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J. Micromech. Microeng. 6 (1996) 157176. Printed in the UK
Equivalent circuit representation of
electromechanical transducers: I.
Lumped-parameter systems
Harrie A C Tilmans
Katholieke Universiteit Leuven, Departement ESAT-MICAS, Kardinaal Mercierlaan94, B-3001 Heverlee, Belgium
Received 7 December 1995, accepted for publication 28 December 1995
Abstract. Lumped-parameter electromechanical transducers are examinedtheoretically with special regard to their dynamic electromechanical behaviour andequivalent circuits used to represent them. The circuits are developed starting frombasic electromechanical transduction principles and the electrical and mechanicalequations of equilibrium. Within the limits of the assumptions on boundaryconditions, the theory presented is exact with no restrictions other than linearity.Elementary electrostatic, electromagnetic, and electrodynamic transducers areused to illustrate the basic theory. Exemplary devices include electro-acousticreceivers (e.g., a microphone) and actuators (e.g., a loudspeaker),
electromechanical filters, vibration sensors, devices employing feedback, and forceand displacement sensors. This paper forms part I of a set of two papers. Part IIextends the theory and deals with distributed-parameter systems.
1. Introduction
Electromechanical transducers are used to convert electrical
energy into mechanical (or acoustical) energy, and vice
versa. They are utilized for electrical actuation and sensing
of mechanical displacements and forces in a wide variety
of applications (see e.g. [13]). An illustrative example of
a sensing device is a microphone in which a sound pressure
is converted into an electrical signal. In a microphone, thepressure acts upon a spring-supported mass, which usually
consists of a stretched diaphragm. The generated mass
(diaphragm) displacement is next converted into an electric
output signal by means of an electromechanical transducer.
In a loudspeaker, on the other hand, an electromechanical
transducer is used to convert the electrical output signal
of an audio amplifier into a force acting on the speaker
diaphragm. This results in a displacement of the diaphragm,
thereby generating sound waves.
The behaviour of electromechanical transducers can
be described by the differential equation(s) of motion of
the structural member(s), by the characteristic equations
of the transducer element(s), and by a set of boundaryconditions. A very explanatory and quick way of gaining a
deeper insight into the dynamic behaviour of the transducer
is the equivalent circuit approach, in which both the
electrical and mechanical portions of the transducer (or
system) are represented by electrical equivalents. The
approach is based on the analogy that exists between
Present address: CP Clare Corporation, Overhaamlaan 40, B-3700
Tongeren, Belgium. e-mail: [email protected]
electric and mechanical systems [14]. In this method, the
transducer is no longer described by complex differential
equations and boundary conditions, but by a lumped-
element electrical circuit in which the elements are
physically representatives of the transducers properties
such as its mass, stiffness, capacitance, and damping.
The circuits implicitly contain, because of the way they
are constructed, all the equations governing the system
represented. To the extent that the original assumptions
are valid, the equivalent circuit can be considered an exact
representation of the electromechanical transducer. The
practicality of the equivalent circuit approach stems from
the field of electricity where it is unthinkable that the
design and analysis of electrical systems is carried out
on the basis of Maxwells equations. The applications
of lumped-element circuits are numerous nowadays, and
their use is strongly justified by modern electric network
theory which provides us with powerful mathematical
techniques and network analysis programs, such as SPICE.
Equivalent circuits are now also implemented for analysing
electromechanical systems, where one of their strengths is
that they provide a single representation of devices thatoperate in more than one energy domain. Noteworthy is
their proven indispensable value in the development of
piezoelectrically driven resonators for use as mechanical
sensors, timebases, and electromechanical filters [3, 5, 6],
and further also in the field of electroacoustics [2].
Equivalent circuits are particularly useful for the analysis
of systems consisting of complex structural members and
coupled subsystems with several electrical and mechanical
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H A C Tilmans
ports. Not only is the strict use of differential equations
very difficult for these cases, but this method also often
obscures the solution [3]. The equivalent circuit method
lends itself to a better visualization of the system, and, once
the basic circuit is constructed, it may be used in further
analyses to investigate the effects of connecting subsystems
or of making modifications to the structure.
The purpose of this paper is to lay a mathematical
foundation for developing equivalent circuit representations
of lumped-parameter electromechanical transducers and
for interconnecting the obtained circuits to the outside
world and further to illustrate their applicability in real
systems. The paper forms part I of a set of two papers.
Whereas part I deals with lumped-parameter systems, in
part II [7] the theory is extended to include distributed-
parameter systems as well. Only the steady-state small-
signal dynamic frequency response is considered, but the
circuits are equally well suited to analyse the transient
behaviour of the transducer system. Large-signal non-
linear behaviour cannot be analysed in a straightforward
manner using equivalent circuits and will not be dealt with
here. Therefore, the transducers that are intrinsically non-
linear are linearized around a bias point. The circuits
are constructed starting from basic electromechanical
transduction principles and the electrical and mechanicalequations of equilibrium. The focus will be on systems
with a single electrical and a single mechanical port.
Simple examples of how to account for more than one
port are given. The basic principles of electromechanical
transduction are not a subject of this paper. For a
description of this topic reference is made to the literature
(see e.g., the books by Woodson and Melcher [8] or Neubert
[1]). The analysis is limited to reversible or bilateral
transducers, i.e., those which give rise to mechanical
motion from electrical energy or the other way round
[1]. These include electrostatic, electromagnetic, and
electrodynamic and evidently exclude thermal transducers.
Further, only rectilinear or translational mechanical systems
are considered, although rotary systems can be described in
a similar way. Acoustic systems are briefly introduced and
are accounted for in the analysis. Finally it is pointed out
that generally applicable assumptions, such as negligible
fringing fields, small velocities compared with the velocity
of light, and perfect conductors (see e.g., [8]), are implicitly
understood.
