Ultra Low Frequency Infrasonic Measurement System · An infrasonic measurement system is built capable of sensing acoustic signals down to 10 mHz which is advantageous for measurements

Rasmus Trock Kinnerup, s052256

Ultra Low Frequency InfrasonicMeasurement System

Master’s Thesis, July 2011

Abstract

An infrasonic measurement system is built capable of sensing acoustic signals down to 10mHz which is advantageous for measurements of wind farm noise or sonic boom shapers.The system consists of an electric preamplifier built into a housing and a G.R.A.S. 40AZ12 -inch prepolarized condenser microphone with a closed vent configuration. The totalsystem has a dynamic range of 94 dB and a lower limiting -3 dB cutoff frequency of 8mHz. The preamplifier connects the microphone signal directly to the input of an op-ampwith an input resistance of 10 TΩ, one of the industry’s highest, which forms a high passfilter with the microphone capacitance of 20 pF. The bias current is supplied to the inputnode by two diode-connected FETs. The big challenge has been to sense the sound signalfrom the capacitive microphone with a high enough input impedance of the preamplifierto avoid an inherent cutoff of frequencies of interest. Being able to measure down to ultralow frequencies in the infrasonic frequency range will aid actors in the debate on windturbine noise. Sonic booms from supersonic flights include frequencies down to 10 mHzand this measurement system will aid scientists trying to modify the N-shaped shock waveat high level which prohibits flights in land zones.

To my wife Cathrine and our children Alfred and Carla

Preface

This report is a Master’s Thesis in Electrical Engineering at the Department of ElectricalEngineering at the Technical University of Denmark.

I have chosen the subject of this project because it deals with an electro acoustic problem.During my previous studies I have had exciting courses in both the acoustic and the elec-tric domain. I found it natural to utilize my knowledge from both domains and to workwith something that was of interest.

The project has been carried out in cooperation with G.R.A.S. Sound & Vibration A/Ssituated in Holte, Denmark. Working with a company have been very valuable for theproject process and they have shown great interest to my project which have been a largemotivating factor.

I would like to acknowledge the many individuals who have supported me during my stud-ies. The employees in the development department of G.R.A.S. Kresten Marbjerg and PerRasmussen have been very supportive. Also my supervisors Arnold Knott from Electron-ics Group have been encouraging and of great help.

Last but not least, appreciation goes to my family. Without their understanding, supportand motivation the project would not have been the same.

Contents

1 Introduction 1

1.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 3

2.1 Infrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Occurrences of Infrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Condenser Microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Infrasonic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 Measurements with Capacitive Sensors . . . . . . . . . . . . . . . . . . . . . 10

2.7 Preamplifiers for Condenser Microphones . . . . . . . . . . . . . . . . . . . 11

3 Electronic Design 13

3.1 Design Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Low Leakage Op-amp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Bias Current Circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 First Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Guarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.6 Analyzing Peaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.7 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.7.1 Circuit b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7.2 Circuit c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.7.3 Circuit d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.7.4 Choosing a Feedback Circuit . . . . . . . . . . . . . . . . . . . . . . 28

3.8 Capacitive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.9 Start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.10 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.11 Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.12 Production Component Variations . . . . . . . . . . . . . . . . . . . . . . . 34

3.13 PCB Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.13.1 Improvements for Next Version . . . . . . . . . . . . . . . . . . . . . 37

3.14 Final Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Acoustic Design 41

4.1 Choice of Microphone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Capacitance and Voltage Variation . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Infrasound Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Leakage and Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5 Modeling of Vent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.6 New Vent Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.7 Consequences of an Airtight Microphone . . . . . . . . . . . . . . . . . . . . 49

5 Measurements 515.1 Electric System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.2 Start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.3 THD and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Frequency Response of Entire System . . . . . . . . . . . . . . . . . . . . . 585.2.1 Microphone Mounting . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Conclusion 636.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

References 65

Appendix 69A Various Matlab scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B Microphone Calibration Chart . . . . . . . . . . . . . . . . . . . . . . . . . 72C Circuit Analysis with feedback b . . . . . . . . . . . . . . . . . . . . . . . . 73D Circuit Analysis with feedback d . . . . . . . . . . . . . . . . . . . . . . . . 74

List of Figures

2.1 Normal equal-loudness-level contours (ISO 226:2003) showing the threshold curve of

human hearing in the lowest frequencies along with measurements from studies by

Watanabe and Møller [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Infrasound is produced by a variety of natural and man-made sources: exploding

volcanoes, earthquakes, meteors, storms and auroras in the natural world; nuclear,

mining and large chemical explosions, as well as aircraft and rocket launches in the

man-made arena [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 The noise shape generated at the aircraft is like the shape of the aircraft but nonlinear

propagation makes the sound wave at ground look like an N-shape. . . . . . . . . . . 6

2.4 F-5E modified Shaped Sonic Boom Demonstration aircraft used to explore supersonic

booms from aircrafts. The N-wave mentioned is painted on the side of the aircraft

with a red line. The blue line painted on top is the shape of the wave from this aircraft. 6

2.5 Generic 3D model of a condenser microphone [3] . . . . . . . . . . . . . . . . . . . . 8

2.6 Generic drawing of cross section of a condenser microphone [3] . . . . . . . . . . . . 8

2.7 Equivalent electric circuit of a condenser microphone. . . . . . . . . . . . . . . 9

2.8 Voltage sensing method measuring direct dc. . . . . . . . . . . . . . . . . . . . 10

2.9 Simple Capacitive Bridge circuit where the capacitance, Cx, is measured incomparison to a known capacitor C1 and with precision adjustable resistors R3

and R4. The AC null detector, D, reads 0 V when the the bridge is in balance. 11

3.1 Simplified schematic of circuit. V represents the acoustic sound pressure, Cm

the variating microphone capacitance, Zb the bias circuitry supplying the biascurrent to the amplifier, Zf1 and Zf2 are the feedback circuitry and A theoperational amplifier coupled as an impedance buffer. . . . . . . . . . . . . . . 13

3.2 I-V characteristics of a P-N junction diode (not to scale) . . . . . . . . . . . . . . . 15

3.3 First prototype built into a homemade Faraday cage to protect from outside noise . . 16

3.4 The mock up of the circuit of the first prototype . . . . . . . . . . . . . . . . . . . . 16

3.5 Full schematic of the first prototype. The resistor Zb shown is replaced withalternative bias circuits as shown in Figure 3.6. . . . . . . . . . . . . . . . . . . 17

3.6 The different bias circuits replacing the impedance Zb from Figure 3.5. Imple-mentation a realize the high resistance with a resistor, b with two diodes inopposite direction and c with two FETs in opposite direction using the gateleakage current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.7 Schematic of the simulated circuit in PSpice. Zin,cmm (1013 Ω || 1 pF) is thecommon mode input impedance of the op-amp which is modeled along with thedifferential input impedance, Zin,dif (1015 Ω || 2 pF) to get a correct simulationin the low frequency range. Both impedances are listed in the data sheet ofOPA129 as a resistance and a capacitance in parallel. . . . . . . . . . . . . . . 18

3.8 Simulated and measured frequency response of first prototype (see Figure 3.7) with

the different bias circuits from Figure 3.6. . . . . . . . . . . . . . . . . . . . . . . . 19

3.9 Simulation of the frequency response with a simple RC filter in the feedback. Peaking

is inevitable. Sweeping the value of Cf shows larger amplitude peaks for lower cut-off

frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.10 Poles have very little imaginary and real values. Furthermore the stability is clearly

achieved since the curve does not go beyond the point (-1,0) marked red. Component

values are Cm = 20 pF, Rb = 1000 GΩ, Cf = 16 µF, Rf = 10 MΩ. . . . . . . . . . . 23

3.11 Schematic of the alternatives to the feedback circuit. . . . . . . . . . . . . . . 23

3.12 Simulation showing the frequency response of the preamplifier with feedback circuit b

and varying Cf2. The variation shows significant damping of the amplitude peak. . . . 24

3.13 Simulation showing the frequency response of the preamplifier with feedback circuit c

and varying Rf1. The variation shows an extra cutoff frequency introducing a minor

attenuation before completely dropping down. . . . . . . . . . . . . . . . . . . . . . 26

3.14 Comparing simulation with circuit b with the combination of circuit a and c. Both

feedback circuits in total provide a high pass filter and a voltage divider. In circuit b

it’s a capacitive and in the combination of a and c it’s resistive. The simulations show

comparable results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.15 Simulation showing the frequency response of the preamplifier with feedback circuit

d, Cf2 = 0.8 µF and varying Rf1. The variation shows significant damping of the

amplitude peak and no attenuation for all especially higher frequencies. . . . . . . . . 27

3.16 Comparison of the frequency responses from the discussed 4 alternatives to a feedback

circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.17 Output signal of op-amp showing that the too large capacitive load in the circuit makes

the op-amp reach its limitations at frequencies above 5.8 kHz. The capacitive load is

16 µF and the rated load capacitance stability of OPA129 is 1 nF up to a bandwidth

of 1 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.18 The implemented circuit minimizing the start-up period in which the ampli-fier seeks its equilibrium potential. The switch is mechanically activated andconnects a small resistance which lowers the time constant of the system. . . . 31

3.19 The time at which the input finds its DC level is decreased when the value of resistor

Rf is decreased. It also affects the frequency response, so it’s only for start-up purposes

before real measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.20 The noise of the transistor of the preamplifier in this case an op-amp OPA129. The

noise is dominated at low frequencies by pink noise (flicker noise or 1/f noise) and

above the noise corner frequency by white noise. . . . . . . . . . . . . . . . . . . . . 33

3.21 Simulation of the frequency response of the final prototype version 1. Increased values

of Rf2 results in less peaking. A component tolerance of ±30 % results in ±0.2 dB . . 34

3.22 Simulation of the frequency response of the final prototype version 1. Increased values

of Cf1 results in less attenuation for all frequencies. A variation of +5 % equals +19

mdB for frequencies above 100 mHz. Not shown is the variation of Cf2 which behaves

opposite. This is due to the voltage division introduced by the two capacitances. . . . 35

3.23 The non-standard pinout of the chosen op-amp OPA129 makes it easy to im-plement a guard trace and separation of inputs and supplies minimizes leakage. 35

3.24 PCB Top layer. The board measures 44.6 mm times 10.3 mm and is 1.5 mm thick. . . 36

3.25 PCB Bottom layer. It is mirrored for easy comparison to the top layer in Figure 3.24 . 37

3.26 PCB and components in a 3D visualization. This was used to see whether the compo-

nents would fit into the microphone housing. . . . . . . . . . . . . . . . . . . . . . . 37

3.27 Picture of un-soldered PCB with rounded inner corners due to the process by which

the board is cut using a 2 mm drill. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.28 Picture of the soldered PCB with copper guard ring and microphone housing. . . . . . 38

3.29 Picture of the entire preamplifier including LEMA plug. . . . . . . . . . . . . . . . . 38

3.30 Full schematic of the final prototype version 1 implementing a feedback circuitwith a high pass filter and a capacitive voltage divider. . . . . . . . . . . . . . . 39

3.31 Full schematic of the final prototype version 2 implementing a resistive voltagedivider. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 G.R.A.S. Type 40AZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 The vent of 40AZ is made by putting in a spacer (green) on top of the insulator and

cutting a slit in the spacer (see Figure 4.3). Equalization occurs through the slit (blue). 44

4.3 A close look of the spacer with a slit which makes equalization occur from the inner

diameter to the outer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Model of the acoustic system (a) simplified to an acoustic volume (CA) and avent for equalization of low frequencies (RA). The model is converted to theanalogous electric circuit (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 -3 dB cutoff frequency of vent as function of width and length of the slit in the spacer.

The current vent/slit dimensions represent the data point in the very top. . . . . . . 47

4.6 A modified vent proposal where the vent is cut skew close to a tangent of the inner

radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.7 A new vent proposal where the vent length is increased by letting the slit run along

the circumference of the spacer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Frequency response of preamplifier with 20 pF input adapter. The simulation is in-

cluded for comparison. The electric lower limiting -3 dB corner frequency is clearly

around 10 mHz for both versions of the preamplifier. . . . . . . . . . . . . . . . . . 52

5.2 Input adapter for supplying electric input to the preamplifier. A 15 pF version is

depicted but a 20 pF also exist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Vent adapter used to seal or equalize the microphone. Both constructions are airtight

from front to back. The adapter on the left has a hole which equalizes the microphone

whereas the adapter to the right seals the microphone. . . . . . . . . . . . . . . . . 52

5.4 DC offset of preamplifier version 1 with step on input through 20 pF input adapter

shows the system’s settle time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.5 Frequency response measurement of first prototype showing 2 sweeps, one with the

switch off (normal operation) and one with the switch on, meaning a lower value

resistor is connected in the feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.6 FFT of output signal with 1 V sine at 1 kHz as input. . . . . . . . . . . . . . . . . . 55

5.7 FFT of output signal with 1 V sine at 100 Hz as input. The signal is marked red to

aid the calculation of the dynamic range. . . . . . . . . . . . . . . . . . . . . . . . 57

5.8 FFT of output signal of version 1 with 1 kHz sine as input. 3 % distortion on the

output is reached at output voltage of 7.6 Vpp . . . . . . . . . . . . . . . . . . . . . 57

5.9 FFT of output signal of version 2 with 1 kHz sine as input. 3 % distortion on the

output is reached at output voltage of 28 Vpp . . . . . . . . . . . . . . . . . . . . . 57

5.10 G.R.A.S. 42AE low frequency calibrator with the microphone including vent inside the

coupler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.11 Measurements with version 1 showing consistency across several microphone cartridges

and no or little difference whether the microphone is loosely mounted on the closed

vent adapter or mounted with the open vent adapter. Lastly the measurements verify

the entire system’s lower limiting frequency of around 190 mHz with a standard off-

the-shelf 40AZ microphone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.12 Measurements with version 2 and 40AZ microphone. With open vent adapter the -3

dB cutoff frequency is 190 mHz and with closed vent adapter and oil it is 8 mHz. . . . 605.13 Frequency response of entire system consisting of preamp v1 and 40 AZ showing the

importance of ventilation. The measurements are conducted with the closed vent adapter. 615.14 Overpressure inside the microphone cavity results in 4 dB attenuation. The over-

pressure occurred when oil was applied along the thread and not only on the contact

surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

List of Tables

3.1 Specifications of chosen amplifier OPA129 along with its rivals. . . . . . . . . . 153.2 New component values in feedback circuit b which meet the requirements of

the maximum load capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.1 Target specifications and obtained specifications of the measurement system . . 64

1

Introduction

A popular interpretation of infrasound is that it is sound at frequencies below the lowerfrequency limit of hearing [1]. But this implies that our hearing has a limit, which is notthe case. If the sound level is high enough the sound even at frequencies below 5 Hz isaudible. The lower the frequency the higher level is required for audibility.

The ability to measure acoustic noise is an important part of engineering. The noise levelgenerated by a product has direct impact on the perception and user experience and insome cases regulations on acceptable noise levels make noise measurements inevitable.Most quality noise measurements are made with condenser microphones due to their ex-cellent all-round performance. They excel in dynamic range, frequency response, linearity,long-term stability among others [4].

Infrasonic noise has received more attention over the last few decades, especially in con-text of wind farms which produce infrasound. To measure the level of infrasound noisemeasurement systems must be able to cover the frequencies of interest ranging below 1Hz. Measurement systems exist capable of measuring down into the ultra low frequencyinfrasonic frequency range e.g. [5], but the dynamic range is typically much less thansystems operating in the traditional audio frequency range from 20 Hz to 20 kHz. Experi-ence with the B&K 2631 Microphone Carrier System capable of measuring to DC shows adynamic range of 40-60 dB. And other similar approaches with demodulation of a carrierfrom Norsonic in Norway shows similar degraded results. In comparison ordinary audiomeasurement systems have a dynamic range of 140-150 dB.

1.1 Problem Definition

This project will deal with the design and implementation of a system for measuringinfrasonic noise going down to very low frequencies. Traditional infrasonic microphonesystems typically cover a frequency range from 1 Hz to 20 kHz, but this system shouldextend down to 0.01 Hz (10 mHz). Existing systems typically have a dynamic range of 40dB but this should be increased to 80 dB.

• Sensor: Condenser microphone

• Frequency range: 10 mHz - 20 kHz

• Dynamic range: 80 dB

• Output: analog

1. Introduction 2

1.2 Thesis Structure

After this introduction relevant background information will be presented. After readingthat, one should be better prepared for reading the following two design chapters. Thefirst design chapter describes the design from an electrical point of view, whereas the sec-ond design chapter describes the system and the design process with an acoustical view.Following the design chapters is a chapter verifying the final prototype of the system in-cluding measurement results and comparison to simulation. Last chapter of the thesis isthe conclusion including discussion of future work.

2

Background

This chapter will present background information which will put the project and its chal-lenges in perspective.

2.1 Infrasound

Infrasound is in popular terms defined as sound at frequencies below human hearing thresh-old. The audible frequency range is usually defined as the range from 20 Hz to 20 kHz.In other words infrasound is sound at frequencies below 20 Hz. The IEC standard definesinfrasound as

acoustic oscillation whose frequency is below the low-frequency limit of audiblesound (about 16 Hz), IEC 1994

But the problem with this way of defining infrasound as sounds below such frequency isthat sound below 20 Hz and even 16 Hz is audible. In Figure 2.1 part of a standard humanthreshold curve is depicted along with measurements by Watanabe and Møller indicatingthe extension of the curve below 20 Hz which is the lower limit of the ISO standard. Thismeans that lower frequencies require higher level to be perceived. Frequencies down to afew hertz are proven audible under certain conditions [1]. Frequencies below 20 Hz andeven 16 Hz are audible and therefore defining infrasound as what humans can not hear issomewhat wrong. Even though no fixed frequency exists where audibility suddenly stopsor begins for that matter, infrasound is in the following discretely defined as sound atfrequencies below 20 Hz.

interpretation is that it is sound of such low frequency that it is below the lower frequency limit of hearing, generally taken to be around 20Hz. A definition of infrasound, found in Standards, is:

Acoustic oscillations whose frequency is below the low frequency limit of audible sound (about 16Hz). (IEC, 1994)

However, sound at frequencies below 16Hz is clearly audible if the level is high. The hearing threshold has been measured reliably down to 4Hz for listening in an acoustic chamber (Watanabe and Møller, 1990) and down to 1.5 Hz for earphone listening (Yeowart et al., 1967). Fig 1 shows the hearing threshold measurement from Watanabe and Møller between 4Hz and 125Hz together with the low frequency end (20Hz to 200Hz) of the standardized hearing threshold (ISO:226, 2003). (The full range of measurements in ISO 226 is from 20Hz to 12.5kHz) There is good correspondence between the two sets of measurements of hearing threshold in the overlap region in Fig 1. Rounded values are in Table 1:

Freq Hz

4

8

10

12.5

16

20

25

31.5

40

50

63

80

100

125

160

200

Level dB

107

100

97

92

88

79

69

60

51

44

38

32

27

22

18

14

Table 1 Hearing threshold levels

There is continuity of perception throughout the frequency range and no evidence for splitting into “infrasound” and” not infrasound” at around 16Hz to 20Hz. However, there is a reduction in slope of the hearing threshold below about 15Hz from approximately 20dB/octave above 15 Hz to 12dB/octave below 15Hz. (Yeowart et al., 1967). There is also a change in perception of tonality, occurring around 18Hz. The common assumption that “infrasound” is inaudible is incorrect.

