

Electrical Requirements for Piezo Operation
General When operated well below the resonant frequency, a piezo actuator behaves as a capacitor: The actuator displacement is proportional to stored charge (first order estimate). The capacitance of the actuator depends on the area and thickness of the ceramic, as well as on its material properties. For piezo stack actuators, which are assembled with thin, laminar wafers of electroactive ceramic material electrically connected in parallel, the capacitance also depends on the number of layers.
The smallsignal capacitance of a stack actuator can be estimated by:
(Equation 14)
Where:
C = capacitance [F (As/V)]
n = number of layers =
ε_{33}T = dielectric constant [As/Vm]
A = electrode surface area of a single layer [m^{2}]
d_{S} = distance between the individual electrodes (layerthickness) [m] I_{0} = actuator length The equation shows that for a given actuator length, the capacitance increases with the square of the number of layers. Therefore, the capacitance of a piezo actuator constructed of 100 µm thick layers is 100 times the capacitance of an actuator with 1 mm layers, if the two actuators have the same dimensions. Although the actuator with thinner layers draws 100 times as much current, the power requirements of the two actuators in this example are about the same. The PI highvoltage and lowvoltage amplifiers in this catalog are designed to meet the requirements of the respective actuator types.
Static Operation When electrically charged, the amount of energy stored in the piezo actuator is E = (1/2) CU^{2} Every change in the charge (and therefore in displacement) of the PZT ceramics requires a current i:
(Equation 15)
Relationship of current and voltage for the piezo actuator
Where:
i = current [A]
Q = charge [coulomb (As)]
C = capacitance [F]
U = voltage [V]
t = time [s]
For static operation, only the leakage current need be supplied. The high internal resistance reduces leakage currents to the microamp range or less. Even when suddenly disconnected from the electrical source, the charged actuator will not make a sudden move, but return to its uncharged dimensions very slowly.
For slow position changes, only very low current is required.
Example: An amplifier with an output current of 20 µA can fully expand a 20 nF actuator in one second. Suitable amplifiers can be found using the “Control Electronics Selection Guide”.
Note The actuator capacitance values indicated in the technical data tables are smallsignal values (measured at 1 V, 1000 Hz, 20 °C, unloaded) The capacitance of piezoceramics changes with amplitude, temperature, and load, to up to 200 % of the unloaded, smallsignal, roomtemperature value. For detailed information on power requirements, refer to the amplifier frequency response curves in the “Piezo Drivers & Nanopositioning Controllers” section.
Dynamic Operation (Linear) Piezo actuators can provide accelerations of thousands of g’s and are ideally suited for dynamic applications.
Several parameters influence the dynamics of a piezo positioning system:
 The slew rate [V/s] and the maximum current capacity of the amplifier limit the operating frequency of the piezo system.
 If sufficient electrical power is available from the amplifier, the maximum drive frequency may be limited by dynamic forces (“Dynamic Operation”).
 In closedloop operation, the maximum operating frequency is also limited by the phase and amplitude response of the system. Rule of thumb: The higher the system resonant frequency, the better the phase and amplitude response, and the higher the maximum usable frequency. The sensor bandwidth and performance of the servocontroller (digital and analog filters, control algorithm, servobandwidth) determine the maximum operating frequency of a piezoelectric system.
 In continuous operation, heat generation can also limit the operating frequency.
The following equations describe the relationship between amplifier output current, voltage and operating frequency. They help determine the minimum specifications of a piezo amplifier for dynamic operation.
(Equation 16)
Longterm average current required for sinusoidal operation
(Equation 17)
Peak current required for sinusoidal operation
(Equation 18)
Maximum operating frequency with triangular waveform, as a function of the amplifier output current limit
Where:
i_{a}* = average amplifier source/sink current [A]
i_{max}* = peak amplifier source/sink current [A]
f_{max} = maximum operating frequency [Hz]
C** = piezo actuator capacitance [Farad (As/V)]
U_{pp} = peaktopeak drive voltage [V]
f operating frequency [Hz]
The average and maximum current capacity for each PI piezo amplifier can be found in the product technical data tables.
Example Q: What peak current is required to obtain a sinewave displacement of 20 µm at 1000 Hz from a 40 nF HVPZT actuator with a nominal displacement of 40 µm at 1000V?
A: The 20 µm displacement requires a drive voltage of about 500 V peaktopeak. With Equation 17 the required peak current is calculated at » 63 mA. For appropriate amplifiers, see “Piezo Drivers & Nanopositioning Controllers”.
The following equations describe the relationship between (reactive) drive power, actuator capacitance, operating frequency and drive voltage.
The average power a piezo driver has to be able to provide for sinusoidal operation is given by:
(Equation 19)
Peak power for sinusoidal operation is:
(Equation 20)
Where:
P_{a} = average power [W] P_{max} = peak power [W] C** = piezo actuator capacitance [F] f = operating frequency [Hz] U_{pp} = peaktopeak drive voltage [V] U_{max} = nominal voltage of the amplifier [V]
It is also essential that the power supply be able to supply sufficient current.
* The power supply must be able to provide enough current. ** For largesignal conditions a margin of 70% of the smallsignal value should be added.
Dynamic Operating Current Coefficient (DOCC) Instead of calculating the required drive power for a given application, it is easier to calculate the drive current, because it increases linearly with both frequency and voltage (displacement). For this purpose, the Dynamic Operating Current Coefficient (DOCC) has been introduced. The DOCC is the current that must be supplied by the amplifier to drive the piezo actuator per unit frequency (Hz) and unit displacement. DOCC values are valid for sinewave operation in openloop mode. In closedloop operation the current requirement can be up to 50% higher.