2. Lumped-parameter electromechanical systems
2.1. General
The essential characteristic of lumped-parameter (or
discrete) systems is that the physical properties ofthe system, such as mass, stiffness, capacitance, and
inductance, are concentrated or lumped into single physical
elements. Thus, elements representing mass are perfectly
rigid, and conversely elastic elements have no mass. This
idealization is similar to electric circuit theory where
inductors are considered to have no capacitance, capacitors
no inductance, and resistors are purely ohmic. As such,
a lumped-parameter electromechanical system consists of a
finite number of interconnected masses, springs, capacitors,
inductors, resistors, etc. Lumped-parameter modelling is
valid as long as the wavelength of the signal is greater than
all dimensions of the system [2, 8]. The dynamic behaviour
of these systems can be described by ordinary differential
equations with time t being the only independent variable.
The analysis described below will focus on lumped-
parameter systems with a single degree of freedom (SDOF)
in the mechanical domain, implying that they display a
single mechanical resonance [9]. The study of SDOF
systems is of importance since (i) many real systems displaya behaviour sufficiently close to the behaviour of an SDOF
system, (ii) they improve the understanding of real systems,
whilst not having to deal with tedious mathematics, and
(iii) very often in a limited frequency range distributed-
parameter systems can be treated as SDOF systems, as will
be further clarified in part II [7].
2.2. An example
An example of a real electromechanical system that behaves
sufficiently like an SDOF lumped-parameter system is
shown in figure 1(a). The structure shown illustrates thebasic configuration of an electrostatic transducer that can
be used as a force gauge, e.g., a microphone [1, 2, 8]. It
consists of a doubly clamped beam (or diaphragm) with
a rigid mass at the centre. The mass is electroded on
one face which defines one plate of the capacitor used for
the electrostatic transduction. The other plate is formed
by a stationary surface which is in close proximity to the
mass. The mass is effectively supported by two adjacent
beam elements that can be approximated by lumped springs,
each with a constant k/2. The total spring constant is
therefore equal to k. The mass moves in response to a
pure mechanical force Fmtand/or a force of electric origin
Fet which is due to the attractive electrostatic force ofthe electrodes (nomenclature is explained below). The
circuit comprising the decoupling capacitor Cd and the
resistanceRL, which may represent the the input resistance
of an amplifier, is used to isolate the output terminals
from the bias voltage v0, and is not intended to affect
the dynamic behaviour of the system in normal operation.
If the voltage ve is taken as the output signal and with
driving frequency this means that RLCd 1 andRL R0[8]. Assuming that the total mass of the two beamsegments is small relative to the rigid mass and, further, if
only incremental signal variations around a biasing point
(further explained below) are considered, the system can
be modelled as the lumped-parameter SDOF system offigure 1(b). Here,m represents the rigid mass, k is the total
stiffness of the sections of the beam supporting the mass,
c is a parameter representing viscous energy losses, and
Cp denotes a (parasitic) parallel capacitance, e.g., due to
the stray capacitance of the interconnecting wires. Further
analysis of the transducer, including the development of
its equivalent circuit, will be presented in the following
subsections.
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Lumped-parameter electromechanical transducers
Figure 1. An example of an electromechanical transducerconsisting of a beam with a rigid mass at the centre, whichis subjected to a forceFetof electric origin and a puremechanical forceFmt. (a) A schematic representation; (b) alumped-parameterincrementalmodel. The transducer canbe used as a force (Fm) or displacement (xm)sensor, withoutput voltageve or currentie.
Figure 2. A schematic representation of a two-portelectromechanical transducer.
2.3. Energy exchange
Exchange of energy of a transducer and the outside world is
achieved through ports (depicted as a pair of terminals).
This is illustrated in figure 2, showing a block diagram of
the often encountered linear electromechanical transducer
with a single electrical and a single mechanical port. A
port is defined by a pair of conjugate dynamic variables
called effortor intensive variable and flow. The power
exchange through the port is given by the product of
effort and flow. The flow is given by the time derivative
of the corresponding state or extensive variable. The
transducer depicted in figure 2 is a two-port energy storage
element, emphasizing the number of ports and the fact that
transducers store energy. Electrical ports are defined by the
{voltage (v), current (i)} pair, and mechanical ports by the{force (F), velocity (u)} pair. Two-port storage elementsare completely characterized by an energy function of the
two independent state variables [8]. The state variables
associated with the mechanical and the electrical ports aredisplacement x and the electric charge q, respectively.
In describing transducers that are partly operating in the
magnetic domain it proves convenient to define a magnetic
port, characterized by the{current (i), flow of flux linkage()} pair. The flux linkagedefines the corresponding statevariable.
The purpose of this section is to derive equivalent
circuit representations of the two-port transducer in figure 2.
Although the transducer in figure 2 consists of only one
electrical and one mechanical port, the discussion below
can easily be generalized to any arbitrary number of ports
(see also [8]).
