EXTENDED OPERATIONAL RANGE OF A FEEDBACK-CONTROLLED OPTICAL
MICROPHONE FOR AEROACOUSTICS RESEARCH
By
Sara Feeney

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Mechanical Engineering
2012

ABSTRACT
EXTENDED OPERATIONAL RANGE OF A FEEDBACK-CONTROLLED
OPTICAL MICROPHONE FOR AEROACOUSTICS RESEARCH
By
Sara Feeney
This study is an extension of a previous study conducted at MSU on a feedback-controlled
optical microphone for potential use in “beam-forming” arrays employing anywhere from
thirty to one hundred microphones in aeroacoustics research. The membrane of the microphone is made from PVDF (polyvinylidene-fluoride), a material with piezoelectric properties,
giving a means for opposing the sound-produced deflection of the membrane using a feedback loop. The feedback capability of the microphone allows it to self-calibrate and gives it
response-matching capabilities in array applications—a unique advantage over current microphone technology, as manual, individual calibration of each microphone in an array can
be time-consuming.
The present research goal is to extend the operational range of the microphone. Two
provisions were proposed to accomplish this goal: the addition of a microphone back-plate
with small holes to create a means to damp the membrane motion; also, by employing an
optical sensor that measures membrane displacement averaged over an area rather than at a
point, as used in the previous study, to attenuate the high-order resonance modes which cause
instability in the control. The improved feedback microphone will ultimately enhance the
capability of “beam-forming” arrays used for research. The results show that individually,
each of the damping and area-averaged sensing was able to improve the operational range of
the microphone, and together, they combined for an even greater improvement in stability,
leading to tripling of the bandwidth of the baseline open-loop microphone.

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

v

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction

vii
1

2 Methods and Modeling
2.1 Microphone Components . . . . . . . . . . . .
2.1.1 PVDF Film Microphone Capsule . . .
2.1.2 Pressurization System . . . . . . . . .
2.1.3 Fiber-Optic Lever Displacement Sensor
2.1.4 Piezo Driver . . . . . . . . . . . . . . .
2.1.5 Feedback Controller . . . . . . . . . . .
2.2 Experimental Procedure . . . . . . . . . . . .
2.2.1 Acoustic Calibration . . . . . . . . . .
2.2.2 Electrical Calibration . . . . . . . . . .
2.3 Analytical Methods . . . . . . . . . . . . . . .
2.3.1 Digital Signal Processing . . . . . . . .
2.3.2 Membrane Model . . . . . . . . . . . .
2.3.3 Closed-Loop Modeling . . . . . . . . .
2.4 Baseline Results . . . . . . . . . . . . . . . . .
2.4.1 Internal Static Pressurization . . . . .
2.4.2 Open-Loop Characterization . . . . . .
2.4.3 Closed-Loop Characterization . . . . .

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3 Damping Provisions in the Microphone Back-Plate to Extend Operational
Range
3.1 Analytical Modeling of a Perforated Back-Plate . . . . . . . . . . . . . . . .
3.2 Damping Back-Plate Fabrication . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Design of Hole Layouts . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Laser Ablation Method . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Damped Membrane Calibration Results . . . . . . . . . . . . . . . . . . . . .
3.3.1 Open-Loop Characterization . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Feedback-Controlled Microphone Calibration Results . . . . . . . . .
3.4 Evaluation of Feedback-Controlled Microphone Results and Extension of Operational Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii

4
4
4
5
6
8
9
15
15
21
24
24
25
27
28
28
30
33

42
42
44
45
49
49
49
55
59

4 Area-Averaging Optical Displacement Sensor to Extend Operational Range
4.1 Modeling of a Displacement-Averaging Optical Sensor . . . . . . . . . . . . .
4.2 Characterization Results of the Feedback-Controlled Microphone with AreaAveraged Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Open-Loop Characterization . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Feedback-Controlled Microphone Calibration Results . . . . . . . . .
4.3 Evaluation of the Extension of the Operational Range of the Feedback-Controlled
Microphone with Area-Averaging Sensing . . . . . . . . . . . . . . . . . . . .
4.4 Characterization Results of the Feedback-Controlled Microphone with AreaAveraged Sensing and Damped Membrane . . . . . . . . . . . . . . . . . . .

63
63

5 Summary and Recommendations

82

A Hole Coordinates of Fabricated Damping Back-Plates

86

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

iv

67
67
70
70
75

LIST OF TABLES

2.1

Electrical components used in the phase-lead compensator depicted in Figure
2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Minimum and maximum values for the user-defined controller variables . . .

11

2.3

Natural frequencies for the undamped microphone in closed-loop operation
for different values of fc . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.1

Damping back-plate designs for the half-inch microphone capsule

. . . . . .

47

3.2

Membrane parameters estimated from open-loop calibration . . . . . . . . .

50

3.3

Actual and predicted damping ratios and natural frequencies for back-plates
with different hole arrangement . . . . . . . . . . . . . . . . . . . . . . . . .

55

Natural frequencies and damping ratios for the undamped microphone in
open- and closed-loop operation . . . . . . . . . . . . . . . . . . . . . . . . .

59

Natural frequencies and damping ratios for the damped microphone in openand closed-loop operation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.6

Feedback gains reached for the damped and undamped microphones . . . . .

60

4.1

Membrane parameters estimated from open-loop calibration of the microphone with the area-averaged displacement sensor . . . . . . . . . . . . . . .

67

Natural frequencies and damping ratios for the undamped microphone with
the area-averaging sensor in open- and closed-loop operation . . . . . . . . .

71

Feedback gains reached for the microphone with the area-averaging optical
sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Natural frequencies and damping ratios for the damped microphone with the
area-averaging sensor in open- and closed-loop operation . . . . . . . . . . .

80

3.4

3.5

4.2

4.3

4.4

v

4.5

Feedback gains reached for the damped microphone with the area-averaging
optical sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

A.1 Polar coordinates of the holes in the damping back-plate with the 91-hole
pattern and circular arrangement . . . . . . . . . . . . . . . . . . . . . . . .

87

A.2 Polar coordinates of the holes in the damping back-plate with the 121-hole
pattern and circular arrangement . . . . . . . . . . . . . . . . . . . . . . . .

88

A.3 Cartesian coordinates of the holes in the damping back-plate with the 91-hole
pattern and hexagonal arrangement . . . . . . . . . . . . . . . . . . . . . . .

89

vi

LIST OF FIGURES

2.1

Exploded view (left) and assembled view (right) of the microphone capsule .

5

2.2

Schematic diagram of the system used to pressurize the microphone cavity .

6

2.3

Schematic drawing of fiber-optic lever probe; red and blue lines indicate incident and reflected light respectively; for interpretation of the references to
color in this and all other figures, the reader is referred to the electronic version
of this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.4

Typical Philtec fiber-optic sensor response . . . . . . . . . . . . . . . . . . .

7

2.5

Block diagram of the microphone in closed-loop operation. Note that P , Zc ,
Zcr , and V are the Laplace transforms of the acoustic pressure, actual and
reference central membrane deflection, and output voltage respectively. . . .

9

2.6

Circuit schematic for the phase-lead compensator . . . . . . . . . . . . . . .

10

2.7

Controller calibration for low proportional gain Kp = 2 and fc = 6000;
Kd = 0.0001 (top); Kd = 0.001 (center); Kd = 0.008 (bottom) . . . . .

13

Controller calibration for high proportional gain Kp = 105 and fc = 6000;
Kd = 0.0001 (top); Kd = 0.001 (center); Kd = 0.008 (bottom) . . . . .

14

Schematic of the plane wave tube experimental setup (figure taken from [3])

16

2.8

2.9

2.10 Schematic of the open-loop acoustic calibration experimental setup . . . . .

18

2.11 Schematic of the closed-loop acoustic calibration experimental setup . . . . .

20

2.12 Diagram of the open-loop electrical calibration experimental setup . . . . . .

22

2.13 Diagram of the closed-loop electrical calibration experimental setup . . . . .

23

vii

2.14 Electrical response of PVDF membrane for various internal static pressurizations; actuation sensitivity of PVDF membrane (top); fundamental resonant
frequency (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.15 Open-loop characterization for the undamped microphone; magnitude (top);
phase (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.16 Open-loop electrical characterization for the undamped microphone with model;
magnitude (top); phase (bottom) . . . . . . . . . . . . . . . . . . . . . . . . 32
2.17 Closed-loop characterization for the undamped microphone increasing Kp first
(fc = 6000 Hz); magnitude (top); phase (bottom) . . . . . . . . . . . . . . .

34

2.18 Closed-loop characterization for the undamped microphone increasing Kp first
(fc = 6000 Hz); magnitude (top); phase (bottom) . . . . . . . . . . . . . . .

35

2.19 Closed-loop characterization for the undamped microphone increasing Kd first
(fc = 6000 Hz); magnitude (top); phase (bottom) . . . . . . . . . . . . . . .

36

2.20 Closed-loop characterization for the undamped microphone increasing Kp and
Kd simultaneously (fc = 6000 Hz); magnitude (top); phase (bottom) . . . .

37

2.21 Closed-loop characterization for the undamped microphone increasing Kp first
(fc = 3000 Hz); magnitude (top); phase (bottom) . . . . . . . . . . . . . . .

39

2.22 Closed-loop characterization for the undamped microphone increasing Kd first
(fc = 3000Hz); magnitude (top); phase (bottom) . . . . . . . . . . . . . . .

40

2.23 Closed-loop characterization for the undamped microphone increasing Kp and
Kd simultaneously (fc = 3000Hz); magnitude (top); phase (bottom) . . . . .

41

3.1

Perforated-insert-plug design (left) and microphone capsule with plug inserted
(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2

“Perforated back-plate design” MATLAB GUI for designing hole layouts . .

