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A theoretical model for the evaluation of measurement uncertainty of a sound level meter calibration by comparison method in an anechoic room
Applied Acoustics 65 (2004) 967–984 www.elsevier.com/locate/apacoust
A theoretical model for the evaluation of measurement uncertainty of a sound level meter calibration by comparison method in an anechoic room Y.T. Kim *, Y.B. Lee, M.J. Jho, S.J. Suh Acoustics and Vibration Group, Division of Physical Metrology, Korea Research Institute of Standards and Science, P.O. Box 102 Yusong, Daejeon 305-600, Republic of Korea Received 17 September 2001; accepted 19 March 2004 Available online 19 June 2004
Abstract A theoretical model for the evaluation of measurement uncertainty of a sound level meter (hereafter as ‘SLM’) calibration by comparison method in an anechoic room was developed. Through this model, the uncertainties in the semi-automatic calibration and that in the fullautomatic calibration were estimated for the recently developed SLM calibration system. In order to estimate the standard uncertainty against the SLM positioning, which is a significant uncertainty component, the sound field curve-fitting formulae were adopted. The validity of the curve-fitting method was proven by the similarity of the spatial distributions of radiation sound field produced by the plane circular piston source and that by the cone shape source. A linear equation was used to fit the measurements of the sound field distribution along the radiation axis. A quadratic equation was used to fit the measurements along the radial axis normal to the radiation axis. The fitting parameters gave us the sensitivity coefficients of the propagation of the uncertainty. In addition, one of the quadratic fitting parameters was found to be a positional uncertainty itself. Using this model, the expanded uncertainties were evaluated for the semi-automatic and full-automatic calibration of SLM. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Uncertainty; Calibration; Sound level meter; Field distribution; Curve-fitting; Sensitivity coefficient
*
Corresponding author. Tel.: +82-42-868-5301; fax: +82-42-868-5643. E-mail address: [email protected] (Y.T. Kim).
0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.03.007
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1. Introduction Whilst there is already a published guide for the expression of uncertainty in measurement [1], it contains only general descriptions on the measurement uncertainty. A theoretical model to estimate the uncertainty for the calibration of the SLM, however, is not yet published. The SLM is the most widely used device to measure noise levels in the industrial field because of the low cost, the compactness of its size, and the relative ease to handle. Demands for the SLM calibration are on the increase, and an automated calibration system [2] was recently developed to reduce the calibration time, and is presently used in our laboratory. Besides developing a theoretical model to estimate the uncertainty of the automated calibration system of the SLM, this paper also estimated the expanded uncertainties of the calibration system for the ‘semi-automatic’ and the ‘full-automatic’ calibration procedures. In order to include all the uncertainty components for the development of this theoretical model, a careful analysis of the measurement system and procedure has been made. The theoretical models for the estimation of the expanded uncertainties corresponding to each calibration procedure are similar, but not identical. Among the uncertainty components in each calibration procedure, the standard uncertainties in SLM positioning are common and significant. In order to include them to the theoretical model, the spatial distribution of the sound field radiated from a cone type speaker must be known, which was calculated and measured in the present work. Sound field curve-fittings are employed to get the positional uncertainties. Due to the axial symmetry of the sound field, there is no azimuth-angle dependency on the positional uncertainties. One of the three parameters for the quadratic fitting of the radial distribution of the sound field is found to be the positional uncertainty for the radial orientation of the SLM. Based on the calibration system construction described in Section 2, the details of the theoretical model for the evaluation of the SLM-calibration are described in Section 3, and the uncertainty estimation results are given in Section 4.
2. Construction of the calibration system Fig. 1 shows the calibration system of the SLM automated in the present work. The sound-generating part was composed of a signal generator (B&K 1049), a digital voltmeter (hereafter as ‘DVM’) I (Keithley 2000) with a frequency counting unit, a power amplifier (B&K 2706), and a speaker (Forstex FE208) which cabinet measures 440(L) 390(W) 490(H) in millimeter. The sound-detection part is composed of a microphone (B&K 4165) coupled with a preamplifier (B&K 2639), a measuring amplifier (B&K 2636), a DVM II (Keithley 2000), and an oscilloscope (HP 54601A). The signal generator, the DVM I in the generating part and the DVM II in the receiving part, were connected with a console PC (Pentium II, 200 MHz) through the GPIB interfaces. The spatial distance between the location of the sound source and
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Fig. 1. Schematic diagram of the automated calibration system of sound level meters.
the location of the microphone or the SLM was maintained at a 1-m separation from each other.
