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3.5. Electromechanical transducers

Two types of electromechanical transducers, electromagnetic and electrostatic, are commonly employed in loudspeakers and microphones. Both may be represented by transformers with properties that permit the joining of mechanical and electrical circuits into one schematic diagram.

Electromagnetic-mechanical transducer

This type of transducer can be characterized by four terminals. Two have voltage and current associated with them. The other two have velocity and force as the measurable properties. Familiar examples are the moving-coil loudspeaker or microphone and the variable-reluctance earphone or microphone.

The simplest type of moving-coil transducer is a single length of wire in a uniform magnetic field as shown in Fig. 3.34. When a wire is moved upward with a velocity

\tilde{u}

$\tilde{u}$

as shown in Fig. 3.34(a), a potential difference

\tilde{e}

$\tilde{e}$

will be produced in the wire such that terminal 2 is positive. If, on the other hand, the wire is fixed in the magnetic field (Fig. 3.34(b)) and a current

\tilde{i}

$\tilde{i}$

is caused to flow into terminal 2 (therefore, 2 is positive), a force

\tilde{f}

$\tilde{f}$

will be produced that acts on the wire upward in the same direction as that indicated previously for the velocity.

The basic equations applicable to the moving-coil type of transducer are

\tilde{f} = B l \tilde{i},

$\tilde{f} = B l \tilde{i},$

(3.27a)

\tilde{e} = B l \tilde{u},

$\tilde{e} = B l \tilde{u},$

(3.27b)

where

Figure 3.34 Simplified form of moving-coil transducer consisting of a single length of wire cutting a magnetic field of flux density B. (a) The conductor is moving vertically at constant velocity so as to generate an open-circuit voltage across terminals 1 and 2. (b) A constant current is entering terminal 2 to produce a force on the conductor in a vertical direction.

$\tilde{i}$ $\tilde{i}$ is electrical current in A,
$\tilde{f}$ $\tilde{f}$ is “open-circuit” force in N produced on the mechanical circuit by the current $\tilde{i}$ $\tilde{i}$ ,
B is magnetic flux density in T,
l is effective length in m of the electrical conductor that moves at right angles across the lines of force of flux density B
$\tilde{u}$ $\tilde{u}$ is velocity in m/s
$\tilde{e}$ $\tilde{e}$ is “open-circuit” electrical voltage in V produced by a velocity $\tilde{u}$ $\tilde{u}$ .

The right-hand sides of Eq. (3.27) have the same sign because when

\tilde{u}

$\tilde{u}$

and

\tilde{f}

$\tilde{f}$

are in the same direction, the electrical terminals have the same sign.

One analogous symbol for this type of transducer is the “ideal” transformer given in Fig. 3.35(a). The “windings” on this ideal transformer have infinite impedance, and the transformer obeys Eq. (3.27) at all frequencies, including steady flow. The mechanical side of this symbol necessarily is of the admittance type if current flows in the primary. The other analogous symbol is the “ideal” gyrator given in Fig. 3.35(b). It is customary to define the mutual conductance g _m of a gyrator, which is the same in both directions, as the ratio of the flow on one side to the drop on the other. The mechanical side of this symbol necessarily is of the impedance type if current flows in the primary. The invariant mathematical operations which these symbols represent are given in Table 3.1. They lead directly to Eq. (3.27). The arrows point in the directions of positive flow or positive potential.

Figure 3.35 Analogous symbols for the electromagnetic transducer of Fig. 3.34. (a) The mechanical side is of the admittance type. (b) The mechanical side is of the impedance type.

Electrostatic-mechanical transducer

This type of transducer may also be characterized by four terminals. At two of them, voltages and currents can be measured. At the other two, forces and velocities can be measured.

