# A Soldier's Microphone/Hydrophone

### 1996 (Note: No figures yet, and biblio. incompl.-corr. coming)

Executive Summary

Two prototype instruments were built and studied for battlefield use. Each was based on the author's SDC transducer, which was patented in 1995. Since SDC devices are not limited by a low frequency cutoff (unlike conventional microphones) an inexpensive sensor of this type could be especially attractive for both detection and study of infrasonic acoustic signatures in the neighborhood of 1 to 20 Hz. From earlier studies, using non-fieldable (expensive) microphones, this regime is known to harbor strong emissions from helicopters. It may also contain machine specific emissions, indicative of type, for other machines of war. One objective achieved during the study, was an extension of the high frequency cutoff of the SDC pressure sensor on which the microphone is based. An additional achievement was the construction of a support electronics package for SDC devices, based on inexpensive components. The final hardware achievement was the construction of a prototype SDC hydrophone; which was used to acquire human heart and respiration sounds, the data of which have expanded the numerical base of a recently awarded patent to the U.S. government (M. Scanlon, inventor). In addition to describing construction techniques and the experiments which were performed to characterize the aforementioned hardware components, theory is first provided to compare these SDC devices against other common instruments. Finally, some additional future developmental efforts are indicated-ones which must be performed if the prototypes are to be transformed into fieldable sensors.

*Visiting Professor, on leave from Texas Tech University.

## 1  Preface

This document is divided into two main parts. In the first part, sections 2 - 6, theoretical comparison is made of the SDC sensor microphone against conventional condenser microphones (including electret types). In the second part, sections 7 - 12, there is first a discussion of the prototype instruments that were built. Then are provided results of experiments which were performed with these prototypes to confirm some of the theoretical predictions. Finally, some conclusions are noted, along with recommendations for future work.

## 2  Introduction

The human ear is an extraordinary sensor, considered in relationship to ruggedness as well as sensitivity and dynamic range. The average (youthful) ear is able to detect pressure variations (sound waves) in the frequency range from about 20 Hz to 20,000 Hz. This response is very nonlinear, with peak sensitivity occurring at 1000 Hz, in terms of sound pressure level at the eardrum. Ear canal resonance causes the total system sensitivity to be maximum at about 4000 Hz[].

The ear's importance to scientific work is reflected in the fact that the reference for acoustic measurements is the decibel (dB), which specifies sound level relative to the pressure standard of Pref  =  20 mPa. The reference level (0 dB) corresponds to the 1 kHz threshold of the ear (faintest detectable sound). In this paper, the dB is defined by 20 log10[P/( Pref)].

By definition, the International System (SI) unit, the pascal, is equal to 1 N/m2. For those who are more familiar with other units, the average atmospheric pressure at sea level is 1.01 × 105 Pa  =  14.7 lb/in2  =  1.00 bar  =
39.4 in of Hg.

### 2.1  Condenser Microphone

One of the most important types of commercial microphones is that which uses a capacitor. In the early days of electronics, this component went by the name, condenser; and that term has persisted in many circles of acoustics to this day. The conventional form of this instrument is one in which the charge on the capacitor is essentially constant. This may be realized by connecting a voltage source to one of its two electrodes through a large resistor (the other electrode being grounded). Alternatively, the fixed external charge, Q, stored by the capacitor may derive from the permanent electric polarization of a dielectric material, such as teflon (metallized on one side). Instruments of this type are referred to as electret microphones. Because of their extensive employment in hearing aids, they have become commonplace and inexpensive. Inexpensive units, however, have a low frequency cutoff greater than  100 Hz.

Changes in spacing between the diaphragm and a backing plate, due to impinging sound waves, give rise to a voltage variation across the terminals of the capacitor. Through the defining relationship of capacitance, Q  =  C V, the fractional change in voltage is seen to be the negative fractional change in capacitance

 DV V =  - DC C
(1)
The sensitivity of the instrument is governed by those system parameters which determine C and its variation. The calculation of C is straightforward when the spacing between the planar electrodes is small enough that the electric field between these plates can be assumed uniform (insignificant fringing). Then the capacitance is given by the well known expression
 C  =  eA/d
(2)
where e is the dielectric constant of the material" between the plates, A is the area of either plate (or smaller of the two if they're unequal), and d is the spacing (gap) between the parallel plates.

The most common research grade (expensive) microphone has been the condenser instrument sold by the Danish company, Bruel & Kjaer (B & K). It uses a grounded diaphragm of nickel (thickness 1.6 mm to 6.5 mm), which is separated by air from a backing plate (nominally 20 mm). The insulated backing plate is charged with 200 V DC through a large resistor[]. Since the gap is air, the constant in Equation(2) is essentially that of free space, e ® e0  =  8.85 x 10-12 C2/Nm2. For the B& K 1in microphone, C » 50 pF.

To understand both the simple result of Equation (2) and the more complicated case of the electret microphone which follows, it is convenient to consider the field between the electrodes. Using Gauss' law, one finds that the electric field (directed from the positive electrode toward the negative one) has a magnitude of

 E  =  s/ e0
(3)

where s is the magnitude of the charge per unit area on either plate. Because the field is uniform, the potential difference between the plates, ò[E\vec]·d[s\vec] ® [E\vec] ·[s\vec] ® E d, is therefore

 V  =  sd / e0  = Q d A e0
(4)

Using the defining relationship for C, one then readily obtains Equation (2) (with subscript 0 on e)

#### 2.1.1  Electret Microphone

The thin dielectric diaphragm of this microphone possesses a permanently polarized, internal electric field. It may be produced by cooling the film from an elevated temperature in a large external DC field; or conversely, it may be exposed to an electron beam in vacuum []. The latter technique is useful in that the required thin layer of metal which must be placed on one side of the dielectric when it is used in a microphone, can be applied while the film is still in the vacuum. This is easily done using common techniques of vacuum evaporation. Either aluminum or gold is easily deposited to the required uniform thickness of » 2 × 10-8 m.

The following description of the principles of operation of this microphone are based on the parameters indicated in Figure 1.

Picture Omitted
Figure 1: Cross section of an electret microphone (not to scale).

Of thickness, s » mm¢s and under tension, the diaphragm is deformed by changes in pressure of the sound wave. This causes modulation of the separation distance, d, between the diaphragm and the metal plate (cross hatched lower electrode). Being polarized, there's a permanent field in the material of the diaphragm. For the figure, this field is downward, so that there's an assumed bound (negative) surface charge density, -sb on the bottom surface of the film. (The field in the air gap is oppositely directed.) A corresponding +sb is located on the upper surface of the film. Once metallized, any potential difference between the two electrodes will tend toward zero since the electrical resistance between them can not be infinite. Thus the upper electrode takes on a charge density -s, and the lower one +s.

Analysis of this system is facilitated by Gauss' law

 ó(ç)õ ®D ·d ®A =  Q
(5)
where Q is the amount of charge inside the user selected closed surface over which the displacement flux density, [D\vec] is integrated. This permits the evaluation of [E\vec], under conditions of high symmetry, since
 ®D =  e ®E =  e0 ®E + ®P
(6)
where P  =  sb.

By Gauss' law using a right circular cylinder pillbox", the field in the air gap is found to be s/e0. Similarly, the field inside the dielectric, due to excess charge on the metal electrodes, is given by s/e  =  [(s)/( e0er)]; where er is the relative permittivity of the film (dimensionless parameter >  1).

Via superposition, one obtains the total field inside the dielectric by subtracting from s/e the amount, sb/e. It is subtracted because the fields produced by the two distributions are oppositely directed.

Assuming uniform fields (valid if both d and s are small compared to the smallest lateral dimension of the film), the potential difference between the electrodes is determined to be

 V  = s e0 (d + s er ) - sb s e0er
(7)
For quiescence (d = constant), V ® 0, so the relationship which exists between the charge densities is
 s  =  sb s/(s + erd)
(8)

One can now readily calculate the change in voltage across the capacitor as a function of gap spacing change, assuming constant s. (This is a good assumption when the sensor is connected to a high resistance amplifier). Using Equation (7), DV  =  [(s)/( e0)]Dd, which when combined with Equation (8) yields

 DV  = sb e0 s Dd/(s + er dm)
(9)
where dm is the mean gap spacing.

### 2.2  Sensitivity

The derivative of voltage with respect to gap spacing determines the sensitivity of a condenser microphone. For the electret type, it is seen from Equation (9) that this sensitivity is directly proportional to the polarization, sb and inversely dependent on the mean gap spacing, dm. (We'll later see that there's also an important limitation imposed by the magnitude of the sensor capacitance, because of the amplifier to which it connects.) It is very difficult to work with gap spacings less than 10 mm, and the upper limit on the polarization is determined not only by the ability of the dielectric to retain a field; but also by its dielectric strength (breakdown potential). Judging from the performance of the common Knowles microphones [], it appears that the maximum operational polarization is in the neighborhood of 105 V/m · e0  »  10 mC/m2. This corresponds to about 10-3 of the dielectric strength of teflon.

