1. Introduction
The circuit shown in
Figure 1 is very common, and it might be introduced in every textbook about circuit theory. When the operational amplifier is ideal, the voltage
VOUT can be expressed as:
It is obvious that the ideal operational amplifier cannot be found, so that Equation (1) is an approximation. When VIN is realized by voltage references, the circuit can be considered as a voltage source. It is common that the voltage of voltage sources drops while the load resistance of the voltage sources is made smaller, but Equation (1) does not offer the information. It is valuable to explore new method to analyze such circuits deeply.
Some studies have done such work. Reference [
1] proposes a new operational amplifier model for evaluating test strategies at a behavioral level, and the simulation results show that the model presents a good agreement with the transistor-level version of the amplifier. Reference [
2] proposes a method to fully characterize noise in the operational amplifier; the method allows the extraction not only of the spectra of the equivalent input current noise and equivalent input voltage noise generators but also of their cross-correlation coefficients, and a key finding is that neglecting the cross-correlation coefficient between the two equivalent input current noises can lead to severe errors in noise analysis. In reference [
3], a high input impedance circuit to convert a single ended voltage to its differential counterpart is proposed, and the design makes full use of the characteristics of operational amplifiers and well applies closed-loop feedback control. In reference [
4], the equivalent small parameter method is used to establish the nonlinear mathematical model of the fractional-order buck–boost converter in continuous current mode, and the method has achieved satisfying results. In reference [
5], a unified model is proposed for small-signal modelling in current controlled converters using the discrete-time analysis method, and simulations and experiments are committed to verify the proposed model. Reference [
6] introduces an additional operational amplifier that is utilized as a buffer to improve the Howland voltage controlled current source, a complete analysis including a new two-port analysis of the circuit is presented, and the results indicate that the method increases output impedance, improves noise performance and achieves stability easily. In reference [
7], the hysteresis-PI control algorithm is used in shunt active power filter design, and the effectiveness of the filter in minimizing the current harmonics has been evaluated under balanced and unbalanced nonlinear load conditions. Reference [
8] goes further, the paper applies sliding-mode controller in the design of fourth-order class-D Amplifier, and experimental results reveal that the proposed controller effectively flattens the frequency response of the fourth-order amplifier and results in THD and voltage overshoot of 0.6% and 1 V respectively. The paper [
9] applies cascode and Miller compensation for three-stage amplifiers to drive a pF-to-nF capacitive load, the simulation results show at least 0.88 MHz GBW is achieved under 4 pF to 1.5 nF capacitor load while the on-chip compensative capacitance is only 1.05 pF. These works have achieved brilliant results. However, as the methods of these works can only be utilized in specific circuit designs, they can hardly be applied to other fields directly.
It is more sensible to explore the philosophical thoughts of these works rather than to study their methods, and the philosophical thoughts lie in the way these papers have applied system modelling or control theory in the design of the circuit. This idea may enlighten the design of the power amplifiers for driving piezoelectric stack actuators, which are widely applied in micromachining, industrial precision positioning systems and biomedical engineering but have problems in being driven, as they always have large capacitance [
10,
11,
12].
Some studies have done basic research. Reference [
13] proposes a 15 W power amplifier to drive piezoelectric stack actuators; in order to save energy, the power supply of the high-voltage operational amplifier is controlled by the input signal, and the power amplifier has about 40 mV output voltage ripples. In reference [
14], high-voltage operational amplifiers are utilized to design power amplifiers for driving piezoelectric stack actuators directly, high-voltage operational amplifiers need dissipating heat well and the output bandwidth of the proposed power amplifier can reach 20 kHz. Applying the schematic described in
Figure 1, reference [
15] designs a power amplifier for driving piezoelectric stack actuators; the reference does not offer much theoretical analysis, the design relies on simulation and the output voltage ripple of the designed power amplifier is less than 20 mV. Reference [
16] uses an isolation amplifier and subtracting amplifier to design an innovative power amplifier; the merits of the amplifier include a wide bandwidth and high potential power, and experiments using a six-level arrangement demonstrate a 100 kHz bandwidth with ±200 V output swing for different capacitive loadings. Reference [
17] designs a high-voltage operational amplifier by discrete electronic elements; a power amplifier is proposed using the operational amplifier as a noninverting amplifier, and the ripple and bandwidth are about 2 mV and 57 kHz, respectively. There are also some works about charge drivers for piezoelectric stack actuators. In reference [
18], a new charge driver circuit and electrical configuration are implemented which allows commonly available piezoelectric bimorphs to be linearized; this circuit consists of four major components: a high-voltage amplifier, a differential amplifier, a piezoelectric load and a PI feedback controller, and experiments show a significant improvement of the hysteresis of the bender when compared to a typical voltage driver. Reference [
19] presents a novel controller circuit to overcome the issues of low-frequency performance, long settling time, floating-load and loss of stroke, and experimental results show that the presented charge controller can effectively reduce more than 88% of the hysteretic nonlinearity even when the operating closes to the transition frequency. These works in the mentioned references are innovative, and they all fully explore the advantages of the operational amplifier which are high input impedance and high magnification. Based on these researches and the idea applying control theory in circuit design, it is liable to propose a unified method to design power amplifiers for driving piezoelectric stack actuators, and the method is simple and flexible to be applied in engineering.
