**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, *Vpea*, and record the model output, *xpzt*. This data can then be compared with the same data acquired during the experiments described in Section 3.3. This comparison is made in **Figure 9** which shows good agreement between the experimental data and model data. In addition, comparisons of the experimentally measured current and modeled current signal also demonstrated similar agreement. Note the hysteresis percentage in **Figure 9**, as defined by the vertical thickness of the hysteresis loops relative to the peak displacement, is lessened for the output signal of the model relative to the percentage of hysteresis demonstrated in **Figure 7**. This is most likely *Modeling of Piezoceramic Actuators for Control DOI: http://dx.doi.org/10.5772/intechopen.96727*

#### **Figure 9.**

*Experimental and modeled input/output data for the full PZT model.*

due to the voltage feedback terms, *Vp* and *Vr*, in **Figure 3** that act to reduce the uncertainty caused by the capacitive hysteresis, *Ch*ð Þ� .

## **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.,

$$R\_{m1} = R\_{W-Y} \| R\_{Y-G} + R\_{G-B} + R\_{W-B} \tag{14}$$

$$R\_{m2} = R\_{W-B} \| R\_{W-Y} + R\_{Y-G} + R\_{G-B} \tag{15}$$

$$R\_{m3} = R\_{G-B} \| R\_{W-B} + R\_{W-Y} + R\_{Y-G} \tag{16}$$

$$R\_{m4} = R\_{Y-G} \| R\_{G-B} + R\_{W-B} + R\_{W-Y}.\tag{17}$$

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

#### **Figure 10.**

*Four bridge resistances as a function of PZT displacement. This data is from 1 of 3 PZTs tested. Note the difference in slopes of the four resistors, one negative slope for the two horizontally placed bridge elements and one positive slope for the two vertically placed bridge elements. Initial resistances at zero elongation reveals the intrinsic bridge imbalance due to manufacturing tolerances.*

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

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, *Wrest=Lrest*, where *Wrest* is the rest width of the PZT and *Lrest* is the rest length of the PZT. Also note that at zero elongation of the PZT, the actual resistance of each bridge element deviates from the nominal value of 350 Ohms. For all three full bridge strain gauges tested the variation from the nominal resistance for each bridge element was found to be 0.224 Ohms, 1-*σ* with a maximum error bounded by �0.4 Ohms. This was consistent with the spec. Sheet tolerance [16].

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,

$$R\_{W-Y} = \left(R\_{nom} + \Delta R\_{nom}^{W-Y}\right) - k\_{\rm gf} \cdot \left(R\_{nom} + \Delta R\_{nom}^{W-Y}\right) \cdot \left(\frac{1}{L\_{rest}}\right) \cdot \left(\frac{W\_{rest}}{L\_{rest}}\right) \cdot s\_{pxt} \tag{18}$$

$$R\_{G-B} = \left(R\_{nom} + \Delta R\_{nom}^{G-B}\right) - k\_{gf} \cdot \left(R\_{nom} + \Delta R\_{nom}^{G-B}\right) \cdot \left(\frac{1}{L\_{ret}}\right) \cdot \left(\frac{W\_{ret}}{L\_{ret}}\right) \cdot s\_{pzt} \tag{19}$$

$$R\_{Y-G} = \left(R\_{nom} + \Delta R\_{nom}^{Y-G}\right) + k\_{\mathcal{g}f} \cdot \left(R\_{nom} + \Delta R\_{nom}^{Y-G}\right) \cdot \left(\frac{1}{L\_{rest}}\right) \cdot s\_{pxt} \tag{20}$$

$$R\_{W-B} = \left(R\_{nom} + \Delta R\_{nom}^{W-B}\right) + k\_{\rm gf} \cdot \left(R\_{nom} + \Delta R\_{nom}^{W-B}\right) \cdot \left(\frac{1}{L\_{rest}}\right) \cdot s\_{pxt},\tag{21}$$

where *kgf* is the *gauge factor* of the bridge element (*kgf* ¼ 2*:*05 [11]) which relates the change in resistance to the strain, (*spzt=Lrest*). *Rnom* is the nominal bridge resistance and Δ*R<sup>W</sup>*�*<sup>Y</sup> nom* , Δ*RG*�*<sup>B</sup> nom* , Δ*R<sup>Y</sup>*�*<sup>G</sup> nom* , Δ*R<sup>W</sup>*�*<sup>B</sup> nom* are the variations from this nominal value. In the following treatment of bridge sensitivity, these intrinsic imbalance terms can be ignored since they are a small portion of the total resistance.

