The ambiguous frequency response: The BDPR, not the axialresponse, depicts how a given loudspeaker will perform in an actual listening environment.
Jul 1, 1997 12:00 PM, Jeffrey A. Rocha
One of the most important considerations in sound-system design is consistency of coverage throughout the entire venue. This consistency is governed by both the three-dimensional radiation characteristics of the sound source and the reflective characteristics of the acoustical environment. Provided that the former criterion is well-defined and well-behaved, the latter's influence can be minimized. Three-dimensional polar patterns and modeling tools can assist in evaluating this coverage over the critical octave bands. A further step is the determination of the full-range, frequency-dependent sound pressure at every seat in the house. The typical axial frequency response can guide the designer in this regard but only when viewed in conjunction with three-dimensional polar plots and beamwidth graphs. Fortunately, there is an easier way to determine whether a given loudspeaker meets the criteria of spatially consistent response.
The ambiguous frequency response The frequency response of a loudspeaker is most commonly depicted as the axial pressure response that results from 1 W of electrical input observed at a distance of 1 m. In fact, the term frequency response is almost exclusively used to describe such a graphical representation. The actual definition of a frequency response curve, however, reveals that the term is much more ambiguous than one might expect.
"A frequency response curve of a loudspeaker is defined as the variation in sound pressure [axial response] or acoustic power [power response] as a function of frequency, with some quantity such as voltage or electrical power [in watts] held constant." (Beranek)
By definition, therefore, quantifying the frequency response actually requires the measurer to make two choices regarding the bounds of the measurement. The argument will be made that the audio industry has historically chosen unwisely on both counts. By electing to hold electrical power constant while measuring sound pressure (as is typically done when measuring frequency response), information that is of little use to consultants and contractors is produced, at least when taken on its own. Further, by measuring axial sound pressure alone, the industry has deprived the design community of a truly useful indication of a loudspeaker's real world performance.
Constant input: voltage or electrical power? The frequency response of a loudspeaker is commonly thought of as a quantity that varies with frequency while the electrical power delivered to the loudspeaker is held at a constant 1 W. The choice of constant electrical input power for this representation is a poor one. It is almost impossible to measure any frequency-dependent loudspeaker parameter while maintaining a constant input power. In practice, this is never done.
Electrical power is by definition load-dependent. In other words, the amount of electrical power that is delivered to a loudspeaker is dependent upon the loudspeaker's impedance. Not only do loudspeakers have varying nominal impedance values (typically 4 V, 8 V or 16 V), but the actual impedance of a loudspeaker varies dramatically with frequency. An 8 V nominal loudspeaker may actually range from 6 V to 40 V over its entire bandwidth. (See Figure 1.)
What is measured in practice is the frequency response of the loudspeaker as the input voltage is held constant. Much the same as a 9 V battery is a 9 V battery regardless of what it is connected to, the input voltage presented to a loudspeaker is independent of impedance. Therefore, if a loudspeaker is swept with a 1 V sine wave of varying frequency, the power being delivered to the loudspeaker will vary greatly, but the voltage willalways remain 1 V. Thus, the constant-voltage frequency response provides a bett er means for the evaluation of a given loudspeaker's response, particularly when viewed in comparison to other devices having dissimilar impedances.
If it is essentially impossible to measure a loudspeaker while maintaining a constant input power, what then is meant by the 1 W designation on most frequency response graphs? If an 8 V nominal loudspeaker is depicted as having a given frequency response measured with 1 W of input power, this translates into a 2.83 V input voltage (P=V2/R, where P is power, V is voltage and R resistance). Therefore, the manufacturer commonly uses the nominal impedance to determine what input voltage should be used to deliver approximately 1 W to the loudspeaker. In essence, the inconsistency of the fictitious constant-power frequency response can be overcome simply by understanding precisely what is really being measured.
Although this input distinction is somewhat academic provided that the intention of the data is completely understood, one cannot underestimate the importance of recognizing the frequency dependence of loudspeaker impedance. Here is a case in point: Ideally, amplifiers are constant-voltage sources. If one attempts to power an 8 V nominal loudspeaker with 500 W of amplifier output (about 63 V), but the loudspeaker actually has a minimum impedance of 6 V at 200 Hz, then the loudspeaker will be attempting to draw a much higher current (10.S A instead of the expected 7.88 A) from the amplifier at 200 Hz. This is all based on Ohm's law, V=IR. This condition can quite readily damage both amplifier and loudspeaker. The situation is of particular concern when the end user attempts to parallel two or more nominal 8 V devices together off of a single amplifier output. In this case the amplifier might be seeing less than a 3 V load over some frequency range. Depending on the amplifier, this problem can be quite detrimental to the system, particularly when the input level over the low-impedance bandwidth is excessive.
