A High-Voltage and Low-Noise Power Amplifier for Driving Piezoelectric Stack Actuators

In this paper, based on the principles of general operational amplifiers, a high-voltage operational amplifier is developed. Considering the influences of piezoelectric stack actuators on the circuit, a novel structure using the high-voltage operational amplifier as a noninverting amplifier is proposed. Because of the simple circuit principles and the voltage feedback control structure, the proposed power amplifier has the advantages of low noise and small size, and it can be realized by discrete electric elements easily. In the application of precision positioning, a power amplifier using the proposed circuit principles for driving piezoelectric stack actuators is designed, simulated, and tested. The simulated results show that the proposed power amplifier could conform to the theory of the circuit. The experimental results show that the designed power amplifier conforms to the simulation, the bandwidth of the power amplifier is about 57 kHz, and the ripple of the power amplifier is less than 2 mV. Furthermore, the output of the proposed power amplifier maintains the same type of wave within in a large range of frequency, while the input is the sinusoidal or square wave, and the resolution of the mechanism which the power amplifier is applied in is about 4.5 nm. By selecting the critical electronic elements and using feedback control, the proposed circuit structure is able to realize a low-cost and high-performance power amplifier to drive piezoelectric stack actuators flexibly, which is the novel work of the paper.


Introduction
Piezoelectric stack actuators have been widely used in robotics, scanning probe microscopies, industrial precision positioning systems, and biomedical engineering because of their small size, high displacement resolution, and lack of electromagnetic interference [1][2][3][4][5][6][7][8]. In these applications, power amplifiers are important subsystems, and low-noise and high-voltage power amplifiers are needed to achieve the performance of piezoelectric stack actuators. There are usually two types of power amplifiers: voltage amplifiers and charge amplifiers [9][10][11][12][13][14]. Because voltage amplifiers have the advantages of simple structure and high bandwidth, they have been widely adopted for commercial power amplifiers [13,15]. Many such voltage amplifiers have been proposed by References [16][17][18][19][20][21][22][23], and the power amplifier can also be used to estimate the force and displacement of the piezoelectric stack actuator [24]. Reference [25] studies and implements a 15 W driver for piezoelectric stack actuators; the power losses of the driver can be reduced because of the proposed circuit structure, and the output voltage ripples are about 40 mV. Reference [13] proposes a power amplifier based on the error-amplified

The input stage
The input stage of the high-voltage operational amplifier is a differential amplifier circuit. The circuit is composed by Q 2-5 , D 1-3 and some resistances, and the pair transistor Q 4 is the critical element of the circuit. The pair transistor and Q 2-3 realize a CC-CE combined amplifier; R 1 and D 3 realize a voltage clamp circuit, which makes the voltage between base of Q 2 and emitter of Q 4 constant. The current source of the differential amplifier circuit realized by Q 5 and R 9 ensures the symmetry of the circuit and provides an appropriate operating point for stage 2. The relationships between the input and output can be expressed as follows: where A 1 is the magnification of the circuit, V Q is the output bias, and V 1 (t) is the voltage signal described in the Figure 1. V Q can be expressed as Equation (2), and the needed operation point V Q2 for Sensors 2020, 20, 6528 3 of 20 stage 2 can be expressed by Equation (3). In Equations (2) and (3), V BE is the drop voltage of the diode and VCC is the voltage of the positive power supply.
Making I a = I b and R 3 = 2R 2 , and then V Q = V Q2 , the designed parameters can satisfy the requirement of the operational point of the stage 2 perfectly, which is the important role of the stage 1. The CC-CE combined amplifier has excellent frequency characteristics, and the magnification A 1 is a constant value during DC to thousands Hz.
Sensors 2020, 20, x FOR PEER REVIEW 3 of 20 Making Ia = Ib and R3 = 2R2, and then VQ = VQ2, the designed parameters can satisfy the requirement of the operational point of the stage 2 perfectly, which is the important role of the stage 1. The CC-CE combined amplifier has excellent frequency characteristics, and the magnification A1 is a constant value during DC to thousands Hz.

The amplify stage
Stage 2 of the circuit is configured as a common-emitter BJT with a current source load. The circuit always needs a stable operating point, which demands that the input signal is very small and has no effect on the operating point. The input voltage V1(t) described in Figure 1 can be expressed as follows: where VQ2 is the operation point and VS(t) is the input signal. Based on the circuit theory, it is proved that the circuit only amplifies the input signal, and the relationship between the input and output signal can be expressed by Equation (5), where A2 is the magnification of the circuit and V2(t) is the voltage signal described in Figure 1.
An ideal current source has infinite internal resistance, but the characteristic of the current source realized by depletion MOSFETs is shown as Figure 2. During the operating range, the actual current source has a dynamic internal resistance, which is expressed in Figure 2.

