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Computer Simulation of Spatial Characteristics of a Loudspeaker System E. Hojan, M. Wojtczak, M. Niewiarowicz Institute of Acoustics, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznafi, Poland

A. N o g a l a & A. Polifiski Loudspeaker Factory "TONSIL', Wrzegnia, Poland (Received 27 November 1989; revised version received 9 May 1990; accepted 6 June 1990)

ABSTRACT The computer-aided methodfor determining the directional characteristics of a loudspeaker system is presented. Considerations were carried out for the three-way loudspeaker system assuming the transition of both the structure of crossover networks and location of loudspeakers in the front panel of the enclosure. The directional characteristics of respective loudspeakers were determined on the basis of measurement data. The calculations of directional characteristics for some specific frequencies from the cut-off region of crossover networks were performed. A comparison of directional characteristics obtained in calculations and measurements was carried out and an angle between the direction of maximal radiation and the symmetry axis of the loudspeaker system has been used as a criterion of compatibility of results.

1 INTRODUCTION

An electrical signal fed to the input of a loudspeaker system, having been changed to an acoustic signal, reaches an observation point in the radiation area of the system. During the process the signal is transmitted by crossover 179

Applied Aco.stics 0003-682X/91/$03"50 ~ 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

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networks (separating filters), loudspeakers and a medium in which the signal propagates from the loudspeaker membrane to the observation point. The filters and loudspeakers modify the signal amplitude and phase depending on their amplitude and phase responses in the frequency function. In the medium, on the other hand, there is interference of signals from specific loudspeakers of the system. The result of this interference depends on the parameters of the crossover networks used, the loudspeakers and the arrangement of loudspeakers in the enclosure. In the radiation area of the system lying beyond its symmetry axis, an uneven amplitude response of the signal is often obtained despite the fact that it is even on the system's axis. This unevenness is caused by an interference of signals from specific loudspeakers of the system for frequencies lying within the cut-off area of the crossover networks, t'a A question arises whether and to what extent amplitude responses of a loudspeaker system can be shaped by changing, for example, parameters of the crossover networks, loudspeakers and arrangement of the loudspeakers in the front panel of the enclosure. 3 An attempt has been made to first calculate and then measure vertical directional characteristics of a loudspeaker system, given a change in the structure of the crossover networks and the location of specific loudspeakers on the front panel of the system. An angle between the axis of maximal radiation, i.e. the direction in which pressure generated by the loudspeaker system is greatest, and the symmetry axis of the system has been assumed to be a measure of compatibility between results obtained in calculations and measurements.

2 D I R E C T I O N A L C H A R A C T E R I S T I C OF A L O U D S P E A K E R SYSTEM The pressure of a tone generated by the ith point source, observed at a point distant r i from the source, can be defined as follows: P/=--Biexp ri

cot

,:.

~bi

(1)

where Bi is the amplitude of acoustic pressure at a distance of 1 m, co is the circular frequency of the signal, 2 is the length of the signal wave, and ~b~is the phase shift. Pressure generated by n point sources will be the sum of pressures from specific sources. Following Meyer, '~'5 to simplify our considerations, a set of Cartesian coordinates has been introduced, with the beginning in the

Spatial characteristics of a loudspeaker system

18 t

Xo'Yo'Zol' ii

I

¥~

I

I

X

Fig. 1. Position of the front panel of the loudspeaker system in the X, E Z coordinates.

symmetry centre of the front panel o f t h e system. It has been assumed that all loudspeakers lie in plane 2"1" and the Z-axis is the symmetry axis of the system (Fig. 1). Having introduced a set of spherical coordinates r i, 0~, Oi with the beginning in the centre of the ith loudspeaker (considered as a directional source), we can state that Bi depends on angular coordinates of the observation point, i.e. for the ith loudspeaker:

Bi =

(2)

Bi(Oi, ~Ji,f)

Hence, acoustic pressure generated by the ith loudspeaker, emitting a signal with f r e q u e n c y f is '~

~ = Bi(Oi,ri~i,f)exp lj(2nft - 2cfrl)]

(3)

where ri = [(Xo- x~): + ( Y o - - Y f + (Zo - - : f ] ~/2 r; = {(x o - xi)" + (Yo --Yi) 2 + [Zo -- zi- c(di(f)

+ Di(f))]2} t z

(4)

given that d~(f) is the delay characteristic of the filter feeding the signal to the ith loudspeaker; D;(f) is the delay characteristic of the ith loudspeaker; x~, y~ and zi are the coordinates of the position of the ith loudspeaker in the X, Y, Z system (Fig. 1); x o, ),'o and Zo are the coordinates of the observation point in the X, Y, Z system; and c is the sound velocity. Taking into account the amplitude response of the filter which feeds the