As compared to the electret, the analysis of the B & K microphone is trivial, for the open circuit case. From Equation (4) it is seen that

 DV  =  E Dd
(10)
As with the electret, maximum sensitivity is realized by operating with the largest safe field that can be established in the gap. For air, this normally will not be greater than 3 × 106 V/m. It is possible to exceed this value by nearly an order of magnitude by going to very narrow gaps, as the author did while researching ultrasound with a capacitive sensor [].

To theoretically calculate the sensitivity in its final desired form, one has to ultimately determine Dd as a function of DP, where DP is the variation of pressure caused by the sound which is driving the microphone. To model the bowing of the diaphragm in the presence of an air mass is not trivial and will not be done here. Results from the work of others will be used (and referenced) as necessary.

## 3  When the Amplifier is connected

Both the B & K and the electret microphones have an electrical equivalent circuit as illustrated in Figure 2, where Cs is input stray capacitance of the amplifier and its input leads.

Picture Omitted
Figure 2: Equivalent circuit of a condenser microphone.

The voltage, Vi(t), corresponds to either Equation (9) or Equation (10) with D® Vi(t). The capacitance, C, is that which is seen when looking from the amplifier back into the terminals of the sensor with its voltage source shorted (Thevenin's technique). For the B & K microphone, it is simply e0  A/dm.

The effect of the amplifier is to cause a voltage division", so that only a fraction of Vi(t) gets amplified. The first case to be considered, involving this divider, is that of an assumed steady state sinusoidal voltage. Later, it will be shown how the divider can cause both attenuation and distortion of a pulse. The steady state case is straightforward, using AC circuit phasor concepts. The pulse treatment, involving transients, will be treated by time domain analysis using the fast Fourier transform (FFT) and the convolution theorem.

### 3.1  Steady State Response

The sensor sees two impedances in series: (iwC)-1 and the parallel equivalent of R and (iwCS)-1, where i  =  (-1)1/2. Typically, C is an order of magnitude greater than CS, when the amplifier is connected close to the microphone; so that CS can be ignored. (If a long coaxial line between sensor and amplifier were used, the signal would be severely attenuated because of the resulting large CS [R then being ignorable].) Thus, the portion of Vi(t) which gets amplified is the voltage across the resistor of the series combination of R and C. The output is then given by
 Vo(t)  =  I(t) R  =  Vi(t) R/Z,        Z  =  R + 1/(iwC)
(11)
which gives
 Vo(t) Vi(t) = wRC(wRC + i) 1 + (wRC)2
(12)
This equation can be rewritten as
 Vo(t) Vi(t) = wRC [1 + (wRC)2)]1/2 eif,      f  =  tan-1 1 wRC
(13)
Notice that the output is shifted in phase with respect to the input; i.e., it leads by the angle f. Additionally, as the frequency goes toward zero, the output also goes to zero; and this simple lead" circuit acts to differentiate the input. As such, it is not a good differentiator because of severe attenuation.

### 3.2  Frequency Response of the Condenser Microphone

Figure 3 is a log-log plot of the lead circuit output as a function of frequency for constant input voltage. The numbers correspond to
 Voltage  = 50 ·f [1 + (50·f)2]1/2
(14)
where f represents Frequency. The rolloff" frequency is defined according to the expression 50·fR  =  1. Thus, at f  =  fR  =  0.02, the output is 3 dB down from its value at f  =  1.

Figure

Figure 3: Illustration of low frequency loss in sensitivity.

## 4  Time Domain Analysis

Transient response of the condenser microphone is best treated by the Green's function technique. The Green's function corresponds to the impulse response of the system. This may be obtained by considering the response of the lead circuit to a step pulse of constant height, Vc, in the limit as the pulse width, Dt, goes to zero.

The equation of the lead circuit is

 Vi(t) - 1 C óõ I(t)dt - I(t)R  =  0,     Vo(t)  =  I(t)R
(15)
which responds to the leading edge of the pulse according to
 Vo(t)  =  Vc e-[t/ RC],      t < Dt
(16)

Following the trailing edge of the pulse, the output is given by

 Vo(t > Dt)  =  Vc(e-[(Dt)/( t)] - 1)e-[((t - Dt))/( t)]
(17)
where the substitution, RC® t, has been made. This can be rewritten as
 Vo Vc =  e-[t/( t)](1 - e[(Dt)/( t)])
(18)

In the limit as Dt® 0, e[(Dt)/( t)] » 1 + [(Dt)/( t)], so that

 Vo Vc =  - Dt t e-[t/( t)],         t > Dt
(19)
For Vo to remain finite as Dt® 0, Vc® ¥ accordingly, which is the essence of an impulse (total area under òVi(t) dt  =  VcD® 1). Thus the Green's function of the system is realized by considering Equations (16) and (19) together. Equation(16) yields a part, g1  =  d(0) where d(0) is the Dirac delta function positioned at t  =  0; and Equation(19) yields g2  =  -[1/( t)]e-[t/( t)]. Thus the Green's function, g  =  g1 + g2 is given by
 g(t)  =  d(0) - 1 t e-[t/( t)]
(20)

### 4.1  Convolution Integral

The convolution theorem is the powerful mathematical basis for time domain analysis using the Green's function. The output of the lead circuit to any arbitrary input waveform, Vi(t), is given by the convolution integral []
 Vo(t)  = óõ t -¥ Vi(x)g(t-x)dx
(21)
Causality requires that the upper limit be t rather than ¥; but analysis (after the fact) usually is concerned with a fully specified time history (so that t ® ¥).

It is seen from Equation(21) that for every desired value of the time in Vo(t), a separate integral is required, unless Vi(t) is simple enough to yield a complete analytic solution. In practice, numerical processing is usually necessary. The only practical way to generate Vo(t) is by means of the convolution theorem. Consider, for example, the present paper's use of 2048 sampling points for each time record that is to be presented. Since it is unrealistic to integrate Equation(21) 2048 times in a brute force" manner, the following elegant methodology is used, based on the fast Fourier transform (FFT). (As compared to the discrete Fourier transform, the FFT is faster by a factor of 190, when dealing with 2K = 2048 point record lengths.)

#### 4.1.1  Convolution Theorem

The convolution theorem [involving Equation (21)] says that
 L[Vo]  =  L[Vi]L[g]
(22)
where L[f] is the transform" of f. This may be any of several transforms, such as Laplace's.

(Note: It is the author's experience that engineers are primarily trained with the Laplace transform, whereas physicists are trained with the Fourier transform, which is a subset of the former and therefore less general. The Laplace transform is more elegant when dealing with analytic solutions, whereas the FFT, developed by Cooley and Tukey [], makes the Fourier transform more useful in numerical work.)

Equation(22) makes for an extremely powerful numerical tool, which is used as follows. From the time record of Vi(t), one obtains its Fourier transform, using the FFT. This transform is then multiplied by the FFT of the Green's function (a product involving 2 x 2048 real and imaginary pairs in the present paper). Finally, the inverse FFT is computed, which gives Vo(t) for the assumed Vi(t). Although the reader may perceive this operation to be cumbersome and time consuming, the entire process takes just a few seconds on a 90 MHz Pentium based PC. It's also a user friendly process because of the extensive algorithm development that has been performed for doing FFT's and their inverse [].

### 4.2  Transient Response-Condenser Microphone

For the examples that follow, the Green's function of Equation(20) was used. The Fourier transform of g could have been readily evaluated analytically, since
 L[g]  =  G(w)  = óõ ¥ -¥ e-iwt(d(0) - 1 t e-[t/( t)])dt  = iwt 1 + iwt
(23)
the right hand side of which is equivalent to Equation(12), as it must be.

Note: Although the Fourier transform range of integration is +¥, it should be noted that the integrand is zero for negative times, since g(t)  =  0 for t < 0. It's also worth noting that there are six different ways to specify the transform and its inverse, all of them acceptable. Some users put [1/( 2p)] in front of one of the pair of equations, others put it with the opposite equation; still others go symmetric" by placing [1/( Ö{2p})] with each equation. Also, the -iwt term may be replaced with iwt. The only necessary condition is that the signs be different in the pair.

The present algorithms used the time form of g(t) and computed the transform numerically with the FFT, rather than using the result given in Equation(23)]. Although more work is thus required of the computer, the software accommodates any Green's function the user should want to use.

#### 4.2.1  Example Waveforms

In the examples which follow, several cases are considered, covering a wide range of the time constant spectrum". Understanding these cases will provide an appreciation for how the low frequency rolloff may or may not be important to a given problem.

Because of the properties of the Dirac delta function, it is readily seen from Equation(20) that the output signal will faithfully reproduce the input in the limit as t® ¥. Steady state analysis in such a case is not meaningful, since a steady condition could never be realized. Figure 4 is an approximation to this case, and is seen to be consistent with expectations. On the other hand, when t is very small, the differentiator characteristic of the lead circuit is evident, as shown in Figure 6. Shape distortion is severe for this case. The intermediate time constant case shown in Figure 5 illustrates not only some distortion, but another property of the lead circuit as well-that the long term time average of the output signal must be zero, once steady state has been reached. Those familiar with electronics will not find this result surprising, since capacitors are used to block" DC, while passing" AC. Such a blocking capacitor is even used with non-electret condenser microphones, so that the large polarizing voltage is not present on the input to the amplifier. It is large enough to be inconsequential to the present analysis.

