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Diagnostic measurements in capacitive transducers
Diagnostic measurements in capacitive transducers H. C a r r a n d C. W y k e s Department of Manufacturing Engineering and Operations Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK Received 18 October 1991; revised 29 June 1992 The capacitive transducer can be used to transmit ultrasound more effectively into air than the more conventional piezoelectric transducer because of its better impedance matching. The behaviour of such transducers is not, however, well understood. An extensive investigation of the way in which the performance of such transducers is affected by variations in backplate, membrane and drive voltage is reported and significant parameters for the construction of such transducers are defined.
Keywords: ultrasonic transducers; capacitive transducers; airborne ultrasonic transducers
Ultrasound is widely used in non-destructive testing and medical imaging and a wide variety of piezoelectric transducers are available at a variety of frequencies, While a variety of single point ultrasonic systems for ranging in air are available, there has been very little progress in producing arrays which can be used for 2-D and 3-D ranging and this is largely due to the lack of suitable transducers. When piezoelectric transducers are used to transmit ultrasound in air, there is a large mismatch in acoustic impedance between the transducer face and the air which means that they are not very efficient. An alternative is the capacitive transducer. There is a substantial body of existing work on these transducers 1-~1 but there is not, as yet, a clear set of guidelines as to how a transducer with a specific resonant frequency, bandwidth and sensitivity can be designed and constructed. This is undoubtedly due to the complicated nature of the interactions which occur in the transducer. The work reported here is primarily an empirical study of the behaviour of such transducers which was aimed at arriving at design rules for the construction of capacitive ultrasonic transducers - it is discussed in more detail in Reference 12. The next section looks at the factors which affect the behaviour of the capacitive transducer and discusses previous work in terms of these factors. The section after outlines the experimental methods used, while in the following sections, the effects of bias voltage and membrane properties are outlined. The performance of transducers which use random and grooved backplates is described and other systems which have been developed based on this work are also briefly described.
as an acoustic detector and reported as early as 19171 . A thin diaphragm is located close to a conducting backplate, a dc voltage is applied between the backplate and the diaphragm, and when the diaphragm is set into vibration by a sound wave, an alternating voltage is produced which is proportional to the incident sound wave. Acoustic microphones use a metallic diaphragm to give a frequency response which has a resonant frequency beyond the audible spectrum so that its response is quite flat at acoustic frequencies. To produce a transducer which resonates at ultrasonic frequencies, it is necessary to use a thin plastic film ( ~ 2 - 1 5 / t m ) which is coated with a conducting layer and which is placed in contact with the backplate - these transducers can be used to transmit ultrasound by driving them with an appropriate alternating voltage. Figure 1 shows the components which make up the capacitive transducer, namely, the transducer body, the backplate and the membrane attached, under tension, to a ring. The response of the transducer is critically dependent on the structure of the backplate. Ultrasonic capacitive transducers can be broadly divided into two categories, those which use backplates with a random surface finish and those which use grooved backplates.
The capacitive transducer
Transducers with random backp/ates Some workers, using backplates whose surfaces are microscopically rough, have found that the resonant frequency of the transducers increase as the roughness decreases 3'4'6. This can be explained by assuming that the membrane acts as a frictionless piston whose motion is constrained by the air-gap which is effectively a spring - see Figure 2. The resonant frequency of the system is given by
The capacitive ultrasonic transducer is a particular form of the condenser microphone used widely for many years
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Ultrasonics 1993 Vol 31 No 1
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Capacitive transducers: H. Carr and C. Wykes
Figure 1 Componentsof a capacitivetransducer Membrane
I
Backplate
[
Air gap
1
~)
I
Figure 2 Simplemodelof a capacitivetransducer 7 = adiabatic constant of air P, = atmospheric pressure p = density of membrane d m = thickness of membrane d, = thickness of air-gap For a Mylar or a K a p t o n membrane, and normal atmospheric pressure, the equation becomes f=
1
1.6 x 1 0 3 _ kHz (d,dm) 1/2
(2)
where da and d= are in micrometres. When the surface finish of the backplate is random, is it not clear how the thickness of the air-gap can be quantified though it is clearly related to surface roughness. Other authors 2's'1°'11 have considered the capacitive transducer primarily as a flexible supported membrane. The resonant frequency of such a system is determined by the tension of the membrane and its dimensions a circular membrane of diameter D and tension T has a fundamental resonant frequency given by Reference 13 as 'f=
2"405(Ty/2 ~D \ P m J
(3)
and an infinite set of related harmonic frequencies. This neglects any effect due to air-gap resistance and stiffness. Merhaut 8 and Warren ~ give theoretical analyses which include the resistive and stiffness forces of the air-gap as well as the m e m b r a n e resonance. They both assume that the membrane is supported uniformly either by a circular or a rectangular frame. The real capacitive transducer, when used to produce ultrasonic waves, is much more complex than this. To produce the small air-gap which is required to produce an ultrasonic resonant frequency, the membrane has to be stretched across the backplate so that it is supported by the high points on the backplate - see Figure 3. When the bias voltage is applied, the membrane is further stretched and distorted. It has been shown using optical interferometry 14 that the surface of the membrane has high points of up to 5 pm at the points of support and also that the membrane resonates at a
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Ultrasonics 1 9 9 3 Vol 31 No 1
series of widely different frequencies in a single transducer. A useful model would need to be able to include both the random points of support, the varying tension away from those points, and the variation of depth of the air-gap across the surface. In addition, it has not been possible, to date, to devise a method by which the membrane tension can be measured though it is likely that it varies significantly in different regions of the membrane, so that even with a theoretical model, it would not be possible to predict precisely the behaviour of any specific configuration of membrane and backplate. The first model suggests that resonant behaviour is determined by the air-gap and membrane thickness, while the second model suggests that it is determined by the separation of points of support, membrane thickness and tension. In practice, the present authors as well as several other authors have found that the air-gap model predicts the resonant behaviour of random-backplate transducer reasonably well and in a later section a method for predicting resonant frequency is given. The work described there also suggests, as would appear intuitively correct, that if the resonant frequency given by the air-gap and that determined by the membrane resonance are matched, the transducer efficiency is likely to be maximized. A quantitative prediction of the sensitivity and bandwidth is not, however, possible, in the absence of a suitable theoretical model.
Grooved transducers The use of grooved backplates has been reported by several authors 2'4'5'9. Several different factors may determine the behaviour of such transducers. The membrane mounted on a grooved backplate has a fundamental resonance whose wavelength is equal to twice the groove width so that the frequency, J'~, is inversely proportional to the groove width. There are also a series of harmonics which are related to the fundamental frequency by f, = nJ'~,
n = 3,5,7 ....
