Design of a Line Array Point Source Loudspeaker System

Design of a Line Array Point Source Loudspeaker System -by Charlie Hughes 6430 Business Park Loop Road Park City, UT 84098-6121 USA // www.soundtube.com // 435.647.9555

22 May 2013 Charlie Hughes The Design of a Line Array Point Source Loudspeaker System 0. Introduction This paper will describe the design and functionality of a fixed length, column line array loudspeaker system. We will detail the initial project constraints and objectives to be achieved. We will then review the physics governing line arrays of loudspeakers and how these principles may be used advantageously. We will also explore some of the seldom discussed disadvantages of line arrays and present solutions to them. Lastly, we will discuss the specific details used in the design of a Line Array Point Source loudspeaker system. We refer to this particular loudspeaker design as a Line Array Point Source (LAPS) for one primary reason: it is comprised of line arrays of loudspeaker drivers, yet behaves as a point source for all listeners no closer than about 2 meters from the loudspeaker system. 1. Project Constraints and Objectives As with any design project, there were a set of goals to be achieved as well as certain constraints that had to be maintained. Without being overly specific, some these are listed below. 1) Passive, non steered line array 2) Use a ribbon HF driver 3) Broad horizontal coverage 4) Relatively narrow vertical coverage 5) Constant directivity with respect to frequency 6) Good frequency response 7) Consistent frequency response 8) Front dimensions not to exceed approximately 42 inches tall and 8 inches wide (1,067 x 203 mm) The loudspeaker system had to be passive; capable of being driven by a single channel of amplification. It would not be steerable, but rather have a fixed directivity. It had to use a ribbon HF driver. The horizontal coverage had to be very wide and the vertical coverage relatively narrow. The coverage angles needed be fairly constant over as wide a frequency range as possible. The frequency response had to be good, both on axis and off axis (i.e. the device must be well behaved). 2013, Excelsior Audio Design & Services, LLC Page 1 of 16

There were size limitations for the loudspeaker enclosure. 2. Physics of Line Arrays 2.1. Line Length Governs Directivity As with any acoustical source, the relationship between the size of the source and the frequency being radiated determine the directivity of the source at that frequency. Perhaps this is better stated by saying the directivity of a source is determined by the relationship between the size of the source and the size of the wavelength being radiated. Quite a few years ago Don Keele wrote an AES paper which contained an empirically derived equation describing the frequency at which a horn would begin to lose directivity control. From this equation (shown below) we can see that this frequency ( ) is a function of both the size of horn s mouth ( ) and the coverage angle ( ) for that dimension of the mouth. We should note that the coverage angle is defined here as the angle between the 6 db points (relative to the on axis SPL) of the sound radiated from the source. The dimension is given in inches. 10 One of the nice things about this relationship is that it seems to hold true not just for horns, but for a wide variety of acoustical sources. We can certainly use it as a reasonably good approximation for the directivity of a cone loudspeaker driver or a line array of drivers. We can rewrite this equation so that we can calculate the coverage angle ( ) at any frequency for a line array of length. 10 From this equation we can see that for a fixed line length,., as frequency increases (gets higher) the coverage angle decreases. We can also see that at a fixed frequency as the line length increases the coverage angle decreases. When we graph this equation we get the curves shown in Figure 2 1. It should be noted that we are only referring to the coverage angle along the same dimension as the length of the line array. Specifically, if we have a vertically oriented line array, as is typically seen, then we are looking only at the vertical coverage angle. 2013, Excelsior Audio Design & Services, LLC Page 2 of 16