A new perspective to view the analog design is thoroughly described in this paper. The perspective creates the model of the designed circuit and analyses the essential performances by classical control theory. Based on the method, two kinds of power amplifiers for driving piezoelectric stack actuators are proposed, one uses high-voltage operational amplifiers as the controller, and the other applies general operational amplifiers. The two kinds of power amplifiers are analyzed by the proposed method which focuses on the stability of the circuit, and prototype circuit of the proposed power amplifiers is simulated and tested. Aiming to evaluate the designed power amplifiers, the power amplifiers are used to drive a 1 dimension stage during the motion range of the mechanism. Those works prove that the proposed method is a systematic approach to the design of power amplifiers and make the design of the power amplifier simple. The novelty of the paper is that the control model of the circuit is set up by the classical control theory, the performances of the circuit are analyzed by the control model qualitatively and the control model of the circuit is used to adjust the structure of the circuit and provide direction for simulation.
The organization of this paper is as follows.
Section 2 presents the method based on the problem described in the introduction. In
Section 3, a power amplifier using high-voltage operational amplifiers is designed, analyzed, simulated and tested. In
Section 4, considering the high price and little choice of high-voltage operational amplifiers, a power amplifier using general operational amplifiers is designed, analyzed, simulated and tested. In
Section 5, the designed two power amplifiers are tested with the mechanism. Finally, the main conclusions are described in
Section 6.
2. The Proposed Approach
Operational amplifiers generally have outstanding amplified ability on differential-mode signals, their inputs have higher input impedance and they have excellent ability to suppress common mode signals. Therefore, their control block diagram can be shown in
Figure 2. In reference [
1], the transfer function
A(s) can be expressed as:
where
S1 and
S2 are poles of the operational amplifier and
A is the open loop voltage gain. Based on this characteristic of the operational amplifier, the operational amplifier can be used as the controller of the circuit and the classical feedback control theory can be applied in the analog circuit design. The schematic and the control block diagram of the analog circuit can be shown in
Figure 3 and
Figure 4, respectively.
In the figures,
G(s) is the transfer function of the forward network and
H(s) is the transfer function of the feedback network. The forward network can use inner-loop feedback control to further weaken the nonlinearity of the circuit model, which makes the setting of the system easier. There are some methods to realize the inner-loop control, and the common and sensible method is utilizing the characteristics of electronic components. In the introduction, it is such an example that applies the MOSFET as the output stage of the circuit, and the circuit principle and the control block diagram of the MOSFET are shown in the
Figure 5.
In the control block diagram,
B(s) is the
RC filter composed by
Rg and the input capacitor of the gate of the MOSFET. The relationship between
Id and
V is generally a nonlinear function, and the voltage
V is the voltage difference between
Vg and
Vs. The local linearization of the function can be expressed by Equation (3):
where
V0 is the bias voltage. The Equation (3) can be further expressed by Equation (4):
Based on Equation (4), the model of the MOSFET can be expressed by
Figure 6. In the model,
K is the mutual conductance parameter of the MOSFET, which can be obtained from the manual of the MOSFET, and the parameter
P can be abstracted as a disturbance.
Based on the models of the MOSFET and the operational amplifier, the control block diagram of the circuit mentioned in the introduction can be expressed as
Figure 7.
Based on the control block diagram, the transfer function of the system can be described as:
In Equation (5), the parameter
K1 can be expressed by Equation (6):
where
RL is the resistance value of the load. Based on the control block diagram shown in
Figure 7, the load regulation rate of the voltage source in the introduction is caused by the steady-state error of the control system as the load
RL is changed. In order to reduce the steady-state error, it is liable to increase the open-loop gain of the whole system as much as possible when the system is stable.
The proposed method, exploiting the characteristics of the operational amplifier and some electronic components, transforms the design of the analog circuit into the design and calibration of the control systems, and the method can be effectively applied in the design of power amplifiers for driving piezoelectric stack actuators.
5. Tests with Mechanical Stage and Some Comparisons
The tests with the mechanical stage are shown in
Figure 30. The mechanical stage is driven by a piezoelectric stack actuator, whose type and producer are P887.51 and Physik Instrumente [
23], respectively, and the maximum displacement of the mechanical stage of 1 degree of freedom is about 13 μm. A grating ruler (Heidenhain LIP281) installed in the mechanical stage can measure the displacement of the mechanical stage, and the displacement can be acquired and stored by the grating ruler acquisition circuit. The designed two power amplifiers are used to drive the piezoelectric stack actuator, respectively, and the input signal of the power amplifiers is generated by a waveform generator (Keysight 33622B). The tests are done in a constant temperature laboratory, and the laboratory is controlled by a precise environmental control system whose ambient temperature is 22 ± 0.2 °C.
The input signal is shown in
Figure 31, and the step voltage and the step time of the input signal are 0.4 V and 10 ms, respectively. The displacements of the mechanical stage excited by the designed two power amplifiers are shown in
Figure 32. According to these figures, the output displacements of the mechanical stage driven by the proposed two power amplifiers are both about 13 μm, and the positive and negative steps of the displacement are both about 1.3 μm. The data demonstrate that the proposed two power amplifiers can drive the piezoelectric stack actuator well. Although the designed two power amplifiers have different electronic performances on ripples and bandwidth, there are few differences on the mechanical performances between the designed two power amplifiers.
The comparisons between the power amplifiers proposed in this paper and some references are listed in
Table 3. The main performances of the power amplifiers listed in the table are ripple, bandwidth, cost, size and circuit complexity. The circuit complexity of power amplifiers is evaluated by the structure of power amplifiers and the number of electric elements used in the circuit. The designed power amplifiers in this paper using high-voltage operational amplifiers and general operational amplifiers are referenced as work A and work B, respectively. Considering the power amplifiers in reference [
14], reference [
15] and the work A, they all reduce the circuit complexity by the integrated device (high-voltage operational amplifiers), which makes the price higher. The power amplifier B possesses the qualities of low ripple, small size, low cost and simple circuit structure, which reveals the superiority of the proposed method.