47

3.3

Sketches of damping back-plates with a circular and a hexagonal hole layout
pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

viii

3.4

Calibrated and predicted frequency response of the microphone with 91 holes,
hexagonal-arrangement damping back-plate: magnitude (top) and phase (bottom) plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Calibrated and predicted frequency response of the microphone with 91 holes,
circular-arrangement damping back-plate: magnitude (top) and phase (bottom) plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Calibrated and predicted frequency response of the microphone with 121 holes,
circular-arrangement damping back-plate: magnitude (top) and phase (bottom) plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Closed-loop characterization for damped microphone increasing Kp first; magnitude (top); phase (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Closed-loop characterization for damped microphone increasing Kd first; magnitude (top); phase (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Closed-loop characterization for damped microphone increasing Kp and Kd
simultaneously; magnitude (top); phase (bottom) . . . . . . . . . . . . . . .

58

3.10 Phase delay comparison for the undamped microphone. Broken lines represent
the models for the linear portions of the phase traces. . . . . . . . . . . . . .

61

3.11 Phase delay comparison for the damped microphone. Broken lines represent
the models for the linear portions of the phase traces. . . . . . . . . . . . . .

62

3.5

3.6

3.7

3.8

3.9

4.1

4.2

4.3

4.4

The first four axisymmetric deflection modes of a circular membrane (figure
taken from [3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Relative average displacement of the membrane as a function of the sensor
probe radius, with the lever sensor-to-membrane diameter of the prototype
marked with an arrow (figure taken from [3]) . . . . . . . . . . . . . . . . . .

66

Comparison of the magnitude frequency response for the microphone with
point and area-averaging displacement sensors. The first three axisymmetric
modes are identified based on circular membrane vibration theory: 2.3fr (9150
Hz), 3.6fr (14300 Hz), and 4.9fr (19500 Hz) for modes 1 to 3 respectively,
based on fr = 3980 Hz for the microphone with the point sensor . . . . . . .

68

Open-loop characterization for the undamped microphone with area-averaging
displacement sensor; magnitude (top); phase (bottom) . . . . . . . . . . . .

69

ix

4.5

Closed-loop characterization for the undamped microphone with area-averaging
displacement sensor increasing Kp first; magnitude (top); phase (bottom) . .

71

Closed-loop characterization for the undamped microphone with area-averaging
displacement sensor increasing Kd first; magnitude (top); phase (bottom) . .

72

Closed-loop characterization for the undamped microphone with area-averaging
displacement sensor increasing Kp and Kd simultaneously; magnitude (top);
phase (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Phase delay comparison for the undamped microphone with the area-averaging
optical sensor. Broken lines represent the models for the linear portions of
the phase traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Open-loop characterization for the damped microphone with the area-averaging
displacement sensor; magnitude (top); phase (bottom) . . . . . . . . . . . .

76

4.10 Closed-loop characterization for the damped microphone with area-averaging
displacement sensor, increasing Kp first; magnitude (top); phase (bottom) .

77

4.11 Closed-loop characterization for the damped microphone with area-averaging
displacement sensor, increasing Kd first; magnitude (top); phase (bottom) .

78

4.12 Closed-loop characterization for the damped microphone with area-averaging
displacement sensor, increasing Kp and Kd simultaneously; magnitude (top);
phase (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.13 Phase delay comparison for the damped microphone with the area-averaging
optical sensor. Broken lines represent the models for the linear portions of
the phase traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.6

4.7

4.8

4.9

5.1

A comparison of frequency and phase response for several of the tested microphone designs; magnitude (top); phase (bottom). Broken lines on top plot
represent the ± 3dB boundaries for identifying the bandwidth . . . . . . . .

x

84

Chapter 1
Introduction
Modern aeroacoustics measurements rely on the use of a large number of microphones in
“beam-forming” arrays. Such arrays are used systematically by NASA, as well as other
agencies and industry, to characterize, and hence utlimately mitigate acoustic noise emanating from aircraft during takeoff and landing [1]. Typically, a beam-forming array employs
anywhere from thirty to one hundred microphones that must have a known calibration and
matched phase response. The basic operating principle of the array involves using slight
differences in the phase of the measured waveforms by the microphones in the beam-forming
array to calculate the location of the acoustic point source.
Various microphones are currently used in beam-forming arrays, including instrument
grade condenser microphones and electret microphones. Condenser microphones’ operating
principle is that the stretched metallic microphone membrane and a conductive back-plate
situated a small distance behind it act as the two plates of a capacitor. Acoustic pressure
causes the diaphragm to deflect, leading to change in the capacitance with the membrane
motion. In a DC-biased condenser microphone, an approximately fixed charge is applied to
1

the plates and the voltage across the plates changes with the capacitance change. Commercial
condenser microphones are available which are individually calibrated and have a high degree
of phase uniformity within ±10◦ at 10 kHz 1 . At a cost of approximately $2000 per channel,
typical arrays (containing 30 or more microphones) using this microphone technology can
be very expensive. Furthermore, the calibration of the array microphones before tests is
essential for accurate measurements. Given the large number of microhpones in an array,
the calibration is time-consuming. Moreover, if the array is used for measurement in the
field, the laboratory-determined calibration may be altered by environmental factors.
An inexpensive option is to use the electret microphone. An electret is a material with
an embedded electric charge. In an electret microphone, the electret is attached to the backplate. Operation is similar to a condenser microphone without the need for a high-voltage
power supply. While this lowers the cost, electret microphones are known to vary widely with
respect to mechanical and electrical sensitivity and phase. Moreover, their response tends
to drift due to environmental effects, and the polarized electret material loses its charge over
time.
An alternative to the above microphone technologies is the concept of a feedback-controlled
microphone. In this technology, membrane motion is detected and nullified using some
means, for example an opposing pressure produced electrostatically or electromagnetically,
through a feedback control loop integrated into the sensor. A feedback-controlled microphone
has been developed recently at MSU. A proof of concept study of this new microphone type
was successfully undertaken [2], [3]. The feedback provision of the microphone allows it to
“self-calibrate” and gives it response-matching capabilities—a unique advantage over current
1 Datasheet - Br¨el & Kjær Type 4958 20kHz Precision Array Microphone
u
2

microphone technology which results in considerable time and cost savings for aeroacoustics tests. Previously existing feedback-controlled microphones are summarized in [3]. The
microphone described in [2] is unique in that the combination of actuation and sensing techniques is being used for the first time and data exists to demonstrate the advantages of the
feedback-controlled microphone, whereas other studies lacked such evidence.
The focus of the present thesis is to improve the design and extend the operational range
of the feedback-controlled microphone developed in [2]. The baseline design as well as the
proposed improvements are outlined in the following chapters. Specifically, perforated backplates were fabricated and incorporated into the microphone design as a means of mechanical
damping of the membrane motion. The microphone response was then characterized to show
the damping of all membrane resonance modes. Secondly, the previous design utilized a
fiber-optic lever displacement sensor that averages displacement over a small point located
at the center of the membrane. By employing a commercially available sensor that averages
displacement over a larger area, the observability of higher-order membrane resonance modes
decreases relative to the fundamental resonance mode. The increase in operational range
was examined through the frequency response for this microphone design. The microphone
was also characterized with the two above provisions implemented simultaneously.

3

Chapter 2
Methods and Modeling

2.1
2.1.1

Microphone Components
PVDF Film Microphone Capsule

The capsule design of the microphone prototype is shown in Figure 2.1. A 28µm PVDF
(polyvinelidine-fluoride) film constitutes the diaphragm. The piezoelectric property of the
membrane provides the actuation means for the feedback. The membrane is clamped between
two aluminum pieces—the capsule body and the clamping ring, fabricated on a lathe from
a piece of stock. The lip around the center hole puts tension in the membrane. And the
rubber gasket between the pieces provides a seal; it also insulates the aluminum pieces from
each other as they act as electrical conductors to the two metallized sides of the PVDF film.
The screws attaching the two aluminum pieces are nylon so as to be nonconducting. On the
opposite end of the microphone, a second rubber gasket between the capsule body and the
acrylic plate further seals the cavity behind the membrane for the purposes of pressurization;
the capsule is internally pressurized to improve electrical actuation sensitivity (in membrane
4

deflection per unit voltage), as will be seen in section 2.4.1. Pressurization is fulfilled through
connecting a pressurized air source to the pressure tap in the acrylic back-plate.

Figure 2.1: Exploded view (left) and assembled view (right) of the microphone capsule

2.1.2

Pressurization System

Figure 2.2 shows the schematic of the pressurization system used to induce curvature on the
PVDF membrane. It is a well-established fact that curvature of the membrane will result in
an increase of actuation sensitivity [4], [5], [6]. Air is directed from a high pressure air system
in the building into the microphone back chamber through the pressure tap. Pressurization
level is manually controlled by a series of valves and monitored with a pressure transducer.
The integrated pressure sensor is model MPX4010DP and it has 0.45 V/kPa sensitivity with
a 0.14V offset for input up to 10kPa. A one-gallon tank is connected to the pressurization
line to reduce any line pressure fluctuations.
5

Figure 2.2: Schematic diagram of the system used to pressurize the microphone cavity

2.1.3

Fiber-Optic Lever Displacement Sensor

The displacement of the PVDF diaphragm in response to incident acoustic pressure is sensed
using a fiber-optic lever displacement sensor. The baseline design utilizes a Philtec model
D20 optical sensor, which measures deflection at virtually a point (0.51mm tip diameter) at
the center of the membrane. Figure 2.3 shows a schematic of the fiber-optic sensor.

Figure 2.3: Schematic drawing of fiber-optic lever probe; red and blue lines indicate incident
and reflected light respectively; for interpretation of the references to color in this and all
other figures, the reader is referred to the electronic version of this thesis.

Light is emitted from the transmitting fiber and reflected off the membrane surface
back to receiving fibers. A photodetector detects the intensity of the reflected light. The
6

corresponding voltage-to-displacement relation is such that the voltage first increases with
distance between the membrane and sensor tip before reaching a peak and then beginning
to decrease. A typical voltage-to-displacement relation is shown in Figure 2.4. As seen from
the figure, the sensor’s response has two relatively linear portions in the “near-” and “farside” operating ranges. For accurate calibration, the peak voltage is always set to five volts
before a microphone characterization is carried out. This adjustment may be necessary due
to fluctuations in temperature or humidity in the room. Because the fiber-optic lever sensor
is more sensitive in the “near-side” range, the operating point is set in the middle of this
range.