Figure 2 1 Coverage angle of two different length line arrays (theoretical) Hopefully this illustrates how the size (length) of a line array controls the directivity of the line array. In order to maintain a relatively narrow coverage angle at low frequencies a relatively long line array is required. The potential disadvantage of this is that a long line array may have too narrow of a coverage angle at higher frequencies. 2.2. Driver Spacing Governs Upper Frequency Limit When dealing with arrays of discrete sources one must always consider the spacing between the individual sources that make up the array. This is true for the design of all loudspeaker systems. For line arrays it s a good idea to keep the spacing of drivers to no more than one quarter of a wavelength apart for the highest frequency at which the drivers will be used. In other words, for a given driver spacing, the drivers should not be used to reproduce frequencies higher than that for which their spacing is one quarter of wavelength. We can show why this is by using four sources (drivers) spaced apart 5 inches (127 mm) centerto center. This distance is a one quarter wavelength at about 675 Hz. It is a half wavelength at about 1.35 khz and a full wavelength at about 2.7 khz. Figure 2 2 shows the polar response graphs for frequencies very close to these (e.g. 630 Hz, 1.25 khz, and 2.5 khz). We can see that there is good directivity control at 630 Hz with a null at 90 up & down. However, at 1.25 khz we see some off axis lobing at about 50 up & down (50 down is 310 ). At 2.5 khz we see the off axis lobe at 90 up & down is about the same level as the on axis SPL. Looking at Figure 2 3 we can see how the off axis lobe at 90 initially forms. At 1.25 khz (Figure 2 2) this lobe has split into two lobes on either side of 90. 2013, Excelsior Audio Design & Services, LLC Page 3 of 16

Figure 2 2 Polar response of four point sources spaced 5 inches center to center Figure 2 3 Polar response of four point sources spaced 5 inches center to center 2013, Excelsior Audio Design & Services, LLC Page 4 of 16

Instead of looking at polar response graphs, another way to view directivity is the directivity map. This is shown in Figure 2 4. Here we can see the directivity across the entire spectrum (all frequencies). Frequency is denoted on the horizontal axis of the graph, while the off axis angle is denoted on the vertical axis of the graph. The level at a particular frequency and angle is denoted by color. The level legend is shown to the right of the map. Figure 2 4 Directivity map of four point sources spaced 5 inches center to center Here we can see dark blue at 630 Hz and 90 telling us that there is very low level there ( 20 db), a null just like the polar graph showed. We can also see the faint green at 1 khz and 90 telling us the level is about 12 db. This is our initial off axis lobe. We can also see that this faint green area follows a trend of getting closer to on axis as frequency increases. This is the lobe splitting at 1.25 khz seen in Figure 2 2. Finally at about 2.5 khz and 90 we can see the lobe with very high level. There is another one at about 5 khz. From this we can conclude that, as a general rule, we don t want to use the sources (drivers) spaced at 5 inches (127 mm) higher than about 700 Hz. There are exceptions to this. Other considerations come into play that may allow this frequency to increase or decrease. 2.3. Near-Field of the Array There is one item often cited when discussing line array that needs to be addressed. This is the claim (both misunderstood and partially erroneous) that the SPL of a line array falls off at 3 db per doubling of distance. This level fall off rate is only true within the near field of the array. In the far field of the array, just as in the far field of other acoustical devices, the fall off rate is 6 db per doubling of distance. Note that this is only the case under free field conditions as well. 2013, Excelsior Audio Design & Services, LLC Page 5 of 16

What this means is that once the distance from a line array is sufficiently far (which we will discuss shorty) the SPL will drop at 6 db every time the distance is doubled, not 3 db. The next thing to understand about the near field of a line array is that the transition from nearfield to far field does not occur at a fixed distance. The extent or distance of the near field from the array changes with frequency. Like almost everything in acoustics, the extent of the nearfield is related to both the overall size (length) of the array and the wavelength of the sound being radiated by the array. The equation that describes this is given below, where λ is the wavelength. 2 λ We can also write this in terms of frequency, instead of wavelength, where c is the speed of sound. 2 c This equation tells us that as the frequency is increased the extent of the near field also increases. It also tells us that the extent of the near field increases with the square of the length of the line array. So we now know three things to be true. 1. The SPL fall off of a line array is 3 db per distance doubling only in the near field. 2. The SPL fall off of a line array is 6 db per distance doubling in the far field. 3. The transition from near field to far field depends on both the size of the array and frequency. These items taken together have a profound implication. At distances closer to the line array than the longest near field extent, the frequency response of the line array will be different at different distances! By rewriting the equation above we can determine the approximate frequency above which we are still in the near field of the array for a specified distance,. 2 By making some simplifications and approximations we can see this changing frequency response in Figure 2 5. Here we start out with a perfect line source about 38 inches (965 mm) long. We have equalized this source to have a perfectly flat frequency response at 1 meter. When we move to 2 meters from the array the frequency response has changed. This is because at frequencies above 1.5 khz, 2 meters is still in the near field. These higher frequencies have a fall off rate of 3 db / DD (Distance Doubling). At frequencies well below this the fall off rate is 6 db / DD. This now gives us about a 3 db variation in our frequency response at 2 meters, whereas the response was flat at 1 meter. 2013, Excelsior Audio Design & Services, LLC Page 6 of 16