0

20

40

60

80

100

120

0 20 40 60 80 100 120 140 160 180 200

Frequency Hz

So

un

d p

ress

ure

leve

l dB

ISO226:2003

Watanabe and Moller 1990

Fig 1. Low frequency hearing threshold

2

Figure 2.1: Normal equal-loudness-level contours (ISO 226:2003) showing the threshold curve ofhuman hearing in the lowest frequencies along with measurements from studies by Watanabe andMøller [1]

2. Background 4

Looking at the wavelength of infrasound elaborates one of the most obvious character-istics. Under normal conditions a sound at 20 Hz has a wavelength of 17 m and at 1Hz it is 343 m. Going all the way down to 10 mHz the wavelength is impressive 34 km.The attenuation of infrasounds differ from their higher frequency neighbors by not beingaffected by viscous dissipation. This means that infrasonic waves can travel for very longdistances (> 100 km) and still be very measurable [6].

Now that infrasound by some has been defined as something humans can not hear one couldask why to bother measuring it. Along with the auditory perception through the ear, soundcan to some degree be sensed by the vestibular balance system and the resonant excitationof body cavities [7]. Altogether infrasound can be perceived. Many misunderstandingsabout infrasound have been developed over time [1] e.g. that infrasound should be a causeof death and possessing the ability to knock down buildings. Even though these beliefsare extreme the impact of infrasound on humans is somewhat unclear. Complaints oninfrasound and low frequency noise (LFN) have been made with an increasing rate andinvestigated in numerous articles [8, 9]. The victims search for answers of their symptomsand scientists have no straightforward answers. In some cases victims do hear soundsin the infrasonic range, but other times it seems that the sounds origin from the victimhimself with some kind of tinnitus. What is proven is that some people are affected byinfrasound and to help solve the problem scientists need to serve methods to measure andinvestigate it.

Figure 2.2: Infrasound is produced by a variety of natural and man-made sources: explodingvolcanoes, earthquakes, meteors, storms and auroras in the natural world; nuclear, mining andlarge chemical explosions, as well as aircraft and rocket launches in the man-made arena [2]

2. Background 5

2.2 Occurrences of Infrasound

As just discussed some people hear or sense infrasound but most of-ten the source of the sound is unknown. The location of infrasonicsources can be difficult to find because sounds with such long wave-lengths behave differently than high frequency sources with regards topropagation over distance and damping through different media. Nev-ertheless some sources of infrasound are known. Wind farms and windturbines are one example, and an important source of LFN which hasattracted a lot of attention in the media in recent years. The windturbine produce noise given by

f =RZ

60(2.1)

where f is the fundamental frequency, R is the rotor speed in RPM and Z is the numberof blades [10]. For a classic 3 blade geared wind turbine at 10 RPM the frequency is 0.5Hz. Talking about noise from a wind turbine can be confusing since the audible noisegenerated by the wind blades is different than noise at the fundamental frequency foundwith (2.1).

Other sources of infrasound created by mother nature include meteorological phenomenalike earthquakes, volcanic eruptions, water falls and avalanches. Infrasound is generatednaturally by the environment here on earth. For concrete numbers on natural occurringevents, wind flowing over a mountain top or massive volcanic plume injections can producebuoyancy waves with dominant very low frequencies < 0.01 Hz [6]. But also man madeprocesses like supersonic jets, explosions both nuclear and chemical produce infrasound.And with advancing technology these man made sound sources seem to become more andmore and produce more and more infrasound. And it is for the purpose of measuring theseoccurrences a measurement system down to 10 mHz becomes obvious.

Supersonic flight creates a sonic boom which in fact is a system of shock waves reachingground. It is not a sound generated at the transition into supersonic speed like an ex-plosion, but a continuous effect as long as the flight is at supersonic speeds. These shockwaves are at a high level so that supersonic flights are restricted from land zones. Thenoise generated at the aircraft is shaped like the body of the aircraft but nonlinear dis-tortion while propagating to ground makes the shape of a sonic boom look like an N (seeFigure 2.3). Many studies have been made to know more about these sonic booms, andthere exist a theory on shaping the aircraft in a clever way so that the sound propagatedto ground is not seen as an N-wave. Projects like Supersonic Business Jet (SSBJ) startedin th 1990s pursued an alternative to the sonic boom which should sound more like apuff. In 2000 a program called Quiet Supersonic Platform (QSP) was started and in 2003the first demonstration was built - Shaped Sonic Boom Demonstrator (SSBD) depicted inFigure 2.4. This demonstration showed the positive result that the N-wave was changed.The program manager commented: ”In 1947 Chuck Yeager broke the sound barrier. Wejust fixed it.” [11]. A measurement system capable of measuring 10 mHz would aid thedevelopment of other shaped sonic booms and may help make supersonic flights possiblethrough land zones which is of great interest to airline companies.

2. Background 6

Figure 2.3: The noise shape generated at the aircraft is like the shape of the aircraft but nonlinearpropagation makes the sound wave at ground look like an N-shape.

Figure 2.4: F-5E modified Shaped Sonic Boom Demonstration aircraft used to explore supersonicbooms from aircrafts. The N-wave mentioned is painted on the side of the aircraft with a red line.The blue line painted on top is the shape of the wave from this aircraft.

2. Background 7

2.3 Applications

A system for measuring infrasound down to 10 mHz can be of great interest for bothmanufacturers and actors in the wind farm debate. To name a few Vestas A/S, DanishMinistry of Environment (and internationally equivalent) and environmental movementssuch as Greenpeace and The Danish Society for Nature Conservation. If not interesteddirectly in a system capable of measuring ultra low frequency infrasound, they will forsure be interested in the results from measurements with it.

In areas with frequent meteorological phenomena of great magnitude an alert system wouldbenefit greatly from a low frequency infrasound system like this. The system can help di-agnose the occurrences like the size, location and impact of an earthquake, avalanche orvolcanic eruption [6].

And lastly a clear application of this system is actors in the development of sonic boomshapers. To be able to test the theoretical solutions in practice they would need a systemcapable of measuring very low infrasonic frequencies. Supersonic flights crossing USA arepresumed to have good economy thus shaping the sonic boom is of interest for aircraftmanufacturers like Airbus and Boeing.

2.4 Condenser Microphones

A condenser microphone is a transducer converting acoustical energy to electrical energy.A generic model of a condenser microphone is shown in Figure 2.5 and a drawing showingthe cross section in Figure 2.6. The diaphragm and backplate form a capacitor and whensound makes the diaphragm move in and out the capacitance change. The rigid backplatehas holes so that the air can move back and forward between the cavities; front cavitybeing between the diaphragm and backplate and the much larger back cavity inside thehousing of the microphone. This type of capacitive sensor is called a spacing-variationsensor because the signal represents a spacing variation. Another type of sensor is anarea-variation sensor, where the plates forming the capacitance slide in a parallel directionincreasing or decreasing the effective capacitive area [12].

The inside of the microphone is vented so that changes in the surrounding atmosphericpressure does not make the diaphragm place itself in a outward or inward direction. TheDC pressure should be equal inside and outside the microphone. The vent is typicallya small tube or opening acoustically connecting the inside of the microphone with theoutside.

The capacitance of the microphone is given by

Cm = ε0A

d(2.2)

where ε0 is the permittivity of vacuum, A the capacitor plate area and d the distancebetween the plates. A typical 1

2 -inch microphone with a plate separation d = 20 µm andan effective diaphragm area A = 45 µm2 yields a microphone capacitance Cm = 20 pF.The microphone can be seen as an almost pure capacitance meaning that is has a veryhigh parallel resistance in the order of 5 · 1015 Ω. The high leakage resistance ensures noattenuation of the polarization voltage of the backplate [13].

2. Background 8

Chapter 2 — Microphone TheoryMeasurement Microphone Design

Microphone HandbookVol.1

Brüel & Kjær2− 8

cal tension in the foil gives the diaphragm the required mechanical stiffness. Thedistance between the backplate and the diaphragm is typically 20 µm (± 0.8 µm).The nominal distance may vary between microphone types from about 15 to 30 µm.

The thickness of the diaphragm may vary from about 1.5 to 8 µm depending on themicrophone type. The tolerance is typically less than 10 % of the nominal thickness.

The diaphragm and the front of the back-plate form the plates of the active capaci-tor which generates the output signal of the condenser microphone (see below). Thiscapacitance which is typically between 2 and 60 pF (10-12F), depends mainly on thediameter of the back-plate. The stray capacitance or the passive capacitance be-tween the back-plate and the housing is kept as small as possible, as this makes anundesired load on the active capacitance. The back-plate is connected to the exter-nal contact which together with the housing make the concentric output terminalsof the microphone. An alternative, microphone design is widely applied byBrüel & Kjær. This patented design employs an integrated backplate and insulator,see Fig.2.2. In contrast to the first mentioned conventional design of microphone

Fig.2.1 Classic Design of a Condenser Measurement MicrophoneFigure 2.5: Generic 3D model of a con-denser microphone [3]



BE 1447 –11 2− 9

which is mainly assembled by screwing the parts together, the integrated back-plateand insulator version is assembled by pressing the parts into each other. This de-sign also deviates from the conventional design by applying a backplate consistingof a metal thin-film placed directly on the surface of the insulator.

In practice, the first mentioned type implies more freedom for the designer to opti-mise the frequency response, while the second is advantageous during production.The main choice which must be made in respect to the two different design types isone of more narrow frequency response tolerances offered by the conventional de-sign, as opposed to reduced production costs for the alternative design.

2.3.3 Material and Process Requirements

A microphone which is to be used for measurements must be stable over time andits properties should preferably not vary with variations in ambient temperature,pressure and humidity. Therefore, carefully selected, high quality materials must beused, even if they are relatively difficult to machine.

The sensitivity of the microphone is inversely proportional to the diaphragm ten-sion. The tension must therefore be kept stable. Normally it is a requirement that ameasurement microphone has a broad frequency range and a high sensitivity. Thiscreates a requirement for light-weight diaphragms with high internal tension andthus a very high loading of the diaphragm material. This is achieved by applying atension of up to 600 N/mm2 (which would break most materials) to the diaphragms

Fig.2.2 Cross-Sectional view of microphone types. The classic type (left) is assembled by screwing theparts together. The new type (right) is assembled by pressing components together. The de-sign is patented by Brüel & Kjær

950573/1e

Diaphragm

Backplate

Housing

Insulator

Diaphragm

Backplate

Housing

(b)(a)

Insulator

Figure 2.6: Generic drawing of cross sectionof a condenser microphone [3]

The voltage V on the capacitor can with a given charge Q be expressed with

V =Q

Cm(2.3)

This relation makes is clear that when the charge is kept constant a change in microphonecapacitance, ∆Cm, will result in a change in voltage, ∆V . It is usually this sensing ap-proach which is used to measure a sound pressure as input to a microphone. This will beelaborated in Section 2.6.

The charge, Q, on the microphone is kept constant by applying a high bias voltage. Eitherthe voltage is applied externally which is the original and classic construction, but alsoprepolarized condenser microphones exist. They are also called electret condenser micro-phones and the bias voltage is provided by a permanently electrically charged or polarizedferroelectric material. This makes the microphone independent of an external high voltagesource, which can be a good thing especially with mobile applications.

The sensitivity is a key parameter of a microphone and it expresses the change in voltagein response to a given sound pressure. A typical sensitivity of a 1

2 -inch microphone is 50mV/Pa. Also the sensitivity is usually given with reference to a specific frequency, becausethe sensitivity is not constant for all frequencies. 250 Hz is often used as reference becausethe microphone frequency response is usually most flat in this region.

The dynamic range of a microphone is limited by the inherent noise in the lowest endand in the highest end by distortion. With condenser microphones it is usually not themicrophone in a measurement system that limits the system. A typical 1

2 -inch microphonehas thermal noise of 14 dB referenced to 20 µ Pa and 3 % distortion upper limit of 146dB which results in a dynamic range of 132 dB.

Compared to e.g. a dynamic microphone the condenser microphone is favorable on manyaspects [4]. Linearity, high and low frequency response, dynamic range, working temper-atures are among the most significant.

An equivalent electric circuit of a microphone is illustrated in Figure 2.7. The model in-cludes a voltage source representing the sound pressure source and a variable capacitorrepresenting the capacitance of the moving and separated plates. The model can be ex-tended with a resistor in parallel with the capacitor, but for most practical simulations it

2. Background 9

does not influence the behavior.

V

Cm

Figure 2.7: Equivalent electric circuit of a condenser microphone.

2.5 Infrasonic Measurements

Measurements in the infrasonic range are different in many ways from measurements inthe ordinary auditory frequency band. First of all the nature of the infrasonic frequen-cies differ from frequencies between 20 Hz and 20 kHz as described in Section 2.1. Butwith all measurements comes a preceding calibration which is described in standards. Un-fortunately the standards for calibration of infrasonic measurement systems are not wellestablished [14].

The measurements themselves are also different. Variations in the atmospheric pressurebegins to play a role since they are not necessarily filtered out by the vent. Also airturbulence or rather the pressure differences causing the air turbulence begin to affectthe measurement. Air turbulence can occur where sources of heat exhaust their excessiveheat. That could be fans, computers or even the human body. Wind will also affect themeasurement and this is again obvious since wind is caused by pressure differences guidingthe air molecules from a high pressure zone to a low pressure zone.

Trying to get rid of noise in the measurement many mechanical setups have been suggested.In [15] a mechanical setup of 32 low-impedance air inlets arranged in a circle with a di-ameter of 16 m is proposed. Also windscreens of different sizes and shapes are proposed.It is obvious that some kind of filtering of wind disturbances will be of great importanceto a good infrasonic measurement. In the IEC 61400-11:2002 standard a measurementtechnique for acoustic noise measurements on wind turbines is described. The standard isa part of a larger standard on wind turbine generator systems. The standard instruct themicrophone to be placed on a acoustically hard sphere (Ø > 1 m) and protected by a halfsphere of cell foam (Ø ≈ 90 mm). Even though this standard is well established it onlymeasures down to around 50 Hz and it is only designed for measurements on wind turbines.

Also the data is treated differently. Specifications in audio equipment are most typicallystated A-weighted which is an international standard used to relate measurement of soundpressure level to the human hearing. Unfortunately A-weighting is designed for low levelsounds and for frequencies in the auditory frequency band which is not applicable here.The A weighting curve approximately follows the equal loudness curve of 40 phons. In 1995a G-weighting was standardized (ISO 7196) designed for infrasound. Unfortunately thisstandard only covers from 1 Hz to 20 Hz. The curve is defined to have a gain of zero dB at10Hz. Between 1 Hz and 20 Hz the slope is approximately 12 dB per octave. The cut-off

2. Background 10

below 1Hz has a slope of 24 dB per octave, and above 20 Hz the slope is -24 dB per octave.

All in all it is clear from previous findings and standards that this measurement systemshould also include some shielding of wind. And even with a properly designed equalizationvent the data will contain data which is not the signal hence noise. It will be a challengingtask to filter the signal from the many other noise sources like wind, atmospheric pressurevariations and naturally present infrasound sources.

2.6 Measurements with Capacitive Sensors

Capacitive sensors can be sensed in several ways. A thorough evaluation is found in [16].As described in Section 2.4 the signal can be sensed by having a constant charge on thecapacitor and sensing the voltage across the capacitor. This voltage reflecting the soundsignal connects to an amplifier which serves the purpose of an impedance buffer and op-tional amplification (see Figure 2.8). The same node is connected to ground through aresistor to make sure the DC level does not float. And because the microphone capaci-tance and the input impedance of the connected amplifier forms a high pass filter, the biasresistor to ground needs to be very large. This will be discussed in details in Section 3.3.The amplifier can be a single transistor in applications where low noise is most important,but this configuration is known to have low power supply rejection. The amplifier can alsobe an operational amplifier (op-amp) or instrumentation amplifier (in-amp). As will beseen later these types of preamplifier circuits have a long start-up due to the large resistorvalues.

V

C

R−

+

AOUT

Figure 2.8: Voltage sensing method measuring direct dc.

The capacitive sensor can also be sensed with a constant voltage across meaning the mi-crophone inputs are short circuited. This approach implies a charge amplifier sensing thecharge change or a current sensing transistor. Compared to the previous mentioned volt-age sensing this charge sensing or current sensing method have a lower signal to noise ratio(SNR) [16].

A third possibility is to use the capacitance of the microphone in an oscillating circuita so called oscillator. This can be implemented either by an RC circuit or LC circuit,where the microphone capacitance is the tuning element in the oscillator. This kind of asignal represented by a frequency is called frequency modulation (FM). The FM signal issensed with a demodulator e.g. a frequency counter to linearize spacing-variation sensors.By using an RC circuit the frequency is proportional to 1

RC and with an LC circuit itsproportional to 1√

RCwhich is harder to linearize [12].

Arranging the microphone capacitance in a bridge circuit like a simple capacitance bridgewas also investigated. But that kind of circuit implies that the comparing component in

2. Background 11

the other leg of the bridge is like the device under test. In other words the bridge has bestperformance when the microphone capacitance and the compared capacitance are equalwith respect to series resistance and capacitance value [17]. In mathematical terms thecomplex values of the two comparing capacitances must be equal in real and imaginaryvalues [18]. Ideally this would only be possible if another similar microphone would beused as reference. If the two legs in the bridge are not equal the signal would be out ofphase and magnitude. This is not possible for all input levels and frequencies, so thissolution was trashed.