(4)
Only odd harmonics occur because the membrane is being driven synchronously at both rail edges, so that all even harmonics will cancel. There is a layer of air trapped between the rails and the membrane with a characteristic stiffness, and another air-layer in the grooves with a different characteristic stiffness. Each of these air-gaps has an associated resonant frequency. The resonant frequency of a transducer with a grooved backplate may be governed by any or all of these factors. Reported results give conflicting evidence on which effects are important 2'4'5'9. The appearance of several peaks in the frequency response (referred to here as a 'multipeaked response') indicates that the first of these effects
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Figure3 Detailedconstructionof a capacitivetransducer
Capacitive transducers. H. Carr and C. Wykes is significant, although this does not exclude the air-gap effects, while the absence of such a response indicates that one or both air-gaps are the main determinant of the resonant response. Kuhl et al. 2, Matsuzawa 4 and Morris 5 reported a multi-peaked response. Matsuzawa concluded that such microphones are 'practically useless'. Kuhl et al. 2 and Morris 5 believe that the resonant frequency of such transducers is determined by the resonant frequency of the m e m b r a n e suspended above the groove, while Martin 9 claimed that the resonant behaviour is determined by the finish on the rails and that the only function of the grooves is to increase the sensitivity of the transducer. The only commercially available capacitive transducer, produced by Polaroid as a range finder for their camera, is a grooved transducer which does not have a multi-peaked response. It has been shown 15 that the resonant frequency of this transducer is critically dependent on the microstructure of the rails since, when these are lightly polished, the resonant frequency of the transducer changes from 40 kHz to 100 kHz. The work described here using grooved backplates found that the transducers involved generally showed a multi-peaked frequency response but other work by one of the authors 15 has shown that the resonant frequency of transducers made using v-grooved backplates had a resonant frequency which could be predicted from the groove air-gap. At present, it is not possible to predict quantitatively the sensitivity or bandwidth of transducers which use grooved backplates because of the complicated processes governing their behaviour.
Experimental
methods
used
A system has been constructed which enables the resonant frequency, bandwidth, and sensitivity of a transducer to be measured and recorded. The system enables the transducer to be driven with a sine wave of 10 ms duration at a range of frequencies between 10 k H z and 5 MHz. The response of the transducer can be detected in several ways with the system: (i) (ii) (iii) (iv)
a calibrated microphone with a flat response up to 150 k H z can be located in the transmitted field; a pair of matched transducers can be used as transmitter/receiver; a large flat reflector can be used so that a single transducer acts as both transmitter and receiver; a laser interferometer can be used to measure the motion of the surface of the transducer at a specific point.
The optical method has the advantage that transducer performance can be studied without the results being affected by attenuation of the ultrasonic waves in air. The system is driven under computer control enabling the results to be stored on disc and plotted automatically. It determines the resonant frequency as the frequency at which the m a x i m u m returned signal is obtained. As noted previously the 'optical' resonant frequency was found to vary at different points on the membrane, so that this measurement had to be made at a series of points across the surface and an average value found. Measurements of the surface topography of the membrane were made using an interference microscope.
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Bias voltage (V) Figure 4 Variation in resonant frequency and sensitivity with increasing bias voltage
Investigation
of bias
voltage
effects
These tests were conducted using a random backplate with an R a value of 2.5/tm and a membrane thickness of 6 ~m. The bias voltage was varied from 0 to 250 V. It was found that both the resonant frequency and the sensitivity increased quite rapidly up to 75 V, increased more slowly after that and then flattened off at about 180 V - see Figure 4. The variation in resonant frequency with variation in bias is probably caused by the reduction in the air-gap thickness as the membrane is pulled in. Above a certain voltage, the air-gap does not increase any further so the frequency becomes constant. The variation in sensitivity is likely to occur because when the membrane tension is insufficient, the membrane is unable to exert sufficient force on the air-gap to resonate. The behaviour of this transducer appears to be dominated by the air-gap effect. It was also found that the bias voltage could induce a polarization voltage across the membrane which could increase or decrease the bias voltage by up to 25 V. This polarization voltage decays after the removal of the bias voltage. Investigation
of membrane
effects
These tests used the same backplate as used in the tests discussed in the previous section. M e m b r a n e materials Only two suitable materials were found Mylar and Kapton, manufactured by D u p o n t - which gave very similar transducer performance. Gold and aluminium were used as coating materials and again were found not to give any significant difference in transducer performance. Membrane thickness Membranes of several thicknesses were tested. A linear relationship was found between the resonant frequency and the inverse square root of the membrane thickness as indicated by Equation (1) see Figure 5. The sensitivity at the resonant frequency was found to decrease linearly with increasing membrane thickness. This would suggest that the membrane used should be as thin as possible; however, problems arise in producing crease-free membranes at thicknesses below 5 pm and, in addition, the membranes become very fragile.
U l t r a s o n i c s 1 9 9 3 V o l 31 N o 1
15
Capacitive transducers." H. Carr and C. Wykes The surfaces which were used were (i) (ii) (iii)
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0
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0.3
r
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0.6
1/~--~, Figure 5 Resonant frequency as a function of the inverse square root of membrane thickness
Membrane tension A working capacitive transducer was produced by pre-tensioning a membrane, locating it in front of the backplate, and then moving the backplate forward into the membrane. No significant resonance was obtained until the membrane and backplate were in contact. The sensitivity changed significantly as the backplate was forced harder into the membrane altering its tension. A jig was constructed which enables a pre-calibrated tension to be applied. The position of the backplate relative to the membrane was measured using a micrometer. Measurements were made with pre-tension values from 25 200 g. It was found that the resonant frequency and bandwidth were not significantly affected by varying the tension (as expected from the air-gap model). However, the sensitivity peaked as the backplate position was varied to produce a change in the tension. The peak value was highest for the 75 g pre-tension setting. Thus, both the pre-tension and the backplate tensions need to be optimized to get the best performance from the transducer. In later investigations of transducer performance, a simple pre-tensioning method was devised which gives consistent optimized results 15 Random
surface
roughness average
Rpm: maximum height of surface high points da: mean depth found from a capacitance measurement 2~:
16
surface wavelength
Ultrasonics
Typical surface profiles for each of these surfaces are shown in Figure 6. The R a values varied from 15/~m down to 0.02 #m (this was an aluminium coated optical flat) giving resonant frequencies measured optically from 70 kHz to 3.3 MHz. No significant differences could be discerned in the behaviour of transducers made with the differing types of random backplate. The values of R a, Rpm and d, were found, not surprisingly to be strongly correlated to one another. The value of )os was also found to be correlated with the R a, Rpm and da values though less strongly. The relationship of the lateral structure of a surface to its roughness variation is normally determined by the process by which the surface has been produced and, usually, the 'wavelength' increases with increasing roughness so that the correlation between surface roughness and surface wavelength is to be expected. Measurements of resonant frequency, sensitivity and Q-factor were made ultrasonically using two matched transducers and optically by measuring the amplitude of oscillation with the Michelson interferometer; measurements of the surface topography were made using the interference microscope.
Resonant frequency Equation (2) predicts that the resonant frequency should 25
=L
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5 10 Relative lateral distance (/~m)
backplates
Parameters and surfaces used The discussion in the second section showed that the behaviour of a transducer which has a random backplate may be determined by the properties of the surface in the horizontal (air-gap model) a n d / o r the vertical direction (membrane model). The vertical properties (surface roughness) determine the depth of the air gap between membrane and backplate, while the horizontal structure (surface 'wavelength') determine the distance between the points at which the membrane is supported and therefore the resonant frequencies of the individual resonators within the membrane. The properties of a random surface can be characterized in a variety of ways. The parameters which were used here were Ra:
E D M (electro-discharge machined) Brass turned Polished
1 9 9 3 V o l 31 N o 1
=L
8,,
2.5
U3.c
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Relative lateral distance (#m) 1
=L
8+.
