Model of a piezoelectric microphone
by Willem A. Hol, 23-01-1999, E-mail: email@example.com
Beranek  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 , 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 , 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:
f0: force of the mechanical (in fact acoustical) source
f : force (over primary side of transformer)
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 f0 and the mechanical Lm, Cm and Rm are replaced by the electrical elements e0, L, C and R:
e0 = f0 / (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 ωo 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 .
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 E0:
E = E0.Z2 / (Z1 + Z2) (7) with
Z1 = R + j.ω.L + 1/(j.ω.C) (8)
and in case of absence of Zload:
Z2 = 1/(j.ω.Cp) (9)
where : e(t) = E.exp(j.ω.t) and e0(t) = E0.exp(j.ω.t).
E reaches a maximum response Emax at the series resonance frequency ω1 of the series combination of L, C and Cp:
ω12 = 1 / (L.Ceff) (10) with
Ceff = C.Cp / (C + Cp) (11) and
Emax = E0 / ( j.ω1.R.Cp) = F0 / ( j.ω1.R.C2p.Tau) (12)
The modulus of Emax will be much bigger than the modulus of E0 when:
1/(ω1.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 .
The tuned frequency response can be modified significantly by adding a coil-resistor combination as shown in Fig.3. This phenomenon is described in  and worked out with frequency responses and characteristic impedance characteristics for a number of ultrasonic transducers in .
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 = E0.Z2 / (Z1 + Z2) = E0 / (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 Emax and E0 is reduced to less than 1 (0dB).
 Beranek, Leo L.(1954), Acoustics, McGraw-Hill.
 website of Bertrik Sikken: http://enterprise.student.utwente.nl/~bertrik/bat/index.html
 website of Massa: http://www.massa.com/air.shtml