QNP3 Hardware Manual
Piezo Engineering Tutorial
C.3.5 Capacitance
PZT actuators can be modeled electrically as a capacitor. The principle equation that describes a capacitor
in terms of geometry and material properties is:
Equation 5
where
C
is capacitance (units of F),
A
is the cross-sectional area of the capacitor perpendicular to the
direction of the electric field (units of m
2
),
T
is the thickness of the dielectric material separating the charge
(units of m), and
ε
is the material permittivity of the dielectric material separating the charge. The material
permittivity is described as:
Equation 6
where
ε
0
is the permittivity of a vacuum (~8.85 x 10
-12
F/m), and
ε
r
is the relative permittivity of the material
(also called the dielectric constant).
Low-voltage, multi-layer actuators are generally used for nanopositioning because they allow for 0.1% to
0.15% nominal strains with low voltages (<200 V). The maximum applied electric field across these
actuators are in the range of 1-4 kV/mm. Because these actuators are constructed from thin layers (typically
50 to 200 μm thick) separated by electrodes, the resulting applied voltages are lower (<200 V) compared to
high-voltage actuators (~1000 V) where layer thickness is ~1 mm. The thickness of each layer (
T
layer
) can be
defined as the overall active length of the piezo actuator (
L
o
) divided by the number of layers (
n
). The piezo
stack capacitance of a multi-layer actuator can then be expressed as a function of the number of layers (
n
)
and the overall active length (
L
o
), as follows:
Equation 7
Typical capacitances of low-voltage, multi-layer piezo actuators used in nanopositioning applications are
between 0.01 to 40 μF. The capacitances specified in Aerotech data sheets are measured at small signal
conditions (1 V
rms
at 1 kHz). For larger signal operation (100-150 V), an increase in capacitance by as much
as 60% should be expected. This capacitance increase should be used when performing sizing calculations
(see Section 5).
The current (
i
) flowing through a capacitor (
C
) is proportional to the change in voltage with respect to time.
This is mathematically represented as:
Equation 8
This simple relationship will be needed to adequately size amplifiers required to drive piezoelectric stages
(see Section 5).
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Appendix C
49