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Relative lateral distance (/~m) F i g u r e 6 Typical surface profiles for (a) EDM; (b) brass-turned and (c) polished backplates
Capacitive transducers: H. Cart and C. Wykes Table I Theoretical and empirical straight line coefficients relating resonant frequency to surface finish
Intercept (kHz) Theoretical Calculated using Rpm Calculated using d a
0
Slope (kHz/~m 1/2)
- (90 _+ 20)
650 770 _+ 50
- (390 _+ 30)
1100 _+ 50
vary in inverse proportion to the square root of the air-gap depth. Each of the parameters, Ra, Rpm and d, is likely to be related to the thickness of the air-gap. The resonant frequency, as measured both ultrasonically and optically, was plotted against the inverse square root of each of these parameters. In the case of the ultrasonic measurements, the curves appeared reasonably linear, in each case, up to about 250 kHz and fell off after this; it is thought that the fall-off occurs because of the absorption of the ultrasound by air. It does not occur when the resonant frequency is measured optically since it is not affected by air attenuation. The Rpm results gave a reasonable straight line up to frequencies of 600 kHz while the d a values gave a linear fit over the whole range. The values of the slopes and intercepts for these results are shown in Table 1. The Rpm values show better agreement with the theoretical model but are valid only up to 600 kHz. The d a values can be used over the whole range. However, the first of these is more likely to be useful in specifying the performance of a transducer since the Rpm value can be predicted before constructing the transducer while the d a value can only be found when the transducer is assembled. The membrane model of the transducer predicts that the resonant frequency of a component sub-membrane should be proportional to the inverse of the distance between the points where it is held fixed against the backplate; this was investigated by plotting optical resonant frequency against (1/2s). The correlation coefficient in this case was 0.77; since 2~ is, in any case, correlated with roughness, and the correlation is less than that obtained for the air-gap model, it can be deduced that the prime determinant of resonant frequency is the air-gap thickness.
Sensitivity Sensitivity was measured both by measuring the received signal level and also by measuring the amplitude of the vibration of the membrane at several points on the membrane surface. While it is not possible to predict the sensitivity of the transducers, it seems reasonable to assume that it is optimized when the average 'size' of the sub-membranes matches the resonant frequency of the air-gap. The ultrasonic sensitivity was plotted against 1/x/d,, 1/2s and 1/[2sx/d,]. Correlation coefficients 0.59, 0.88 and 0.96 respectively were obtained; this appears to confirm the hypothesis. In practice, it would be difficult to design and construct a transducer to satisfy this condition, firstly because the required wavelength is not known, since the tension of the assembled membrane cannot be easily measured, and secondly because it would be very difficult to control the wavelength of a random surface produced by a conventional process. The values of the amplitude of oscillation as measured
by the interferometer were plotted against R,, 2 s, and sensitivity as measured ultrasonically, and a linear regression performed. The correlation coefficients were 0.58, 0.46 and 0.36 respectively, so that it appears that there is no useful information to be gleaned from these measurements.
Q-[actor It is not possible to make any quantitative pedictions of the Q-factor or bandwidth of the transducers. It is expected that the Q will increase (i.e. bandwidth will decrease) with increasing sensitivity since both are determined by the damping in the system; this was indeed found to be the case. The actual values of Q varied between 1 and 8. Topography of the membrane An interference microscope was used to determine the surface topography of the membrane this is discussed in detail in Reference 15. Figure 7 shows interferograms of the membrane before and after the bias voltage has been applied. They show that the membrane is significantly deformed by the application of the bias voltage, changing from a reasonably linear interference pattern, implying flatness of the order of 0.5/~m to sharply curving, and in some cases closed fringes, patterns indicating variations of surface height of several micrometres the bias voltage clearly causes the membrane to conform much more closely to the backplate topography.
Conclusions The work described here confirmed that the resonant frequency of a transducer constructed using a random backplate is governed by the roughness of the surface and the best prediction of the resonant frequency can be made by finding the air-gap thickness from a measurement of the Rpm of the backplate or d, found from a capacitance measurement using the values in Table 1. Resonant frequencies of up to 3 MHz have been obtained using a mirror as a backplate with a 2.5/~m membrane. The sensitivity of the transducer is optimized by matching the 'wavelength' of the surface to that which gives component membranes whose frequencies match that defined by the air-gap thickness. The Q-factor varies significantly between values of one and eight, and increases with increasing sensitivity so that it is likely to be governed by the same parameters as govern the sensitivity.
l a
/
b
F i g u r e 7 Tracings of interferograms of the surface of the membrane (a) before and (b) after the application of the bias voltage
Ultrasonics 1993 Vol 31 No 1
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Capacitive transducers. H. Cart and C. Wykes
Transducers using grooved backplates
250
Backplates used
200
A series of plates with circular grooves was constructed and the performance of transducers using them as backplates measured. The following parameters can be varied; groove width, groove depth and rail width. The values can be varied relative to one another, or the overall pitch can be varied keeping the relative values constant so that an enormous number of varying backplates could be tested. In this work, four sets were used as follows: Set A
equal values for groove width, and depth and rail width; pitch varied between 0.25 and 1 m m ; Set B depth 0.5 mm, rail width 0.5 mm, groove width 0.25 to 1 mm; Set C - - depth 0.5 mm, groove width 0.5 mm, rail width 0.25 to 1 mm; Set D - - g r o o v e width 0.5 mm, rail width 0.5 mm, depth from 0.01 to 1 mm. The discussion here suggests that the behaviour of such transducers may be determined by one or more of the following:
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" 100 = E
g
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a
Resonant frequency The response of all the grooved transducers, as measured ultrasonically, was found to be multi-peaked - Figure 8
0.
-10 "o
--20 -& E < --30
--40
I 20
50
I 100
I 200
Frequency (kHz)
Figure 8 Measured frequency responses of transducers with backplates B showing the multi-peaked response indicated by Equation (4) with peaks occurring at approximately fo, 3fo, 5fo
18
Ultrasonics 1993 Vol 31 No 1
Inversegroove width
250
~'125 u_
LL
a
The first of these effects predicts a multi-peaked frequency response, whereas the second would give the same response in each case as all the backplates have the same surface finish on the rails. The third predicts that the resonant frequency should increase with decreasing air-gap. The value of the resonant frequencies varied by only about +_ 1 k H z when a transducer using a given backplate was assembled several times. Measurements with nominally identical r a n d o m backplates suggest that the resonant frequency varies by up to 10% so that the error in measured frequencies is of this order.
