Electroacoustics and Audio Engineering: Paper ICA2016-193

Modeling of the electroacoustic coupling of electrostatic microphones including the preamplifier circuit

Bernardo Henrique Pereira Murta(a), Eric Brandão(b), Julio Cordioli(c), William D’A. Fonseca(d), Paulo H. Mareze(e)

(a, b, d, e)Federal University of Santa Maria, Acoustical Engineering, Santa Maria, RS, Brazil, [email protected], [email protected] (c)Universidade Federal de Santa Catarina, Florianópolis, Brazil, [email protected]

Abstract: This research aims to study tools to model and design electrostatic microphones coupled with its preamplifier circuits. The outcome is the access to their combined sensitivities curves, which allows the design of microphones with a wider and flat bandwidth. Analytical and numerical mod- eling techniques are explored and compared. On one hand, the lumped parameters approach is the basis of the analytical modeling of acoustic transducers. That is, this technique allows the engineer to design the transducer and its preamplifier circuit by predicting its sensitivity changes due to variations of model properties with low computational cost. On the other hand, numerical analysis is carried out using the Finite Element Method with a multiphysics approach, which is able to solve both the transducer model and the coupled electrical circuit. Two microphones with different complexities and constructive characteristics are studied. For validation of the proposed techniques, the behavior of a commercial measurement microphone model that has been well studied in the literature is considered. Once the validation of the modeling approach is satisfac- tory, one can use the same methodology to study a piezoelectric microphone for hearing aid applications, for instance. Its frequency response requires a designed preamplifier which should be able to make its sensitivity flatter over the audio bandwidth and to improve its output level. The research also objectifies to analyze the whole chain of energy transducing and signal conditioning in order to prepare the fully coupled model for optimization procedures. The goal is to conceive efficient high-performance systems with low cost hardware. Keywords: electroacoustics, microphone modelling, finite element method Modeling of the electroacoustic coupling of electrostatic microphones including the preamplifier circuit

1 Introduction Computational modeling of microphones has been extensively used during design stages of transducers, mainly, in order to reduce unnecessarily prototyping costs. Thus, it is possible to select arbitrary geometries and material compositions to achieve microphones with specific characteristics as wide and flat bandwidth or particular sensitivities. Optimization methods, as the genetic algorithm, can be implemented to solve transducer models aiming to select the best fit for shapes and materials, searching for desired responses [1,2]. Robust numeric methods as the Finite Element Method (FEM) and the Boundary Element Method (BEM) are implemented in several commercial software, allowing the multiphysics modeling of complex geometry transducers with different materials [3]. However, as an outcome, a high computational cost is a common obstacle in the application of optimization techniques to such models. An alternative is the development of analytical methods to solve simplified situations. Generally, it is possible to achieve smaller solution time with a satisfactory representation of the designed transducer. A common approach to solve such devices is the equivalent circuit method, which was extensively used during the 60’s and 70’s, especially due to the advent of computers [4]. This approach consists of replacing the differential equations that represent each relevant characteristic of the transducer by analog electrical circuit elements, such as , and . For each of the physics that comprehend the transducer (acoustical, mechanical and electrical) a sub-circuit is derived and coupled by means of to the adjacent physics. Therefore, a single circuit representing the whole acoustic-mechanic-electrical transducing chain of a microphone can be constructed and solved via the Kirchhoff Laws [5]. Many limitations and simplifications are required to apply correctly the analog circuit technique. On one hand, analyzed wavelengths must be larger than the transducer’s dimensions. Thus, the energy will be evenly (or with low variations) spatially distributed [6, 7], meaning that the microphone’s diaphragm moves as a rigid body. On the other hand, the complexity of the transducer’s elements as multi-material diaphragms or non-basic and asymmetric geometries cannot be well represented. To circumvent those limitations, new researches aim to develop higher complexity methods considering boundary conditions, geometries and materials [6, 8, 9]. Based on different models, it is possible to forge small details of microphones, for example, the influence of a perforated backplate facing the transducer’s diaphragm [10] in its frequency response [11]. All that brings an important role in computational models in the industry [12] may be a relevant tool to decrease design costs. There are still several limitations when considering the development of modern transducers such as microelectromechanical systems (MEMS). They may present multi-material non-usual geometries [7, 9] when compared to regular microphones as well when modeling anisotropic piezoelectric materials [13]. Thus, the development of new studies to include such conditions in equivalent circuit approach is desired Another step in the modeling process is considering integrated designed preamplifiers capable

2 of providing adequate signal conditioning for the studied transducer. Such circuits are required, for example, to couple the high output impedance of capacitive and piezoelectric microphones with the small input impedance of the following stages such as analog-to-digital converter (ADC) and digital signal processing devices. Furthermore, analog filters and gain stages to adjust the frequency response of the modeled transducer may be necessary when low-cost devices are the target of a study.