Figure

Figure 4: Response to a pulse train, long time constant.

Figure

Figure 5: Response to a pulse train, intermediate time constant.

Figure

Figure 6: Response to a pulse train, extremely short time constant.

Relative to the total record length of 2048, the time constant was set respectively at 2000, 200, and 2 in Figures 4 through 6. Each of the pulses, in the 4 pulse input train, was of width 50 (every figure). Thus the pulse width was respectively 0.025t, 0.25t, and 25t. These numbers relate to the corresponding rolloff frequencies for the three cases, according to

 fR  = 1 2pt
(24)

A pulse of sound input to the microphone is distorted greatly when the pulse width, D >  2.5t. This is illustrated in Figure 7, for the case of an input pulse, of width 50; whose shape is one cycle of an offset cosine. This shape was chosen to approximate a low frequency blast for which there is molecular flow (not a shock of the characteristic N-wave" type). (Some experiments with an SDC pressure sensor seem to support this case for some conditions, although it may not be representative of battlefield conditions.) The offset cosine was chosen instead of one half a sine wave, because the latter has infinite slope at the beginning and end points.

Figure

Figure 7: Response to a pulse for which Dt  =  2.5t.

Stated in terms of rolloff frequency, pulse distortion begins to become significant as fR® 1/(2pDt) and is severe for fR  >  (2.5 Dt)-1. Many bullets create a shock whose width is in the neighborhood of Dt » 1 - 2 ms . Thus, shape distortion is serious for rolloff frequency of the order of 200 - 400 Hz; which is where many inexpensive electret microphones experience significant low frequency cutoff. The problem with trying to use these microphones in reconstruction of pressure profiles is thus obvious.

Figure 8 illustrates pulse" distortion with a more likely shock input, in the form of an N-wave. For comparison with Figure 7, the ratio of pulse width to time constant was retained at 2.5; however, the ratio used to generate Figure 9 was 25. From this figure, it can be seen how the complications due to natural ringing" (echos) that result from reflection of a shock from various obstacles, will be exacerbated when the low frequency cutoff of the microphone is severely limited. Any process to estimate direction to a shock source, using arrival time data, will be made more difficult because of the distortion.

Figure

Figure 8: Response to an N-wave, Dt  =  2.5t.

Figure

Figure 9: Response to an N-wave pulse, extremely short time constant.

### 4.3  Physical basis for fR

The basis for low frequency cutoff in the condenser microphone is easy to understand. The value of R in t  =  RC is collectively determined by all the components to which the backing plate of the microphone is connected. In times past [before the field effect transistor (FET)], the dominant factor was oftentimes the input resistance of the amplifier. For example, a microphone with 50 pF capacitance connected to an amplifier whose input resistance is 109 W, would have a cutoff frequency of only fR  =  20 Hz. Since extraordinary measures are required to increase R significantly above 1000 MW, the lowering of fR has required artistry as well as engineering. In the B & K case, some of the measures employed have included: (1) a quartz, sapphire, or ruby insulator to hold the backing plate, and (2) deposition of a thin layer of quartz on the diaphragm to prevent growth of fungus. The latter measure speaks to the issue of water vapor, since a large relative humidity contributes to reduced R and thus increased fR. Some of the B & K instruments even use dehumidifiers as a countermeasure.

Such sophisticated measures for maintaining the integrity of R are not feasible with the miniaturized hearing aid (electret) condenser microphones. Even if the techniques could be used, they would still have limited benefit because of the smaller value of C for these units. The approximate order of magnitude greater fR of the electrets is a reflection, largely, of their smaller size (and thus smaller C).

### 4.4  Conclusions: Influence of low frequency cutoff in a condenser microphone

The previous sections have considered not only the steady state response of the condenser microphone, but they have also rigorously treated the transient response. The level of analysis chosen was deemed necessary for two reasons: (1) the author's limited experience suggests that the majority of acousticians have not been trained in the time domain techniques necessary to treat transient (pulse) behavior; and (2) a credible theoretical foundation was essential for weighing the relative advantages and disadvantages of the new SDC microphone against the conventional condenser microphone.

The accuracy of some claims based on the models must await hardware tests, some of which could not be completed within time constraints of the summer program. It is hoped that the preliminary results, both theoretical and experimental, will be of sufficient interest to warrant further investigations in the future. It is also hoped that the tutorial features of this paper will be beneficial to the reader whose experience has been limited in any of the aforementioned areas.

The bottom-line" conclusions from the preceding theoretical analysis are as follows: (1) Based on the steady state analysis, the inexpensive electret microphones are incapable of monitoring infrasonic emissions. This fact was already well established before initiation of any of the present study. Although several of the B & K condenser microphone models would perform well in this regime (because of superior low frequency cutoff), the cost per microphone is too great to permit widespread utilization. (2) The present modelling predicts that low frequency limitations can cause significant pulse distortion. This distortion complicates the use of timing data in determining direction to a shock producing threat. The analysis shows that the amount of distortion is sensitive to the product Dt ·fR, where Dt is the width of the shock and fR is the rolloff frequency (3 dB point) of the microphone. Distortion results when this product exceeds a threshold of (Dt ·fR)min  =  1/(2p)  =  0.16. For Dt ·fR  >  0.4, it is predicted that the distortion can become severe enough to interfere with time of flight algorithms.

## 5  Symmetric Differential Capacitive (SDC) Pressure Sensor

The sensor which was modified for use as a microphone is illustrated in Figure 10.

Picture Omitted
Figure 10: Illustration of the SDC pressure sensor.

It is just one sensor, among several of SDC type; the broad class of which was patented in 1995 []. The straight lines shown bisecting the circular electrodes of radius a, represent insulator strips that electrically separate the electrodes-1 from 2 etc. Although there are 4 physical pieces to the drive (outside) structure, these comprise only 2 equipotentials, because of the external cross coupling indicated. The drive is shown applied across equipotentials 1 & 2. The pickup to the instrumentation amplifier is from the middle (inside) plate pair (moving diaphragm) labeled 3 & 4. The lines" from the amplifier to 3 & 4 are for illustration purposes only. The diaphragm is aluminized mylar and electrical contact is outside the region of diaphragm bowing.

This instrument was originally designed to measure slow pressure variations, such as those associated with respiration []. It is a capacitive bridge type sensor that is doubly differential as compared to the usual differential capacitive sensor []. As such, it is at least twice as sensitive, for the same electrode areas (for small stray capacitance of the connecting amplifier). Unlike the condenser microphones treated earlier, this sensor does not have a low frequency cutoff associated with the electronics. This property results from the high frequency (compared to audible) of the drive signal applied to the bridge. In this respect, it is similar to the B & K carrier based microphones, which also are not low frequency limited. The detection electronics is significantly different, however. The B & K microphone (Model 2631) uses frequency modulation; i.e., capacitance change of the microphone causes variation in the frequency of an oscillator of which it is a part. One of the earliest reported uses of a capacitor for measuring small mechanical variations was of this type []. The SDC sensors, on the other hand, cause essentially an amplitude modulation" of the drive (carrier) to the bridge.

(Note: There may be a temptation at this point to want to compare these systems to AM and FM receivers (radio), in which the superiority of FM is well known in a noise sense. Such a comparison is not presently meaningful. The amplitude" modulating property of the SDC detectors is dramatically different from AM broadcast receivers, in that it uses synchronous detection-a powerful noise reducing technique.)

To understood the SDC detectors, one should refer to the equivalent circuit(s) shown in Figure 11.

Picture Omitted
Figure 11: Equivalent circuit(s) of the SDC sensor.

The capacitance, C0  =  e0A/d, where e0 is the permittivity of free space (really air), A  =  pa2, and dm is the mean gap spacing between adjacent electrodes. All expressions for capacitance ignore fringe (edge) effects.

All four capacitors of the bridge change in a symmetric fashion, so that C3  =  C1 and C4  =  C2 where

 C1  = e0A 2(dm-y)
(25)
 C2  = e0A 2(dm+y)
(26)
and y is the amount of motion of the moving plate. (Note: the diaphragm does not really move uniformly, but rather experiences bowing; so these expressions are rough approximations.)

The lower part of Figure 11 is the Thevenin equivalent, which accounts for influence of the amplifier on the detector. Looking back into the sensor from the amplifier, the voltage source (open terminal) is

 Vo  =  Vi C1-C2 C1+C2 =  Vi y dm
(27)

An advantage over a half bridge is worth noting. For the half bridge, the right hand side of Equation(27) would be [(Viy)/( dm(1-[(y2)/( dm2)]))], which is seen to be nonlinear and a cause for harmonic distortion in the output as compared to the SDC microphone.

The output (impedance) properties of the sensor are ascertained by looking into the device with its voltage source shorted. From the upper part of Figure 11, this is seen to be capacitive, in the amount 1/2(C1 + C2)  =  1/2C0. For symmetry, two capacitors are chosen, each of magnitude C0; which combine to give the single value C0/2, when the voltage source is shorted.