3. Theoretical background A free-filed calibration of SLM by comparison method is made by comparing the sound pressure levels detected by a standard reference microphone to that by a target SLM, in the same acoustic pressure field by a single-tone sinusoidal wave at the frequencies prescribed in the standards [3–6]. The direct output of the comparison of the two sound pressure levels is the difference designated by D, as defined in the following equation: D ¼ SPLs SPLm ;
ð1Þ
where SPLs and SPLm are the sound pressure levels measured by a calibration system connecting the SLM and the microphone, respectively. Since the sound pressure level is defined as SPL 20 logðp=pref Þ, the direct output of the comparison of the two sound pressure levels in Eq. (1) can be written as: ps d¼ ¼ 10D=20 ; ð2Þ pm
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where ps and pm are the acoustic pressures corresponding to SPLs and SPLm , respectively. Hence, the estimation of the expanded uncertainty of Eq. (1) is equivalent to that of Eq. (2). In this paper, the measurement uncertainty of Eq. (2) is mainly discussed, and is used to reduce in dB scale, which is equivalent to the estimation of the expanded uncertainty in Eq. (1). It is practically impossible to measure under the same environmental conditions (temperature, humidity, and ambient pressure), but with the assumption that the quantities ps and pm are measured under similar conditions, the uncertainty components related to these environmental conditions are neglected here. The combined uncertainty for the uncorrelated input quantities is defined as the sum of the standard uncertainties corresponding to each input quantity by the law of the propagation of uncertainty [1]. The ratio of acoustic pressures d is determined by measuring ps (using the SLM) and pm (using the reference microphone). These measurements are uncorrelated with each other. Hence, 2 the combined uncertainty of d can now be written as u2c ðdÞ ¼ ðod=ops Þ 2 u2 ðps Þ þ ðod=opm Þ u2 ðpm Þ, where the partial derivatives ðod=ops Þ and ðod=opm Þ are called the ‘‘sensitivity coefficients’’. The combined relative standard uncertainty of the calibration system for the SLM in an anechoic room (or free field) can be written, as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uc ðd Þ u2 ðps Þ u2 ðpm Þ ¼ : ð3Þ þ 2 d ps2 pm Since the measurements of ps and pm are not made by direct reading of pressures, thus the uncertainties uðps Þ and uðpm Þ are also the combined uncertainties of other input quantities. For the semi-automatic measurement, the direct reading quantity for the measurement of ps is the sound pressure level, and that for the measurement of pm is the electric voltage. While in the full automatic measurement, the direct reading quantities for the measurement of ps and pm are the electric voltages. For the semi-automatic method, the direct reading of sound pressure level indication is converted to the acoustic pressure by: ps ¼ pref 10ðSPL=20Þ , where pref is 20 lPa. In this case, the standard uncertainty uðps Þ is composed of three uncertainties: one in repeated measurements, in the resolution of the SLM, and in the SLM positioning. The position of the SLM ½Rslm ; zslm is assumed to locate at the original position ½R; z of the reference microphone with uncertainties ½uðRÞ; uðzÞ. In here, Rslm and zslm are the radial normal distance from the radiation axis to the acoustical center of the SLM and the axial distance from the acoustical center of the source speaker to that of the SLM, respectively. Hence, the standard uncertainty uðps Þ can be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 rp dp pffiffiffi þ pffiffiffiffiffi þ c2R u2 ð RÞ þ c2z u2 ð zÞ; uðps Þ ¼ ð4Þ n 12 where cR ops =oðoRÞ, and cz ops =oz are the sensitivity coefficients, which will be discussed later. The uðRÞ, and uðzÞ are the standard uncertainties in the SLM positioning. The rp is the standard deviations of the acoustic pressure measurements and