Frequency dependence: sound pressure or acoustical power? It is perhaps easiest to answer this question with a question: Is the intention of the sound system to provide highly intelligible, dynamic, full-range sound to one very important member of the audience who is consistently seated in a specially located chair, while the rest of the audience resides in sonic mediocrity? If so, then the axial pressure response of the loudspeaker will suffice for performance evaluation. If, however, it is required that every ticket holder or congregation member observes the same high caliber of sound reproduction, then a more inclusive measurement is required. This measurement is the power response or, stated more precisely, the power response as observed over the intended coverage area of the sound source.
The power response of a loudspeaker portrays the sound pressure averaged over all directions of radiation rather than just one. Because professional loudspeakers are typically designed to achieve broad-band pattern control, it is actually more informative to average this data for only the radiation directions contained within the intended coverage of the device, henceforth referred to as the beamwidth delimited power response (BDPR).
In other words, if the loudspeaker is nominally 90 degrees horizontally by 40 degrees vertically, then all of the energy radiated through an area defined by these angles as a function of frequency would be included in the BDPR. In this way, the response of the loudspeaker at any off-axis point within the intended pattern of the system is given as much importance as the axial response. This point is critical because loudspeakers operate in three dimensions, and all directions of radiation contribute to the net observed sound quality of a system.
Beamwidth defined The beamwidth of a system governs the consistency of the response throughout the coverage area. Beamwidth is most commonly defined as the angular distance between two points on either side of the primary axis where the sound pressure is 6 dB below the maximum pressure contained within tha t angle. This maximum is typically, but not always, found on the primary axis. (See Figure 2.)
It is critical to observe not only the point at which the pressure falls to 6 dB below the maximum, but also the smoothness and consistency with which the pressure falls to this -6 dB point. These factors can be seen in the polar plots. Ideally the progression is gradual and consistently decreasing. The beamwidth, therefore, is both frequency and angle dependent. In other words, for every device there is a unique beamwidth that changes with frequency not only in the horizontal and vertical planes, but for every angle in between. Therefore, the ability of the system to maintain a constant beamwidth for all frequencies within its passband and over the entire intended coverage area will have a direct impact on the power response. If the axial response is flat and the beamwidth is consistent, then the resulting BDPR will greatly resemble the axial response.
The power response Most have observed that virtually any loudspeaker can be made to sound or at least measure quite nicely at one point in space. If the measurement microphone (or calibrated ear) is then moved to another area within the coverage of the device, however, the pressure response often looks quite different. In such a case, the loudspeaker diverges from the critical criterion of spatial consistency. But why? It is easiest to explain this phenomena when studying a typical two-way system crossed over in the range of 1 kHz to 2 kHz.
In the two-way system we are not interested in the pattern control of the woofer or horn loaded compression driver alone, but in the beamwidth that is realized by the complete system. The beamwidth of a woofer decreases as frequency rises. This occurs as the dimensions of the wavelength begin to approach and fall below the diameter of the woofer. In a similar fashion, the beamwidth of a horn-loaded compression driver typically increases as frequency drops. Similarly, this occurs as the wavelength approaches and exceeds the dimensions of the horn mouth height. Unfortunately, the height of the horn mouth is typically significantly smaller than the diameter of the cone. As a consequence, the beamwidth of the horn at crossover is significantly larger than that of the cone. (See Figure 3.)
If a large woofer is used in conjunction with a relatively small horn in a two-way system, the beamwidth will typically narrow as the frequency approaches crossover. Once the system has been crossed over into the horn, the beamwidth immediately expands due to the size of a wavelength relative to the mouth dimensions. (See Figure 4.) As frequency continues to rise, beamwidth finally decreases to the nominal horn coverage. This beamwidth discontinuity results in a similar discontinuity in the power response.
It should be noted that it is quite simple to manipulate the output of these devices such that the axial response looks perfectly flat. As one walks off the primary axis of the loudspeaker, however, the output from the upper range of the woofer (l kHz to l.5 kHz) would diminish rapidly, while the lower horn output (1.5 kHz to 2 kHz) would remain excessive over an area much larger than the intended coverage. Systems that exhibit these undesirable characteristics are nearly impossible to array and do not provide consistent sound throughout a given venue.
The BDPR would readily point out this discontinuity because of the presence of a dramatic dip and then peak in the response near crossover. In this particular case, the system's shortcomings would also be apparent in the horizontal or vertical beamwidths. If, however, the beamwidth inconsistencies occur more toward the corners of the horn, the horizontal and vertical beamwidths would not exhibit the flaw, yet it would still remain readily apparent in the BDPR.