The amplifier's influence derives from the components to the right of the dashed line; i.e., stray capacitance Cs of the amplifier plus leads, shunted by a large value of resistance, R. Typically, for FET transistors used in an instrumentation amplifier, R is many megohms and may be ignored. (Note that this was not the case for the condenser microphones.)

The series combination of capacitors is seen to be a voltage divider. Thus the final signal that actually gets amplified is

Vof  =   y
dm

 C0 2

 C0 2 + Cs
Vi
(28)

For greatest sensitivity, it is seen from Equation(28) that Cs should be made as small as possible. The standard means for accomplishing this (as with all capacitive detectors), is to place the amplifier as close to the sensor as possible.

### 5.1  Demodulation

If one assumes that the stray capacitance of the amplifier connection, Cs, is small compared to the sensor capacitance, C0, then Equations (27) & (28) are identical (Vo  =  Vi [y/( dm)]). The sympathetic response of the diaphragm to a steady state sound of angular frequency w is given by y  =  y0 cos wt. The amplitude, y0, is approximately proportional (low levels) to the sound pressure amplitude, PS, and depends on a variety of diaphragm factors, as indicated in Figure 12.

Picture Omitted
Figure 12: Microphone Diaphragm Response to Sound.

Denoting the angular frequency of the drive (carrier) to the bridge by wc; the signal into the amplifier, at low frequencies, can be represented as

 Vo  »  KPS cos wt cos wc t
(29)
where K is a constant. The manner in which the voltage decreases for f  >  fR (due to both mechanical properties of the diaphragm (Figure 12), and electronics filtering of wc during demodulation) will be treated later. (Note: In the case of the condenser microphone, the rolloff (fR) caused low frequency cutoff. Because of the difference in the electronics, the SDC microphone is of opposite type; i.e., fR causes high frequency cutoff, whereas response on the low end goes all the way to DC.)

#### 5.1.1  Synchronous Detection

The lock-in amplifier, or synchronous detector, is the ideal electronics for bridge type sensors. As a bridge is varied slowly through its null (balanced) position, the phase changes by 180 degrees. By utilizing this phase change, the lock-in causes the polarity of the output voltage [multiplier on cos wct in Equation (29)] to reverse sign algebraically across the null. Even if the lock-in did not provide additional advantages of signal to noise ratio (SNR) improvement, this would still constitute a significant improvement over simply connecting a high impedance meter" across the output of the bridge. With a meter, there is a discontinuity in the slope of the output as the bridge is tuned through null. The lock-in thus increases the range of linearity by a factor of two.

The primary secret of the lock-in amplifier's success has to do with coherence, as implied by the term, synchronous", that is used to describe its operation. A reference signal to the demodulator, in phase with the sensor drive source is utilized, as illustrated in Figure 13.

Picture Omitted
Figure 13: Block Diagram, SDC Electronics.

Picture Omitted
Figure 14: Operations performed by the synchronous detector.

Mathematically, the synchronous detector operates on the signal and reference signals in the manner indicated by Figure 14. In most actual systems, as opposed to this idealized example, the multiplication is not analog, but rather is accomplished by a switching network. The filter must then remove odd harmonics as well as performing its other essential function described in the mathematics which follows. [Note: To simplify the mathematics, it is convenient to work initially with a reference frequency that is different from the signal frequency. At the end of the analysis, they will be equated. The signal" frequency, wS, in the following equations is the same as wc in Equation (29), and VS  =  2-1/2KPS cos wt. The Ö2 factors in the analysis are used because it's convenient to work with rms rather than peak values.]

For the mathematical treatment, assume that the signal and reference are harmonic

 S(t)  = Ö2 VS cos (wS t + fS)
(30)
 R(t)  =  Ö2 VR cos (wR t + fR)
(31)
 N(t)  =  uncorrelated to R(t)
(32)

The product is given by

 VP(t)  =  2 VS VR cos (wS t + fS) cos(wR t + fR) + R(t) N(t)
(33)
If we use the trigonometric identity 2cosAcosB  =  cos(A+B)+cos(A-B), then we obtain
 VP = VSVRcos[(wS+wR)t+fS+fR]+VSVRcos[(wS-wR)t+fS-fR]+R(t)N(t)
(34)
We follow the multiplication circuit with a filter (low pass) whose cutoff is wF. Since wS+wR >> wF, and since wS = wR, we obtain the output result
 Vfd(t)  =  k VS cos (fS - fR)
(35)
so that
 Vfd(t)  a   VS cos f
(36)

where the constant, k, embodies the filter attenuation; it also is determined by the reference signal voltage , VR, supplied to the demodulator. Notice that the term, R(t)N(t) has disappeared because the reference signal and the noise are not correlated. For maximum sensitivity, cos f is adjusted to unity. In any case, the output polarity is seen to reverse in going through bridge null.

Combining all the various parts, one obtains the output from the filtered synchronous detector, due to sound on the SDC sensor, given by

 VLF(t)  =  KD Vi dm PS cos wt
(37)
where the subscript LF implies low frequency response of the system. (There is no electronic rolloff as w® 0; however, sections to follow deal with the high frequency cutoff.) The constant, KD, is primarily determined by the mechanical properties of the diaphragm, the generic features of which are illustrated in Figure 12. Remember that Vi is the drive voltage applied to the sensor; increasing it is seen to increase the sensitivity of the microphone accordingly. The output proportionality to [(Vi)/( dm)] is the same form as was determined earlier for the condenser microphone; except there, the polarizing voltage was constant (DV  =  E Dd ® [(V0y)/( dm)]).

There is a significant feature of SDC sensors which results from their symmetry and concomitant differential output. The electronics associated with these sensors (instrumentation amplifier to which the balanced output is connected) typically has a very high common mode rejection ratio (CMRR). Since noise from the environment couples primarily to the amplifier in common mode fashion, it is thus severely attenuated as compared to the differential signal from the sensor. Thus the signal to noise ratio (SNR) is inherently greater than that of an unbalanced detector. The same is true of the popular linear variable differential transformer (LVDT). The SDC device is electrically similar to the LVDT, except that it uses capacitors rather than inductors.

All other factors being equal", the noise figure of an SDC microphone should be greater than that of an unbalanced condenser microphone. However, equality" is not easily realized. Whereas it is relatively easy (and therefore common) to put large DC voltages across the condenser microphone (200 V, B & K), it is not easy to drive the SDC microphone with a correspondingly large AC voltage. The voltage could be increased using a step-up transformer, but the cost would be prohibitive. Rather than try to improve upon performance (midrange frequencies), as compared to the condenser microphone; it is recommended that the SDC drive voltage be left at a convenient lower value. Comparable (rather than improved) performance should then be strived for on the basis of the improvements in SNR borne of symmetry. There is also a further electronics advantage, which derives from the use of synchronous detection. [This presupposes that the current electronics developmental effort will result eventually in an electronics support for the sensor that will be inexpensive. The cost for an SDC control unit of earlier design (general purpose laboratory) has been too great to allow battlefield utilization.]

There are two primary advantages of synchronous detection. The detector's use of an ultrasonic carrier (1) reduces 1/f noise by forcing the critical 1st stage amplifier to operate at a higher frequency, and (2) even the noise that remains at that frequency is further attenuated, because it is not correlated with the reference signal that is derived from the carrier. These advantages can be quite impressive and are well known to those who employ commercial lock-in amplifiers. For example, many experiments in optics that would otherwise require very low ambient light conditions, can be conducted in a lighted room by using a chopper and a lock-in amplifier.

### 5.2  Frequency Dependence of Sensitivity

As previously noted, the proposed SDC microphone will not be afflicted with a low frequency cutoff. An approximate equivalent circuit of the sensor, for describing frequency response, is shown in Figure 15. If the cutoff of the low-pass filter of the synchronous detector is significantly below that of the diaphragm, then the filter will be dominant and determine Req and Ceq in a straightforward manner involving gain of the amplifier. If the converse is true, then Req and Ceq become more difficult to express in terms of diaphragm parameters, such as radius, thickness, tension, mass density, and modulus.

Figure 16 illustrates the steady state features of the Figure 15 circuit. To determine the transfer characteristics,

 Vo  = 1 Ceq óõ Idt  = I iwCeq = 1 iwCeq Vi Z
(38)
where Z  =  Req + [1/( iwCeq)], and from which
 Vo Vi = 1 - iwReqCeq 1 + (wReqCeq)2 = 1 (1 + (wReqCeq)2)1/2 eif
(39)
 f  =  - tan-1wReqCeq
(40)
and it is seen that the output lags the input by the angle |f|. Whereas the lead circuit of the condenser microphone at low frequencies was a differentiator, this lag circuit becomes an integrator when wReqCeq  <<  1 (high frequencies). This permits a conceptual understanding for the difference in influence on a pulsed input.

Picture Omitted
Figure 15: Equivalent circuit for SDC microphone cutoff.

Figure

Figure 16: Illustration of high frequency loss in sensitivity.

In graphing Figure 16, the following Equation was used.