An example is a piezoelectric crystal microphone such as is shown in Fig. 3.36. A force

\tilde{f}

$\tilde{f}$

applied uniformly over the face of the crystal causes an inward displacement of magnitude

\tilde{ξ}

$\tilde{ξ}$

in meters. As a result of this displacement, a voltage

\tilde{e}

$\tilde{e}$

appears across the electrical terminals 1 and 2. Let us assume that a positive displacement (inward) of the crystal causes terminal 1 to become positive. For small displacements, the induced voltage is proportional to displacement. The inverse of this effect occurs when no external force acts on the crystal face but an electrical generator is connected to the terminals 1 and 2. If the external generator is connected so that terminal 1 is positive, an internal force

\tilde{f}

$\tilde{f}$

is produced which acts to expand the crystal. For small displacements, the developed force

\tilde{f}

$\tilde{f}$

is proportional to the electric charge

\tilde{q}

$\tilde{q}$

stored in the electrodes.

Using the above relationships, we can write

\tilde{q} = C_{E} \tilde{e} - d_{31} \tilde{f}

$\tilde{q} = C_{E} \tilde{e} - d_{31} \tilde{f}$

(3.28a)

Figure 3.36 Piezoelectric crystal transducer mounted on a rigid wall.

\tilde{ξ} = d_{31} \tilde{e} - C_{M} \tilde{f}

$\tilde{ξ} = d_{31} \tilde{e} - C_{M} \tilde{f}$

(3.28b)

where

$\tilde{q}$ $\tilde{q}$ is electrical charge in C stored in the electrodes of the piezoelectric device,
$\tilde{e}$ $\tilde{e}$ is “open-circuit” electrical voltage in V produced by a displacement $\tilde{ξ}$ $\tilde{ξ}$ ,
$\tilde{f}$ $\tilde{f}$ is “open-circuit” force in N produced by an electrical charge $\tilde{q}$ $\tilde{q}$ ,
$\tilde{ξ}$ $\tilde{ξ}$ is displacement in m of a dimension of the piezoelectric device in m,
d ₃₁ is piezoelectric strain coefficient with dimensions of C/N or m/V. It is a real number when the network is linear, passive, and reversible. (The subscripts denote the relative directions of the applied field and resulting movement or vice versa. In this case, the two are at right angles. If they were in the same direction, for example, we would use d ₁₁, d ₂₂, or d ₃₃, where 1, 2, and 3 can be regarded as denoting the x, y, or z directions.)

and the electrical capacitance C _E and mechanical compliance C _M are given by

C_{E} = \frac{ε_{0} ε_{r} h w}{d}

$C_{E} = \frac{ε_{0} ε_{r} h w}{d}$

(3.29)

C_{M} = \frac{h}{Y d w}

$C_{M} = \frac{h}{Y d w}$

(3.30)

where

ε ₀ is permittivity of free space in F/m,
ε _r is relative permittivity of the free (nonblocked) piezoelectric dielectric (dimensionless),
Y is Young's modulus of elasticity in N/m² with electrical short-circuited.

In reality, C _E and C _M vary with displacement

\tilde{ξ}

$\tilde{ξ}$

, but it is assumed that the displacement is very small, so these are linearized equations. If the material shows no piezoelectric effect, applying an external force

\tilde{f}

$\tilde{f}$

simply leads to a deflection

\tilde{ξ}

$\tilde{ξ}$

according to Hooke's law. Because of the piezoelectric effect, the displacement also leads to an induced charge

\tilde{q}

$\tilde{q}$

on the electrodes, which in turn leads to a voltage (electrical force)

\tilde{e}

$\tilde{e}$

. Conversely, applying an electrical voltage leads to a mechanical force. Solving Eqs. (3.28a) and (3.28b) for

\tilde{e}

$\tilde{e}$

and

\tilde{f}

$\tilde{f}$

gives

\tilde{e} = \frac{1}{1 - k_{31}^{2}} (\frac{1}{C_{E}} \tilde{q} - \frac{d_{31}}{C_{E} C_{M}} \tilde{ξ})

$\tilde{e} = \frac{1}{1 - k_{31}^{2}} (\frac{1}{C_{E}} \tilde{q} - \frac{d_{31}}{C_{E} C_{M}} \tilde{ξ})$

(3.31a)

\tilde{f} = \frac{1}{1 - k_{31}^{2}} (\frac{d_{31}}{C_{E} C_{M}} \tilde{q} - \frac{1}{C_{M}} \tilde{ξ})

