ELECTROMECHANICAL TRANSDUCERS
17
Table 1.2 Electromechanical mobility analogies [42]
Mechanical parameter Electrical parameter
Variable Velocity, angular velocity Voltage
Force, torque Current
Lumped network elements Damping Conductance
Compliance Inductance
Mass, mass moment of inertia Capacitance
Transmission lines Compliance per unit length Inductance per unit length
Mass per unit length Capacitance per unit length
Characteristic mobility Characteristic impedance
Immitances Mobility Impedance
Impedance Admittance
Clamped point Short circuit
Free point Open circuit
Source immitances Force Current
Velocity Voltage
ABCD matrix form as:

˙x
1
F
1

=


cos βx jZ
0
sin βx

p
(1.3)
v
p
=

E
ρ
=
1
√
C
l
M
l
(1.4)
In these equations j =
√
−1, ˙x
1
and ˙x
2
are velocities, F
1
and F
2
forces at two ends of
a transmission line, Z
0
, β and v

V
2
I
2

(1.5)
18
MEMS AND RF MEMS
Table 1.3 Direct analogy of electrical and mechan-
ical domains
Mechanical quantity Electrical quantity
Force Voltage
Velocity Current
Displacement Charge
Momentum Magnetic flux linkage
Mass Inductance
Compliance Capacitance
Viscous damping Resistance
Source: Tilmans, 1996.
In Equation (1.5) V and I are the voltage and current on the transmission line (with
subscripts representing its ports). The other quantities in the matrix are also represented
by equivalent electrical parameters as:
Z
0
=

µ
ε
=


An understanding of the underlying operational principle is essential in obtaining the
equivalent circuit of these transducers. A brief description of the operational principles
of some of these common transduction mechanisms used in electromechanical systems is
provided below.
1.4.1 Piezoelectric transducers
When subjected to mechanical stress, certain anisotropic crystalline materials generate
charge. This phenomenon, discovered in 1880 by Jaques and Pierre Curie, is known
as piezoelectricity. This effect is widely used in ultrasonic transducers. Lead zirconate
titanates (PZTs) are the most common ceramic materials used as piezoelectric transducers.
These crystals contain several randomly oriented domains if no electric potential is applied
during the fabrication process of the material. This results in little changes in the dipole
moment of such a material when a mechanical stress is applied. However, if the material
is subjected to an electric field during the cooling down process of its fabrication, these
domains will be aligned in the direction of the field. When external stress is applied
to such a material, the crystal lattices get distorted, causing changes in the domains
ELECTROMECHANICAL TRANSDUCERS
19
≈
f
p
C
0
C
1
L
1
jX
(c)
C
0


1983 Wiley
and a variation in the charge distribution within the material. The converse effect of
producing strain is caused when these domains change shape by the application of an
electric field.
The development of the equivalent circuit for a piezoelectric bar is illustrated in
Figure 1.7 (Johnson, 1983). The bar vibrates in the direction (with force F and veloc-
ity ˙x) shown in the figure, by the application of an applied voltage (V ). The reactance
(j X) curve in Figure 1.7(b) can be obtained by ignoring higher order modes of vibration,
and the losses. One circuit configuration that results in similar reactance characteristics
is shown Figure 1.7(c). The electromechanical equivalent circuit can be constructed from
this, incorporates a gyrator with a resistance A and an inverter of reactance jκ in addi-
tion to the corresponding spring constant K and mass M. The gyrator represents the
nonreciprocal nature of the piezoelectric transducer. The inverter is required here since
the gyrator converts the parallel resonant circuit to a series circuit (Johnson, 1983). The
series combination of inverter and gyrator functions as a transformer with an imaginary
turns ratio jκ/A.
In general the piezoelectric transduction phenomenon is quadratic in nature, but may
be assumed to be linear for small deformations. The electromechanical coupling can then
be written as
Q = d
1
F (1.8)
x = d
2
V (1.9)
In these equations, d
1
and d
2

permittivity of the material.
PZT thin films have been developed using standard thin-film deposition techniques such
as sputtering, and physical or chemical vapor deposition. Their use in sensors and actuators
is inherently limited by the quality and repeatability of thin films obtained by these
techniques. Compared with bulk material processing techniques thin-film performance is
severely hampered by the surface properties where the film is deposited (Muralt, 2000).
Nonferroelectric AlN thin films are also explored, for sensor applications where voltage
output is required. However, PZT thin films are still preferred as actuators. Compared
with other electromechanical conversion schemes these require low voltage input but have
generally low electromechanical conversion efficiency.
1.4.2 Electrostrictive transducers
Electrostriction is the phenomenon of mechanical deformation of materials due to an
applied electric field. This is a fundamental phenomenon present to varying degrees in
all materials, and occurs as a result of the presence of polarizable atoms and molecules.
An applied electric field can distort the charge distribution within the material, resulting
in modifications to bond length, bond angle or electron distribution functions, which in
turn affects the macroscopic dimensions of the material.
The electric field E and the electric displacement D in a material are related by
D = ε
0
E + P (1.13)
where ε
0
is the free space permittivity (= 8.85 × 10
−12
Fm
−1
)andP is the polarization
of the material.
Using conservation of energy, the first law of thermodynamics for a electrically deform-