Figure 2.4: Typical Philtec fiber-optic sensor response

Chapter 4 describes results obtained using an area-averaging optical sensor, Philtec model
D169, with a tip diameter of 4.29mm. As will be explained in Chapter 4, in utilizing the
7

area-averaging sensor, the sensed membrane deflection due to high-order resonance modes is
attenuated by spatial averaging of those modes relative to that produced by the fundamental
resonance. This type of attenuation is one of the two mechanisms explored in this study
as ways of increasing the range of controller gains/sensor response over which stable sensor
operation is maintained.
The optical sensor transfer function is given in equation (2.1) and is based on a manufacturer given cut-off frequency of 20kHz and a first-order dynamical system model.

Go (s) =

2.1.4

79400
(V /m)
s
+1
125600

(2.1)

Piezo Driver

Figure 2.5 shows a block diagram of the microphone in feedback operation. The piezoelectric
system represents the piezoelectric driving circuit employed to drive the PVDF membrane
in response to the output of the lever sensor. For these experiments, a ThorLabs model
MDT691 is used; it provides a gain of 15 and a maximum output between 0 and 150V. During
operation, a DC offset of 75V is imposed using a knob on the front panel of MDT691. This
centers the PVDF driving voltage in the middle of the operating range, enabling attainment
of membrane driving voltage swings of up to the full 150V range, peak to peak. The transfer
function of the peizo driver is taken simply as a flat magnitude with no phase delay. This
decision is based on the insignificant value of the phase of the piezo driver relative to the
amount of phase delay caused by the dynamic response of the fiber-optic sensor.
8

Figure 2.5: Block diagram of the microphone in closed-loop operation. Note that P , Zc ,
Zcr , and V are the Laplace transforms of the acoustic pressure, actual and reference central
membrane deflection, and output voltage respectively.

2.1.5

Feedback Controller

For the feedback control, a new compensator was built modeled after the one used in [3]. The
phase-lead compensator (see compensator in Figure 2.5) has three user-specified parameters
to achieve the desired transfer function: proportional gain Kp , derivative gain Kd , and corner
frequency fc . Varying these parameters allow the effective membrane’s mass (M), stiffness
(K), and damping (C) to be altered. This enables tuning of the frequency response of the
microphone, and hence, in a beam-forming array, several capsules can be phase matched (a
crucial characteristic for the accuracy of beam-forming measurements).
Figure 2.6 shows a schematic of the operational-amplifier-based circuit for the compensator. Terminology for the components for the rebuilt controller are kept consistent with
their counterparts in previous research [3]. Resistors are represented by Rp1 , Rp2 , Rd2 , and
Rf,sum , while digital potentiometers Rp , Rd , and Rc control proportional gain, derivative
gain, and corner frequency. A LabVIEW program called “DigitalControllerNew.vi” is used
9

to set the digital potentiometers. A fixed capacitance is represented by C. The rationale for
creating a new controller was to have the ability to swap out capacitances for the case when
higher derivative gains (Kd ) were needed. The new controller was built on a breadboard,
thus giving ease in switching out components.

Figure 2.6: Circuit schematic for the phase-lead compensator

The transfer function for the controller GC (s) is given in equation (2.2)

Kd s
GC (s) = Kp +
τd s + 1
Rp Rf,sum
Rp1 Rp2

(2.2b)

Rd Rf,sum C
Rd2

(2.2c)

1
2πRc C

(2.2d)

Kp =

Kd =

(2.2a)

fc =

10

The resistance and capacitance values employed are listed in Table 2.1. The corresponding minimum and maximum Kp , Kd , and fc values are listed in Table 2.2.
Table 2.1: Electrical components used
Symbol
OA1
OA2
OA3
OA4
Rp1
Rp2
Rd2
Rf,sum
Rtest
Rp
Rd
Rc
C

in the phase-lead compensator depicted in Figure 2.6
Type
Value
Op-amp
UA741CN
Op-amp
UA741CN
Op-amp
UA741CN
Op-amp
UA741CN
Resistor
1.794 kΩ
Resistor
2.202 kΩ
Resistor
2.208 kΩ
Resistor
21.96 kΩ
Resistor
22.04 kΩ
Digital Pot MCP42050
Digital Pot MCP42100
Digital Pot MCP42010
Capacitor
0.022 µF

Table 2.2: Minimum and maximum values for the user-defined controller variables
Kp
Kd (s) fc (Hz)
minimum 0.36 1.61E-5
1500
maximum 143 9.59E-3 222000

Before using the new compensator, it had to be calibrated to verify its operation. To
do so, a sinusoidal voltage sweep from 0 to 25kHz was connected to the compensator input,
voltage vi (Figure 2.6), using an HP model 33120A function generator. The resulting output
voltage (vo ) as well as vi were recorded digitally in order to obtain the magnitude and
phase response of the compensator. The data acquisition and processing details employed
are similar to those used for acoustic and electrical calibration of the microphone, which are
described in detail in section 2.2.
Figure 2.7 shows some calibration plots for a low Kp value, while Figure 2.8 shows some
11

calibration plots for a high Kp value within the available range (see Table 2.2). All results
are obtained for fc = 6000, and a low, intermediate, and high value of Kd . Figures 2.7 and
2.8 show that except for an early high frequency roll-off, the actual and expected controller
responses are consistent. The early roll-off is believed to be due to the limitation of the gainbandwidth product of operational amplifiers in the circuit. Attempts to remedy this issue
were made by using some higher bandwidth op-amps. However, these chips were electrostatic
sensitive and prone to noise in the signal. Recommendations for future work are to employ
thin film resistors and special capacitors in conjunction with the higher bandwidth op-amps
in the circuit.

12

Figure 2.7: Controller calibration for low proportional gain Kp = 2 and fc = 6000;
Kd = 0.0001 (top); Kd = 0.001 (center); Kd = 0.008 (bottom)

13

Figure 2.8: Controller calibration for high proportional gain Kp = 105 and fc = 6000;
Kd = 0.0001 (top); Kd = 0.001 (center); Kd = 0.008 (bottom)

14

2.2

Experimental Procedure

Two primary experimental methods are carried out for characterization of the frequency
response of the microphone capsule. The first, acoustic calibration, is the common method
for determining the frequency response of a sensor. In addition, the piezoelectric property
of the membrane allows it to be electrically actuated, which yields a second method of
membrane characterization, electrical calibration. It is this provision that would enable the
microphones to “self-calibrate” in a beam-forming array.

2.2.1

Acoustic Calibration

Acoustic calibration takes place in a plane wave tube apparatus. Sound waves propogate
down the tube perpendicularly to the tube’s axis. This guarantees that the reference and prototype microphones, each mounted at the same distance down the tube, will be excited with
identical acoustic pressures (in both magnitude and phase). Figure 2.9 shows a schematic
of the plane wave tube setup. The plane wave tube consists of a tube with a square cross
section of side length 12.7mm. This corresponds to a cut-off frequency of approximately 13.7
kHz (equivalent to an acoustic wavelength of twice the side length [8]).
A Dayton RS150S-8, 40 Watt speaker, driven by a Hafler P1000 audio amplifier, is placed
at the end of the tube. The B&K reference microphone and prototype capsule are mounted
in the tube so their faces are flush with the tube walls. The fiber-optic lever sensor enters
the tube through a small hole opposite the prototype capsule and points at the membrane’s
center. The sensor is mounted on a micrometer traverse table for precise adjustment.

15

Figure 2.9: Schematic of the plane wave tube experimental setup (figure taken from [3])

16

For calibration of the microphone in open-loop operation, the plane wave tube is incorporated into the experimental setup shown in Figure 2.10. The optical sensor output is fed to a
Larson Davis preamplifier model 2200C. The output of the amplifier is displayed on an oscilloscope, model Instek OS-653. The reference microphone used is B&K type 4938A 1/4” and
is channeled through the B&K conditioning amplifier. The reference microphone signal is
also viewed on the oscilloscope; and the reference microphone and feedback microphone voltages are recorded by a data acquisition board, PC-based National Instruments PCI-6024E
12-bit analog-to-digital converter (ADC), in analog input channels ai0 and ai1, respectively.
The ADC is capable of acquiring up to 8 multiplexed channels in differential mode at a
maximum sampling frequency of 200,000Hz. The speaker used to inject sound into the plane
wave tube is driven by a Hafler amplifer, model P1000, coupled with a Hewlett-Packard
function generator, model 33120A, which outputs white noise with 15MHz bandwidth.
A typical characterization experiment is initiated by turning on the function generator
and amplifier and increasing the volume level to a reasonably loud setting without any
clipping of the amplified signal (detected when an LED indicator light on the amplifier front
panel is lit). The Philtec optical sensor is brought close to the microphone membrane in its
near-field range to a value of around 2.5 volts, near the center of its linear range.
Data

are

then

recorded

using

a

LabVIEW

program

called

“Mic-Plate

Calibration V3 SimpleDAQ.vi”. The recorded output of the B&K and feedback microphones, which consists of 100 records of 16,384 points each, acquired at 50,000 samples
per second, are analyzed as described below in section 2.3.1 to obtain the magnitude and
phase response of the feedback microphone over the frequency range of interest.

17

Figure 2.10: Schematic of the open-loop acoustic calibration experimental setup

18

The acoustic calibration setup for the microphone in closed-loop operation is similar to
that of the open-loop and is shown in Figure 2.11. The main additional pieces of equipment
relative to the open-loop setup are the controller and piezo driver, which are described in
sections 2.1.4 and 2.1.5. During closed-loop operation, feedback gains are varied through the
LabVIEW interface controlling the digital potentiometers. These gains are systematically
increased until instability is reached. Three different methods of increasing gains are typically
used: increasing Kd first until instability and backing off a bit followed by increasing Kp to
just before instability, increasing Kp first until instability and backing off a bit followed by
increasing Kd to just before instability, and increasing Kp and Kd simultaneously until just
before instability. Throughout all testing, fc is kept at a predetermined fixed value.