$\tilde{f} = \frac{1}{1 - k_{31}^{2}} (\frac{d_{31}}{C_{E} C_{M}} \tilde{q} - \frac{1}{C_{M}} \tilde{ξ})$

(3.31b)

where k ₃₁ is the dimensionless piezoelectric coupling coefficient which is related to the piezoelectric strain coefficient d ₃₁ by

k_{31} = \frac{d_{31}}{\sqrt{C_{E} C_{M}}} = d_{31} \sqrt{\frac{Y}{ε_{0} ε_{r}}}, 0 < k_{31} < 1

$k_{31} = \frac{d_{31}}{\sqrt{C_{E} C_{M}}} = d_{31} \sqrt{\frac{Y}{ε_{0} ε_{r}}}, 0 < k_{31} < 1$

(3.32)

Another commonly used parameter is the piezoelectric stress coefficient g ₃₁ in Vm /N or m²/C, which is defined by

g_{31} = \frac{d_{31}}{ε_{0} ε_{r}} = \frac{k_{31}}{\sqrt{ε_{0} ε_{r} Y}}

$g_{31} = \frac{d_{31}}{ε_{0} ε_{r}} = \frac{k_{31}}{\sqrt{ε_{0} ε_{r} Y}}$

(3.33)

Eq. (3.31) is often inconvenient to use because they contain charge and displacement. One prefers to deal with current and velocity, which appear directly in the equation for power. Conversion to current and velocity may be made by the relations

\tilde{u} = \frac{d ξ}{d t} = j ω \tilde{ξ},

$\tilde{u} = \frac{d ξ}{d t} = j ω \tilde{ξ},$

(3.34a)

\tilde{i} = \frac{d q}{d t} = j ω \tilde{q},

$\tilde{i} = \frac{d q}{d t} = j ω \tilde{q},$

(3.34b)

so that Eq. (3.31) becomes, in z-parameter matrix form,

[\begin{matrix} \tilde{e} \\ \tilde{f} \end{matrix}] = [\begin{matrix} \frac{1}{j ω C_{E}^{'}} & \frac{d_{31}}{j ω C_{E}^{'} C_{M}} \\ \frac{d_{31}}{j ω C_{E}^{'} C_{M}} & \frac{1}{j ω C_{E}^{'}} \end{matrix}] [\begin{matrix} \tilde{i} \\ - \tilde{u} \end{matrix}] .

$[\begin{matrix} \tilde{e} \\ \tilde{f} \end{matrix}] = [\begin{matrix} \frac{1}{j ω C_{E}^{'}} & \frac{d_{31}}{j ω C_{E}^{'} C_{M}} \\ \frac{d_{31}}{j ω C_{E}^{'} C_{M}} & \frac{1}{j ω C_{E}^{'}} \end{matrix}] [\begin{matrix} \tilde{i} \\ - \tilde{u} \end{matrix}] .$

(3.35)

The elements of Eq. (3.35) are related by the equations

C_{E}^{'} = (1 - k_{31}^{2}) C_{E}

$C_{E}^{'} = (1 - k_{31}^{2}) C_{E}$

(3.36)

C_{M}^{'} = (1 - k_{31}^{2}) C_{M}

$C_{M}^{'} = (1 - k_{31}^{2}) C_{M}$

(3.37)

Note in particular that

$C_{E}^{'}$ $C_{E}^{'}$ is electrical capacitance measured with the mechanical “terminals” blocked so that no motion occurs $(\tilde{u} = 0)$ $(\tilde{u} = 0)$ .
C _E is electrical capacitance measured with the mechanical “terminals” operating into zero mechanical impedance so that no force is built up $(\tilde{f} = 0)$ $(\tilde{f} = 0)$ .
$C_{E}^{'}$ $C_{E}^{'}$ is mechanical compliance measured with the electrical terminals open-circuited $(\tilde{i} = 0)$ $(\tilde{i} = 0)$ .
C _M is mechanical compliance measured with the electrical terminals short-circuited $(\tilde{e} = 0)$ $(\tilde{e} = 0)$ .

Figure 3.37 Two forms of analogous symbols for piezoelectric transducers. The mechanical sides are of the impedance type.

