A game of numbers
Apr 1, 1999 12:00 PM, Joe Brusi
Loudspeaker directivity is expressed in many different ways on spec sheets and marketing materials provided by loudspeaker manufacturers. Until there is a benchmark for expressing this data, it will continue to be difficult to compare products produced by companies who use varying methods of quantifying loudspeaker performance. This article will offer an explanation of these methods, thereby allowing you to make a more informed decision when specifying equipment.
Measuring directivity Complete directivity measurements start with the measurement of the sound pressure level (SPL) at all points on a sphere surrounding the loudspeaker system. In other words, the loudspeaker is at the center of that sphere, and the distance between the loudspeaker and the surface of this sphere would be equal at all points and should be large when compared to the loudspeaker's dimensions. The setup for this type of measurement is shown in Figure 1. These measurements are made at all frequencies, thereby allowing us to ascertain the frequency response at any angle. This measurement is typically determined by placing a mic at a practical distance-around 13 ft (4 m)-and then rotating the loudspeaker to achieve the different angles. Most often, this rotation happens around one axis, thus necessitating two measurements-one for the vertical and one for the horizontal, in between which the loudspeaker is manually rotated. The most sophisticated systems can rotate around two axes so that all points of the sphere can be measured in the same run. The two-axis method provides not only vertical and horizontal polar plots, but also the polars at oblique angles as well.
The end result is a set of frequency response curves for each point of the measurement sphere with resolution anywhere from 1/24-octave to 1/3-octave and angular intervals anywhere from 1degrees to 10degrees. A set of these for the horizontal slice can be plotted as a waterfall display (see Figure 2), where the high-resolution data shows the transition from a flat on-axis frequency response (0degrees) to a response dominated by bass frequencies behind the cabinet (180degrees). One way to visualize the measurement points is as the equator on a globe.
When the SPL at all points of the sphere is plotted in 3-D for a given frequency, we get what is commonly referred to as the directivity balloon. The reasons for that name are clear when we see Figure 3. The plot looks lopsided-like a squashed balloon-because it is displaying the effects of a horn that provides narrower dispersion vertically than horizontally. We will use the most common loudspeaker configuration in professional sound reinforcement, a two-way 15 inch (381 mm) cabinet, as the primary example.
If we divide the measurement sphere into slices, we end up with a set of polar plots, the result of 360degrees rotation around one axis. The horizontal polar would be similar to walking around a loudspeaker placed on the floor. Only vertical and horizontal polar plots are customarily included in spec sheets. An example of these can be seen in Figure 4.
If we join all the points where the level is the same on the measurement sphere, we end up with an isobar plot. Figure 5 shows a plot with -3 dB, -6 dB, and -9 dB isobars.
The coverage angle is defined as the angle enclosed by the -6 dB points on a loudspeaker polar plot. The -6 dB point is used to avoid overlap when a number of sources are splayed. Ideally, having the splay angle between adjacent boxes within an array the same as the coverage angle of a single box, resulting in seamless coverage between loudspeaker systems.
There is some disagreement among spec writers as to what the 0 dB reference in the calculation of the coverage angle should be. Some manufacturers take the on-axis level, and some use the maximum output level exhibited on the polar. If the polar is smooth, both references will be the same, but such irregular responses as those measured from horns at high frequencies and those in the crossover regions of multi-way passive crossovers usually cause significant differences.
Figure 4 shows graphically how the coverage angle is calculated. In this case, we have about an 80degrees x 35degrees pattern for the 6.3 kHz 1/3-octave band polar shown. Vertical and horizontal coverage angles with frequency can be seen in Figure 6. This two-way system has a 360degrees coverage at low frequencies, which gradually decreases until the crossover frequenc y where the horn and driver take over to achieve fairly constant coverage up to 16 kHz. Both graphs reflect the same set of data, but the top one uses a linear vertical scale while the bottom one uses a logarithmic scale. The latter tends to be the most common, and it reveals the horn's performance a bit better visually. Because the coverage angle varies with frequency, spec sheets normally provide a value that has been averaged over a specified frequency range, typically that of the high-frequency horn. This value often conforms to nominal values, such as 90degrees, 60degrees or 40degrees. Our example loudspeaker would most likely be rated as a 90degrees x60degrees (horizontal x vertical) system.