The amplify stage
Stage 2 of the circuit is configured as a common-emitter BJT with a current source load. The circuit always needs a stable operating point, which demands that the input signal is very small and has no effect on the operating point. The input voltage V 1 (t) described in Figure 1 can be expressed as follows: where V Q2 is the operation point and V S (t) is the input signal. Based on the circuit theory, it is proved that the circuit only amplifies the input signal, and the relationship between the input and output signal can be expressed by Equation (5), where A 2 is the magnification of the circuit and V 2 (t) is the voltage signal described in Figure 1.
An ideal current source has infinite internal resistance, but the characteristic of the current source realized by depletion MOSFETs is shown as Figure 2. During the operating range, the actual current source has a dynamic internal resistance, which is expressed in Figure 2. Taking the model of the BJT into account, the model of the circuit within the bandwidth is shown as Figure 3. Ignoring R5 and C1, the magnification of the circuit is expressed as follows: The dynamic internal resistance of the current source implemented by depletion MOSFETs is numerically big, so that the magnification of the circuit is also very big. The high-frequency attenuation is implemented by the network of R5 and C1, which improves the frequency characteristics of the circuit and enhances the stability of the circuit.

The output stage
The MOSFETs M1 and M2 are the critical elements of the output stage, which is a general class B amplifier circuit. In the circuit, R7 is used to eliminate crossover distortion; R6 and R8 are used to reduce the effect of the input capacitance of M1 and M2. The circuit has advantages of a large output current and high bandwidth, and its magnification can be expressed as A3(s) ≈ 1.
When the power supply for the amplifier has the function of current protection or the output current of the amplifier is moderate, the circuit in Figure 1 can be applied directly. The practical circuit is shown in Figure 4 for when the demanded currents exceed the maximum currents of M1 and M2. As the parallel circuits of the class B amplifier are identical, the output current of the circuit can be expressed as Equation (7), which has enlarged the output current N times. Taking the model of the BJT into account, the model of the circuit within the bandwidth is shown as Figure 3. Ignoring R 5 and C 1 , the magnification of the circuit is expressed as follows: Sensors 2020, 20, x FOR PEER REVIEW 4 of 20 Taking the model of the BJT into account, the model of the circuit within the bandwidth is shown as Figure 3. Ignoring R5 and C1, the magnification of the circuit is expressed as follows: The dynamic internal resistance of the current source implemented by depletion MOSFETs is numerically big, so that the magnification of the circuit is also very big. The high-frequency attenuation is implemented by the network of R5 and C1, which improves the frequency characteristics of the circuit and enhances the stability of the circuit.

The output stage
The MOSFETs M1 and M2 are the critical elements of the output stage, which is a general class B amplifier circuit. In the circuit, R7 is used to eliminate crossover distortion; R6 and R8 are used to reduce the effect of the input capacitance of M1 and M2. The circuit has advantages of a large output current and high bandwidth, and its magnification can be expressed as A3(s) ≈ 1.
When the power supply for the amplifier has the function of current protection or the output current of the amplifier is moderate, the circuit in Figure 1 can be applied directly. The practical circuit is shown in Figure 4 for when the demanded currents exceed the maximum currents of M1 and M2. As the parallel circuits of the class B amplifier are identical, the output current of the circuit can be expressed as Equation (7), which has enlarged the output current N times. The dynamic internal resistance of the current source implemented by depletion MOSFETs is numerically big, so that the magnification of the circuit is also very big. The high-frequency attenuation is implemented by the network of R 5 and C 1 , which improves the frequency characteristics of the circuit and enhances the stability of the circuit.

The output stage
The MOSFETs M 1 and M 2 are the critical elements of the output stage, which is a general class B amplifier circuit. In the circuit, R 7 is used to eliminate crossover distortion; R 6 and R 8 are used to reduce the effect of the input capacitance of M 1 and M 2 . The circuit has advantages of a large output current and high bandwidth, and its magnification can be expressed as A 3 (s) ≈ 1.
When the power supply for the amplifier has the function of current protection or the output current of the amplifier is moderate, the circuit in Figure 1 can be applied directly. The practical circuit is shown in Figure 4 for when the demanded currents exceed the maximum currents of M 1 and M 2 . As the parallel circuits of the class B amplifier are identical, the output current of the circuit can be expressed as Equation (7), which has enlarged the output current N times.
Considering that the parameters of discrete electronic elements are inconsistent, the currents of Mni and Mpi need to be limited, so the MOSFETs are not damaged by transient current. In the circuit, the current of Mpi is limited by Qpi and Rpi, and the current of Mni is limited by Qni and Rni. The parallel connection of the class B amplifier is a major design method in the design of power amplifiers for piezoelectric stack actuators, which is different from a general operational amplifier, as the output current of a general operational amplifier is always about several mA.
In the actual model of MOSFETs, there are input capacitances, which are shown in Figure 5. As the number of the parallel connections of the class B amplifier increases, the bandwidth of the circuit reduces, which would reduce the stability of the high-voltage operational amplifier, and this is the cost of increasing the output current. Generally, the number of parallel circuits is determined by simulation.