2
b
Figure 9 Fundamental resonant frequency for transducers with backplates (a) A and (b) B
I
i
I
I
E
I
I
0.5
groove width rail surface finish depth of grooves
~
4
Inverse pitch (ram-1 )
I
(i) (ii) (iii)
/
u_
Rail width (mm)
I
I
1
-"
0 b
I
0.1
I °•
I
I
I
r
I
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r
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1
Groovedepth (mm)
Figure 10 Fundamental resonant frequency for transducers using backplates (a) C and (b) D
shows the frequency responses of all the backplates in set B and the 'multi-peaked' response can be clearly seen. Most have three peaks, but in some cases only two peaks occur. The ratio of the second to the first frequency varied between 2.5 and 3.2 and of the third to the first varied between 4.0 and 5.2 this is in reasonably good agreement with Equation (4) so that it appears that the behaviour of the transducers is primarily governed by the membrane resonance. According to the membrane model, the fundamental frequency should increase inversely with groove width. This means that sets A and B should show a linear increase in resonant frequency with the inverse of pitch and groove width respectively; this was found to be the case Figure 9. The fundamental membrane resonance should not vary with either rail width or groove width (sets C and D). This is clearly the case with set C - see Figure lO(a). It is approximately true for set D except for the shallowest grooves where the resonant frequency increases significantly and there is some evidence of a decrease in resonant frequency for the deepest grooves. Thus, it is clear that membrane resonance is the main factor determining the resonant behaviour of these transducers. A transducer made using a grooveless backplate and with the same surface finish as the grooved backplates, had a resonant frequency of 310kHz; the relative strengths' of the different peaks - see Figure 8 in the frequency response curve suggest that the overall frequency response is at least, to some extent, a convolution of the rail air-gap response and that of the membrane. The groove air-gaps varied from 0.01 to l mm; the predicted resonant frequencies for these gaps would be from 2 0 k H z to 200kHz. The fact that the resonant frequency did not vary significantly with groove depth except for very shallow grooves suggests that in most
Capacitive transducers." H. Carr and C. Wykes cases, this factor was not important for most of the transducers. Optical measurements were made on one of the transducers (this transducer had a two-peaked response). Points on the membrane which were over a rail and also over the groove were examined. The fundamental resonance, as measured ultrasonically, was observed in both cases while the third harmonic was observed only on the groove. However, points on the rail were observed to vibrate at frequencies of up to 900 kHz. In addition, the m a x i m u m amplitude of vibration on the rail, which occurred at the first peak, was only half that on the groove. This suggests that at the lower frequencies the membrane is primarily driven by the annular resonance of the membrane, and points on the rails are vibrating due to transfer of motion from points on the groove, while at the higher frequencies, the vibration of the rail is caused primarily by excitation from the air gaps due to the microstructure on the rail. The implications of this are discussed further in the next section.
Sensitivity For set A (varying pitch ), it was found that the sensitivity increased with decreasing pitch - this probably occurs because the transducer becomes more efficient as the resonant frequency of the annular membrane approaches that of the rail air-gap; this is the same effect as discussed earlier. In addition, as the pitch decreases, the same rail area drives annular membranes which are narrower and lighter. For set B (varying groove width), the intensity of the lowest frequency peak again increased as the groove width was decreased, while the intensity of the middle peak decreased with decreasing rail width. The first of these effects is again due to improved matching between membrane resonance and rail surface finish resonance; the second is probably due to the attenuation by air of the higher frequencies. The intensity of the peaks in set C (varying rail width) showed little variation but had a small peak when the rail and groove widths were matched. The membrane located over the rail may also act as an annular membrane and the best response is likely to be obtained when the two widths are equal. The variation of intensities of transducers made with set D was quite complex. As the depth was decreased, the intensity of the peaks increased, then decreased and then appeared to oscillate. N o coherent explanation for this behaviour could be found.
Conclusions The behaviour of transducers constructed using the backplates described here is governed primarily by the width of the grooves and secondly by surface finish on the rails; such transducers give a multi-peaked response. The most sensitive transducer is obtained when the rail width equals the groove width and the groove depth is such that the influence of the groove bottom surface is negligible. A multi-peaked response is not always given by a grooved transducer the ultrasonic transducer marketed by Polaroid has a single resonance of 40 kHz; it has a rail width of 0.25 ram, a groove width of 0.25 mm, and a groove depth of 0.5 mm. The surface finish on the rails is random; the peaks are separated by about 0.5 m m and
are very 'spiky' with a mean height of about 1/tm. The resonant frequency appears to be determined by the depth of the grooves; however, it is critically dependent on the microstructure of the rails. It has been shown 12 that when the rails are lightly polished, the resonant frequency of the transducer changes to 100 kHz. It appears that the resonance generated by the unpolished spiky surface must be of considerably less strength than that due to the air gap but when the rails are polished, this resonance predominates. The multi-peaked response due to annular resonance does not occur because the resonances due to both rail surface finish and groove depth do not overlap with the annular frequencies. There have been m a n y contradictory reports of how grooved transducers behave, and the work described here goes some way to explaining these contradictions. Thus, while it is difficult to argue with Matsuzaka 3 that multi-peaked transducers are unlikely to have much practical application (though specific problems, for example, surface finish measurement, which use the variation of some specific property with wavelength might provide a use for them) they have provided an insight into how transducers with grooved backplates work.
O t h e r s y s t e m s based on this w o r k
V-grooved transducers In order to obtain more predictable transducers, a series of transducers were tested using v-grooved backplates. The use of a v-groove meant that the membrane had contact with the rail along a single line only, so that the surface finish on the tops should not affect the resonance of the system. In this case, only the air thickness a n d / o r the membrane width should affect the resonant frequency. It was found in practice that only the first parameter was important - the results of this work are described in detail in Reference 12.
Arrays One of the aims of the research p r o g r a m m e in which this work was done was the development of phased arrays for airborne ultrasound. Eight and 16 element receiver arrays with separate transmitters have been constructed at frequencies of 50 and 100 k H z respectively. The first of these gives a returned signal-to-noise ratio of 2:1 from a cylinder located at a distance of 5 m and the second gives a signal-to-noise ratio of 5:1 from an edge located at a distance of 1 m. These are discussed in detail in References 16 and 17. A system currently under development at Nottingham TM enables the range and bearing of targets to be measured in a processing time of 30 ms with range and bearing resolutions of 1.5 m m and 2.5 ° respectively.
Conclusions A wide-ranging investigation of the behaviour of capacitance transducers has been conducted. It has been established that the resonant frequency is dependent on the membrane thickness. Appropriate bias voltage values have been defined. A better understanding has been obtained of the behaviour of transducers made with random and grooved backplates - these have been summarized. The work has led to the construction of more predictable transducers and also to phased arrays using the capacitive method.
Ultrasonics 1993 Vol 31 No 1
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Capacitive transducers." H. Carr and C. Wykes
Acknowledgements
7
The authors would like to thank other members of the ultrasonics research group, including W.S.H. Munro, S. Pomeroy, R. Williams, M. Wybrow and B. Pancholi for help and advice. They would also like to thank the department of Manufacturing Engineering and Operations Management for providing the facilities to do the work. The research programme was supported financially by the ACME committee of SERC.