The equivalent circuit shown in Fig. 3.37(a) is essentially a two-port network defined by the z-parameters in the matrix of Eq. (3.35), although z-parameter matrices will be discussed in greater detail in Section. 3.10. Noting from Eqs. (3.36) and (3.37) that

C_{E}^{'} C_{M} = C_{E} C_{M}^{'}

$C_{E}^{'} C_{M} = C_{E} C_{M}^{'}$

, Eq. (3.31) can alternatively be written as

[\begin{matrix} \tilde{e} \\ \tilde{f} \end{matrix}] = [\begin{matrix} \frac{1}{j ω C_{E}^{'}} & \frac{d_{31}}{j ω C_{E} C_{M}^{'}} \\ \frac{d_{31}}{j ω C_{E} C_{M}^{'}} & \frac{1}{j ω C_{M}^{'}} \end{matrix}] [\begin{matrix} \tilde{i} \\ - \tilde{u} \end{matrix}] .

$[\begin{matrix} \tilde{e} \\ \tilde{f} \end{matrix}] = [\begin{matrix} \frac{1}{j ω C_{E}^{'}} & \frac{d_{31}}{j ω C_{E} C_{M}^{'}} \\ \frac{d_{31}}{j ω C_{E} C_{M}^{'}} & \frac{1}{j ω C_{M}^{'}} \end{matrix}] [\begin{matrix} \tilde{i} \\ - \tilde{u} \end{matrix}] .$

(3.38)

which is represented by the equivalent circuit as shown in Fig. 3.37(b). The mechanical sides of Fig. 3.37(a,b) are of the impedance-type analogy. Let us discuss Fig. 3.37(a) first.

Looking into the electrical terminals 1 and 2, the element

C_{E}^{'}

$C_{E}^{'}$

is the electrical capacitance of the transducer. To measure

C_{E}^{'}

$C_{E}^{'}$

, a sinusoidal driving voltage

\tilde{e}

$\tilde{e}$

is applied to the transducer terminals 1 and 2, and the resulting sinusoidal current is measured. During this measurement, the mechanical terminals 3 and 4 are open-circuited (motion blocked,

\tilde{u}

$\tilde{u}$

= 0). A very low driving frequency is used so that the mass reactance and mechanical resistance can be neglected. The negative capacitance −

C_{E}^{'}

$C_{E}^{'}$

represents the force of attraction between the electrodes which varies with the displacement. Hence, it can be thought of as a negative stiffness which can be subtracted from the natural stiffness of the material.

Looking into the mechanical terminals 3 and 4 of Fig. 3.37(b),

C_{M}^{'}

$C_{M}^{'}$

is the mechanical compliance of the transducer measured at low frequencies with the electrical terminals 1 and 2 open-circuited

(\tilde{i} = 0)

$(\tilde{i} = 0)$

. A sinusoidal driving force

\tilde{f}

$\tilde{f}$

is applied to terminals 3 and 4 of the transducer and the resulting sinusoidal displacement is measured. Again, the negative compliance −

C_{M}^{'}

$C_{M}^{'}$

is because of the force of attraction between the electrodes. Eliminating

\tilde{q}

$\tilde{q}$

and

\tilde{ξ}

$\tilde{ξ}$

between Eq. (3.31) leads to the following simplified equations

\tilde{f} = \frac{d_{31}}{C_{M}} \tilde{e} - \frac{1}{C_{M}} \tilde{ξ}

$\tilde{f} = \frac{d_{31}}{C_{M}} \tilde{e} - \frac{1}{C_{M}} \tilde{ξ}$

(3.39a)

\tilde{e} = \frac{d_{31}}{C_{E}} \tilde{f} + \frac{1}{C_{E}} \tilde{q}

$\tilde{e} = \frac{d_{31}}{C_{E}} \tilde{f} + \frac{1}{C_{E}} \tilde{q}$

(3.39b)

In the steady state

\tilde{u} = j ω \tilde{ξ}

$\tilde{u} = j ω \tilde{ξ}$

and

\tilde{i} = j ω \tilde{q}

$\tilde{i} = j ω \tilde{q}$

so that

\tilde{f} = \frac{d_{31}}{C_{M}} \tilde{e} - \frac{1}{j ω C_{M}} \tilde{u}

$\tilde{f} = \frac{d_{31}}{C_{M}} \tilde{e} - \frac{1}{j ω C_{M}} \tilde{u}$

(3.40a)

\tilde{e} = \frac{d_{31}}{C_{E}} \tilde{f} + \frac{1}{j ω C_{E}} \tilde{i}

$\tilde{e} = \frac{d_{31}}{C_{E}} \tilde{f} + \frac{1}{j ω C_{E}} \tilde{i}$

(3.40b)

from which we obtain the two simplified equivalent electrical circuits as shown in Fig. 3.38.

