US EuroTek, Inc.

2. Electro-Mechanical relations of piezoelectric actuators

2.1. Principle structures of Piezoelectric actuators
Generally, all piezoelectric devices/transducers such as stacks, bimorphs, tubes can be described as a kind of capacitor with an electromechanically active dielectric medium:
the PZT ceramic. Therefore, the electrical capacitance of such devices is an important operating parameter, especially when adapting the supply electronics for dynamic operation. The electrical capacitance of Piezoelectric actuators is shown in the data sheet. The strain within the PZT-medium is related to the internal electrical field strength when a voltage is applied to the element. An important consequence for practical consideration is, that the thinner the PZT-layers are, the lower can be the driving voltage. Furthermore the degree of lamination determines the electrical capacitance of Piezoelectric actuators.

fig-2a
Fig. 2a:
Schematic representation of the capacitive layer structure of a Piezoelectric stack

fig-2b
Fig. 2b:
Discretely built-up piezoelectric stack (high voltage type), external contact electrodes visible

Low voltage actuator types are operated with maximum voltages ranging from 50 V to 150 V, whereas the high voltage elements require hundreds of volts up to 1000 V (and more). For standard stacks the achieved maximum strain is about 1-1.5%  of the stack length. There exist highstrain stacks based on optimized PZT-materials showing a strain of 2% and more at fieldstrength of 3 kV/mm.

2.2. Polarity of piezoelectrical elements
For piezoelectrical components an electrical polarity is usually defined. Piezoelectrical actuators e.g. stacks can only achieve their maximum response by applying the maximum voltage with correct polarity. Operation with counterpolarity voltage although possible is limited to remarkably lower ratings. A stack actuator shrinks under these conditions, increasing thereby to some  extent the total moving range of the stack (see brochure "Piezoelectric stackactuators ). Bare Piezoelectric stacks without casing are usually electrically insulated at the mechanical mounting points. They are supplied with pigtails showing the polarity by red(+) and black(-) insulation. Such elements can be combined therefore with positive or negative voltage supplies without difficulty. The situation changes, when actuators with casing are used. Here, the ground is defined by the coax-cable and therefore the polarity of the supply voltage is fixed.
Piezoelectric actuators with casing and the supply electronics from US EuroTek, Inc. are designed for positive polarity both for low voltage and high voltage actuators. This supports the easy combination of higher voltage actuators with lower voltage supplies, which is an important aspect for dynamic operation of actuators (see section 2.5.).

2.3. Operating characteristics of Piezoelectric actuators
The expansion of piezoelectric actuators is illustrated by voltage/expansion diagrams showing the well-known hysteresis (fig.3).

fig-3.jpg (18381 bytes)
Fig. 3:
Relative voltage/expansion diagram of a free running Piezoelectric actuator for different voltage reversal points

Actuators are normally classified by the maximum applicable voltage for maximum stroke, and characterized as low voltage and high voltage types. For newcomers in piezotechnology, this sometimes gives the impression, that the voltage rating of an actuator is the sole criterion for selecting a proper electronic supply. This is however not correct.

For any application of Piezoelectric actuators the electrical power/ current balance for charging and discharging the Piezoelectric actuators capacitance has to be kept in mind. The variety of electrical supplies on offer is due mainly to the different power/current ratings of these devices.

The charge/current balance during operation is related to the capacitive nature of actuators as shown below:
Basic capacitor equation
0(t) = C U(t)
C - actuator's capacitance
Q - actual electrical charge
U - applied voltage

Obviously the expansion of an actuator is also related to the quantity 0 of electrical charge stored in the actuator's capacitance C, when a voltage U is applied.
From this charge balance, the kinetic parameters of motion like speed and acceleration can be derived. These relations are the base for specifying the necessary current/power for distinct driving conditions.
Actuator's position I - charge  = Q(t)
Speed v ~ current I=dQ/dt = Q(t)
Acceleration b ~ variation of current = dl/dt Q(t)

