Low-Frequency Loudspeaker Assessment by Nearfield Sound-Pressure Measurement
D. B. KEELE, JR.
Electro-Voice, Inc., Buchanan, Mich. 49107
Introduction
This paper describes a technique for assessing loudspeaker performance at low frequencies using nearfield sound-pressure measurements. This method can be performed in any environment, unlike traditional methods requiring anechoic chambers or costly outdoor testing sites. It allows for simple measurements of frequency response, power response, distortion, and electroacoustical efficiency.
The technique builds upon work by Small [1], which demonstrated that valid measurements could be made at low frequencies by sampling pressure inside an enclosure. This paper extends this by using nearfield measurements outside the enclosure, requiring less complex signal processing and offering accuracy over a wider frequency range.
Glossary of Symbols
| Symbol | Definition |
|---|---|
| a | radius of circular radiator |
| aD | radius of diaphragm, = √SD/π |
| aV | radius of circular vent, = √SV/π |
| c | velocity of sound in air, = 343 m/s |
| ein | voltage applied to driver input |
| f | frequency, in Hz |
| fB | Helmholtz resonance frequency of vented box |
| f3 | low-frequency cutoff (-3 dB) of speaker system |
| Io | acoustic intensity, in power per unit area, = p2/(2 ρoc) for a plane wave |
| k | wave number, = 2π/λ = ω/c |
| P | peak sound pressure |
| PF | peak sound pressure in farfield of acoustic radiator |
| PN | peak sound pressure in nearfield of acoustic radiator |
| PNrms | root mean square sound pressure in nearfield of radiator, = PN/√2 |
| PR | peak sound pressure on axis of piston at distance r |
| PA | acoustic output power |
| PE | nominal electrical input power |
| Q, QB | ratio of reactance to resistance (series circuit) or resistance to reactance (parallel circuit) or cabinet at fB considering all system losses |
| r | distance from pressure sample point to center of piston |
| RE | dc resistance of driver voice coil |
| S | surface area |
| SD | effective projected surface area of driver diaphragm |
| SV | cross-sectional area of vent |
| SPL | sound pressure level, in dB re 20 μN/m2 |
| Uo | output volume velocity of acoustic radiator |
| λ | wavelength of sound in air, = c/f |
| η | nominal power transfer efficiency, = PA/PE |
| ηo | reference efficiency defined for radiation into a half-space free field |
| ρo | density of air, = 1.21 kg/m3 at 20° C |
| ω | radian frequency variable, = 2πf |
Theory
Pressure on Axis
Consider a rigid flat circular piston mounted in an infinite flat baffle (half-space) generating peak sinusoidal acoustic volume velocity Uo. The nearfield and farfield sound pressures can be derived from the following equation, which gives the pressure magnitude along the piston axis for measurement distances r varying over the complete range from zero to infinity [2, p. 175]:
PR = (2 ρoc Uo / πa2) ⋅ sin(k/2 ⋅ √r2 + a2 - r)
where PR is the peak pressure magnitude measured at distance r from the piston, a is the piston radius, c is the velocity of sound in air (343 m/s), k is the wave number (2π/λ = ω/c), r is the distance from the measuring point to the piston center, Uo is the piston peak output volume velocity, and ρo is the density of air (1.21 kg/m3 at 20° C).
Figure 1 Description: A diagram illustrating a rigid circular piston of radius 'a' radiating into a half-space (2π steradians). The piston vibrates with peak volume velocity Uo and generates peak farfield pressure PF at a distance 'r' away from the center of the piston.
Farfield Pressure
At points far from the piston where r >> a and for low frequencies such that ka < 1, Eq. (1) converges to:
PF = (ρoc Uo / 2πr)
This is the familiar equation for farfield low-frequency sound pressure for a generalized simple sound source radiating from an infinite baffle. It shows an inverse relationship between pressure and distance.