2 Equivalent circuit method The equivalent circuit of a transducer depends on its geometrical properties, composition of materials and the boundary conditions of its mounting. The analysis is subdivided into three domains, each one containing its own circuit and coupled by means of ideal transformers. There is the acoustic circuit, in which the incident sound pressure and the volume of air around the diaphragm are modeled. Then, the mechanical domain models the diaphragm as a one degree of freedom system, computing its velocity and the associated . Finally, the electrical domain solves the electromechanical coupling and the voltage across the microphone’s terminal.

2.1 Acoustical domain (A.D.) The acoustical equivalent circuit is dependent of the transducer’s geometry, as shown in Figure1. Thus, to each element interacting with the sound pressure waves an impedance is attributed. The acoustical impedances are defined as the ratio between the incident pressure Pe and the produced volume velocity Qe˙, thus Zea = Pe/Qe˙ [kg/sm4]. Here the tilde notation represents the complex harmonic amplitude and the dot is the time derivative. Passive circuit elements are used to model those impedances. Thus, resistors are used to represent the elements while the reactive parts of the impedances are modeled by ca- pacitors (associated to acoustical compliance) and inductors (corresponding to acoustical ). Specific properties and analytical equations describing some commonly found impedances in equivalent acoustical circuits of microphones are presented in the following. The incident pressure over the transducer’s diaphragm is modeled by an ideal AC voltage source, which magnitude is defined as the sound pressure in Pascal (Pa). Cavities delimiting air volumes V are represented by an acoustical compliance, as the capaci-  3  tance Ca m /Pa [6], given as: V Ca = 2 , (1) ρ0c0 where ρ0 is the air density and c0 the speed of sound in the medium for the ambient conditions.

Tube elements with radius at smaller than at < 0,4 mm are considered small ducts [14]. This impedance is commonly present in measurement microphones as the pressure equalization vent. Their acoustical impedance Zat is given by:

8η0l 4ρ0l Zat = 4 + jω 4 , (2) πat 3πat

3 Rabp Mabp

P Raslit Maslit ˙ e Zeraddia P2 Qe 1:Sd Xe˙ P1 Mateq

Rateq Ca2 Ca1 ... M.D.,E.D.

Zeradteq

P0 A.D. Figure 1: Equivalent circuit of the capacitive B&K 4134 microphone (right) and its schematic (left). where ω = 2π f is the , η the air’s viscosity and l tube’s length. The small duct   acoustical resistance Rat kg/m4s is obtained by

8η0l Rat = 4 , (3) πat where η0 is air viscosity and l tube length. The reactive part, with inductive behavior, is modelled   by the acoustical mass Mat kg/m4 calculated by

4ρ0l . Mat = 4 (4) 3πat

The slits can be found between the microphone’s backplate and√ its carcass, for example. A narrow slit is defined when its thickness tslit is smaller than 0,003/ f or tslit <0,6 mm. According to Beranek and Mellow [4], its acoustical resistance Rslit can be estimated by

12η0l Rslit = 3 , (5) tslitwslit where wslit is the slit’s width and lslit its length. The acoustical mass Mslit is given by

6ρ0lslit Mslit = . (6) 5tslitwslit

A rigid disk is considered micro-perforated when the diameter (2aholes) of its holes is sub- millimetric, 2a < 1 mm, typically used in capacitive measurement microphones’ backplate [15].

4 Thus, such elements’ impedances presents inductive and resistive behavior. The parameter y relates the diameter of the holes and the boundary layer inside it [14], hence, r ωρ0 y = 2aholes . (7) 4η0

Thus, its resistance Rbp is given by: √ r 2  32η0lholes y 2 2aholes Rbp = 2 1 + + y , (8) Sdiskψ(2aholes) 32 32 lholes where lholes is the hole’s length and ψ the perforated area ratio considering the disk’s area Sdisk. Its acoustical mass estimated by

 r 2  ρ0lholes y 2aholes Mbp = 1 + 9 + + 0,85 . (9) Sdiskψ 2 lholes

The radiation impedance of a piston with radius a at the end of a tube is chosen to model such physical phenomena. Beranek and Mellow [4] derive an equivalent circuit in the acoustical domain, presented in Figure2, to describe the frequency dependent behavior. According to the authors, components shall be obtained as:

3 0,1977ρ0c0 ρ0c0 πa Ra1 = 2 , Ra2 = , Ca1 = 2 and (10) a Sd ρ0c0

A

Ca1 Ra1

Ma1

Ra2

B

Figure 2: Equivalent circuit (in the acoustical domain) of the radiation impedance Zerad defined as the equivalent impedance between nodes A and B.

