Instron testing data with and without load resistor. For each PZT, SN637, SN629 and SN618 the load test was repeated three times.

## Abstract

In this chapter a full electromechanical model of piezoceramic actuators is presented. This model allows for easy integration of the piezo stack with a structural finite element model (FEM) and includes the flow of energy into and out of the piezo element, which is governed by the transducer constant of the piezo element. Modeling of the piezo stack capacitive hysteresis is achieved using backlash basis functions. The piezo model can also be used to predict the current usage of the PZT which depends on the slew rate of the voltage applied to the PZT. Data from laboratory experiments using a load frame and free response tests is used to estimate the PZT model parameters. In addition, a simplified model of a modulated full bridge strain gauge is derived based on test data which includes the effect of intrinsic bridge imbalance. Sensors of this type are often used with feedback control to linearize the behavior of the device. Taken together, the actuator and sensor model can be used for the development of piezo actuated control algorithms.

### Keywords

- PZT
- transducer constant
- strain gauge
- hysteresis
- charge feedback

## 1. Introduction

Lead zirconium titanate (PZT) was first developed around 1952 at the Tokyo Institute of Technology. Due to its physical strength, chemical inertness, tailorability and low manufacturing costs, it is one of the most commonly used piezo ceramics. It is used in various applications as an actuator including micromanipulation and ultrasonics. Due to the piezoelectric effect [1], it can also be used as a sensor of force or displacement or for energy harvesting since current is produced by the device in response to an applied strain. Three main issues, however, compromise the utility of these devices. The first is that the stroke of these devices is limited to about 40

Many models for PZT stacks have been developed in the literature, such as those in [8, 9, 10], but few incorporate the transfer of energy into a structure and out of the structure back into the device. Of those that do describe this important detail, [1, 11, 12, 13], the treatment of piezoelectric hysteresis in the model is often problematic in that either friction elements, such as the Maxwell resistive capacitor (MRC) are used, which are difficult to simulate, or nonlinear differential equations are used which have very heuristic fitting functions [1]. The hysteresis model described here overcomes these difficulties by using backlash basis functions which are relatively easy to fit to experimental data and are numerically efficient to simulate. The hysteresis model does not included creep, but it is possible to add springs and dashpots to the backlash elements which will capture the creep behavior.

The PZT model presented in this chapter can also be used to predict current usage which is important in terms of amplifier design and slew rate limitations. It can also be easily integrated into a FEM of the structure that the PZT is embedded in. The model in this chapter is based on the work in [1, 12]. Unfortunately, these papers offer little in terms of determining the parameters of the model. In particular, the internal capacitances and transducer constant vary from one type of PZT to another. As a result, this chapter describes experimental techniques for determining these parameters regardless of the particular PZT configuration. We also validate proper operation of our test equipment and compare the simulated model output to test data.

For control design purposes it is also important to model the sensor in addition to the actuator. Various types of sensors have been used for PZT control such as optical encoders with 1.2 nm resolution, laser metrology gauges, and more commonly, strain gauges. Strain gauges offer good performance for the price and can be directly bonded to the sides of the PZT. A model of a modulated full strain gauge bridge is given which includes the effect of intrinsic bridge imbalance and bridge sensitivity. A simplified version of this model, excluding all exogenous signals, is given to facilitate loop shaping of a control loop.

PZTs are often used in optical instrument such as coronagraphs, interferometers, spectrometers and microscopes where micro-manipulation below the scale of optical wavelengths is required. Given the small wavelengths of visible and infrared light (350 nm - 14

The PICMA P-888.51 can be ordered with a full bridge strain gauge bonded directly to the sides of the PZT ceramic. As mentioned above, strain gauges are used to linearize the hysteretic behavior of PZTs and to mitigate their drift or creep, which can be substantial. To determine the bridge response, we analyze the modulating electronics used to read the Wheatstone bridge. The modulation is a technique commonly used for rejection of noise picked up along the cabling between the PZT and electronics board. The strain gauge bridge we analyze is composed of two Vishay half bridges (Part number N2A-XX-S053P-350) bonded to opposite sides of the PZT as shown in Figure 9. Each bridge resistor has a nominal resistance of 350 Ohms.

Three identical PZTs were tested which we refer to by their serial numbers, SN637, SN629 and SN618.

