Journal of Sound and Vibration (1991) 149(l), 125-136

OPTIMIZATION OF THE SOUND PRESSURE LEVEL

PATTERN FOR A CURVED ARRAY SONAR TRANSDUCER

J. P. HUISSOON AND D. M. MOZIAR

Department of Mechanical Engineering, University L$ Waterloo, Waterloo, Ontario, Canada

(Received 11 August 1989, and accepted in revised form 18 September 1990)

A technique is described whereby the characteristic dimensions of a sonar array trans-

ducer may be numerically optimized. The technique is based on the computation of a
rating parameter that reflects specific features of the beam pattern. A numerical solution
for computing the beam pattern of a curved array is given. Optimization of the beam
pattern, with the rating parameter used as the objective function for array element phase
selection, is demonstrated. The effects of characteristic array dimensions on both the rating
parameter and beam pattern are presented.

1. INTRODUCTION

Non-contact range sensors are frequently required in manufacturing applications. Object

detection in assembly cells, boundary location and obstacle avoidance for automated
vehicles, and safety zone scanning in hazardous areas are typical examples. For such
varied purposes, sonar is frequently used since the transducers are inexpensive, relatively
robust and easily interfaced to computers and programmable controllers. Other advantages
include the ability to operate regardless of lighting conditions, and the relatively rapid
and accurate detection capability over short to intermediate range.
The need for directional sensing may be addressed by the use of array transducers.
The operating principle is that the array elements, usually arranged in a planar fashion,
are driven by signals of the same frequency but at different phases (and possibly different
amplitudes). By interference, a beam pattern in the desired direction is generated. One
consequence of this mode of operation (for planar arrays) is the variation in beam pattern
when the direction is altered. For short arrays (in terms of wavelengths) with relatively
few elements, the main lobe width and side lobe levels are so affected by steering that
deflections of more than ~30” from the normal to the array result in an unacceptable
beam pattern [l-3].
To enable a much larger angular range to be obtained, a curved array transducer has
been devised [4]. This design provides a compact transducer with almost 360” field of
view, having a narrow main lobe and low side lobe levels, both unaffected by beam
steering. This is accomplished by driving only a selected subset of the total number of
array elements. The transducer is designed for airborne sonar and is of the electrostatic
(or solid dielectric) type [5,6]. A similar design, for endoscopic applications and with
piezoelectric elements, has also been reported [7].
The characteristics of such a transducer, namely main lobe width, side lobe levels and
transmitted power, are dependent on the dimensions of the array and the operating
conditions (i.e., amplitude, frequency and phase of the signals applied to the array
elements). While the measurement and analysis of the beam pattern for (planar) array
transducers has been reported [8,9], these have relied on a visual assessment of the beam
125
0022-460X/91/160125+12$03.00/0 @ 1991 Academic Press Limited
126 J. P. HUISSOON AND D. M. MOZIAR

pattern to determine the relative quality of competing designs and operating conditions.
In order to optimize the design and operating conditions, a parametric assessment of the
beam pattern is required. Descriptions are presented in the following sections of the
numerical technique used to compute the sound pressure distribution and of a method
whereby a rating parameter, that reflects the “quality” of this distribution, may be
calculated. Results are presented that show how this rating parameter may be used to
achieve an optimized transducer design.

2. NUMERICAL SOLUTION
The curved array consists of a number of rectangular elements, as illustrated in Figure
1. Array element n is driven by a harmonic signal with phase 4,. With this transducer
design, the beam is steered by driving a set of adjacent array elements, and hence the
phases are symmetrical about the centre of this set.

Figure 1. View of curved array (not to scale). The six unshaded elements are driven.