R4R3

C1 Cx

DV

Figure 2.9: Simple Capacitive Bridge circuit where the capacitance, Cx, is measured incomparison to a known capacitor C1 and with precision adjustable resistors R3 and R4.The AC null detector, D, reads 0 V when the the bridge is in balance.

2.7 Preamplifiers for Condenser Microphones

The most commonly seen circuitry to interface a condenser microphone is by far the volt-age sensing method also called Direct DC. It is very simple and thereby very small insize. The purpose of the preamplifier is to convert the impedance level from a very highimpedance microphone to the low impedance cable which transmits the signal to whateverinstrument waiting to receive.

It is usually strived to have unity gain in the preamplifier. More specifically the pream-plifier is constructed to fit the application, the microphone specifications and frequencyrange of the system. When using an externally polarized microphone the preamplifier ofcourse needs to be able to deliver that. Also many preamplifiers are powered by a constantcurrent source through the signal line which is called by many names. G.R.A.S. calls itCCP (Constant Current Power supply), Bruel & Kjær calls their version DeltaTron, inpiezoelectric electric domains its often called IEPE and more generally it is called CCLD(Constant Current Line Drive).

The impedance buffer made up by the preamplifier makes sure that the signal is notattenuated by the attached cable which can be long. Without the buffering a long cableand the resulting large capacitance would attenuate the signal which can not be accepted.

3

Electronic Design

In the following sections the design process will be described from an electrical point ofview, finishing off with the presentation of the final prototype design. The initial designconsiderations will be elaborated moving on to pre-verification with simulation, measure-ments on the first prototype and lastly manufacturing of the final prototype.

V

Cm

Zb

−

+A

OUT

Zf1

Zf2

Figure 3.1: Simplified schematic of circuit. V represents the acoustic sound pressure, Cm

the variating microphone capacitance, Zb the bias circuitry supplying the bias currentto the amplifier, Zf1 and Zf2 are the feedback circuitry and A the operational amplifiercoupled as an impedance buffer.

3.1 Design Topology

As described in Section 2.6 and 2.7 many topologies were considered for the design of thepreamplifier. A bridge circuit was dismissed due to the complexity of having a perfectlymatched ’other leg’ in the bridge, which is needed for a good accuracy but impractical dueto the variating microphone capacitance.

A frequency modulation using the microphone capacitance in an oscillator was also dis-missed. Mainly due to the disadvantages over the chosen direct DC topology which issimpler and smaller in size. Furthermore a contact was established to Norsonic, a Nor-wegian company specialized in applications for measurement of sound and vibration [19].They shared their experience with the design of an FM system where a dynamic range ofonly 40 dB was achieved.

The chosen topology is a simple direct DC detection circuit. The change in capacitanceis measured by charging the capacitance and connecting it to the input of an amplifier.

3. Electronic Design 14

In fact the microphone is charged on forehand via the polarized material, so simply byconnecting the microphone to an amplifier input gives a voltage output proportional tothe spacing variation between the capacitive plates.

Different amplifiers can be chosen. Most commonly found in the industry is a JFET sinceit has low input leakage and low noise specifications. But also op-amps and in-amps canserve the purpose of amplifying.

3.2 Low Leakage Op-amp

The importance of the input leakage becomes clear when seeing that the microphonecapacitance forms a high pass filter with the input resistance of the circuitry connectedto it. And low input leakage is equivalent to a high input resistance. The electric -3 dBcutoff frequency is calculated using

f =1

2πCmRin(3.1)

where Cm is the microphone capacitance and Rin is the input resistance of the pream-plifier. With the goal of a lower frequency limit of 10 mHz and a nominal Cm = 20 pFthe required Rin is in the order of 800 GΩ. The search for an amplifier began with theprimary requirement of at least 1000 GΩ or 1012 Ω.

The amplifier can be implemented in many ways and as mentioned a JFET is usually seenin microphone preamplifiers. But investigations went through the market of op-amps invarious configurations and even in-amps which are the more complex solution. It becameclear that the amplifier using an op-amp should be in a non-inverting configuration sinceit has the important property of high input resistance [20]. The in-amp has the advantageof a high common mode rejection (CMR) which is good for extracting a weak signal in anoisy environment and to minimize offsets [21]. National Semiconductor has an LMP7721Precision Amplifier which has the industry’s lowest input bias current of 3 fA (20 fA atmax). It was deselected because it only operates with a single supply voltage of maximum5 V, which in Section 4.1 will become clear to be insufficient. Texas Instruments hasproduced an Ultra-Low Bias Current Difet Op-amp called OPA129 which was comparedto its brother IN116 which is the same as an in-amp. Even though the IN116 had higherinput resistance the choice fell on OPA129 which had better noise specifications and asimpler layout. The three rivals and their specifications are listed for comparison in Table3.1.

3.3 Bias Current Circuitry

Into any transistor and therefore also the chosen op-amp is a positive or negative leakagecurrent. It is there because the input impedance of the terminals have a finite value andis not infinity as assumed when operating with ideal op-amps and the concept of virtualground. Even though the leakage current of OPA129 and the other candidates is in orderof fA or 10−15 A it is sufficient to cause a floating voltage on the input node. The leakagecurrent which must flow in or out of the inputs must be supplied from a source. That is thereason why a bias current path most be established on the positive input of the amplifier.This is typically done with a high ohmic resistor connected to ground or a voltage supply.


Table 3.1: Specifications of chosen amplifier OPA129 along with its rivals.

OPA129 IN116 LMP7721

Texas Texas NationalInstruments Instruments Semiconductor

Topology Op-amp In-amp Op-ampInput bias current, typical [fA] 30 3 3Input bias current, max [fA] 100 25 20Input impedance, differential [Ω] 1013 1015

Noise at 1 kHz [nV/√Hz] 17 28 6.5

Supply voltage [V] ±15 ±15 5.5

A problem arises because the total input impedance of whatever connected circuitry seenby the microphone must be at least 1000 GΩ. Otherwise the 10 mHz lower frequency limitis not obtained. That high a ohmic value is impractical for a traditional resistor whether itis a thin, thick or metal film resistor. Ohmite, a company specialized in manufacturing ofresistors, has a 100 GΩ resistor which is a metal film resistor vacuum sealed in a long glasstube. Thin film resistors are in general not available above a couple of GΩs and thick filmresistors move up in the range of 50 GΩ. It could be possible to connect a bunch of either ofthem in series but other solutions where sought in order to keep down the size of the design.

The leakage of the OPA129 is 100 pA at most but typical 30 pA. The bias circuitry shouldbe able to supply that. A way to supply the leakage current is to use the leakage currentof another component.

A diode has a reverse current which for specific low leakage diodes is very small. Thisregion is active when the diode is reverse biased (see Figure 3.2) and not beyond the break-down voltage Vbr. In the forward direction the diode begins to conduct when the forwardvoltage Vd reaches around 0.8 V. Two diodes connected in opposite direction and in serieswill serve the purpose of a very large resistance. But it is only possible if the voltage acrossthe diodes swing within Vbr and Vd. Since the input node swings according to the soundpressure the diodes can not be connected directly to ground. As calculated in Section4.2 the voltage swing on the amplifier input can be around ±12 V at 138 SPL. There-fore the diodes are connected from the input terminal of the amplifier to a node whichhas the same amplitude (or close) as the input node. The output signal is used througha high pass filter which cuts off any DC present. The filter will be presented in Section 3.7.

Figure 3.2: I-V characteristics of a P-N junction diode (not to scale)


Several diodes came into play. NXP Semiconductors have a low leakage diode BAS116which has a typical reverse current, IR, of 3 pA. Recalculating the reverse current to aequivalent resistance is not straight forward since the characteristics is not well defined andnonlinear around the transition between reverse and forward region around 0 V. Anotheraspect of choosing a diode as a low leakage component is that diodes essentially is designedfor rectifying and not for low leakage. Some transistors on the other hand are designedspecifically for low leakage through the gate. This property was looked into. Especiallythe NXP Semiconductors BFR31 which has a gate cut-off current of 200 pA. The gatecut-off current is the reverse current of the internal gate-source diode when drain-sourceis shorted. One argument for looking at FETs over diodes is that FETs are available atmuch lower prices than diodes, the reason being that they are produced in far larger num-bers than diodes and the production costs thereby minimized. Even though the leakagecurrent of BFR31 is larger than the reverse current of BAS116 the designed purpose oflow leakage and not rectifying makes better for the application.

Measurements where conducted on the first prototype with the mentioned diodes and FETand will be presented in the next section. While working with the different bias circuits itbecame clear that the voltage across Zb indeed has an effect. Because the preamplifier hasgain of about -1 dB (a factor 0.89) the feedback voltage is always less in amplitude thanthe input voltage. At 10 V input the feedback voltage is 8.9 V and thus the voltage acrossZb is 1.1 V. This is beyond the forward voltage of the diode which starts to conduct hencethe high ohmic resistance it emulates decreases. Using leakage currents of FETs does notseem to fall for the same issue since it is designed for low leakage. This problem will bediscussed further in Section 5.1.

3.4 First Prototype

To be able to build a working prototype a small box of aluminium was made. The enclosureworks in practice as a Faraday cage which blocks out external static electric fields. The boxthereby work as a filter for the noise present in the world around us. In cooperation withthe mechanical department at G.R.A.S. the box was equipped with a thread arrangementfor mounting of a 1

2 -inch microphone, two BNC connectors, a power switch and a secondswitch which will be presented later. The box measures 188x120x57 mm and is providedwith rubber studs on the bottom side. A top-view picture of the box is shown in Figure 3.3.

Figure 3.3: First prototype built into ahomemade Faraday cage to protect from out-side noise

Figure 3.4: The mock up of the circuit ofthe first prototype


The circuit shown in Figure 3.5 is implemented in a quick and dirty mock up on a copperplate. The feedback impedances Zf1 and Zf2 are chosen as an RC circuit with a cutofffrequency of 1 mHz. The component values are Zf1 = Cf = 16µF and Zf2 = Rf = 10MΩ.The power supply for the circuit is two 9 V batteries each with a decoupling capacitor of100 nF placed close to the pins on the op-amp.

V

Cm

Zb

−

+A

V+ 100 nF

V− 100 nF

OUT

Cf

16 µFRf 10 MΩ

Figure 3.5: Full schematic of the first prototype. The resistor Zb shown is replaced withalternative bias circuits as shown in Figure 3.6.

The first prototype was fixed with respect to feedback circuitry and was used to test thecircuit in general, the performance of the op-amp and the bias circuits shown in Figure 3.6.The initial step was to simulate the circuit using PSpice. The simplified model build inPSpice is shown in Figure 3.7 and differs from the already presented schematic by havingan ideal op-amp. To get a correct response of the model in the low frequency range theinput impedance of the op-amp is modeled as a finite impedance. This impedance is statedin the data sheet of OPA129 both as a differential input impedance, Zindiff

and a commonmode input impedance, Zincmm . A spice model of OPA129 was tried implemented butit turned out to give very incorrect results. It became obvious that the spice model is abehavioral model and not necessarily correct for all applications. Thus it was decided towork with an ideal op-amp and modeling the input impedance manually.

a b c

Figure 3.6: The different bias circuits replacing the impedance Zb from Figure 3.5. Im-plementation a realize the high resistance with a resistor, b with two diodes in oppositedirection and c with two FETs in opposite direction using the gate leakage current.

In Figure 3.8 the result of the simulations are shown. For all measurements the feedbackcircuit is as illustrated in Figure 3.11. As mentioned the feedback is a simple RC circuitwith low cutoff frequency of 1 mHz. The difference between the measurements/simulations


V

Cm

Zb

−

+

AZincmm

Zindif

OUT

Zf1

Zf2

Figure 3.7: Schematic of the simulated circuit in PSpice. Zin,cmm (1013 Ω || 1 pF) is thecommon mode input impedance of the op-amp which is modeled along with the differentialinput impedance, Zin,dif (1015 Ω || 2 pF) to get a correct simulation in the low frequencyrange. Both impedances are listed in the data sheet of OPA129 as a resistance and acapacitance in parallel.

is the bias circuitry.

Initially a BAS216 diode was tested even though the reverse current is 30 nA which is fartoo much to meet the requirements. It is worth mentioning that the component modelsused in the simulation are spice models downloaded directly from the manufacturer’s web-sites. Simulation and measurement shows similarities with a peak around 0.4 and 0.7 Hz.The remarkable attenuation of around 8 dB for the measurement is due to wrong guardingwhich will be described later.

Then a BAS116 low leakage diode was tested. Unfortunately the match between simula-tion and measurement is not as good as with the BAS216. Simulation shows a peak below10 mHz whereas the measurement shows one at 150 mHz. It is not investigated furtherwhy this mismatch occurred. Either the spice model of the diode is not perfect for thisusage or the measurement has introduced some kind of error. Maybe the previously men-tioned problem with voltage across the diodes due to attenuation of the feedback voltagecauses the higher cutoff frequency.

Lastly the FET BFR31 was tried. This shows a very good match with a peak in simulationat 5.5 mHz and a measured at 8 mHz. Again the attenuation in the measurement is dueto introduced capacitances as will be elaborated in the next section. The results showthat using two FETs coupled in opposite direction to supply the bias current is sufficientto achieve a high enough resistance and thereby a low enough cutoff frequency.

3.5 Guarding

Ultra-low input bias current op-amps introduce the need for extra careful layout to achievethe documented performance. The output signal of the microphone needs to be guarded onits way to the input of the op-amp. A guard is a low impedance conductor that surroundsan input line and the potential of this guard must be similar to the input line’s voltage.Without guarding large stray capacitances will be introduced which will attenuate the sig-nal. The BFR31 FETs have an input capacitance of 4 pF. The input capacitance of a FET


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101

−40

−30

−20

−10

0

10

20

30

40Frequency response of first prototype with different bias circuits

Frequency [Hz]

Am

plitu

de [d

B V

RM

S]

Simulation: BAS216Simulation: BAS116Simulation: BFR31Measurement: BAS216Measurement: BAS116Measurement: BFR31

Figure 3.8: Simulated and measured frequency response of first prototype (see Figure 3.7) withthe different bias circuits from Figure 3.6.

is defined as the sum of the gate-source capacitance and the gate-drain capacitance. Andas mentioned the OPA129 has a differential input capacitance of 2 pF. These introducedcapacitances can not be removed but stray capacitance like introduced by wiring withoutguarding can be dealt with.

In Figure 3.8 the attenuation in the measurements can be explained by stray capacitancein the wiring. For the BAS216 case the guard following the microphone signal from theinside wall of the microphone thread to the input of the op-amp on the copper board wasconnected to ground. This correspond to no guarding since the guard’s voltage level isnot equal to the input signal. This introduces a lot of the -8 dB gain which is seen on theBAS216 measurement. After connecting the guard correctly to the output signal of theop-amp the gain in the same setup was -2.5 dB which is more like the other measurements.

The guard also needs to be present on the PCB since leakage current on the surface of theboard can exceed the leakage current of the pin. For example, a circuit board resistanceof 1012 Ω from a power supply pin to an input pin produces a current of 15 pA - morethan 100 times the input bias current of the op-amp [22]. A guard trace is surroundingthe input pins on the amplifier. This ensures that a current will not as likely flow from theinput pin to somewhere else since the voltage level everywhere else is at the same potential.


3.6 Analyzing Peaking

Now with a first prototype which seems to be able to meet the requirement of measuringdown to 10 mHz another problem arises. The peaking at the low cutoff frequency is notacceptable. In Figure 3.9 it is seen that decreasing the cutoff frequency by variating thefeedback capacitor will only make the matter worse. To better understand what causesthis an analytical expression of the transfer function is sought.

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5

10

15

20

Frequency response with variation of Cf in the feedback

Frequency [Hz]

Am

plitu

de [d

BV

]

33 pF59 pF104 pF186 pF330 pF587 pF1 nF1.9 nF3.3 nF5.9 nF10 nF19 nF

Figure 3.9: Simulation of the frequency response with a simple RC filter in the feedback. Peakingis inevitable. Sweeping the value of Cf shows larger amplitude peaks for lower cut-off frequencies.

First Kirchhoff’s Current Law (KCL) is used on the node connected to the positive inputof the amplifier. For simplicity the concept of virtual ground is assumed meaning that thevoltage on the two inputs are equal and no current flow in or out of the inputs. Furthermorethe bias circuitry is assumed to be a very large resistor, Zb.

0 = (VO − VA)1

Zb+ (VO − VIN )

1

Zm⇔ (3.2)

VA1

Zb= VO

(1

Zm+

1

Zb

)− VIN

1

Zm(3.3)

where VA is the voltage at the node connecting the bias circuitry to the feedback circuitry.And Zm is the impedance of the microphone. And now using KCL on the node VA yields

0 = (VO − VA)1

Zb+ (VO − VA)

1

Zf1− VA

1

Zf2(3.4)


Rearranging coefficients of VO on one side and of VA on the other

VO

(1

Zb+

1

Zf1

)= VA

(1

Zb+

1

Zf1+

1

Zf2

)Isolating VA

VA = VO

1Zb

+ 1Zf1

1Zb

+ 1Zf1

+ 1Zf2

(3.5)

Substituting (3.5) into (3.3) yields

VO

( 1Zb

+ 1Zf1

1Zb

+ 1Zf1

+ 1Zf2

)1

Zb= VO

(1

Zm+

1

Zb

)− VIN

1

Zm

Rearranging all coefficients of VO on one side and VIN on the other

VIN1

Zm= VO

(1

Zm+

1

Zb− 1

Zb

( 1Zb

+ 1Zf1

1Zb

+ 1Zf1

+ 1Zf2

))

Rearranging and isolating VOVIN

on one side

VOVIN

=Zb

Zm

(ZbZm

+ 1−1Zb

+ 1Zf1

1Zb

+ 1Zf1

+ 1Zf2

) (3.6)

Now it’s time to insert Laplace transformed expressions instead of complex impedances.They are transformed as follows

Zb = Rb Zf1 =1

sCf(3.7)

Zm =1

sCmZf2 = Rf (3.8)

Substituting the expressions from (3.7) and (3.8) in (3.6) yields

VOVIN

=Rb

1sCm

(sRbCm + 1−

1Rb

+sCf

1Rb

+ 1Rf

+sCf

)=

sRbCm

sRbCm + 1−1Rb

+sCf

1Rb

+ 1Rf

+sCf

=sRbCm( 1

Rb+ 1

Rf+ sCf )

sRbCm( 1Rb

+ 1Rf

+ sCf ) + 1Rb

+ 1Rf

+ sCf − 1Rb− sCf

=s2RfCfRbCm + s(Rf +Rb)Cm

s2RfCfRbCm + s(Rf +Rb)Cm + 1(3.9)


The expression in (3.9) is the transfer function of the circuit and relates the output voltageto the input voltage. It is a second order system which is normally written on the standardform:

Y (s)

U(s)=

b

s2 + 2ζωns+ ω2n

=

bω2n

s2

ω2n

+ 2ζ sωn

+ 1(3.10)

where Y (s) is the output, U(s) the input, b the static gain, ζ the damping ratio, ωn thenatural frequency [23].