Nearfield Pressure
At points very close to the center of the piston where r << a, Eq. (1) simplifies to:
PN = PR (r=0) = (2 ρoc Uo / πa2) ⋅ sin(ka/2)
If the frequency is low enough such that ka < 1, this further reduces to:
PN = (ρoc k Uo / πa)
Figure 2 Description: A plot showing the normalized frequency dependence of nearfield sound pressure for a rigid piston in constant acceleration mode. The pressure remains constant up to a frequency where a/λ = 0.26 (ka = 1.6), with a 1 dB fall at this frequency. Nulls occur when the piston radius equals a wavelength or multiple thereof. Above a/λ = 0.5 (ka > π), the pressure envelope falls at 6 dB per octave.
Near-Far Pressure Relationships
Dividing the nearfield equation (Eq. 4) by the farfield equation (Eq. 2) and solving for PN yields:
PN = (2r / a) ⋅ PF
This shows that for low frequencies (ka < 1), nearfield sound pressure is directly proportional to farfield sound pressure, depending only on the ratio of piston radius to sample distance, and is independent of frequency. Crucially, nearfield measurements are independent of the acoustic environment, allowing valid inferences about farfield anechoic operation.
Measuring Distance
Examining Eq. (1) in detail reveals the axial sound pressure dependence on measuring distance. For distances less than approximately 0.75 a2/λ, plane waves are radiated within a cylinder of diameter 2a. Beyond this, spherical divergence occurs, and pressure falls inversely with distance. At frequencies where a = λ (ka ≥ 2), pressure exhibits maxima and nulls as distance increases. For low frequencies (ka < 2π), a null occurs at r = ∞. Figure 3 plots Eq. (1) normalized to maximum axial pressure for various a/λ values.
Figure 3 Description: Plots showing sound pressure along the axis of a rigid circular piston radiating into a half-space for several values of a/λ. The plots illustrate how pressure varies with distance at different frequency-to-size ratios.
To be within 1 dB of the true nearfield pressure, the microphone must be no farther than 0.11a from the piston's center. For low frequencies, farfield conditions exist beyond 2a.
Figure 4 Description: A plot of sound pressure along the axis of a rigid circular piston radiating into a half-space at low frequencies (ka < 1). It shows the pressure variation with distance, indicating that measurements within 0.11a are within 1 dB of the true nearfield pressure.
Flat Piston Pressure Distribution
The analysis so far focused on the piston's axis. The nearfield pressure distribution over the surface of a piston is complex, especially at higher frequencies. However, in the low-frequency range (ka < 1), the distribution is smooth and well-behaved. Figure 5 shows the radial dependence of pressure magnitude for ka = 0.5 and 2, indicating that the low-frequency nearfield pressure varies gradually across the surface, peaking at the piston center.
Figure 5 Description: Normalized nearfield sound pressure distribution on the surface of a rigid circular piston in an infinite baffle, for ka = 0.5 and 2. The distribution is circularly symmetric and depends on the radial distance from the center.
Conical Piston Pressure Distribution
Real loudspeakers often use conical diaphragms. The paper notes that while the theory for flat pistons is well-established, there are no documented studies on the nearfield sound distribution of vibrating cones. However, practical measurements on cone systems have correlated well with flat piston theory when measurements are taken near the cone's apex or dust dome, where pressure is maximum.
Radiated Sound Power
The total radiated sound power (PA) into a half-space for an omnidirectional source (ka < 1) is given by integrating intensity over a hemisphere. Using the relationship between nearfield and farfield pressure (Eq. 5), the power can be calculated from nearfield measurements:
PA = (SD / 4 ρoc) ⋅ PNrms2
This equation allows assessment of total radiated sound power using a simple nearfield pressure measurement at the piston's center. Figure 6 plots this relationship for acoustic power output versus nearfield SPL for various piston sizes.
Figure 6 Description: A graph showing the total radiated sound power (PA) of a rigid circular piston into a half-space, plotted against nearfield sound pressure level (PNrms) for different piston sizes (e.g., 10-in2 effective area, 8-in, 12-in, 15-in advertised diameters).