If each array element is represented as a rectangular piston of width 6 and height h,

the pressure distribution at a point (r, 0) in the far field of the array (in the plane containing
the normals through the centre of the array elements) may be written as [lo]

exp j(& - kri + 40, (1)

where f, = pocokhbA,/2nri, Ui= (kb/2) sin +i, bi is the phase of element i, I,$ is the angle
subtended at the centre of element i between the normal to the element and the line to
the field point (I, f3), ri is the distance from the centre of element i to the field point
(r, e), k is the wavenumber (w/c), N is the number of elements in the array, p0 is the
density of the transmission medium, co is the speed of sound in the medium, A0 is the
amplitude of oscillation of the element surface, and w is the angular frequency of
oscillation (a list of symbols and notation is given in the Appendix).
 OPTlMlZATlON OF CURVED ARRAY TRANSDUCER 127

For an electrostatic array, the assumption that the amplitude of oscillation is uniform
across the element width will not be entirely valid. With each element considered as a
rectangular membrane, simply supported along two opposite sides and vibrating in its
fundamental mode, the displacement function may be approximated by

A(x) = A,(1 -4x:/b’), -b/2 s x s b/2. (2)

The directivity function, (sin u/u), for each element in equation (1) then becomes [ 1 I]

3%
D(u,) =--$ -cosv,
,( u, >

For an arbitrary displacement function u(x), for which no closed form solution of the
directivity function is readily available, the pressure distribution may be computed by
numerically subdividing each element into m parallel strips. The amplitude and phase
of the pressure at field point (r, 0) is then computed as

+(r, 0) =f,lmm, arg P(r, e) = arctan (&l&J, (4,5)

where f2=Cz, C,?, uii cos (kr, -+i), f--=x:, C,“=, uci sin (krq -&), uij is the relative
amplitude of strip j on element i w.r.t. A,,, and r,, is the distance from stripj on element
i to the field point (r, 0).
A discrete approximation to the beam pattern is obtained by computing the sound
pressure at a large number of field points about the array. It has been found that using
900 points over one quadrant provides sufficient accuracy for this purpose.

3. TRANSDUCER RATING PARAMETER

In assigning a value to a particular beam pattern that represents its “quality”, it is
necessary to define the characteristics that constitute a quality criterion. For rangefinding
applications, these may be expressed as (i) main lobe width, (ii) on-axis sound pressure,
(iii) significant side lobe levels, and (iv) side lobe directionality.
For most purposes, the width of the main beam is desired to be as small as possible.
Simultaneously, the side lobe levels should be small. Furthermore, the proximity of
significant side lobes to the main lobe is important; the further removed these are, the
worse is the effective directionality of the transducer. These requirements suggest that a
comparison of beam patterns should be based on the pressure distribution within the
main lobe with respect to that of the side lobes.
To determine the width of the main lobe, a number of criteria were examined. Initially,
the main lobe limit was taken at the first occurrence of a relative phase difference in
pressure of 90” with respect to the on-axis pressure. For beam patterns with well defined
main and side lobes, this occurs at the first local minimum in the beam pattern, as shown
in Figure 2. However, this definition has little meaning when the 90” phase difference
does not correspond to a local minimum, such as shown in Figure 3.
Two alternative criteria are to define the main lobe limit at the first off-axis minimum
in sound pressure, or to use the conventional main lobe width definition as the first -3 dB
point in the beam pattern. However, when computing the main lobe width (or limit) as
a function of one or more variable parameters, discontinuities arise with both of these
criteria, when the main lobe subsides into dominating side lobes, also shown in Figure 3.
Consequently, a rating that relies on the definition of a main lobe is undesirable for
numerical optimization.
128 J. P. HUISSOON AND D. M. MOZIAR

-10 dB

-20 dB

Figure 2. Beam pattern for curved array; R, /b = 7, b/A = 1.05, 4, = 53.2”, I#Q= 159.2” 0, 90” phase with
respect to P,,

Figure 3. As Figure 2, but 4, = 55.5” and I#I~= 165.3”.