The overall magnification
Based on the analysis of the three stages, the input-output relationship of the proposed high-voltage operational amplifier is described by the Equation (8), where A is the magnification of the proposed high-voltage operational amplifier and A1-3 are the magnification of the three stages.
Cgd Cgs Figure 4. The power-amplifier circuit with multi-level parallel connection.
Considering that the parameters of discrete electronic elements are inconsistent, the currents of Mni and Mpi need to be limited, so the MOSFETs are not damaged by transient current. In the circuit, the current of Mpi is limited by Qpi and Rpi, and the current of Mni is limited by Qni and Rni. The parallel connection of the class B amplifier is a major design method in the design of power amplifiers for piezoelectric stack actuators, which is different from a general operational amplifier, as the output current of a general operational amplifier is always about several mA.
In the actual model of MOSFETs, there are input capacitances, which are shown in Figure 5. As the number of the parallel connections of the class B amplifier increases, the bandwidth of the circuit reduces, which would reduce the stability of the high-voltage operational amplifier, and this is the cost of increasing the output current. Generally, the number of parallel circuits is determined by simulation.
Considering that the parameters of discrete electronic elements are inconsistent, the currents of Mni and Mpi need to be limited, so the MOSFETs are not damaged by transient current. In the circuit, the current of Mpi is limited by Qpi and Rpi, and the current of Mni is limited by Qni and Rni. The parallel connection of the class B amplifier is a major design method in the design of power amplifiers for piezoelectric stack actuators, which is different from a general operational amplifier, as the output current of a general operational amplifier is always about several mA.
In the actual model of MOSFETs, there are input capacitances, which are shown in Figure 5. As the number of the parallel connections of the class B amplifier increases, the bandwidth of the circuit reduces, which would reduce the stability of the high-voltage operational amplifier, and this is the cost of increasing the output current. Generally, the number of parallel circuits is determined by simulation.

The overall magnification
Based on the analysis of the three stages, the input-output relationship of the proposed high-voltage operational amplifier is described by the Equation (8), where A is the magnification of the proposed high-voltage operational amplifier and A1-3 are the magnification of the three stages.
Cgd Cgs Figure 5. The input capacitance of MOSFETs.

The overall magnification
Based on the analysis of the three stages, the input-output relationship of the proposed high-voltage operational amplifier is described by the Equation (8), where A is the magnification of the proposed high-voltage operational amplifier and A 1-3 are the magnification of the three stages.

The Principles Applying the High-Voltage Operational Amplifier as a Power Amplifier for Piezoelectric Stack Actuators
When applying the high-voltage operational amplifier to form a power amplifier, it is general to use feedback control, so that the magnification of power amplifiers is determined by the feedback Sensors 2020, 20, 6528 6 of 20 network and the value of the magnification is constant during the designed frequency range. The feedback network is always formed by high-precision resistances and capacitances, which reduces the influence induced by discrete electronic elements of the high-voltage operational amplifiers. There are two methods to use the proposed high-voltage operational amplifier for piezoelectric stack actuator: the noninverting amplifier and the inverting amplifier. The noninverting amplifier used as a power amplifier for piezoelectric stack actuator is shown in Figure 6; it can isolate the input and output well. The transfer function of the circuit can be expressed as follows: Sensors 2020, 20, x FOR PEER REVIEW 6 of 20

The Principles Applying the High-Voltage Operational Amplifier as a Power Amplifier for Piezoelectric Stack Actuators
When applying the high-voltage operational amplifier to form a power amplifier, it is general to use feedback control, so that the magnification of power amplifiers is determined by the feedback network and the value of the magnification is constant during the designed frequency range. The feedback network is always formed by high-precision resistances and capacitances, which reduces the influence induced by discrete electronic elements of the high-voltage operational amplifiers. There are two methods to use the proposed high-voltage operational amplifier for piezoelectric stack actuator: the noninverting amplifier and the inverting amplifier. The noninverting amplifier used as a power amplifier for piezoelectric stack actuator is shown in Figure 6; it can isolate the input and output well. The transfer function of the circuit can be expressed as follows: A(s) Figure 6. The noninverting amplifier is used as a power amplifier.
In Equation (9), A(s) is the transfer function of the operational amplifier, and H(s) is the transfer function of the feedback network, which is always realized by resistances and capacitances. As long as A(s)H(s) >> 1, Equation (9) can be expressed as follows: The stability of the power amplifier using feedback control needs to be analyzed and verified by simulations, especially when piezoelectric stack actuators are used as the load.
Considering the noise of circuit as disturbances, the model of the amplifier is shown as Figure 7. In the figure, N1-3(s) are the noise of the input stage, the amplifier stage, and the output stage, respectively, and the influences of the noise can be expressed by Equation (11). According to the equation, the influences of N2(s) and N3(s) would be attenuated as A1(s) >> 1 and A2(s) >> 1, so that only the noise of the input stage decides the output noise of the circuit. The input stage is composed by low-noise discrete elements, making the output noise of the circuit very low. In Equation (9), A(s) is the transfer function of the operational amplifier, and H(s) is the transfer function of the feedback network, which is always realized by resistances and capacitances. As long as A(s)H(s) >> 1, Equation (9) can be expressed as follows: The stability of the power amplifier using feedback control needs to be analyzed and verified by simulations, especially when piezoelectric stack actuators are used as the load.
Considering the noise of circuit as disturbances, the model of the amplifier is shown as Figure 7. In the figure, N 1-3 (s) are the noise of the input stage, the amplifier stage, and the output stage, respectively, and the influences of the noise can be expressed by Equation (11). According to the equation, the influences of N 2 (s) and N 3 (s) would be attenuated as A 1 (s) >> 1 and A 2 (s) >> 1, so that only the noise of the input stage decides the output noise of the circuit. The input stage is composed by low-noise discrete elements, making the output noise of the circuit very low.