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dp is the acoustic pressure resolution corresponding to the resolution of sound pressure level d. The relationship between d and dp can easily be derived from ps dp ¼ pref 10ðSPLdÞ=20 dp ¼ ps 10d=20 1 : ð5Þ For the full automatic method, the electric voltage of the measuring amplifier output signal vds , whose input is then the output of the SLM vs , is measured by a DVM, so that ps ¼ vs =ss , where ss is the sensitivity of the SLM in mV/Pa scale and vs ¼ vds =ges is the output signal voltage of the SLM. The ges and vds are the effective gain of the amplifying network and the voltage-indication of the DVM II, respectively. The sensitivity of the SLM can be determined by comparing the DVM readout with the acoustic pressure of the sound field applied to the SLM in an anechoic room. The standard uncertainty for the SLM can then be written including the terms relating to the SLM orientation, as: u2c ðps Þ ¼ c2s u2 ðss Þ þ c2v u2 ðvds Þ þ c2g u2 ðges Þ þ c2R u2 ð RÞ þ c2z u2 ð zÞ; vds =s2s ges ,
ð6Þ
2 vds =ss ges ,
where cs ops =oss ¼ cg ops =oges ¼ cv ops =ovds ¼ 1=ss ges , cR ops =oðdRÞ, and cz ops =oz are the sensitivity coefficients. The standard uncertainties, such as uðss Þ, uðvds Þ, uðges Þ, uðRÞ, and uðzÞ, and the sensitivity coefficients cR and cz will be discussed after the standard uncertainty of the reference microphone. To determine the acoustic pressure incident to the reference microphone pm , the electric voltage of the measuring amplifier output signal, whose input is the reference microphone output, is measured by DVM II, so that pm ¼ vm =sm , where sm is the sensitivity of the microphone and vm ¼ vdm =gem the signal voltage of the reference microphone output. gem and vdm are the gain of the amplifier and the indication of DVM II, respectively. The gain of the amplifier can be measured by connecting the signal generator output directly to the preamplifier instead of applying an acoustic field to the microphone, as well as by comparing readout of the DVM with the electric input voltage. The reference microphone sensitivity can be measured by the reciprocity calibration method in the coupler [7], and the nominal uncertainty given in the calibration certificate can then be used to estimate the standard uncertainty of the acoustic pressure. The orientational effect of the reference microphone in the sound field on the uncertainty is associated with only the distance between the microphone position and the SLM position. Hence, the consideration of the reference microphone orientation will be redundant, and the estimation of uncertainty caused by the reference microphone alignment is not necessary. Therefore, the standard uncertainty for the reference microphone can be written as: u2 ðpm Þ ¼ c2m u2 ðsm Þ þ c2g u2 ðgem Þ þ c2v u2 ðvdm Þ; vdm =s2m gm ,
2 vdm =sm gm ,
ð7Þ
where cm ¼ cg ¼ and cv ¼ 1=sm gm . For the semi-automatic calibration, the direct reading of the SLM indicator is made instead of the DVM readout through the amplifying network in the calibration system. Hence,
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Eq. (4) together with Eq. (7) is used to estimate the uncertainty of the SLM calibration. For the full-automatic calibration, as seen in Eq. (6) and Eq. (7), the uncertainty components uðsÞ , uðvd Þ , and uðge Þ are common. The sensitivities ss and sm are the quantities of different devices, such as the SLM and the reference microphone. However, the gains ges and gem are of the same amplifying network, and vds and vdm are the readout voltages of the same DVM. Hence, the uncertainties uðvd Þ and uðge Þ appeared in Eq. (6) are not needed to be in Eq. (7) for the full automatic calibration [2], and the last two terms on the right hand side of the Eq. (7) can be eliminated, uðpm Þ ¼ jcm uðsm Þj:
ð8Þ
Hence, Eq. (6) through Eq. (8) is used in the estimation of the uncertainty of the full-automatic SLM calibration. The sensitivities of the SLM or the reference microphone given in the dB scale are commonly defined as sdB 20 logðs=sref Þ, where the reference sensitivity sref is 1 V/Pa and s is the sensitivity in the V/Pa scale. The standard uncertainty for the reference microphone sensitivity can be transformed from the nominal value un , given in the dB (re 20 lPa) scale by a reciprocity calibration method with a 95% confidence level, to an expression of un ðsmdB Þ s 10 20 1 : uðsm Þ ¼ ð9Þ 1:96 Since the SLM sensitivity was determined by comparing the DVM readout voltage of the direct output of SLM to the acoustic pressure amplitude corresponding to the sound pressure level readout of the SLM, the standard