This response measure serves as an accurate representation of the performance of a loudspeaker over its intended beamwidth, but it also can be viewed in conjunction witha full three-dimensional power response to determine the frequency content of any dominant lobes that are projected outside of the intended coverage area. These lobes will interfere (frequency-dependent cancellation or summation will be observed) with the radiation from additional nearby sources, significantly minimize gain before feedback when projected in the vicinity of live microphones, and contribute greatly to direct reflections when projected off of rigid boundaries.
In general, we like our loudspeakers to sound great everywhere that they are supposed to and to be silent everywhere else. Although this goal is somewhat elusive, the power response indicates how closely any device approaches this desired perfection. It should, therefore, be apparent that power response is indeed a far better performance indicator than axial sound pressure. But how is it measured?
Power response measurement One method to arrive at the BDPR of a loudspeaker is to generate the response from the data that is required for most modeling software (EASE or CADP2, for example). In order to generate a three-dimensional model of performance, the output from the loudspeaker must be acquired for all directions of radiation. It follows logically that if one were to select only the measurements that were made within the intended coverage of the device and include them in the finished response measure, then the task would be complete. The problem here is that, as a result of the manner in which the data is acquired, the calculations involved in manipulating the data are quite complex.
The data currently required for modeling is gathered at points located at the intersections of what can be visualized as lines of latitude and longitude that lie on an imaginary unit sphere that surrounds the loudspeaker. These lines are arranged in 5 degrees increments. (See Figure 5.) As can be observed on a world globe, the areas that are formed by these intersecting lines on a sphere are much larger at the equator than they are at the poles. As a result, a measurement that is taken at a point near this fictitious equator represents the loudspeaker's performance over a larger area than data gathered near the pole. Consequently, the data gathered at every point must be weighted by a number equivalent to the ratio of its area to the area of the complete sphere. As it is a unit sphere, the total area is 1. These data manipulations can get quite involved.
In the frequent case when this data is not provided by the manufacturer, there is an easier way to quickly estimate the power response of an individual loudspeaker. The only requirement is a real time analyzer (RTA) that possesses the capability to continually average the microphone input. Several PC-based RTAs have this capability, and the author has typically used the stand-alone dbx RTA-1. The only sacrifice is one of resolution. Most RTAs are limited to some finite percent octave resolution. The loudspeaker should be driven with a pink-noise source. Actual program material may also be used, but this is best done with program material that contains a lot of broadband energy. The measurement microphone should then be slowly moved throughout the intended coverage of the device as defined by the nominal beamwidth at a distance of several meters until the entire coverage area has been traversed a couple of times. The resulting averaged response serves as a remarkably accurate depiction of the BDPR. The smoothness of this response will dictate how the loudspeaker will perform in a three- dimensional sense. For a more in-depth look at the benefits of this performance measure, it will be advantageous to walk through an example.
To demonstrate the effectiveness of the power response in depicting spatial consistency, two systems were equalized for nearly perfect axial response. One of these systems had an undersized high-frequency horn, about 7 inches (178 mm) tall, paired with a single 15 inch (381 mm) woofer.
The second system had a much larger high-frequency horn with a mouth height nearly equal to the 15 inch (381mm) diameter of the woofer accompanying it. The axial pressure response and BDPR of each system were measured and are depicted in Figures 6 and 7. In general, the power response of a loudspeaker should be linear, but will typically exhibit a negative slope because of the rising Q of the system. Even in a properly designed system, the Q into three-dimensional space will increase slightly with increasing frequency because the output through the corners of the horn falls off more rapidly than the output in both the horizontal and vertical planes. The graphs below have been normalized to one another so that the initial level of each is matched to the axial level of the loudspeaker.
In Figure 6 the beamwidth discontinuity results in a dip caused by the narrowing of the 15 inch (381 mm) woofer polars followed by a peak created by the wide-open horn coverage. Figure 7, depicting the matched system, exhibits a continuously decreasing response that is the hallmark of a well-designed, spatially consistent loudspeaker.
In typical literature, the technical specifications are accompanied by a depiction of the axial response. The preceding example illustrates that either of the measured systems would undoubtedly be deemed excellent loudspeakers because of their tremendously linear axial responses. It is only when one digs deeper and observes the true three-dimensional performance that the superior loudspeaker shines. The BDPR serves as a true measure of performance that is enormously meaningful to consultants and contractors. It depicts how a given loudspeaker will perform in an actual listening environment when in the presence of thousands of listeners and many reflective surfaces. It holds the key to successful high-performance installations.
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