Figure 2 5 Change in frequency response due to near field to far field transition Moving to 4 meters from the array the changing frequency response has gotten even worse. For frequencies above about 3 khz this distance is in the near field. They have only dropped 3 db relative to the SPL at 2 meters. At slightly lower frequencies the level has decreased a bit more than 3 db since they are beginning to be in the far field at distances from 2 4 m. At even lower frequencies the SPL has dropped 6 db as they are in the far field at the 2 meter distance. We now have about a 6 db variation in our response at 4 meters. At 8 meters we have a 9 db variation in the response. All frequencies above about 6 khz are still in the near field at this distance. Thus, they have only fallen off another 3 db relative to the level at 4 meters. This trend continues for the other distances shown in the graph. For frequencies below 20 khz the response does not change (other than a constant 6 db drop in level) from 32 meters to 64 meters. From this we can tell that we are in the far field at 32 meters. If we were only to consider frequencies below 10 khz could say that we are in the farfield at 16 meters. This is because the frequency response below 10 khz does not change from 16 to 32 meters. There is no way to correct this naturally occurring behavior of a line array with equalization. It must be addressed in the design of the line array itself. 3. Design of the LA-880i With all of the above in mind, let s now consider the design of a fixed length, column line array loudspeaker. Because the frequency response of a line array can change until we are in the far 2013, Excelsior Audio Design & Services, LLC Page 7 of 16

field of the line array, we should now understand the importance of item 7) Consistent frequency response from our list in Section 1. 3.1. Target Vertical Directivity & Line Length of Each Pass Band A target vertical directivity of approximately 40 was selected for the design. Since the vertical directivity is governed by the length of the line, we want the line array to be as long as possible (within the size constraint of the enclosure). This would yield directivity control to as low of a frequency as possible. A low frequency line length of approximately 38 inches (965 mm) would fit within the maximum enclosure size and should yield a 40 beamwidth at about 650 Hz. This could be made up of eight 4.5 inch (115 mm) woofers with a center to center spacing of 4.88 inches (124 mm). This spacing corresponds to one quarter wavelength at about 700 Hz. Ideally, this would be the maximum frequency to which the low frequency line would be used. However, if the crossover frequency is not too much higher than this and if the midrange line uses a different driver spacing (which it certainly will) we can use the low frequency line a bit higher in frequency. A target crossover frequency of about 1 khz was chosen. The midrange line would need to have a beamwidth of approximately 40 in this crossover region. Using the equation from Section 2.1 and solving for L we find that we should have a line about 25 inches (635 mm) long. The midrange array will need to reproduce frequencies up to about 4 5 khz to crossover with the ribbon high frequency driver. This translates to a midrange driver spacing of less than 1 inch (25 mm) to not exceed the one quarter wavelength spacing criterion. This would be difficult to achieve for a driver that could also have relatively high SPL output and an 800 1, 000 crossover point to the low frequency array. A custom neodymium motor, 2 inch (55 mm) midrange driver was developed as the optimum compromise for the conflicting requirements. Eight of these drivers in a proprietary spacing, split array gives the desired line length. The rather unusual spacing of the midrange drivers help to minimize the lobing that would normally occur as a result of exceeding the one quarter wavelength spacing criterion. The high frequency ribbon driver selected has a radiating length of just over 2.5 inches (66 mm). Its vertical directivity in the crossover region to the midrange array would be a bit broader than desired. However, considering the other design constraints and potential compromises, this was the best option. 3.2. Near-Field / Far-Field Considerations Section 2.3 detailed the theoretical extent of the near field of an ideal sound source. In practice, there is a transition region from the end of the near field to the beginning of the true far field. For our discussion from this point forward we will consider the far field to begin at twice the extent of the near field to account for this transition region. 2013, Excelsior Audio Design & Services, LLC Page 8 of 16