19

Figure 2.11: Schematic of the closed-loop acoustic calibration experimental setup

20

2.2.2

Electrical Calibration

A second means for calibrating the microphone is made possible by the piezoelectric capability of PVDF. By applying a sinusoidal sweep voltage across the membrane over the desired
calibration frequency range, the membrane deflection amplitude and phase can be obtained
at different frequencies yielding the open-loop characterization. The sweep signal represents
an “equivalent” piezoelectric pressure, and hence it replaces the B&K microphone output
as the reference signal in the calibration process. The setup for this test is diagrammed
in Figure 2.12. For this test, the microphone is mounted outside the plane wave tube to
avoid exciting tube resonances, which could affect the membrane deflection and hence the
calibration, by the sound generated from membrane movement.

21

Figure 2.12: Diagram of the open-loop electrical calibration experimental setup

22

For closed-loop electrical calibration, the setup in Figure 2.13 is used. Here, a sinusoidal
sweep voltage from the function generator is input as a test voltage to the controller (input
vtest in Figure 2.6) and is recorded in addition to the input from the optical sensor fed
through the Larson-Davis preamplifier using the ADC.

Figure 2.13: Diagram of the closed-loop electrical calibration experimental setup

23

2.3

2.3.1

Analytical Methods

Digital Signal Processing

In all of the calibration experiments, two signals are sampled—that of the reference signal (either the reference microphone’s output voltage for the acoustic tests or the function
generator’s output for the electrical tests), and the voltage from the fiber-optic lever displacement sensor measuring the prototype microphone membrane’s deflection. These signals are
digitized using an analog-to-digital converter (ADC).
Through analysis in the frequency domain, the cross power spectrum for the two signals
√
as well as the auto power spectrum are each computed. If X = |X|ejφx (j = −1) is the
Fourier transform of the reference signal x and Y = |Y |ejφy is the Fourier transform of the
microphone signal y, the transfer function of the prototype sensor is given by the division of
the cross spectrum and auto spectrum, demonstrated in equation (2.3).

H=

XY ∗
|Y | j(φx −φy )
∗ = |X| e
XX

(2.3)

When the complex result of equation (2.3) is broken into polar components, the magnitude and phase of the prototype sensor are obtained and can be plotted for a range of
frequencies. Note that in practice, the cross and auto spectrum are obtained from averaging those obtained from 100 records with each record containing 16,384 points sampled at
50,000Hz. This yields a frequency resolution of 3.05Hz, Nyquist frequency of 25,000Hz, and
random uncertainty of 10%.
24

2.3.2

Membrane Model

The PVDF membrane can be modeled with the linear wave equation for a stretched membrane, given in equation (2.4).

2
2z + p = ρ ∂ z
T
s 2
∂t

(2.4)

where T , z, p, and ρs represent membrane tension, displacement, acoustic pressure, and
surface density. After an assumption that the membrane undergoes axi-symmetric parabolic
deflection (up to the fundamental resonance frequency) and knowing that displacement is
measured at the center, zc , equation (2.5) can be used to describe the motion, where a is
the membrane radius.

ρs zc +
¨

4T
zc = p
a2

(2.5)

In the Laplace domain, taking center displacement as output and acoustic pressure as
input, the transfer function H is given in equation (2.6), where M = ρs , K0 = 4T , and a
a2
generalized damping term C0 is added.

H(s) =

Zc (s)
1
=
2+C s+K
P (s)
Ms
0
0

(2.6)

The surface density M is taken as a known constant, taken from the properties of the
PVDF material. Stiffness, K0 , is related to the tension T , though T is difficult to estimate.
So K0 is calculated from the natural frequency ωn (2πfn ) using equation (2.7).

2
K0 = ωn M
25

(2.7)

Determination of the natural frequency is accomplished by observing the fundamental
resonant frequency fr and half-power frequencies on either side of the resonant frequency,
f1 and f2 , from the plot of the experimentally acquired frequency response. Equation (2.8)
shows how the damping ratio ζ can be determined from these values.

f − f1
2ζ = 2
fr

(2.8)

Knowing ζ, the natural frequency ωn can be calculated using equation (2.9)

ωr = ωn

1 − 2ζ 2

(2.9)

M K0

(2.10)

and C0 can be determined:

C0 = 2ζ0

During experimentation, multiple resonant peaks are observed, occurring at frequencies
predicted for the axi-symmetric modes of a circular membrane. The model includes these
modes since they are a cause of instability in the feedback system. A modified membrane
transfer function is given in equation (2.11) with Cm = (ωm /ω0 )2 C0 , Km = (ωm /ω0 )2 K0 ,
and ωm /ω0 representing the resonant frequency ratios for the higher order axi-symmetric
modes calculated from vibration theory and listed, for example, in [8].

H(s) =

Zc (s)
=
P (s)

N −1
1
(−1)m
M s 2 + Cm s + Km
m=0
26

(2.11)

2.3.3

Closed-Loop Modeling

One advantage of the feedback-controlled microphone is the ability to tune the microphone’s
response. Given that one use of the microphones is in array applications where response
matching is crucial, being able to predict the closed-loop response from given open-loop
parameters and controller gain settings is important.

The closed-loop transfer function, based on the block diagram depticted in Figure 2.5, is
given is follows:

Wcl (s) =

H(s)GO (s)
V (s)
=
P (s)
1 + H(s)GO (s)GC (s)GP Z (s)

(2.12)

Employing the denominator of equation (2.12) as the characteristic equation, a pole
placement method was used in [3] to design the controller to achieve a certain desired fundamental resonant frequency and damping for the closed-loop response. A similar method
was not utilized here because the main interest was in determining the limits of instability
of the sensor rather than arriving at a desired response.

The following results show how the original microphone design performs. Chapters 3 and
4 on the added provisions of damping and area-averaged sensing respectively will demonstrate
the improvement in the operational range of the microphone that each of these methods
offers.
27

2.4
2.4.1

Baseline Results
Internal Static Pressurization

An induced curvature in the membrane is known to improve the piezoelectric response of
the membrane (i.e. deflection per unit applied voltage) [4], [5], [6]. In the present work,
this is achieved by pressurizing the back chamber of the microphone as described in section 2.1.2. To select a value for the chamber pressure, internal static pressurization tests
were performed. The changes in actuation sensitivity (Se ) and resonant frequency (fn ) of
the PVDF membrane were determined for several internal pressurizations using electrical
calibration and the results are plotted in Figure 2.14.
Based on the responses shown in Figure 2.14, an internal static pressurization of 2kPa is
chosen for all characterization tests of the microphone. This pressurization yields a significant
increase in the actuation sensitivity with no substantial increase in resonant frequency.

28

Figure 2.14: Electrical response of PVDF membrane for various internal static pressurizations; actuation sensitivity of PVDF membrane (top); fundamental resonant frequency
(bottom)

29

2.4.2

Open-Loop Characterization

Figure 2.15 shows the open-loop characterization for the undamped microphone, both acoustic and electrical calibration. The two methods of characterization yield similar responses, as
expected. Therefore the electrical open-loop calibration is used from this point forward. It
gives a smoother response, eliminating the small steps seen in the acoustic response, believed
to be due to acoustic reflections in the plane wave tube.
Figure 2.16 shows the electrical characterization alone with the open-loop model determined using the method in section 2.3.2.

30

Figure 2.15: Open-loop characterization for the undamped microphone; magnitude (top);
phase (bottom)

31

Figure 2.16: Open-loop electrical characterization for the undamped microphone with model;
magnitude (top); phase (bottom)

32

2.4.3

Closed-Loop Characterization

As with the open-loop characterization, calibrations were performed acoustically and electrically for the closed-loop case. As described in section 2.2, the gains were systematically
increased in three different patterns. Figure 2.17 compares the acoustic and electrical responses. In this case, the electrical and acoustic calibrations were both performed with the
microphone mounted in the plane wave tube. It can be seen from Figure 2.17 that the two
methods of characterization yield similar results, just as with the open-loop. Therefore, focus
will be on using the electrical calibration method because it is simpler and can be used over
a wider bandwidth than the plane wave tube.
Figures 2.18–2.20 show the electrical response plotted with the model curve predicted
using equation 2.12 coupled with the knowledge of the open-loop parameters, feedback gains,
and transfer function of the lever sensor. The figures correspond to the three methods of
increasing the gains, described in the closed-loop calibration procedure of section 2.2, while
setting fc to a fixed 6000 Hz. This value for fc is somewhat arbitrary; it was chosen to be
slightly higher than the open-loop microphone’s natural frequency so the influence of Kd
would be limited to the frequency range up to fc . Overall, the results in Figures 2.18–2.20
show an alteration in the microphone’s response, as demonstrated, for example in the increase
in the natural frequency of the closed-loop microphone. The specific values of the natural
frequencies can be found in Table 2.3. For reference, the natural frequency of the open-loop
microphone is 3980 Hz. It should be noted that the results in Figures 2.18–2.20 represent the
baseline closed-loop microphone response against which the response of the microphone with
added damping provision and with an area-averaging sensor will be compared in Chapters
3 and 4, respectively.