The Current Requirements of the Power Amplifiers
Piezoelectric stack actuators have complex electrical and mechanical models [29], and the models have influences on the power amplifier. In general, the model of piezoelectric stack actuators is a capacitor in series with a resistor, which makes the power amplifier need the ability of high output current. In Reference [13], the model of piezoelectric stack actuators is proposed, as shown in Figure 8. As demonstrated in the figure, Cx is the capacitance value of piezoelectric stack actuators, and Rx is the dynamic power loss of piezoelectric stack actuators. The value of Cx is almost several µF, and the value of Rx is several Ohm, and these values can be acquired by impedance analyzer or get from the manufacturer. Taking the model of piezoelectric stack actuators into account, the whole circuit model of the power amplifier and piezoelectric stack actuators is shown in Figure 9. The current I(s) can be expressed as Equation (12): where Rs is the output resistor of the power amplifier, the value is always less than 1 Ω, F(s) is the magnification of the power amplifier, and Lx is the inductance of the wire. The value of Lx is generally very small, so that its influence can be neglected. Then the output current I(s) can be expressed as follows: When the input signal is explicit, the current of the power amplifier can be calculated by Equation (13), which determines the selection of the electronic elements and the parallel quantity of the current amplification stage of the power amplifier. Periodic sinusoidal tracking is common in the application of piezoelectric stack actuators. In that situation, the out current is calculated easily; let s = wj, and Equation (13) can be expressed as follows: Let R = Rx + Rs and the amplitude of V(jw) be A. When only amplitude is calculated, the Equation (14) is changed as follows:

The Current Requirements of the Power Amplifiers
Piezoelectric stack actuators have complex electrical and mechanical models [29], and the models have influences on the power amplifier. In general, the model of piezoelectric stack actuators is a capacitor in series with a resistor, which makes the power amplifier need the ability of high output current. In Reference [13], the model of piezoelectric stack actuators is proposed, as shown in Figure 8. As demonstrated in the figure, Cx is the capacitance value of piezoelectric stack actuators, and Rx is the dynamic power loss of piezoelectric stack actuators. The value of Cx is almost several µF, and the value of Rx is several Ohm, and these values can be acquired by impedance analyzer or get from the manufacturer. Taking the model of piezoelectric stack actuators into account, the whole circuit model of the power amplifier and piezoelectric stack actuators is shown in Figure 9. The current I(s) can be expressed as Equation (12): where Rs is the output resistor of the power amplifier, the value is always less than 1 Ω, F(s) is the magnification of the power amplifier, and Lx is the inductance of the wire. The value of Lx is generally very small, so that its influence can be neglected. Then the output current I(s) can be expressed as follows:   When the input signal is explicit, the current of the power amplifier can be calculated by Equation (13), which determines the selection of the electronic elements and the parallel quantity of the current amplification stage of the power amplifier. Periodic sinusoidal tracking is common in the application of piezoelectric stack actuators. In that situation, the out current is calculated easily; let s = wj, and Equation (13) can be expressed as follows: Let R = R x + R s and the amplitude of V(jw) be A. When only amplitude is calculated, the Equation (14) is changed as follows: According to Equation (15), the current is determined by the values of the RC and the input frequency when parameter A is determined. As the values of the RC are determined, the current increases by the increasing of the input frequency, and the maximal value of the current is AR −1 . Let w = 2πf and A = 100; as the amplitude of output voltage is 100 V, the relationship between the input frequency and the current as the RC has a different value, as shown in Figure 10. According to Figure 10, the capacitance values of piezoelectric stack actuators and the maximum operating frequency are crucial parameter to design power amplifiers.
Sensors 2020, 20, x FOR PEER REVIEW 8 of 20 According to Equation (15), the current is determined by the values of the RC and the input frequency when parameter A is determined. As the values of the RC are determined, the current increases by the increasing of the input frequency, and the maximal value of the current is AR −1 . Let w = 2πf and A = 100; as the amplitude of output voltage is 100 V, the relationship between the input frequency and the current as the RC has a different value, as shown in Figure 10. According to Figure  10, the capacitance values of piezoelectric stack actuators and the maximum operating frequency are crucial parameter to design power amplifiers.

Setup of Prototype Circuit Board
In the theoretical analysis of Section 2, the design method of the power amplifier is proposed. Based on different type of piezoelectric stack actuators, it is easy to design the power amplifier by choosing appropriate electronic elements and applying feedback control. In a research of the projection lithography, the requirements are as follows: 1.

2.
The output voltage range is −100 to +100 V, and the magnification is 16 times. 3.
The piezoelectric stack actuator works at an operating point; it is needed to track the input signal near the operating point, and the highest moving frequency of the mechanism is less than 100 Hz. 4.
The power amplifier, which should be easy to be integrated with the mechanism, must reserve negative voltage for other piezoelectric stack actuators from Physik Instrumente [30], such as P-143.05.
Based on the above requirements, the demanded output current is not large, as the piezoelectric stack actuators do not need to follow the input signal in the full range, so that, in stage 3, the supplied output current is composed by single MOSFETs, and the design could reduce the size of the power amplifier. Taking all the factors into account and referring to the circuit diagram in Figure 1, the results of the design are as follows: 1.
In the first and second stage, Q 4 is SSM2212, Q 2-3 are CZTA44, and Q 1 is FTZ560, while the current source is realized by CPC5603 that is Q 5 or Q 6 in Figure 1. The selection of Q 1-3 and Q 5-6 is based on the maximum Collector-Emitter Voltage or the maximum Drain-Source Voltage of these elements. Q 4 is dual-matched NPN transistor whose input noise is low, and SSM2212 matches it well.