The generation for example of a sine-wave oscillation by a Piezoelectric actuator requires a defined supply current depending on actuator's capacitance and moving amplitude. Therefore an amplifier has to be selected for both criteria: voltage and current.
Another consequence of the above is that, during a steady state of the actuator (constant position, constant force) no current is flowing, therefore no power is required. When a charged actuator is disconnected from the supply, it holds its position. This is an important difference to electromagnetic systems, where a constant position requires constant electrical power due to the sustaining current.
The speed of an actuator cannot be increased infinitely even by very high currents, but is limited by the elastic properties of the stack. The maximum speed of stacked elements is in the range of a few m/sec.
Because of the very limited moving range of Piezoelectric actuators the generation of above speeds requires high acceleration rates up to 104-105g.
During operation of a piezodriven mechanical setup for highly dynamic application, it has to be verified that the mechanics coupled to the actuator shows a sufficiently high stiffness/ resonant frequency, otherwise the mechanics cannot follow actuator's motion and it is fruitless to optimize the drive for high speed/acceleration.

2.4. Peak current, average current
Piezoelectric actuators require electrical power/current only during dynamic operation. Expansion and contraction are characterized by charging/discharging currents.
The short term available maximum peak current of a supply determines the minimum risetime/maximum speed of an actuator. Amplifiers of the series LE provide a special booster stage for high peak currents to get minimum risetimes. The average current of a supply determines the long-term cw-repetition rate of charging/discharging an actuator.
For cw sine oscillation of an actuator, the required peak and average currents show a fixed ratio of approx. 3:1. Therefore, the selection of a supply to obtain a distinct cw-actuator frequency has to consider both, peak and average current data.

2.5. Power efficiency
This section will lead on the first glance to the (surprising) result, that it is sometimes very reasonable and necessary to combine a high voltage actuator with a low voltage supply, where only a fraction of the actuator's maximum amplitude can be achieved.
The reason for this strategy are twofold:
· optimizing power efficiency of a dynamically operated actuator system
· minimizing selfheating of a dynamically operated actuator.
The basic idea is easily demonstrated with the following example, where the task requires the generation e.g. of a +/-2,5 pm sine oscillation with a distinct frequency:
The first example uses an actuator type PSt 500/5/5, where 500 V has to be applied to get the full stroke of 5 pm. A second example is to use the longer stack PSt 500/5/15 capable for a 15 pm motion at 500 V, showing an actuator's capacitance 3 times larger than in the 1st case.
The important fact is, that with the longer stack only 150 V are needed to get the desired 5 pm stroke. Comparing the actuators' energy content 1/2 CU2 respectively, despite its larger capacitance the longer stack is favored regarding power efficiency as only 1/3 of the power necessary to drive the shorter PSt 500/5/5 with full strain is required. It is obvious, that a 150 V system's total power efficiency is further improved by using a 150 V supply showing higher current output compared to a 500 V supply operated at reduced voltage rating.
In the above described strategy, the problem of selfwarming under dynamic operating conditions is minimized by the reduced power input and by distribution of the dissipated energy over a larger volume/surface of the longer actuator. This is a powerful method to extend the application range of Piezoelectric actuators to high frequency cw-operation without the risk of overheating.
This strategy of dynamic operation of actuators with reduced strain shows restrictions in other operating parameters: A longer stack has a lower stiffness and resonance, and it has to be determined, whether this is acceptable for a distinct application.
Finally, an important contribution to the overall power efficiency of an actuator system is the use of recharger amplifiers (switched amplifiers).
In most applications, Piezoelectric actuators display mainly a reactive load, where the energy content of a charged actuator flows back to the amplifier during the discharging cycle. Switched amplifiers RCV are able to recycle this energy with high efficiency, so that the needed linepower for a dynamically operated system has only to cover the (much smaller) active part of the power balance.
This active power is drawn from the system as mechanical power or dissipated by the selfheating of the actuators. This technique shows the optimum of system's overall power efficiency, and favors actuator applications, where high power levels are required e.g. for active vibration cancellation in heavy mechanical structures (vehicles, airplanes etc.) or anywhere, where the power consumption from the power supply is restricted i.e. battery operated systems. Power efficiency n is defined as
n= (Pr-Pal)  Pr = reactive power output from amplifier
       (Pr)    Pal = active power consumption from line

An ideal amplifier without internal losses shows an efficiency 1.