33

Figure 2.17: Closed-loop characterization for the undamped microphone increasing Kp first
(fc = 6000 Hz); magnitude (top); phase (bottom)

34

Figure 2.18: Closed-loop characterization for the undamped microphone increasing Kp first
(fc = 6000 Hz); magnitude (top); phase (bottom)

35

Figure 2.19: Closed-loop characterization for the undamped microphone increasing Kd first
(fc = 6000 Hz); magnitude (top); phase (bottom)

36

Figure 2.20: Closed-loop characterization for the undamped microphone increasing Kp and
Kd simultaneously (fc = 6000 Hz); magnitude (top); phase (bottom)

37

The closed-loop calibrations were repeated for two other values of fc . Because the value
of 6000 Hz was somewhat arbitrary, both higher and lower values (3000 and 10,000 Hz
respectively) were tested. The higher value of 10,000 Hz did not produce substantial change
in the microphone’s response, and hence the results are not included here. On the other
hand, the calibration results for fc = 3000 Hz are shown in Figures 2.21–2.23 for the three
methods of increasing gains. Lowering the value of fc produced some improvement in the
results, when improvement is defined as higher closed-loop natural frequencies.
Table 2.3: Natural frequencies for the undamped microphone in closed-loop operation for
different values of fc
Increase
Increase
Increase
Kp first
Kd first
together
4414
4309
4336
Natural frequency for fc = 6000 Hz
4490
4506
4417
Natural frequency for fc = 3000 Hz

38

Figure 2.21: Closed-loop characterization for the undamped microphone increasing Kp first
(fc = 3000 Hz); magnitude (top); phase (bottom)

39

Figure 2.22: Closed-loop characterization for the undamped microphone increasing Kd first
(fc = 3000Hz); magnitude (top); phase (bottom)

40

Figure 2.23: Closed-loop characterization for the undamped microphone increasing Kp and
Kd simultaneously (fc = 3000Hz); magnitude (top); phase (bottom)

41

Chapter 3
Damping Provisions in the
Microphone Back-Plate to Extend
Operational Range
In Chapter 2, a feedback-controlled microphone was discussed. Though the microphone has
the unique advantages of ‘self-calibration’ and response matching to other similar types of
microphones, controller instability caused by high-order resonance peaks limits the ability
to increase the fundamental resonance frequency through feedback. To address this, a microphone with mechanical damping provisions was constructed and tested for increase in
stability and operational range of the microphone.

3.1

Analytical Modeling of a Perforated Back-Plate

A method for decreasing the strength of the membrane resonances is the addition of perforated back-plates. These are similar to the concept of back-plates found in commercial
42

condenser microphones. A rigid back-plate with small diameter holes has been incorporated
into the microphone design. This back-plate sits a distance of 1mm behind the membrane,
as seen in Figure 3.1. As acoustic pressure from an incoming sound causes the membrane
to oscillate, air is forced through the holes and the associated viscous friction causes losses
which in turn results in damped membrane motion.

Figure 3.1: Perforated-insert-plug design (left) and microphone capsule with plug inserted
(right)

Holes with known diameter and depth are used because a model is available to give
the predicted damped response for the membrane. This model is based on work done by
Zuckerwar [7] with additional assumptions and simplifications used by Radcliffe [3]. Details
of the model can be found in these references; however, essential traits are summarized
here for completeness. The model is based on electrical-circuit analogy, in which each hole is
modeled as a resistance and an inductance in series. The parallel network of such elements is
then connected in series with a capacitor representing the air chamber behind the perforated
plate. The simplifying assumptions introduced in [3] include assuming that the membrane
has piston-like motion and that air movement through all holes is identical. Through the
43

analysis outlined in [3], it is shown that the resisting pressure due to the influence of the backplate can be modeled as an added effective inertial mass, M0b , damping, C0b , and stiffness,
K0b . For a hole length of lk and a hole radius of rk (see Figure 3.1), these expressions are
given in equations (3.1), (3.2), and (3.3):

ρ (l + 1.7rk )
M0b = 0.4314 · A 0 k
qπr2
k
(8ω0 ρ0 µ)1/2
C0b = 0.4314 · A
qπr2
k

K0b = 0.4314 · A

l
1+ k
2rk

γρ0 c2
T
Vb

(3.1)

(3.2)

(3.3)

where A is the membrane surface area, ρ0 is air density, q is the number of holes, ω0 is
the membrane natural frequency, µ is the air viscosity, γ is the specific heat ratio, cT the
isothermal speed of sound through air, and Vb is the capsule back chamber volume. When
these terms are added to the membrane transfer function, they modify it as follows:

H(s) =

3.2

1
2 + (C + C )s + K + K
(M + M0b )s
0
0
0b
0b

(3.4)

Damping Back-Plate Fabrication

The damping back-plates are thin, disc-like components that are mounted in the capsule so
that they sit approximately 1mm back from the membrane. The current design involves a
cylindrical plug on which the damping back-plate disc is mounted using superglue and the
44

resulting assembly is inserted into the microphone capsule design depicted in Figure 3.1. A
mechanical stop for the top of the plug is used to position the damping back-plate at the
correct distance from the membrane. The plug is easily removed from the capsule, giving
the ability to switch between back-plates with different hole layout patterns efficiently.

3.2.1

Design of Hole Layouts

Using the model described in section 3.1, some designs for the damping back-plate’s layout
were created. Based on analysis in [3], in order to effectively design a perforated back-plate
with minimal effective mass, a very small hole radius, rk , is ideal. In this way, one can
increase the damping while minimizing the reduction of the resonance frequency. Accordingly, different ways of making very small holes (smaller than 0.25mm diamter, the limit
with conventional drilling techniques) were investigated. Limitations based on manufacturability forced the hole depth-to-diameter ratio (lk /2rk ) to a maximum of 1. This limitation
is for a laser ablation technique, determined as the most cost-effective available method for
fabricating damping back-plates and described in 3.2.2. Other methods considered included
microfabrication, but available facilities were tooled for thin-film microfabrication, not for
bulk micromachining applications as necessary for the back-plate fabrication. Also, using a
porous material instead of creating holes in a solid material was considered, but this method
would not allow the model described in section 3.1 to be utilized and a systematic design
approach would not have been possible.
A MATLAB program with GUI interface created by Radcliffe [3] was utilized in determining the size and number of holes for the back-plate designs. Back-plates were designed
for half-inch, three-eighths-inch, and quarter-inch microphone membrane sizes. However, in
45

the present study, only half-inch sensors are characterized. Figure 3.2 shows a screenshot
of the GUI, with the design for the half-inch diameter membrane. The membrane parameters on the lefthand side of the GUI—Diameter, Surface Density, Damping Ratio, and Edge
Tension—as well as the back-plate parameters—Hole Diameter, Hole Length, # of Holes,
and Back. Volume—are all input parameters. The membrane parameters M, C, and K as
well as the air layer parameters are all GUI output values. The input membrane parameters
are found from electrical calibration of the open-loop response as described in Chapter 2.
The plot shown at the bottom of the GUI is the magnitude frequency response plot. The
black line is the calibration of the membrane without damping, used to obtain the membrane
input parameters to the model, and the red line is the prediction of the model with the added
damping.
The hole dimensions were chosen as 0.1mm diameter and 0.1mm hole depth, based on the
laser ablation process. To yield visibly damped resonance peaks while keeping the natural
frequency near the open-loop for the undamped microphone, the targeted damping ratio
was 0.1. The holes were laid out in two different configurations: concentric circles and
a hexagonal layout. Letting the geometry govern the design, back-plates with 91 and 121
holes were created. Table 3.1 summarizes the back-plate designs for the half-inch microphone
sensor size, including the resulting damping ratio based on the predictions of the MATLAB
code. The coordinates of the hole centers for the three back-plate layouts are included
in Appendix A for reference. Figure 3.3 shows a sketch of back-plates with circular and
hexagonal hole layout geometry.

46

Figure 3.2: “Perforated back-plate design” MATLAB GUI for designing hole layouts

Table 3.1: Damping back-plate designs for the half-inch microphone capsule
Half-Inch Diameter Capsule
Number of holes (q)
91
91
121
Hole layout geometry
hexagonal circular circular
Predicted damping ratio ζ
0.1161
0.1161
0.0946

47

Figure 3.3: Sketches of damping back-plates with a circular and a hexagonal hole layout
pattern

48

3.2.2

Laser Ablation Method

Laser ablation is a method by which solid material is removed by a laser. This was done at
NASA Langley Research Center (LaRC) in the Fabrication Technology Development Branch.
The material, aluminum in the case of the back-plates, absorbs the energy and sublimates.
For the damping back-plates, the hole outlines were traced by the laser, and afterward, a
circular outline was drawn around the outside to cut out a disc-shaped back-plate from the
aluminum sheet. The pattern was read from an AutoCAD .dwg file. The laser ablator
follows the lines in the order they are drawn, so care must be taken to create the holes in
the AutoCAD drawing in the correct sequence. Also, the beamwidth of the laser must be
taken into account; for a beamwidth of 0.025mm and desired hole diameter of 0.1mm, holes
should be specified to 0.075mm diameter in the .dwg file.

3.3
3.3.1

Damped Membrane Calibration Results
Open-Loop Characterization

The open-loop parameters for the membrane are determined by the method described in
section 2.3.2 using the baseline calibration shown in Figure 2.16. These parameters are
summarized in Table 3.2 and are then used in the model from section 3.1 to determine the
predicted frequency response for the half-inch microphone with the three damping inserts.
Figures 3.4–3.6 show the open-loop characterizations for the microphone with each of
the three damping inserts described in Table 3.1. Plotted also are the predicted responses.
These figures show that the constructed back-plates work well in damping the resonance
peak substantially. It is clear that there is a deviation from the model and that the actual
49

Table 3.2: Membrane parameters estimated from open-loop calibration
Parameter
Symbol
Value
Unit
Mass
M
0.0493
kg/m2
Damping
Stiffness

C0
K0

36.7960
30.841 · 106

P a · s/m
P a/m

Damping Ratio
Natural Frequency

ζ0
f0

0.0149
3.98

kHz

damping is stronger than predicted. The model, however, gave good predictions in [3]. Some
sources of lack of agreement in the current work may include that the very small hole sizes
require some modification to the model. Also, perhaps the model needs to be modified to
account for the small air gap between the back-plate and the membrane.