2.
In the third stage, M 1 and M 2 are IXFR15N100Q3 and IXTH10P60P, which both have more than 45 A instantaneous current and more than 10 A continuous current. The most powers of the MOSFETs are 400 and 460 W, respectively. The selection of M 1 and M 2 is mainly based on V DS and I D , which are the maximum Drain-Source Voltage and the maximum Drain Current, respectively, and the heat-dissipation performance of these elements should be taken into account.

3.
To achieve 16-times magnification, the structure of the noninverting amplifier is used, and the values of resistors are 10 and 150 k, respectively. In order to enhance the stability of the power amplifier, the resistor of 150 k is paralleled with a capacitor of 15 pF.
In order to confirm the effectiveness of design, the simulation of the power amplifier is done in the next section.

The Simulated Frequency Response of the Power Amplifier
In order to analyze the characteristic of the proposed power amplifier, the simulation of the power amplifier is implemented with the software Cadence 16.6. In the simulation, the positive and negative DC power supplies are set as +110 and −110 V, respectively.
Based on the circuit theory shown by Figure 6 in Section 2, it is convenient to form a power amplifier to drive piezoelectric stack actuators, and the simulated amplitude frequency response of the power amplifier is shown in Figure 11. According to the figure, the following conclusions could be drawn: The power amplifier is stable; the magnification of the power amplifier in the passband is 24.082 dB, which agrees with the theoretical magnification (16 times); the bandwidth is about 68 kHz, and the 0 dB frequency is about 3.5 MHz; and the magnification is flat during the passband of the power amplifier, which is suitable to form a high-precision and highly dynamic piezoelectric positioning system. Sensors 2020, 20, x FOR PEER REVIEW 10 of 20 Figure 11. Simulated frequency response of the power amplifier.

The Simulated Response of the Power Amplifier with Load
The simulated waveform of the power amplifier as sinusoidal input with piezoelectric stack actuators is to verify the ability to respond to the sinusoidal signal. In the simulation, a capacitor of 3.1 µF in series with a 10 Ω resistor is used to model the piezoelectric stack actuator, as the capacitor of the piezoelectric stack actuator (P887.51) is 3.1 µF. Using sinusoidal wave with amplitude of 6 V and frequency of 100 Hz as input, which conforms to the working conditions of the power amplifier, the simulated input and output waveforms are shown in Figure 12, the currents of the MOSFETs are shown in Figure 13, and the power consumptions of the MOSFETs are shown in Figure 14. According to the simulated results, the power amplifier has an excellent ability to respond to sinusoidal input, and the current and power consumption are in a safe work area of the MOSFETs.

The Simulated Response of the Power Amplifier with Load
The simulated waveform of the power amplifier as sinusoidal input with piezoelectric stack actuators is to verify the ability to respond to the sinusoidal signal. In the simulation, a capacitor of 3.1 µF in series with a 10 Ω resistor is used to model the piezoelectric stack actuator, as the capacitor of the piezoelectric stack actuator (P887.51) is 3.1 µF. Using sinusoidal wave with amplitude of 6 V and frequency of 100 Hz as input, which conforms to the working conditions of the power amplifier, the simulated input and output waveforms are shown in Figure 12, the currents of the MOSFETs are shown in Figure 13, and the power consumptions of the MOSFETs are shown in Figure 14. According to the simulated results, the power amplifier has an excellent ability to respond to sinusoidal input, and the current and power consumption are in a safe work area of the MOSFETs.

