Modeling of Piezoceramic Actuators for Control

*Joel Shields and Edward Konefat*

### **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 *μm*, which for some applications, necessitates the use of dual stage actuation [2, 3] or inchworm actuators [4] which combine brake PZTs with actuation PZTs to extend the dynamic range of these devices. In addition, PZTs suffer from significant amounts of hysteresis (15 percent of full stroke) and creep which causes drift of the device over time. These two issues make open loop operation of these devices problematic since multiple displacements are possible for the same input and due to the fact that creep dissipates at very slow rates. To overcome these problems, it is possible to invert the hysteresis nonlinearity [5] but this can only improve accuracy by one order of magnitude and leaves the creep issue unresolved. This has led many to employ servo control with displacement sensing of the PZT elongation [6–8]. This will linearize the device to the accuracy of the sensor, and address creep, but the hysteresis nonlinearity has an impact on the loop stability which must be analyzed. In a linear sense, the hysteresis has an effect on the both the loop gain and phase margin. These impacts necessitate high fidelity modeling of the PZT behavior, which can be quite rich and faceted. The model described in this chapter is intended to be used as a necessary precursor for control design and simulation.

Many models for PZT stacks have been developed in the literature, such as those in [8–10], 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, 11–13], 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 *μm*), subnanometer positioning is required for these applications. PZTs come in a wide variety of shapes and configurations. Here we focus on piezo stack actuators which are the most commonly used. In particular, we tested and modeled the P-888.51 PICMA stack multilayer piezo actuator [14] which is ubiquitous in its use for various missions at JPL. This PZT has dimensions of 10 mm 10 mm 18 mm and has a nominal stroke of 15 *μm* at 100 Volts across its terminals. It has a blocking force of up to 3600 Newtons and a mechanical stiffness of 200 *N=μm*. Nominal voltage range applied to the PZT is 0–100 Volts, but it is possible to apply between 20 - 120 Volts for extended stroke. These devices are operated with large preloads to keep the device in compression during operation. Dynamic loads that put the ceramic device in tension need to be avoided since they are brittle and fracture easily in tension. The recommended preload is 15 MPa which for the 10 mm 10 mm cross sectional area of the device is 1500 Newtons. The first mode of this particular PZT is 70 kHz which is far greater than the first mode of any structure it is generally incorporated into.

*Modeling of Piezoceramic Actuators for Control DOI: http://dx.doi.org/10.5772/intechopen.96727*

**Figure 1.**

*From left to right, circuit diagram of strain gauge Wheatstone bridge showing resistive elements labeled by the color of wires on each side of the resistor, physical layout of resistive elements and wiring connections, and strain gauge elements bonded to the PZT.*

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 1**. 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 (*Vpea*) history due to the nonlinear capacitance, *Ch*ð Þ� . As current, *q*\_, flows into the circuit charge develops across the working

#### **Figure 2.**

*Nonlinear electro-mechanical PZT model. Features of this model include the nonlinear capacitance, Ch*ð Þ� *, and transducer constant, Tem, which governs the flow of energy into and out of the PZT.*

capacitor, *Cp*, which due to the *inverse piezoelectric effect* produces a force, *Fp* in the piezoelectric stack. Conversely, any environmental force, *Fe*, which causes velocity of the PZT stack generates a current, *q*\_ *<sup>p</sup>*, in the circuit due to the *piezoelectric effect*.

The unknown model parameters in **Figure 2** include the two capacitances, *Cp* and *Ch*ð Þ� , and the transducer constant, *Tem*. The resistor value, *R*, is known by design (80 Ohms) and can be easily measured. The mechanical stiffness of the PZT, *kp*, can be obtained from Physik Instrumente (PI) spec. Sheets [14]. The damping coefficient, *dp*, of the PZT ceramic is a structural parameter of the PZT and not part of this investigation. It is primarily a parameter that effects the high frequency behavior of the PZT in terms of changing the modal response. Here we are more concerned with the low frequency, nonlinear behavior of the device.

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

$$V\_{ped} = V\_r + V\_h + V\_p \qquad \qquad (\text{KVL}) \tag{1}$$

$$
\dot{q} = \dot{q}\_c + \dot{q}\_p \tag{1CL}
\\
\tag{2}
$$

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

$$q = \mathbf{C}\_h(\mathbf{V}\_h) \qquad \qquad q\_c = \mathbf{C}\_p \mathbf{V}\_p \tag{3}$$

$$W\_r = i\_r R \qquad \text{with} \qquad q(t) = \int\_0^t i\_r(t)dt. \tag{4}$$

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, *M*ð Þ� , and hysteretic capacitance, *Ch*ð Þ� , using primitive Simulink components. Looking at **Figure 3** KVL is represented with the first two summing nodes. The center summing node is depicting an integrated version of KCL, Eq. (2). Since the voltage across the resistor, *Vr*, is proportional to the current, *q*\_, the charge, *q*, needs to be differentiated, thus we use the Laplace operator, *s*, in **Figure 3** to represent this. The saturation block on

