**Model of a
piezoelectric microphone**

by
Willem A. Hol, 23-01-1999, E-mail: willem.hol@wxs.nl

**Functional
diagram**

Beranek
[1] provides a simplified functional diagram for a piezoelectric transducer
consisting of a mechanical compliance, a transformer (mechanical <->
electrical),

an
electrical capacitance and input / output (Please refer to [1], page 73).

At
higher frequencies the mass and the mechanical resistance must be considered.
These elements can be added in series with the mechanical compliance (See [1], page 75 and 86).

A
mechanical source, which linearly relates to the acoustical pressure, is placed
at the mechanical side and an electrical impedance as load at the electrical
side resulting in the functional diagram of Fig.1. In this diagram with a mechanical
side of the impedance type the following notations are used:

f_{0}: force
of the mechanical (in fact acoustical) source

f : force (over
primary side of transformer)

u: velocity

Lm: mass

Cm: mechanical
compliance

Rm: mechanical
resistance

e: electrical
voltage (over secondary side of transformer)

I: electrical
current

Cp: electrical
capacitance of the microphone

Tau: coupling
coefficient of the piezoelectric device

Zload: electrical load impedance

The
mechanical - electrical transformer has the ratio Cp.Tau : 1, so:

f = Cp.Tau.e (1) and

I = Cp.Tau.u (2)

The
transformer can be eliminated by using (1), (2) and Kirchhoff. This results in
the equivalent functional diagram of Fig.2. The mechanical source f_{0}
and the mechanical Lm, Cm and Rm are replaced by the electrical elements e_{0},
L, C and R:

e_{0} = f_{0} / (Cp.Tau) (3)

L = Lm / (Cp.Tau)^{2} (4)^{ }

C = Cm. (Cp.Tau)^{2} (5)

R = Rm / (Cp.Tau)^{2} (6)

Some
important characteristics of the LCR-combination, like the series resonance
frequency ω_{o} and quality Q, do
not change.

The
characteristic impedance of the microphone consists of the serial LCR
combination in parallel with Cp as described by Bertrik Sikken [2].

This
can also be observed in Fig.2.

The
complex output voltage E of the piezoelectric microphone is directly related to
the complex source voltage E_{0}:

E = E_{0}.Z_{2} / (Z_{1} + Z_{2}) (7)
with

Z_{1 }= R + j.ω.L + 1/(j.ω.C) (8)

and
in case of absence of Zload:

Z_{2} = 1/(j.ω.Cp) (9)

where
: e(t) = E.exp(j.ω.t) and e_{0}(t)
= E_{0}.exp(j.ω.t).

E
reaches a maximum response E_{max} at the series resonance frequency ω_{1 }of
the series combination of L, C and Cp:

ω_{1}^{2} = 1 /
(L.Ceff) (10) with

Ceff = C.Cp / (C + Cp) (11)
and

E_{max} = E_{0} / ( j.ω_{1}.R.Cp) = F_{0}
/ ( j.ω_{1}.R.C^{2}p.Tau) (12)

The
modulus of E_{max} will be much bigger than the modulus of E_{0}
when:

1/(ω_{1}.Cp)
>> R (13)

The
above described functional diagrams provide a good approximation for the
relation between impedance curves and frequency responses of piezoelectric
microphones with one resonance peak, like the Massa TR89B type 40, E-152/40 and
E-152/75. Please refer to [3].

**Detuning**

The
tuned frequency response can be modified significantly by adding a
coil-resistor combination as shown in Fig.3. This phenomenon is described in
[2] and worked out with frequency responses and characteristic impedance
characteristics for a number of ultrasonic transducers in [3].

With
Yp as the admittance (inverse of impedance) of the combination of Lp, Cp and Rp
and Zs as the impedance of the combination of L, C and R, equation (7) can be
worked out as:

E = E_{0}.Z_{2} / (Z_{1} + Z_{2}) = E_{0}
/ (1 + Zs.Yp) (14)

A
relatively wide band response can be realised by selecting Lp in such a way
that the parallel resonance frequency of the combination Lp, Cp and Rp equals
the series resonance frequency of the combination of L, C and R:

Lp.Cp = L.C (15)

The
Q of the parallel combination has to be much smaller than the Q of the series
combination. With this detuned frequency response, the resulting modulus ratio
of E_{max} and E_{0} is reduced to less than 1 (0dB).

**References:**

[1]
Beranek, Leo L.(1954), Acoustics, McGraw-Hill.

[2]
website of Bertrik Sikken: http://enterprise.student.utwente.nl/~bertrik/bat/index.html

[3]
website of Massa: http://www.massa.com/air.shtml