Bertet, Stéphanie

Campmans, Theo

Consten, Niels

Wijnant, Ysbrand

English

Recently a new acoustic measurement device, the SonoCat, has become available. The probe consists of 8 MEMSmicrophones, embedded in a small rigid sphere. By accurately calibrating the amplitude and phase mismatch betweenthe microphones, the probe is able to measure the sound intensity accurately. In this paper, we will explain the methodthat has been used to derive the sound intensity from the measured pressure signals and how the effect of the presenceof the sphere has been accounted for. In addition, we will show experimental results, obtained in the anechoic room ofthe University of Twente, comparing the sound intensity, as measured by the probe, with a benchmark sound intensityprobe. The accuracy and precision of the new probe will be discussed

University of Twente Research Information

On the Measurement of Sound Power using aCubical Arrangement of Microphones in a Small Rigid SphereNiels Consten1,2, Theo Campmans3, Stéphanie Bertet1, Ysbrand Wijnant1,21Soundinsight, Enschede, The Netherlands, Email: {n.consten, s.bertet, y.h.wijnant}@soundinsight.nl2University of Twente, Enschede, The Netherlands, Email: {n.consten, y.h.wijnant}@utwente.nl3LBP Sight, Nieuwegein, The Netherlands, Email: t.campmans@lbpsight.nlIntroductionSound intensity probes of closely spaced microphoneswere first introduced in the early 1980s and are oftenused for sound power determination. These conventionalsound intensity probes consist of two axially and face-to-face arranged microphones and can estimate the co-axial component of the sound intensity vector. By doingeither point or scan measurements, the sound power canbe determined from the measured (time-averaged) soundintensity. Several papers, books [1] and even standards[2, 3, 4] have been written that discuss this principle andits accuracy.Lately, 3D sound intensity probes, measuring three com-ponents of the sound intensity vector, have entered themarket consisting of either 4 (e.g. Siemens SoundBrush),6 (e.g. G.R.A.S. 50VI-1) or 8 microphones (e.g. Soundin-sight SonoCat). These are either ’transparent’ or ’rigid’probes. In the former case, the microphones interferewith the sound field. In the latter case, the micropho-nes are embedded in for example a rigid sphere causingeven more scattering. However, this scattering is inde-pendent of the angle of incidence allowing us to correctfor this. The focus in this paper is on the comparisonof a conventional and a rigid spherical probe for soundpower determination.This paper first explains the methods for the conventionaland spherical probe to derive the sound intensity from themeasured pressure signals. In latter case, we show howto infer the sound field in case the rigid sphere wouldnot be present. In addition, we present experimentalresults, obtained in an anechoic room at the Universityof Twente, comparing the sound powers as measured byboth probes. Finally, the effect of a background noisesource on the measurements will be discussed.TheoryConventional probeConventional sound intensity probes consist of two mi-crophones Pi separated by a distance s, see figure 1.Applying the finite sum and difference approximationon the two microphone signals enables estimating thesound pressure and particle velocity component in thePPsSn21Figure 1: Conventional sound intensity probe equipped withtwo microphones. Reused and modified from [5].SABApproximated fieldActual fieldPnUSSpnUPABnUPAB195Figure 2: Sound pressure and particle velocity field. Reusedand modified from [5].n-direction at the probe centre, see figure 2, where:P (x, k) ≈ 12[P1(x, k) + P2(x, k)] , (1)U(x, k) · n(x) ≈ iZ0ks[P2(x, k)− P1(x, k)] . (2)Here x is the position vector of the probe centre, P1 andP2 are the sound pressure signals of both microphones,Z0 = ρ0c0 is the characteristic impedance of the acou-stic medium, k = ω/c0 is the wave number and s is thedistance between both microphones. The sound