#### **Figure 3.**

*Block diagram of PZT model showing the inverse piezo effect and piezo effect with charge feedback. The nonlinear capacitance function, Ch*ð Þ� *, is modeled with backlash operators. The mechanical impedance, M*ð Þ� *, can be as simple as a spring representing the mechanical stiffness of the PZT material or a FEM of the structure the PZT is embedded into.*

*Modeling of Piezoceramic Actuators for Control DOI: http://dx.doi.org/10.5772/intechopen.96727*

current is included to capture current limits of the PZT amplifier. As an example of this, the flight amplifier for the Nancy Grace Roman Space Telescope has a limit of +2 mA for sourcing of current into the PZT and � 14 mA for sinking of current out of the PZT.

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, *Fp*.), the mechanical impedance of the PZT, *M*ð Þ� , reduces to 1*=kp*. The displacement of the PZT can then be written as,

$$\propto\_{pzt} = \frac{1}{k\_p} \left( F\_\varepsilon + F\_p \right), \tag{5}$$

with *Fp* ¼ *TemVp*. With the terminals of the PZT open circuit *Vp* ¼ *Vpea* assuming the device has been shorted and drained of all charge prior to applying an environmental force. Since *Vpea* is a measurable signal this allows us to solve for *Tem* using Eq. (5),

$$T\_{em} = \frac{k\_p \varkappa\_{pxt} - F\_e}{V\_{pea}},\tag{6}$$

where both *xpzt* and *Fe* can also be measured. This equation forms the basis of the experiments described in Section 3.2.

To determine the nonlinear capacitance, *Ch*ð Þ� , and working capacitance, *Cp*, we collect input/output data while applying a voltage, *Vpea*, to the PZT with the environmental force, *Fe*, set to zero. To determine the hysteresis of this block over its full range of inputs, we apply a sinusoidal *Vpea* voltage that goes from 0–100 Volts. This voltage is applied slowly so that we can assume that *M*ð Þ� ≈1*=kp*. The output signal of *Ch*ð Þ� , *q*, can be obtained by integrating measurements of the current flowing into the PZT as a result of the voltage, *Vpea*, applied to the PZT terminals. The input signal to *Ch*ð Þ� , *Vh*, can be solved for using KVL,

$$V\_h = V\_{\text{peak}} - V\_r - V\_p,\tag{7}$$

where *Vr* ¼ 0 since during this test the resistor *R* is not used. *Vp* in Eq. (7) can be solved for by using Eq. (5) with *Fe* ¼ 0 and *Fp* ¼ *TemVp*. This leads to,

$$\mathbf{V}\_p = \frac{\mathbf{x}\_{p\mathbf{z}t}\mathbf{k}\_p}{T\_{em}} \tag{8}$$

Eqs. (7) and (8) together with the measured charge, *q*, form the basis of the experiments described in Section 3.3 for determining the nonlinear capacitance, *Ch*ð Þ� .

From the block diagram in **Figure 3** we have,

$$V\_p = \frac{1}{C\_p} \left( q - T\_{em} \mathbf{x}\_{pxt} \right). \tag{9}$$

Plugging in *Fe* ¼ 0 and *Fp* ¼ *TemVp* with *Vp* given by Eq. (9) into Eq. (5) we have,

$$\kappa\_{pxt} = \underbrace{\frac{T\_{em}}{k\_p C\_p + T\_{em}^2}}\_{k\_{q2x}} q \,. \tag{10}$$

We can solve for the gain, *kq*2*x*, in a least squares sense,

$$k\_{q2\text{c}} = \left(\mathbf{q}^T\mathbf{q}\right)^{-1}\mathbf{q}^T\mathbf{x}\_{\text{ppt}} \tag{11}$$

where **q** and **x***pzt* are vectors of the sampled signals *q* and *xpzt*. The working capacitance, *Cp*, is then,

$$C\_p = \frac{T\_{em} - T\_{em}^2 k\_{q2\kappa}}{k\_p k\_{q2\kappa}}\tag{12}$$

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

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 (*μm=V*), 3.183 (*μm=V*) and 3.246 (*μm=V*) for SN637, SN629 and SN618 respectively. Calibrated steady state measurements of the strain gauge had an RMS of 22 nm which was more than sufficient for our testing campaign.

#### **3.2 Experiment 1: instron testing**

As discussed above we use the force balance Eq. (6) to determine the transducer constant, *Tem*. The environmental force, *Fe*, in this equation was applied using an Instron 8801 Servohydraulic load frame which is part of the JPL Materials Testing and Failure Facility. Refer to **Figure 4** for a picture of the PZT test jig. To avoid potential damage to the load frame the hydraulic press was operated in displacement mode which left force an uncontrolled, but measured parameter (see **Figure 5**). To measure the back EMF term, *Vpea*, in this equation a voltmeter with high input impedance is required. Without a high impedance voltmeter the RC decay time constant would be much faster than the duration of the experiment and compromise the data. We used a Keithley 617 high resistance meter with 200 TeraOhm input impedance to keep charge from bleeding off the PZT leads. To measure the PZT displacement an Agilient 34401A DMM was used to monitor the amplified bridge output voltage. The applied compressive load to the PZT was measured with the load cell that is incorporated into the load frame.

