Idealized Loudspeaker Geometry

Loudspeaker directional geometry is of interest to the audio engineer because it allows the develop-ment of relative areas associated with different C_∠. The basic formula for finding Q in the idealized case of all energy passing through C_∠ is:

(9-9)

Since Q is the inverse of area, we can then write:

(9-10)

These geometrical equations are useful in deter-mining the minimum apparent Q that could theoreti-cally be associated with a given requirement of C_∠.

One of the authors was instrumental in the first Q measurements ever published by a commercial

sound manufacturer. The methods available today include:

1. The equal-area, multiple-microphone method.

2. The equal-angle, weighted-area method.

3. The critical-distance method.

9.4.1 Classic Method of Obtaining Axial Q In the noise-measurement field, a relatively standard measurement procedure has been in effect since 1953 (first outlined by Gross and Peterson in the 1953 edition of the Noise Measurement Handbook).

This method calls for a series of measuring points spaced about the sound source so as to allow each measuring point to represent an equal area on the surface of the sphere. Because of the nature of such geometric patterns, only six such sets of uniformly distributed points are possible. These six sets have 2, 4, 6, 8, 12, and 20 uniformly distributed points.

Fig. 9-8 illustrates plane views of such points. The coordinates are given in terms of distances from the center along three mutually perpendicular axes (x, y, z). The “+” refers to the existence of two points, one above the x-y reference plane and one below. When measurements are to be made on a hemisphere, only the four points above the plane are used. Fig. 9-9 shows how such coordinates are utilized to find the desired points.

The length of the vector to the point is found by:

(9-11) The angle between the z-axis and the vector is found by:

(9-12) The L_P measured at each of the equal-area points is averaged by converting to power ratios, adding them (dividing them by 2 if only hemispherical measurements are taken, as is often the case), taking 10 times the logarithm of the sum of the powers, and subtracting 10 times the logarithm of the number of points sampled. This gives the average L_P around the sound source being measured; this is identified as .

(9-13) Figure 9-7. Coverage angles, directivity ratio, and

directivity index.

β α

α–Degrees Directivity index–Decibels β–Degrees

Ideal Sound Column

Ideal Multicell Horn

D1 Q

B. C_∠, Q, and D_I compared in idealized case A. Angular distribution

Q 180

arc θ

2

---⎝sin ⎠

⎛ ⎞ φ

2

---⎝sin ⎠

⎛ ⎞

× sin

---=

Relative area

arc θ

2

---⎝sin ⎠

⎛ ⎞ φ

---2

⎝sin ⎠

⎛ ⎞

× sin

---180

=

Vector length r = x²+y²+z²

Angle = acos( )z

L_P L_P

10log(10^{LP 1( )⁄10}+10^{LP 2( )⁄10}+… 10+ ^{LP n( )⁄10}) 10log(Number of points)

–

=

Loudspeaker Directivity and Coverage 161

Then Q is found by taking the point of the highest level (usually the on-axis point) and subtracting from it the . The power antilog of this becomes Q.

(9-14) This is called the axial Q. If some other point instead of the on-axis point is chosen, then the calculation becomes the relative Q. This may be

further modified into apparent Q by multipliers that will be introduced later.

Manufacturers of loudspeakers have not used this method but rather have concentrated over the years on gathering polar response data, usually in the hori-zontal and vertical planes only. Various methods of utilizing such data in order to obtain Q have been tried over the years. While recognizing that the first attempts were crude, it should also be recognized that at the time the cruder methods were used, the alternative was no Q data at all.

9.4.2 Equal-Angle, Weighted-Area Method Real-life loudspeakers radiate sound over the C_∠ and out of the sides, top, and bottom. In order to find the axial Q, it is necessary to average the SPL over the entire space surrounding the source. The method proposed by the author is one derived out of work done by Ben Bauer, C. T. Molloy, and Bob Beavers.

This method requires a horizontal and vertical polar plot for each of seven octave bands—125Hz, 250 Hz, 500Hz, 1000Hz, 2000Hz, 4000Hz, and 8000Hz.

Fortunately, octave intervals offer more than enough detail to allow accurate planning of the effect of Q on such variables as gain, articulation loss of consonants, etc. Also, most commercial loudspeakers are sufficiently symmetrical in their polar responses to allow the use of a horizontal and vertical polar plot at each octave interval. In some Figure 9-8. Plan views of points uniformly distributed

on the surface of a sphere of unit radius.