The Simulated Response of the Power Amplifier with Load
The simulated waveform of the power amplifier as sinusoidal input with piezoelectric stack actuators is to verify the ability to respond to the sinusoidal signal. In the simulation, a capacitor of 3.1 µF in series with a 10 Ω resistor is used to model the piezoelectric stack actuator, as the capacitor of the piezoelectric stack actuator (P887.51) is 3.1 µF. Using sinusoidal wave with amplitude of 6 V and frequency of 100 Hz as input, which conforms to the working conditions of the power amplifier, the simulated input and output waveforms are shown in Figure 12, the currents of the MOSFETs are shown in Figure 13, and the power consumptions of the MOSFETs are shown in Figure 14. According to the simulated results, the power amplifier has an excellent ability to respond to sinusoidal input, and the current and power consumption are in a safe work area of the MOSFETs.   Just as the above setup, the simulated waveforms of the power amplifier as square input are shown in Figures 15-17. The input square wave has the amplitude of 6 V, the frequency of 100 Hz, and the rising time and failing time of 1 µs. The input and output waveforms of the simulation are shown in Figure 15, which contains the rising and failing edge details. The current of the MOSFETs is shown in Figure 16, and the power consumptions of the MOSFETs are shown in Figure 17. According to the simulated results, the power amplifier has an excellent ability to respond to step input, the most instantaneous currents of the MOSFETs are less than 15 where P is the average power, T is the period, PI is the instantaneous power, and TI is the instantaneous time. In Equation (16), the instantaneous power is idealized as a narrow pulse. Based on Equation (16), the average power is about 30 W, as the period is 10 ms. The average power of the MOSFETs is less than the maximum powers of the MOSFETs, i.e., 400 and 460 W, respectively. However, the instantaneous power consumptions of the MOSFETs are so big that the MOSFETs should well dissipate heat to avoid damage.  Just as the above setup, the simulated waveforms of the power amplifier as square input are shown in Figures 15-17. The input square wave has the amplitude of 6 V, the frequency of 100 Hz, and the rising time and failing time of 1 µs. The input and output waveforms of the simulation are shown in Figure 15, which contains the rising and failing edge details. The current of the MOSFETs is shown in Figure 16, and the power consumptions of the MOSFETs are shown in Figure 17. According to the simulated results, the power amplifier has an excellent ability to respond to step input, the most instantaneous currents of the MOSFETs are less than 15 where P is the average power, T is the period, PI is the instantaneous power, and TI is the instantaneous time. In Equation (16), the instantaneous power is idealized as a narrow pulse. Based on Equation ( Figure 14. Simulated power consumptions of the MOSFETs. Just as the above setup, the simulated waveforms of the power amplifier as square input are shown in Figures 15-17. The input square wave has the amplitude of 6 V, the frequency of 100 Hz, and the rising time and failing time of 1 µs. The input and output waveforms of the simulation are shown in Figure 15, which contains the rising and failing edge details. The current of the MOSFETs is shown in Figure 16, and the power consumptions of the MOSFETs are shown in Figure 17. According to the simulated results, the power amplifier has an excellent ability to respond to step input, the most instantaneous currents of the MOSFETs are less than 15 A, and the most instantaneous power consumptions are about 1500 W. The instantaneous currents of the MOSFETs are less than the maximum instantaneous currents of the MOSFETs, which are about 40 A. The instantaneous power consumptions of the MOSFETs are about 1500 W, and the impulse time is about 200 µs. The average power of the MOSFETs in a period can be expressed by Equation (16): where P is the average power, T is the period, P I is the instantaneous power, and T I is the instantaneous time. In Equation (16), the instantaneous power is idealized as a narrow pulse. Based on Equation   The simulated results show that the designed power amplifier is consistent with the analysis in Section 2, and the power amplifier is stable and has excellent waveform response. The design results can be further verified by experiments.

Experimental Setup
The test setup of the power amplifier is shown in the Figure 18. The main instruments are as follows: The power supply, which is customized from Chaoyang Power Supply Corporation, has the ability of current protection, whose maximum continuous current is 4 A and instantaneous current is 20 A, as the instantaneous time is 2 ms; the waveform generator is 33522B from Keysight Technologies (USA) [31]; the oscilloscopes measuring the waveforms is MSO5204 from Tektronix Corporation (USA) [32]; the multi-meter measuring DC voltage is 34470A form Keysight Technologies [31]; the network analyze testing the system frequency response characteristic is E5061B from Keysight Technologies [31]; and the piezoelectric stack actuator is P887.51 from Physik Instrumente [30]. The simulated results show that the designed power amplifier is consistent with the analysis in Section 2, and the power amplifier is stable and has excellent waveform response. The design results can be further verified by experiments.

Experimental Setup
The test setup of the power amplifier is shown in the Figure 18. The main instruments are as follows: The power supply, which is customized from Chaoyang Power Supply Corporation, has the ability of current protection, whose maximum continuous current is 4 A and instantaneous current is 20 A, as the instantaneous time is 2 ms; the waveform generator is 33522B from Keysight Technologies (USA) [31]; the oscilloscopes measuring the waveforms is MSO5204 from Tektronix Corporation (USA) [32]; the multi-meter measuring DC voltage is 34470A form Keysight Technologies [31]; the network analyze testing the system frequency response characteristic is E5061B from Keysight Technologies [31]; and the piezoelectric stack actuator is P887.51 from Physik Instrumente [30].

Frequency-Amplitude Characteristic
The frequency-amplitude characteristic of the power amplifier is shown in Figure 19. According to the figure, the passband gain of the system is 24.026 dB, −3 dB bandwidth is about 57 kHz, and 0 dB bandwidth is about 2.388 MHz. The overall shape of the frequency-amplitude characteristics is consistent with the results of simulation analysis; there are a few differences between the tested frequency and the simulated frequency at the key frequency points, but the differences are not significant. The reasons for the differences are that the simulation uses an ideal electronic elements model which could not be found, and the simulation does not consider the distributed parameter of the print circuit board. As the differences are slight, especially in the low-frequency range, the simulation results can be used as references in the design.

The Ripple of the Power Amplifier
The out ripple is the important performance of various power amplifiers. As to the linear power supply, such as the designed power amplifier, the ripple is the output noise. The output ripple of the power amplifier would influence the output displacement accuracy of the piezoelectric stack actuator, especially in high-precision applications. Oscilloscope MSO5204 from Tektronix Corporation [32] was used for ripple test, with P2221 from Tektronix Corporation [32] as probe. We connected the input to ground, and the achieved output ripple is shown in Figure 20. According to the figure, the ripple of the power amplifier is less than 2 mV.

Frequency-Amplitude Characteristic
The frequency-amplitude characteristic of the power amplifier is shown in Figure 19. According to the figure, the passband gain of the system is 24.026 dB, −3 dB bandwidth is about 57 kHz, and 0 dB bandwidth is about 2.388 MHz. The overall shape of the frequency-amplitude characteristics is consistent with the results of simulation analysis; there are a few differences between the tested frequency and the simulated frequency at the key frequency points, but the differences are not significant. The reasons for the differences are that the simulation uses an ideal electronic elements model which could not be found, and the simulation does not consider the distributed parameter of the print circuit board. As the differences are slight, especially in the low-frequency range, the simulation results can be used as references in the design.