50

Figure 3.4: Calibrated and predicted frequency response of the microphone with 91 holes,
hexagonal-arrangement damping back-plate: magnitude (top) and phase (bottom) plot

51

Figure 3.5: Calibrated and predicted frequency response of the microphone with 91 holes,
circular-arrangement damping back-plate: magnitude (top) and phase (bottom) plot

52

Figure 3.6: Calibrated and predicted frequency response of the microphone with 121 holes,
circular-arrangement damping back-plate: magnitude (top) and phase (bottom) plot

53

As indicated in section 2.3.2, a lumped-parameter, second-order model represents the
response of the microphone through the first resonance frequency very well and can be used
to determine the stiffness, damping ratio, and resonance frequency of the membrane. To this
end, such a model was fit to the responses shown in Figures 3.4–3.6. Because the resonance
frequency is highly damped, the natural frequency and damping ratio estimation methods
of section 2.3.2 are invalid. Instead, the second-order equation model is fit directly to the
response data in order to determine the parameters of interest. To facilitate the fitting
process, the expression for the amplitude for a second-order system is rearranged and a
linear fit is performed on the data. Equation (3.5) gives the expression for the magnitude A
of the frequency response normalized by the response at zero frequency.

A=

1
ω
ω
[1 − ( ωn )2 ]2 + 4ζ 2 ( ωn )2

(3.5)

Setting x = ω 2 , the expression may be rearranged as follows:

1 1
(
− 1) = mx + b
x A2
1
4
ωn

(3.6b)

2(2ζ 2 − 1)
4
ωn

(3.6c)

m=

b=

(3.6a)

Based on equation 3.6, a linear fit can be performed on data in Figures 3.4–3.6 after
the dependent variable has been manipulated to match the expression on the left hand
side of equation (3.6a) and the independent variable is set to ω 2 . The membrane’s natural
frequency and damping ratio can then be determined from the slope and intercept of the
54

fit using equations 3.6b and 3.6c. The curve fit is done over the frequency range up to
4000Hz to avoid the inclusion of data corresponding to high-order resonances, which are not
represented by the second-order model. The resulting curve fits can be seen in Figures 3.4
and 3.6, drawn in red. Table 3.3 compares the damping ratio obtained from the fit to that
predicted by the design model for the three cases investigated. The best fit model yielded
complex numbers for the natural frequency and damping ratio of the microphone response
with the back-plate with 91 holes in circular arrangement.
Table 3.3: Actual and predicted damping ratios and natural frequencies for back-plates with
different hole arrangement
91 holes hexagonal 91 holes circular 121 holes circular
ζpredicted
0.1161
0.1161
0.0946
ζactual
0.5697
?
0.4739
fn,predicted (Hz)
3721
3721
3846
fn,actual (Hz)
3541
?
3330

The values in Table 3.3 confirm the discrepency noted in the response plots earlier with
the actual damping being larger than the damping ratio predicted from the model. The
121-hole circular pattern insert was used for further closed-loop testing. Of the three damping back-plates, it had the lowest value for the actual damping ratio, closest to the target
damping value of around 0.1.

3.3.2

Feedback-Controlled Microphone Calibration Results

Figures 3.7–3.9 show the closed-loop responses for the damped microphone. Figure 3.7 shows
the characterization with Kp increased first. Figure 3.8 shows the characterization with
Kd increased first. And Figure 3.9 shows the characterization with Kp and Kd increased
simultaneously in small increments. Here, all the results are done for fc = 3000.
55

Figure 3.7: Closed-loop characterization for damped microphone increasing Kp first; magnitude (top); phase (bottom)

56

Figure 3.8: Closed-loop characterization for damped microphone increasing Kd first; magnitude (top); phase (bottom)

57

Figure 3.9: Closed-loop characterization for damped microphone increasing Kp and Kd
simultaneously; magnitude (top); phase (bottom)

58

3.4

Evaluation of Feedback-Controlled Microphone Results and Extension of Operational Range

The increase of operational range for the closed-loop operation of the microphone is examined
and compared. Specifically, the extent of change that can be exerted on some metric of
the microphone’s response through the feedback is assessed quantitatively. One method to
compare stability range is by measuring the increase in natural frequency of the membrane.
For the undamped-microphone case with a clear fundamental resonance peak, the method
described in section 2.3.2 is used to extract the natural frequency and damping ratio from
the data plotted in Figures 2.18–2.20, and the damping ratios and natural frequencies are
summarized in Table 3.4. For the damped-microphone case, there is not a strong resonant
peak and a best fit model is used, as described in section 3.3.1. The models were developed
from the closed-loop, damped microphone electrical calibration results plotted in Figures
3.7–3.9. Damping ratio and natural frequency values are summarized in Table 3.5.
Table 3.4: Natural frequencies and damping ratios for the undamped microphone in openand closed-loop operation
Closed Loop
Open Loop
Increase
Increase
Increase
Kp first
Kd first
together
3862
4490
4506
4417
Natural frequency fn (Hz)
0.0149
0.0219
0.0327
0.0091
Damping ratio ζ

In Tables 3.4 and 3.5, it is observed that the closed-loop method of increasing Kd first has
the highest natural frequency for both the undamped- and damped-microphone cases. For
the undamped microphone, closed-loop operation is able to increase the natural frequency
by up to 16.7%, while for the damped microphone, the natural frequency increases up to
59

Table 3.5: Natural frequencies and damping ratios for the damped microphone in open- and
closed-loop operation
Closed Loop
Open Loop
Increase
Increase
Increase
Kp first
Kd first
together
3432
4278
5181
4659
Natural frequency fn (Hz)
0.4885
0.3187
0.7911
0.6170
Damping ratio ζ

51.0% during closed-loop operation. This is a significant improvement in the microphone
stable operational range due to the addition of the damping back-plate.
Another mode of comparison between the damped and undamped microphones is to
compare the controller gains attained before the operation became unstable in each situation.
Table 3.6 shows the data pertinent to this comparison.
Table 3.6: Feedback gains reached for the damped and
Undamped
Kp Kd (s)
Increase Kp first
4 0.0011
Increase Kd first
1 0.0013
IncreaseKp and Kd together 10 0.0008

undamped microphones
Damped
Kp Kd (s)
62 0.0001
1 0.0017
13 .0014

As seen from the table, the controller gains can be increased to a higher level before reaching instability for the damped microphone. This gives another indication of the increased
operational range through the addition of membrane damping.
A third method of comparison for the damped and undamped microphones is to demonstrate the amount of control possible of the phase response for each case. This measure
is particularly beneficial to beamforming array applications where being able to match the
time delays for several sensors is very important. To avoid signal distortion, the microphone
phase response corresponding to this time delay, τ , should vary linearly with frequency;
sensors with ζ = 0.6 to 0.7 have phases which are nearly straight for the widest frequency
60

range. Assuming, however, that the linear range is wide enough for the chosen application
regardless of the damping ratio, being able to manipulate τ using the feedback over a wide
range of values is ideal.
When Kp is increased, the damping is decreased and the phase response slope at zero
frequency decreases, yielding a smaller τ . Conversely, when Kd is increased, the damping is
increased and the phase slope increases, yielding a larger τ . Figure 3.10 shows the open-loop
and the closed-loop phase response for when either Kp or Kd are taken to their exteme
values just before instability is reached for the undamped microphone. Figure 3.11 shows
similar results for the damped microphone.

Figure 3.10: Phase delay comparison for the undamped microphone. Broken lines represent
the models for the linear portions of the phase traces.

To calculate τ , the value of the phase at 1000Hz was retrieved from the data. Using
this value and an assumed value of zero degrees at zero frequency, a line connecting the
two points to model the linear portion of the phase could be determined. The linear range
extends further than 1000Hz for many cases, but fitting over a smaller range ensured that no
61

Figure 3.11: Phase delay comparison for the damped microphone. Broken lines represent
the models for the linear portions of the phase traces.

nonlinear phase behavior would be influencing the calculation of τ . The τ values range from
0.8591ms (open-loop) to 6.0678ms (increase Kd ) for the undamped microphone and from
9.4347ms (increase Kp ) to 23.3166ms (increase Kd ) for the damped microphone. The range
of phase stability is extended by over 250%. The delay has a 13.9ms range for the damped
microphone, an improvement over the 5.2ms range for the undamped baseline microphone
design.

62

Chapter 4
Area-Averaging Optical Displacement
Sensor to Extend Operational Range
In addition to the damping back-plates described in Chapter 3, another method for improving the operational range of the microphone is the employment of an optical sensor that
averages the membrane displacement over an area larger than that of the ‘point’ sensor used
previously. This is expected to decrease the observability of higher-order membrane resonance modes relative to the fundamental resonance mode (as rationalized in section 4.1) and
thus increase the gain margin in closed-loop operation leading to enhanced stability.

4.1

Modeling of a Displacement-Averaging Optical Sensor

As described in Chapter 2, the membrane dynamics can be modeled with the linear wave
equation. Because membrane deflection is measured at the center of the membrane, only the
63

axi-symmetric modes of vibration are detected by the optical sensor. The non-axisymmetric
resonances have zero deflection at r = 0. Theoretical analysis was employed in [3] to assess
the expected attenuation of higher-order axisymmetric membrane resonances (which are
believed to lead to instability, and hence reduced operational range, of the microphone in
operation) through the measurement of an area-averaged displacement. The axisymmetric
modes for a circular membrane clamped at its edges take the shape of Bessel functions of
the zeroth order [9], as depicted in Figure 4.1 for the first four modes, where r is the radial
coordinate in the undeflected membrane plane, z is the radial mode shape of the membrane
deflection, zc is the deflection at the membrane’s center (i.e. at r = 0), and a is the radius
of the membrane.