3. Elementary lumped-parameter transducers
3.1. Basic configurations
Four examples of elementary electromechanical lumped-
parameter transducers are shown in figure 3. Their op-
eration is respectively based on electrostatic (with out-of-plane motion), electrostatic (with in-plane motion), elec-
tromagnetic, and electrodynamic transduction principles,
each of which is extensively described in the literature
(see e.g., [1, 2,8, 10]). The transducer of figure 3(a) is
termed a transverse electrostatic transducer, emphasizing
that the plates move transverse or perpendicular to each
other, as opposed to the parallel electrostatic transducer of
figure 3(b) in which the plate surfaces stay parallel dur-
ing motion. Apart from the electrodynamic transducer, the
transducers shown all display non-linear behaviour. For
instance the electrostatic force of the transducers in fig-
ure 3(a) and (b) shows a quadratic dependence on the
charge or the voltage (see appendix A). It is evident thatlinear transducers are mathematically more tractable, but,
furthermore, linear transducers are also of great practical
importance. For instance, a condenser microphone is pur-
posely operated to behave as a linear device, since nonlin-
ear effects cause distortion and loss of fidelity [2]. Linear
behaviour is achieved for incremental or small-signal vari-
ations around bias or equilibrium levels. In fact, if only the
first two terms in a Taylor series expansion about a static
equilibrium point are included, the total signal, which is in-
dicated with a subscript t, can be written as the sum of an
equilibrium signal, which is indicated with a subscript 0,
and the incremental signal, which is indicated without any
subscript, e.g.,xt(t)=
x0+
x(t). Transducers can be biased
in several ways. For instance, an electrostatic transducer
can be electrically biased by applying a d.c. bias voltage
v0, or by introducing a bias charge q0, e.g., by means of an
electret. The electromagnetic transducer can be biased by
applying a bias current i0 in the transducer coil or by plac-
ing a permanent magnet in the magnetic circuit. The type
of biasing will not change the analysis of the system signif-
icantly (see e.g., the book by Rossi [2]). Therefore, without
loss of generality, in this paper the analyses are limited to
voltage biasing for the electrostatic transducer and current
biasing for the electromagnetic transducer. These biasing
schemes are also most often employed in practice. Biasing
of the electrodynamic transducer is always done with a per-
manent magnet. It is pointed out that the electrodynamictransducer is inherently linear and biasing is here imple-
mented to attain any electromechanical transduction at all.
It is further assumed that the incremental signals are sinu-
soidal with driving frequency , e.g., x(t)= x exp(it ),where x denotes a phasor [11]. Note that this is not a
real limitation, since for a linear system the steady-state
response to an arbitrary signal can be synthesized from the
response to sinusoidal driving signals using the techniques
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Table 1. Constitutive and transfer equations of the lumped-parameter transducers shown in figure 3, completed with astabilization springk at the mechanical port (as illustrated by figure 4). The matrix equations describe the relations betweenthe phasor quantities of the sinusoidal signals. The meaning of the symbols is explained in appendix B and in table 2.
Transducer State Efforts Flows
type q1,q2 e1,e2 f1,f2
e1e2
= B
q1q2
e1f1
= T
e2f2
Electrostatic
(transverse, q(t),x(t) v(t),F(t) i(t)= q(t), x(t)
1C0
C0
C0k
1
1i
(k 2C0
)
iC0
kC0
figure 3(a))Electrostatic
(parallel, q(t),x(t) v(t),F(t) i(t)= q(t), x(t)
1C0
C0
C0k+
2
C0
1
ki
iC0
k
C0+
2
k
figure 3(b))
Electromagnetic (t),x(t) i(t),F(t) (t)=v(t), x(t)
1L0
L0
L0k
1
1i
k 2
L0
iL0
kL0
(figure 3(c))
Electrodynamic (t),x(t) i(t),F(t) (t)=v(t), x(t)
1L0
L0
L0k+
2
L0
1
ki
iL0
k
L0+
2
k
(figure 3(d))
Table 2. Parameters used in table 1, expressed in terms of the dimensional parameters, the bias conditions, and physicalconstants.
Transduction factors [N V1 = A (m s1)1] and Static( 0)
Tranducer type Static components [N A1 = V (m s1]1) coupling factor
Electrostatic C0 q0v0 = 0Aed+x0
= q0d+x0
= 0Aev0(d+x0)
2
2
C0k
(transverse, figure 3(a))
Electrostatic C q0v0
= 0(l0x0)hd
= q0l0x0 =
0hv0d
1
1+C0k
2
(parallel, figure 3(b))
Electromagnetic L0 0i0 = N20Ae
d+x0 = 0
d+x0= N
20Aei0(d+x0)
2
L0k
(figure 3(c))
Electrodynamic L0 =Ls= coil seriesconductance = B0l
1
1+L0k
2
(figure 3(d))
domain. In fact the electromagnetic transducer and the
transverse electrostatic transducer are dual to each other
with respect to the electric domain. The same can
be said for the parallel electrostatic transducer and the
electrodynamic transducer. Dual systems are described by
equations of the same form, but in which the coefficients
(e.g., capacitance and inductance) and effort-flow variables
(e.g. voltage and current) are interchanged. The observed
electrical duality is physically described by Amperes
circuital law and Faradays law of magnetic induction,
conveniently expressed asv
i
=
0 N
1/N 0
m.m.f.
=
0 1
1 0
i
or :
i
=
0 1
1 0
v
i
(1)
where m.m.f. denotes the magnetomotive force, is themagnetic flux flow in the magnetic circuit, and N is given
by the number of active turns of the transducer coil coupled
with the magnetic field of the transducers in figure 3(c) and
(d). The above matrix equation clearly shows that the effort
variable in the electric domain becomes the flow variable
in the magnetic domain, and vice versa.
The coupling factor , also indicated in table 2, is
an important characteristic of electromechanical transducers
as it provides a measure for the electromechanical energy
conversion which takes place in the lossless transducer
[6,10, 12]. For a two-port storage element the coupling
factor can be found from the constitutive matrix B
as the following ratio: (coefficients product of the
interaction terms)/(coefficients product of the principal(diagonal) terms). A coupling factor of zero means no
electromechanical interaction. It can be shown that a
stable equilibrium exists for 0 < < 1 [10,12]. Typical
values for are in between 0.05 and 0.25. Furthermore,
the coupling factor provides an elegant way to relate the
parameter values, e.g., the spring constant, measured at one
of the ports to the conditions, e.g., v= 0, at the other port.This will be further explained in the next subsection.
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Table 3. Direct electromechanical analogies for lumpedtranslational systems [1, 2].
Mechanical quantity Electrical quantity
ForceF VoltagevVelocityu= x Currenti =qDisplacementx ChargeqMomentump Magnetic flux linkage Massm InductanceLCompliance 1/ka CapacitanceCViscous resistancec ResistanceR
a
k represents the spring constant.