#### **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,

$$\mathbf{V}\_{mod}^+ - \mathbf{V}\_{sig}^+ = \mathbf{i}\_L \cdot \mathbf{R}\_{Y-G} \tag{22}$$

$$V\_{mod}^{+} - V\_{sig}^{-} = i\_R \cdot R\_{W-Y} \tag{23}$$

$$V\_{sig}^{+} - V\_{mod}^{-} = i\_L \cdot R\_{G-B} \tag{24}$$

$$V\_{s\dot{g}}^{-} - V\_{mod}^{-} = \dot{\imath}\_{L} \cdot R\_{Y-G},\tag{25}$$

where *iL* and *iR* are the currents indicated in **Figure 1**. Solving this system of equations for *iL*, *iR*, *V*<sup>þ</sup> *sig* and *V*� *sig* as functions of the two excitation voltages *V*<sup>þ</sup> *mod* and *V*� *mod* gives the output voltage, *Vout* ¼ *V*<sup>þ</sup> *sig* � *V*� *sig* , as,

$$\begin{split} V\_{\text{out}} &= V\_{\text{sig}}^{+} - V\_{\text{sig}}^{-} = \frac{R\_{W-B}}{R\_{W-B} + R\_{W-Y}} V\_{\text{mod}}^{+} + \frac{R\_{W-Y}}{R\_{W-B} + R\_{W-Y}} V\_{\text{mod}}^{-} \\ &- \frac{R\_{G-B}}{R\_{G-B} + R\_{Y-G}} V\_{\text{mod}}^{+} - \frac{R\_{Y-G}}{R\_{G-B} + R\_{Y-G}} V\_{\text{mod}}^{-} .\end{split} \tag{26}$$

Substituting the resistances, Eqs. (18)–(21), into Eq. (26) and letting *V*<sup>þ</sup> *mod* ¼ *Vmod* and *V*� *mod* ¼ �*Vmod* we can express the output voltage of the bridge as a function of the displacement *xpzt*,

$$W\_{out} = f(s\_{p\text{pt}}) = \frac{2V\_{mod}k\_{\text{gf}}(L\_{rest} + W\_{rest})\mathbf{x}\_{p\text{pt}}}{\left(2L\_{rest}^2 + k\_{\text{gf}}L\_{rest}\mathbf{x}\_{p\text{pt}} - W\_{rest}k\_{\text{gf}}\mathbf{x}\_{p\text{pt}}\right)}.\tag{27}$$

Note that the output voltage of the Wheatstone bridge is actually weakly nonlinear in the PZT displacement, *xpzt*, which is a bit surprising. Also note that the signal level of the output is directly proportional to the modulation voltage, *Vmod*. This implies that the resolution of the device can be arbitrarily increased with larger excitation voltages. The cost of this increased resolution would be an increased thermal signature. The thermal signature can be an issue in some applications since it can cause excessive drift of the electronics and warping of any optics near the bridge.

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

$$\frac{\partial V\_{\text{out}} - f\left(\mathbf{x}\_{\text{pzt}\_o}\right)}{\mathbf{x}\_{\text{pzt}} - \mathbf{x}\_{\text{pzt}\_o}} = \frac{\partial f}{\partial \mathbf{x}\_{\text{pzt}}}\bigg|\_{\mathbf{x}\_{\text{pzt}} = \mathbf{x}\_{\text{pzt}\_o}} = \frac{\left(4L\_{\text{rest}}^2 V\_{\text{mod}} k\_{\text{gf}} \mathbf{x}\_{\text{pzt}} \left(L\_{\text{rest}} + W\_{\text{rect}}\right)\right)}{\left(2L\_{\text{rest}}^2 + k\_{\text{gf}} L\_{\text{rest}} \mathbf{x}\_{\text{pzt}} - k\_{\text{gf}} W\_{\text{rect}} \mathbf{x}\_{\text{pzt}}\right)^2 \Big|\_{\mathbf{x}\_{\text{pzt}} = \mathbf{x}\_{\text{pzt}\_o}}}.\tag{28}$$

Evaluating this derivative at *xpzto* ¼ 7*:*5ð Þ *μm* gives a gain of 354.18 (V/m) between the change in PZT elongation from *xpzto* and the change in differential bridge output voltage, *Vout* � *f xpzto* .

#### **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

#### **Figure 11.**

*Circuit schematic (top subfigure) and simplified block diagram of the strain gauge processing electronics (bottom subfigure). Modulation of the Wheatstone bridge is done with a 10 kHz square wave with 2 Volts applied to the reference excitation terminal and* � *2 Volts applied to the reference ground terminal. These voltages are swapped every 50 usec. The bridge voltages are sampled at 800 kHz and demodulated in firmware.*

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

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.

### **Acknowledgements**

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.