Frequency-Amplitude Characteristic
The frequency-amplitude characteristic of the power amplifier is shown in Figure 19. According to the figure, the passband gain of the system is 24.026 dB, −3 dB bandwidth is about 57 kHz, and 0 dB bandwidth is about 2.388 MHz. The overall shape of the frequency-amplitude characteristics is consistent with the results of simulation analysis; there are a few differences between the tested frequency and the simulated frequency at the key frequency points, but the differences are not significant. The reasons for the differences are that the simulation uses an ideal electronic elements model which could not be found, and the simulation does not consider the distributed parameter of the print circuit board. As the differences are slight, especially in the low-frequency range, the simulation results can be used as references in the design.

The Ripple of the Power Amplifier
The out ripple is the important performance of various power amplifiers. As to the linear power supply, such as the designed power amplifier, the ripple is the output noise. The output ripple of the power amplifier would influence the output displacement accuracy of the piezoelectric stack actuator, especially in high-precision applications. Oscilloscope MSO5204 from Tektronix Corporation [32] was used for ripple test, with P2221 from Tektronix Corporation [32] as probe. We connected the input to ground, and the achieved output ripple is shown in Figure 20. According to the figure, the ripple of the power amplifier is less than 2 mV.

The Ripple of the Power Amplifier
The out ripple is the important performance of various power amplifiers. As to the linear power supply, such as the designed power amplifier, the ripple is the output noise. The output ripple of the power amplifier would influence the output displacement accuracy of the piezoelectric stack actuator, especially in high-precision applications. Oscilloscope MSO5204 from Tektronix Corporation [32] was used for ripple test, with P2221 from Tektronix Corporation [32] as probe. We connected the input to ground, and the achieved output ripple is shown in Figure 20

Tests with Piezoelectric Stack Actuator
In order to verify the stability of the power amplifier, a piezoelectric stack actuator (P887.51) is connected to the power amplifier. Using 100 Hz sinusoidal wave or 100 Hz square wave whose amplitude are both 6 V as input, respectively, the responses are shown in Figures 21 and 22, which agree with the simulation results. The reason of choosing 100 Hz as the frequency point is that the highest moving frequency of the mechanism is 100 Hz.

Tests with Piezoelectric Stack Actuator
In order to verify the stability of the power amplifier, a piezoelectric stack actuator (P887.51) is connected to the power amplifier. Using 100 Hz sinusoidal wave or 100 Hz square wave whose amplitude are both 6 V as input, respectively, the responses are shown in Figures 21 and 22, which agree with the simulation results. The reason of choosing 100 Hz as the frequency point is that the highest moving frequency of the mechanism is 100 Hz.

Tests with Piezoelectric Stack Actuator
In order to verify the stability of the power amplifier, a piezoelectric stack actuator (P887.51) is connected to the power amplifier. Using 100 Hz sinusoidal wave or 100 Hz square wave whose amplitude are both 6 V as input, respectively, the responses are shown in Figures 21 and 22, which agree with the simulation results. The reason of choosing 100 Hz as the frequency point is that the highest moving frequency of the mechanism is 100 Hz.

Tests in the System
The mechanical stage for which the piezoelectric stacks actuator (P887.51) is applied is shown in Figure 23. The mechanical stage of 1 degree of freedom has a nominal displacement of 0-12 µm, which can be measured by Heidenhain LIP281 integrated in the stage. The stage is used to move required displacement and do some local adjustments at the position, and it is electrically excited by the designed power amplifier. The power amplifier uses the waveform generator (Keysight 33522B) to produce the input signal, and the power amplifier is powered by the customized power supply. The laboratory is under precise environmental control, with its ambient temperature kept at 22 ± 0.2 °C. The noise of the used displacement sensor is shown in Figure 24a. According to the figure, the noise of the sensor is about 1 nm. The resolution of the mechanical stage is shown in Figure 24b, as the input signal is a square wave whose period and amplitude are 500 ms and 3 mV, respectively. According to Figure 24, the resolution of the mechanism is about 4.5 nm. The input-output relationship of the mechanical stage is shown in Figure 25; the step voltage and the step time of the input signal are 1.2 V and 200 ms, respectively. According to Figure 25, the output step displacement is about 2.5 µm, and the rise and fall characteristics are not consistent, due to the hysteresis effect of Figure 22. Measured waveforms of input (c1) and output (c2), as the input square wave has amplitude of 6 V and frequency of 100 Hz.

Tests in the System
The mechanical stage for which the piezoelectric stacks actuator (P887.51) is applied is shown in Figure 23. The mechanical stage of 1 degree of freedom has a nominal displacement of 0-12 µm, which can be measured by Heidenhain LIP281 integrated in the stage. The stage is used to move required displacement and do some local adjustments at the position, and it is electrically excited by the designed power amplifier. The power amplifier uses the waveform generator (Keysight 33522B) to produce the input signal, and the power amplifier is powered by the customized power supply. The laboratory is under precise environmental control, with its ambient temperature kept at 22 ± 0.2 • C.