A. Eight points

B. Twelve points

C. Twenty points

L_P

Q = 10^{(LP on axis( )–LP) 10⁄}

Figure 9-9. Locating measuring points on a spherical surface.

r has been normalized to unity so that arccos of z is arccos of z/r

162 Chapter 9 rare cases, additional diagonal plots have to be taken, see Fig. 9-10.

Since both a vertical and a horizontal polar response are taken at each frequency, the manufac-turer must then process fourteen polar plots in order to obtain the desired data for a particular loud-speaker.

9.4.3 Processing the Polar Plots

The method of processing the polar plots is illus-trated in Table 9-2. Starting at the 0° on-axis point of the polar plot, assign an arbitrary value of 100dB to the 0° point. Tabulate the relative differences in level, referred to this level, for each 10° point all the way around the horizontal plot. Continue on the

Table 9-2. Weighting Polar Data Taken at 10° Intervals to Correspond to Measurements Taken from Points Surrounded by Equal Surface Areas

Angles L_P Rel L_P Weighting L_W

0° (on axis) 100 1.000000 × 10¹⁰ 0.002418 2.418000 × 10⁷

10° & 350° 99 7.943282 × 10⁹ 0.004730(2) 7.514345 × 10⁷

20° & 340° 97 5.011872 × 10⁹ 0.008955(2) 8.976263 × 10⁷

30° & 330° 96 3.981072 × 10⁹ 0.012387(2) 9.862707 × 10⁷

40° & 320° 94 2.511886 × 10⁹ 0.014990(2) 7.530636 × 10⁷

50° & 310° 93 1.995262 × 10⁹ 0.016868(2) 6.731217 × 10⁷

60° & 300° 93 1.995262 × 10⁹ 0.018166(2) 7.249187 × 10⁷

70° & 290° 92 1.584893 × 10⁹ 0.019007(2) 6.024813 × 10⁷

80° & 280° 91 1.258925 × 10⁹ 0.019478(2) 4.904270 × 10⁷

90° & 270° 88 6.309574 × 10⁸ 0.019630(2) 2.477139 × 10⁷

100° & 260° 86 3.981072 × 10⁸ 0.019478(2) 1.550866 × 10⁷

110° & 250° 85 3.162278 × 10⁸ 0.019007(2) 1.202108 × 10⁷

120° & 240° 85 3.162278 × 10⁸ 0.018166(2) 1.148919 × 10⁷

130° & 230° 83 1.995262 × 10⁸ 0.016868(2) 6.731217 × 10⁶

140° & 220° 82 1.584893 × 10⁸ 0.014990(2) 4.751510 × 10⁶

150° & 210° 81 1.258925 × 10⁸ 0.012387(2) 3.118862 × 10⁶

160° & 200° 80 1.000000 × 10⁸ 0.008955(2) 1.791000 × 10⁶

170° & 190° 80 1.000000 × 10⁸ 0.004730(2) 9.460000 × 10⁵

180° (off axis) 80 1.000000 × 10⁸ 0.002418 2.418000 × 10⁵

Total 6.934851 × 10⁸

Example Vertical Polar Data at 1000 Hz

Angles L_P Rel L_P Weighting L_W

10° & 350° 100 1.000000 × 10¹⁰ 0.004730(2) 9.460000 × 10⁷

20° & 340° 100 1.000000 × 10¹⁰ 0.008955(2) 1.791000 × 10⁸

30° & 330° 100 1.000000 × 10¹⁰ 0.012387(2) 2.477400 × 10⁸

40° & 320° 99 7.943282 × 10⁹ 0.014990(2) 2.381396 × 10⁸

50° & 310° 96 3.981072 × 10⁹ 0.016868(2) 1.343054 × 10⁸

60° & 300° 94 2.511886 × 10⁹ 0.018166(2) 9.126186 × 10⁷

70° & 290° 93 1.995262 × 10⁹ 0.019007(2) 7.584790 × 10⁷

80° & 280° 92 1.584893 × 10⁹ 0.019478(2) 6.174110 × 10⁷

90° & 270° 91 1.258925 × 10⁹ 0.019630(2) 4.942541 × 10⁷

100° & 260° 89 7.943282 × 10⁸ 0.019478(2 3.094385 × 10⁷

110° & 250° 87 5.011872 × 10⁸ 0.019007(2) 1.905213 × 10⁷

120° & 240° 85 3.162278 × 10⁸ 0.018166(2) 1.148919 × 10⁷

130° & 230° 79 7.943282 × 10⁷ 0.016868(2) 2.679746 × 10⁶

140° & 220° 75 3.162278 × 10⁷ 0.014990(2) 9.480509 × 10⁵

150° & 210° 72 1.584893 × 10⁷ 0.012387(2) 3.926414 × 10⁵

Loudspeaker Directivity and Coverage 163

vertical plot in the same manner but skipping the 0°