Zerad can be calculated as: 1 1 1 = + . (11)  −1 Zerad jωMa1 1 + jωC + R Ra1 a1 a2

5 2.2 Mechanical domain The mechanical circuit can be derived intuitively according to Beranek and Mellow [4]. Consider- ing the hypothesis of a simple diaphragm vibrating as a piston, a single degree of freedom with mass, stiffness and damping is produced. The impedance of a component Zem is obtained by the ratio between the Fe applied over it and its velocity Xe˙ , which yields   Fe Ns Zem = . (12) Xe˙ m

Cm Mm Xe˙

Fe Rm

Figure 3: Left: Detail of the important diaphragm parts; Right: Generic example of the equivalent mechanical circuit of a diaphragm in impedance analogy. Mass is modeled by an , stiffness by a and damping elements are represented by resistors. As the diaphragm moves as a piston, all mechanical components are connected in series.

The circuit is constructed considering the impedance analogy. Thus, forces are represented by a and as currents. The effective moving mass (Mm [kg]) is modeled by an inductor. Homentcovschi and Miles [12] derive the equivalent mass of a tensed membrane as: 4 M = ρ πa2 , (13) m 3 m   where ρm is the membrane superficial density in kg/m2 .

The mechanical compliance of such diaphragms (Cm [m/N]) is related to the conservative energy. According to the same cited authors [12] it may be modeled as 1 C = , (14) m 8πT where T is the membrane tension. Lastly, the mechanical resistance Rm [Ns/m] models the system’s losses. It may be obtained by means of fitting techniques using adjusted models or experimental data.

6 3 Capacitive microphone modelling The equivalent circuit of the capacitive microphone is then derived as shown in Figure4. Open circuit FRF is extracted by means of the voltage across nodes A and B and the input pressure. This lumped model will be later compared with numeric data simulated by Jensen and Olsen [3]. On one hand, in the referred study, the vent duct is modeled exposed to the pressure field. On the other hand, the lumped model described solves it for the unexposed vent situation, resulting in disagreement at low frequencies.

Rabp Mabp

P Ce Raslit Maslit e Zeraddia Rm Mm 1: E0∗Cm 1:Sd Xe˙ h0 A

Mateq Cm

B Rateq Ca2 Ca1 M.D. E.D.

Zeradteq

A.D. Figure 4: Complete equivalent circuit of the capacitive transducer. The open circuit sensitivity is given by He = EeAB/Pe .

The open circuit sensitivity FRF, shown in Figure5, is derived as:  −1 EeAB E0 jωMm Rm 1 Hemic = = Zeeq3 + Zeraddia 2 + 2 + 2 , (15) Pe jωSd Sd Sd jωCmSd where the equivalent impedances are  −1 −1 Zeeq1 = Zeradteq + Rateq + jωMateq + jωCa2 , (16)

 −1 1 −1 Zeeq2 = + Rslit + jωMslit , (17) Rabp + jωMabp and  −1 −1 Zeeq3 = Zeeq1 + Zeeq2 + jωCa1 . (18)

7 ]

1Pa −38 . −40 dBref [ −42

−44 2 2 FEM - ρm = 0,0445 kg/m Lumped - ρm = 0,0445 kg/m 2 2 −46 FEM - ρm = 0,0550 kg/m Lumped - ρm = 0,0550 kg/m

Sensitivity 100 101 102 103 104 Frequency [Hz] Figure 5: Comparison between B&K 4134 capacitive microphone models by means of equiv- alent circuit and FEM simulations normalised by the sensitivity at 1 kHz. Both lumped and numeric models behaved similarly for variations in the membrane’s density.

4 Piezoelectric microphone modelling The studied piezoelectric microphone’s geometry is given in Figure6. It consists of a circular buzzer bimorph diaphragm fixed on an air cavity, with one metallic disk mounted under a PZT-5A disk with a smaller diameter. This transducer is modeled in FEM and its equivalent circuit is derived and shown in Figure7.

PZT-5A Pressure in face Brass

Fixed Rigid walls Air volume constraint Figure 6: Left: Detailed diaphragm; Right: Properties of the FEM model for the piezoelectric microphone.