8 9
10 11 12
References 1 2 3 4 5 6
20
Wente, E.C. Phys Rev (1917) 10 39-63 Kuhl, W., Schodder, G.R. and Sehroder, F.K. Condenser transmitters and microphones with solid di-electric diaphragms for air-borne ultrasonics Acustica (1954) 4 519-532 Matsuzawa, K. Condenser microphones with plastic diaphragms for air-borne ultrasonics II J Phys Soc Japan (1959) 15 167 174 Matsuzawa, K. Condenser microphones with plastic diaphragms for airborne ultrasonics J Phys Soc Japan (1958) 13 1533-1544 Morris, J.C. Broad band echo location. D. Phil. Thesis, University of Birmingham Wright, W.M. Tech. Memo. no 47, Acoutics Research Laboratory, Harvard University (1962)
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13 14 15
!6 17 18
Massa, F. Ultrasonic transducers for use in air Proc IEEE (1965) 53 1363 1371 Merhaut, J. A contribution to the theory of electrostatic transducers based on the electrostatic principles Acustica (1967) 17 283-293 Martin G. ME thesis, Canterbury, New Zealand (1969) Warren, J.E. and Brezezinski, A.M. Capacitance-microphone static membrane deflections J Acoust Soc Am (1972) 52(3) 711 719 Warren, J.E., Brzezinski, A.M. and Hamilton, J.F. Capacitance microphone dynamic membrane deflections J Acoust Soc Am (1973) 54(5) 1201-1213 Cart, H. Electrostatic transducers for air-borne ultrasound, PhD thesis, University of Nottingham (1989)
Kinsler, L.E., Frey, R.S., Cooper, A.B. and Sander, J.W. Fundamentals of Acoustics Wiley, New York (1982) Carr, H., Rafiq, R. and Wykes, C. The use of interferometry in the investigation of ultrasound transducers J Phys D: Appl Phys (1989) 22 495 499 Rafiq, M. and Wykes, C. The performance of capacitive ultrasonic transducers using v-grooved backplates Meas Sci Technol (1991) 2 168-174
Munro, W.S.H., Pomeroy, S., Rafiq, M., Williams, H.R., Wybrow, M.D. and Wykes, C. Ultrasonic vehicle guidance transducer Ultrasonics (1990) 28 350 354 Munro, W.S.H. Ultrasonic phased arrays for imaging and automatic vehicle guidance PhD thesis, University of Nottingham (1990) Webb, P. Private communication
Keywords: ultrasonic transducers; capacitive transducers; airborne ultrasonic transducers
Ultrasound is widely used in non-destructive testing and medical imaging and a wide variety of piezoelectric transducers are available at a variety of frequencies, While a variety of single point ultrasonic systems for ranging in air are available, there has been very little progress in producing arrays which can be used for 2-D and 3-D ranging and this is largely due to the lack of suitable transducers. When piezoelectric transducers are used to transmit ultrasound in air, there is a large mismatch in acoustic impedance between the transducer face and the air which means that they are not very efficient. An alternative is the capacitive transducer. There is a substantial body of existing work on these transducers 1-~1 but there is not, as yet, a clear set of guidelines as to how a transducer with a specific resonant frequency, bandwidth and sensitivity can be designed and constructed. This is undoubtedly due to the complicated nature of the interactions which occur in the transducer. The work reported here is primarily an empirical study of the behaviour of such transducers which was aimed at arriving at design rules for the construction of capacitive ultrasonic transducers - it is discussed in more detail in Reference 12. The next section looks at the factors which affect the behaviour of the capacitive transducer and discusses previous work in terms of these factors. The section after outlines the experimental methods used, while in the following sections, the effects of bias voltage and membrane properties are outlined. The performance of transducers which use random and grooved backplates is described and other systems which have been developed based on this work are also briefly described.
as an acoustic detector and reported as early as 19171 . A thin diaphragm is located close to a conducting backplate, a dc voltage is applied between the backplate and the diaphragm, and when the diaphragm is set into vibration by a sound wave, an alternating voltage is produced which is proportional to the incident sound wave. Acoustic microphones use a metallic diaphragm to give a frequency response which has a resonant frequency beyond the audible spectrum so that its response is quite flat at acoustic frequencies. To produce a transducer which resonates at ultrasonic frequencies, it is necessary to use a thin plastic film ( ~ 2 - 1 5 / t m ) which is coated with a conducting layer and which is placed in contact with the backplate - these transducers can be used to transmit ultrasound by driving them with an appropriate alternating voltage. Figure 1 shows the components which make up the capacitive transducer, namely, the transducer body, the backplate and the membrane attached, under tension, to a ring. The response of the transducer is critically dependent on the structure of the backplate. Ultrasonic capacitive transducers can be broadly divided into two categories, those which use backplates with a random surface finish and those which use grooved backplates.
The capacitive transducer
Transducers with random backp/ates Some workers, using backplates whose surfaces are microscopically rough, have found that the resonant frequency of the transducers increase as the roughness decreases 3'4'6. This can be explained by assuming that the membrane acts as a frictionless piston whose motion is constrained by the air-gap which is effectively a spring - see Figure 2. The resonant frequency of the system is given by
The capacitive ultrasonic transducer is a particular form of the condenser microphone used widely for many years
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0041-624X/93/010013-08 (~ 1993 Butterworth-HeinemannLtd
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Ultrasonics 1993 Vol 31 No 1
13
Capacitive transducers: H. Carr and C. Wykes
Figure 1 Componentsof a capacitivetransducer Membrane
I
Backplate
[
Air gap
1
~)
I
Figure 2 Simplemodelof a capacitivetransducer 7 = adiabatic constant of air P, = atmospheric pressure p = density of membrane d m = thickness of membrane d, = thickness of air-gap For a Mylar or a K a p t o n membrane, and normal atmospheric pressure, the equation becomes f=
1
1.6 x 1 0 3 _ kHz (d,dm) 1/2
(2)
where da and d= are in micrometres. When the surface finish of the backplate is random, is it not clear how the thickness of the air-gap can be quantified though it is clearly related to surface roughness. Other authors 2's'1°'11 have considered the capacitive transducer primarily as a flexible supported membrane. The resonant frequency of such a system is determined by the tension of the membrane and its dimensions a circular membrane of diameter D and tension T has a fundamental resonant frequency given by Reference 13 as 'f=
2"405(Ty/2 ~D \ P m J
(3)
and an infinite set of related harmonic frequencies. This neglects any effect due to air-gap resistance and stiffness. Merhaut 8 and Warren ~ give theoretical analyses which include the resistive and stiffness forces of the air-gap as well as the m e m b r a n e resonance. They both assume that the membrane is supported uniformly either by a circular or a rectangular frame. The real capacitive transducer, when used to produce ultrasonic waves, is much more complex than this. To produce the small air-gap which is required to produce an ultrasonic resonant frequency, the membrane has to be stretched across the backplate so that it is supported by the high points on the backplate - see Figure 3. When the bias voltage is applied, the membrane is further stretched and distorted. It has been shown using optical interferometry 14 that the surface of the membrane has high points of up to 5 pm at the points of support and also that the membrane resonates at a
14
Ultrasonics 1 9 9 3 Vol 31 No 1
series of widely different frequencies in a single transducer. A useful model would need to be able to include both the random points of support, the varying tension away from those points, and the variation of depth of the air-gap across the surface. In addition, it has not been possible, to date, to devise a method by which the membrane tension can be measured though it is likely that it varies significantly in different regions of the membrane, so that even with a theoretical model, it would not be possible to predict precisely the behaviour of any specific configuration of membrane and backplate. The first model suggests that resonant behaviour is determined by the air-gap and membrane thickness, while the second model suggests that it is determined by the separation of points of support, membrane thickness and tension. In practice, the present authors as well as several other authors have found that the air-gap model predicts the resonant behaviour of random-backplate transducer reasonably well and in a later section a method for predicting resonant frequency is given. The work described there also suggests, as would appear intuitively correct, that if the resonant frequency given by the air-gap and that determined by the membrane resonance are matched, the transducer efficiency is likely to be maximized. A quantitative prediction of the sensitivity and bandwidth is not, however, possible, in the absence of a suitable theoretical model.
Grooved transducers The use of grooved backplates has been reported by several authors 2'4'5'9. Several different factors may determine the behaviour of such transducers. The membrane mounted on a grooved backplate has a fundamental resonance whose wavelength is equal to twice the groove width so that the frequency, J'~, is inversely proportional to the groove width. There are also a series of harmonics which are related to the fundamental frequency by f, = nJ'~,
n = 3,5,7 ....