and 180°points (already recorded). Convert each L_P level to a relative power ratio (Rel L_P). Multiply the ratio by a weighting factor proportional to the area surrounding the measuring point in terms of a sphere with a surface of unity. Total all the weighted power ratios (L_PW). Then subtract 10 times the logarithm of the sum from the on-axis reading of 100dBand take the power antilog. This is the axial Q. Figs. 9-11 and 9-12 depict two methods of dividing a sphere into relative areas surrounding each 10° point. Fig. 9-11 shows the zonal method (best for a cone loudspeaker or exactly symmetrical one-cell horns). Fig. 9-12 shows the quadrangle method (best for loudspeaker types other than a single cone or an exactly symmetrical one-cell horn). The dash lines are an extension of the great circles forming the bound-aries of the area under consideration. To avoid an overcrowded diagram, only a few such areas are shown in either view. Table 9-3 shows spherical areas.

Obviously, the same polar charts used to calculate Q can also be used to obtain C_∠ (the 6dB-down points from the on-axis reading expressed as an angle). During the calibration of the equipment for the polar responses, the on-axis sensitivity can be measured at the 1m–1W, 4ft–1W, etc. This should then be translated into the EIA rating, 30ft at 0.001W.

The four primary measurements are:

160° & 200° 74 2.511886 × 10⁷ 0.008955(2) 4.498789 × 10⁵

170° & 190° 72 1.584893 × 10⁷ 0.004730(2) 1.499309 × 10⁵

Total 1.238267 × 10⁹ 10log [6.934851 × 10⁸ + 1.238267 × 10⁹] = 92.86 L_P

10(100–92.86)/10 = 5.18 = Q at 1000Hz 10log 5.18 = 7.14dB = D_I at 1000Hz

Table 9-2. (cont.) Weighting Polar Data Taken at 10° Intervals to Correspond to Measurements Taken from Points Surrounded by Equal Surface Areas

Angles LP Rel LP Weighting LW

Figure 9-11. Sphere divided into polar lunes. (Courtesy The Audio Engineering Society).

φ = 0°

φ = 90°

θ = 0°

sinθdθ = cosθ1− cosθ2 θ2

θ1

Figure 9-10. Sphere divided into zones.

90° Vert

90° Horiz 270°

Horiz

270° Vert

180°

"Off axis"

0°

"On axis"

Horizontal polar plot Vertical

polar plot

90° Vert

90° Horiz 270°

Horiz

270° Vert

180°

"Off axis"

0°

"On axis"

Horizontal polar plot Vertical

polar plot

45° left

45° right A. Vertical and horizontal only

B. Diagonal plots added

164 Chapter 9 1. Q in octave bands.

2. C_∠ in octave bands.

3. Axial sensitivity in octave bands.

4. Power handling (program levels).

9.4.4 The Critical Distance Method

The technique most widely used in the actual testing of arrays in large auditoriums and arenas is the measurement of the critical distance, D_c, on the axis of interest. If the engineer has a calibrated Q loud-speaker on hand the measurement of L_D and L_R allows

(9-15) and

(9-16) where,

D_c is the critical distance in ft or m,

Ref distance is a distance in ft or m from the speaker in the free field of the loudspeaker that is ≥10dB above the reverberant sound field,

L_D is the sound level at the reference distance, L_R is the sound level in the, hopefully, steady

rever-berant sound field,

S is the total absorption in ft² or m² (this will auto-matically include any modifiers that may be present as well),

Q is the directivity factor of the test loudspeaker at that frequency, octave band, etc.

Having obtained the needed absorption figure, measure the D_c of the array in the same way and use the S obtained from the first measurement to calculate:

(9-17) Figure 9-12. Polar response alignments.