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signal to the ith loudspeaker, quantity B~ for a loudspeaker working in the system can be expressed by the formula

(5)

Bi(f 0,, ~,) = a,(f)K~( f 0,, t~i)

where Ki(f 0~, ~) is the directional characteristics of the ith loudspeaker for frequencyf defined as the dependence of acoustic pressure on angles, and a~(f) is the amplitude response of the filter. Assuming that the cross-section of the loudspeaker directional characteristic resembles, at any place, an ellipse, characteristic Ki(f 0~, ~) can be expressed through a vertical (V) and horizontal (H) loudspeaker directional characteristic by means of the equation K i ( f Oi, ~'i) = f~[ I4..,(fOOsin~,,] z

+ [H~(fO,)cos~,,]'-} v2

(6)

The acoustic pressure generated by the loudspeaker system is the sum of pressures from specific loudspeakers. An absolute value of the pressure is needed to determine the directional characteristic of the system, given by ?I

IPl =

ai(f)Ki(--f' dpi' ~i) COS (rr tLL,~ ri

r;

i=1 n

+

rr

LL-~ i=l

~

ai(f')Ki(f ri

0,- sin

r[

1/2

(7)

where n is the number of loudspeakers. With eqn (7) it is possible to determine the pressure distribution of the tone with frequencyf generated by the loudspeaker system for any arrangement of loudspeakers in the plane of the system's front panel and for any configuration of crossover networks. The directional characteristic of the system was measured at a constant distance from the symmetry axis of its front panel. Hence, it seems justified to introduce to the calculations another system of spherical coordinates ro, 0o, ~9o,with its beginning in the symmetry centre of the front panel (where 0o~(0; rt), ~,o~(0; 27z)). The system is shown in Fig. 2. Assuming that all the loudspeakers are in the plane of the front panel of the loudspeaker system, for a given arrangement of loudspeakers it is possible to reduce the number of variables (spherical coordinates) occurring in eqn (7) from n x 3 (where n is the number of loudspeakers) to 3, i.e. we obtain

IPI = IPl(ro, 0o, ~o)

Spatial characteristics of a loudspeaker system

/

183

I

Fig. 2. A three-way system with loudspeakers placed on one line.

It suffices to find geometric relations between coordinates r o, 0 o, ¢o and ri, Oi, ~bi for each i. Moreover, by determining distance r o for which the directional characteristic of the system is calculated and by selecting only one plane (qJo = const), we obtain

IPI = IPl(Oo) DISTRIBUTION OF ACOUSTIC PRESSURE WITHIN THE RADIATION AREA OF SYSTEMS WITH A GIVEN ARRANGEMENT OF LOUDSPEAKERS Let us assume that a given system is c o m p o s e d of three conical loudspeakers (woofer, mid-range and tweeter) arranged in the plane of the front panel, as shown in Fig. 2. Let roi, 0oi, 4Jo~be the coordinates of the position of the ith loudspeaker in the system to, 0o, 4%. F o r loudspeakers lying in the plane of the front panel 0o~ = 7z/2 for all i. Angle ~Oot = ~z, ~o2 = 0 , and angle ~0o3 has any constant value. Let ro~ = R~. Using the dependence between coordinates of the spherical system and the coordinates of the Cartesian system, and considering the coordinates of the position of loudspeakers, eqns (4) will assume the forms, respectively, {(ro sin 0o sin ~Oo-- Ri sin I//0i) 2 "[- (r o sin 0o cos ~o - Ri C O S I / / 0 i ) 2 + (to cos 00) 2} t/2 r; = {(ro sin 0 o sin ~o - Ri sin ~oi) 2 + (ro sin 0o cos ~o - Ri cos ¢oi): + (r o cos 0 o - c[di(f) + ai(f)]) 2 } I,:2 ri =

(8)

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184

Hence. the distances r i and rl have been expressed through coordinates of the spherical system r 0, 00, 0o. Coordinates of angles 0i and ¢~, which are either overt (Oi) or covert (0i) in eqn (6), should be determined in a similar way. It is easy to get the following geometric relations:

)

0 i = arc cos

cos 0 o

¢i = arc sin ( v / r

[ __ (ro cos Oo)2J

_-__R_, n £o,

(9)

(10)

Equations (8)-(10) help obtain the required relation

IPl = IPl(ro, 0o, In this paper we calculated only the vertical directional characteristic of the system at constant distances of 1 and 3 m from its front panel, hence the dependence on P(O o) has been calculated. In the case where all the loudspeakers are on the vertical symmetry line of the system, i.e. ~0o3= 0, the argument of function arc sin in eqn (10) will be equal to zero. Having substituted these two values to eqn (6) we get the same numerical values. Arrangement of all the loudspeakers on the symmetry axis of the front panel is therefore a special case for consideration discussed in this chapter.