With this in mind, the far field of the 38 inch (965 mm) low frequency array is not reached at 1 khz until about 8.9 feet (2.7 m). The far field of the 25 inch (635 mm) midrange array is not reached at 5 khz until about 19.2 feet (5.9 m). Because of the small size of the ribbon high frequency driver its far field is reached at only about 0.8 feet (0.23 m) at 20 khz. Unless the near field / far field issue is addressed in the design of the loudspeaker, it will have a changing frequency response at distances closer than about 6 meters (20 feet). Listeners can easily be within this distance in many applications. Fortunately, a solution is not too difficult to implement. We can t lower the upper frequency limit of each pass band. Our only other option is to make the length of the line shorter. This is actually a better option because the extent of the near field is proportional to the line length squared. Shortening the line length will reduce the near field extent much more rapidly than lowering the upper frequency limit. How can we make the line length of each pass band shorter and still have the directivity control we want? The overall line length governs the directivity primarily in the low frequency region of each pass band. We need to shorten the line length in the high frequency region of each pass band. Applying an additional low pass filter to the outer drivers in each pass band will effectively shorten the line length only in the high frequency region of each pass band, leaving the full length of the line in the low frequency region. The driver positions in the loudspeaker are shown in Figure 3 2 & Figure 3 4. The drivers are color coded to help illustrate that the outer set of drivers don t reproduce frequencies as high as do the inner drivers. The crossover outputs that feed each set of drivers are shown in Figure 3 1 & Figure 3 3. This technique is known as frequency tapering. The length of the line is tapered in a frequency dependent manner. By frequency tapering the low frequency array, the inner 4 drivers have a line length of only about 18 inches (457 mm). At 1 khz the far field of an array of this length is reached at about 2.2 feet (0.7 m). Tapering of the midrange array yields a line length of about 10 inches (254 mm) for the inner four drivers. At 5 khz the far field of this array is reached at about 3.1 feet (0.9 m). Since we have a gradual reduction in the output of the outer drivers, not a sharp cutoff, we will consider the far field to be twice the distance from our previous determinations. This puts the far field for the low frequency array at about 4.4 feet (1.4 m) and the midrange array at about 6.1 feet (1.9 m). 2013, Excelsior Audio Design & Services, LLC Page 9 of 16

Figure 3 1 Crossover output to LF drivers Figure 3 2 Drivers in LF array 2013, Excelsior Audio Design & Services, LLC Page 10 of 16

Figure 3 3 Crossover output to MF drivers Figure 3 4 Drivers in MF array 2013, Excelsior Audio Design & Services, LLC Page 11 of 16

3.3. Modified Vertical Directivity Shortening the line length in the higher frequency region of each pass band also helps maintain a more constant vertical beamwidth. Referring to Figure 2 1 the beamwidth is not constant. It gets narrower and narrower at progressively higher frequencies. The frequency tapering that is applied to the low frequency array helps to increase its vertical beamwidth at the higher frequencies. This is illustrated in Figure 3 5. We can see that with all 8 low frequency drivers the beamwidth is much narrower at 1 khz than with just the 4 inner drivers. The additional low pass filter on the outer drivers should result is a smooth transition between these two beamwidth curves above about 800 Hz. Figure 3 5 Figure 3 6 2013, Excelsior Audio Design & Services, LLC Page 12 of 16

The same thing occurs with the frequency tapering of the midrange array (Figure 3 6). Above about 2 khz there should be a smooth transition between these two beamwidth curves. We can show the predicted beamwidth for each pass band, only over the frequency region which it covers, to get a better idea of the overall beamwidth of the loudspeaker system. This is shown in Figure 3 7 along with the measured beamwidth of the loudspeaker. The measured beamwidth agrees very closely with a smooth transition between the segments of the predictions. Figure 3 7 Figure 3 8 Measured vertical directivity map of the line array loudspeaker 2013, Excelsior Audio Design & Services, LLC Page 13 of 16