## 2. Full electro-mechanical PZT model

The PZT model we analyze is depicted in Figure 2. It is taken from the model proposed in [1, 12, 13] with a few modifications. We have added a resistor on the input since this is part of the amplifier used to drive the device. We depict the resistor in Figure 2 as having a switch since most of our experiments were conducted without this resistor. This resistor and the equivalent capacitance of the circuit determines the RC time constant of the actuator model. This time constant can vary based on the voltage input (* inverse piezoelectric effect*produces a force,

*.*piezoelectric effect

The unknown model parameters in Figure 2 include the two capacitances,

Writing some simple circuit equations based on Figure 2, we have Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL),

We also have the three constitutive relations for each of the components in this circuit,

Using Eqs. (1)–(4) we can convert the circuit diagram in Figure 2 to the block diagram in Figure 3. Figure 3 is easier to simulate since the mixed electrical and mechanical domains are irrelevant. Few simulation software applications allow for mixed domain signals. SPICE is a typical example of this. Simscape™ does allow for mixed domain signals but nonlinear multivalued capacitors are not supported. Putting the model in block diagram form allows for easy integration in the Matlab® and Simulink® programming environment which supports simulation of both a structural model for the mechanical impedance,

At steady state and with the boundary conditions at both ends of the PZT free (i.e. no structural stiffness to oppose the PZT produced force,

with

where both

To determine the nonlinear capacitance,

where

Eqs. (7) and (8) together with the measured charge,

From the block diagram in Figure 3 we have,

Plugging in

We can solve for the gain,

where

Eqs. (10)–(12) form the basis of the experiments described in Section 3.3 for determining the working capacitance,

The strain gauge voltage change is the difference between the voltage at the start of the test and at the time that the peak force was applied.

## 3. Experiments

### 3.1 Displacement sensing for test campaign

Both experiments described below require sensing of the PZT elongation. To do this we use the strain gauge bonded to the PZT. For testing purposes we applied a DC voltage (10 Volts) as the excitation input to the bridge and configured a differential amp on the output of the bridge. The gain of this amplifier was set to 311.4 which amplified the millivolt signal of the bridge output to the 2–3 Volt range which was easily measurable by our test equipment. The amplified voltage was calibrated to the actual displacement of the PZT using a micrometer with 0.5 micron accuracy (Mitutoyo MDH high accuracy digimatic micrometer, series 293.). A linear fit of the calibration data, measured displacement vs. measured voltage, revealed a fit error of 0.71 microns for the worst case PZT with scale factors of 3.077 (

### 3.2 Experiment 1: instron testing

As discussed above we use the force balance Eq. (6) to determine the transducer constant,

Each of the three PZTs was tested with and without the load resistor to see how much resistive force is created by the charge feedback. Tests done with the resistor bleeds off any charge that would otherwise develop and leaves only the mechanical stiffness to oppose the applied compressive force. In addition, to verify repeatability of the experimental results each tested configuration was repeated three times. Table 1 summarizes the results of these tests. Note that with the load resistor the peak displacements are substantially greater than the cases without the load resistor even though the peak loads applied are similar. This is the result of charge feedback generating an opposing force beyond that provided by the mechanical stiffness alone. It may be surprising to find that the electrical force produced is close to or even greater than the force produced by the mechanical stiffness when the resistor is not used. Plugging in the peak force,

R In | SN637 | SN637 | SN637 | SN629 | SN629 | ||||
---|---|---|---|---|---|---|---|---|---|

(Run 1) | (Run 2) | (Run 3) | (Run 1) | (Run 2) | (Run 3) | (Run 1) | (Run 2) | (Run 3) | |

Strain Gauge Voltage Change (V) | −0.925 | −1.040 | −0.958 | −1.112 | −0.918 | −0.944 | −0.885 | −0.914 | −0.926 |

Peak Displacement (um) | |||||||||

Peak Force (N) | −772.2 | −778.1 | −774.3 | −799.5 | −738.6 | −773.3 | −792.8 | −798.8 | −809.8 |

Peak Back EMF (V) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

(Run 1) | (Run 2) | (Run 3) | (Run 1) | (Run 2) | (Run 3) | (Run 1) | (Run 2) | (Run 3) | |

Strain Gauge Voltage Change (V) | −0.415 | −0.413 | −0.426 | −0.382 | −0.381 | −0.384 | −0.334 | −0.334 | −1.078 |