3.3. Equivalent circuit representations
3.3.1. Analogies. The development of equivalent
circuit representations is based on the analogy in the
mathemtical descriptions that exists between electric and
mechanical (including acoustical) phenomena [1,2, 11].
The analogies are a result of the formal similarities of
the integrodifferential equations governing the behaviour of
electric and mechanical systems. For instance, Newtons
second law of motion relating the force F and velocity
u for a rigid mass m, F =
m du/dt =
m d2x/dt2, is
mathematically analogous to the constitutive equation of
an electric inductor, v= L di/dt= L d2q/dt2. In thisanalogy, the force F plays the same role as the voltage
v, the velocity u as the current i, and the displacement
x as the charge q. The mass m in mechanical systems
corresponds to the inductance L in electrical circuits. The
foregoing examples illustrate the so-called directanalogy,
summarized in table 3. It is pointed out that equivalent
systems that are constructed based on this type of analogy
display the duality property in the sense that across
or between variables are equated to through variables,
and, conversely, through variables are equated to across
variables. This means that force (a through variable) is
analogous to voltage (an across variable), and velocity (an
across variable) to current (a through variable). Hence, this
implies that the network topologies of the mechanical and
electrical circuits are not the same. A series connection in
the mechanical circuit becomes a parallel arrangement in
the equivalent electrical circuit, and vice versa. This will be
further elucidated by the examples presented in section 5.
The direct analogy was in fact implicitly understood in
the foregoing. The governing equations, however, can also
be written in a form that suggests an analogy between the
force and the current, between the velocity and the voltage,
between the mass and the capacitance, etc. This analogy
is called theinverseor mobility-typeanalogy (see e.g. [1]).
In order to avoid any confusion in this paper no furtherreference will be made to this latter form of analogy.
3.3.2. Equivalent networks. The construction of the
equivalent networks starts with the transfer matrices given
in the last column of table 1. This becomes clear after the
matrices are split into their constituent transfer matrices.
For instance, the transfer matrix of the electrostatic
transducer of figure 4 can be split as follows:v
i
=
1
1i
(k k)iC0
kC0
F
u
=
1 0
iC0 1
1/ 0
0
1 1i
(k k)0 1
F
u
(2)
where k = 2/C0 (see also table 4). In the matrixequation above, the centre matrix represents the transducer,
flanked by the matrices of the electrical admittance and the
mechanical impedance. Each of the constituent transfer
matrices can be represented by an equivalent network. Theoveral equivalent network consists of a cascade connection
of these networks and is shown in figure 5(a). It can
easily be shown that the network in figure 5(a) forms an
exact representation of the transfer matrix equation (2).
According to the aforementioned analogy a spring is
represented by a capacitor. The impedance (force/velocity)
of the spring k in figure 5(a) therefore is equal to k/i.
The electromechanical coupling is modelled through an
ideal electromechanical transformer with a transformer ratio
given by , called the transduction factor, which was
introduced in tables 1 and 2. The transformer relations,
are given by F = v and i = u, conform tothe sign conventions indicated in figure 5(a). Note the
existence of a spring with a negative constantk =2/C0 = 0Aev20/(d+ x0)3. The spring is a resultof the electromechanical coupling and apparently leads to
a lowering of the overal dynamic spring constant. This
is easily seen by combining the two springs into a single
spring with constant k= k k= k(1 2) (see alsotable 4). As long as k > 0, which is equivalent to thecondition
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Table 4. Circuit elements introduced as a result of electromechanical coupling. The elements are used in the equivalentcircuits of figures 58. Expressions for , ,C0, andL0 in terms of the physical parameters can be found in table 2.
Transducer type C0 ,L0 C
0,L
0 k
k
Electrostatic C0 = C012 =C0+ C
0 C
0 =
2
12 C0 = 112
2
k k =k(1 2)=k k 2k = 2
C0(transverse, figure 5)
Electrostatic C0 = C012 =C0+ C
0 C
0 =
2
12 C0 = 2
k k = k
12 =k+ k 2
12 k= 2
C0(parallel, figure 6)
Electromagnetic L0 = L012 =L0+ L
0 L
0=
2
12 L0 = 112
2
k k =k(1 2)=k k 2k = 2
L0(figure 7)
Electrodynamic L0 = L01
2 =L0+ L
0 L
0=
2
1
2L0 =
11
22
k k = k
1
2 =k+ k
2
1
2k=
2
L0
(figure 8)
Figure 5. Possible equivalent circuit representations of the transverse electrostatic transducer with a single electric port anda single mechanical port as depicted in figure 4. The meaning of the symbols is explained in tables 2 and 4. Thetransformers model the electromechanical coupling. The transformer relations given byF= v and i = u conform to thesign conventions given in (a).