E
k
− TS (1.15)
Taking the derivative of Equation (1.15) and making use of Equation (1.13) we get:
dG = dU −
T
ij
d
S
ij
−
S
ij
d
T
ij
− E
k
(dD
k
− dP
k
) − T dS − S dT(1.16)
Substituting for dU from Equation (1.14), this simplifies to:
dG =−
S
ij
d
T
ij

∂G
∂
T
ij
(1.19)
E
k
=
∂G
∂P
k
(1.20)
S =−
∂G
∂T
(1.21)
Assuming isotropic dielectric behavior, the Gibbs energy function for an elastic material
is given by (Hom et al., 1994):
G =−
1
2
s
P
ij kl
T
ij
T
kl
− Q
mnpq


|P|
P
s

2

(1.22)
The first term on the right-hand side describes the elastic behaviour of the material, s
P
being its elastic compliance at constant polarization. The electromechanical coupling is
denoted in the second term with the electrostrictive coefficients forming the matrix Q.The
last term is the dielectric behaviour of the material. P
s
is the spontaneous polarization,
and k is a material constant related to its dielectric constant. Since the material is assumed
to be isotropic, the magnitude of polarization is given as:
|P|=

P
k
P
k
(1.23)
22
MEMS AND RF MEMS
Temperature-dependent material coefficients used in Equation (1.22) such as s
P
, Q, P
s

(Chen and Gururaja, 1997).
Material compositions based on lead magnesium niobate [Pb(Mg
0.33
,Nb
0.67
)O
3
(PMN)]
are commonly used as electrostrictive transducers. Their properties have been studied
extensively (Pilgrim, 2000). Practical thin-film transducers using this approach are yet to
be realized. However, polymeric thin-film materials with compliant graphite electrodes
are shown to have excellent electrostrictive properties (Pelrine, Kornbluh and Joseph,
1998). These materials are capable of efficient and fast response with high strains, good
actuation pressures (up to 1.9 MPa), and high specific energy densities. In this case, the
electrostriction phenomenon is not due to the molecular dipole realignment (Heydt et al.,
1998). In these silicone film actuators, the strain results from external forces caused by
electrostatic attraction of their graphite compliant electrodes. Although their mechanism is
electrostatics based, these actuators are shown to produce much larger effective actuation
pressure than conventional air-gap electrostatics with similar electric field.
1.4.3 Magnetostrictive transducers
Certain ferromagnetic materials show deformation when subjected to a magnetic field.
This phenomenon, commonly known as magnetostriction, is reversible and is also called
the Joule and Villari effect. In their demagnetized form, domains in a ferromagnetic
material are randomly oriented. However, when a magnetic field is applied these domains
gets oriented along the direction of the field. This orientation results in microscopic
forces between these domains resulting in the deformation of the material. By reciprocity,
mechanical deformation can cause orientation of domains, resulting in induction at the
macroscopic level (Rossi, 1988). The elongation is quadratically related to the induced
magnetic field and hence is strongly nonlinear.
Apart from the ferroelectric bar, the magnetostrictive transducer consists of a coil and

∞
L
0
L
1
C
1
L
0
h : 1
Figure 1.8 Equivalent circuit for a magnetostrictive transducer. Reproduced from R.A. Johnson,
1983, Mechanical Filters in Electronics, Wiley Interscience, New York, by permission of Wiley,

1983 Wiley
in the direction shown with force F and velocity ˙x. The development of the equivalent
circuit of such a transducer is shown schematically in Figure 1.8. The reactance (j X)
diagram shown in Figure 1.8(b) is measured with no load. The pole and zero frequencies
in this curve correspond to parallel and series resonances of the system. It is not very
hard to obtain the component values of an LC circuit shown in Figure 1.8(c) which result
in the same pole and zero frequencies as with the system in Figure 1.8(a). Therefore
Figure 1.8(c) is an idealized electrical equivalent circuit for the transducer shown in
Figure 1.8(a). This is an idealized model as it does not take into consideration the losses
in the system.
It is now possible to translate this electrical equivalent circuit to the electromechanical
circuit shown in Figure 1.8(d). This has electrical and mechanical components (mass M
and spring K) connected with an electromechanical transformer. The turns ratio of this
transformer is decided by the amount of coupling, known as the electromechanical cou-
pling coefficient. This is defined as the ratio of the energy stored in the mechanical circuit
to the total input energy.
The electromechanical coupling for a magnetostrictive transducer shown in Figure 1.8(a)

used in magnetostrictive transducers. Some of these materials can also be deposited as
thin films thus making it possible to fabricate microactuators and sensors with them.
Amorphous thin films such as TbFe
2
,Tb
0.3
Dy
0.7
Fe
2
and DyFe
2
have been reported
in the literature (Body, Reyne and Meunier, 1997). The realization of such thin films
is more process dependent than their bulk counterparts, as the preparation conditions
affect the homogeneity and growth process of the film as well as its stoichiometry. RF
magnetron sputtering of METGLAS

2605-SC ribbon with a chemical composition of
Fe
81
Si
3.5
B
13.5
C
2
on a GaAs substrate has been used in a pressure sensor with figure of
merit comparable with that of conventional piezoresistive strain gauges (Karl et al., 2000).
Microelectromechanical filters using this technology have not been reported so far in the

C =
εA
d
=
εA
d
0
− x
= εA

d
0

1 −
x
d
0

−1
= C
0

1 −
x
d
0

−1
(1.29)