Directly comparing (3.10) to (3.9) yields the system’s natural frequency

ωn =

√1

RfCfRbCm(3.11)

And its damping ratio

ζ =1

2

√1

RfCfRbCm(Rf +Rb)Cm (3.12)

For all practical component values ζ is below 1 which makes it an under-damped systemwith complex poles with a negative real value. With Cm = 20 pF, Rb = 1000 GΩ, Rf =10 MΩ and Cf = 16 µF the damping ratio is ζ = 0.18. This implies that the system isstable. For a better understanding on the location of poles and zeros a Nyquist diagramis made. See Figure 3.10.

Returning to the origin of this analysis, it was suppose to help understand why the systemhas an amplitude peak and how to avoid it. Introducing damping will minimize the peakand with a critical damped system (ζ = 1) it would disappear. Unfortunately the expres-sions derived in (3.12) shows that it is not possible with only two degrees of freedom beingthe feedback components, Rf and Cf . But when the cutoff frequency is to be maintainedand Cm and Rb are rather fixed it is actually only Rf that is adjustable to maximize thedamping. But as explained in Section 3.3 the resistor values quickly become impractical.Especially when Rf occur as a sum together with Rb which is in the order of 1000 GΩ. Soto have a say in this matter it should be very large and impractical.

To round up, this analysis show that another approach to the circuit must be considered.

3.7 Feedback

Different variations of feedback circuits can be implemented and to deal with the am-plitude peak as just described, some alternatives are investigated. The alternatives areshown in Figure 3.11. Circuit a shows the simple RC filter which proved to be insufficient.Circuit b is one alternative where a capacitor is put in parallel with the feedback resistor.The capacitor introduces a capacitive voltage divider together with the existing feedbackcapacitor, Cf , now named Cf1. Another possibility is circuit c which is a resistive voltagedivider. The last investigated alternative is a filter modifying circuit b with an extra re-sistor connected in parallel with the original feedback capacitor.


−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−5

−4

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0

1

2

3

4

5

0 dB

−10 dB−6 dB

−4 dB

−2 dB

10 dB6 dB4 dB

2 dB

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

PolesZeros

Figure 3.10: Poles have very little imaginary and real values. Furthermore the stability is clearlyachieved since the curve does not go beyond the point (-1,0) marked red. Component values areCm = 20 pF, Rb = 1000 GΩ, Cf = 16 µF, Rf = 10 MΩ.

a

Rf2

Cf1

b

Rf2

Cf1

Cf2

c

Rf2

Rf1

d

Rf2

Cf1

Cf2

Rf1

Figure 3.11: Schematic of the alternatives to the feedback circuit.

Simulations where made in PSpice to see what results could be achieved with the 4 differ-ent alternative feedback circuits. The simulations are conducted with a circuit model asshown in Figure 3.7 and the bias circuit of Figure 3.6 c, and the variating feedback circuitsof Figure 3.11.


3.7.1 Circuit b

The first simulation involves circuit b where a capacitive voltage divider is added by con-necting a capacitor in parallel with the existing resistor in the feedback. In Figure 3.12simulation results from this setup is shown. The effect of the voltage divider is clearly seenby the increasing attenuation at higher frequencies. With the chosen interval of values forCf2 the amplitude varies from -0.9 dB to -3.5 dB. When trying to make sure not to com-promise the low frequency cutoff frequency it is obvious that the best curve without peakis with 4 µF. This value correspond to a fourth of the original feedback capacitor, Cf1.This choice introduces -0.6 dB of gain relative to the dB level without the extra capacitor.With the chosen extra capacitor the total gain is -1.5 dB. Introducing more attenuationleads to another issue. Loading the microphone introduces distortion and nonlinearitieswhich is why preamplifiers are usually designed to have a gain as close to 0 dB as possible.But some attenuation is acceptable and is seems a good trade off to accept some attenu-ation and get rid of the peak.

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5

10

Frequency response with variation of Cf2

in the feedback

Frequency [Hz]

Am

plitu

de [d

BV

]

128 nF405 nF1.28 µF4.05 µF12.8 µF40.5 µF128 µF

Figure 3.12: Simulation showing the frequency response of the preamplifier with feedback circuitb and varying Cf2. The variation shows significant damping of the amplitude peak.

As in the analysis in Section 3.6 the expressions for the cutoff frequency of circuit variationb is

ωn =

√1

Rf2(Cf1 + Cf2)RbCm(3.13)


and its damping ratio

ζ =1

2

√1

Rf2(Cf1 + Cf2)RbCm(Cf2Rf2 + (Rf2 +Rb)Cm) (3.14)

The equations deriving these expressions can be found in Appendix C.

Lets try to put in real numbers to see that components can alternate the damping

The frequency ωn is fixed to 2π · 0.01 Hz

ζ =1

2· 2π · 0.01 Hz · (Cf2Rf2 + (Rf2 +Rb)Cm) (3.15)

The bias resistance, Rb, is 1000 GΩ and the microphone capacitance, Cm, is 20 pF

ζ =1

2· 2π · 0.01 Hz · (Cf2Rf2 + (Rf2 + 1000 GΩ) · 20 pF) (3.16)

Simplifying by saying Rf2 Rb

ζ =1

2· 2π · 0.01 Hz · (Cf2Rf2 + 1000 GΩ · 20 pF) (3.17)

Rearranging to isolate the unknown with ζ < 1

Cf2Rf2 <1

π · 0.01 Hz− 1000 GΩ · 20 pF (3.18)

Cf2Rf2 < 11.831s (3.19)

Using Rf2 = 10 MΩ as in the original circuit a

Cf2 < 1.2 µF (3.20)

which is not far from what the simulations in Figure 3.12 shows (damped scenario withCf2 = 4.05 µF). If Rb is not quite as large as 1000 GΩ and the cutoff frequency is a bitlower than 1 mHz the value of Cf2 will increase to something closer to 4 µF which is whatthe simulation shows.

3.7.2 Circuit c

The result from the previous section with circuit b were done by introducing a capacitivevoltage divider. Circuit c has a resistive voltage divider and no frequency dependence.The simulations with varying attenuation in the voltage divider is shown in Figure 3.13.It shows that with a voltage divider feeding back half the amplitude of the output voltage(Rf1 = 100 GΩ) the response drops 1 dB at 0.1 Hz and cuts off completely above 10mHz. Referring back to Section 3.3 the diodes inside the FETs require the voltage nodeVA to be equal or close to equal in magnitude and phase. The extra cutoff introducedin this scenario might be due to the internal diodes beginning to conduct larger currentbeyond the voltage limits Vbr and Vd. In the other 10 scenarios in Figure 3.13 the extra


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101

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0

Frequency response with variation of Rf1

in the feedback

Frequency [Hz]

Am

plitu

de [d

BV

]

1 GΩ1.6 GΩ2.5 GΩ4.0 GΩ6.3 GΩ10 GΩ16 GΩ25 GΩ40 GΩ63 GΩ100 GΩ

Figure 3.13: Simulation showing the frequency response of the preamplifier with feedback circuitc and varying Rf1. The variation shows an extra cutoff frequency introducing a minor attenuationbefore completely dropping down.

cutoff is also present but not as profound. With the choice of Rf1 = 10 GΩ the extra cut-off introduces only 0.3 dB of attenuation and still has a -3 dB frequency limit of a 2.5 mHz.

Circuit b has a filter and a capacitive voltage divider. Circuit c has only the voltagedivider but by combining circuit a and c the total feedback circuit should treat the signalthe same. A comparison of the two cases is shown in Figure 3.14 and it is clearly seenthat the results are similar. So it is just as good with a capacitive voltage divider as witha resistive voltage divider.

3.7.3 Circuit d

The last variant of a feedback circuit is that of Figure 3.11 d. The analytic expression ofthe transfer function is a third order system as seen in Appendix D. This makes is it a lotmore difficult to extract useful tuning parameters. So for this feedback circuit only PSpicesimulations are used to validate the performance.

Varying Cf2 changes the overall gain in the transfer function as was seen with feedbackcircuit b in Figure 3.12. A higher value introduces more attenuation. Varying Rf1 changesthe peak and also the roll off below the cutoff frequency. For lower values the peak dis-appears but the slope below the cutoff frequency gets smaller. It makes sense that for aninfinite resistance the peak is as with circuit b and with no resistance the high pass filteris not existing meaning direct feedback. In Figure 3.15 the variation of the resistor Rf2 isshown. This is with a fixed value of Cf2 = 0.8 µF.


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101

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−8

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−6

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−2

−1

0Comparing frequency response of circuit b with combination of circuit a and c

Frequency [Hz]

Am

plitu

de [d

BV

]

Circuit bCombination of Circuit a and c

Figure 3.14: Comparing simulation with circuit b with the combination of circuit a and c. Bothfeedback circuits in total provide a high pass filter and a voltage divider. In circuit b it’s a capacitiveand in the combination of a and c it’s resistive. The simulations show comparable results.

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0

1

2

3

4


in the feedback

Frequency [Hz]

Am

plitu

de [d

BV

]

100 kΩ159 kΩ251 kΩ398 kΩ631 kΩ1.0 MΩ1.6 MΩ2.5 MΩ4.0 MΩ6.3 MΩ10 MΩ

Figure 3.15: Simulation showing the frequency response of the preamplifier with feedback circuitd, Cf2 = 0.8 µF and varying Rf1. The variation shows significant damping of the amplitude peakand no attenuation for all especially higher frequencies.

Introducing the resistor Rf1 does not seem to have the great effect. Minimizing Cf2 de-creases the overall gain but reveals the amplitude peak, and by adjusting the resistor Rf1

to a no-peak situation there is nothing achieved compared to the feedback circuit b. Onthe contrary the slope below the cutoff frequency is decreased compared to circuit b. And


this is not desired.

With Cf2 = 0.4 µF (10 times smaller than circuit b) there is -0.9 dB in overall gain com-pared to -1.5 dB in circuit b. But the amplitude has only dropped to -5 dB at 1 mHzwhere the choice of resistor results in a no-peak situation. In comparison the amplitudehas dropped to -25 dB in a similar situation with circuit b. With Cf2 = 0.8 µF (5 timessmaller than circuit b) there is -1.0 dB in overall gain and the amplitude has dropped to-7 at 1 mHz. With Cf2 = 2 µF (2 times smaller than circuit b) there is -1.2 dB in overallgain and the amplitude has dropped to -13 dB at 1 mHz.

3.7.4 Choosing a Feedback Circuit

In the previous couple of subsections 4 variants of a feedback circuit are presented. Tomore easily compare the performance they can be seen together in Figure 3.16. Circuit ais clearly not desired and the arguments already presented says enough. Circuit c has asomewhat mysterious second cutoff frequency which for a relatively small frequency rangedrops the amplitude. But for the chosen component values this drop is very small. Circuitd has has the best gain performance for the flat part of the frequency response, but likecircuit c the slope below the cutoff frequency is too small. Circuit b has a desired largeslope below the cutoff frequency and a flat response for higher frequencies even thoughit introduces a bit of attenuation. For the final prototype there will be presented twoversions. Version 1 implements circuit b and Version 2 implements circuit c. The reasonfor making two different versions is to be able to test both and to have two differentalternatives to match the simulations.

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0

5

10

Frequency response with all the proposed feedback circuits

Frequency [Hz]

Am

plitu

de [d

BV

]

a − simple RCb − parallel capacitorc − resistive voltage dividerd − cap and res in parallel

Figure 3.16: Comparison of the frequency responses from the discussed 4 alternatives to a feedbackcircuit.


3.8 Capacitive Load

The op-amp is specified to have a load capacitance stability of 1 nF with a gain of 1. Thisparameter lead to a problem which was noticed when looking at the output signal witha sinusoid at different frequencies as input. The graphs in Figure 3.17 show 5 differentscenarios measured on the first prototype with circuit a in the feedback. The graphs havebeen scaled along the frequency axis for easy comparison. They have also been shiftedalong the amplitude axis to better distinguish then from one another. The top graph showsa 100 mHz perfect sinusoid which is also what is fed to the input. But as the frequencyis increased the sinusoid is distorted and at 7.1 kHz it looks like a triangular wave. Thedeformation of the sinusoid happens because the op-amp is not rated to operate with theimplemented capacitive load at those frequencies. It is the slew-rate limitation which isreached meaning that the op-amp can not change the voltage quick enough.

Relative time

Vol

tage

(sh

ifted

leve

l)

Output signal with sinus input

100 mHz10 Hz1 kHz5.8 kHz7.1 kHz

Figure 3.17: Output signal of op-amp showing that the too large capacitive load in the circuitmakes the op-amp reach its limitations at frequencies above 5.8 kHz. The capacitive load is 16 µFand the rated load capacitance stability of OPA129 is 1 nF up to a bandwidth of 1 MHz.

The rated maximum load capacitance of 1 nF is given that the full bandwidth of 1 MHz isutilized or under full power response only 47 kΩ. This is not the case for this applicationwhich is why it could be of interest to calculate the maximum load capacitance for thisapplication. The maximum output current of the op-amp, Imax is 10 mA and the maximumload capacitance can be calculated with

Cload,max =1

2π ·BW · VmaxImax

(3.21)


where BW is the bandwidth and Vmax is the maximum voltage on the load.

(3.22)

When this application needs performance up to 20 kHz and can expect voltages of 26 Vthe resulting maximum load capacitance is

Cload,max =1

2π · 20 Hz · 26 V10 mA

(3.23)

Cload,max = 3.06 nF (3.24)

This leads to a problem with the presented feedback components. In feedback circuit athe load capacitance is 16 µF and for circuit b 20 µF. Therefore the new values with theimplemented circuits is tuned to continue providing the same results but also to meet therequirement of the maximum load capacitance. The new component values for circuit bare listed in Table 3.2.

Table 3.2: New component values in feedback circuit b which meet the requirements ofthe maximum load capacitance.

Cf1 Rf2 Cf2

Circuit b 3.2 nF 50 GΩ 470 pF

3.9 Start-up

When operating a system with a very large time constant it takes a long time for thesystem to stabilize after turning on. This is inherent in such a system. When the op-ampis turned on and the input nodes are floating at unknown potentials there is only a littlechance that the input nodes will be 0 V. In fact the voltages will most likely be differentthan 0 V. And when powered on the op-amp will have a period where it will try to stabilizeand reach 0 V or the potential of the DC offset.

This start-up period is large due to the time constant which is between 100 and 1000seconds. The system could be left like that but it would require the user to wait for a verylong time before every measurement. That is not a desired feature of any measurementsystem so to avoid the waiting period the system is provided with a circuit to decrease thetime constant briefly.

The circuit implemented is illustrated in Figure 3.18 and connects a resistor in parallel tothe existing resistor in the feedback circuit. The added resistor has a much lower resis-tance which makes the total feedback resistance decrease to a level close to the resistoradded. By connecting a 100 kΩ resistor through a switch to the existing 50 GΩ the totalequivalent resistance is 100 kΩ. And with a new smaller feedback resistor the new timeconstant of the feedback circuit becomes a much more friendly value.

The effect of the circuit is clearly seen in simulation in Figure 3.19 where the blue curveshows start-up without using the switch and the green curve shows the start-up activatingthe switch. For the situation without activating the switch the voltage reaches 0 V afterapproximately 300 s and when activating the switch it decreases to 25 s. The simulationis made by applying a step function on the input and looking at the voltage level at the


100 kΩ50 GΩ470 pF

3.3 nF

Figure 3.18: The implemented circuit minimizing the start-up period in which the amplifierseeks its equilibrium potential. The switch is mechanically activated and connects a smallresistance which lowers the time constant of the system.

output.

0 50 100 150 200 250 300−0.5

0

0.5

1Simulated offset with step on input

Time [s]

Vol

tage

[V]

R

f2 = 50 GΩ

Rf2

= 100 kΩ

Figure 3.19: The time at which the input finds its DC level is decreased when the value of resistorRf is decreased. It also affects the frequency response, so it’s only for start-up purposes beforereal measurements.

It was also considered adding the switch and the resistor directly on the input node inparallel with the bias current circuit. But it was unclear what consequences it might intro-duce. The amplifier circuit is much more sensitive on that node and anything connected tothe input node will have to have the leakage or high resistance. Various data sheets fromdifferent switch manufacturers where examined but none promised an insulation resistancehigh enough. They all write ’minimum 100 MΩ’ which was quite odd. It might be thatmechanical engineers thinks such a resistance value is rocket high and therefore does notbother to state a real value. Or else all switches does in fact only insulate that value.


Either way connecting a switch to the input node was not done. The switch might alsohave parasitics which could attenuate the signal or distort it.

3.10 Noise

The preamplifier has inherent noise which will be discussed in this section. Normally themost dominant noise sources of a preamplifier are the bias resistor and the transistors(JFET, op-amp or similar). But as we will see in a bit the dominant source of noise forthis preamplifier is the op-amp. Noise of the microphone will be dealt with in Section 4.1.In general the noise in an audio measurement system is dominated at high frequencies bythe microphone and at low frequencies by the preamplifier [3].

In [24] noise from a similar direct DC circuit as this is calculated and the bias resistornoise is given as

V 2n,bias =

4KT

Rbias

1

(ω · Cm)2(3.25)

where K is Boltzmann’s constant 1.38−23 and T is the temperature in Kelvin. This re-lation shows that with increasing bias resistance the noise is moved to lower frequencies.Rbias in this preamplifier is as discussed not straight forward since the reverse current oftwo diodes are used and not a resistance. But the relation also holds for decreasing reversecurrent resulting in noise at lower frequencies. Furthermore the noise from the resistoris shunted by the microphone capacitance which forms a low pass filter which ensuresdecreasing resistor noise above the cutoff frequency [3]. So in this circuit the bias circuitrydoes not contribute significant to the total noise.

The noise of the transistor on the other hand is responsible for most of the noise of theelectric circuit. At low frequencies flicker noise or 1/f noise is dominant and at higherfrequencies white noise establishes a noise floor. The noise corner frequency, fnc, is wherethe two different noise colors (pink and white) are equal and in this preamplifier it is justbelow 1 kHz.