transducer (including a springk), the decomposition can be
expressed asv
i
=
0 1
1 0
1 0
iL0 1
1/ 0
0
1 1i
(k k)0 1
F
u
(3)
wherek = 2/L0 (see also table 4). The first constituentmatrix represents (1) and can be modelled using an ideal
gyrator with gyrator resistance equal to unity. As a result
of the aforementioned duality (see subsection 3.2), the
equivalent circuit representing the cascade of the remainingthree matrices can easily be derived from the circuit in
figure 5(a). The overall equivalent circuit is obtained
by placing the gyrator in cascade with this circuit. The
result is shown in figure 6(a). It can easily be shown that
the circuits shown in figure 6(b)(d) also form equivalent
circuits of the same electromagnetic transducer. In the
latter two representations the gyrator and the transformer
are combined into a single gyrator, defined by the following
relations: F= i and v= u , conforming to the signconventions indicated in figure 6(c). The circuit parameters
indicated with a are clarified in table 4.A similar approach can be used to construct the
equivalent circuits for the parallel electrostatic and the
electrodynamic transducer. The results are shown in
figures 7 and 8, respectively. The two transducers are
clearly dual to each other. Also note the small, but
important, differences between the electromagnetic and the
electrodynamic transducer on the one hand, and between
the transverse and the parallel electrostatic transducer
on the other hand. For instance, the equivalent circuit
representation of the electrodynamic transducer shown infigure 8(a) can be obtained from the equivalent circuit of
the electromagnetic transducer of figure 6(c) by replacing
the compliance 1/k by a compliance 1/k.The parameters indicated with a are a result of
the electromechanical coupling. The circuit parameters
indicatedwithouta are the usually observed parameters,i.e., in the absence of electromechanical coupling (= 0).For zero coupling, there will be only one spring constant k,
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meaning that the relation between the incremental variables
can be described by linear (differential) equations with
constant coefficients. The equilibrium conditions at the
electrical site are governed by Kirchhoffs voltage (KVL)
and current (KCL) laws. At the mechanical site the
governing laws are Newtons second law of motion or,
more appropriately dAlemberts principle, expressed as
Fi= 0, and the geometric compatibility or the continuityof space, formulated as ui = 0 [8]. The lattercondition is seldom used for the analysis of mechanical
systems, but must at all times be satisfied. Note that
the latter two conditions are directly obtained by invoking
Kirchhoffs laws to the mechanical part in the equivalent
circuit representation, thereby illustrating once more the
analogy between electric and mechanical systems. As a
matter of fact, this mathematical correspondence of laws
is an essential requirement for using equivalent circuit
representations for the analysis of mechanical systems.
The foregoing forms the fundamental basis for devel-
oping equivalent circuits for electromechanical transducers
with arbitrary electric and mechanical loads and/or sources.
As an illustration, consider the system shown in figure 1(a)
and its lumped-parameter model of figure 1(b). The elec-
trical part is in fact already represented by an equivalent
network and therefore needs no further explanation. Forthe mechanical part the conditions may be derived as fol-
lows. Applying dAlemberts principle to the mass m leads
to
Fm(t ) Fe(t ) kx(t) cx(t) mx(t) = 0Fm Fe
k
iu cu imu = 0 (4)
where the second of the above equations is given in terms
of the phasors of the respective quantities. Recalling that
dAlemberts principle is the electromechanical analogue
of KVL, it is easy to show that the circuit shown in
figure 9 defines a possible equivalent circuit of the system
in figure 1. The equivalent circuit of figure 5(a) (enclosed
by the dashed box in figure 9) is chosen to represent
the transducer (plus stabilization spring k), because of its
direct physical significance. Note, however, that the circuit
enclosed by the box can be replaced by any of the other
circuits presented in figure 5. This may appear a little
awkward. For instance if the circuit of figure 5(b) is
used, the velocity through the compliance (capacitor) 1/k
as predicted by the equivalent circuit is not the same as
the velocity of the mechanical resistance c and the mass
(inductor) m. In practice however (see figure 1(b)) the
velocities through the aforementioned components are the
same. The answer to this apparent paradox is simple: the
compliance 1/k in the equivalent circuit is not the same
as the compliance 1/k in figure 1(b). They only happento be numerically equal. The internal configuration of the
circuit in figure 5(b) is reorganized at the expense of giving
up the practical meaning. It is evident that the physical
link is lost even further for the circuit in figure 5(d).
This emphasizes once more that the equivalent circuits
very often only serve an algebraic purpose. Based on the
foregoing, it is concluded that the circuit representations of
figures 5(a), 6(c), 7(a) and 8(a) have a true physical link,
thus minimizing the chance for misinterpretation as much
as possible. It is for this reason that emphasis will be on
these circuit representations while discussing the examples
presented in section 5.
5. Examples of lumped-parameter
electromechanical systems
5.1. Force and displacement transducers (microphone)
The basic operating principles and configuration of an
electrostatic force or displacement transducer have alreadybeen presented in subsection 2.2. A schematic diagram
is shown in figure 1. In operation, the force Fm(t) to be
measured (defined as positive in the positive x direction) is
exerted on the mass, or the mass is displaced by an amount
xm(t) in the case where the displacement must be measured.
As explained before a motion of the mass is converted
into an electrical signal, e.g., a current, which flows in
part through the resistors R0 and RL, thereby producing
an output voltage ve(t ). This voltage is a measure for
the applied force or displacement. Using the equivalent
circuit shown in figure 9, extended with the appropriate
mechanical sources (Fm or xm) which have to be connected
to the mechanical terminals, the steady-state analysis forsinusoidal signal operation becomes very straightforward.
The transfer function describing the relation between the
input variable and the output voltage is easily obtained from
the equivalent circuit. For the force transducer this results
in
ve
Fm= iRp
(k + ic 2m)(1 + iRp(C0 + Cp)) + iRp2(5)
and for the displacement transducer
ve
xm= iRp
1 + iRp(C0 + Cp)
=v0
d+ x0C0
C0 + CpiRp (C0 + Cp)
1 + iRp(C0 + Cp)(6)
where Rp R0/RL. It is noted that the current ie canalso be taken as the output signal. The respective transfer
functions for this case are easily obtained from the above
equations by noting that ie= ve/RL.
5.1.1. The condenser microphone. If the applied force
is the result of an acoustic pressure pm(t) the transducer can
be used as an electrostatic or condenser microphone. For
instance, consider the condenser microphone as depicted
in figure 10(a). The electric terminals are biased with
a voltage v0 and a bias resistor R0 in the same way
as shown in figure 1(a.) The input signal is the sound
generator pressure pg (t) which is applied to a movablerigid front plate with area A and mass m. The plate
is mounted on a peripheral spring with an equivalent
constant k. The pressure produces a small signal voltage
vout(t) at the electric terminals that can be connected to
a (pre-)amplifier (not shown). A Thevenin equivalent
circuit [11] of the transducer is shown in figure 10(b).