2.6. Frequency response
The performance of an amplifier is characterized by its frequency response, describing what cw-frequency/amplitude relations that can be achieved for a defined capacitive load. The achievable maximum frequencies of an actuator/supply- system depend both on the output power of the supply, the capacitance of the driven actuator and the oscillation amplitude. To make the selection of an amplifier/actuator combination with respect to frequency response easier, some response curves for different capacitive loads are shown in the data sheet for distinct amplitudes. The response for intermediate capacitances are achieved by simple interpolation. An additional figure for an amplifier's performance is the achievable minimum risetime, which is tabulated for some load capacitances (see section 2.4.).

2.7. Voltage stability, noise
One of the most striking features of Piezoelectric actuators is their unlimited positioning sensitivity, which explains the sub-nanometer resolution for example scanning tunnel microscopes:

A infinitely small voltage step triangle.GIF (53 bytes) U is transformed into infinitely small mechanical shift triangle.GIF (53 bytes) I.
triangle.GIF (53 bytes) I = I triangle.GIF (53 bytes) U/U = actuator's shift for signal voltage U
Neglecting external influences, the positioning sensitivity of an actuating system is limited only by the stability of the electronic supply (noise).

Example:
The amplifiers SQV 150 show a noise of approx. 1 mV equivalent to a S/N ratio of about 105. A 100 µm actuator such as the PSt 150/7/1 00 VS 12 operated with the SQV 150 shows a variation in position of only 1 nm.

2.8. Pulsed operation of Piezoelectric actuators
An important feature of Piezoelectric actuators is their capability to produce extreme forces and acceleration rates, which can be used for fast switching of valves or to produce mechanical shocks. In such cases, the actuator should switch in as short time as possible between 2 distinct levels, whereas the exact motion profile between these levels is not important. The minimum risetime of an actuator can derived from its elastic properties:
A short electrical pulse excites the resonant oscillation of the actuator and the minimum risetime Tp can be estimated by

Tp ~ Tr/3
Tr = period time of actuator's resonance
Tp = minimum risetime in pulsed operation
Example:
A systems's resonant frequency of 3 kHz results in a minimum mechanical risetime of about 100 µsec.

A simple calculation shows, that above shown pulse generation requires peak powers up to the kilowatts range with currents of 10 to 100 Amperes. In these cases it is reasonable not to use analogue amplifiers but electronic pulse switches. The common design of a HV-pulse generator consists of a high voltage supply, which continuously charges at a defined low power (i.e. 50 Watt) a large internal charge storing capacitor.
This capacitor delivers short term the very high currents to the external Piezoelectric actuator capacitance, when it is switched by transistors via a load resistor P. The load resistor P acts as current limiter to avoid electrical overpowering and defines the time constant RC of the pulser (rise/fall-time) according the well-known relation

Ua = U0 (1-e-t/RC)
R = internal load resistor of switch
C = capacitance of external load (Piezoelectric actuator)
Ua = voltage level at actuator
Uo = supply voltage from internal charge storing capacitor

For the operation of the HVP's 3 time constants have to be distinguished:
· Switching time of output transistors:
order of magnitude: 1 µssec
defines the minimum electrical pulsewidth
· Time constant RC:
Defines the signal/voltage risetime at the actuator. Pulsewidths shorter RC lead to a partial charging of the actuator and thereby to intermediate positions between "low and "high
· Period time of actuator/actuated system (fig. 4.):
This time constant defines the minimum mechanical rise/fall-time of the system.
To excite the minimum mechanical risetime Ip of an actuator, the PC time constants of the pulsed system has to be shorter than Tp.

fig-4
Fig. 4:
Excitation of a mechanical pulse by a voltage step OV/Uo; Io final static position of actuator

2.9. Feedback controlled systems
Piezoelectric actuators are well-suited for setting up electronically controlled systems for fast and precise handling of mechanical parameters such as position, speed and force. Because of the hysteretic and slightly nonlinear behavior of Piezoelectric actuators, and nonpredictable external influences, this has to be done by feedback control. A sufficiently fast and sensitive transducer picks up the actual position or other parameter of interest and the signal is evaluated by feedback control electronics, producing the control signal for the actuator.
The overall efficiency e.g. precision of such a system is determined by the transducer and electronics and not by the actuator. The high performance of feedback controlled systems is demonstrated by the atomic resolution of the scanning tunnel microscopes (STMs).

A further application is the active stabilization of mechanical arrangements e.g. laser resonators against misalignment due to the thermal drifts or mechanical shocks.

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