(4)
Only odd harmonics occur because the membrane is being driven synchronously at both rail edges, so that all even harmonics will cancel. There is a layer of air trapped between the rails and the membrane with a characteristic stiffness, and another air-layer in the grooves with a different characteristic stiffness. Each of these air-gaps has an associated resonant frequency. The resonant frequency of a transducer with a grooved backplate may be governed by any or all of these factors. Reported results give conflicting evidence on which effects are important 2'4'5'9. The appearance of several peaks in the frequency response (referred to here as a 'multipeaked response') indicates that the first of these effects
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Metalliclayer Dielectriclayer Air layer Backplate
Figure3 Detailedconstructionof a capacitivetransducer
Capacitive transducers. H. Carr and C. Wykes is significant, although this does not exclude the air-gap effects, while the absence of such a response indicates that one or both air-gaps are the main determinant of the resonant response. Kuhl et al. 2, Matsuzawa 4 and Morris 5 reported a multi-peaked response. Matsuzawa concluded that such microphones are 'practically useless'. Kuhl et al. 2 and Morris 5 believe that the resonant frequency of such transducers is determined by the resonant frequency of the m e m b r a n e suspended above the groove, while Martin 9 claimed that the resonant behaviour is determined by the finish on the rails and that the only function of the grooves is to increase the sensitivity of the transducer. The only commercially available capacitive transducer, produced by Polaroid as a range finder for their camera, is a grooved transducer which does not have a multi-peaked response. It has been shown 15 that the resonant frequency of this transducer is critically dependent on the microstructure of the rails since, when these are lightly polished, the resonant frequency of the transducer changes from 40 kHz to 100 kHz. The work described here using grooved backplates found that the transducers involved generally showed a multi-peaked frequency response but other work by one of the authors 15 has shown that the resonant frequency of transducers made using v-grooved backplates had a resonant frequency which could be predicted from the groove air-gap. At present, it is not possible to predict quantitatively the sensitivity or bandwidth of transducers which use grooved backplates because of the complicated processes governing their behaviour.
Experimental
methods
used
A system has been constructed which enables the resonant frequency, bandwidth, and sensitivity of a transducer to be measured and recorded. The system enables the transducer to be driven with a sine wave of 10 ms duration at a range of frequencies between 10 k H z and 5 MHz. The response of the transducer can be detected in several ways with the system: (i) (ii) (iii) (iv)
a calibrated microphone with a flat response up to 150 k H z can be located in the transmitted field; a pair of matched transducers can be used as transmitter/receiver; a large flat reflector can be used so that a single transducer acts as both transmitter and receiver; a laser interferometer can be used to measure the motion of the surface of the transducer at a specific point.
The optical method has the advantage that transducer performance can be studied without the results being affected by attenuation of the ultrasonic waves in air. The system is driven under computer control enabling the results to be stored on disc and plotted automatically. It determines the resonant frequency as the frequency at which the m a x i m u m returned signal is obtained. As noted previously the 'optical' resonant frequency was found to vary at different points on the membrane, so that this measurement had to be made at a series of points across the surface and an average value found. Measurements of the surface topography of the membrane were made using an interference microscope.
100
.
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I
t
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I
200
Bias voltage (V) Figure 4 Variation in resonant frequency and sensitivity with increasing bias voltage
Investigation
of bias
voltage
effects
These tests were conducted using a random backplate with an R a value of 2.5/tm and a membrane thickness of 6 ~m. The bias voltage was varied from 0 to 250 V. It was found that both the resonant frequency and the sensitivity increased quite rapidly up to 75 V, increased more slowly after that and then flattened off at about 180 V - see Figure 4. The variation in resonant frequency with variation in bias is probably caused by the reduction in the air-gap thickness as the membrane is pulled in. Above a certain voltage, the air-gap does not increase any further so the frequency becomes constant. The variation in sensitivity is likely to occur because when the membrane tension is insufficient, the membrane is unable to exert sufficient force on the air-gap to resonate. The behaviour of this transducer appears to be dominated by the air-gap effect. It was also found that the bias voltage could induce a polarization voltage across the membrane which could increase or decrease the bias voltage by up to 25 V. This polarization voltage decays after the removal of the bias voltage. Investigation
of membrane
effects
These tests used the same backplate as used in the tests discussed in the previous section. M e m b r a n e materials Only two suitable materials were found Mylar and Kapton, manufactured by D u p o n t - which gave very similar transducer performance. Gold and aluminium were used as coating materials and again were found not to give any significant difference in transducer performance. Membrane thickness Membranes of several thicknesses were tested. A linear relationship was found between the resonant frequency and the inverse square root of the membrane thickness as indicated by Equation (1) see Figure 5. The sensitivity at the resonant frequency was found to decrease linearly with increasing membrane thickness. This would suggest that the membrane used should be as thin as possible; however, problems arise in producing crease-free membranes at thicknesses below 5 pm and, in addition, the membranes become very fragile.
U l t r a s o n i c s 1 9 9 3 V o l 31 N o 1
15
Capacitive transducers." H. Carr and C. Wykes The surfaces which were used were (i) (ii) (iii)
200 "l-
I O0 u_
~
0
I
I
0.3
r
I
0.6
1/~--~, Figure 5 Resonant frequency as a function of the inverse square root of membrane thickness
Membrane tension A working capacitive transducer was produced by pre-tensioning a membrane, locating it in front of the backplate, and then moving the backplate forward into the membrane. No significant resonance was obtained until the membrane and backplate were in contact. The sensitivity changed significantly as the backplate was forced harder into the membrane altering its tension. A jig was constructed which enables a pre-calibrated tension to be applied. The position of the backplate relative to the membrane was measured using a micrometer. Measurements were made with pre-tension values from 25 200 g. It was found that the resonant frequency and bandwidth were not significantly affected by varying the tension (as expected from the air-gap model). However, the sensitivity peaked as the backplate position was varied to produce a change in the tension. The peak value was highest for the 75 g pre-tension setting. Thus, both the pre-tension and the backplate tensions need to be optimized to get the best performance from the transducer. In later investigations of transducer performance, a simple pre-tensioning method was devised which gives consistent optimized results 15 Random
surface
roughness average
Rpm: maximum height of surface high points da: mean depth found from a capacitance measurement 2~:
16
surface wavelength
Ultrasonics
Typical surface profiles for each of these surfaces are shown in Figure 6. The R a values varied from 15/~m down to 0.02 #m (this was an aluminium coated optical flat) giving resonant frequencies measured optically from 70 kHz to 3.3 MHz. No significant differences could be discerned in the behaviour of transducers made with the differing types of random backplate. The values of R a, Rpm and d, were found, not surprisingly to be strongly correlated to one another. The value of )os was also found to be correlated with the R a, Rpm and da values though less strongly. The relationship of the lateral structure of a surface to its roughness variation is normally determined by the process by which the surface has been produced and, usually, the 'wavelength' increases with increasing roughness so that the correlation between surface roughness and surface wavelength is to be expected. Measurements of resonant frequency, sensitivity and Q-factor were made ultrasonically using two matched transducers and optically by measuring the amplitude of oscillation with the Michelson interferometer; measurements of the surface topography were made using the interference microscope.