In the data sheet of OPA129 the noise of the op-amp is given both in numbers in thespecification but more accurately in a graph shown in Figure 3.20.

To calculate the total voltage noise of the op-amp, the spectral density is integrated overthe bandwidth. This is done with (3.26) [25].

Vn,FET = Vn,floor

√fnc · ln

fmax

fmin+ (fmax − fmin) (3.26)

where Vn,floor is the noise density of the white noise above fnc, fmax and fmin are thefrequencies in each end of the bandwidth of interest. For this application Vn,floor = 15nV√Hz

, fnc = 500 Hz, fmax = 20 kHz and fmin = 0.01 Hz. This yields a Vn,FET = 2.47

µVRMS .

Very often the peak-to-peak value of the voltage noise is of interest and in the data sheet ofOPA129 this value is also stated to 4 µV pp for the bandwidth 0.1 Hz to 10 Hz. To compare


OPA1294SBOS026Awww.ti.com

0

Frequency (Hz)

FULL-POWER OUTPUT vs FREQUENCY

Out

put V

olta

ge (

VP

P)

10k 100k1k 1M

30

20

10

10

Frequency (Hz)

INPUT VOLTAGE NOISE SPECTRAL DENSITY

Vol

tage

Den

sity

(nV

/√H

z)

1 10 100 1k 10k 100k

1k

100

10

1

0.1

0.01

–15 –10 –5 5 10 15

Common-Mode Voltage (V)

BIAS AND OFFSET CURRENT vs INPUT COMMON-MODE VOLTAGE

Nor

mal

ized

Bia

s an

d O

ffset

Cur

rent

0

BIAS AND OFFSET CURRENT vs TEMPERATURE

Ambient Temperature (°C)

Bia

s an

d O

ffset

Cur

rent

(fA

)

100pA

10pA

1pA

100

10

1

–50 50 125–25 0 25 75 100

IB and IOS

1001 1M 10M1k 10k 100k10

COMMON-MODE REJECTION vs FREQUENCY

Frequency (Hz)

Com

mon

-Mod

e R

ejec

tion

(dB

)

140

120

100

80

60

40

20

0

COMMON-MODE REJECTION vs INPUT COMMON-MODE VOLTAGE

Common-Mode Voltage (V)

Com

mon

-Mod

e R

ejec

tion

(dB

)

70

15 1510 105 0 5

120

110

100

90

80

TYPICAL PERFORMANCE CURVES (Cont.)At TA = +25°C, +15VDC, unless otherwise noted.

Figure 3.20: The noise of the transistor of the preamplifier in this case an op-amp OPA129. Thenoise is dominated at low frequencies by pink noise (flicker noise or 1/f noise) and above the noisecorner frequency by white noise.

the result of (3.26) which is in RMS the expected peak-to-peak value can be calculated.The instantaneous value of noise will be equal to or less than 6 times the RMS value 99.7% of the time [26]. For the calculated RMS value the peak-to-peak voltage noise is 14.9 Vpp.

Now the noise is discussed from a theoretical view and it has been explained that thedominant source of noise in the preamplifier is the op-amp. No simulations have beenmade with respect to noise, and the measurements will be presented in Section 5.1.

3.11 Dynamic Range

The dynamic range of the preamplifier is the ratio of the maximum output voltage thepreamplifier can handle to the lowest output voltage. The maximum output voltage islimited by the maximum input voltage of the op-amp which is again limited by the supplyvoltage. Furthermore the maximum output voltage can be limited by maximum currentand maximum slew rate, but that is mostly important when combining high frequenciesand high signal levels [3]. In acoustic systems it is common to state the upper limit of thedynamic range by the level at which a certain percentage of distortion is present. For thispreamplifier 3 % distortion is defined as the upper limit.

The lowest output voltage is set by the noise which has been discussed in Section 3.10. Thehighest output voltage is 13 V stated by the op-amp when supplied with ± 15 V. It is ex-pected that some attenuation in the op-amp configuration is inevitable, an estimate is 1 dBwhich yields an output voltage of 11.6 Vp. With a maximum of 23.2 Vpp and a minimum of14.9 µVpp the dynamic range is theoretically 124 dB, which is quite impressive if succeeded.


Everything until now has been un-weighted numbers. As described in Section 2.5 it isbetter to use no weighting than some weighting not designed for the specific purpose.

3.12 Production Component Variations

Even though simulations have already been presented showing the effect of variating com-ponent values, it is still of interest to see how the component tolerances impact the system.In Figure 3.21 the simulation of the circuit is shown with variation of the resistor Rf2 whichhas a tolerance of ±30 %. It can be seen that the smallest component value is least desiredsince the peak appears. To avoid this possible variation in a production a resistor with alower tolerance must be chosen.

10−3

10−2

10−1

100

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0


in the feedback

Frequency [Hz]

Am

plitu

de [d

BV

rms]

35 GΩ50 GΩ65 GΩ

Figure 3.21: Simulation of the frequency response of the final prototype version 1. Increasedvalues of Rf2 results in less peaking. A component tolerance of ±30 % results in ±0.2 dB

In Figure 3.22 the simulation of a variating feedback capacitor, Cf1 which have a toler-ance of ±5 %. This possible variation of capacitance will result in a ±19 mdB differencein amplitude. The variation will also affect the cutoff frequency which will decrease forhigher values of Cf1.

All in all the tolerances pointed out do not have huge impact on the performance of thecircuit.


10−3

10−2

10−1

100

−2

−1.9

−1.8

−1.7

−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1

Frequency response with variation of Cf1

in the feedback

Frequency [Hz]

Am

plitu

de [d

BV

rms]

3.135 nF3.3 nF3.465 nF

Figure 3.22: Simulation of the frequency response of the final prototype version 1. Increasedvalues of Cf1 results in less attenuation for all frequencies. A variation of +5 % equals +19 mdBfor frequencies above 100 mHz. Not shown is the variation of Cf2 which behaves opposite. This isdue to the voltage division introduced by the two capacitances.

3.13 PCB Layout

OPA129 3SBOS026A www.ti.com

1001 1M 10M1k 10k 100k10

POWER SUPPLY REJECTION vs FREQUENCY

Frequency (Hz)

Pow

er S

uppl

y R

ejec

tion

(dB

)

140

120

100

80

60

40

20

0

+PSRR

–PSRR

OPEN-LOOP FREQUENCY RESPONSE

Frequency (Hz)

Vol

tage

Gai

n (d

B)

140

120

100

80

60

40

20

0

1001 1M 10M

θ

45

90

135

180

Pul

se S

hift

(deg

rees

)Gain

1k 10k 100k10

PhaseMargin≈90°

Power Supply Voltage ...................................................................... ±18VDifferential Input Voltage ............................................................ V– to V+Input Voltage Range .................................................................... V– to V+Storage Temperature Range ......................................... –40°C to +125°COperating Temperature Range ...................................... –40°C to +125°COutput Short Circuit Duration(1) .................................................................. ContinuousJunction Temperature (TJ) ............................................................ +150°C

ABSOLUTE MAXIMUM RATINGS

NOTE: (1) Short circuit may be to power supply common at +25°C ambient.

PACKAGE INFORMATION(1)

PRODUCT PACKAGE-LEAD PACKAGE DESIGNATOR

OPA129P DIP-8 POPA129PB DIP-8 POPA129U SO-8 DOPA129UB SO-8 D

NOTE: (1) For the most current package and ordering information, see thePackage Option Addendum at the end of this data sheet, or see the TI websiteat www.ti.com.

CONNECTION DIAGRAM

ELECTROSTATICDISCHARGE SENSITIVITY

Any integrated circuit can be damaged by ESD. TexasInstruments recommends that all integrated circuits behandled with appropriate precautions. Failure to ob-serve proper handling and installation procedures cancause damage.

ESD damage can range from subtle performance deg-radation to complete device failure. Precision inte-grated circuits may be more susceptible to damagebecause very small parametric changes could causethe device not to meet published specifications.

Top View DIP/SO

TYPICAL PERFORMANCE CURVESAt TA = +25°C, +15VDC, unless otherwise noted.

1

2

3

4

8

7

6

5

Substrate

V+

Output

V–

NC

–In

+In

NC

OPA

NC: No internal connection.Figure 3.23: The non-standard pinout of the chosenop-amp OPA129 makes iteasy to implement a guardtrace and separation of in-puts and supplies minimizesleakage.

The final prototype circuit shown in Figure 3.30 and3.31 have been implemented on a PCB and there area couple of important design considerations regard-ing the layout which will be discussed in this sec-tion.

As already mentioned guarding of the input signal is veryimportant. The guard also needs to be present on thePCB which is easily done with the non-standard pinout ofthe op-amp shown in Figure 3.23. The guard is connectedto the output in order to have a low voltage difference be-tween the input and the guard it self. The guard traceconnects pin 1, pin 4, pin 6 and pin 8 and surrounds theinput pins 2 and 3. The guard copper tube connects tothis node and thereby the input signal is guarded all theway to the input of the op-amp. This minimizes the stray capacitance which is not wantedin the circuit.

Placing of the components have also been considered and especially the components con-nected to high impedance nodes like the op-amp and the bias circuitry. These componentsare placed under the guard ring which helps to ensure the high impedance.


Another issue is the board surface resistance which is usually not an issue. But whendealing with resistances in the 1000 GΩ range it can be something creating troubles. Forinstance with the normally used board material FR4 the surface resistivity is 30 GΩm [27].This means that implementing a 50 GΩ resistor on the board needs careful placement nottoo close to nodes different in potential. Another possible issue with the FR4 materialis its volume resistivity which has the same effects and need the same precautions. Formoisture tolerant applications FR4 board can also be troublesome since it has only got amoisture absorbency of .05 to .07 (% 24h). Other board materials exist which are moreexpensive but will minimize these problems. The author of [27] has developed an extremeboard material with superb parameters for high demanding applications. Actually it isa dielectric film which have low moisture absorption and a surface resistance of 40 PΩ(40 · 1015). Ceramic materials can also be used to achieve higher performance than thestandard FR4 material. For extreme moisture tolerant applications the entire board canbe molded in anti-moisture-epoxy or something similar. Even though the problem withsurface resistivity exist and moisture can be a problem this prototype is manufactured onstandard FR4 material and measurements does not show signs of problems.

Regarding the placement of the wires on the PCB special care have been taken to thepads and how wires connects to them. Wires can not have sharp angles of incidence toa pad because the manufacturing process can possibly break the connection. The padsfor the feedback components are arranged so that all the proposed feedback circuits canfit to them. And since the 50 GΩ resistor is a size 0805 the pads for this component aremodified so both a 0805 component and a size 1206 can fit. With respect to the previouslymentioned problem with surface resistivity the distance from Cf2 pads to other nodes aremaximized. It was also considered to drill out the space between the two pads which wouldmake the board leakage path longer and thereby the board resistance larger.

A practical issue in the manufacturing process is the shape of the board which has 4 innercorners making it fit into the microphone housing. The board is cut out by a millingmachine which in a standard setup has a drill with a diameter of 2 mm. This leaves noroom for sharp inner corners which is why they where filed by hand after receiving thePCBs.

The final PCB layout is shown in Figure 3.24 and 3.25 top layer and bottom layer re-spectively. In Figure 3.26 a 3D model of the PCB including components is shown. Thismodel was used to see if the components especially the bias FETs and the switch couldfit into the housing and under the guard ring. And in Figure 3.27 the manufactured PCBis shown.

Figure 3.24: PCB Top layer. The board measures 44.6 mm times 10.3 mm and is 1.5 mm thick.


Figure 3.25: PCB Bottom layer. It is mirrored for easy comparison to the top layer in Figure3.24

Figure 3.26: PCB and components in a 3Dvisualization. This was used to see whetherthe components would fit into the micro-phone housing.

Figure 3.27: Picture of un-soldered PCBwith rounded inner corners due to the pro-cess by which the board is cut using a 2 mmdrill.

3.13.1 Improvements for Next Version

When soldering a prototype something often shows to be inconveniently designed. Hereare listed some of the suggested improvements that would make the next version of thePCB a better one.

• The text labels (V+, V-, GND and OUT) are too small in font size.

• The pads for contacts (V+, V-, GND and OUT) have solder mask on them bymistake which should be removed.

• That also applies for the guard trace (reference to data sheet of 3 fA Input BiasOp-amp LMP7721)

• The pads for supply voltages are moved to the bottom layer because the wires collidewith the switch on the center of the PCB. This could also be solved by having shorterwires, but they are advantageous when assembling the housing.

• Mounting the microphone spring contact to the PCB was tricky because the contactwas made to fit a board of 1 mm in thickness. Using a 1 mm thickness instead of a1.5 mm would also make the guard ring fit more easily.


3.14 Final Prototype

The components where soldered to the PCB and two versions of the final prototype wasmade. Version 1 implements feedback circuit b and Version 2 implements feedback circuitc - a resistive voltage divider. The guard ring is soldered directly to the board connectedto the guard trace on the PCB like shown in Figure 3.28. Also a spring connector for themicrophone is mounted. It is located inside the guard ring and interface the microphonewith a copper pad on a spring.

Figure 3.28: Picture of the soldered PCB with copper guard ring and microphone housing.

To be able to activate the switch used during or after start-up a drill hole is made in thehousing. This makes it possible with a pen or a pair of tweezers to push the button andthereby quickly discharge the input node of the op-amp. In a production the switch wouldbe equipped with some kind of nob which would stick out the hole for activation withoutfurther tools.

After soldering the PCB it was fitted into a housing like shown in Figure 3.29. The housingis of aluminium which gives the preamplifier a nice feel but also shields from electric fieldslike in the first prototype. In the left of the picture you see the thread for mounting amicrophone and the guard ring protrude the housing for correct guarding all the way fromthe microphone to the op-amp.

The electric interface for power supplies and output signal is done through a high qualitypush-pull circular connector made by LEMA. They are commonly used for audio equip-ment and measurement systems in other domains.

Figure 3.29: Picture of the entire preamplifier including LEMA plug.

The complete schematics of both versions of the preamplifier is seen in Figure 3.30 and3.31.


V

Cm

BFR31Zb

−

+A

OPA129

V+ 100 nF

V− 100 nF

OUT

Cf1

3.3 nF

Rf2

50 GΩ

Cf2470 pF

Figure 3.30: Full schematic of the final prototype version 1 implementing a feedback circuitwith a high pass filter and a capacitive voltage divider.

V

Cm

BFR31Zb

−

+A

OPA129

V+ 100 nF

V− 100 nF

OUT

Rf1

10 GΩ

Rf2

100 GΩ

Figure 3.31: Full schematic of the final prototype version 2 implementing a resistive voltagedivider.

4

Acoustic Design

This chapter deals with the acoustic design of the measurement system. To begin withdetails on the microphone will be elaborated and from there how the system acousticallyperforms at ultra low frequencies. In the end proposals for modifications of the presentmicrophone will be presented.

4.1 Choice of Microphone

Product Data

SOUND & VIBRATIONG R A S. . . . Skovlytoften 33,

2840 Holte, Denmarkwww.gras.dk - [email protected]

The G.R.A.S. ½″ Free-fi eld microphone Type 40AZ is a precision condenser microphone for low-fre-quency (incl. infra-sound) measurements in open acoustic fi elds. It is a pre-polarized free-fi eld micro-phone with a large dynamic range and a wide fre-quency response.

As a free-fi eld microphone, Type 40AZ is designed essentially to measure the sound pressure as it would appear if the microphone were not present, the sound fi eld pointing towards the microphone.

At low frequencies, the disturbing effects of its pres-ence in the sound fi eld are minimal (large wave-lengths compared to the size of the microphone).

At higher frequencies (>1 kHz), the effects of dif-fractions generally cause microphones to measure sound pressure levels increasing with frequency. Fig. 2 shows corrections to be made for various angles of incidence. In a free-fi eld microphone, the effects of diffraction are compensated for to provide a fl at frequency response in a free-fi eld for 0º inci-dence. For Type 40 AZ, this compensation is shown in Fig. 3.

The G.R.A.S. 1/4″ High Impedance preamplifi er Type 26CG (input impedance = 40 GΩ)* is avail-able for use with Type 40AZ (see data sheet for Type 26CG). The mounting thread (11.7 mm - 60 UNS-2) is compatible with microphone preamplifi ers for WS2P/F microphones.

All G.R.A.S. microphones comply with the specifi -cations of IEC 1094: Measurement Microphones, Part 4: Specifi cations for working standard micro-phones.

Non-corrosive, stainless materials are used in manu-facturing these microphones to enable them to with-stand rough handling and corrosive environments.

G.R.A.S. WS2P/F microphones are guaranteed for 5 years, and each microphone is individually cali-brated, calibration chart supplied.

* For the microphone capsule capacity of Type 40AZ (20 pF), the high input impedance of Type 26CG (40 GΩ) leads to a low-frequency 3-dB limit fL (40 GΩ) leads to a low-frequency 3-dB limit fL (40 GΩ) leads to a low-frequency 3-dB limit f = 0,2 Hz.

Typical applications Precision acoustic measurements

Type 0 and 1 SPL measurements

Free-fi eld measurements

Low-frequency measurements

Infra-sound measurements

½″ Free-fi eld Microphone, Infra-sound Type 40AZ

Fig. 1 ½″ Free-fi eld Microphone Type 40AZ (inset shows true size)

Specifi cations

...continued overleaf

Vers

. 18.

03.2

009

Frequency response:0.5 Hz - 20 kHz: . . . . . . . . . . . . . . . . . . . . . . . . .± 2.0 dB1 Hz - 10 kHz: . . . . . . . . . . . . . . . . . . . . . . . . . ± 1.0 dB

Resonant frequency:90 ° phase shift:. . . . . . . . . . . . . . . . . . . . . . . . . . 4 kHz

Nominal sensitivity:at 250 Hz:. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 mV/Pa

Polarization voltage: . . . . . . . . . . . . . . . . . . . . . . . .0 VDynamic range:

Upper limit (3% distortion): . . . . . . . .146 dB re. 20 µ Pa Microphone thermal noise . . . . . . . . 14 dB re. 20 µ Pa

Capacitance:Polarized:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 pF

Figure 4.1:G.R.A.S.Type 40AZ

The microphone chosen for this measurement system is a 12 -inch free-field

microphone, Type 40AZ made by G.R.A.S. shown in Figure 4.1. It ismade specifically for infrasound measurements but unfortunately the acous-tic lower limiting frequency is not stated in the data sheet directly. Thedata sheet states the -3 dB low-frequency limit of 0.2 Hz when the 40AZ isconnected to a preamplifier type 26CG with an input impedance of 40 GΩ.0.2 Hz could be the electric -3 dB low frequency limit because (3.1) yieldsexactly that value with Cm = 20 pF and Rin = 40 GΩ. But it is for surethat the acoustic frequency limit is equal to or below 0.2 Hz. The lowerlimiting frequency of a complete measurement chain is determined by thehighest of either the electric or acoustic lower limiting frequency. Anothercharacteristic of the 40AZ is that it is prepolarized i.e. the polarization volt-age is supplied internally by a charged material. This implies that the terminals wherethe preamplifier connects are 0 V with a sound signal on top.