intensitycomponent in the n-direction at the probe centre can nowbe calculated at each measurement point byIac(x, k) · n(x) =12<[P (x, k)U(x, k) · n(x)]. (3)Finally, one can determine the sound power by integra-ting the sound intensity over the measurement surfaceS:Wac(k) =∫SIac(x, k) · n(x) dS′. (4)Spherical probeT. Sondergaard and M. Wille presented a way to me-asure sound intensity with a tetrahedral arrangement ofmicrophones in a small rigid sphere [6]. They used spheri-cal harmonics to estimate the sound pressure and particlevelocity at the sphere centre for the case the sphere wasnot in the field. This paper extents this method for acubical arrangement of microphones.Accounting for the presence of the sphere, the sound fieldcan be described as:P (r, k) = Pin(r, k) + Psc(r, k) (5)where the incident and scattered field around the sphereare defined as, respectively:Pin(r, k) =∞∑n=0n∑m=−nAmn(k)jn(kr)Ymn (θ, φ), (6)Psc(r, k) =∞∑n=0n∑m=−nBmn(k)h(2)n (kr)Ymn (θ, φ). (7)Here r = [r, θ, φ] is the position vector in spherical coor-dinates with its origin at the sphere centre, jn(kr) isthe n’th order spherical Bessel function; h(2)n (kr) is then’th order spherical Hankel function of the second kind;Y mn (θ, φ) is the spherical harmonic of degree n and orderm; and Amn(k) and Bmn(k) are the spherical coefficientsof, respectively, the incident and scattered field. For thespherical harmonics, we use the following convention:Y mn (θ, φ) =√2n+ 14π(n−m)!(n+m)!Pmn (cos θ)eimφ, (8)with Pmn (cos θ) the associated Legendre function of de-gree n and orderm. Furthermore, since the radial compo-nent of the particle velocity should be zero on the spheresurface, i.e. at r = a with a the radius of the sphere, onecan derive that:Bmn(k) = −Amn(k)j′n(ka)h(2)′n (ka), (9)where the apostrophe denotes the derivative. Theunknown coefficients Amn can now be calculated by usingthe orthogonality of Y mn (θ, φ), i.e.:Amn(k) =∫ 2π0∫ π0P (a, k)Y mn (θ, φ)∗ sin θdθdφjn(ka)− j′n(ka)h(2)′n (ka)h(2)n (ka), (10)where a = [a, θ, φ] is the position vector on the sphere.For a cubical arrangement of microphones, we can ap-proximate the integral as follows:Amn(k) ≈π2∑8i=1 P (ai, k)× Y mn (θi, φi)∗jn(ka)− j′n(ka)h(2)′n (ka)h(2)n (ka). (11)with ai = [a, θi, φi] the position vector of microphone i.Once we have determined Amn, we can evaluate Eq. (6)at two points close to the sphere centre, e.g. r1 = [ε, 0, 0]and r2 = [ε, π, 0] with ε → 0 and substitute thesesound pressures into Eqs. (1) and (2), where P1(x, k) =Pin(r1, k), P2(x, k) = Pin(r2, k), s = 2ε and r = 0 coi-ncides with x. We can now evaluate the sound inten-sity using Eq. (3) and subsequently the sound powerusing Eq. (4). Note that we align the z-axis with then-direction and that we ignore the contribution of thescattered field (Eq. (7)) to the sound field.ExperimentSetupsTwo experimental setups were considered in an anechoicroom (above 300Hz) at the University of Twente. Fi-gure 3 shows the first experimental setup (A): A JBLFlip 3 speaker in the middle of a cart emits Gaussianwhite noise, such that sound waves flow into the roomthrough the open surfaces of the cart. For the secondsetup (B), the speaker was positioned approximately 1moutside the cart to mimic a background noise source, seefigure 4. The speaker had the same output sound level inboth setups. The dimension for the open surfaces were0.47m x 0.55m for S1 and S3, and 1.03m x 0.55m for S2and S4.Figure 3: Experimental setup A: Cart with JBL Flip 3 spea-ker emitting white noise in an anechoic room at the Universityof Twente.S1S2S3S4ABFigure 4: Speaker location for setup A and B and measure-ment surfaces S1 to S4.EquipmentThe B&K 2260 sound intensity meter with a spacer of12 mm acted as the conventional probe. For the spheri-cal probe, The Soundinsight SonoCat P08R015M003B01(see figure 5) having a cubical arrangement of micropho-nes in a small rigid sphere was used. Table 1 presents theexact microphone positions. The z-axis (θ = 0) is usedas the