Table 9-3. The 10° Spherical Areas for Zones and Quadrangles

10° Intervals Area of Spherical Zone

Area of Spherical Quadrangle

0° (on axis) 0.003805302 0.004835888

10° 0.030268872 0.037841512

20° 0.059618039 0.071640216

30° 0.087155743 0.099098832

40° 0.112045263 0.119916888

50° 0.133530345 0.134945232

60° 0.150958174 0.145327696

φ = 0°

φ = 90°

θ = 0°

A₁ A₂ A₃ β

α

A₁ =

arcsin

(

sin × sinθ

2 2

φ

)

180

φ = 0°

φ = 90°

θ = 0°

A. Side view

B. Front view

70° 0.163799217 0.152053952

80° 0.171663302 0.155822296

90°* 0.087155744 0.078517492

Total = 1.000000001 Total = 1.000000004

* The 90° area is to the hemisphere dividing point only. For horizontal and vertical plots there are two on-axis areas, eight areas for angles between 0° and 90°, and four areas at 90°. When right and left diagonal polar plots are added, then there are two on-axis areas, 16 areas for angles between 0° and 90°, and eight areas at 90°.

Table 9-3. (cont.) The 10° Spherical Areas for Zones and Quadrangles

10° Intervals Area of Spherical Zone

Area of Spherical Quadrangle

D_c Ref distance for L_D 10

L_D–L_R

( )

---20

×

=

Sa D_c²

0.019881Q

---=

a

a

Q D_c²

0.019881Sa

---=

Loudspeaker Directivity and Coverage 165 When measuring D_c, the first measurement is

made as far into the reverberant sound field as it is convenient to get—2 to 3 times D_c is ideal.

The second measurement is made by walking toward the loudspeaker (usually on the axis) until the sound level is a minimum of 10dB higher than the reverberant sound field measurement—again, typically 10 to 15dB. This insures that the rever-berant sound field does not significantly influence the direct sound field measurement. An excellent idea of how steady the reverberant sound field is can be quickly reached during the sampling of it for the reverberant measurement.

The accuracy of this method is defined by precisely the same constraints that apply to the use of Sabine’s reverberation equations. This technique should not be used wherever Sabine’s reverberation equations cannot be applied.

9.4.5 Architectural Mapping

Since the Pharaoh Zoser and his architect-astron-omer-scientist-magician-visier, Imhotep, first built pyramids approximately 6000 years ago (i.e., 3000–

4000B.C.), architectural renderings have changed little. The floor plan and section view are still the architectural mainstays.

Unfortunately, attempting to distort the essen-tially spherical wavefronts of various kinds as they intersect with linear dimensions on drawings into readily recognizable patterns has, for that same period, defied solution thus leading to Astrolabes’, Orrery’s, and other spherical devices that allow accurate measurements to be undertaken with visu-alization of the answer.

Over the centuries, map-makers have exhibited great ingenuity in creating flat maps of spherical areas but always with serious distortion hidden somewhere in the rendering.

The design of the loudspeaker coverage of an audience area has traditionally followed this centu-ries old pattern to the consequent discomfort of the listener located in one of the hot or dead spots over-looked by these “flat earth” techniques. Today repu-table companies provide remarkably useful data and design programs that allow full manipulation of such data in 3-D visual presentations.

9.4.6 The Dangers

It is the temptation to become so involved in obtaining superior coverage that you overlook the very real problems of:

1. When to vary the Q of a device and when you can vary its L_W instead.

2. The cumulative “N” factor.

3. The necessity to use the L_W of only the devices supplying L_D to a point of measurement or observation vs. the totalL_W supplying L_R. 4. When to turn from the Peutz equation using Q,

V, RT₆₀, etc., to the Peutz equation using L_D, L_R, L_N, and RT₆₀. (This is, in our opinion, one of the most serious flaws in several of the most promoted “flat earth” techniques and one, we fear, not even understood by those advocates.) 5. Solving the %AL_CONS and PAG = NAG before N

is accurately determined leads to nonsense such as %Al_CONS predicted from relative dB values obtained from range and device coverage but divorced from the shifting L_W due to N.

You will find in other chapters of this book the constraints placed on how coverage is achieved by the necessity to achieve useful intelligibility, acoustic gain, and the utilization of existing devices. Often beautiful coverage can be achieved by a multiple driver array only to find that the number of sources has reduced intelligibility to an intolerable level, which results in a shortening of the distance that sound can be successfully projected and which, in turn, leads to a whole new approach to coverage—

perhaps from high density overhead distribution rather than a multi-driver single source array.

In document Sound System Engineering.pdf (Page 177-182)