**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:*
Schematic representation of the capacitive layer structure of a Piezoelectric stack
*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:*
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 10^{4}-10^{5}g.
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 CU^{2 }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 ** **U is transformed into infinitely
small mechanical shift I.**
**I = I
** **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 = U_{0 }(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:*
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. |