uncertainty for the SLM sensitivity can be written with its nominal uncertainty component, as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 h i2 r s un ðssdB Þ pffiffiffi þ 10 20 1 ; uðss Þ ¼ ð10Þ 1:96 n where a 95% confidence level is assumed for the nominal uncertainty. The standard uncertainty of the measurement by DVM uðvds Þ appeared in Eq. (6) is composed of an uncertainty by the repeated measurement of voltages and the nominal uncertainty in the certificate of the DVM. Assuming that the nominal uncertainty in the certificate had a uniform or rectangular distribution of possible values, the standard uncertainty of the DVM measurement can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 rv vd r pffiffiffi þ pffiffiffi : ð11Þ uðvs Þ ¼ n 3 Here, r is the nominal relative uncertainty of the DVM in the certificate, rv is the standard deviation pffiffiffi of the DVM measurement, n is the number of measurement repetition, and 3 is due to the assumption of a uniform or rectangular distribution of possible values. The gain of the amplifying network ges in Eq. (6) can be changed by different settings of the level range attenuator of the measuring amplifier and is an effective
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value that should be measured. This gain is not equal to the nominal value of the measuring amplifier gain. The standard uncertainty of the effective gain of amplifying network can then be written as: rg uðge Þ ¼ pffiffiffi ; ð12Þ n where rg and n are the standard deviation and the number of measurements, respectively. To determine the sensitivity coefficients of cR and cz , and the standard uncertainties uðRÞ and uðzÞ in Eq. (6), the spatial distribution of the radiation field produced not by an ideal piston source, but by a real speaker of practical use must be known. The field distribution can be given by the Rayleigh–Sommerfeld equation [7], either by a direct numerical double-integration or by an exact analytical solution of it, with the proper geometrical boundary of the real speaker source used. The exact analytical solution is somewhat difficult to obtain. It can, however, easily be estimated and verified that the radiation field from the real speaker is similar to that from the ideal piston source. For the verification of the similarity of the radiation fields, a simple geometry of a cone is assumed as shown in Fig. 2. To this geometry, the original form of the Rayleigh–Sommerfeld equation is applicable Z Z jkr0 q0 cu0 k e pðr; hÞ ¼ j dS 0 ; ð13Þ 2p r0 s where q0 , c, u0 , and k are the density of air, the sound speed in air, the membrane excitation velocity, and the wave number. The distance r0 from the surface element dS 0 to the field point is altered from the original form for the plane piston to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 ¼ r2 þ rs2 2rrs cos /s ; ð14Þ where r, rs , and /s are the distance from the origin to the field point, the distance from the origin to the surface element, and the angle between r and rs . The projection x
x
dS cosϕ
φs θ
rs zs
ϕ
p ( r ,θ )
r′ r
dS
Rs z
p ( R, Θ
R′
Φs
(
dS ′ =
R Θ
y
z
y (a)
(b)
Fig. 2. Configuration of coordinate system of the cone-shape radiator for the numerical prediction of radiation sound field.
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dS on the xy-plane of the surface element dS 0 on the cone is equal to that of the plane circular piston [7], as dS ¼ Rs dRs dUs , where Rs and Us are the radial distance on the xy-plane from the origin to dS and the angle between Rs and the projection R on the xy-plane of r. As seen in the Fig. 2, the cone-angle is designated as u, so the surface element can be written as: dS 0 ¼
1 Rs dRs dUs : cos u
ð15Þ
To use this projected coordinate system in a numerical calculation of Eq. (13), r0 must be rewritten as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 ¼ R02 þ z02 ; ð16Þ where R0 is the projection of r0 on the x–y plane and z0 ¼ z zs is the axial distance corresponding to r0 . Fig. 3 shows the radiation field spatial distributions of sound at 1.0 kHz and that at 6.3 kHz of frequency from the speaker modeled in a cone shape together with that
Fig. 3. Examples of the prediction results of the sound pressure level field distribution for the 1 and 6.3 kHz frequency with cone-shape and plane-shape models.
