DC 2 additionally possesses the merit of significantly limited current ripples. To sum up, DC 2 can provide a relatively stable current to drive AMB coils with a much shorter rise time and relatively minor ripples.

Conclusions
A novel compact hybrid power amplifier applied for active magnetic bearings (AMBs), named as dual cooperative drive circuit (DC 2 ), is proposed and verified by intensive simulations and experiments. DC 2 can conduct under any of the two operation modes: digital driving mode (DDM) and analog driving mode (ADM). As the rotor/AMB dynamics is at transient state, DC 2 operates under DDM to provide quick energization to AMBs by higher current slew rate. On the other hand, as the rotor/AMB dynamics is at steady state, DC 2 operates under ADM to provide an output current with relatively mild current ripples so that the steady state of the rotor can be well-retained, without significant reciprocated fluctuation away from the neutral position. It is noted that all the signals fed to DC 2 are of pulse width modulation (PWM) fashion. As a matter of fact, DC 2 can be directly driven by a digital signal processing (DSP) chip. From the experimental results, compared to a traditional two-level power amplifier, the improvement degree of reducing the amplitudes of peak-to-peak current ripples is up to 62% by DC 2 . From the viewpoint of response, the rise time by DC 2 is much shorter than that under ADM. By taking advantages of complementary cooperation between DDM and ADM, the proposed DC 2 manifests its superiorities on both high current slew rate, low tracking error, and significantly suppressed current ripples.  Acknowledgments: The authors would like to express their appreciations to Ministry of Science and Technology (Taiwan).

Conflicts of Interest:
The authors declare no conflicts of interest.

Conclusions
A novel compact hybrid power amplifier applied for active magnetic bearings (AMBs), named as dual cooperative drive circuit (DC 2 ), is proposed and verified by intensive simulations and experiments. DC 2 can conduct under any of the two operation modes: digital driving mode (DDM) and analog driving mode (ADM). As the rotor/AMB dynamics is at transient state, DC 2 operates under DDM to provide quick energization to AMBs by higher current slew rate. On the other hand, as the rotor/AMB dynamics is at steady state, DC 2 operates under ADM to provide an output current with relatively mild current ripples so that the steady state of the rotor can be well-retained, without significant reciprocated fluctuation away from the neutral position. It is noted that all the signals fed to DC 2 are of pulse width modulation (PWM) fashion. As a matter of fact, DC 2 can be directly driven by a digital signal processing (DSP) chip. From the experimental results, compared to a traditional two-level power amplifier, the improvement degree of reducing the amplitudes of peak-to-peak current ripples is up to 62% by DC 2 . From the viewpoint of response, the rise time by DC 2 is much shorter than that under ADM. By taking advantages of complementary cooperation between DDM and ADM, the proposed DC 2 manifests its superiorities on both high current slew rate, low tracking error, and significantly suppressed current ripples.