Being a free-field microphone means that it is designed to have an electrical output volt-age that is proportional to the acoustic pressure which would exist at the position of thediaphragm in the absence of the microphone. The microphone is designed to compensatefor the effects of acoustic reflections at its diaphragm at an angle of incidence equal to zerodegree, which begin to matter above 1 kHz. Below that frequency wavelengths are longcompared to the microphone dimensions and therefore diffractions do not play a role.

The nominal capacitance of the microphone is 20 pF and it is rear-vented i.e. the equal-ization of atmospheric pressure occurs backwards or in the direction of the preamplifierconnected to it. The dynamic range of the microphone alone is limited by 3 % distortionat 146 dB and by thermal noise at 14 dB. This yields a dynamic range of 132 dB.

Each microphone has a serial number and the primarily used 40AZ in this project has111305. Each microphone also comes with a calibration chart which shows the frequencyresponse for frequencies above 250 Hz along with the sensitivity, S, which for 40AZ serial

4. Acoustic Design 42

111305 is 52.57 mV/Pa or -25.59 dB re. 1V/Pa. The calibration chart is found in AppendixB.

4.2 Capacitance and Voltage Variation

The microphone is a spacing-variation capacitive transducer meaning that the capacitancechange when the diaphragm moves in and out with respect to the rigid backplate. It is ofinterest to know how much the capacitance change and to begin with the voltage signalgenerated by a given sound pressure level (SPL) is calculated

Vs = 20 µPa · 10dBSPL/20 · S ·√

8 (4.1)

where dBSPL is the upper and lower limits of the dynamic range (146 dB and 14 dB) and√8 converts RMS values to peak-to-peak assuming sinusoids. This yields Vs ranging from

14 µVpp at 14 dB SPL to 56 Vpp at 146 dB SPL.

It is realized that an SPL of 146 dB is quite a lot for an audio measurement systemespecially one designed for noise measurements. Furthermore the chosen electrical designis limited by the maximum input voltage of the op-amp about ±12 V or 24 Vpp . Thiscorrespond to a maximum of 138 dB SPL which should be more than enough.

To calculate the change of capacitance, the relation from (2.3) is initially used to calculatethe charge of the microphone

Q = Vpol · Cm (4.2)

where Vpol is the polarization voltage of the permanently electrically charged material.With Cm = 20 pF and Vpol = 200 V (estimated)

Q = 4 nC (4.3)

Now the capacitance can be calculated using the relation

Cm =Q

Vpol ± 12Vs

(4.4)

The relation states that the capacitance to varies between 17 and 23 pF at 146 dB SPLresulting in a ∆Cm around 3 pF at max input. At 14 dB SPL the change of capacitanceis so small that it only makes sense to state the ∆Cm which is 0.7 aF (10−18 F).

4.3 Infrasound Calibration

Like any other measurement system a calibration is needed to prove a system’s perfor-mance compared to a known system. For infrasound measurement systems this is nodifferent, and the procedure is as follows. The ventilation which equalizes the slow vari-ations in atmospheric pressure has a great influence on the low frequency response. Thetime constant of the ventilation is a compromise between the frequencies you would liketo measure and the frequencies you do not (noise). Also the placement of the vent isimportant e.g. if the vent is exposed to the sound field or not.


A minor effect on the low frequency response is due to the ambient pressure causing theimpedance of the cavity to vary. The air stiffness of the microphone is relatively low com-pared to the stiffness of the diaphragm which causes the change of impedance to alter thelow frequency response [28]. This effect is not as pronounced on ground as when measuringat other altitudes like diving tanks or aircrafts.

Lastly the compression of air is usually assumed to be adiabatic meaning no heat is trans-ferred between the system and the surroundings. But at low frequencies it actually changesinto an isothermal compression which increases the sensitivity [28]. This effect is very im-portant below 1 Hz but not so often taken into consideration. Normally the transducer isthought as linear and inverse with respect to SPL but that implies that it is an adiabaticcompression of air which occurs in the microphone.

Also the preamplifier plays a significant role in the measurement chain and thereby thecalibration. The acoustic lower limiting frequency is dependent on the microphone, butthe system’s total lower limiting frequency contains both the acoustic and the electriclower limiting frequency. A way to extract the two frequencies is to measure the totalresponse and then compare to the electric which can reveal the acoustic lower limitingfrequency.

4.4 Leakage and Equalization

As mentioned Section 4.3 the lower limiting frequency is primarily determined by the ven-tilation. However, equalization can also occur through leaks in the construction. Leakagethrough the diaphragm seems strange but nevertheless diaphragms can have pin holes ofthe size of large molecules. A standard way to test the diaphragm is to flush it with heliumwhereby the very small molecules will penetrate any pin holes revealing them. Hydrogencan be used for better results due to the smaller molecular size but the risk of explosionis high. Also the assembly of the different parts of the microphone (see Figure 2.6) area great source of possible leakage. To ensure a completely airtight microphone all partsare carefully checked for flaws and oil is applied on the contact surfaces before assembly.These precautions will minimize leakage of air. However, it is not possible to make a 100% airtight construction.

Without ensuring a very airtight microphone the effect of a designed vent is difficult tocontrol. And a well controlled ventilation is required when operating at that low frequen-cies. The more controlled the vent is the more accurately the results will be since you knowexactly what frequencies you have measured. Sealing the construction of the microphoneby minimizing leakage makes the design and manufacturing of a vent more easy since ittakes less effort to ensure a certain total leakage. If a designer seeks a total equalizationof 1 unit, the amount left for the vent it self is made smaller and thereby more difficultwhen leakage elsewhere in the microphone is present. Designing a vent for frequencies inthe order of 10 mHz, is in such a small physical scale compared to the size of molecules.

In the chosen 40AZ microphone the vent is made by putting a spacer on top of the insula-tor and cutting a slit in the spacer which is illustrated in Figure 4.2 and 4.3. The spaceris 20-25 µm thick and measures 1.5 mm from the from inner to outer radius. In the endof the slit at the inner radius the back cavity is acoustically connected. And at the outer


radius the preamplifier connects and by its leaky construction the atmospheric pressure.The length of the slit equals the length of the vent and the cross section is determined bythe thickness of the spacer times the width of the slit. With a width of 90 µm the slit hasa cross section of 1.8 · 10−9 m2.



BE 1447 –11 2−9

which is mainly assembled by screwing the parts together, the integrated back-plateand insulator version is assembled by pressing the parts into each other. This de-sign also deviates from the conventional design by applying a backplate consistingof a metal thin-lm placed directly on the surface of the insulator.

In practice, the rst mentioned type implies more freedom for the designer to opti-mise the frequency response, while the second is advantageous during production.The main choice which must be made in respect to the two dierent design types isone of more narrow frequency response tolerances oered by the conventional de-sign, as opposed to reduced production costs for the alternative design.

2.3.3 Material and Process Requirements

A microphone which is to be used for measurements must be stable over time andits properties should preferably not vary with variations in ambient temperature,pressure and humidity. Therefore, carefully selected, high quality materials must beused, even if they are relatively dicult to machine.

The sensitivity of the microphone is inversely proportional to the diaphragm ten-sion. The tension must therefore be kept stable. Normally it is a requirement that ameasurement microphone has a broad frequency range and a high sensitivity. Thiscreates a requirement for light-weight diaphragms with high internal tension andthus a very high loading of the diaphragm material. This is achieved by applying atension of up to 600 N/mm 2 (which would break most materials) to the diaphragms

950573/1e

Insulator

Diaphragm

Backplate

Housing

(b)(a)

Insulator

Spacer

Air in/out

Air in/out

Figure 4.2: The vent of 40AZ is made by putting in a spacer (green) on top of the insulator andcutting a slit in the spacer (see Figure 4.3). Equalization occurs through the slit (blue).

wh

l

Figure 4.3: A close look of the spacer with a slit which makes equalization occur from the innerdiameter to the outer.


4.5 Modeling of Vent

The influence of the vent just presented will now be calculated. To start with the physi-cal or acoustic system parts are converted into analogous electric circuit elements wherevoltage is equal to pressure and current to volume velocity. This is called impedance anal-ogous circuits [29]. A simplified acoustic model of the vent system includes a volume anda resistance. Normally a tube (in this case the slit) models as a resistance and a mass butthe mass term is inverse proportional to frequency so for this ultra low frequency scenarioit is neglected. The acoustic system is illustrated in Figure 4.4 a. And the convertedanalogous electric circuit is illustrated in Figure 4.4 b.

RA

CACA

RA

a b

Figure 4.4: Model of the acoustic system (a) simplified to an acoustic volume (CA) anda vent for equalization of low frequencies (RA). The model is converted to the analogouselectric circuit (b).

The resistance of the vent, RA, is calculated with

RA =12 · η · lw3 · h

(4.5)

where η is the viscosity coefficient (ηair = 1.86 · 10−5 N ·sm2 ) and l, w and h are the length,

width and height of the vent respectively. The acoustic compliance of the air inside thevolume is calculated with

CA =V

ρ0 · c2(4.6)

where V is the volume of the cavity, ρ the density of air (ρ0 = 1.204 kg · m−3 at 20C)and c is the velocity of sound at 20C.

With the component values of the electric circuit defined it is easy to calculate the cutofffrequency with (3.1). The two missing values are the width of the slit and the volume ofthe cavity. None of them are known. But the data sheet of 40AZ states the volume of thefront cavity to 50 mm3 and a rough estimate is that the back cavity is 100 times larger[29] which yields V = 5 · 10−6 m3. From the measurements which will be presented inSection 5.2 the acoustic lower limiting frequency is 190 mHz. Reverse engineering yieldsa width of the slit of 0.09 mm.


4.6 New Vent Proposal

As mentioned designing a vent to achieve a well controlled cutoff frequency at ultra lowfrequencies is a very delicate business. In this section the present type of construction willbe analyzed with the purpose of finding dimensions which will meet the requirement whichis a -3 dB cutoff frequency below 10 mHz. An overlap of the low pass filter of the ventand the lower limiting frequency of the preamplifier is not desired. Therefore the designparameter for the lower limiting frequency of the vent is set to 5 mHz. In the end of thissection other possible ways of constructing a vent will be discussed.

An obvious way to start searching for modifications of the existing construction that willlead to a lower cutoff frequency is to start with the parts that are easiest to modify. Todecrease the cutoff frequency either the resistance of the vent or the compliance of thevolume needs to increase. The volume of the cavity is difficult to alter since it will requirea completely new microphone design. The vent on the other hand can be modified and forincreased resistance either the length of the vent can increase or the cross-section decrease(see (4.5)). And since the width is in power of 3 it should be clear that modifying this willhave greatest impact.

In Figure 4.5 the cutoff frequency is plotted with two changing parameters namely thewidth and length of the vent. The width is linearly decreased to 10 µm and the lengthincreased linearly to 33.5 mm. The choice of these numbers are not random. Laser cuttingis a common manufacturing process employed to cut many types of materials e.g. metal.The process can produce cuts with a kerf down to 1 µm in organic materials [30, 31] or 2µm in steel [32]. The kerf is the material removed by the laser beam hence the dimensionof the smallest possible slit. These specifications on laser cutting and a belief that lasercutting the spacer is possible support the choice of the variation of the width. The varia-tion of the length becomes clear when presenting the new vent in Figure 4.7.

Minimizing only the width of the slit to 26 µm shows that the -3 dB cutoff frequency isdecreased to 4.7 mHz (see data mark in Figure 4.5). This suggests that laser cutting thevent and maintaining the very simple construction will make it possible to alter the 40AZwith an acoustic lower limiting frequency which will match the electric lower limiting fre-quency. But it depends on the manufacturing process.

If the vent length is thought as a better parameter to tune an easy way to make it longeris to make a skew cut across the spacer illustrated in Figure 4.6. The maximum lengthachievable with the given dimensions of the spacer are calculated using the Pythagoreantheorem as follows

l =√r22 − r21 (4.7)

where r1 and r2 are the inner and outer radii of the spacer respectively

r2 =1

2· 11.7 mm = 5.9 mm (4.8)

r1 = r2 − 1.5 mm = 4.4 mm (4.9)

which yields

l = 3.9 mm (4.10)


10.026.0

42.058.0

74.090.0

1.57.9

14.320.7

27.133.5

10−5

10−4

10−3

10−2

10−1

100

X: 2.6e−005Y: 0.0015Z: 0.004734

l [mm]w [µm]

f [H

z]

Figure 4.5: -3 dB cutoff frequency of vent as function of width and length of the slit in the spacer.The current vent/slit dimensions represent the data point in the very top.

wh

l

Figure 4.6: A modified vent proposal where the vent is cut skew close to a tangent of the innerradius.


Another new vent proposal increasing the length of the vent tremendously is the construc-tion illustrated in Figure 4.7. A circular cut around the spacer would make it possibleto increase the length to a value close to the circumference of the spacer. The maximumlength is then close to

l = lold + c (4.11)

where c is the circumference of the spacer

c = 2πr = 2πr1 + r2

2= 2π · 5.1 mm = 32 mm (4.12)

which yields

l = 33.5 mm (4.13)

wh

l

Figure 4.7: A new vent proposal where the vent length is increased by letting the slit run alongthe circumference of the spacer.

This proposal depends even more on the manufacturing process because it is thought as achallenging task to cut a slit around the surface of such a small spacer. Without knowingthe manufacturing process by which the current slit is made it might be possible to usethe same process and hence the same slit width and cut the circular vent in the spacer.From Figure 4.5 in the most left data point an unchanged width and a length increasedto 33.5 mm yields a cutoff frequency of 8.8 mHz.

Designing vents on standard microphones with cutoff frequencies in the order of 1 Hz hasan important rule to obey. The vent must be shorter than one quarter of a wavelength.But with the low frequencies dealt with in this case this will never be a problem. Thewavelength of a 10 mHz tone is 100 m.

To round up this section it must be said that designing an acoustic vent for equalizationof the microphone is more a mechanical problem than an acoustic. That is especiallytrue with respect to the challenges. The acoustic challenge is rather straight forward; theequations yield the design parameters. The true challenge at this point is the physicalconstruction and what is possible and not. Engineers dealing with nanotechnology mighthave valuable information regarding this. Using laser cutting to make the slit in the spacerwill probably be able to lower the cutoff frequency sufficiently.


4.7 Consequences of an Airtight Microphone

Working with a microphone with a very slow equalization like the one being designedcomes with some precautions which must be known by the user. It is related directlyto the large time constant and the equalization of the pressure inside the cavity. Whenmeasuring, the atmospheric pressure must not change to fast. If this is the case thenthe diaphragm will find an inbound or outbound resting position which will compromisethe measurement system e.g. by introducing distortion. This requirement entails thatmeasuring while moving up or down with high speed is not allowed. This could be in anelevator, on an escalator, or during liftoff or landing with a spacecraft. The atmosphericpressure might change too fast for the vent to cope with the pressure difference. This mustbe considered when relevant.

This altitude problem also exists when the measurement system is not in use. If theoutside pressure is increasing very fast the pressure inside might pop out the diaphragmbefore the vent is able to equalize it. This means that when transporting the measurementsystem in fast changing pressure environments it needs to be more ventilated or sealed. Asolution to the problem could be a second vent which is activated during transportation.Or it could be an airtight transportation box. If the issue is not handled correctly onecould imagine that the microphone would break by running up a few stairs or taking afast elevator.

The precautions mentioned in Section 2.5 regarding infrasonic measurements will of coursealso be relevant for this system and for some of them more pronounced with an almostairtight microphone.

5

Measurements

This chapter will verify the performance of the design by presenting measurements bothelectric and acoustic and elaborating on these. The measurements will be compared tothe specifications stated in Section 1.1.

For most measurements both versions of the preamplifier are measured, and to summarizeversion 1 incorporates a high pass filter and a capacitive voltage divider in the feedbackwhereas version 2 has a resistive voltage divider alone. Besides the feedback circuit bothversions are alike.

5.1 Electric System

The most important design parameter of the measurement system is the lower limitingfrequency. Electrically this is verified by measuring the frequency response with a sinusoidsweep on the input. The measurement is conducted on a Dynamic Signal Analyzer SR785by Stanford Research Systems which is noted for having a generator able to produce sig-nals down in the ultra low infrasonic frequency range. The output signal of the analyzer isconnected through an input adapter serving as a capacitive microphone (see Figure 5.2).The input adapter has a BNC connector at one terminal, a female microphone thread atthe other and a 20 pF capacitor in between which is equal in value as the nominal ca-pacitance of the microphone. Even though the capacitance does not change with voltagelevel like the microphone does the measurement gives a good approximation to the electriclower limiting frequency.

In Figure 5.1 the measurements are shown along with the simulations. Version 1 of thepreamplifier shows a -3 dB cutoff frequency of 10 mHz and version 2 something similar.Unfortunately the number of measured data points are in some cases kept at a minimumand in this case not far enough down in frequency. This is because every data point takesa very long time to measure. The analyzer uses one period of the signal to settle the leveland at least one period to measure. Ideally more periods should be averaged to makethe measurement more precise, but at 10 mHz one period is 100 seconds resulting in 3minutes for one single data point. Doing a sweep from 100 Hz to lets say 1 mHz with 30data points will take hours.

5.1.1 Damping

Another important parameter of any preamplifier is the overall attenuation or the damp-ing. Ideally the gain should be 0 dB meaning no damping as the op-amp is coupled as avoltage follower. But as described the gain is not 0 dB due to feedback and nonidealities.

5. Measurements 52

10−2

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0Frequency response of preamplifier with 20 pF input adapter

Frequency [Hz]

Am

plitu

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RM

S]

Measurement: Version 1Measurement: Version 2Simulation: Version 1Simulation: Version 2

Figure 5.1: Frequency response of preamplifier with 20 pF input adapter. The simulation isincluded for comparison. The electric lower limiting -3 dB corner frequency is clearly around 10mHz for both versions of the preamplifier.

The evaluation of the damping is done with a pistonphone, a calibrated microphone andcalibrated preamplifier.