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from the plane circular piston model [8]. As seen in Fig. 3, the spatial distribution of the sound field pattern radiated from the speaker is similar to that from the plane piston. Hence, one may take an approximated form the radiated field from the speaker as that from the plane circular piston. The acoustic pressure along the radiation axis by the plane circular piston source is inversely proportional to the axial distance z in the free-field (far-field) region [7]. Hence, the inverse of the acoustic pressure along the radiation axis can now be fitted by the linear equation: 1 ¼ az þ b ¼ a z b0 ; ð17Þ p where a and b are the fitting parameters, and especially the fitting parameter b0 indicates an effective acoustic center of the sound source. The sensitivity coefficients in Eq. (6) can be written as: cð zÞ ¼
op a ¼ : oz ð az þ bÞ2
ð18Þ
For the precise determination of the axial distance z, the time delay dt between the moment of electric signal applying to the speaker and that of electrical signal received by the SLM can be used. Although small, the possibility of SLM missorientation along the radiation axis still exists, so that the axial distance can be written as: z cdt, where c is the speed of sound. Hence, the standard uncertainty for the SLM positioning can be written as: r uð zÞ ¼ cuðdtÞ ¼ c pffiffiffi : ð19Þ n The acoustic pressure distribution along the radial axis normal to the radiation axis can be approximated as a Bessel function [7]. Moreover, in a small range around the axial point in the free-field region, a quadratic approximation can be appropriate, p ¼ AR2 þ BR þ C;
ð20Þ
where A, B, and C are the fitting parameters. Eq. (20) can be rewritten as: p ¼ AðR B0 Þ2 þ E, where B0 ¼ B=2A and E ¼ C B2 =4A. Since the fitting result B0 is equal to the distance from the center of the spatial distribution along the radial axis to the microphone center, and the average value of B0 turned out to be the positional standard uncertainty uðRÞ, i.e. n P B0 i¼1 u ð RÞ ¼ : ð21Þ n Now, the sensitivity coefficient cR in Eq. (6) is deduced from Eq. (20), cR ¼ 2AR þ B:
ð22Þ
The expanded uncertainty is determined by the relation: U ¼ kuc ðd Þ;
ð23Þ
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where k is the coverage factor. For normal distribution, the coverage factor is 2, which gives a 95% confidence level. The expanded uncertainty conversion from the ratio d to that of the difference D can be obtained by: d þU UdB ¼ 20 log : ð24Þ d Consequently, the expanded uncertainty in the dB scale for the semi-automatic calibration can be obtained from Eq. (24) by substitution of Eqs. (3)–(5), (7) and (18)–(23). The expanded uncertainty in the dB scale for the full-automatic calibration can be obtained from Eq. (24) by substituting the Eqs. (3), (6), (8)–(12), and (18)–(23).
4. Results The SLM calibration by a comparison method requires the placement of the reference microphone at certain position and exchange of it to the SLM at the same position in an anechoic room. The distance between the position of the previously placed reference microphone and that of the lately exchanged SLM is found to be an error of the SLM position. For the uniformly and equally distributed sound field, there is no propagation of the positional uncertainty to the uncertainty of the acoustic pressure measurement. Generally, the sound field radiated from the speaker is spatially distributed with various magnitudes, so that the reference microphone and the SLM measure the different acoustic pressures according to their positions. This means that the uncertainty of the SLM position propagates to the uncertainty of the measurement values of the acoustic pressure. Although the spatial distribution of the sound field radiated from the speaker is similar to that from the plane piston, those are not same. Because the prediction through the Rayleigh–Sommerfeld equation gives only the relative values of the acoustic pressure amplitude in the sound field, the predicted results cannot be used as a real field to reduce the positional uncertainty to the effect on the acoustic pressure measurements. The best way is to use the equation determined through the curve-fitting method of the measured spatial distribution of the sound field radiated from the real speaker source. Fig. 4 is composed of the result measuring the sound pressure level by using the reference microphone with the variation of the frequency and the axial distance (Fig. 4(a)) and its comparison results with numerical predictions and linear curve-fitting values. Where, Fig. 4(b), (c) and (d) are the examples at 1.0 kHz frequency results represented as an inverse of acoustic pressure, an acoustic pressure, and a sound pressure level, respectively. As seen in Fig. 4(a), the sound fields radiated from the speaker and predicted by numerical method are commonly normalized to 85 dB at the axially 1-m position from the speaker. As seen in figures Fig. 4(b), (c) and (d), the measurement result agrees well with the linearly fitted curve, and differs infinitesimal but distinct from the prediction result. This difference reveals an increasing tendency as far off from the normalized point, and its degree depends on the frequency.
