Lumped Parameters

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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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H A C Tilmans

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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H A C Tilmans

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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