A Soldier's Microphone/Hydrophone

(Design Considerations for a New Technology)

Dr. Randall D. Peters^*

Department of Physics
U.S. Military Academy
West Point, New York

ARL Sponsor: M. V. Scanlon
Sensors Directorate
U.S. Army Research Laboratory
Adelphi, MD

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

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

2  Introduction

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 P_ref  =  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/( P_ref)].

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

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

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

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

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

2.1.1  Electret Microphone

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

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, -s_b on the bottom surface of the film. (The field in the air gap is oppositely directed.) A corresponding +s_b 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

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)/( e₀e_r)]; where e_r 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, s_b/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

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

2.2  Sensitivity

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

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

The voltage, V_i(t), corresponds to either Equation (9) or Equation (10) with DV ® V_i(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/d_m.

The effect of the amplifier is to cause a ``voltage division", so that only a fraction of V_i(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

3.2  Frequency Response of the Condenser Microphone

4  Time Domain Analysis

The equation of the lead circuit is

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

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

4.1  Convolution Integral

It is seen from Equation(21) that for every desired value of the time in V_o(t), a separate integral is required, unless V_i(t) is simple enough to yield a complete analytic solution. In practice, numerical processing is usually necessary. The only practical way to generate V_o(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

(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 V_i(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 V_o(t) for the assumed V_i(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

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

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.

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

A pulse of sound input to the microphone is distorted greatly when the pulse width, Dt  >  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.

Stated in terms of rolloff frequency, pulse distortion begins to become significant as f_R® 1/(2pDt) and is severe for f_R  >  (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.

4.3  Physical basis for f_R

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 f_R 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 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 ·f_R, where Dt is the width of the shock and f_R is the rolloff frequency (3 dB point) of the microphone. Distortion results when this product exceeds a threshold of (Dt ·f_R)_min  =  1/(2p)  =  0.16. For Dt ·f_R  >  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

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.

The capacitance, C₀  =  e₀^A/_d, where e₀ is the permittivity of free space (really air), A  =  pa², and d_m 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

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

An advantage over a half bridge is worth noting. For the half bridge, the right hand side of Equation(27) would be [(V_iy)/( d_m(1-[(y²)/( d_m²)]))], 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 C_s 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

For greatest sensitivity, it is seen from Equation(28) that C_s 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

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

5.1.1  Synchronous Detection

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.

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, w_S, in the following equations is the same as w_c in Equation (29), and V_S  =  2^-1/₂KP_S 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

The product is given by

where the constant, k, embodies the filter attenuation; it also is determined by the reference signal voltage , V_R, 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

5.1.2  Advantages & Disadvantages

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

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

In graphing Figure 16, the following Equation was used.

5.3  Transient Response-SDC microphone

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 f_R, 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

6  Diaphragm Modes

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

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

6.2  Plate Modes

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

7  Prototype Microphone Mechanical Design

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 1_T and 1_B and likewise between 2_T and 2_B.

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.

8  Experimental Results

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

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".

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

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(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

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

8.5  Temporal data

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 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 (1_T, 1_B, 2_T, and 2_B). 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 1_T and 2_T.

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 (2_B and 1_B) set. The top moving electrodes (1_T and 1_B) 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 1_T - 3 and 2_T - 4 gap space to increase, at the same time as the 2_B - 3 and 1_B - 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

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 [].

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10  Electronics

(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).

11  Conclusions

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

13  Acknowledgements

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

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P. Morse, Vibration and Sound, 2nd Ed., New York, McGraw-Hill (1948).

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Condenser Microphones and Microphone Preamplifiers for acoustic measurements Data Handbook, Bruel & Kjaer, p. 26 (1982).

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J. Fraden, AIP Handbook of Modern Sensors, Amer. Inst. of Phys. 1993.

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Data Sheets, Knowles Electronics Inc, 1151 Maplewood Dr., Itasca, Ill 60143.

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R. Peters, Temperature dependence of the nonlinearity parameters of copper single crystals, PhD dissertation, Univ. of Tennessee, 1968.

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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.

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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).

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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).

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R. Peters, ``Remote respiratory monitor", 8th IEEE Symp. on Comp. Based Medical Sys., p. 204 (1995).

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See, for example, R. Pallas-Areny & Webster, Sensors and Signal Conditioning, Wiley, New York (1991).

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R. Whiddington, ``The ultra-micrometer; an application of the Thermionic Valve to the measurement of very small distances", Phi. Mag. 40, 634 (1920).

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This technique was developed to calibrate pressure sensors sold by TEL-Atomic, Inc., Jackson, Mich., for physics laboratory and demonstration purposes.

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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.

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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.

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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.

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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).