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110 25 cm 75 cm
50 cm 100 cm
100
-1
95 90
Experimental Numerical method Linear fit
3 1/p [Pa ]
SPL [dB]
105
2 1
85 80
100
(a)
2.0
0.5
(c)
20
40 60 80 100 Axial distance [cm]
0
20
(b)
SPL [dB]
1.0
0.0
0
Experimental Numerical method Linear fit
1.5
p [Pa]
1000 10000 Frequency [Hz]
100 98 96 94 92 90 88 86 84
Experimental Numerical method Linear fit
20 (d)
40 60 80 100 120 Axial distance [cm]
40 60 80 100 Axial distance [cm]
Fig. 4. Selected examples of the measurement results of the radiation field distribution along the radiation axis and three-type of the fitting result expression of the 1 kHz sound field.
In this study, the expanded uncertainties are estimated for two cases of the SLM calibrations, one for semi-automatic calibration, and the other for full automatic calibration. For the semi-automatic calibration, the expanded uncertainty in the dB scale is given by Eq. (24) with the comprising standard uncertainty components listed in Table 1. The first two terms in the right-hand side of Eq. (4) are also given in Table 1. The standard uncertainty elements for the calculation of the reference microphone uðpm Þ, the standard uncertainty elements used in the calculation for the positioning of the SLM along the radiation axis uðzÞ, and that along the radial axis uðRÞ in Table 1 are separately presented in Tables 2–4, respectively. Table 2 is a representation of the results estimated through Eq. (7), which corresponds to the reference microphone standard uncertainty for the semi-automatic calibration. The standard uncertainty of the source sound field calibration by the reference microphone with comprising uncertainty components and sensitivity coefficients are given. As seen in the Table 2, the absolute values of the products jcm uðsm Þj, cg uðgem Þ, and jcv uðvdm Þj are found to have values of the order of 109 , 104 , and 103 , respectively. Since jcv uðvdm Þj is the greatest among them, the uncertainty relating to the measurement by a DVM is the most significant for the source sound field calibration measurement. The uncertainties of the reference microphone sensitivities uðsm Þ with the frequency variation listed in Table 2 is the reduction of the nominal uncertainties given in the dB re 1 V/Pa scale by a reciprocity
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Table 1 Uncertainty estimation result for the semi-automatic calibration of the sound level meter pffiffiffiffiffi pffiffiffi f (Hz) uðpm Þ cc uðzÞ cR uðRÞ rp n dp 12 uðps Þ uðdÞ UdB 3 3 6 3 ( 10 Pa) ( F 10 Pa) ( 10 Pa) ( 10 Pa) ( 104 Pa) ( 103 Pa) ( 103 ) (dB) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
2.57 2.05 1.72 2.30 2.43 1.58 2.37 2.64 2.37 2.49 4.06 3.70 2.92 3.61 5.02 3.68 3.97 5.22 3.05