Resonant frequency Equation (2) predicts that the resonant frequency should 25
=L
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2.5
ej~.~
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I
j
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I
5 10 Relative lateral distance (/~m)
backplates
Parameters and surfaces used The discussion in the second section showed that the behaviour of a transducer which has a random backplate may be determined by the properties of the surface in the horizontal (air-gap model) a n d / o r the vertical direction (membrane model). The vertical properties (surface roughness) determine the depth of the air gap between membrane and backplate, while the horizontal structure (surface 'wavelength') determine the distance between the points at which the membrane is supported and therefore the resonant frequencies of the individual resonators within the membrane. The properties of a random surface can be characterized in a variety of ways. The parameters which were used here were Ra:
E D M (electro-discharge machined) Brass turned Polished
1 9 9 3 V o l 31 N o 1
=L
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2.5
U3.c
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i
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Relative lateral distance (#m) 1
=L
8+.
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,
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Relative lateral distance (/~m) F i g u r e 6 Typical surface profiles for (a) EDM; (b) brass-turned and (c) polished backplates
Capacitive transducers: H. Cart and C. Wykes Table I Theoretical and empirical straight line coefficients relating resonant frequency to surface finish
Intercept (kHz) Theoretical Calculated using Rpm Calculated using d a
0
Slope (kHz/~m 1/2)
- (90 _+ 20)
650 770 _+ 50
- (390 _+ 30)
1100 _+ 50
vary in inverse proportion to the square root of the air-gap depth. Each of the parameters, Ra, Rpm and d, is likely to be related to the thickness of the air-gap. The resonant frequency, as measured both ultrasonically and optically, was plotted against the inverse square root of each of these parameters. In the case of the ultrasonic measurements, the curves appeared reasonably linear, in each case, up to about 250 kHz and fell off after this; it is thought that the fall-off occurs because of the absorption of the ultrasound by air. It does not occur when the resonant frequency is measured optically since it is not affected by air attenuation. The Rpm results gave a reasonable straight line up to frequencies of 600 kHz while the d a values gave a linear fit over the whole range. The values of the slopes and intercepts for these results are shown in Table 1. The Rpm values show better agreement with the theoretical model but are valid only up to 600 kHz. The d a values can be used over the whole range. However, the first of these is more likely to be useful in specifying the performance of a transducer since the Rpm value can be predicted before constructing the transducer while the d a value can only be found when the transducer is assembled. The membrane model of the transducer predicts that the resonant frequency of a component sub-membrane should be proportional to the inverse of the distance between the points where it is held fixed against the backplate; this was investigated by plotting optical resonant frequency against (1/2s). The correlation coefficient in this case was 0.77; since 2~ is, in any case, correlated with roughness, and the correlation is less than that obtained for the air-gap model, it can be deduced that the prime determinant of resonant frequency is the air-gap thickness.
Sensitivity Sensitivity was measured both by measuring the received signal level and also by measuring the amplitude of the vibration of the membrane at several points on the membrane surface. While it is not possible to predict the sensitivity of the transducers, it seems reasonable to assume that it is optimized when the average 'size' of the sub-membranes matches the resonant frequency of the air-gap. The ultrasonic sensitivity was plotted against 1/x/d,, 1/2s and 1/[2sx/d,]. Correlation coefficients 0.59, 0.88 and 0.96 respectively were obtained; this appears to confirm the hypothesis. In practice, it would be difficult to design and construct a transducer to satisfy this condition, firstly because the required wavelength is not known, since the tension of the assembled membrane cannot be easily measured, and secondly because it would be very difficult to control the wavelength of a random surface produced by a conventional process. The values of the amplitude of oscillation as measured
by the interferometer were plotted against R,, 2 s, and sensitivity as measured ultrasonically, and a linear regression performed. The correlation coefficients were 0.58, 0.46 and 0.36 respectively, so that it appears that there is no useful information to be gleaned from these measurements.
Q-[actor It is not possible to make any quantitative pedictions of the Q-factor or bandwidth of the transducers. It is expected that the Q will increase (i.e. bandwidth will decrease) with increasing sensitivity since both are determined by the damping in the system; this was indeed found to be the case. The actual values of Q varied between 1 and 8. Topography of the membrane An interference microscope was used to determine the surface topography of the membrane this is discussed in detail in Reference 15. Figure 7 shows interferograms of the membrane before and after the bias voltage has been applied. They show that the membrane is significantly deformed by the application of the bias voltage, changing from a reasonably linear interference pattern, implying flatness of the order of 0.5/~m to sharply curving, and in some cases closed fringes, patterns indicating variations of surface height of several micrometres the bias voltage clearly causes the membrane to conform much more closely to the backplate topography.
Conclusions The work described here confirmed that the resonant frequency of a transducer constructed using a random backplate is governed by the roughness of the surface and the best prediction of the resonant frequency can be made by finding the air-gap thickness from a measurement of the Rpm of the backplate or d, found from a capacitance measurement using the values in Table 1. Resonant frequencies of up to 3 MHz have been obtained using a mirror as a backplate with a 2.5/~m membrane. The sensitivity of the transducer is optimized by matching the 'wavelength' of the surface to that which gives component membranes whose frequencies match that defined by the air-gap thickness. The Q-factor varies significantly between values of one and eight, and increases with increasing sensitivity so that it is likely to be governed by the same parameters as govern the sensitivity.
l a
/
b
F i g u r e 7 Tracings of interferograms of the surface of the membrane (a) before and (b) after the application of the bias voltage
Ultrasonics 1993 Vol 31 No 1
17
Capacitive transducers. H. Cart and C. Wykes
Transducers using grooved backplates
250
Backplates used
200
A series of plates with circular grooves was constructed and the performance of transducers using them as backplates measured. The following parameters can be varied; groove width, groove depth and rail width. The values can be varied relative to one another, or the overall pitch can be varied keeping the relative values constant so that an enormous number of varying backplates could be tested. In this work, four sets were used as follows: Set A
equal values for groove width, and depth and rail width; pitch varied between 0.25 and 1 m m ; Set B depth 0.5 mm, rail width 0.5 mm, groove width 0.25 to 1 mm; Set C - - depth 0.5 mm, groove width 0.5 mm, rail width 0.25 to 1 mm; Set D - - g r o o v e width 0.5 mm, rail width 0.5 mm, depth from 0.01 to 1 mm. The discussion here suggests that the behaviour of such transducers may be determined by one or more of the following:
"lv
"I-
~z 125
" 100 = E
g
x.. u_
2
a
Resonant frequency The response of all the grooved transducers, as measured ultrasonically, was found to be multi-peaked - Figure 8
0.
-10 "o
--20 -& E < --30
--40
I 20
50
I 100
I 200
Frequency (kHz)
Figure 8 Measured frequency responses of transducers with backplates B showing the multi-peaked response indicated by Equation (4) with peaks occurring at approximately fo, 3fo, 5fo
18
Ultrasonics 1993 Vol 31 No 1
Inversegroove width
250
~'125 u_
LL
a
The first of these effects predicts a multi-peaked frequency response, whereas the second would give the same response in each case as all the backplates have the same surface finish on the rails. The third predicts that the resonant frequency should increase with decreasing air-gap. The value of the resonant frequencies varied by only about +_ 1 k H z when a transducer using a given backplate was assembled several times. Measurements with nominally identical r a n d o m backplates suggest that the resonant frequency varies by up to 10% so that the error in measured frequencies is of this order.