The Thevenin equivalent circuit is very simple and its
elements can be determined (experimentally) as follows.
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Figure 9. An equivalent circuit representation of the electromechanical transducer system shown in figure 1. The circuitenclosed by the dashed box represents the transducer of figure 4 (see also figure 5(a)).
Figure 10. The condenser microphone. (a) A schematic diagram; (b) the incremental signal Thevenin equivalent circuit; (c)a detailed equivalent circuit illustrating the interaction between the variables in the three signal domains of interest, i.e.,electric, mechanical, and acoustic.
The Thevein voltage vT h is given by the open-circuit
voltage and the Thevenin impedanceZT h is the impedance
seen at the electric terminals if all the independent
sources, here only the pressure generator, are set to
zero. However, the circuit is not at all convenient for
attaining a better understanding of, and for analysing and
optimizing, the microphone behaviour, as the shape of the
frequency response and the microphones sensitivity are
determined by the bias conditions, geometrical parameters,
and damping and dynamic characteristics of the specificmicrophone structure, including the electrical, mechanical,
and acoustical parts. Damping for instance is due to the
air-streaming resistance of the air gap, and can be strongly
reduced by introducing acoustic holes in the backplate.
Also, a narrow gap is of importance to attain a high
sensitivity. The stray capacitance Cp between the metal
case (which is electrically connected to the front plate)
and the back plate results in a capacitive attenuation of
the transducer signal and thus a lowering of the sensitivity
[2]. None of these individual components or others are
reflected in the Thevenin equivalent circuit. Therefore, a
more detailed equivalent circuit similar to the one shown
in figure 9 is required. For this purpose, the acoustic
signal domain must be included. The effort, flow, and state
variables in the acoustic domain are respectively given by
the acoustic pressure pa (N m2), the volume velocity
(m3 s1), and the volume displacement (m3) [2]. It can
be shown [2] that, for a system in equilibrium, equalityof the acoustic pressures pi at either side of an acoustic
junction applies, mathematically formulated as pi= 0.Further, the continuity law states that for the incident
volume velocities i at a junction the following relationapplies: i = 0 [2]. At this point it is evident thatthe latter two relations form the analogies to KVL and
KCL, respectively. In the example discussed here, the
mechanical and the acoustic domain are linked via the
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Figure 11. An equivalent circuit representation of an electrodynamic (moving-coil) loudspeaker. The acoustic power istransmitted into an acoustic impedanceZa; details ofZadepend on the specific speaker construction and can be found in [2].
Figure 12. A schematic diagram of an electrostatic comb-driven micromechanical resonator according to Tang et al [14]. (a)A typical layout of a linear resonant plate. (b) An equivalent circuit representation for a zero externally applied mechanicalforce,F
m= 0. The system behaves like an electrical two-port network. (c) Typical amplitude response plots of the
transadmittancei2/v1 evaluated for a short-circuited output (v2 = 0), as expressed by (7).
5.4. Electromechanical series filters employing in-plane
parallel microresonators
Electromechanical filters are used for signal processing
in for instance telecommunication systems which require
narrow bandwidth (high Q), low loss, good signal-to-
noise ratio, and stable temperature and aging characteristics
[3]. Lin et al [15] have recently described a new
class of passive bandpass electromechanical filters that
employ laterally driven in-plane resonators (of the type
that were introduced in the previous subsection), linked
through coupling springs. The passband characteristics
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Figure 13. (a) A schematic diagram of a series two-resonator electromechanical filter using electrostatic comb-drivenmicromechanical resonators, according to Linet al [15]. (b) An equivalent circuit representation. Cb represents a parasiticfeedthrough capacitor and/or an intentionally added bridging capacitor. (c) Typical amplitude response plots of thetransadmittancei2/v1 evaluated for a short-circuited output (v2 = 0) and for different values of the coupling spring, asexpressed by (8).
(centre frequency, shape, and bandwidth) are determined
by the specific design of the individual filter components.
Very important in this respect are also the number of
resonators and coupling springs used [3,15]. In fact, afilter configuration consisting of a single resonator and no
coupling springs has already been described in the previous
subsection. A typical response is given by (7) and is
graphically displayed in figure 12(c). The graph clearly
shows that the passband characteristics are far from being
the ideal square shape. Using several resonators that are
linked through coupling springs is a well known method
for obtaining better passband characteristics [3].
An example of a so-called series two-resonator
filter, with a square truss coupling spring, is shown in
figure 13(a). The filter consists of two spring-suspended
masses m1 and m2 that are coupled through a relativelyweak coupling spring kc. Electromechanical coupling to
each of the masses is accomplished using comb-shaped
electrostatic transducer elements. The system defines an
electrical two-port network, that behaves like an electrical
bandpass filter. The mechanical part of the filter behaves
like a two-degrees-of-freedom lumped-parameter system,
having two (closely separated) mode frequencies that are
determined by the masses m1 and m2 and by the three
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springs k1, k2, and kc. Usually, the structure is designed
such that k1 = k2 = k and m1 = m2 = m. Thetwo mode frequencies are then given by
(k/m) and
[(k+ 2kc)/m]. An equivalent circuit representation ofthe filter is shown in figure 13(b). The transduction factors
and the static capacitors of the transducer elements are the
same as explained in section 5.3. Damping is represented
by the viscous resistors c1 and c2, each one associated
with one of the resonators. Note that the coupling spring
experiences a displacement (and therefore a velocity) that
is given by the difference in displacements (velocities)
of the two masses. This example also illustrates theduality that exists between the equivalent circuit diagram
of figure 13(b) and the mechanical diagram of figure 13(a)
(see also subsection 3.3). For instance theseriesconnection
of the two springmass systems and the coupling spring in
the mechanical diagram becomes a parallelarrangement in
the equivalent circuit. The equivalent circuit also includes
a capacitor Cb which may either represent an unwanted
feedthrough capacitance between the two electrical ports or
it may represent an intentionally added bridging capacitor
used to further shape the passband characteristics [3].