)3.62 )3.60 )3.63 )3.55 )3.31 )3.44 )3.15 )3.31 )3.08 )3.50 )3.81 )3.70 )3.31 )3.33 )3.56 )2.89 )3.50 )3.63 )3.12
1.58 0.55 1.23 2.36 )0.26 0.23 0.74 0.54 0.49 0.28 )0.66 0.15 )0.54 )0.09 )1.49 )0.81 0.24 1.33 )0.10
2.59 2.07 1.81 2.32 2.59 1.64 2.59 2.87 2.59 2.59 4.51 4.10 3.14 3.75 5.33 3.82 3.82 4.51 2.32
1.20 1.18 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.18 1.20 1.18
4.45 4.15 4.06 4.24 4.20 3.81 4.08 4.38 4.02 4.36 5.90 5.52 4.57 5.02 6.41 4.80 5.18 5.79 3.89
1.45 1.31 1.24 1.36 1.37 1.16 1.33 1.44 1.32 1.41 2.01 1.87 1.52 1.74 2.29 1.70 1.85 2.20 1.40
0.25 0.22 0.21 0.23 0.23 0.20 0.23 0.24 0.22 0.24 0.34 0.32 0.26 0.29 0.39 0.29 0.31 0.37 0.24
calibration method to the mV/Pa scale by Eq. (9). The nominal sensitivity with its uncertainty in the calibration certificate of the reference microphone is shown in Fig. 5. Although the sensitivity coefficient cm ¼ vdm =s2m gm involves the effective gain of Table 2 The standard uncertainty of the reference microphone used in the semi-automatic calibration of SLM f (Hz) cm cg cv uðsm Þ uðgem Þ uðvdm Þ uðpm Þ ( 104 Pa2 /V) ( 101 Pa) ( 10þ1 Pa/V) ( 105 V/Pa) ( 103 ) ( 104 V) ( 103 Pa) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
)7.89 )7.91 )7.90 )7.88 )7.88 )7.93 )7.89 )7.95 )7.91 )7.99 )7.99 )8.02 )8.08 )8.18 )8.30 )8.57 )8.95 )10.12 )10.66
)3.52 )3.52 )3.52 )3.51 )3.51 )3.53 )3.51 )3.53 )3.51 )3.53 )3.53 )3.53 )3.53 )3.53 )3.52 )3.53 )3.53 )3.52 )3.52
2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.21 2.21 2.22 2.22 2.23 2.25 2.27 2.31 2.38 2.48 2.82 2.97
7.95 7.94 7.94 7.94 7.94 7.94 7.93 7.92 7.91 7.88 7.88 7.84 7.78 7.70 7.56 7.35 9.39 10.36 11.78
2.90 2.00 0.19 2.43 1.64 0.11 0.18 1.38 0.12 2.08 1.25 0.64 1.26 2.93 2.52 0.61 2.49 2.15 1.14
1.03 0.83 0.73 0.93 1.03 0.66 1.03 1.14 1.03 1.03 1.80 1.63 1.25 1.50 2.12 1.52 1.52 1.79 0.93
2.57 2.05 1.72 2.30 2.43 1.58 2.37 2.64 2.37 2.49 4.06 3.70 2.92 3.61 5.02 3.68 3.97 5.22 3.05
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Table 3 The fitting parameter in Eq. (17), the standard uncertainty for the positioning of the SLM on the radiation axis, and the sensitivity coefficient f (Hz)
a (Pa/m)
b ( 102 Pa)
cz ( 101 Pa/m)
uðzÞ (cm)
cz uðzÞ ( 103 Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
2.89 2.87 2.87 2.84 2.60 2.73 2.49 2.65 2.45 2.79 3.02 2.94 2.65 2.65 2.82 2.30 2.78 2.90 2.47
)6.33 )0.04 )0.06 )0.01 0.20 0.09 0.32 0.18 0.37 0.03 )0.21 )0.12 0.18 0.17 )0.01 0.52 0.04 )0.08 0.34
)3.62 )3.60 )3.63 )3.55 )3.31 )3.44 )3.15 )3.31 )3.08 )3.50 )3.81 )3.70 )3.31 )3.33 )3.56 )2.89 )3.50 )3.63 )3.12
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
)3.62 )3.60 )3.63 )3.55 )3.31 )3.44 )3.15 )3.31 )3.08 )3.50 )3.81 )3.70 )3.31 )3.33 )3.56 )2.89 )3.50 )3.63 )3.12