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 *Modeling of Piezoceramic Actuators for Control DOI: http://dx.doi.org/10.5772/intechopen.96727*

#### **Figure 4.**

*PZT installed in the Instron jig between two compression platens. The load frame is used to apply a compressive load of 800 (N) while the PZT displacement, back EMF, and applied load are measured. The PZT is compressed with and without a 200 Ohm resistor applied across the leads of the PZT. This resistor is shown on the right terminal block in the open circuit configuration.*

#### **Figure 5.**

*PZT signal set recorded during each free response test. The charge signal was obtained by integrating the current. The position signal was obtained by sampling the strain gauge voltage and applying the calibration described in Section 3.1.*

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, *Fe*, the peak displacement, *xpzt*, and peak back EMF, *Vpea*, from **Table 1** into Eq. (6), and using the spec. Sheet value [14] of the mechanical stiffness, *kp* ¼ 200 ð Þ *N=μm* , we can solve for the transducer


#### **Table 1.**

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

constant, *Tem*. The mean value of *Tem* over each PZT and over the three runs for each PZT was 46.71 (N/V) or (C/m). The variance of this estimate was minimal.

#### **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 *μF* capacitor against a SPICE simulation with the same capacitor value. The peak charges agreed to within 2 percent providing confidence to the current measurements.

Using the PZT position and charge measurements shown in **Figure 6** along with the previously estimated value of the transducer constant, *Tem*, and stiffness, *kp*, taken from the PI spec. Sheet [14], we can use Eqs. (10)–(12) to solve for the working capacitance, *Cp*. This resulted in a value of 5.802 *μF* for SN637 which is close to the small signal (1 *Vpp*) effective capacitance value listed in the PI spec. Sheet of 6.0 *μF* 20%. Note the effective capacitance in our model is the series

*Modeling of Piezoceramic Actuators for Control DOI: http://dx.doi.org/10.5772/intechopen.96727*

#### **Figure 6.**

*Force trajectories during operation of the load frame in displacement mode. Each of the three PZTs was tested three times with the load resistor, Rin cases, and three times without the load resistor, Rout cases. Since force was not a controlled variable during the test it had a noticeable but tolerable amount of variation or jitter.*

combination of *Cp* with *Ch*ð Þ� which is *CpCh*ð Þ� *<sup>=</sup> Cp* <sup>þ</sup> *Ch*ð Þ� � �. Thus, whatever the value of *Ch*ð Þ� it can only act to reduce the effective capacitance from the baseline value of *Cp*.

Before determining the nonlinear function, *Ch*ð Þ� , the input signal, *Vh*, is generated using Eqs. (7) and (8). This revealed that the voltage drop across the hysteretic capacitance, *Vh*, was approximately one-third of the applied voltage, *Vpea*. This leaves two-thirds of the applied voltage across the working capacitor to produce work. The output of the nonlinear function, *Ch*ð Þ� , is given by the charge measurements, *q*. The input and output data is shown in **Figure 6** which demonstrates a great deal of hysteresis. Note that creep between the two applied sinusoids infects the data as can be seen in the lower left of **Figure 7**. This portion of the data was not fit by the model described below.

#### *3.3.1 Hysteresis model*

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

$$q = \sum\_{i=1}^{N} B\_i \left( V\_h, \boldsymbol{\varkappa}\_i^o, \boldsymbol{w}\_i \right) \cdot k\_i \tag{13}$$

where *Bi Vh*, *xo <sup>i</sup>* , *wi* � � is a backlash operator with input variable, *Vh*, initial condition, *x<sup>o</sup> <sup>i</sup>* , and deadband width, *wi*. Each backlash output is weighted with a gain *ki* which changes the slope of the backlash operator when the deadband is engaged. Details of fitting the parameters of this model to experimental data are given in [5, 15] but, to summarize, the gains *ki* are used to fit the curvature of the hysteresis data, the schedule of deadband widths, *wi*, are used to accurately capture regions of the hysteresis that have high curvature and the initial conditions are used to capture the uniqueness of the initial loading curve. A diagram of the hysteresis model is provided in **Figure 8**.

#### **Figure 7.**

*Capacitive hysteresis data and model fit. Linearized approximations are shown with the large signal approximation of 30.5 μF.*

#### **Figure 8.**

*Diagram of capacitive hysteresis model used to fit the data in Figure 7. Initial state and deadband width are indicated for the Nth element of the model. The output arm displacement of each backlash element is multiplied by gain ki and summed with the other backlash elements to form the output variable q.*