Figure 5.2: Input adapter for supplyingelectric input to the preamplifier. A 15 pFversion is depicted but a 20 pF also exist.

Figure 5.3: Vent adapter used to seal orequalize the microphone. Both constructionsare airtight from front to back. The adapteron the left has a hole which equalizes themicrophone whereas the adapter to the rightseals the microphone.

The arrangement is that a known and calibrated preamplifier and microphone is used asreference and then the same calibrated microphone is used on the unknown preamplifier(both version 1 and 2). The pistonphone produces a 250 Hz tone at 114 dB re. 20 µPa

5. Measurements 53

equal to 10 Pa. The sound signal is connected to the microphone through the couplerof the pistonphone and is converted into a voltage by the microphone. This voltage iscalculated by the sensitivity stated in the calibration chart and fed to the input of thepreamplifier. The output of the preamplifier is connected to the SR785 analyzer whichyields a total damping at 250 Hz (third octave band) of the total measurement chain.

The pistonphone is a G.R.A.S. 42AP (serial 68449) with build-in barometer showing thenecessary correction for the atmospheric pressure, which for all mentioned measurementsis -0.04 dB. The known preamplifier is a G.R.A.S. 26CA (serial 122012) and with thesetup just described the analyzer reads -5.9 dB at 250 Hz. The calibration chart of themicrophone states a sensitivity of -25.59 dB re. 1 V/1 Pa equal to -5.59 dB re. 1 V/10 Pa(see Appendix B) which is exactly as referenced to the output of the pistonphone. Theknown preamplifier is therefore accounting for −5.9 − (−5.59) − 0.04 = −0.35 dB gainwhich is close to -0.25 dB stated in its calibration chart.

The preamplifier version 1 is then placed in the measurement chain replacing the knownpreamplifier and the analyzer yields a total damping at 250 Hz of -6.35 dB. That leavesthe preamplifier with a gain of -0.80 dB.

For version 2 the damping of the total measurement chain reads -6.64 on the analyzerwhich yields a gain of -1.09 dB.

5.1.2 Start-up

Another important parameter of the preamplifier is the start-up described in Section 3.9.The switch connects a resistor minimizing the total resistance in the feedback circuit. Toverify the effect of this start-up circuit the output signal during start-up is measured andcompared to simulations.

The setup was quite easy but the execution of the measurement was difficult. To be ableto compare one measurement to another it would be favorable for the voltage level to startat the same value every time. Unfortunately the level at the input reaches a random levelat start-up. A rather fixed voltage was introduced to the input through the 20 pF inputadapter (see Figure 5.2). Measuring the time signal on a Tektronix TDS2024B digital os-cilloscope shows the measurements in Figure 5.4. Without the switch being activated thesystem show a very long settle time of 220 seconds. Simulations show a bit longer whichis possibly due to the real resistances implemented by the feedback circuit and the biascircuitry. Simulations conducted but not shown show faster settling times with decreasingfeedback resistance. The scenario where the switch is activated on the other hand showsvery fine correspondence between simulation and measurement. Both have a settling timeof around 25 s which proofs the effect of the start-up circuit.

The frequency response is dramatically altered when the switch is activated which is shownin Figure 5.5. When the switch is activated the lower limiting frequency of the preamplifierincreases to around 1 Hz. If the switch did not have an effect on the frequency responseit could be argued why not connecting the resistor permanently. The measurement isconducted on the first prototype of the preamplifier which has a 16 µF capacitor and a10 MΩ resistor in the feedback. The switch interconnects a 10 kΩ resistor in parallel tothe existing causing the cutoff frequency of the feedback circuit to increase from 1 mHz to1 Hz. At frequencies below the cutoff frequency of the feedback circuit the bias circuitry

5. Measurements 54

0 50 100 150 200 250 300−0.5

0

0.5

1

1.5

2

Time [s]

Vol

tage

[V]

Offset with step on input, preamp version 1

Measurement: switch offMeasurement: switch onSimulation: switch offSimulation: switch on

Figure 5.4: DC offset of preamplifier version 1 with step on input through 20 pF input adaptershows the system’s settle time.

does not behave like a resistance because the voltage across it is beyond the diode limita-tions mentioned in Section 3.3.

10−4

10−2

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102

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−9

−8

−7

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−3

Frequency [Hz]

Am

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B]

Frequency response with switch on and switch off

Switch offSwitch on

Figure 5.5: Frequency response measurement of first prototype showing 2 sweeps, one with theswitch off (normal operation) and one with the switch on, meaning a lower value resistor is con-nected in the feedback.

5. Measurements 55

5.1.3 THD and Noise

Measuring the output signal for a given input signal tells a lot about the performance ofthe preamplifier. The noise can be estimated along with total harmonic distortion (THD).In the following measurements a Fast Fourier Transform (FFT) is done on the output sig-nal when a sinusoid is on the input through the 20 pF input adapter. The measurementsare done with a Rohde & Schwarz UPV Audio Analyzer which generates the sinusoid,feeds it to the input adapter and reads the output signal from the preamplifier. The UPVAudio Analyzer is made for regular audio and hence the generator is not designed to pro-duce ultra low frequencies, but it extends down to 100 mHz. The analyzer can measuredown to DC but unfortunately the measurements where conducted measuring only downto 200 mHz. And also the acquisition time was to short resulting in the data points at lowfrequencies being of little value. At the time when this was realized the UPV had beenreturned to Rohde & Schwarz from whom it had been borrowed. The Stanford SR785was used but trouble with windowing and the fact that the number of FFT lines is lim-ited to 800 made the measurements difficult to conduct and the results not convincing.The measurements in Figure 5.6 show the output of the preamplifier with a 1 V sinusoidat 1 kHz on the input. Both version 1 and 2 of the preamplifier are measured with the UPV.

100

101

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0

Frequency [Hz]

Am

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FFT of output signal with 1 kHz sine as input

Version 1Version 2

Figure 5.6: FFT of output signal with 1 V sine at 1 kHz as input.

From both measurements the noise of the preamplifier discussed in Section 3.10 can beseen. The noise corner frequency is about 500 Hz, where pink noise (below fc) is replacedby white noise (above fc). Pink noise has a slope of approximately -10 dB/decade andwhite noise is flat. Also the harmonics of the input signal can be seen with the levels beingquite small for this input voltage level.

The measurements can also be used to estimate the dynamic range of the preamplifier.

5. Measurements 56

Figure 5.7 will support the explanation on how to calculate the signal to noise ratio (SNR)from an FFT. The dynamic range is comparable to the maximum achievable SNR becausethe noise is constant with respect to input voltage and with the signal increasing to itsmaximum leads to the definition of the dynamic range being the ratio of the maximumoutput to the minimum output.

The amplitude of the frequency bins containing the signal are squared and summed inorder to get the power of the signal, Ps. The remaining frequency bins are also squaredand summed to get the power of the noise, Pn. The FFT gain, AFFT, in dB also needs tobe subtracted which is equal to dividing by the number of frequency bins in the sample.In math this looks like

Pn =

f2∑k=f1

A2k (5.1)

Ps =

fs+b∑k=fs−b

A2k (5.2)

where f1 and f2 is the frequency range, Ak the amplitude of frequency bin k and b indicatesthe number of neighboring bins to include in the signal to get all the power. To calculatethe SNR

SNR = 10 · log(Ps

Pn

)−AFFT (5.3)

where

AFFT = 10 · log(ktotal) (5.4)

With the data from Figure 5.7 and the relatively low input voltage of 1 V the SNR is -95dB. Which is an approximation because the noise power is not sufficient due to the lowacquisition time and lower frequency limit of 200 mHz.

To get a better estimation of the dynamic range of the preamplifier the output voltageis measured when it reaches 3 % distortion. This along with a better estimation of thetotal noise voltage will result in the dynamic range. The input voltage is increased until3 % distortion is reached on the output voltage. For version 1 the output voltage is 7.6Vpp and for version 2 it is 28 Vpp. The measurements are shown in Figure 5.8 and 5.9. Itcame as a surprise that version 1 reached distortion at so low voltages. And the reasonis probably that the attenuation in the preamplifier makes the voltage across the biascircuitry increase and thereby the equivalent resistance decreases. From Figure 5.8 and5.9 the output voltages are indicated, and with the input voltages being 8.6 Vpp and 33Vpp respectively, the voltage across the bias circuit is more than 1 V. The reason why itdoes not happen with version 2 is that the feedback capacitor in version 1 of 3.3 nF has aresistance of 48 kΩ at 1 kHz. That resistance is much smaller than the 10 GΩ which is inthe feedback circuit of version 2.

The noise voltage is estimated by a measurement on the SR785. The white noise voltagedensity is around -125 dBVrms/

√Hz equal to 560 nVRMS/

√Hz. The noise corner fre-

quency is estimated to 500 Hz from Figure 5.7. The noise of the preamplifier is calculated

5. Measurements 57

100

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Am

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FFT of output signal with 100 Hz sine as input

Version 1Version 2Signal

Figure 5.7: FFT of output signal with 1 V sine at 100 Hz as input. The signal is marked red toaid the calculation of the dynamic range.

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20

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Am

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Version 1

Figure 5.8: FFT of output signal of ver-sion 1 with 1 kHz sine as input. 3 % dis-tortion on the output is reached at outputvoltage of 7.6 Vpp

102

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20

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Version 2

Figure 5.9: FFT of output signal of ver-sion 2 with 1 kHz sine as input. 3 % dis-tortion on the output is reached at outputvoltage of 28 Vpp

using (3.26) which yields 92 nVRMS. Calculating the ratio of the maximum output voltageand the noise voltage yields the dynamic range of version 1 of 84 dB and 94 dB for version 2.

To round up this section on measurements of dynamic range it would be of interest tomake a new measurement with the UPV. The acquisition time should be at least 100 s forthe slow 10 mHz signals to be sensed, and the analyzer should measure down to 10 mHz.But the two estimates of the dynamic range of the preamplifier show believable values yetfar from the theoretical value of 124 dB calculated in Section 3.10.

5. Measurements 58

5.2 Frequency Response of Entire System

Measuring the frequency response of both the microphone and preamplifier is the mostobvious and important verification of the system. The measurement system is supposedto operate in its entirety and not in parts separately. The measurement is done with a lowfrequency calibrator G.R.A.S. 42AE which is an acoustic source using a constant force toensure a constant sound pressure level inside the calibration chamber. The microphoneattached to the preamplifier is fitted into one of the holes down into the coupler, so boththe diaphragm and the vent of the microphone are exposed to the sound field inside thecoupler. This ensures that the measurement setup (see Figure 5.10) includes the totalsystem and measures both the electric and the acoustic lower limiting frequencies. Thecalibrator ensures a constant sound pressure level down to very low frequencies about 0.01Hz and is in theory only limited by any air leakage of the system. The sound pressure levelis determined by the input signal which is supplied by the SR785 Analyzer. The outputsignal of the preamplifier is also returned to the SR785.

Figure 5.10: G.R.A.S. 42AE low frequency calibrator with the microphone including vent insidethe coupler.

The first measurements presented in Figure 5.11 show the low frequency response withseveral setups. To make sure the vent of the microphone is exposed to the sound pressurein the coupler and not equalized to the ambient pressure a vent adapter like shown inFigure 5.3 was mounted between the microphone and the preamplifier. The first adapterused is the closed vent adapter which seals the measurement setup. The inside of themicrophone is equalized to the sound pressure of the coupler through the thread connect-ing the microphone to the preamplifier which is not tightened. The equalization occursinto the microphone through the vent and not through the preamplifier because the ventadapter is sealed. To verify that venting through the thread was sufficient the open ventadapter was made. This adapter is like the other airtight back towards the preamplifier,but it has a hole on the side (a large vent) equalizing the cavity.

5. Measurements 59

Also shown in Figure 5.11 are measurements with two other 40AZ microphones. One ofthe three 40AZ microphones show a -3 dB lower limiting frequency of 70 mHz. The micro-phone used through the entire project along with one other 40AZ have a -3 dB frequencyof 190 mHz. This cutoff frequency is determined by the vent in the microphone and as weshall see in the next section the mounting of the microphone to the preamplifier can havesignificant impact on the total response.

10−2

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0Frequency response of preamplifier version 1 with 40AZ

Frequency [Hz]

Am

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Mic not tightened to closed vent adapterMic not tightened to closed vent adapterAnother 40AZ (Serial: 111320) not tightenedAnother 40AZ (Serial: 111312) not tightenedMic on open vent adapter

Figure 5.11: Measurements with version 1 showing consistency across several microphone car-tridges and no or little difference whether the microphone is loosely mounted on the closed ventadapter or mounted with the open vent adapter. Lastly the measurements verify the entire system’slower limiting frequency of around 190 mHz with a standard off-the-shelf 40AZ microphone.

The measurements of version 2 of the preamplifier are shown in Figure 5.12 and againthe acoustic lower limiting frequency is 190 mHz. When sealing the setup with oil inthe thread the cutoff frequency lowers to 8 mHz. The ripple on the closed vent adaptermeasurement can be due to overload. Auto range of sensitivity is disabled on the SR785analyzer which might allow low frequency spikes and other disturbances.

5.2.1 Microphone Mounting

The mounting of the microphone to the preamplifier has a great impact on the response.Like discussed in Section 4.4 leakage can occur many places in the microphone construc-tion but also the connection to the preamplifier is of importance since the vent equalizedbackwards into the cavity between the microphone and the preamplifier. The more air-tight the connection is the lower the acoustic lower limiting frequency becomes. This isillustrated by the measurements in Figure 5.13.

5. Measurements 60

10−3

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Am

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Frequency response of preamplifier version 2 with 40AZ

Open vent adapterClosed vent adapter and oil

Figure 5.12: Measurements with version 2 and 40AZ microphone. With open vent adapter the -3dB cutoff frequency is 190 mHz and with closed vent adapter and oil it is 8 mHz.

When the microphone is loosely mounted to the preamplifier the ventilation occurs throughthe vent and through the thread. This results in the cutoff frequency just below 200 mHz.Tightening the thread as much as possible by hand results in a cutoff frequency decreasedto 90 mHz. The last measurement shows the lower limiting frequency when the contactsurface between the microphone and the vent adapter is lubricated with oil to ensure noventilation. This decreases the -3 dB frequency to 6 mHz. With this measurement allfrequencies are attenuated 0.3 dB which is due to unequalized pressure explained next.

Before the just mentioned lubrication with oil another measurement was made where theoil was applied on the thread and not on the contact surface. This had the effect that thecavity of the microphone was sealed before the thread was screwed on all the way. Andwhen screwing on the microphone the enclosed air inside the cavity made the pressure riseand the diaphragm establish a new equilibrium position. The phenomenon is illustratedin Figure 5.14 where this introduces 4 dB attenuation because of overpressure in the mi-crophone cavity.

To summarize the results of the entire measurement system it is clear that the 10 mHzlower limiting frequency can be achieved when the microphone is sealed. The electriclower limiting frequency is in this case the limiting parameter. When the microphoneis vented the limiting parameter of the entire measurement system is the acoustic lowerlimiting frequency which is 190 mHz. Sealing of the microphone introduces precautionsto be taken with respect to overpressure in the microphone cavity. This occurs whenthe microphone is screwed on with oil in the contact surface and when the atmosphericpressure changes due to altitude changes or weather changes.

5. Measurements 61

10−3

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0Frequency response of preamplifier version 1 with 40AZ

Frequency [Hz]

Am

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RM

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Loose mountedTightenedSealed with oil

Figure 5.13: Frequency response of entire system consisting of preamp v1 and 40 AZ showing theimportance of ventilation. The measurements are conducted with the closed vent adapter.

10−3

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0Introduced attenuation due to overpressure in the microphone cavity

Frequency [Hz]

Am

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Oil along threadOil on contact surface

Figure 5.14: Overpressure inside the microphone cavity results in 4 dB attenuation. The over-pressure occurred when oil was applied along the thread and not only on the contact surface.

6

Conclusion

Measurement of infrasound is important because noise at frequencies below 20 Hz is au-dible thus most commonly tried avoided at high levels. And to avoid noise in a designprocess or in an established setup it is advantageous to be able to measure it. Sources ofinfrasound include wind turbines which are best placed where the generated noise leveldoes not disturb or damage the environment and nearby living people. Other infrasonicsources containing ultra low frequencies are the sonic boom - an N-shaped shock waveformed from a supersonic flight. Modifications of the aircraft shape makes it possible toreshape the shock wave so flights in land zones will be allowed.

The noise measurement systems widely available typically measures down to 1 Hz whichis not sufficient for measurement of ultra low frequency infrasound. Some systems existmeasuring down into the mHz range but they have very poor dynamic range of about40 dB. The design of a measurement system capable of measuring down in the ultra lowfrequencies with good dynamic range has been presented. The system consisting of a con-denser microphone and a preamplifier will aid stakeholders in the wind farm noise debateand sonic boom shapers.

The acoustic part of the measurement system has been analyzed and a new vent configu-ration of the G.R.A.S. 40AZ 1

2 -inch microphone has been proposed. Acoustically the lowerlimiting frequency can be pushed to 10 mHz by either sealing the microphone resultingin numerous precautions to take when measuring. Another solution is the modify theequalization vent by increasing its length or decreasing its cross section area.

Electrically the system connects the capacitive microphone directly to an op-amp resultingin an electric lower limiting frequency determined by the capacitance of the microphoneand the input impedance of the preamplifier. The design parameter of 10 mHz and a20 pF microphone capacitance yields an input resistance of 1 TΩ which is obtained bya OPA129 op-amp. The bias circuit preventing a floating input voltage due to leakagecurrent in or out of the op-amp is constructed with two diode-connected FETs. They em-ulate a sufficiently large resistance which in parallel with the input impedance of OPA129determines the total input impedance of the preamplifier and thereby the electric lowerlimiting frequency of the system.

The challenges encountered in the design was the mentioned bias circuitry which plays animportant role on the lower limiting frequency. Also the feedback circuit was presented infour alternative ways which had disadvantages as peaking below the cutoff frequency andunwanted attenuation through the preamplifier.

6. Conclusion 64

The final prototype have been measured and the target specifications of the design statedin Section 1.1 have been fulfilled. For comparison they are stated in Table 6.1.

Table 6.1: Target specifications and obtained specifications of the measurement system

Target Obtained Note-3 dB lower limiting frequency 10 mHz 8 mHz When microphone is sealedDynamic range 80 dB 94 dB With version 2

6.1 Future Work

During the project a number of aspects have come up which were out of the scope of thisproject. Further research could go in the direction of any of them which would eitherbenefit this specific measurement system or benefit the concept of measuring ultra lowfrequency infrasound in general.