Peak Displacement (um) | |||||||||

Peak Force (N) | −798.8 | −797.1 | −834.3 | −807.2 | −795.8 | −800.0 | −808.8 | −815.4 | −844.0 |

Peak Back EMF (V) | 11.68 | 12.32 | 12.55 | 11.98 | 12.51 | 12.63 | 12.20 | 12.21 | 12.15 |

### 3.3 Experiment 2: free-free test

To perform the free-free test on the PZT, the applied voltage, resulting current applied to the PZT and PZT displacement all need to be measured synchronously. To do this we used a Keysight Technologies B2902A precision source to sample the applied voltage and current. This instrument was designed to measure I/V measurements easily and accurately. To measure the PZT displacement the output of the bridge differential amp was measured with a Tektronics oscilloscope. To synchronize the two data sets we used the peak value in each data record. These signals are shown in Figure 5 together with the applied charge which is just the integrated current. Before each test a relay was used to temporarily short the leads of the PZT to drain any charge. This put the PZT in its rest state prior to each test.

To validate our instrumentation we compared the current and charge reported by our test setup using a simple 5.5

Using the PZT position and charge measurements shown in Figure 6 along with the previously estimated value of the transducer constant,

Before determining the nonlinear function,

#### 3.3.1 Hysteresis model

To fit the capacitive hysteresis data a neural network of backlash basis functions is employed of the form,

where

## 4. Complete model

To validate the PZT model described in this chapter we exercise the model in Figure 3 by applying a full scale (0–100 Volt) sinusoidal input voltage,

## 5. Modulated strain gauge model

We have seen from the test data that the behavior of the PZT is nonlinear due to its hysteresis. Many applications of PZT actuators require greater positioning accuracy than what can be achieved with open loop operation of the PZT. To overcome this accuracy issue, it is common to use strain gauges bonded directly to the face of the PZT. These sensors are used with feedback control to greatly improve the positioning accuracy of the actuator. Typically the accuracy can be improved from 12–15 percent down to 0.1 percent by using strain gauge sensors. Strain gauge sensors also suffer from hysteresis (This is what limits their accuracy.) but the amount of hysteresis is much less than the actuator.

Before designing any control loop that uses strain gauge feedback, it is necessary to understand the output response of the strain gauge voltage versus the input strain. To do this, we first characterize the resistive changes of the four Wheatstone bridge resistors as a function of the PZT elongation. The results of this analysis can then be used to derive the sensitivity of the bridge output. This analysis assumes that the strain gauge bridge is modulated with a square wave reference at its excitation terminals. The modulation is done in order to reduce noise pickup in the cabling between the bridge location and PZT processing electronics board which can be separated by several meters in typical applications. Any noise with a frequency content well below the modulation frequency of 10 kHz can be eliminated with this technique. Demodulation can be done in analog electronics but here we describe a case where the demodulation is done in FPGA firmware.

### 5.1 Bridge resistance changes

The stain gauge configuration we study is a full Wheatstone bridge, meaning that all four resistive elements of the bridge see a change in strain due to the elongation of the PZT. Two of the bridge elements, on opposite diagonals of the Wheatstone bridge, are bonded to the PZT along the length of the PZT which is the direction of intended length change. The remaining two bridge elements are bonded to the PZT perpendicular to this, along the width of the actuator. These two bridge elements will also see a resistance change when the PZT elongates due to the fact that as the PZT lengthens it also gets more narrow. The circuit diagram of the Wheatstone bridge and layout of the bridge elements when bonded to the PZT is shown in Figure 1.

To characterize the exact relationship between the PZT elongation and resistive change of each bridge element we conducted a test where the resistance measurement across each strain gauge resistor was measured for several increments of PZT displacement from its rest length to its maximum displacement. Each resistance measurement is the equivalent resistance of the element across the probe terminals in parallel with the other three elements, i.e.,

These measurements must be solved for the individual bridge elements by solving the system of equations in (14)–(17) for each of the four bridge element resistances. This system of equations is nonlinear but can be solved uniquely using a nonlinear equation solver. Mathematica was used for this purpose. Solving this system of equations resulted in three complex solutions and one real solution for each of the resistances on the right hand side. Since the resistance must be real, the real solution was taken as the answer.