The filter action is reflected in the transadmittance
Y (i), defined asi2/v1, which can easily be obtainedfrom the equivalent circuit, either from a direct circuitanalysis or by using circuit simulation software. For a
symmetrical design (m1 = m2 = m, k1 = k2 = k,c1= c2= c , and 1= 2= ), assuming short-circuitedconditions at the output (i.e., v2= 0) and further in theabsence of capacitor Cb, the transfer admittance can easily
be obtained directly from figure 13(b), resulting in
Y (i) ioutvin
= i2v1
= i2
k
kc/k
1 + 2kc/k 1
1 + ick + (i)2m
k
1 + ic
k+2kc + (i)2m
k+2kc
. (8)
The two mode frequencies ((k/m) and[(k + 2kc)/m)indicated before are clearly reflected in the above equation.The frequency response Y (i) of this symmetrical design
is determined by the magnitude of the coupling spring as
illustrated by the graph of figure 13(c) showing typical
amplitude response plots. For kc > 0c, the response
displays two distinct resonant peaks, whereas forkc < 0c,
only a single resonance is observed. For the transition
value, kc= 0c, a flat passband is obtained. The lattercondition is generally preferred in practical filter designs.
The response achieved for k0c = c clearly shows that thepassband more closely resembles the ideal square shape as
compared to the passband obtained for a single resonator as
shown in figure 12(c). For more details, reference is made
to the literature, e.g. [3, 15].
5.5. Vibration sensors
Vibration sensors are employed for measurements on
moving vehicles, on buildings, or on machinery or
as seismic pickups [1]. The basic principle of
vibration measurements is simply to measure the relative
displacement of a mass connected by a (soft) spring to
the vibrating body. An example of a vibration sensor
employing an electrodynamic displacement transducer is
shown in figure 14(a). The transducer detects the mass
displacement xm relative to the displacement xin of the
vibrating body. In the equivalent circuit the input motion
is modelled using an ideal velocity source uin . The
displacement xin and acceleration ain can be directly
obtained from the velocity as follows: xin = uin /iand ain = iuin . An equivalent circuit representationof the system in figure 14(a) is shown in figure 14(b).
The resistor Rs represents the total series resistance of
the coil and interconnecting wires. Note that the input
velocity source, the mass m, and the network consisting
of the series connection of the damper c, the spring k,
and the mechanical port of the transducer, are subject to
the same force. In the equivalent circuit this means that
these three networks are placed in parallel. It is evident
that the velocity (and thus the displacement) experienced
by the spring, the damper, and the mechanical port of
the transducer is given by the difference of the mass
displacement and the vibrational input, um uin . Fromthe equivalent circuit, the frequency response for velocity
measurements can now easily be obtained:
ve
uin =imRL
2 + ki
1 + 1
Q
i0
+
i0
2RL + Rs+ iL0
i0
2
1 + 1Q
i0
+
i0
2 = B0l (9)
where the first approximation applies for very large load
resistors RL and the second approximation is valid at
high frequencies, 0. Further, 0 =
k/m and
Q = m0/c denote the undamped resonant frequencyand quality factor of the springdampermass system,
respectively. Typical amplitudefrequency response plots
are graphically depicted in figure 14(c). The equation above
illustrates that at very high frequencies ( 0) accuratevelocity measurements are possible since the output voltage
is directly proportional to the input velocity. At these high
frequencies the mass practically stays at rest, um 0.For a good design thereof displaying a large bandwidth,
a low resonant frequency is desired, which can be achieved
by choosing a soft spring and/or a large (seismic) mass.
Signals from such transducers can be readily integrated
electrically to obtain displacement information.
Finally, it is pointed out that the electrodynamic
transducer can be replaced by any of the other transducers
shown in figure 3. The equivalent circuit of figure 14(b) is
easily adapted to the new configuration by carrying out theappropriate replacements of the transducer circuit enclosed
by the dashed box in figure 14(b).
5.6. Systems employing electromechanical feedback
Electromechanical feedback (or force balancing) is often
employed for applications requiring a great accuracy [1].
Instruments employing feedback are even often considered
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Figure 15. (a) A schematic diagram of a electrostatic force-balanced transducer for the measurement of forces (includingpressures)Fm or accelerationsain. (b) An equivalent circuit representation. The subscripts s and f refer to the sensingcapacitor (the upper one) and the feedback or actuating capacitor (the lower one), respectively. (c) Typical amplituderesponse curves, clearly displaying low-pass second-order filter characteristics.
parameterKs= 1/CF. Note that the output voltageof the charge amplifier, va= s Ks (xm xin ), is directlyproportional to the relative mass displacement. The analysis
does not change significantly if the upper-capacitor/charge
amplifier combination is replaced by another displacement
sensor, e.g., a differential capacitive detector [1, 16, 17, 19].