Looking into the mechanical terminals 3 and 4 of Fig. 3.38(a), the element C _M is the mechanical compliance of the transducer but measured in a different way. A sinusoidal driving force

\tilde{f}

$\tilde{f}$

is applied to terminals 3 and 4 of the transducer at a very low frequency so that the mass reactance and mechanical resistance can be neglected, and the resulting sinusoidal displacement is measured. During this measurement, the electrical terminals 1 and 2 are short-circuited

(\tilde{e} = 0)

$(\tilde{e} = 0)$

. Looking into the electrical terminals 1 and 2 of Fig. 3.38(b), the element C _E is the electrical capacitance measured at low frequencies with the mechanical terminals 3 and 4 short-circuited

(\tilde{f} = 0)

$(\tilde{f} = 0)$

A sinusoidal driving voltage e applied to the terminals 1 and 2 of Fig. 3.38(a) produces an open-circuit force

Figure 3.38 Two simplified forms of analogous symbols for piezoelectric transducers. The mechanical sides are of the impedance type.

\tilde{f} = \frac{d_{31}}{C_{M}} \tilde{e} .

$\tilde{f} = \frac{d_{31}}{C_{M}} \tilde{e} .$

(3.41)

A sinusoidal driving force

\tilde{f}

$\tilde{f}$

applied to the terminals 3 and 4 of Fig. 3.38(b) produces an open-circuit voltage

\tilde{e} = \frac{d_{31}}{C_{E}} \tilde{f} .

$\tilde{e} = \frac{d_{31}}{C_{E}} \tilde{f} .$

(3.42)

The choice between the alternative analogous symbols of Fig. 3.38 is usually made on the basis of the use to which the transducer will be put. If the electrostatic transducer is a microphone, it usually is operated into the gate of a field-effect transistor so that the electrical terminals are essentially open-circuited. In this case, the circuit of Fig. 3.38(b) is the better one to use because C _E can be neglected in the analysis when

\tilde{i} = 0

$\tilde{i} = 0$

. On the other hand, if the transducer is a loudspeaker, it usually is operated from a low-impedance amplifier so that the electrical terminals are essentially short-circuited. In this case, the circuit of Fig. 3.38(a) is the one to use because

C_{E}^{'}

$C_{E}^{'}$

ω is small in comparison with the output admittance of the amplifier.

The circuit of Fig. 3.38(a) corresponds more closely to the physical facts than does that of Fig. 3.38(b). If the device could be held motionless

(\tilde{u} = 0)

$(\tilde{u} = 0)$

when a voltage was impressed across terminals 1 and 2, there would be no stored mechanical energy. All the stored energy would be electrical. This is the case for circuit (a), but not for (b). In other respects the two circuits are identical.

At higher frequencies, the mass M _M and the resistance R _M of the crystal must be considered in the circuit. These elements can be added in series with terminal 3 of Fig. 3.38.

These analogous symbols indicate an important difference between electromagnetic and electrostatic types of coupling. For the electromagnetic case, we ordinarily use an admittance-type analogy, but for the electrostatic case, we usually employ the impedance-type analogy.

In the next part we shall introduce a different method for handling electrostatic transducers. It involves the use of the admittance-type analog in place of the impedance-type analog. The simplification in analysis that results will be immediately apparent. By this new method, it will also be possible to use the impedance-type analog for the electromagnetic case.

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Table of Contents for Acoustics

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3.5. Electromechanical transducers

Electromagnetic-mechanical transducer

Electrostatic-mechanical transducer

Table of Contents for
Acoustics