Choosing a condenser microphone as the sensor is obvious since they exhibit great specifica-tions with respect to e.g. humid conditions. These specifications characterizing condensermicrophones would be natural to test on the preamplifier. It would be good to performenvironmental tests such as performance under humid conditions. This way not only themicrophone but the entire measurement system can be characterized with high quality.

Originally the system was supposed to deliver a digital output signal for easy connectionto e.g. a computer. This was given a low priority on the cost of unforeseen challenges.It could be a plus to the measurement system to incorporate an analog to digital converter.

The offset performance of the preamplifier has not been investigated. A DC offset candegrade the performance of both the microphone linearity and the preamplifier maximumallowed output voltage thus the dynamic range. Using an in-amp e.g. IN116 show betteroffset performance and this should be investigated. An offset correction circuitry couldalso be designed for the current OPA129.

The start-up circuit of the preamplifier ensuring quick discharge of the input nodes of theop-amp is implemented using a switch. No indication exist to show when the measurementsystem is stabilized and ready for use. This could be solved by a light emitting diode andmaybe a sensing circuit indicating when the system is ready.

Lastly specifically on this design it would be of interest to see the performance of theentire measurement system on real infrasound sources. It was originally intended to es-tablish a setup at a wind turbine and measure real data. But realizing that infrasoundmeasurement is a challenging task in it self and the fact that the system only measuresdown to 10 mHz when the vent is closed with precautions in result, the field measurementwas deemphasized.

On a more general level looking into the manufacturing processes involved with the de-sign of microphone vents would be interesting. Laser cutting has been mentioned but itshould be investigated what it is capable of manufacturing. To a start the proposed ventconfigurations presented should be tested.

References

[1] Geoff Leventhall. Low frequency noise. what we know, what we do not know, andwhat we would like to know. Low Frequency Noise and Vibration and its Control,2008.

[2] Preparatory Commission for the Comprehensive Nuclear-Test-Ban Or-ganization. http://www.ctbto.org/verification-regime/monitoring-technologies-how-they-work/infrasound-monitoring/page-1/.

[3] Bruel & Kjær A/S. Microphone handbook, volume 1: Theory, 1996. BE 1447 –11.

[4] P. V. Bruel and W. J. Parker. The condenser microphone and some of its uses inlaboratory investigations. Electro-Acoustic Group, 1963.

[5] Allan J. Zuckerwar and William W. Shope. A solid-state converter for measurementof aircraft noise and sonic boom. IEEE Transactions on Instrumentation and Mea-surement, 1974.

[6] Jeffrey B. Johnson, Jonathan M. Lees, and Hugo Yepes. Volcanic eruptions, lightning,and a waterfall: Differentiating the menagerie of infrasound in the ecuadorian jungle.Geophysical Research Letters, 33, 2006.

[7] Howe Gastmeier Chapnik Limited (HGC Engineering). Wind turbines and infrasound.Canadian Wind Energy Association, 2006.

[8] Frits van den Berg. Low frequency noise can be a phantom sound. Low FrequencyNoise and Vibration and its Control, 2008.

[9] Christian Sejer Pedersen, Henrik Møller, and Kerstin Persson Waye. Low-frequency-noise complaints: an investigation of twenty-one cases. Low Frequency Noise andVibration and its Control, 2008.

[10] Takanao Sugimoto, Kenji Koyama, Yosuke Kurihara, and Kajiro Watanabe. Measure-ment of infrasound generated by wind turbine generator. SICE Annual Conference,2008.

[11] Kenneth J. Plotkin. Sonic boom: From bang to puff. Echoes, The newsletter of TheAcoustical Society of America, 20(3), 2010.

[12] Larry K. Baxter. Capacitive Sensors Design and Applications. IEEE Press, 1997.

[13] Bruel & Kjær A/S. Condenser microphones and microphone preamplifiers for acousticmeasurements, data handbook, 1982. BP 0100, 2-044 01 00-2A.

http://www.ctbto.org/verification-regime/monitoring-technologies-how-they-work/infrasound-monitoring/page-1/



References 66

[14] Gunnar Rasmussen and Kim M. Nielsen. Low frequency calibration of measurementmicrophones. Low Frequency Noise and Vibration and its Control, 2008.

[15] Benoit Alcoverro and Alexis Le Pichon. Design and optimization of a noise reduc-tion system for infrasonic measurements using elements with low acoustic impedance.Acoustical Society of America, 2004.

[16] Jelena Citakovic Haas-Christensen. New Technology-Driven Approaches in the Designof Preamplifiers for Condenser Microphones. PhD thesis, Technical University ofDenmark, 2009.

[17] David A. Bell. Electronic Instrumentation and Measurements. Prentice Hall Career& Technology, Englewood Cliffs, New Jersey, USA, 2nd, edition, 1994.

[18] Bernard M. Oliver and John M. Cage. Electronic Measurements and Instrumentation.McGraw-Hill Book Inc., 1971.

[19] M.Sc. Ole-Herman Bjor Senior Scientist. Norsonic as, norway, 2011. http://www.norsonic.com.

[20] Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits, International StudentEdition. Oxford University Press, Inc., New York, NY, USA, 5th edition, 2004.

[21] Charles Kitchin and Lew Counts. A Designer’s Guide to Instrumentation Amplifiers.Analog Devices, Inc., 3rd edition, 2006.

[22] Burr-Brown Products from Texas Instruments. Datasheet of texas instruments opa129ultra-low bias current difet operational amplifier, 2007. SBOS026A.

[23] Ole Jannerup and Paul Haase Sørensen. Reguleringsteknik. Polyteknisk Forlag, 4thedition, 2006.

[24] Claus Erdmann Furst. A low-noise/low-power preamplifier for capacitive micro-phones. IEEE, 1996.

[25] Ron Mancini. Op amps for everyone. Design Reference, Texas Instruments,SLOD006B, August 2002.

[26] Texas Instruments. Noise analysis in operational amplifier circuits. Application Re-port, SLVA043B, 2007.

[27] Robert Tarzwell. Xtreme resistance and impedance circuits. DMR LTD, February2009.

[28] Erling Frederiksen. Low frequency calibration of acoustical measurement systems.Bruel & Kjær Technical Review No. 4 1981, 1981.

[29] Jr. W. Marshall Leach. Introduction to Electroacoustics and Audio Amplifier Design.Kendall/Hunt Publishing, 3rd edition, 2003.

[30] Nick Rau et al. Limits of very small manufacturing processes. http://inst.eecs.berkeley.edu/~ee245/fa97/small.html, 2011.

[31] Wikipedia. Excimer laser. http://en.wikipedia.org/wiki/Excimer_laser,2011.

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http://www.engineersedge.com/manufacturing/laser_cutting.htm

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Appendix

A Various Matlab scripts

cap change.m - calculating change of capacitance of the microphone

1 % How much does the capacitance of the microphone change during operation?2 clc; clear all;3 format longEng;45 % Microphone specific data6 dyn_max = 146 % Maximum dB SPL (where distortion takes over)7 dyn_min = 14 % Minimum dB SPL (where noise floor takes over)8 sens = 50e-3 % Sensitivity in V/Pa9 cap = 20e-12 % Capacitance when charged in F

10 polV = 200 % Polarization voltage in V1112 % Voltage swing (Vpp)13 Vpp_max = 20e-6*10^(dyn_max/20)*sens*sqrt(8) % sqrt(8) requires sinusoid14 Vpp_min = 20e-6*10^(dyn_min/20)*sens*sqrt(8) % sqrt(8) requires sinusoid1516 % Charge when polarized17 Q = cap*polV1819 % Capacitance when max outswing (plus/minus half of Vpp)20 Cmax = [Q/(polV+Vpp_max/2) Q/(polV-Vpp_max/2)]2122 % Capacitance when min outswing (plus/minus half of Vpp)23 Cmin = [Q/(polV+Vpp_min/2) Q/(polV-Vpp_min/2)]2425 % Capacitance change26 deltaC_max = Cmax-cap27 deltaC_min = Cmin-cap2829 %%%% Result %%%%30 % with input of 146 dB SPL cap changes from 17.5 pF to 23.3 pF31 % with input of 14 dB SPL cap changes 0.7 aF - 10^(-18)3233 % Capacitance of a microphone34 eps0=8.85e-12; % Permittivity of vacuum35 d=20e-6; % Plate distance [m]36 A=45e-6; % Capacitor plate area [m^2]37 Cm = eps0*A/d

Appendix 70

mic vent.m - calculations of vent design

1 clc; clear all;2 close all;3 format longEng;45 rho0 = 1.204; % Density of air at 20C6 c = 343; % Velocity of sound at 20C7 eta = 1.86e-5; % Viscosity coefficient of air at 20C and 0.76 m Hg89 Vf = 50e-9; % Volume of front cavity [m^3] (ref. datasheet)

10 V = 100*Vf; % Volume of back cavity [m^3] (ref. Leach)1112 % Spacer dimensions13 w = 90e-6; % Width of slit in spacer [m]14 h = 20e-6; % Thickness of spacer [m]15 l = 1.5e-3; % Length of slit in spacer [m]1617 l = linspace(l,33.5e-3,11)’; % Increasing the length18 w = linspace(w,10e-6,11); % Decreasing the width1920 C = V./(rho0*c.^2); % Acoustic compliance of back cavity [m^5/N]21 % R = [8*eta*l./(pi.*a.^4)]’ % Acoustic resistance of vent (tube)22 % R = [12*eta*l*1./(w.^3*h)]’; % Acoustic resistance of vent (slit)2324 % Plots f as function of w and l25 [W, L] = meshgrid(w,l);26 R = [12*eta*L./(W.^3*h)]; % Acoustic resistance of vent (slit)27 f = 1./(2*pi*R*C);2829 figure1 = figure;30 axes1 = axes(’Parent’,figure1,’ZScale’,’log’);31 view(axes1,[-125 18]);32 box(axes1,’on’);33 grid(axes1,’on’);34 hold(axes1,’all’);35 surf(W,L,f,’Parent’,axes1); % Plots 3D36 colormap hsv % Determines colors37 xlim([min(W(1,:)) max(W(1,:))])38 ylim([min(L(:,1)) max(L(:,1))])39 xlabel(’w [\mum]’)40 ylabel(’l [mm]’)41 zlabel(’f [Hz]’)42 % Defining labels43 xl=fliplr(W(1,:));44 xl=xl(1:2:end); % Selects every second for label45 yl=L(:,1);46 yl=yl(1:2:end); % Selects every second for label47 xl = sprintf(’%3.1f|’,xl*1e6);48 yl = sprintf(’%3.1f|’,yl*1e3);49 if ispc50 xl = strrep(xl, ’e-00’, ’e-’);51 yl = strrep(yl, ’e-00’, ’e-’);52 end53 % Defining ticks54 xt=fliplr(W(1,:));55 xt=xt(1:2:end);56 yt=L(:,1);57 yt=yt(1:2:end);58 set(gca,’XTick’,fliplr(W(1,:)))59 set(gca,’YTick’,L(:,1))60 set(gca,’XTick’,xt)61 set(gca,’YTick’,yt)62 set(gca,’XTickLabel’,xl)63 set(gca,’YTickLabel’,yl)6465 % Max length of skew slit in spacer66 r2 = 11.7e-3/2; % Outer radius of spacer67 l_old = 1.5e-3; % Difference between inner and outer radius68 r1 = r2-l_old; % Inner radius of spacer

Appendix 71

69 l_skew = sqrt(r2^2-r1^2); % Max length of slit with skew cut7071 % Max length of circular slit in spacer72 r = (r1+r2)/2; % Radius in the midle of the spacer73 circ = 2*pi*r; % Circumference of the slit74 l_circ = l_old+circ; % Max length of slit with circular cut

sys analysis v1b.m - transfer function of circuit of first prototype

1 % This circuit is as of April 4th 2011 - version 1. This has a simple high2 % pass filter in the feedback.3 clc; clear all;4 close all;5 format longEng;67 f = [[1e-3:1e-6:1e-1] [1e-1:1e-3:1] [1:1e2:100e3]];8 omega = f*2*pi; % Angular frequency9 s=i*omega; % Laplace transformed frequency

10 Cm=20e-12; % Capacitance of microphone11 Rb=500e9; % Equivalent resistance of bias circuit12 Cf=16e-6; % Feedback capacitor13 Rf=10e6; % Feedback resistor1415 omegaN=(1/(Cf*Rf*Cm*Rb))^(1/2); % Natural frequency of system16 fn=omegaN/(2*pi) % In Hertz17 zeta=0.5*omegaN*Cm*(Rf+Rb) % Damping of system1819 % Poles and zeros of system (calculated using the bottom three lines)20 pole1 = -(Cm*Rb - (Cm^2*Rb^2 + 2*Cm^2*Rb*Rf + Cm^2*Rf^2 - 4*Cf*Cm*Rb*Rf)^(1/2) + Cm*Rf)

/(2*Cf*Cm*Rb*Rf)21 pole2 = -((Cm^2*Rb^2 + 2*Cm^2*Rb*Rf + Cm^2*Rf^2 - 4*Cf*Cm*Rb*Rf)^(1/2) + Cm*Rb + Cm*Rf)

/(2*Cf*Cm*Rb*Rf)22 zero1 = 023 zero2 = -(Rb + Rf)/(Cf*Rb*Rf)2425 % Transfer function written on standard form26 H = (s.^2*Cf*Rf*Cm*Rb + s*Cm*(Rf+Rb)) ./ (s.^2*Cf*Rf*Cm*Rb + s*Cm*(Rf+Rb) + 1);27 % System in form for nyquist plot28 sys=tf([Cm*Rb*Cf*Rf Cm*(Rf+Rb) 0],[Cm*Rb*Cf*Rf Cm*(Rf+Rb) 1])29 nyquist(sys)30 hold all;31 p1=plot(pole1,’bx’, ’LineWidth’,2, ’MarkerSize’,10)32 p2=plot(pole2,’rx’, ’LineWidth’,2, ’MarkerSize’,10)33 z1=plot(zero1,’bo’, ’LineWidth’,2, ’MarkerSize’,10)34 z2=plot(zero2,’ro’, ’LineWidth’,2, ’MarkerSize’,10)35 legend([p2 z2],’Poles’,’Zeros’,4)36 grid37 % Saving a pdf38 set(gcf, ’PaperUnits’, ’centimeters’); set(gcf, ’PaperSize’, [15 10]);39 set(gcf,’PaperPosition’,[0 0 15 10]);40 print(gcf, ’-dpdf’, ’-r300’, ’../Latex/graphics/peaking_nyquist.pdf’);4142 figure()43 subplot(2,1,1);44 semilogx(f,20*log10(abs(H)))45 grid on46 ylabel(’|H(j\omega)|’);47 title(’Magnitude in dB’);48 subplot(2,1,2);49 semilogx(f,unwrap(angle(H))*180/pi);50 grid on51 xlabel(’f [Hz]’);52 ylabel(’\angleH(j\omega) [\circ]’);53 title(’Phase in degrees’);5455 % syms Cm Rb Cf Rf s;56 % poles=solve(’s^2*Cm*Rb*Cf*Rf+s*Cm*(Rf+Rb)+1=0’,’s’) % poles of system57 % zeros=solve(’s^2*Cm*Rb*Cf*Rf+s*Cm*(Rf+Rb)=0’,’s’) % zeros of system

Appendix 72

B Microphone Calibration Chart

Test Frequency

[Hz]

Measured Level

[mV/Pa]

Measured Level

[dB re. 1V/Pa]

Uncertanty

[dB]

250 52.57 -25.59 ±0.06

100 1000 10000Frequency [Hz]

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

[dB

]

20000

Appendix 73

C Circuit Analysis with feedback b

The starting point is (3.6) which is repeated here. Naming is with reference to Figure 3.1and Figure 3.11 b.

VOVIN

=Zb

Zm

(ZbZm

+ 1−1Zb

+ 1Zf1

1Zb

+ 1Zf1

+ 1Zf2

) (1)


Zb = Rb Zf1 =1

sCf1(2)

Zm =1

sCmZf2 = Rf2||Cf2 =

11

Rf2+ sCf2

=Rf2

1 + sRf2Cf2(3)

which by insertion yields

VOVIN

=sCm

sCm + 1Rb−

sCf1+1Rb

Rb(sCf1+sCf2+1Rb

+ 1Rf2

)

(4)

Extending the fraction by multiplying with Rb

VOVIN

=sRbCm

sRbCm + 1−sCf1+

1Rb

sCf1+sCf2+1Rb

+ 1Rf2

(5)

Another extending of the fraction

VOVIN

=sRbCm(sCf1 + sCf2 + 1

Rb+ 1

Rf2)

sRbCm(sCf1 + sCf2 + 1Rb

+ 1Rf2

) + sCf1 + sCf2 + 1Rb

+ 1Rf2− sCf1 − 1

Rb

(6)

Collecting coefficients of s

VOVIN

=s2RbCm(Cf1 + Cf2) + sRbCm( 1

Rb+ 1

Rf2)

s2RbCm(Cf1 + Cf2) + sRbCm( 1Rb

+ 1Rf2

) + sCf2 + 1Rf2

(7)

Writing to the standard form with 1 as coefficient to s0

VOVIN

=s2RbCmRf2(Cf1 + Cf2) + s(Rf2 +Rb)Cm

s2RbCmRf2(Cf1 + Cf2) + s((Rf2 +Rb)Cm + Cf2Rf2) + 1(8)

Now on its standard form (see (3.10)) the natural frequency and damping can easily beread as stated in (3.13) and (3.14).

Appendix 74

D Circuit Analysis with feedback d

The starting point is (3.6) which is repeated here. Naming is with reference to Figure 3.1and Figure 3.11 d.

VOVIN

=Zb

Zm

(ZbZm

+ 1−1Zb

+ 1Zf1

1Zb

+ 1Zf1

+ 1Zf2

) (9)


Zb = Rb Zf1 = Rf1||Cf1 =1

1Rf1

+ sCf1

=Rf1

1 + sRf1Cf1(10)

Zm =1

sCmZf2 = Rf2||Cf2 =

11

Rf2+ sCf2

=Rf2

1 + sRf2Cf2(11)

which by insertion yields...... a very long and nasty 3rd order expression which is not used for further analysis andtherefore omitted.

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Ultra Low Frequency Infrasonic Measurement System · An infrasonic measurement system is built capable of sensing acoustic signals down to 10 mHz which is advantageous for measurements