The vertical directivity map (Figure 3 8) shows good control with some widening in the crossover region to the high frequency ribbon driver. Figure 3 9 Measured frequency response and an average response within the coverage angles of the loudspeaker, no equalization Figure 3 10 Measured frequency response and an average response within the coverage angles of the loudspeaker, with some equalization 2013, Excelsior Audio Design & Services, LLC Page 14 of 16

The frequency response of the loudspeaker is also quite nice, both on and off axis (Figure 3 9). Here we can see the measured on axis response as well as an average frequency response curve calculated for positions within the coverage angles (both horizontal and vertical) of the loudspeaker. An equalized response of both of these is shown in Figure 3 10. 4. Increasing Vertical Directivity at Lower Frequencies Some applications may benefit from increased directivity control at lower frequencies than what the LA 880i can provide. This is addressed with the LA 808i. This is not a subwoofer for the LA 880i. It is a Low Frequency Directivity Extension loudspeaker. This LFDE unit, as the name implies, extends the directivity control of the line array system to a lower frequency. 4.1. Low Frequency Directivity Extension (LFDE) Cabinet The LA 808i, LFDE unit, has the exact same external dimensions as the LA 880i full range unit. The LFDE unit must be placed either above or below the full range unit in order for the system to work properly. This effectively increases the length of the low frequency line to about 78 inches (1.98 m). This should give us a 40 vertical beamwidth down to about 300 Hz (green curve in Figure 4 1). Figure 4 1 Vertical beamwidth of the LA 880i (black) and the LA 880i & LA 808i, LFDE unit (red) We shouldn t use the LFDE to reproduce frequencies all the way up to the crossover point (approximately 1 khz) of the low frequency drivers in the full range unit for the same reason that we needed to frequency taper the low frequency line in the full range unit. A passive low pass filter at about 500 Hz is included in the LFDE unit. This yields a smooth transition in the length of the line with respect to frequency. It also results in a good low frequency extension of the vertical beamwidth (red curve in Figure 4 1). 2013, Excelsior Audio Design & Services, LLC Page 15 of 16

It is also possible to use two LFDE units with a single full range unit; placing one LFDE above and one LFDE below the full range. This increases the low frequency line length to about 114 inches (2.9 m) and should extend the 40 beamwidth to about 200 Hz! The predicted extension is shown as the purple curve in Figure 4 1. 4.2. Affects to System Response & Passive Equalization With the additional eight drivers in a single LFDE unit added to the full range unit the system would probably have too much output in the mid bass frequency region. Eight 2 inch midrange drivers and one 2.5 inch ribbon high frequency driver cannot be expected to keep up with sixteen 4.5 inch low frequency drivers. To correct for the spectral imbalance that would occur as a result of this, passive equalization in included in the LFDE unit. This equalization needs to be applied to not only on the eight low frequency drivers in the LFDE unit, but also to the eight low frequency drivers in the full range unit. A provision for an output was included in the design of the passive electronics for the LFDE. It includes an output connector and jumper cable to route the signal from the LFDE to the fullrange unit. This sends both the equalized signal from the LFDE to just the low frequency drivers in the full range unit as well as sending the raw input (unequalized) signal from the LFDE to the midrange and high frequency drivers. This results in good frequency response when either the full range unit is used alone or when it is used with an LFDE unit. When two LFDE units are used with a full range unit some additional equalization may need to be applied to the system. This is easily accomplished with a single parametric EQ filter. 5. Conclusion We have detailed the design of a three way loudspeaker system made from line arrays of drivers for the low and mid frequency pass bands. The high frequency pass band uses a true line source; a ribbon driver. The line array within each pass band is frequency tapered to yield the desired directivity control to as low a frequency as possible. The frequency tapering also allows the far field of the loudspeaker to be reached by about 2 meters. At this distance and beyond the loudspeaker will behave as a point source but with the directivity control desired. 2013, Excelsior Audio Design & Services, LLC Page 16 of 16