The solution adopted to generate a continuous rating that reflected the above require-
ments was to apply a weighting function to the computed pressure distribution. The
weighting function chosen is defined by a pair of blended curves as

W,(e)=a,+a,e+aze2+a,e” for 0s lels em,

w(e)= (61
{ w,(e) = ~(~/2- ey for e,, s lel s 7r/2 I ’

where O,,,, is half the desired main lobe width, ai are constants defining the cubic spline,
and c and z are constants defining the exponential curve.
 OPTlMIZATlON OF CURVED ARRAY TRANSDUCER 129

The boundary and compatibility conditions applied to these equations are:

W,(O) = 1, (d W,/de)l+=,, = 0, W,(O,,,) = WZ(@“,,) =03 (7a-c)

(d W,/d@l,=,,,, = (d W,ld@l,=,,~,, (d’ W,/d@)l,,ZH,,,i = (d’W,/d8’)1,=,,,,. (76 e)

The blended weighting function, W(O), represents a desired beam pattern, with a main
lobe width of 20,, and an exponentially decreasing side lobe magnitude beyond B,,,,.
While this weighting function could be applied directly to the normalized pressure
distribution, it was found that the selectivity was significantly improved (based on a visual
assessment of optimized beam patterns) by computing the rating parameter as
lr/Z ii’l
R= P’( 0) W( 0) dO/ P’(fI){l- W(e)}de, (8)
I0

or in discrete form as
IIT, ,I,,

R, = C P; W,/ C P:(l- w,), (9)

j=O ,=”

where P: is the normalized sound pressure at angle 0, and nst is the number of points
at which this is evaluated.
The on-axis pressure should also be considered when making a comparison between
arrays. However, its relative importance with respect to the overall pressure distribution
is difficult to assess. Since optimization of the rating parameter tends to concentrate the
pressure distribution in the main lobe, this should indirectly result in a high on-axis
pressure. This will be considered further in the next section.
While amplitude shading is not considered in this optimization of the curved array, it
may be used to demonstrate the characteristics of the rating parameter for certain well
known shading formulae. The beam patterns for a planar six-element line array are shown
in Figure 4. The associated values of the rating parameter are given in Table 1.

40 60
Degrees off-oxis. 8

Figure 4. Beam patterns of a six-element line array of half-wave spacing between elements with amplitude
shading functions. -, Unshaded; - - -, binomial; - - - - - -, Dolph-Chebyshev.
130 J. P. HUISSOON ANI) D. M. MCIZIAR

TABLE 1

Rating parameter values associated with Figure 4

tl,,,, = 10” B,,,, = 20”

Unshaded 0.764 1.440

Binomial 0.956 2.665
Dolph-Chebyshev 1.249 3.344

The relatively close rating values of the binomial and unshaded arrays for 6,,,, = 10”
show the overall description of the beam pattern being reflected in the rating parameter.
However, for 13,,,,= 20”, the significant second side lobe in the unshaded pattern results
in a greater proportional difference in Ra with respect to the binomial shading. For good
selectivity, the value used for em, should be about the width of the main lobe that can
be expected for the array dimensions and operating frequency. However, if the rating
parameter is computed for two beam patterns with identical side lobe distributions and
magnitudes but with different main lobe widths, the pattern with the narrower main lobe
will always receive a higher rating, regardless of the value used for 19~~.It is also important
to note that comparison of rating parameters is only valid if these have been calculated
for the same value of &,.
With the rating parameter used as the objective function, good agreement was obtained
between those phases that maximize this function and those that minimize the main lobe
width and side lobe sound pressure levels. The relative magnitude of the rating parameter
for different beam patterns also complements the subjective “merit” that may be visually
assigned to these [12].
The rating assigned to a particular transducer must represent the optimum beam pattern
that can be obtained for the transducer dimensions and operating frequency: i.e., that
beam pattern resulting from the application of the optimum phases to the active array