The results of these measurements are shown in Figure 10 for PZT SN618. The two strain gauges that are oriented along the length of the PZT increase their resistance as the PZT elongates since this deformation lengthens the gauge and thins the wires of the bridge. The two strain gauges oriented along the width of the PZT decrease in resistance since, in this direction, the PZT narrows. As it narrows the wires of these bridge resistors become shorter and thicker reducing their resistance. Interestingly, the ratio of the slopes of the horizontally mounted gauges to the vertically mounted gauges is equal to the aspect ratio of the PZT,

Taking into account the nominal resistance, variation in nominal resistance and strain induced resistance change, the resistance of each bridge element can be written as,

where * gauge factor*of the bridge element (

### 5.2 Bridge sensitivity

Referring to Figure 9 we can derive the bridge sensitivity using simple circuit analysis. Writing out the constitutive relation for each resistor we have,

where

Substituting the resistances, Eqs. (18)–(21), into Eq. (26) and letting

Note that the output voltage of the Wheatstone bridge is actually weakly nonlinear in the PZT displacement,

To derive the linearized gain of Eq. (27), we can use the first term of a Taylor series,

Evaluating this derivative at

### 5.3 Simplified model

After the bridge, a differential amplifier is used to boost the voltage to the ±2.5 Volt range of the A2D converter (COBHAM RAD1419 with 14 bits). This sampling is done at high rate, 800 kHz, to capture the high and low modulation levels of the 10 kHz square wave. This gives 40 samples during each portion of the modulation signal. These samples are averaged and then differenced before being operated on in firmware. The FPGA firmware applies a linear transformation to this signal in order to map it to the desired DAC range of ±2.6 Volts for the feedback servo. This bias and scale factor trimming can be done in analog electronics but is more accurately done in firmware. The full signal chain from PZT displacement to demodulated firmware voltage is shown in the top subfigure of Figure 11. The gain of the differential amp cannot map its output voltage exactly to the rails of the A2D since the intrinsic imbalance of the gauge and resulting bias voltage at zero PZT displacement prevents this.

For purposes of control loop design a simplified model of the strain gauge response is useful. If we ignore exogenous signals, such as the gauge noise and bias trim, which do not effect the loop gain, a simplified linear model of gauge response is shown in the bottom subfigure of Figure 11. The bridge gain, the differential amp gain of 300, demodulation gain of 2 and trim scale factor are all indicated in this figure. The demodulation gain is 2 since the firmware takes the difference between the same two voltages with opposite sign. Eqs. (27) and (28) give the sensitivity of each of these voltage levels and not their difference.

## 6. Conclusions

In this work we have developed a full electro-mechanical model of piezoelectric actuators and determined the parameters of this model. Unique experiments were designed to determine the transducer constant and capacitances of the model. The hysteretic capacitance was fit with backlash basis functions which was proposed by the first author in [15]. This hysteresis model is numerically efficient and captures the multivalued behavior of hysteresis as well as the curvature of the hysteresis loops. The output of the PZT model agreed well with the experimental data and successfully predicts the current draw of the actuator which is an important feature of the model for comparison against power limits and slew rate requirements.

For control design actuator models are only half of what is required. Models of the sensors used is also important. In this work we focused on the use of strain gauge sensors which are commonly used with PZT actuators. A nonlinear model of the strain gauge full bridge was developed from which a linearized model was generated. This linear model included the effects of Wheatstone bridge sensitivity, differential amplification, demodulation and firmware scaling.

Although not the focus of this work, the PZT model that we have developed and experimentally identified could easily be included into a FEM of the structure that the PZT is intended to move. This has been done for the fast steering mirror (FSM) used by the Nancy Grace Roman Space Telescope. This FSM has three PZTs that are used to actuate special flexures that amplify the PZT elongation. This amplified motion is used to move a mirror flat in tip, tilt and piston.

One issue that is often over looked with piezo devices is the creep that is produced by these devices. This makes open loop operation with these actuators very challenging. With sensing the creep is usually slow enough to be effectively cancelled by using feedback. Nonetheless, a full piezo model with creep has not been successfully developed to the knowledge of the authors. Augmentations to the backlash elements presented in this work have shown promise in this area but further investigations are necessary.

## Acknowledgments

The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The author wishes to thank the Nancy Grace Roman Space Telescope Project for funding this work.

## Conflict of interest

The authors declares no conflict of interest.