n-direction (normal to the surface).Figure 5: Spherical probe: SonoCatTable 1: Radial distance (r), polar angle (θ) and azimuthalangle (φ) (ISO convention) of the microphone positions of thespherical probe relative to the sphere centre .Mic r [mm] θ [◦] φ [◦]1 15 21.3 180.02 15 48.6 0.03 15 81.5 12.54 15 81.5 -12.55 15 98.5 55.46 15 13.1 180.07 15 98.5 -55.48 15 15.9 0.0ProcedureThe measurements with both probes are performed ac-cording to ISO standard 9614-2:1996 [3] for each setup.The four openings of the cart were used as the scanningsurfaces and each surface was scanned twice: white tapewas used to facilitate the scan pattern (see figure 3) suchthat horizontal and vertical scan movements could beperformed. The same person scanned the surfaces withboth probes to minimize possible measurement errors in-troduced by the operator. Furthermore, the dynamic ca-pability, repeatability and extraneous noise criteria weremonitored during the measurements according to [3].ResultsFigures 6 and 7 show the sound power levels per 1/3 oc-tave band in setup A as measured by, respectively, theconventional and spherical probe. Due to errors introdu-ced by phase-mismatch of microphones and the finite dif-ference approximation the frequency range for accuratemeasurements is limited from 100Hz to 5kHz, accordingto the specifications of the conventional probe. Two fre-quency bands (80Hz and 6.3kHz) outside this range wereconsidered for comparison purposes only. For each mea-sured surface and frequency band the sound energy wasflowing out of the cart.The difference of the sound power levels as measured byboth probes (conventional minus spherical) in setup A ispresented in figure 8. The standard deviations for theuncertainty in the determination of sound power levelsfor the engineering grade according to [3] are also pre-sented. The difference of the total sound power levelsbetween both probes (green line) easily fall within thesestandard deviations. For the individual surfaces (colou-red bars), only sound power levels in the frequency bands800Hz and 1.6kHz slightly exceed the standard deviation5055606570758080100125160200250315400500630800100012501600200025003150400050006300Sound Power [dB(Z)]Frequency [Hz]SS1S2S3S4Figure 6: Sound power level per 1/3 octave band as measu-red by the conventional probe through all surfaces individu-ally (Si) and in total (S) for setup A.5055606570758080100125160200250315400500630800100012501600200025003150400050006300Sound Power [dB(Z)]Frequency [Hz]SS1S2S3S4Figure 7: Sound power level per 1/3 octave band as mea-sured by the spherical probe through all surfaces individually(Si) and in total (S) for setup A.for surface S3; sound power levels in all other bands andin the bands of the other surfaces are within the stan-dard deviation. We therefore conclude that both probesmeasure comparable sound power levels in setup A.Tables 2 and 3 show the total A- and Z-weighted soundpower levels in the range of 100Hz to 5kHz as measuredby both probes in setup A (where the speaker is insidethe cart) and B (where the speaker is outside the cart).It is again clear that both probes measure comparablesound power levels (within 0.6 dB) in setup A. In se-tup B we would expect to measure zero sound power(energy flowing in on one side will flow out on anotherside); however, some sound power was still measured dueto non-ideal conditions and equipment limitations. Thisdoes not have to be a problem as long as the effect ofthe background noise source on the sound power deter-mination of the source under test is negligible. To ap-proximate this effect, the sound powers as measured inboth setups are superposed, see setup A+B in tables 2and 3. It is clear that the influence of a background noisesource on the measurements is negligible in this case. An-4.0-3.0-2.0-1.00.01.02.03.04.080100125160200250315400500630800100012501600200025003150400050006300Sound Power Difference [dB]Frequency [Hz]S1S2S3S4SSDSDFigure 8: Sound power level