180

Figure 5. Rating parameter as function of &, and & for array with R, /b = 10, h/A = 1.5.
 OPTIMIZATION OF CURVED ARRAY TRANSDUCER 131

elements. Since an exhaustive search over all possible phase combinations is undesirable,
the Downhill Simplex algorithm [ 131 is used to converge to the optimum phases. Initial
estimates of the optimum phases are obtained by computing the distance from the centre
of each element to the on-axis reference point. The difference in these distances is then
cancelled by the appropriate phase for each element, so that all signals are in phase at
the reference point. A beam pattern obtained by using this geometrical solution is shown
in Figure 3. The optimized beam pattern for this array and operating frequency, with
ornr = 3.75”, is shown in Figure 2. The pronounced differences between these two beam
patterns illustrate the need for phase optimization.
To ensure convergence on the globally maximum value of the rating parameter,
regardless of the initial estimate, it is necessary that the rating should be continuous as
the phases are altered and that a unique maximum exists, with no other local maxima
over the solution space. For a number of transducer configurations (i.e., array element
width, array curvature and operating frequency), the complete rating surface was gener-
ated. A typical surface is shown in Figure 5. It is clear that a single maximum exists with
an absence of other local maxima. All other configurations exhaustively tested produced
similar rating surfaces [ 121.

4. RESULTS

In this section, results are presented for a curved array transducer of which six elements
are driven. Each rating implicitly includes the phase angles c$, and C#Qat which this is
optimal for en!, = 3.75”. The beam patterns have been computed at r = 100 Nb by using
equation (1).
The parameters that will affect the beam pattern (and hence the rating) are (i) radius
of curvature (R,.), (ii) array element width (b), and (iii) operating frequency (or
wavelength, A = 2rrclw).
In Figure 6 is shown the rating parameter as a function of b/A for various ratios of
array curvature, R, / b. It is clear that a planar array consistently yields the highest rating,
although for values of b/A ~0.7, the difference is small. The general reduction in rating

I I I, I I I I I I I I I I I I I I I
0.6 0.8 I.0 I.2 I.4 1.6 I8 2.0 2.2 2,4

Array elemeni width, b/X (wavelengths)

Figure 6. Rating parameter as function of b/h for R,/b =5(O), 7(A). IO(O), 14(O) and UO(*)
132 J. P. HUISSOON AND I>. M. MOZIAR

with decreasing R,./ b is partially caused by an increase in the level of the minima between
the lobes. For b/A < 0.7 and R,.l b 2 7, this is the only significant cause of the reduction
in rating.
As b/h increases from O-7 to 1, a grating lobe gradually develops. This lobe becomes
more pronounced as R,./ b decreases. This is due to the increase in the angle subtended
between adjacent array elements with decreasing R,./b, leading to an increasingly larger
area being insonified by the main lobes of the individual elements. This effect, together
with a narrowing of the main lobe and an increasing number of side lobes as b/h increases,
results in the divergence of the rating curves.
For R,/ b = 10, the increase in rating due to the narrowing of the main lobe approxi-
mately balances the decrease due to increasing side and grating lobe magnitudes, resulting
in the fairly constant rating value observed. This indicates that the rating parameter
describes the overall beam pattern with respect to the weighting function chosen. While
this has been shown to be useful in phase optimization (Figures 2, 3 and 5), specific
features may make one beam pattern preferable to another, although their rating values
may be similar. The proposed rating procedure may be modified by applying prescribed
weighting criteria that enhance or suppress the effect of specific beam pattern features
on the rating parameter. Such criteria might include a reduced weighting with increasing
main lobe width or maximum off-axis SIX. However, the relative weighting of such
features is difficult to generalize. Certain features of the beam pattern should nevertheless
be considered in the optimization.
Since the concept of the curved array design is based on steering the beam by the
selection of a set of adjacent array elements (six in this case), the main lobe width is
important. If this is less than b/R,(radians), the main lobes of adjacent beam positions
do not overlap and blind spots occur. This array characteristic may be defined by the
percentage overlap as
PO= 100(8_~dB-b/Rr)/O_3dB. (10)
As shown in Figure 7, the main lobe width (-3 dB) is almost a linear function of A/b.
The minimal dependence on array curvature is useful since, if the side lobe levels are