Efficiency
Transducer efficiency (η) is the ratio of acoustic output power (PA) to nominal electrical input power (PE). Nominal electrical input power is defined as the power available across the voice coil resistance (RE) for a given input voltage (ein): PE = ein2 / RE.
Efficiency can be computed using nearfield pressure and input voltage:
η = (PA / PE) = (SD RE / 2 ρoc) ⋅ (PNrms / ein)2
This method provides efficiencies within 1 dB of the true value for ka < 1.6. Figure 7 illustrates this relationship for a 1-volt RMS input and a 10-ohm voice coil resistance, showing how to scale for different piston sizes and voice coil resistances.
Figure 7 Description: A graph illustrating the relationship between nominal efficiency of a loudspeaker driver (operating as a piston in a half-space) and nearfield sound pressure level. It is normalized for 1V RMS input and 10-ohm voice coil resistance, with curves for different piston sizes.
Frequency and Power Response
As Eq. (5) shows, the near-far pressure relationship is frequency-independent for ka < 1. Therefore, low-frequency response can be measured by plotting nearfield pressure versus frequency. Acoustic power output versus frequency can then be derived using Eq. (10) or Figure 6.
Distortion
Low-frequency harmonic distortion measurements can be accurately made in the nearfield, correlating well with farfield measurements if distortion components are within the specified frequency limits. The higher SPL in the nearfield can improve the acoustic signal-to-noise ratio, making distortion tests more meaningful even in noisy environments.
Loudspeaker System Measurements
The nearfield pressure technique is highly effective for evaluating assembled loudspeaker systems. Measuring the nearfield pressure frequency response of each driver (in and out of the system) provides data on low-frequency bass response, overall system frequency response, efficiency, relative efficiency, and driver levels.
Closed Box
For a closed-box system, the woofer's nearfield pressure frequency response, measured with constant voltage, directly corresponds to the anechoic chamber (half-space loading) frequency response in the piston range. Figures 6 and 7 can be used to plot system acoustic power output and efficiency against frequency.
In multiway systems, in-box nearfield SPL measurements of all drivers, with the crossover connected, yield data for relative levels, overall response, efficiencies, and crossover frequencies. Eq. (5) helps compute each driver's contribution to the farfield pressure. It's noted that for systems with similar driver characteristics and efficiencies, nearfield SPL is inversely proportional to linear dimensions (e.g., smaller tweeters have higher nearfield SPL).
Vented Box
The nearfield technique is also effective for measuring low-frequency characteristics of vented enclosure systems. The overall system operation can be assessed by measuring each driver's nearfield pressure individually.
The vented-box system frequency response can be evaluated using this method. Figure 8 reproduces theoretical data showing the overall response and individual contributions of the vent and driver for a 4th-order Butterworth alignment. The driver diaphragm response shows a null at the vent resonance frequency (fB), the depth of which relates to cabinet losses (QB). The driver reference efficiency (ηo) can be derived from the nearfield SPL in the level response region above 2fB.
The vent's contribution is measured by placing the microphone at the center of the vent, flush with the surface. Valid vent measurements are limited to frequencies below approximately 1.6fB due to diaphragm crosstalk at higher frequencies.
Figure 8 Description: Theoretical sound pressure frequency response of a vented enclosure system aligned for a 4th-order Butterworth high-pass filter. It shows the overall response and the individual contributions of the vent and diaphragm.
By combining nearfield measurements of the vent and driver, an approximate farfield system response can be constructed. Eq. (5) is used to adjust relative levels based on diaphragm and vent diameters before summing. For example, if the vent diameter is half the driver diameter, the driver output needs a 6 dB increase before summation, considering both magnitude and phase.
Figure 12 Description: Experimental measurements on a 15-inch vented-box system. Includes nearfield frequency response of the diaphragm, nearfield response of the vent, and woofer impedance magnitude.