 Voltage  = 1 [1 + (50·f)2]1/2
(41)
where f represents Frequency as in the earlier treatment of the condenser microphone. As there, the rolloff frequency is defined according to 50 ·fR  =  1.

### 5.3  Transient Response-SDC microphone

The Green's function for the lag circuit is obtained in the same manner as was done earlier for the lead circuit. The result is
 glag(t)  = 1 t e-[t/( t)],     t  =  ReqCeq
(42)
This was used (convolution theorem & FFT) as before to generate some example transient responses. Figures 17 - 19 show the results for three different ratios of Dt to t. Decreasing t has been a primary objective of the present hardware developmental effort. Success therein results in two benefits that can be seen clearly from the indicated figures: (1) there is less distortion to a pulse, and (2) the attenuation of the pulse is decreased. Concerning the latter, as [(Dt)/( t)]®  0, not only does the pulse distortion become increasingly severe, but the sensitivity of the microphone goes toward zero. In terms of the rolloff frequency, this begins to happen when fR  <  (2pDt)-1.

Figure

Figure 17: Response to a pulse for which Dt  =  25t.

Figure

Figure 18: Response to a pulse for which Dt  =  2.5t.

Figure

Figure 19: Response to a pulse for which Dt  =  0.25t.

The pressure sensor that was first tested (for evaluation as a microphone) has an unacceptably low cutoff. Effects from both electronics and mechanical contributions were severe. The 3 dB point of the SDC support electronics had been set at about 10 Hz. Altering the electronics to move this upward several hundred Hz still did not yield acceptable performance, because of the large diameter (2 in) and small tension of the diaphragm. Not only does the increased mass of the diaphragm under these conditions result in an unacceptably low value of fR, but there are other causes for roll-off. If the wavelength of sound in air gets significantly smaller than the diaphragm diameter, then the sensitivity must fall off. For the 2 in diameter pressure sensor, this happens at about (442 m/s)/5.08 × 10-2 m  =  8700 Hz. Since the falloff was seen to occur at a much lower frequency than this, there is a more important effect; which is described later in terms of diaphragm modes.

### 5.4  Calibration

The SDC microphone should be amenable to the same simple calibration technique that was developed for the pressure sensor[]. A smallbore plastic catheter, with one end attached to the sensor, is whirled by hand; so that the open end of the catheter moves around a circle of radius, R. Consider a differential mass of air, dm, situated at a distance, r, from the center. At a steady angular velocity, of magnitude w, the centripetal force on the mass must be balanced by a pressure difference, dP, for which AdP  =  w2rdm; where A is the cross sectional area of the catheter. The total pressure difference along the whirling section is obtained by integration, recognizing that dm  =  rA dr, where the density of air (here reasonably assumed incompressible) is r  =  1.3 kg/m3. The result is
 P  = óõ R 0 w2rrdr  = 1 2 rV2
(43)
where V = wR. This is seen to be identical with the velocity head" of Bernoulli's equation. The speed of the tip can be readily measured with a stopwatch. One observes the output voltage from the SDC sensor electronics, during the time required to complete ten rotations (measured with a stopwatch). During this time, active feedback is provided. If the voltage starts to drop, then the catheter angular speed is increased and vice versa. It is fairly easy, by this means, to regulate the speed to within 5 %. Example results, using the technique, are provided in Figure 20; which was generated with one of the TEL-Atomic pressure sensors. Two different values of R were used- 0.5 m and 1.0 m. The triangles correspond to the former. The sensitivity of the sensor used to generate Figure 20 is only 1.6 mV/Pa [using Equation(43)], which is much too small for use as a microphone. It is a sensor that was designed to measure much larger pressures than those associated with sound waves.

Figure

Figure 20: Results from a Whirling Catheter" calibration run.

## 6  Diaphragm Modes

Depending primarily on the ratio of its thickness to radius, the diaphragm of a capacitive microphone may respond to incident sound in either one or the other (or in some cases as a combination) of two ways: (1) as a stretched membrane, or (2) as a vibrating plate. There's little question that the SDC pressure sensor responds as the former, because of its large (2 in) diameter and small (10 mm) thickness. By contrast, micro-electromechanical-systems (MEMS) sensors, fabricated in silicon, tend to fall in the category of a vibrating plate. It appears that the diaphragms of the B & K microphones are primarily drumhead in character. This conclusion is based on comparative frequency data for the instruments in relationship to the theoretical discussions which follow.

For the membrane case, a microphone's sensitivity tends toward zero as the sound frequency goes much above the lowest resonance frequency of the membrane (n01), which is a breathing" mode with the only nodal curve being the circle of attachment. This can be appreciated by the p phase difference for motions either side of a node; thus the average motion over the whole surface may be small or identically zero.

The mathematical description of a vibrating plate is considerably different than that of a membrane-some of those differences are now described.

### 6.1  Drumhead Modes

Unlike a stretched string, the natural modes of vibration do not form a harmonic series; i.e., the overtones are not integral multiples of the fundamental, n01. Instead, they are determined by the roots to Bessel's equation
Jm(2pna/c)  =  0,     c  =     æ
Ö

 T s

(44)
where a is the radius of the membrane, and c is the propagation speed along the membrane; which depends on its tension, T, and its mass per unit area, s [1].

Using tabulated roots of J0, J1, J2, ..., the following frequencies result:

1. The lowest mode (n01) is that for which the only nodal curve is the circle of attachment of the membrane. Being the 1st root of J0, its value is

 n01  =  0.3827 1 a ( T s )1/2
(45)

2. Higher order resonances occur at

n11  =  1.593n01, n21  =  2.136n01, n02  =  2.295n01, n31  =  2.653n01, n12  =  2.917n01, ....

Because of the p phase difference between adjacent antinodes, the average motion across the full surface of the diaphragm tends toward zero when the high order modes are excited. Actually, there are a series of resonances and antiresonances for the conventional capacitive microphone, as discussed by Morse. By defining a parameter

 m  =  2.405 f n01
(46)
where f is the sound frequency, the first zero (antiresonance) occurs at m  =  5.136; so that the sound frequency for which this occurs is
 far1  =  2.14 n01  =  n21
(47)
At a slightly higher frequency, there is a peaked (2nd resonance response) which occurs when
 fr2  =  2.30 n01  =  n02
(48)
These expressions are not easily applied to the SDC microphone because the electrodes have been split" (from circles into semicircles). In a test of the pressure sensor connected with a catheter to a loudspeaker, a high frequency response peaked at 910 Hz was noted, with a broad region of near zero response centered around 650 Hz. There is some evidence, largely speculative, that (1) the primary resonance (n01) was near 340 Hz, (2) the 910 Hz mode was due to n31, and (3) none of the other modes listed just after Equation(45), that might otherwise be displayed (such as n21), were visible because of the SDC electrode features.

### 6.2  Plate Modes

Morse also treats the case of a vibrating circular plate, under no tension. He obtains the lowest resonance frequency in terms of the plate's modulus, Q, mass per unit volume, r, Poisson ratio, s, half thickness, h, and radius, a. It is given by
 n01  =  0.934( h a2 )[ Q r(1 - s2) ]1/2
(49)

The most significant feature of Equation(49) is the proportionality to a-2 rather than a-1 of the drumhead. It is this dependence that occasioned the author to come to the conclusion about the B & K instruments behaving mainly like a membrane. From their data book, the cutoff frequencies for the 1, 1/2, 1/4, and 1/8 in instruments are respectively 18, 40, 100, and »  140 kHz (free-field, 0 degree incidence). The trend is roughly consistent with Equation(45) but not Equation(49).

### 6.3  Air Damping

Equation(45) explains most of the trends illustrated in Figure 12, but it does not account for damping. The damping is primarily determined by the air mass between the diaphragm and its adjacent static electrodes. In the SDC pressure sensor, this damping was small compared to the usual condenser microphone, because the spacing dm was very large (about 500 mm rather than 20 mm as desired for microphone use). The large spacing was also responsible for a much smaller sensitivity than was sought. Fieldable microphones will require a smaller spacing.

## 7  Prototype Microphone Mechanical Design

Illustrated in Figure 21 is the prototype microphone which was built and tested during this study. Primarily because of time constraints of the abbreviated summer program, it was not possible to build a sophisticated instrument. The expectation that a crudely fabricated prototype would nevertheless yield useful results proved to be true.

The static electrodes shown in Figure 21 were made of 1/16 in printed circuit board, clad with copper on both sides. Not shown in the figure are the external copper circles which were grounded to improve shielding. The 1 in diameter circular PC board central electrodes were formed using a die punch and a hammer. Once stamped, insulator strips were produced by filing away copper along a diameter; and a near uniform distribution of approximately 75 holes of 500 mm diameter each were drilled in each electrode. They were then re-flattened with the hammer on a smooth hard surface. Each electrode was then pressed back into the hole from which it came, the gap spacing being maintained by friction. The root sum square (rss) deviation from flat of each electrode was estimated to be no better than 100 mm. The gap spacing was established by manually pushing at various points around the circumference, while viewing with a magnifying glass. The nominal gap was visually estimated to be 500 mm. This estimate was reasonably supported by measured values of the capacitance.