P Ce e Zerad Rm Mm Qe 1:Sd Xe˙ 1:γ Ie A

Ca Cm ... Preamplifier

B A.D. M.D. E.D. Figure 7: Equivalent circuit of the piezoelectric transducer.

It is possible to couple a preamplifier in both models. To include it in the lumped model it is

8 necessary to use a circuit simulation software to derive its FRF. Then, one can multiply circuit’s FRF to the open circuit sensitivity FRF of the transducer. Note that this coupling considers the use of buffer stages between transducer’s output and preamplifier’s input. The numeric solver is able to simulate the coupled circuit. Thus, it is required to export a SPICE netlist from the circuit simulation software to couple it to the transducer. The designed preamplifier consists in a notch filter tuned to the microphone’s first and a voltage-follower gain stage. Hence, it is possible to reach a wider flat bandwidth and a smaller resonance peak as can be seen in Figure8.

−10 Microphone’s terminal

] Output of filtering stage −15 Full preamplifier 1Pa . −20 dBref [ −25

|FRF| −30

−35 102 103 104 Frequency [Hz] Figure 8: FRFs extracted through the numeric modeling. The microphone’s open-circuit sen- sitivity is compared to the FRF including only the filtering stage and considering the whole preamplifier.

5 Conclusions This work discusses conditions and methods to model transducers and coupled preamplifiers focusing in specific behaviors of the system’s FRF. Thus, optimization methods can be applied to reach the best selection of geometries and materials. While the numeric model is more exact and complex, the lumped approach consists in computationally cheaper models with considerable representative results.

References [1] Gustavo Martins. Modelos Multi-físicos de Transdutores Piezoelétricos para Aparelhos Audi- tivos com Vistas à Aplicação de Técnicas de Otimização. Tese de doutorado, Universidade Federal de Santa Catarina, Florianópolis, SC, Brasil, 2015.

[2] Gustavo Martins and Julio Cordioli. Multiphysical finite element model of a hearing aid

9 piezoelectric loudspeaker for optimization purposes. Acta Acustica united with Acustica, 101 (5):993–1006, 2015.

[3] Mads Jakob Herring Jensen and Erling S. Olsen. Virtual prototyping of condenser mi- crophones using the finite element method for detailed electric, mechanic, and acoustic characterization. In Proceedings of Meetings on Acoustics, volume 19, page 030039. Acous- tical Society of America, 2013.

[4] Leo L. Beranek and Tim Mellow. Acoustics: sound fields and transducers. Academic Press, 2012.

[5] Alexander Sadiku. Fundamentals of electric circuits. McGraw-Hill, New York, 2000.

[6] Mario Rossi. Acoustics and electroacoustics. Artech House Publishers, 1988.

[7] Stephen Horowitz, Toshikazu Nishida, Louis Cattafesta, and Mark Sheplak. Development of a micromachined piezoelectric microphone for aeroacoustics applications. The Journal of the Acoustical Society of America, 122(6):3428–3436, 2007.

[8] W. Marshall Leach. Introduction to electroacoustics and audio amplifier design. Kendall/Hunt Publishing Company, 2003.

[9] Valeriy Sharapov, Zhanna Sotula, and Larisa Kunickaya. Piezo-Electric Electro-Acoustic Transducers. Springer, 2014.

[10] Thomas Lavergne, Stéphane Durand, Michel Bruneau, Nicolas Joly, and Dominique Ro- drigues. Dynamic behavior of the circular membrane of an electrostatic microphone: Effect of holes in the backing electrode. The Journal of the Acoustical Society of America, 128(6): 3459–3477, 2010.

[11] Brüel & Kjær. Condenser microphones and microphone preamplifiers for acoustic measure- ments. B&K, Denmak, 1982.

[12] Dorel Homentcovschi and Ronald N. Miles. An analytical-numerical method for determining the mechanical response of a condenser microphone. The Journal of the Acoustical Society of America, 130(6):3698–3705, 2011.

[13] Israel Pereira. Caracterização Numérica e Experimental de um Protótipo de Microfone Piezoelétrico Visando Uso em Aparelhos Auditivos. Dissertação de mestrado, Universidade Federal de Santa Catarina, Florianópolis, SC, Brasil, 2013.

[14] Eric Brandão. Apostila da dsiciplina de Eletroacústica I. Engenharia Acústica, Universidade Federal de Santa Maria, 2014.

[15] Brüel & Kjær. Microphone handbook, vol. 1: Theory. Technical Documentation be1447, 1996.

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