06 0.6 I.0 1.2 I.4 I.6 I.8 2.0 2.2 2.4

Wevelength/element width, x/6

Figure 7. Main lobe width as function of A/b for different values of R,./b; key as Figure 6.
 OPTIMIZATION OF CURVED ARRAY TRANSDUCER 133

also comparable, no loss in directivity will result on varying the curvature. The increased
main lobe widths for the smallest R,/ b and A/b ratios, are due to the -3 dB point
occurring on the first side lobe, which has blended into the main lobe. In Figure 8 is
shown the data of Figure 7 replotted as percentage overlap. It is seen that specifying the
overlap (which would generally be SO) relates 6/R,, the incremental steering angle, to b/A.
A comparison of side lobe effects may be subdivided into their levels and directions.
Once again, it is difficult to generalize the relative importance of these, although the
maximum off-axis SPL is probably the principal feature to be minimized between compet-
ing beam patterns. In Figure 9, the side lobe rejection (largest off-axis local maximum

70

,m
‘: 50
co
\
2
Q 30
‘8
UT
1 IO
8-

-10

-30

Figure 8. Percentage overlap as function of h/A for different values of R,/ b; key as Figure 6.

0.6 0.8 I.0 I2 1.4 I.6 I4 2.0 2.2 2.4

Array element width, b/X Iwavelengths)

Figure 9. Side lobe rejection as function of h/A for different values of R,/h; key as Figure 6
134 J. P. HUISSOON AND D. M. MOZIAR

SE) is shown as a function of b/A. It is interesting to note that for b/A = 0.75, array
curvature has no effect on side lobe rejection. Of greater importance are the local maxima
at higher b/A. These occur either as the off-axis local sound pressure level maximum
changes from the first side lobe to the grating lobe, or as the first side lobe blends into
the main lobe (which may cause a discontinuity such as occurs in the curve for R,./b = 14).
In Figure 10 is shown the relative reduction in on-axis pressure, referenced at 1 m.
This reduction is caused entirely by cancellation, due to the phases of the array elements
that optimize the rating parameter. The on-axis pressure is referenced to that produced
by an equivalent concave array, focussed at the reference point (i.e., Prr, = Nfi in equation
(1)). The relatively small loss for b/A < 1 and for all but the smallest R,/ b, indicates that
the rating parameter does indeed tend to maximize the on-axis pressure, as suggested in
the previous section. The significant loss for b/A > 1 is primarily due to the large number
of side lobes and grating lobe(s) that the rating procedure attempts to minimize, at the
expense of pressure concentration in the main lobe. The increased loss with decreasing
R,/ b at higher values of b/A is also caused by the negative percentage overlap (Figure 8).

O-6 0.8 I.0 I.2 I.4 I.6 I.8 2.0 2.2 2.4
Array element width, b/x (wavelengths)

Figure 10. On-axis pressure level reduction as function of b/h for different values of R,/h; key as Figure 6.

These results allow some design features and constraints to be identified. It would
generally be useful to take advantage of the optimal side lobe rejection at specific values
of b/A. Minimal overlap (either positive or negative) is also desirable although, in certain
applications, increased steering resolution may be preferred at the expense of overlap.
Reduced on-axis pressure (poor efficiency) should also be avoided. The relative import-
ance of each of these depends on the application. For example, the local maximum in
side lobe rejection for Rc/ b = 7 gives very little overlap. However, as seen in Figure 2,
the associated beam pattern has a wide grating lobe at 55” off-axis. If greater overlap is
acceptable, the maximum side lobe rejection for R,./b = 14, at b/A = 1.82, could be
considered; the beam pattern for this is shown in Figure 11. The blending of the first side
lobe and main lobe is evident and this has resulted in the grating lobe at 30” off-axis
being used to compute the side lobe rejection.
 OPTIMIZATION OF CURVED ARRAY TRANSDUCER 135

-lOdB

-20 d8

Figure11. Optimized beam pattern mith H,,,,= -3.15”for array: R</h = 14, bJA = 1.83.