difference per 1/3 octave bandas measured by the conventional and spherical probe for allsurfaces individually (Si) and in total (S) for setup A.approximation for the case where the sound power radi-ation of the background noise source is 10 dB higher, isalso presented as setup A+(B+10dB) in tables 2 and 3.Here it is assumed that the measured sound power in se-tup B would have been increased by 10 dB. The influenceis within 0.9dB for the conventional probe and 0.5dB forthe spherical probe.Table 2: Total A-weighted (100-5000 Hz) Sound Power Le-vels as measured by both the conventional and spherical probeSetup C-probe[dB(A)]S-probe[dB(A)]A (+) 85.0 (+) 84.4B (−) 67.0 (−) 64.9A+B (+) 84.9 (+) 84.4A+(B+10dB) (+) 84.2 (+) 83.9Table 3: Total Z-weighted (100-5000 Hz) Sound Power Levelsas measured by both the conventional and spherical probeSetup C-probe[dB(Z)]S-probe[dB(Z)]A (+) 85.8 (+) 85.2B (−) 68.4 (−) 64.8A+B (+) 85.7 (+) 85.2A+(B+10dB) (+) 84.9 (+) 84.8DiscussionWe have assumed that the sound powers in both setups(A and B) can be superposed to consider the effect of abackground noise source. However, in practice the sour-ces are measured at the same time and thus the soundfield is more complex than the individual cases. As soundintensity probes are less accurate in complex sound fields,the real effect of such a background noise source can bedifferent than presented here.Furthermore, we assumed that the sound power as me-asured in setup B would increase proportionally to theincrease in sound power emission of the background noisesource. This seems reasonable as the sound pressures atall microphones will increase proportionally. Hence, themeasured sound intensity and thus sound power in setupB would increase with the same ratio, see also Eqs. (1),(2), (3) and (4).ConclusionThis paper presented a method to measure sound powerlevels using a cubical arrangement of microphones in asmall rigid sphere. The underlying model uses spheri-cal harmonics and infers the sound field for the case therigid sphere would not be present by adding scatteringterms to the model. The performance of such a sphericalprobe has been compared to the performance of a con-ventional probe in an experiment. The determination ofthe sound power emission according to the ISO standard9614-2:1996 of a speaker lying in a cart shows comparableresults for both probes. The influence of a backgroundnoise source on the measurements has been investigatedin two cases and has shown to be negligible or small forboth probes.AcknowledgementThe authors would like to express their gratitude andappreciation to Anne de Jong (ASCEE) for his input andsupport during this project.References[1] Fahy, Frank J.: Sound intensity, E & FN Spon, Lon-don, 2nd edition, (1995).[2] ISO 9614-1:1993: Acoustics – Determination of soundpower levels of noise sources using sound intensity –Part 1: Measurement at discrete points, (1993).[3] ISO 9614-2:1996: Acoustics – Determination of soundpower levels of noise sources using sound intensity –Part 2: Measurement by scanning, (1996).[4] ISO 9614-3:2002: Acoustics – Determination of soundpower levels of noise sources using sound intensity –Part 3: Precision method for measurement by scan-ning, (2002).[5] Kuipers, E. R. and Wijnant, Y. H. and De Boer, A.:Measuring sound absorption: Considerations on themeasurement of the active acoustic power, Acta Acus-tica united with Acustica, 100 (2), 193–204, (2014).[6] Sondergaard, Thomas and Wille, Morten: Optimi-zed vector sound intensity measurements with a te-trahedral arrangement of microphones in a sphericalshell, The Journal of the Acoustical Society of Ame-rica, 138 (5), 2820–2828, (2015).

On the Measurement of Sound Power using a Cubical Arrangement of Microphones in a Small Rigid Sphere

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On the Measurement of Sound Power using a Cubical Arrangement of Microphones in a Small Rigid Sphere

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