The subscripts, T and B of Figure 21, are for purpose of identifying the application place of electrical drive. External cross connection (to form 2 equipotentials, characteristic of SDC devices) is established by soldering an external wire between 1T and 1B and likewise between 2T and 2B.

The diaphragm was fabricated from aluminized mylar, of 10 mm thickness. A straight insulator strip of approximately 1 mm width was produced in the otherwise uniform distribution of aluminum, of thickness » 0.02 mm. This was accomplished by dipping a toothpick in concentrated sodium hydroxide and then moving it, in light contact with the metal, along a straight edge. After washing, the film was then stretched in a crochet" like hoop, made of plexiglass turned in a lathe (the newly formed insulator strip passing along a diameter of the hoop). In the stretched state, the membrane was superglued (non-metallic side) to the non-metal bearing (top) surface of the outer ring which supports the lower static electrode of Figure 21. Note that the top and bottom support rings for the static electrodes are oppositely positioned with respect to their single side copper surfaces. The copper of the bottom ring is not indicated. That of the top ring is shown (bold line) because it is the means for making electrical (soldered) contact with the membrane semi-circular electrodes. It can be seen from Figure 21 that electrical connection to the diaphragm electrodes is via mechanical contact between the aluminum of these electrodes and the bottom layer of copper on the upper outer ring that supports the top electrode The drive wire pair were soldered at 3 & 4. It may be useful for the reader to compare this microphone to the SDC pressure sensor (Figure 10). For the prototype, the mechanical contact was maintained by light pressure from external screws and nuts.

Picture Omitted
Figure 21: Illustration of the prototype SDC microphone.

## 8  Experimental Results

The data of this document were obtained with the microphone illustrated in Figure 21, as supported by the Symmetric Differential Capacitive Control Unit sold by Tel-Atomic Inc of Jackson, Michigan[]. This electronics package is too expensive for extensive fielding, so consideration of alternatives became a natural part of the present study. A possible beginning electronics configuration, which showed promise, is discussed later in section 10.

In the figures which follow, temporal data is not provided. However, the study was also concerned with transient time records, derived from shocks and acquired with a storage oscilloscope. These will also be described later in section 8.5.

### 8.1  SDC high frequency response

Partly because of their experience with fluidic amplifiers having similar symmetry, the acoustics personnel headed by Steve Tenney at ARL recognized that the SDC pressure sensor might be convertible to a useful microphone. Since pressure sensors are normally concerned with very low frequencies, as compared to those measured with microphones; a first challenge was to extend the frequency range of SDC operation. This required two modifications: (1) decreasing the physical dimensions of the transducer (specifically the diameter of the mylar membrane diaphragm), and (2) increasing the cutoff frequency of the low pass filter that is used with its synchronous demodulator. The latter modification was trivial-simply changing the size of the capacitor that shunts the feedback gain resistor in the output stage operational amplifier. To accomplish the former, however, required major changes to traditional manufacturing techniques. Methodologies for construction of the crude prototype were described earlier. A major challenge for any future work will be the identification and perfection of assembly line techniques for mass production of fieldable microphones.

Earlier theoretical predictions pertaining to frequency response have been reasonably validated. The circular drumhead" diaphragm should resonate at a frequency which is directly proportional to the tension with which it is stretched, and is inversely proportional to its radius, a. These proportionalities were demonstrated as follows. The first phase measurements of the study were performed with the 2 in pressure sensor, connected to the sealed enclosure of a loudspeaker with a short section of tygon tubing. The tubing was necessary because the diaphragm of the pressure sensor does not couple to the system it is monitoring via multiple holes, as in Figure 21. It could be compared with the follow-on prototype, because both were produced from the same 10 mm mylar. Because of the expected proportionalities, the prototype microphone was made smaller, and its diaphragm was tensioned as greatly as was conveniently possible in the homebuilt crochet" hoop. Although tensions in the two were not measured (either absolutely or in a relative sense), it is thought that the tension in the microphone is greater than that of the pressure sensor by a factor of about 2.

The resonance frequency of the SDC microphone was found to be in the neighborhood of 1 kHz, whereas that of the pressure sensor was about 300 Hz. These numbers are consistent with the earlier statements and with the expectation that the mechanics of vibration should be closer to that of a drumhead" than a plate".

Figure

Figure 22: SDC microphone high frequency cutoff.

Shown in Figure 22 are data that were taken by positioning the prototype microphone close to an inexpensive midrange speaker that was driven by a frequency adjustable sinewave oscillator. The drive voltage was held constant as the frequency was varied from 200 to 2000 Hz. Not shown in the graph are the responses that were seen above the zero" at 1600 Hz. Sharp secondary resonances at 1750 and 2200 Hz were observed. All these results are in reasonable agreement with the earlier treatment of drumhead vibration.

### 8.2  Spectra

The speaker used to generate the data of Figure 22 was not suited to low frequency measurements. For SDC response measurements down to 1 Hz, the microphone was placed in the ARL anechoic chamber. The frequency domain graphs of Figures 23 through 26 were generated using Fast Fourier Transform (FFT) algorithms, that are part of the well known LabView (National Instruments) software. The ARL computer with which these data were acquired and then anaylyzed, is MAC rather than PC based. Because the LaTEX desk top publishing software with which this document was written, is PC based, the laser printed figures were scanned rather than directly imported. By contrast, other figures of this document were directly integrated post script (PS) or encapsulated PS files. The EPS files derive from monitor displayed graphs generated by QuickBasic, which were screen captured using PCXLAB, by Frendsen. The PS files were produced with TEXCAD, a computer aided drawing package which is common to many users of LaTEX shareware produced by Eberhard Mattes.

Figure 23: Microphones' response at 1 Hz.

(This page left blank for replacement with separate figure 23)

Figure 24: Microphones' response at 2 Hz.

(This page left blank for replacement with separate figure 24)

Figure 25: Microphones' response at 5 Hz.

(This page left blank for replacement with separate figure 25)

Figure 26: Microphones' response at 10 Hz.

(This page left blank for replacement with separate figure 26) Sound fields were generated with high quality loudspeakers, driven by a powerful McIntosh vacuum tube, push pull amplifier. Both speakers and microphone were located inside a sophisticated anechoic chamber (manufactured by Eckel-Cambridge, MA). The dichotomy of the equipment collection was striking. On the one hand, test instruments and the reference B & K microphone were state of the art, whereas the prototype SDC microphone was almost as crude as could be built and yet yield significant results. This dichotomy is mentioned for the following reason. It is rarely possible to take a sensor which performs extremely well in one arena and adapt it to another, where design considerations are necessarily quite different. Consider, for example, the B & K microphones. They have been without equal for most applications, largely because of the excellence of their manufacture. Consequently, they have become the primary world standard for acoustic measurements during the last half-century. As with any sensor, however; their regions of acceptable performance are limited. The low frequency cutoff of currently manufactured instruments is one which is especially important to envisioned army applications involving infrasound. This point should become clear to the reader in the discussions which follow.

### 8.3  SDC - B&K comparisons

Earlier theretical results predicted that the SDC microphone should not experience low frequency cutoff, in stark contrast to the conventional condenser microphone. Already well known were the resulting severe limitations that can result when working with inexpensive electret microphones. Reasons for the usually large value of the low frequency cutoff have already been mentioned. Even though the 1/2 in B & K microphone, against which the SDC prototype was compared, is not so severely limited; nevertheless, the extent of limitation had not been previously determined. This was true even though the microphone has served as a standard for acoustics testing at ARL. Lack of data for frequencies below » 5 Hz resulted because of changes in the B & K product line. If the 1/2 in ARL reference microphone were connected to an earlier B & K electronics package that is carrier based (high frequency), then the low frequency cutoff should be consistent with the manufacturer's specified 0.01 Hz for this configuration. The electronics currently produced by B & K and provided with the microphone, yield a much higher cutoff of » 5 Hz (refer to ref. 2). The data of Figures 23 - 26 are consistent with this value.

In all of the Figures 23 through 26, the SDC prototype response is provided in the top graph; and the 1/2 in B & K response in the bottom one. All plots are log-log, with the ordinate units being dB below 1 V. Thus, it is seen that the background noise for the SDC microphone is » 30 mV in the range from 1 -50 Hz. The SNR of the B & K microphone is seen to be less for frequencies below 10 Hz. Although it had been previously thought that the 1/f character of the B & K microphone response was acoustical in nature, the present comparisons show that this cannot be the case. Instead, the 1/f noise derives from the B & K electronics. The distinctly different frequency trends for the two microphones is due to differences in detection technique. In particular, the reduced low frequency noise of the SDC microphone is the direct result of synchronous demodulation; which was described earlier.

In each of Figures 23 - 26, a monochromatic drive was applied to the loudspeaker woofer. The sizeable full width half maximum (FWHM) of the respective acoustic lines" evident in the figures, is a reflection of short record collection times, and not due to a loss in signal generator monochromaticity.