The frequency response function as obtained from the
equivalent circuit can be expressed as
H (i) voutFm main
= As Ksk + As fKs
1
1 + 1Q
i0+
i0
2
1
f
11 + 1
Q
i0
+
i0
2 1f (10)
where the first approximation applies for very high closed-
loop gains, resulting in a large electrical spring constant
ke As fKs k, and the second approximation is
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Figure 16. (a) A schematic diagram showing the implementation ofQ-control for a three-port electrostatic comb-drivenmicromechanical resonator according to Nguyen and Howe [20]. (b) An equivalent circuit representation. The subscripts s,f, and in refer to the sense electrode, the feedback electrode, and the driving or input electrode, respectively.
valid at low frequencies, 0. Further,
0= 0
1 + As fKsk
0
As fKs
k
= 0
ke
k=
k3
m(11)
and
Q= Q1 + As fKsk QAs fKsk= Q
ke
k=
mke
c(12)
where 0=
k/m and Q= m0/c=
km/c denotethe undamped resonant frequency and quality factor of the
springdampermass system, respectively. Note that for
a symmetrical design the bias displacement x0= 0, andthat the transduction factors s andfas well as the static
capacitors C0s and C0f are equal, respectively given by
s = f = = 0Aev0/d2 and C0s = C0f = C0 =0Ae/d, as presented in table 2. Moreover, as indicated
in the equivalent circuit of figure 15(b), it follows that k
is now given by k= k ks kf= k 2k, where thelatter equality applies for a symmetric design with k =2/C0 (compare table 4). Typical amplitudefrequency
response plots are graphically depicted in figure 15( c). The
system displays second-order low-pass filter characteristics.Equation (11) clearly shows that, due to the feedback,
the bandwidth, compared to a system without feedback,
is increased by a factor
ke/k, where ke denotes theelectrical spring constant. Thus, the mechanical spring k
may be considered to be replaced by an electrical spring
ke, which offers a greater linearity and accuracy and lack
of hysteresis. Other advantages of employing feedback
are the already indicated increased bandwidth, and further
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The above equations define the terminal voltage and the
force as being the effort variables at the respective ports.
The equilibrium values are given by the partial derivatives
ofWe with respect to the corresponding state variable [8].
Note that Ft is the externally applied force necessary to
achieve equilibrium. It is evident that in magnitude it
is equal to the electrostatic Coulomb force between the
charged plates of the capacitor, but has opposite direction.
Also note that the force displays a quadratic dependence
on the charge, which makes the system non-linear.
Linearization is commonly accomplished by introducing
bias or equilibrium conditions. The equations whichdescribe the linear relations between the incremental or
small-signaleffort variables and state variables, determined
at the bias point defined by a displacement x0and chargeq0,
are called the constitutive equations. For the electrostatic
transducer these are given by
v(q, x) = v tqt
0
q + vtx t
0
x
= (d+ x0)0Ae
q + q00Ae
x= 1C0
q + v0x0
x (A5a)
and
F(q, x)=
Ft
qt
0
q+
Ft
x t
0
x
= q00Ae
q + 0 x= v0x0
q + 0 x. (A5b)
To obtain the very last expressions on the right hand side,
the following equality was used: q0= C0v0= 0Aev0/x0,wherev0 denotes the bias voltage andC0 denotes the static
or bias capacitance. It is noted that the bias signals are
independent of time since they define the static equilibrium
condition.
The ideal transducer described above is not of great
practical use since it is not stable. This can be reasoned as
follows. Assume the system is in equilibrium, meaning that
the externally applied mechanical force is counterbalanced
by the electrostatic force. Now if, due to some disturbance,the movable plate is displaced a little in the direction of the
fixed plate (i.e., the negative x direction) while keeping
the applied voltage constant, the attractive electrostatic
force will increase a little and becomes somewhat larger
than the external mechanical force. This will increase
the displacement of the plate even further, which in its
turn results in a further increase of the electrostatic force.
This will go on until the gap spacing is reduced to zero.
Similarly, if the movable plate were initially displaced a
little further away from the fixed plate (i.e., in the positive
xdirection), force equilibrium is again destroyed. Now, the
mechanical force exceeds the ever decreasing electrostatic
force and the plate will eventually disappear into infinity.
Stability is easily attained by including a mechanical spring
with constantk, e.g., as shown in figure 4. Analytically this
means that a mechanical energy term must be added to the
energy function of (A1), resulting in
Wem= Wem
qt, xt = q2t
2C(xt)+ 1
2k
xt xr2
= q2t(d + xt)
20Ae+ 1
2k
xt xr2
(A6)
where xr denotes the rest position of the spring.
Equation (A4a) is not affected this way, but (A4b) is:
Ft
qt, xt Wem(qt, xt)
xt
qt=constant
= q2t
20Ae+ kxt.
(A7)
Finally, the second of the constitutive equations, (A5b),
must be replaced by
F(q, x) = Ftq t 0
q + Ftxt 0
x= q00Ae
q + kx= v0x0
q + kx.
(A8)The constitutive equations (A5a) and (A8) are expressed
as a relation of the(q, x) type, whereby the state variables
are chosen as the independent variables. Sometimes it is
more convenient to choose the voltage and the displacement
as the independent variables. The electromechanical
interactions are now described by equations of the (v, x)
type, given by
q(v, x) = 0Aed+ x0
v q0d+ x0
x
= 0Aed+ x0
v 0Aev0(d+ x0)2
x (A9a)
F(q, x) = q0d+ x0
v +
k q20
0Ae(d+ x0)
x
= 0Aev0(d+ x0)2
v +
k 0Aev20
(d+ x0)3
x. (A9b)
It can easily be shown that the equilibrium position of the
transducer, including the spring k , is, apart from excessive
bias loads [10], stable. In fact the system is stable as long
as k > k, where k= 0Aev20 /(d+ x0)3, i.e., the secondterm within parentheses in (A9b).
Appendix B. Nomenclature
A beam or diaphragm area (subjected to
acoustic pressure) [m2]
Ae effective electrode area (also for the
electromagnetic transducer) [m2]
B0 applied bias magnetic induction [T]
c viscous drag parameter [N s m1]C0 static or bias capacitance of the
electrostatic transducers [F]
Cp parasitic capacitance [F]
d gap spacing at rest, also called the
zero-voltage gap spacing [m]
F, F0, Ft incremental, bias (static) and total
applied mechanical force [N]i, i0, it incremental, bias (static) and total
current [A]
k mechanical spring constant [N m1]k spring induced by electromechanical
coupling effects [N m1]k effective dynamic spring constant
as a result of electromechanical coupling
[N m1]
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