The Q factor is a mathematical expression indicating directionality of the source. Larger Q factor values denote more directional sources. The Q factor is calculated by comparing the on-axis level with the average level for all the points in the measurement sphere. In practice, Q is often derived from the horizontal and vertical polar plots. A spherical source (a source that has the same output level at all angles, close to a subwoofer at low frequencies) has a Q factor of 1. A hemispherical source, essentially a sphere that is placed against a wall (this condition is referred to as 2p or half space), has a Q of 2. As with coverage angle, different evaluation methods are used, which can generate slightly different Q results. A more detailed explanation of the Q factor can be found in a number of books dealing with loudspeaker systems. I recommend Glen Ballou's New Audio Cyclopedia as a quick reference.
The directivity index (DI) is the same as Q factor, but expressed in a logarithmic fashion: DI (in dB)=10log Q. Thus, a spherical source has a Q factor of 1 and a directivity factor of 0 dB, and a hemispherical source has a Q factor of 2 and a directivity factor of 3 dB. Typical directivity indexes for a horn would range from 10 dB to 20 dB, corresponding to Q factors of 10 to 100. The DI for the polar plots represented in Figure 4 is 12 dB (Q factor of 16). On-axis sensitivity and DI have a direct correlation. Using a subwoofer as an example, when it is placed against a wall, it will increase in sensitivity by 3 dB (DI will change from 0 to 3 dB, Q factor from 1 to 2) as compared to its free-field sensitivity. Figure 7 shows the DI and Q with frequency for the example two-way system. We can see a rising response from the 15 inch (381 mm) woofer, which corresponds to the narrowing of the cone's directivity (or beaming) as frequency increases. In contrast, the directivity that the horn exhibits is fairly constant with frequency, commonly referred to as a constant directivity (CD) horn. DI (Q) and coverage angles are typically plotted for 1/3-octave bands.
It is increasingly common for manufacturers to provide off-axis frequency response measurements on their data sheets. These measurements are made generally at intervals of 10degrees or 15degrees off-axis from the direct on-axis 0degrees, 0 dB measurement. Generally, spec sheets show plots of only one vertical and one horizontal set of measurements. A bit more insight is provided by data that show multiple intervals left and right (horizontal) in off-axis response and up and down (vertical) off-axis response on the specification sheets to show the asymmetrical response of the device in question. Vertical and horizontal relative frequency responses in 15degrees intervals (0degrees, 15degrees, 30degrees and 45degrees) can be seen in Figure 8 for the example loudspeaker.
When examining these off-axis frequency responses, it is important to realize that they are relative, which means that the 0degrees response would be a straight line going through 0 dB, since it has become the reference for the other responses. If we were able to equalize the loudspeaker flat at 0degrees, these would be our frequency responses at the specified angles.
In order to represent all measured data for all frequencies, some fairly complex plots are required. Another isobaric display can be seen in Figure 9, where we have measurement angle as a function of frequency. In this case, isobars are plotted in 1 dB increments from 0 dB to 6 dB. The latter would represent the coverage angle. The flat portion of the isobars in the 2 kHz to 16 kHz region shows the constant directivity nature of the horn in this system.
Electro-acoustic modeling software such as EASE from Renkus-Heinz, JBL's CADP2 and Bose's Modeler can be used to display directivity information quite effectively. These programs use fairly crude directivity balloons typically with 10degrees resolution in octave bands. Lately, the AES has been trying to standardize directivity data for use in modeling software and increasing data resolution for more accuracy in the predictions that the programs can provide.
The projection of the loudspeaker's directivity onto a coverage area can be seen for EASE and CADP2 in Figures 10 and 11 respectively. These plots are excellent for showing graphically a great deal of information for technical and non-technical people. When a non-technical customer can visualize this information on the modeling software, it creates the understanding necessary to get design ideas across, close the sale and seal the installation of a system for your customer.
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