As can be seen from the SDC graphs, the woofer efficiency degrades significantly for frequencies below 10 Hz. [From studies performed with SDC pressure sensors (such as the whirling catheter calibration scheme mentioned in section 5.4), it can be confidently stated that the loss of signal at low frequencies is due to the loudspeaker and not the microphone.] Moreover, the speaker's harmonic distortion is seen to be significant at the low frequencies. In Figure 24, for example, both microphones show an unusually large 3rd harmonic that is only 10 dB below the fundamental, even though the 2nd harmonic is more than 15 dB down. In the 10 Hz Figure 26 case, the 2nd through 5th harmonics are all quite large. It should also be noted that there is a pronounced 60 Hz pickup in all of the SDC cases. This was largely due to the manner in which the SDC microphone was attached to the electronics hardware. For sake of convenience, the two part chassis box was opened, and the microphone was attached with a screw and nut to one of the two components. In the future, most of this pickup can be eliminated by employing a better mechanical mount.

For all frequencies below 10 Hz, it is seen that the SDC microphone outperforms the B & K microphone. In fact, somewhere in the neighborhood of 2 Hz, the B & K signal is no longer above noise. In contrast, the B & K unit outperforms the prototype for all frequencies above about 10 Hz. It's superior SNR at high frequencies was expected, for the following reason.

### 8.4  SDC prototype limitations

It was noted earlier that the prototype microphone was crudely built with a very wide gap spacing of » 500 mm. Additionally, the drive voltage applied to the SDC microphone in these studies was only 10 V peak to peak. Because the sensitivity of any capacitive microphone is proportional to the magnitude of the electric field in the gap, there is a resulting dramatic reduction in the instrinsic sensitivity, as compared to the B & K instrument. The amount of this reduction can be appreciated by considering the nominal parameter values for the B & K units-20 mm gap spacing, with a bias voltage of 200 V. Thus the electric field is greater than that of the SDC unit by a factor of 500, which corresponds to 54 dB. Any SDC microphone that should become a viable candidate for fielding must have reduced the gap spacing significantly below the 500 mm value of the prototype. By means of standard processing techniques, such as etching, it should be straightforward to reduce the gap by an order of magnitude. As demonstrated by the author during his doctoral work with ultrasonic capacitive microphones, this task can be greatly simplified by polishing the electrode surfaces until they're nearly optically flat in the visible. The microphone may prove to be acceptably sensitive at this point, without having to go to higher drive voltages. In the unlikely event that still greater sensitivity at high frequencies is needed, an inexpensive transformer may be added. A commercial transformer costing less than \$10 (price per unit in small numbers) has been routinely used to provide a seven-fold increase in drive voltage with the TEL-Atomic electronics.

### 8.5  Temporal data

It was mentioned earlier that some time traces were taken with a storage oscilloscope. Although it would have been possible to use an IEEE488 standard hookup to produce records from which figures could have been generated, time constraints did not permit this.

In these shock studies, a pellet gun was used with scotch tape placed across the breech opening, where the pellet is normally inserted. Loud bangs from this gun, subsequent to trigger pulls, were studied with the microphone. In lieu of this schock tube device, ordinary vigorous hand claps are capable of much the same signature, except that the levels are lower and the hands eventually begin to complain. An especially useful source of shocks may be an ordinary hand-cranked Wimshurst machine. The breakdown of air (simulated lightning/thunder), occurs when the Wimshurst operates with its largest capacitors, is quite loud and abrupt. In use, one must be careful to avoid electromagnetic pulse damage to the computer acquisition system. The author discovered this need while using a Wimshurst machine to study field dependent mechanical strains of dielectrics placed between capacitor plates. The mechanisms of charge storage on dielectrics, as influenced by defects (pertinent to electret microphone design), has not been carefully studied. Understanding the long term stability of these states could be important to future improvements of a variety of devices such as electret microphones.

There are several noteworthy results of the shock studies. One, the diaphragm of the SDC prototype microphone is underdamped, as expected. Reduction of the gap spacing from 500 mm to 50 mm should increase the damping to a more desirable level. The free decay period of about 1 ms is consistent with its measured resonance frequency of 1 kHz, discussed earlier.

It was also found, as expected, that the shock signature depends significantly on environment, particularly echo features. Multiple ringing can be readily seen when the shock bounces several times around a room with hard walls and/or ceilings and floors. The value of an anechoic chamber where these reflections are virtually eliminated, becomes quickly appreciated.

## 9  Hydrophone

### 9.1  Introduction

The development of a low frequency hydrophone was motivated by work performed by Michael Scanlon, of the ARL acoustics group (Sensors Directorate). He is the inventor of a recent patent that was assigned to ARL[]. Not only has he shown that a hydrophone placed inside a fluid-filled bag is useful for SIDS monitoring; it could also become commonplace on the battlefield. Credence for such a statement is realized by recognizing that the stethoscope has been used for many decades by physicians to assess health of a patient. Whereas the frequency response in the case of the conventional stethoscope is limited by the quality of the doctor's ears, a properly built hydrophone opens new vistas, because of sensitivity to infrasound. Scanlon has shown, for example, that there are low frequency signatures" associated with complex pulmonary/cardiovascular interactions. For example, there are significant differences in time/spectral plots, depending on whether the patient's breath is held. Also, skipped heartbeats and murmurs can be readily detected in some cases. A full appreciation for the value of this technology must obviously await extensive testing by physicians.

The sensor used in the SIDs monitor work was a water-proofed electret type. As such, it comprises two membranes: (1) one that separates the air cavity of the microphone proper from the water, and (2) the standard diaphragm of the sensor. The extra membrane (water separator) is a cause for reduced sensitivity; and also, the low frequency cutoff of the microphone is greater than desired. Thus it was natural to consider an SDC alternative to this Knowles' unit.

The SDC hydrophone built by the author during this study is illustrated in Figure 27. All of the semi-circular electrodes were cut with scissors from brass shim stock of thickness 250 mm. Their radii are 0.7 cm. The static pair, 3 and 4, were cut with small protruding ears" for solder attachment to the coaxial cable which connects the hydrophone to the TEL-Atomic support electronics input. These were superglued to the housing, which threads into the water bottle. A slit was cut through the flat base of this cylindrical housing, to freely pass the cross coupling links of the moving members of the hydrophone (1T, 1B, 2T, and 2B). Note the difference between this design and that of the microphone. In the microphone, the central electrode pair were the moving ones (diaphragm bowing); whereas, here the central pair are fixed, and the outside set do the moving. Also note that in the microphone case, the drive equipotential pairs were formed by solder attachment of external wires. For the hydrophone of Figure 27, the cross connections are by means of the links. These rectangular links are about 1/3 a diameter wide and positioned on opposite sides of center, so that they're electrically separated. The SDC drive signal is applied from a coaxial cable by soldering to the topside of 1T and 2T.

Picture Omitted
Figure 27: Edge view illustration of the prototype SDC hydrophone.

The narrow rectangular strips (both ears and links) were cut from the same brass stock as the electrodes. They were formed as an integral part of the lower (2B and 1B) set. The top moving electrodes (1T and 1B) were soldered to these strips in the last step of fabrication. The strips provide the structural rigidity with which the electrodes maintain parallelism. The rubber pad shown was cut out of a thin black rubber glove. It provides restoration to the moving electrodes when water pressure is reduced. To provide a water seal, silicone was used to fill the region inside the housing that surrounds the plunger. This silicone also adds to the restoring force. The remaining unlabeled four rectangles" shown in the figure are actually two center slotted circular disks of thin insulating fiber glass to which the moving electrodes were glued. The plunger was turned from plexiglass and glued to the bottom of the sensor. It should be noted that (for purpose of clarity) Figure 27 is not drawn to scale. The spacing between all adjacent electrodes is roughly 1/2 mm in the unstrained state. An increase of pressure on the plunger side of the hydrophone causes the 1T - 3 and 2T - 4 gap space to increase, at the same time as the 2B - 3 and 1B - 4 gap spacings decrease. Thus the device is a full bridge sensor, a hallmark feature of the SDC sensor technology. (Actually, the symmetry is slightly degraded by insignificant dielectric constant mismatches.)

### 9.2  Experimental results

Shown in the top trace of Figure 28 are the results of an experiment in which the author sat quietly in the ARL anechoic chamber, with the hot water bottle containing the SDC hydrophone, placed lightly between his left arm and chest. This spectrum was obtained with the LabView FFT software mentioned earlier. For reference purposes, the background is shown in the lower trace. It was generated by placing the bottle at rest by itself inside the chamber.

One can quickly glean a number of interesting conclusions from Figure 28. The line at 1.3 Hz (upper trace) corresponds to heart beat (78 per min). In the vicinity of 0.2 Hz (broad line") is evidence of respiration (12 per min). The sharpest line in the spectrum is the one whose frequency is 7 Hz. It has been long known that the body exhibits resonances in the vicinity of 10 Hz, of which this is probably an example. It is undoubtedly the means whereby even a deaf person may feel" some sounds.