2
b
Figure 9 Fundamental resonant frequency for transducers with backplates (a) A and (b) B
I
i
I
I
E
I
I
0.5
groove width rail surface finish depth of grooves
~
4
Inverse pitch (ram-1 )
I
(i) (ii) (iii)
/
u_
Rail width (mm)
I
I
1
-"
0 b
I
0.1
I °•
I
I
I
r
I
1
r
0.5
J
1
Groovedepth (mm)
Figure 10 Fundamental resonant frequency for transducers using backplates (a) C and (b) D
shows the frequency responses of all the backplates in set B and the 'multi-peaked' response can be clearly seen. Most have three peaks, but in some cases only two peaks occur. The ratio of the second to the first frequency varied between 2.5 and 3.2 and of the third to the first varied between 4.0 and 5.2 this is in reasonably good agreement with Equation (4) so that it appears that the behaviour of the transducers is primarily governed by the membrane resonance. According to the membrane model, the fundamental frequency should increase inversely with groove width. This means that sets A and B should show a linear increase in resonant frequency with the inverse of pitch and groove width respectively; this was found to be the case Figure 9. The fundamental membrane resonance should not vary with either rail width or groove width (sets C and D). This is clearly the case with set C - see Figure lO(a). It is approximately true for set D except for the shallowest grooves where the resonant frequency increases significantly and there is some evidence of a decrease in resonant frequency for the deepest grooves. Thus, it is clear that membrane resonance is the main factor determining the resonant behaviour of these transducers. A transducer made using a grooveless backplate and with the same surface finish as the grooved backplates, had a resonant frequency of 310kHz; the relative strengths' of the different peaks - see Figure 8 in the frequency response curve suggest that the overall frequency response is at least, to some extent, a convolution of the rail air-gap response and that of the membrane. The groove air-gaps varied from 0.01 to l mm; the predicted resonant frequencies for these gaps would be from 2 0 k H z to 200kHz. The fact that the resonant frequency did not vary significantly with groove depth except for very shallow grooves suggests that in most
Capacitive transducers." H. Carr and C. Wykes cases, this factor was not important for most of the transducers. Optical measurements were made on one of the transducers (this transducer had a two-peaked response). Points on the membrane which were over a rail and also over the groove were examined. The fundamental resonance, as measured ultrasonically, was observed in both cases while the third harmonic was observed only on the groove. However, points on the rail were observed to vibrate at frequencies of up to 900 kHz. In addition, the m a x i m u m amplitude of vibration on the rail, which occurred at the first peak, was only half that on the groove. This suggests that at the lower frequencies the membrane is primarily driven by the annular resonance of the membrane, and points on the rails are vibrating due to transfer of motion from points on the groove, while at the higher frequencies, the vibration of the rail is caused primarily by excitation from the air gaps due to the microstructure on the rail. The implications of this are discussed further in the next section.
Sensitivity For set A (varying pitch ), it was found that the sensitivity increased with decreasing pitch - this probably occurs because the transducer becomes more efficient as the resonant frequency of the annular membrane approaches that of the rail air-gap; this is the same effect as discussed earlier. In addition, as the pitch decreases, the same rail area drives annular membranes which are narrower and lighter. For set B (varying groove width), the intensity of the lowest frequency peak again increased as the groove width was decreased, while the intensity of the middle peak decreased with decreasing rail width. The first of these effects is again due to improved matching between membrane resonance and rail surface finish resonance; the second is probably due to the attenuation by air of the higher frequencies. The intensity of the peaks in set C (varying rail width) showed little variation but had a small peak when the rail and groove widths were matched. The membrane located over the rail may also act as an annular membrane and the best response is likely to be obtained when the two widths are equal. The variation of intensities of transducers made with set D was quite complex. As the depth was decreased, the intensity of the peaks increased, then decreased and then appeared to oscillate. N o coherent explanation for this behaviour could be found.
Conclusions The behaviour of transducers constructed using the backplates described here is governed primarily by the width of the grooves and secondly by surface finish on the rails; such transducers give a multi-peaked response. The most sensitive transducer is obtained when the rail width equals the groove width and the groove depth is such that the influence of the groove bottom surface is negligible. A multi-peaked response is not always given by a grooved transducer the ultrasonic transducer marketed by Polaroid has a single resonance of 40 kHz; it has a rail width of 0.25 ram, a groove width of 0.25 mm, and a groove depth of 0.5 mm. The surface finish on the rails is random; the peaks are separated by about 0.5 m m and
are very 'spiky' with a mean height of about 1/tm. The resonant frequency appears to be determined by the depth of the grooves; however, it is critically dependent on the microstructure of the rails. It has been shown 12 that when the rails are lightly polished, the resonant frequency of the transducer changes to 100 kHz. It appears that the resonance generated by the unpolished spiky surface must be of considerably less strength than that due to the air gap but when the rails are polished, this resonance predominates. The multi-peaked response due to annular resonance does not occur because the resonances due to both rail surface finish and groove depth do not overlap with the annular frequencies. There have been m a n y contradictory reports of how grooved transducers behave, and the work described here goes some way to explaining these contradictions. Thus, while it is difficult to argue with Matsuzaka 3 that multi-peaked transducers are unlikely to have much practical application (though specific problems, for example, surface finish measurement, which use the variation of some specific property with wavelength might provide a use for them) they have provided an insight into how transducers with grooved backplates work.
O t h e r s y s t e m s based on this w o r k
V-grooved transducers In order to obtain more predictable transducers, a series of transducers were tested using v-grooved backplates. The use of a v-groove meant that the membrane had contact with the rail along a single line only, so that the surface finish on the tops should not affect the resonance of the system. In this case, only the air thickness a n d / o r the membrane width should affect the resonant frequency. It was found in practice that only the first parameter was important - the results of this work are described in detail in Reference 12.
Arrays One of the aims of the research p r o g r a m m e in which this work was done was the development of phased arrays for airborne ultrasound. Eight and 16 element receiver arrays with separate transmitters have been constructed at frequencies of 50 and 100 k H z respectively. The first of these gives a returned signal-to-noise ratio of 2:1 from a cylinder located at a distance of 5 m and the second gives a signal-to-noise ratio of 5:1 from an edge located at a distance of 1 m. These are discussed in detail in References 16 and 17. A system currently under development at Nottingham TM enables the range and bearing of targets to be measured in a processing time of 30 ms with range and bearing resolutions of 1.5 m m and 2.5 ° respectively.
Conclusions A wide-ranging investigation of the behaviour of capacitance transducers has been conducted. It has been established that the resonant frequency is dependent on the membrane thickness. Appropriate bias voltage values have been defined. A better understanding has been obtained of the behaviour of transducers made with random and grooved backplates - these have been summarized. The work has led to the construction of more predictable transducers and also to phased arrays using the capacitive method.
Ultrasonics 1993 Vol 31 No 1
19
Capacitive transducers." H. Carr and C. Wykes
Acknowledgements
7
The authors would like to thank other members of the ultrasonics research group, including W.S.H. Munro, S. Pomeroy, R. Williams, M. Wybrow and B. Pancholi for help and advice. They would also like to thank the department of Manufacturing Engineering and Operations Management for providing the facilities to do the work. The research programme was supported financially by the ACME committee of SERC.
8 9
10 11 12
References 1 2 3 4 5 6
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