The peak and longterm average current capacities of the different piezo amplifiers can be found in the technical data tables for the electronics, the DOCC values in the tables for the piezo actuators.
Example: To determine whether a selected amplifier can drive a given piezo actuator at 50 Hz with 30 µm peaktopeak displacement, multiply the actuator’s DOCC by 50 x 30 and compare the result with the average output current of the selected amplifier. If the current required is less than or equal to the amplifier output, then the amplifier has sufficient capacity for the application.
Dynamic Operation (Switched) For applications such as shock wave generation or valve control, switched operation (on / off) may be sufficient. Piezo actuators can provide motion with rapid rise and fall times with accelerations in the thousands of g’s. For information on estimating the forces involved, see “Dynamic Forces”.
The simplest form of binary drive electronics for piezo applications would consist of a large capacitor that is slowly charged and rapidly discharged across the PZT ceramics.
The following equation relates applied voltage (which corresponds to displacement) to time.
(Equation 21)
Voltage on the piezo after switching event.
Where:
U_{0} = start voltage [V]
U_{pp} = source output voltage (peaktopeak) [V]
R = source output resistance [ohm]
C = piezo actuator capacitance [F]
t = time [s]
The voltage rises or falls exponentially with the RC time constant. Under quasistatic conditions, the expansion of the PZT ceramics is proportional to the voltage. In reality, dynamic piezo processes cannot be described by a simple equation. If the drive voltage rises too quickly, resonance occurs, causing ringing and overshoot. Furthermore, whenever the piezo actuator expands or contracts, dynamic forces act on the ceramic material. These forces generate a (positive or negative) voltage in the piezo element which is superimposed on the drive voltage. A piezo actuator can reach its nominal displacement in approximately 30 % of the period of the resonant frequency, provided the controller can deliver the necessary current.
The following equation applies for constantcurrent charging (as with a linear amplifier):
(Equation 22)
Time to charge a piezoceramic with constant current. With lowercapacity electronics, amplifier slew rate can be a limiting factor.
Where:
t = time to charge piezo to U_{pp} [s]
C = piezo actuator capacitance [F]
U_{pp} = voltage change (peaktopeak) [V]
i_{max} = peak amplifier source/sink current [A]
For fastest settling, switched operation is not the best solution because of the resulting overshoot. Modern techniques like InputShaping^{®} solve the problem of resonances in and around the actuator with complex signal processing algorithms.
Note Piezo drives are becoming more and more popular because they can deliver extremely high accelerations. This property is very important in applications such as beam steering and optics stabilization. Often, however, the actuators can accelerate faster than the mechanics they drive can follow. Rapid actuation of nanomechanisms can cause recoilgenerated ringing of the actuator and any adjacent components. The time required for this ringing to damp out can be many times longer than the move itself. In timecritical industrial nanopositioning applications, this problem obviously grows more serious as motion throughputs increase and resolution requirements tighten.
Classical servocontrol techniques cannot solve this problem, especially when resonances occur outside the servoloop such as when ringing is excited in a sample on a fast piezo scanning stage as it reverses direction. A solution is often sought in reducing the scanning rate, thereby sacrificing part of the advantage of a piezo drive.
A patented realtime feedforward technology called InputShaping^{®} nullifies resonances both inside and outside the servoloop and thus eliminates the settling phase. More information here.
Heat Generation in a Piezo Actuator in Dynamic Operation PZT ceramics are (reactive) capacitive loads and therefore require charge and discharge currents that increase with operating frequency. The thermal active power, P (apparent power x power factor, cos j), generated in the actuator during harmonic excitation can be estimated with the following equation:
(Equation 23)
Heat generation in a piezo actuator.
Where:
P = power converted to heat [W]
tan d = dielectric factor (» power factor, cos j, for small angles d and j)
f = operating frequency [Hz]
C = actuator capacitance [F]
U_{pp} = voltage (peaktopeak)
For the description of the loss power, we use the loss factor tan d instead of the power factor cos j, because it is the more common parameter for characterizing dielectric materials. For standard actuator piezoceramics under smallsignal conditions the loss factor is on the order of 0.01 to 0.02. This means that up to 2 % of the electrical “power” flowing through the actuator is converted into heat. In largesignal conditions however, 8 to 12 % of the electrical power pumped into the actuator is converted to heat (varies with frequency, temperature, amplitude etc.). Therefore, maximum operating temperature can limit the piezo actuator dynamics. For large amplitudes and high frequencies, cooling measures may be necessary. A temperature sensor mounted on the ceramics is suggested for monitoring purposes.
For higher frequency operation of highload actuators with high capacitance (such as PICA™Power actuators), a special amplifiers employing energy recovery technology has been developed. Instead of dissipating the reactive power at the heat sinks, only the active power used by the piezo actuator has to be delivered.
The energy not used in the actuator is returned to the amplifier and reused, as shown in the block diagram in Fig. 26. The combination of lowloss, highenergy piezoceramics and amplifiers with energy recovery are the key to new highlevel dynamic piezo actuator applications.
For dynamic applications with low to medium loads, the newly developed PICMA^{®} actuators are also quite well suited. With their high Curie temperature of 320 °C, they can be operated with internal temperatures of up to 150 °C.




Drawings & Images: 

Fig. 25. Design of a piezo stack actuator.
Fig. 26. Block diagram of an amplifier with energy recovery for higher frequency applications.