From Figure 28, one may also readily determine the upper frequency cutoff of the hydrophone. The rolloff at about 15 Hz is consistent with the relatively large mass of the moving parts of the sensor. For production purposes, this mass needs to be reduced (to yield a cutoff near 50 Hz), and the sensitivity should be increased by decreasing the gap spacing between electrodes. For improved sensitivity, an increase of the drive voltage from +5 V could be accomplished using a transformer (as mentioned relative to the SDC microphone).

Whereas Figure 28 was obtained from a fairly short record ( < 1 min), some other data were collected over longer intervals using a B & K spectrum analyzer operating in an averaging mode. Shown in Figure 29 is a case in which the hotwater bottle was simply laid on top of the author's chest (not in the anechoic chamber) Figure 30 is the associated background, taken with the bottle at rest on a chair. Although the 1.3 Hz fundamental frequency of his heartbeat is not readily apparent in this case, the 2nd through 4th harmonics are way above noise, particularly the 4th harmonic. Additionally, the broad body resonance" region centered at 10 Hz appears to accommodate harmonics 6 through 10. With regard to line widths, there are evidently major differences, in general, according to the duration of the analyzed interval. In this case the properties do not derive from the uncertainty principle (which causes line broadening), but rather from physiological variations that appear to correlate with periodicities yet unstudied. The implications should be of great interest to the medical community, particularly physiologists. For the author, these results are not surprising; since years ago he looked at the time delay between the R-wave of the EKG and the arrival time of arterial pulses at various points in the body. In this work he was able to show that these times (associated with a highly nonlinear system) depend on stroke volume of the heart [].

Figure 28: Sample response of the SDC hydrophone.

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Figure 29: Response measured with an analogue spectrum analyzer.

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Figure 30: Background associated with Figure 29.

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## 10  Electronics

A primary factor in the high cost of existing commercial electronics support for the SDC sensors has been the use of a sine wave drive. By integrating the Signetics NE5521 integrated circuit into the TEL-Atomic package, the overall cost was reduced and performance was improved. In spite of this, the cost has remained too high for extensive fielding of an SDC microphone. Thus, the author was led to consider digital circuitry. Two critical factors had to be addressed: (1) inexpensive bipolar square wave generation and (2) switching circuits to provide synchronous demodulation. John Speulstra and Art Harrison provided ready solutions to both these problems. For the drive, one uses a simple opamp oscillator feeding a flip-flop. For the demodulator, a CMOS switch can be configured with an opamp, to do what otherwise has taken a larger number of active components. Their design is shown in Figure 31. The filters at all power points (inductor" & 2 capacitors, each case) may not be necessary to filter out noises, particularly when using battery power. The inductor is in each case a small toroidal ferrite with 3 turns of hookup wire.

Figure 31: Prototype electronics support for SDC sensors.

(This page left blank for replacement with separate figure 31) To test the electronics, an SDC position sensor was fabricated from PC boards. The cardinal" electrode dimension of the sensor was 2.4 by 3.6 cm, so that the maximum range of linearity is +1.8 cm. The spacing between adjacent electrodes is 0.2 cm. This sensor is known from other studies to be very linear []. The results of a linearity test with it are shown in Figure 32. (Note that the negative range exceeds the positive range by 0.5 cm; this is due to stray capacitance unbalance in the electronics-readily fixed.) Although the sensitivity is low, it is seen that the linearity of the electronics is outstanding. The gain of the instrumentation amplifier (AD620) had to be kept small to avoid unpleasant saturation effects, for large deviations from the null position. Time did not allow, but the author is confident that the sensitivity can be made adequate by following the last indicated stage of Figure 31 with an additional opamp (probably only one required).

Figure

Figure 32: Test of electronics linearity.

## 11  Conclusions

This study has demonstrated that the SDC technology is capable of filling an important battlefield sensing void-that of infrasonics. The importance of certain acoustic signatures, from sounds that are both airborne and intrinsic to the human body, has come to be appreciated only recently. The seminal contributions to this field are the result of research done by personnel associated with the Army Research Laboratory at Adelphi, MD. Were cost not a factor, some commercial microphones might fill a limited number of categories of this void. Other categories would remain unfilled, such as those involving robustness.

Various configurations of SDC microphones and/or hydrophones are envisioned for addressing presently known challenges of battlefield deployment. The envisioned instruments are expected to be inexpensive, robust, and suitable for use by soldiers in a variety of applications. In some cases, a single microphone may be all that is required; others might use a large number of sensors in an array.

The preliminary work of this report involves measurements performed with three prototypes. These prototypes addressed both mechanical and electronic issues via: (1) an SDC microphone, (2) an SDC hydrophone, and (3) an electronics support package for both. Further refinements are required, largely ones involving issues of mass production.

## 12  Recommendations

The most cost effective SDC microphones for initial fielding are likely to be of macro" rather than micro" type. This is expected in spite of recent research trends in the direction of micro-electro-mechanical- systems (MEMS). The SDC technology has been recognized for several years as one of the best candidates for MEMS integration on silicon; nevertheless, conventional production methodologies should be the first ones evaluated. Only then can the payoff" of miniaturization be quantified. This recommendation is based partly on the known limitations of a popular commercial device, whose price has dropped because of MEMS type processing in high volume- the electret microphone of hearing aids. These microphones do not respond at all frequencies of interest to the army, partly because of their small size and resulting small capacitance. Since all miniaturized capacitive sensors are subject to the challenges of femtoFarad scale, there is merit to first considering larger instruments for which this constraint does not apply.

## 13  Acknowledgements

The author is appreciative of the support provided by Steve Tenney, Mike Scanlon, and Steve Post, of the Acoustics group, Sensors Directorate, at the Army Research Laboratory, Adelphi Maryland. (This work would not have been funded without Mr. Tenney's recognition of the potential value of the SDC pressure sensor as a microphone.)

Steve Post assembled the hybrid electronics package described in this document and also produced the associated figure. His assistance with other hardware tasks is much appreciated, as are the extra mile" contributions of Mike. The pellet gun shock tube was loaned by Jerry Gerber, who also provided the author with useful shock references.

Two others who deserve special recognition are Art Harrison and John Speulstra, who are primarily responsible for the new electronics design. Being of a different ARL group than acoustics, the author was much impressed with (1) their interest in this work, in spite of their own pressing schedules, (2) their frequent assistance in spite of those pressures, and (3) their outstanding abilities.

## References

[]
P. Morse, Vibration and Sound, 2nd Ed., New York, McGraw-Hill (1948).
[]
Condenser Microphones and Microphone Preamplifiers for acoustic measurements Data Handbook, Bruel & Kjaer, p. 26 (1982).
[]
J. Fraden, AIP Handbook of Modern Sensors, Amer. Inst. of Phys. 1993.
[]
Data Sheets, Knowles Electronics Inc, 1151 Maplewood Dr., Itasca, Ill 60143.
[]
R. Peters, Temperature dependence of the nonlinearity parameters of copper single crystals, PhD dissertation, Univ. of Tennessee, 1968.
[]
A description of Green's functions from a physicist's perspective is found in J. Marion & S. Thornton, Classical Dynamics of Particles and Systems, Saunders College Publishing, 4th ed. Fort Worth , 1995.
[]
J. Cooley & J. Tukey, An algorithm for the machine calculation of complex Fourier series". Math. Comput. 19 , 297-301 (1965). For graphical assistance in understanding the discrete Fourier transform, the reader is referred to: R. Peters, Fourier transform construction by vector graphics", Am. J. Phys. 60, 439 (1992).
[]
W. Press et al, Numerical Recipes, The Art of Scientific Computing Cambridge Univ. Press, 1986.
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R. Peters, Symmetric differential capacitance transducer employing cross coupled conductive plates to form equipotential pairs", U. S. Patent No. 5,461,319 (1995).
[]
R. Peters, Remote respiratory monitor", 8th IEEE Symp. on Comp. Based Medical Sys., p. 204 (1995).
[]
See, for example, R. Pallas-Areny & Webster, Sensors and Signal Conditioning, Wiley, New York (1991).
[]
R. Whiddington, The ultra-micrometer; an application of the Thermionic Valve to the measurement of very small distances", Phi. Mag. 40, 634 (1920).
[]
This technique was developed to calibrate pressure sensors sold by TEL-Atomic, Inc., Jackson, Mich., for physics laboratory and demonstration purposes.
[]
This company markets physics demonstration and laboratory equipment which uses SDC sensors. Instruments which they sell include: (1) Computerized Cavendish Balance, (2) Multipurpose Chaotic Pendulum, (3) Pressure Transducer, and (4) Accelerometer.
[]
Michael V. Scanlon, Sudden Infant Death Syndrome (SIDS) Monitor and Stimulator", U. S. Patent No. 5,515,865. Other pending patents, relevant to battlefield use, are under review.
[]
During the early 1970's at the University of Mississippi, the author measured propagation speeds of arterial pulses using an EKG and a pulse sensor. As inferred by known influences of posture and/or the valsalva maneuver, these measurements showed that the speed is a function of stroke volume.
[]
The planar electrode SDC position sensor used for the linearity tests has been referred to as the author's crucial" invention. It works on the basis of area variation rather than the gap changes of the pressure sensor. It is described in the patent (ref. 9).

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