1
Mode 0
Mode 1
Mode 2
Mode 3

z(r)/zc

0.5

0

−0.5
−1

−0.5

0

0.5

1

r/a
Figure 4.1: The first four axisymmetric deflection modes of a circular membrane (figure
taken from [3])

64

Referring to Figure 4.1, it is evident that the membrane deflection associated with resonances other than the fundamental mode have both positive and negative components which,
when averaged over an area, will at least partially cancel out without substantially affecting the sensed displacement of the first resonance (which is completely in phase over the
averaging area). This can be demonstrated quantitatively by calculating the displacement
obtained from a sensor that responds to the average displacement over an area of radius as
centered on r = 0. The plot in Figure 4.2 displays the magnitude of the resulting average
displacement (zavg ) normalized by the membrane’s center displacement, plotted against the
probe radius divided by the membrane’s radius. It is predicted that using an optical sensor
with a radius that is around 69% that of the microphone membrane would result in a significant increase in closed-loop gain margin. This is because for this ratio, the first-order mode
completely cancels and the second and third resonances have minimal effect. Meanwhile,
the mode of interest, the zeroth resonance, still has significant average displacement at this
sensor-to-membrane radius ratio.
For this research, the area-averaging sensor utilized is lever sensor Philtec model D169
with a fiberoptic diameter of 4.29mm. This is compared to utilizing a point sensor for
previous results, Philtec model D20, with a tip diameter of 0.51mm. The transfer function
for the area-averaged sensor with sensitivity 35500V/m and a 20,000Hz cut-off frequency is
the following:

Go,area (s) =

35500
s
125600 + 1

(4.1)

The ratio of the radius of the Philtec D169 area-averaging sensor to that of the membrane
is 0.34 and is indicated in Figure 4.2 with a bold black arrow along the horizontal axis.
65

Figure 4.2: Relative average displacement of the membrane as a function of the sensor probe
radius, with the lever sensor-to-membrane diameter of the prototype marked with an arrow
(figure taken from [3])

Looking at Figure 4.2, the expected effect from using the D169 sensor is a relative decrease
in the measured strengths of the higher-order frequencies; the design is expected to be most
effective in damping mode 3. It was not possible to build a prototype with the “optimal”
sensor to membrane radius ratio because on one hand, there was limited availability of
optical lever sensor sizes; and also, in trying to construct smaller microphone sensors, there
was difficulty in actuating the PVDF effectively.

66

4.2

Characterization Results of the Feedback-Controlled
Microphone with Area-Averaged Sensing

4.2.1

Open-Loop Characterization

Figure 4.3 compares the open-loop microphone calibration results using the point and the
area-averaged optical displacement sensors. The resonances are all sharper when measured
by the point optical sensor. The fundamental resonance frequency peak is shifted slightly,
perhaps due to changing membrane tension over time. The same microphone and membrane
was used for both experiments, but over a month’s time elapsed between recording each
data set. It is significant to note that the third axisymmetric mode is damped the most,
as expected based on the discussion of Figure 4.2. Comparison between the open-loop
characterization results for the microphone with the area-averaging sensor and the model
predictions are shown in Figure 4.4. The membrane parameters estimated from these results
are listed in Table 4.1.
Table 4.1: Membrane parameters estimated from open-loop calibration of the microphone
with the area-averaged displacement sensor
Parameter
Symbol
Value
Unit
Mass
M
0.0493
kg/m2
Damping
Stiffness

C0
K0

91.195
24.24 · 106

P a · s/m
P a/m

Damping Ratio
Natural Frequency

ζ0
f0

0.0417
3.53

kHz

67

Figure 4.3: Comparison of the magnitude frequency response for the microphone with point
and area-averaging displacement sensors. The first three axisymmetric modes are identified
based on circular membrane vibration theory: 2.3fr (9150 Hz), 3.6fr (14300 Hz), and 4.9fr
(19500 Hz) for modes 1 to 3 respectively, based on fr = 3980 Hz for the microphone with
the point sensor

68

Figure 4.4: Open-loop characterization for the undamped microphone with area-averaging
displacement sensor; magnitude (top); phase (bottom)

69

4.2.2

Feedback-Controlled Microphone Calibration Results

Figures 4.5–4.7 show the closed-loop responses for the microphone with the area-averaging
sensor. Similar to closed-loop testing in previous chapters, three different methods of increasing the controller gains are implemented. In Figure 4.5, initially, the proportional gain
Kp is increased until the microphone becomes unstable. This is followed by reducing Kp
slightly to bring the microphone within the stability limits then increasing the derivative
gain Kd to the highest possible value for stable operation. Figure 4.6 follows the same steps
but in opposite order, so Kd is increased first. Finally, Figure 4.7 shows the closed-loop
response when the gains Kp and Kd are increased together incrementally until just before
instability is reached.

4.3

Evaluation of the Extension of the Operational Range
of the Feedback-Controlled Microphone with AreaAveraging Sensing

As with the damped microphone feedback-controlled results, several methods of comparison
of operational range are utilized. The first method is to measure the increase in natural
frequency of the membrane. The method described in section 2.3.2 is used to determine
the natural frequency and damping ratio from the data plotted in Figures 4.4–4.7 because
they all have a clear fundamental resonance peak. The damping ratio and natural frequency
values are summarized in Table 4.2.

70

Table 4.2: Natural frequencies and damping ratios for the undamped microphone with the
area-averaging sensor in open- and closed-loop operation
Closed Loop
Open Loop
Increase
Increase
Increase
Kp first
Kd first
together
3530
3773
4936
4603
Natural frequency fn (Hz)
0.0410
0.0122
0.0691
0.0108
Damping ratio ζ

Figure 4.5: Closed-loop characterization for the undamped microphone with area-averaging
displacement sensor increasing Kp first; magnitude (top); phase (bottom)

71

Figure 4.6: Closed-loop characterization for the undamped microphone with area-averaging
displacement sensor increasing Kd first; magnitude (top); phase (bottom)

72

Figure 4.7: Closed-loop characterization for the undamped microphone with area-averaging
displacement sensor increasing Kp and Kd simultaneously; magnitude (top); phase (bottom)

73

As with the point optical sensor, the closed-loop method of increasing Kd first has the
highest natural frequency increase, an increase of 39.8%. This is not as high of an increase as
for the damped microphone in chapter 3 (51.0%), but an improvement over the undamped
microphone with the point optical sensor (16.9%). The feedback gains reached for the
microphone with the area-averaging optical sensor are summarized in Table 4.3. The gains
listed in Table 4.3 are higher than those listed in Table 3.6 for the undamped microphone
with the point sensor. All together, the results show that in fact the use of an area-averaging
sensor does increase the gain margin relative to the employment of the single-point sensor.
However, for the specific configuration of damping and area-averaged sensing used here, the
former is more effective in enhancing the operational range relative to the latter.
Table 4.3: Feedback gains reached for the microphone with the area-averaging optical sensor
Kp Kd (s)
Increase Kp first
16 0.0001
Increase Kd first
1
0.003
IncreaseKp and Kd together 23 0.0022

The amount of phase control was also explored for the microphone with the area-averaging
sensor. Figure 4.8 shows the open-loop and the closed-loop phase response for cases where
Kp or Kd are taken to their extreme values just before instability is reached.
Following a similar procedure for demonstrating phase control as used in section 3.4, the
τ values range from 5.2252 ms (increase Kp ) to 15.0285 ms (increase Kd ) for the undamped
microphone with the area-averaging sensor. This is a 9.8 ms range, an improvement of 188%
over the undamped microphone with the point optical sensor. Moreover, the 9.8 ms range
is 140% of the open-loop microphone time delay (7.0253 ms), demonstrating the ability to
tune the phase response of different microphones in an array over a fairly substantial range.
74

Figure 4.8: Phase delay comparison for the undamped microphone with the area-averaging
optical sensor. Broken lines represent the models for the linear portions of the phase traces.

4.4

Characterization Results of the Feedback-Controlled
Microphone with Area-Averaged Sensing and Damped
Membrane

After implementing the damping back-plates and the area-averaging sensor individually,
the next natural step is to combine the two provisions. Figure 4.9 shows the open-loop
microphone calibration for this combination. The closed-loop results are given in Figures
4.10–4.12.

75

Figure 4.9: Open-loop characterization for the damped microphone with the area-averaging
displacement sensor; magnitude (top); phase (bottom)

76

Figure 4.10: Closed-loop characterization for the damped microphone with area-averaging
displacement sensor, increasing Kp first; magnitude (top); phase (bottom)

77

Figure 4.11: Closed-loop characterization for the damped microphone with area-averaging
displacement sensor, increasing Kd first; magnitude (top); phase (bottom)

78

Figure 4.12: Closed-loop characterization for the damped microphone with area-averaging
displacement sensor, increasing Kp and Kd simultaneously; magnitude (top); phase (bottom)

79

Using the method described in section 3.3.1, a best fit model was used to determine
the natural frequency and damping ratio for the responses in Figures 4.9–4.12. This is
summarized in Table 4.4. The closed-loop calibrations for the cases when Kd was increased
first and when Kp and Kd were increased together have no values listed because the best
fit method yielded imaginary numbers. This is believed to be caused by the membrane’s
response becoming over-damped at the first fundamental resonance. One can observe though,
from the plots, that a relatively flat response is obtained for a wide range of frequencies,
especially for the case when Kp and Kd are increased together.
Table 4.4: Natural frequencies and damping ratios for the damped microphone with the
area-averaging sensor in open- and closed-loop operation
Closed Loop
Open Loop
Increase
Increase
Increase
Kp first
Kd first
together
3315
4051
?
?
Natural frequency fn (Hz)
0.4837
0.2606
?
?
Damping ratio ζ

Table 4.5 lists the gains reached before operation became unstable for each situation.
These are the highest gains reached for any of the microphones, an expected result when
combining both methods of improving the operational range.
Table 4.5: Feedback gains reached for the damped microphone with the area-averaging
optical sensor
Kp Kd (s)
Increase Kp first
93 0.0001
Increase Kd first
0.5 0.0042
IncreaseKp and Kd together 30 0.003

As with the damped-membrane microphone and the microphone utilizing the area-averaging
sensor, the phase control was examined for the microphone with both the damped membrane
80

and the area-averaging sensor. Figure 4.13 shows the open-loop and closed-loop phase response when either Kp or Kd are taken to their extreme values just before instability is
reached. Here, the τ values have their widest range, from 10.1107ms (increase Kp ) to
30.2893ms (increase Kd ). This 20.2ms range is a 388% improvement over the baseline design (i.e. the open-loop response of a microphone with undamped membrane and with point
displacement sensor) and is the widest range of phase control for any of the microphone
designs tested.

Figure 4.13: Phase delay comparison for the damped microphone with the area-averaging
optical sensor. Broken lines represent the models for the linear portions of the phase traces.