Figure 13 Description: Photographs showing nearfield measurements being taken on assorted direct radiators in a nonanechoic environment, including a tweeter in a closed-box system and a woofer and vent in a vented-box system.
Figure 14 Description: Approximate overall low-frequency response of a 15-inch vented-box system derived from nearfield measurements. It indicates a slight mistuning from a 4th-order Butterworth alignment at 40 Hz.
Conclusion
The presented theory and experimental measurements demonstrate that loudspeaker system piston-range characteristics can be easily measured using nearfield pressure sampling with a test microphone close to the acoustic radiator. These valid nearfield measurements can be performed in any reasonable environment without an anechoic chamber or large outdoor test site. The experimental results show excellent agreement with traditional test methods.
Appendix
Experimental Measuring Equipment
The following equipment was used:
- Beat frequency audio oscillator: Bruel and Kjaer (B&K) type 1014.
- Power amplifier: 200 watt, McIntosh, model MI-200AB.
- Capacitor microphone: ¼ inch, B&K type 4135 with follower.
- Capacitor microphone: ½ inch, B&K type 4133 with follower.
- Precision measurement amplifier: B&K type 2606.
- Graphic level recorder: B&K type 2305.
Implementation of Box-Pressure Measurements
The frequency equalization network used for Small's box-pressure measurement method [1] was corrected for the 1/ω2 behavior. Box compliance shift and enclosure loss effects were not compensated. A second-order high-pass RC filter with a 1 kHz corner frequency (-3 dB) was used to provide an approximate ω2 response up to about 1 kHz.
Acknowledgment
The author acknowledges Raymond J. Newman (Senior Engineer, Loudspeaker Systems, Electro-Voice) for observing the correlation between nearfield and anechoic measurements. Appreciation is also extended to John Gilliom (Chief Product Engineer, Loudspeakers, EV) and Ray Newman for their criticism and review, and to Dr. Richard H. Small (University of Sydney, Australia) for his comments and suggestions.
References
- [1] R. H. Small, "Simplified Loudspeaker Measurements at Low Frequencies," J. Audio Eng. Soc., vol. 20, pp. 28-33 (Jan./Feb. 1972).
- [2] L. E. Kinsler and A. R. Frey, Fundamentals of Acoustics (Wiley, New York, 1962).
- [3] J. Zemanek, "Beam Behavior Within the Nearfield of a Vibrating Piston," J. Acoust. Soc. Am., vol. 49, pp. 181-191 (1971).
- [4] N. W. McLachlan, Loudspeaker Theory, Performance, Testing and Design (Publications, New York, 1960).
- [5] R. H. Small, "Direct-Radiator Loudspeaker System Analysis," J. Audio Eng. Soc., vol. 20, pp. 383-395 (June 1972).
- [6] J. E. Benson, "Theory and Design of Loudspeaker Enclosures Part I: Electro-Acoustical Relations and Generalized Analysis," Amalgamated Wireless (Australasia) Ltd. Tech. Rev., vol. 14, pp. 1-57 (Aug. 1968).
- [7] A. N. Thiele, "Loudspeakers in Vented Boxes," J. Audio Eng. Soc., vol. 19, pp. 382-392 (May 1971); pp. 471-483 (June 1971).
- [8] J. E. Benson, "Theory and Design of Loudspeaker Enclosures Part III: Introduction to Synthesis of Vented Systems," A.W.A. Tech. Rev., vol. 14, pp. 369–484 (Nov. 1972).
- [9] H. F. Olson, "Direct Radiator Loudspeaker Enclosures," J. Audio Eng. Soc., vol. 17, pp. 22-29 (Jan. 1969).
- [10] R. F. Allison and R. Berkovitz, "The Sound Field in Home Listening Rooms,” J. Audio Eng. Soc., vol. 20, pp. 459–469 (July/Aug. 1972).
Note: Mr. Keele's biography appears in the January/February 1973 issue of the Journal.