4 EXPERIMENTAL INVESTIGATIONS

4.1 Technical data of the loudspeaker systems under investigation The arrangement of the loudspeakers on the front panel and the configuration of crossover networks used are shown in Fig. 3. We have investigated four loudspeaker systems in the following configuration: 1. Z l h (system Z1 2. Z1B (system Z1 3. Z2A (system Z2 4. Z2B (system Z2

+ + + +

filter filter filter filter

A) B) A) B)

The directional characteristics o f the l o u d s p e a k e r systems were calculated, and later measured, for six frequencies each time, from the region of the cut-off frequencies of the crossover networks used.

Spatial characteristics of a loudspeaker system

330

185

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Z2

A)

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B)

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~,TpF o~/o,7m H

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Fig. 3.

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Arrangement of loudspeakers in systems Zl and Z2, and configurations of crossover networks A and B. (Dimensions in millimetres.)

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E. Hojan et al.

4.2 Measurement conditions Measurements o f the directional characteristics of individual loudspeakers and loudspeaker systems were performed in an anechoic chamber. It should be emphasized that when an individual loudspeaker is measured, the remaining ones in the system are disconnected, which is the only difference between the measurement of the directional characteristics of individual loudspeakers and the entire system. Thus, diffraction effects are eliminated. An automatically controlled turntable, permitting measurements every 5 ° in the range - 9 0 ° to + 9 0 ° with respect to the symmetry axis of the loudspeaker or the system, was used. A white noise was used as the measurement signal and the results of the measurements were recorded by means o f a two-channel 2034 B & K analyzer interfaced with a Schneider CPC6128 microcomputer and a printer. Hence, it is possible to obtain a diagram o f the directional characteristics for any frequency as well as to perform calculations of quantitative parameters, discussed in detail in the next section. Measurements were taken for two distances from the loudspeaker systems, namely 1 and 3 m.

4.3 Parameters describing the quantitative evaluation of the compatibility of results between calculations and measurements (a) Maximal radiation angle ~0, where ~ois the angle between the direction in which pressure generated by the system is greatest and the symmetry axis of the loudspeaker system. (b) Maximal angle difference A(p = [(~)mes - - q)cal]

where q~mesis the maximal radiation angle determined in measurements and ~Pca~is the maximal radiation angle determined in calculations. (c) Coefficient o f directivity (for the maximal radiation angle)

Q _ P,I

+

P~o2

2P,o

where P~ is the acoustic pressure for cp, P,~I is the acoustic pressure for q~t = q~ + 15°, and P~: is the acoustic pressure for ~o2 = ~o - 15L (d) Measure o f directivity D = 20 log Q (dB)

Spatial characteristics of a loudspeaker system

187

(e) Differential angular index R (%) R = WA~ W where W is the n u m b e r o f m e a s u r e m e n t cases under investigation and Wa,0 is the n u m b e r of cases for which dependence IcPmcs- ~0c,tl = A(o < q3 is fulfilled, where ~0 designates a set value of angular difference (e.g. 5°; 10°; 20°). (f) Differential index of directivity measure DR = Wa° W

where W is the n u m b e r of measurement cases under investigation and WAOis the n u m b e r of cases for which dependence Ad = IDmcs -- Dc,~l < d is fulfilled, where d is the set value of difference in directivity measures between the measure obtained in calculations D¢,,j and that obtained in m e a s u r e m e n t s Dine s (e.g. d = 2; 4;... dB). 4.4 Results of calculations and measurements

Exemplary directional characteristics of loudspeaker systems Z 1a and Z2 A for the frequency f = 2016 Hz, obtained in calculations and measurements, are shown in Fig. 4. A list of directional characteristics obtained in calculations and measurements helped us c o m p a r e the directions of their maximal radiation. Values of these angles ~0, as well as coefficients of directional characteristic Q and the statistical parameters R and DR, have been given in Table 1 with respect to specific systems and frequencies. Because of insimaificant differences between the results obtained for measurement distances of 1 and 3 m, only that for 1 m are taken into consideration. Evaluations of the compatibility of results between calculations and measurements were m a d e on the basis of a difference in the values of the angles of maximal radiation. Differences smaller, equal to 5 °, 10" and 20 °, have been adopted to be the criterion of evaluation. On the basis of this data, all the cases which met the criterion have been added. Next, the value obtained was referred to the total n u m b e r of angles compared, which helped

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Spatial characteristics o f a loudspeaker system

189

TABLE 1 Set of Results from Measurements and Calculations

Loudspeaker system

f (Hz) Measurements

Calculations

Q

D (dB)

Q

D (dB)

Aqa (:)

R (%) 5~

I0:

DR (%) 20 ° 2 dB 4 dB

ZIA

1024 1 504 2016 6400 7488 8512

0"92 0"60 0'70 0'38 0'64 0"83

--0"75 --4"35 --3"09 --8"49 --3"90 --1"63

0"84 0"62 0"69 0"46 0"65 0"85

--1"55 --4"14 --3"20 --6"73 --3"74 --1"45

15 I0 5 0 0 10

50

83

100

100

100

ZI a

1248 I 504 2016 5504 5984 7008

0"86 0"58 0"59 0"50 0"56 0-72

-1"33 -4"76 -4'53 -6"06 -5"05 -2"84

0"76 0"51 0"51 0"22 0.44 0-79

-2"35 -5"88 -5"81 -13"2 -7-09 -2"05

5 5 0 0 0 5

100

I00

I00

67

83

Z2 A

l 248 1 504 2016 5504 5984 7008

0"66 0"63 0"54 0.64 0-48 0"66

-3"73 -4-05 -5"36 -3.92 -6.44 -3.62

0-75 0"69 0"56 0-62 0.49 0.50

-2"49 -3"23 -5'04 -4.11 -6.26 -6.05

0 5 l0 15 5 25

50

67

83

83

100

Z2 B

1024 1 504 2016 6400 7488 8512

0-89 0"69 0.78 0"56 0-46 0'39

-1.04 -8.20 -2.13 -5.10 -6.74 -8-14

0.82 0'66 0-49 0.50 0-41 0"32

-1.75 -3-05 -6-16 -6.05 -7-75 -9.99

10 35 10 25 10 0

17

67

67

83

100

determine the percentage of cases which met the criterion; also for R and DR parameters. The results of these comparisons are also shown in Table 1. 4.5 Discussion of results The analysis of data in Table 1 helps formulate a few conclusions. (a)

The level of compatibility between results of calculations and measurements depends both on the configuration of the crossover network and the arrangement of loudspeakers on the front panel of the system. (b) For a loudspeaker system situated alongside its vertical axis (Zl), compatibility between measurement and calculation data given by

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the measure of precision to which angle ~max is determined is generally greater than for the loudspeaker system situated beyond this axis (Z2); given the assumed angle differences A~o = 5 :, 10°, 20 °, respectively, R = 50 + 100% for Z1 and R = 50 + 66-7% for Z2. This relation is not as unequivocal when quantity DR, which compares the widths of the leaves of directional characteristics connected with q)m~s and q)c~, is taken to be the compatibility measure. (c) Angle differences of the direction of maximal radiation as great as 20 ° provide for a compatibility between calculations and measurements equal to 100% with respect to the cases with loudspeakers situated on the symmetry axis of the system. (d) The percentage of compatibility between results of calculations and measurements for all cases is greater than 67% for the angle difference > 10 :. (e) A change in the distance between the observation point and the loudspeaker system (from I to 3m) does not affect the mutual relationship between the results of calculations and measurements.

5 CONCLUSION The above theoretical considerations leading to a description of a distribution of acoustic pressure in the radiation area of a loudspeaker system allowed us to compare the directional characteristics on the vertical plane of the loudspeaker systems, obtained in calculations and in measurements. The compatibility level between the results fully confirms our theoretical considerations and helps analytically predict the direction of maximal radiation of the system. This claim is especially very strong when the compatibility criterion is set at the level of 10° of the angle of that direction. If greater compatibility between the results of calculations and measurements is required, of the order of 10 ~, at the present stage of both the theoretical considerations and measurement accuracy, a drop in the compatibility is to be expected. If the percentage of compatibility between the results of calculations and measurements is to be increased, given the compatibility criterion of 5 °, it is necessary to consider especially deeply the influence of both the configuration of the crossover network and real localization of the radiation plane of a single loudspeaker with respect to the front panel of the loudspeaker system. This problem is currently under investigation.

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191

REFERENCES 1. Schaudinischky, L. H., Schwartz, A. & Mashiah, S. T., Sound columns--practical design and applications. J. Audio Eng. Soc., 19 (1971) 36--40. 2. Wojtczak, M., The influence of the arrangement of loudspeakers in a loudspeaker system on its acoustic properties (in Polish). Diploma dissertation, Institute of Acoustics, Adam Mickiewicz University, Poznafi, 1987. 3. Niewiarowicz, M. & Wojtczak, M., A method for determining directional characteristics of loudspeaker systems (in Polish). Proceedings of the X X X V Open Seminar on Acoustics, Biatowie2a, 1988, pp. 162-8. 4. Meyer, G. D., Computer simulation of loudspeaker directivity. J. Audio Eng. Soc., 32 (1984) 294-314. 5. Meyer, G. D., Digital control of loudspeaker array directivity. J. Audio Eng. Soc., 32 (1984) 747-54.