5. CONCLUSIONS

A technique for computing the sound pressure distribution for a curved array sonar
transducer has been presented. A rating parameter has been proposed that describes the
overall pressure distribution with respect to that desired. It has been shown that the beam
pattern may be optimized using this rating parameter, and that such optimization is
desirable to avoid poor beam patterns resulting from phase selection based on a geometric
calculation. By considering specific features of the optimized beam patterns as functions
of array dimensions and frequency, locally optimized designs may be identified.

REFERENCES
1. K. HIGUCHI, K. SUZUKI and H. TANIGAWA 1986 Proceedings of rhe IEEE Ultrasonics
Svmposium 1986, 559-562. Ultrasonic phased array transducer for acoustic imaging in air.
2. S. KURODA, A. JITSUMORI and T. INARI 1984 Rohorica 2(l), 271-285. Ultrasonic imaging
system for robots using an electronic scanning method.
_7 R. S. ELLIOTT 1963 Microwave Journal 6, 53-60. Beamwidth and directivity of large scanning
arrays.
4. J. P. HUISSOON and D. M. MOZIAR 1989 Ulfrasonics 27(4), 221-225. Curved ultrasonic array
transducer for AGV applications.
5. W. KUHL, G. R. SCHODDER and F. K. SCHROU~R 1954 Acustica 4, 519-532. Condensor
transmitters and microphones with solid dielectric for airborne ultrasonics.
6. L. KAY 1985 in Robot Sensor.s (editor A. Pugh) vol. 2, 287-299. Airborne ultrasonic imaging
of a robot work space.
7. H. P. SCHWARTZ, H. J. WELSC‘H, P. BUCKER, M. BI~~INGER and R. M. SCHMITT 1988
Proceedings of‘the IEEE Ultrasonics Symposium 1988,639-642. Development of a new ultrasonic
circular array for endoscopic applications in medicine and NDT.
8. S. A. FAL‘KEVICH and L. V. BURLAKOVA 1986 Souiec Journal o,f‘Nondestructiue Testing 22(7).
437-443. Directivity analysis of ultrasonic phase arrays.
9. H. MURATA and S. HIDAKA 1987 Memoirs ofthe Fnculr,v of‘Engineering, Kobe University 34.
Measurement of directivity of ultrasonic transducer.
10. J. P. HUISSOON and D. M. MOZIAR 1989 Proceedings q/‘/he /EEE Ulrrasonics Symposium
1989, 691-694. Simulationand designof curved array transducers for airborne sonar.
136 J. P. HUISSOON AND D. M. MOZIAR

11. E. SKUDRZYK 1971 The Foundations of Acoustics. New York: Springer-Verlag.

12. D. M. MOZIAR 1989 M.A.Sc. Thesis, University of Waterloo. Design of a curved ultrasonic
array transducer for automated guided vehicles.
13. J. A. NELDER and R. MEAD 1965 Computer Journal 7(4), 308-313. Simplex method for function
minimization.

APPENDIX: LIST OF SYMBOLS AND NOTATION

coefficients in weighting function
2” amplitude of oscillation of array element centre
b array element width
c exponential coefficient in weighting function
CO sound velocity
D(u) directivity function of array element
h array element height
k wavenumber ( ~T/A )
m number of subdivisions of array element
nsr number of field points at which sound pressure is calculated
N number of array elements driven simultaneously
P(r, 0, 1) sound pressure as function of position, time
P’ normalized sound pressure
P a\- on-axis sound pressure
PreJ on-axis sound pressure for equivalent focused concave array
PO percentage overlap of main lobes for adjacent beam positions
radial distance from array centre
Ik rating parameter (continuous)
RI rating parameter (discrete)
R‘ radius of curvature of array
uij discrete element displacement function
w(e) weighting function
X distance across element from centre
Z exponent in weighting function
8 angular position w.r.t. normal through array centre
e half angle of main lobe for weighting function
cm main lobe width (-3 dB) of beam pattern
A wavelength
PO density of transmission medium
= (kb/2) sin $,
Zi phase of array element i w.r.t. centre elements
*, angle subtended at centre of element i between normal
to element and the line to field point (T, 13)
w angular frequency of oscillation