Tests in the System
The mechanical stage for which the piezoelectric stacks actuator (P887.51) is applied is shown in Figure 23. The mechanical stage of 1 degree of freedom has a nominal displacement of 0-12 µm, which can be measured by Heidenhain LIP281 integrated in the stage. The stage is used to move required displacement and do some local adjustments at the position, and it is electrically excited by the designed power amplifier. The power amplifier uses the waveform generator (Keysight 33522B) to produce the input signal, and the power amplifier is powered by the customized power supply. The laboratory is under precise environmental control, with its ambient temperature kept at 22 ± 0.2 °C. The noise of the used displacement sensor is shown in Figure 24a. According to the figure, the noise of the sensor is about 1 nm. The resolution of the mechanical stage is shown in Figure 24b, as the input signal is a square wave whose period and amplitude are 500 ms and 3 mV, respectively. According to Figure 24, the resolution of the mechanism is about 4.5 nm. The input-output relationship of the mechanical stage is shown in Figure 25; the step voltage and the step time of the input signal are 1.2 V and 200 ms, respectively. According to Figure 25, the output step displacement is about 2.5 µm, and the rise and fall characteristics are not consistent, due to the hysteresis effect of The noise of the used displacement sensor is shown in Figure 24a. According to the figure, the noise of the sensor is about 1 nm. The resolution of the mechanical stage is shown in Figure 24b, as the input signal is a square wave whose period and amplitude are 500 ms and 3 mV, respectively. According to Figure 24, the resolution of the mechanism is about 4.5 nm. The input-output relationship of the mechanical stage is shown in Figure 25; the step voltage and the step time of the input signal are 1.2 V and 200 ms, respectively. According to Figure 25, the output step displacement is about 2.5 µm, and the rise and fall characteristics are not consistent, due to the hysteresis effect of the piezoelectric stacks actuator. All of these results show that the designed power amplifier has excellent performances to drive the piezoelectric stacks actuator. Sensors 2020, 20, x FOR PEER REVIEW 17 of 20 the piezoelectric stacks actuator. All of these results show that the designed power amplifier has excellent performances to drive the piezoelectric stacks actuator.

Comparisons
The comparisons between the proposed power amplifier and some existing power amplifier are listed in Table 1, and the comparisons are carried out as the power amplifies are without load. According to the table, the designed power amplifier has the advantages of low noise, small size, and low price. However, every coin has two sides; the design involves much work on the selection of the electronic elements, which increases the difficulty of design. As the paper discussed the principles of the circuit thoroughly in Section 2, the adverse factors in the design are minimized. Table 1. Comparisons between the proposed power amplifier and some existing power amplifiers.

Conclusions
In this article, aiming at low cost, small size, and high precision, a power amplifier which conforms to the design principles of general operational amplifier is explored. Based on the theory of general operational amplifiers, the circuit structure of the power amplifier is proposed, and the method of applying the proposed amplifier to form a piezoelectric driver is described. The unique design requirements produced piezoelectric stack actuators, which are similar to the series connection of capacitance and resistance, were analyzed. In a precision positioning application, using P887.51 from Physik Instrumente [30], core electronic elements were selected. Using the PSPICE models of the elements, the designed circuit was simulated and confirmed to meet the requirements. Then, by testing a prototype circuit, experimental data and curves were achieved, and the experimental results are consistent with the simulation results perfectly. The experimental results demonstrate that the bandwidth of the power amplifier is about 57 kHz, the output ripple is less than 2 mV, the displacement resolution of the mechanical stage which the power amplifier is applied in is about 4.5 nm, and the proposed power amplifier can well driver the piezoelectric stack actuator during the travel range of the mechanical stage.
Author Contributions: L.X., methodology, investigation, experiment data analysis, and writing-original draft, review and editing; H.L., supervision and funding acquisition; P.L., conceptualization; C.G., conceptualization. All authors have read and agreed to the published version of the manuscript.

Comparisons
The comparisons between the proposed power amplifier and some existing power amplifier are listed in Table 1, and the comparisons are carried out as the power amplifies are without load. According to the table, the designed power amplifier has the advantages of low noise, small size, and low price. However, every coin has two sides; the design involves much work on the selection of the electronic elements, which increases the difficulty of design. As the paper discussed the principles of the circuit thoroughly in Section 2, the adverse factors in the design are minimized.

Conclusions
In this article, aiming at low cost, small size, and high precision, a power amplifier which conforms to the design principles of general operational amplifier is explored. Based on the theory of general operational amplifiers, the circuit structure of the power amplifier is proposed, and the method of applying the proposed amplifier to form a piezoelectric driver is described. The unique design requirements produced piezoelectric stack actuators, which are similar to the series connection of capacitance and resistance, were analyzed. In a precision positioning application, using P887.51 from Physik Instrumente [30], core electronic elements were selected. Using the PSPICE models of the elements, the designed circuit was simulated and confirmed to meet the requirements. Then, by testing a prototype circuit, experimental data and curves were achieved, and the experimental results are consistent with the simulation results perfectly. The experimental results demonstrate that the bandwidth of the power amplifier is about 57 kHz, the output ripple is less than 2 mV, the displacement resolution of the mechanical stage which the power amplifier is applied in is about 4.5 nm, and the proposed power amplifier can well driver the piezoelectric stack actuator during the travel range of the mechanical stage.