81

Chapter 5
Summary and Recommendations
This study was undertaken with the intent to extend the operational range of a feedbackcontrolled microphone developed recently at MSU in [2] and [3].
Mechanical damping provisions were introduced to decrease the strength of the microphone membrane’s high-order resonance peaks which limit the stability. Chapter 3 provides
a description of implementing said damping provisions through the use of back-plates with
small holes that were designed and fabricated in the present work using laser ablation at
NASA Langley Research Center. The results demonstrated that, through several methods of quantifying the microphone’s operational range, the damped response was always
an improvement over the undamped response. Natural frequencies reached higher percent
increases; controller gains were pushed to greater limits before instability was reached; and
phase delays were capable of spanning larger ranges for the damped-membrane microphone.
Chapter 4 gives a description of the improvements in microphone operational range expected from using an area-averaging fiber-optic sensor as well as including results demonstrating those improvements. The increases in stability over the baseline design were not
82

as dramatic as those for the damped-membrane microphone as described in Chapter 3, but
there was still a measurable improvement. Also explored in Chapter 4 was the extension of
operational range for the microphone employing both methods of decreasing the strength of
membrane resonances (damping provisions and area-averaging optical sensor). This combination yielded the greatest improvement of all.
Figure 5.1 shows a summary plot demonstrating the relative improvement for the different
microphones tested. It compares the baseline open-loop response with closed-loop responses
for both damped and undamped membrane, and for both point and area-averaging optical
sensor. Using the standard method of identifying bandwidth as the frequency band over
which the magnitude response remains within ± 3dB (broken thick black lines in Figure
5.1), it is seen that the provisions implemented here are capable of approximately tripling the
microphone’s bandwidth. In comparison, in previous studies at Michigan State University,
the best improvement in bandwidth was only an approximate doubling of the bandwidth.
A recommendation to further validate the research pertaining to this feedback-controlled
microphone is to build an array of two or more sensors that is programmed to demonstrate
the ‘self-calibrating’ capability. A frequency sweep can be input to each sensor and the two
microphones can be electrically calibrated. Then, based on the parameters calculated from
this, feedback can be used to tune the response of one sensor to match the other; again, this
ability to use feedback to obtain a predicted response was demonstrated in [2] and [3].

83

Figure 5.1: A comparison of frequency and phase response for several of the tested microphone designs; magnitude (top); phase (bottom). Broken lines on top plot represent the
± 3dB boundaries for identifying the bandwidth
84

Another proposal for improvement is to work with a smaller membrane diameter. This
would give a larger bandwidth. Microphones with membrane diameters of three eighths inch
and one quarter inch were constructed; but they were not tested because sufficient actuation
of the PVDF membrane could not be achieved. The signal-to-noise ratio was too great. A
recommendation is to explore the use of a higher-voltage piezo-driver.
Damping back-plates were also constructed for the smaller microphone sensor sizes, so
once the PVDF actuation issue for these is resolved, the fabricated back-plates can be implemented. Relating to the damping back-plates, it was observed in Chapter 3 that the model
used to determine the amount of damping achieved by using the discs was less accurate than
expected. Modification to the model is necessary. A possible improvement is to include
the air gap between the membrane and the back-plate, previously excluded from the model.
This gap was smaller in the design employed in [2] and [3] but when the microphone capsule
was redesigned to accommodate the damping back-plates, this gap became bigger.
A final recommendation is to construct a microphone with the hypothesized ‘ideal’ optical
sensor-to-membrane diameter ratio. As discussed in Chapter 4, the suggested optimal ratio
is 0.69, thought to be the ratio where the first three high-order resonance modes’ magnitudes
are decreased the most relative to the fundamental resonance. Further experimentation with
other ratios is also recommended.

85

Appendix A
Hole Coordinates of Fabricated
Damping Back-Plates

86

Table A.1: Polar coordinates of the holes in the damping back-plate with the 91-hole pattern
and circular arrangement
Hole # r (µm) θ (deg)
Hole # r (µm) θ (deg)
Hole # r (µm) θ (deg)
1
0
0
42
4000
60
83
5000
252
2
1000
0
43
4000
75
84
5000
264
3
1000
60
44
4000
90
85
5000
276
4
1000
120
45
4000
105
86
5000
288
5
1000
180
46
4000
120
87
5000
300
6
1000
240
47
4000
135
88
5000
312
7
1000
300
48
4000
150
89
5000
324
8
2000
0
49
4000
165
90
5000
336
9
2000
30
50
4000
180
91
5000
348
10
2000
60
51
4000
195
11
2000
90
52
4000
210
12
2000
120
53
4000
225
13
2000
150
54
4000
240
14
2000
180
55
4000
255
15
2000
210
56
4000
270
16
2000
240
57
4000
285
17
2000
270
58
4000
300
18
2000
300
59
4000
315
19
2000
330
60
4000
330
20
3000
0
61
4000
345
21
3000
20
62
5000
0
22
3000
40
63
5000
12
23
3000
60
64
5000
24
24
3000
80
65
5000
36
25
3000
100
66
5000
48
26
3000
120
67
5000
60
27
3000
140
68
5000
72
28
3000
160
69
5000
84
29
3000
180
70
5000
96
30
3000
200
71
5000
108
31
3000
220
72
5000
120
32
3000
240
73
5000
132
33
3000
260
74
5000
144
34
3000
280
75
5000
156
35
3000
300
76
5000
168
36
3000
320
77
5000
180
37
3000
340
78
5000
192
38
4000
0
79
5000
204
39
4000
15
80
5000
216
40
4000
30
81
5000
228
41
4000
45
82
5000
240
87

Table A.2: Polar coordinates of the holes in the damping back-plate with the 121-hole pattern
and circular arrangement
Hole # r (µm) θ (deg)
Hole # r (µm) θ (deg)
Hole # r (µm) θ (deg)
1
0
0
42
3800
60
83
4750
252
2
950
0
43
3800
75
84
4750
264
3
950
60
44
3800
90
85
4750
276
4
950
120
45
3800
105
86
4750
288
5
950
180
46
3800
120
87
4750
300
6
950
240
47
3800
135
88
4750
312
7
950
300
48
3800
150
89
4750
324
8
1900
0
49
3800
165
90
4750
336
9
1900
30
50
3800
180
91
4750
348
10
1900
60
51
3800
195
92
5700
0
11
1900
90
52
3800
210
93
5700
12
12
1900
120
53
3800
225
94
5700
24
13
1900
150
54
3800
240
95
5700
36
14
1900
180
55
3800
255
96
5700
48
15
1900
210
56
3800
270
97
5700
60
16
1900
240
57
3800
285
98
5700
72
17
1900
270
58
3800
300
99
5700
84
18
1900
300
59
3800
315
100
5700
96
19
1900
330
60
3800
330
101
5700
108
20
2850
0
61
3800
345
102
5700
120
21
2850
20
62
4750
0
103
5700
132
22
2850
40
63
4750
12
104
5700
144
23
2850
60
64
4750
24
105
5700
156
24
2850
80
65
4750
36
106
5700
168
25
2850
100
66
4750
48
107
5700
180
26
2850
120
67
4750
60
108
5700
192
27
2850
140
68
4750
72
109
5700
204
28
2850
160
69
4750
84
110
5700
216
29
2850
180
70
4750
96
111
5700
228
30
2850
200
71
4750
108
112
5700
240
31
2850
220
72
4750
120
113
5700
252
32
2850
240
73
4750
132
114
5700
264
33
2850
260
74
4750
144
115
5700
276
34
2850
280
75
4750
156
116
5700
288
35
2850
300
76
4750
168
117
5700
300
36
2850
320
77
4750
180
118
5700
312
37
2850
340
78
4750
192
119
5700
324
38
3800
0
79
4750
204
120
5700
336
39
3800
15
80
4750
216
121
5700
348
40
3800
30
81
4750
228
41
3800
45
82
4750
240
88

Table A.3: Cartesian coordinates of the
pattern and hexagonal arrangement
Hole # x (µm) y (µm)
Hole #
1
0
0
42
2
1000
0
43
3
2000
0
44
4
3000
0
45
5
4000
0
46
6
5000
0
47
7
-1000
0
48
8
-2000
0
49
9
-3000
0
50
10
-4000
0
51
11
-5000
0
52
12
500
866
53
13
1500
866
54
14
2500
866
55
15
3500
866
56
16
4500
866
57
17
-500
866
58
18
-1500
866
59
19
-2500
866
60
20
-3500
866
61
21
-4500
866
62
22
500
-866
63
23
1500
-866
64
24
2500
-866
65
25
3500
-866
66
26
4500
-866
67
27
-500
-866
68
28
-1500
-866
69
29
-2500
-866
70
30
-3500
-866
71
31
-4500
-866
72
32
0
1732
73
33
1000
1732
74
34
2000
1732
75
35
3000
1732
76
36
4000
1732
77
37
-1000
1732
78
38
-2000
1732
79
39
-3000
1732
80
40
-4000
1732
81
41
0
-1732
82

holes in the damping back-plate with the 91-hole
x (µm)
1000
2000
3000
4000
-1000
-2000
-3000
-4000
500
1500
2500
3500
-500
-1500
-2500
-3500
500
1500
2500
3500
-500
-1500
-2500
-3500
0
1000
2000
3000
-1000
-2000
-3000
0
1000
2000
3000
-1000
-2000
-3000
500
1500
2500
89

y (µm)
-1732
-1732
-1732
-1732
-1732
-1732
-1732
-1732
2598
2598
2598
2598
2598
2598
2598
2598
-2598
-2598
-2598
-2598
-2598
-2598
-2598
-2598
3464
3464
3464
3464
3464
3464
3464
-3464
-3464
-3464
-3464
-3464
-3464
-3464
4330
4330
4330

Hole #
83
84
85
86
87
88
89
90
91

x (µm)
-500
-1500
-2500
500
1500
2500
-500
-1500
-2400

y (µm)
4330
4330
4330
-4330
-4330
-4330
-4330
-4330
-4330

BIBLIOGRAPHY

90

BIBLIOGRAPHY

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92