Table 4 The fitting parameters in Eq. (20), the standard uncertainty for the positioning of the SLM on the radial axis, and the sensitivity coefficient f (Hz)
A B C cR jB=2Aj ( 101 Pa/m2 ) ( 103 Pa/m) ( 101 Pa) ( 103 Pa/m) (mm)
uðRÞ (mm)
cR uðRÞ ( 106 Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
)0.17 )0.17 )0.17 )0.17 )0.17 )0.18 )0.19 )0.19 )0.22 )0.25 )0.30 )0.38 )0.51 )0.72 )1.13 )1.87 )2.75 )3.62 )5.46
1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21
1.61 0.56 1.26 2.41 )0.27 0.24 0.76 0.55 0.50 0.29 )0.67 0.15 )0.55 )0.09 )1.52 )0.83 0.24 1.36 )0.10
1.33 0.46 1.04 1.99 )0.22 0.20 0.62 0.45 0.42 0.24 )0.56 0.12 )0.45 )0.08 )1.60 )1.42 0.43 3.43 0.42
0.355 0.354 0.355 0.354 0.355 0.355 0.355 0.354 0.355 0.355 0.354 0.353 0.353 0.351 0.351 0.349 0.349 0.350 0.351
1.33 0.46 1.04 1.99 )0.22 0.20 0.62 0.45 0.42 0.24 )0.56 0.12 )0.45 )0.08 )1.26 )0.69 0.20 1.12 )0.08
3.90 1.39 3.13 5.91 0.65 0.54 1.63 1.16 0.93 0.48 0.93 0.16 0.44 0.05 0.71 0.38 0.08 0.47 0.04
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0.06
Sensitivity [V/Pa]
0.05 0.04 0.03 0.02 0.01 0.00 100
1000 Frequency [Hz]
Fig. 5. Calibration chart of the reference microphone sensitivity with nominal uncertainties represented as a function of frequency.
amplifying network, it looks somewhat independent to the frequency, and is approximated 1.01. Whether the calibration method is automatic or semi-automatic, there may be two different ways to estimate the standard uncertainty uðps Þ in Eq. (3). For the semi-
Table 5 Summary of the uncertainty estimation results for the full-automatic calibration of the SLM f uðpm Þ cz uðzÞ cR uðRÞ cs uðss Þ cv uðvds Þ cg uðges Þ uðps Þ uðdÞ UdB (Hz) ( 104 Pa) ( 103 Pa) ( 105 Pa) ( 105 Pa) ( 105 Pa) ( 106 Pa) ( 103 Pa) ( 102 ) (dB) 125 6.27 160 6.28 200 6.28 250 6.26 315 6.26 400 6.29 500 6.26 630 6.30 800 6.26 1000 6.30 1250 6.30 1600 6.29 2000 6.29 2500 6.30 3150 6.27 4000 6.30 5000 8.40 6300 10.48 8000 12.56
)1.81 )1.80 )1.81 )1.77 )1.65 )1.72 )1.58 )1.65 )1.54 )1.75 )1.90 )1.85 )1.66 )1.67 )1.78 )1.45 )1.75 )1.82 )1.56
6.65 2.31 5.20 9.95 )1.12 0.98 3.12 2.27 2.08 1.20 )2.78 0.62 )2.27 )0.38 )6.30 )3.43 1.01 5.60 )0.42
)8.05 )6.44 )5.64 )7.24 )8.09 )5.08 )8.10 )8.90 )8.11 )8.10 )14.08 )12.87 )9.95 )12.00 )17.40 )12.80 )13.36 )17.92 )9.76
2.64 2.61 2.60 2.63 2.64 2.59 2.64 2.64 2.64 2.63 2.75 2.72 2.66 2.70 2.85 2.72 2.73 2.87 2.67
)1.43 )1.14 )1.00 )1.28 )1.43 )0.90 )1.43 )1.57 )1.43 )1.43 )2.48 )2.25 )1.73 )2.06 )2.94 )2.10 )2.10 )2.48 )1.28
1.81 1.80 1.81 1.77 1.65 1.72 1.58 1.65 1.54 1.75 1.90 1.85 1.66 1.67 1.78 1.45 1.75 1.82 1.56
5.38 5.40 5.39 5.29 4.97 5.14 4.76 4.97 4.67 5.23 5.63 5.49 4.97 5.00 5.30 4.43 5.50 5.90 5.68
0.09 0.09 0.09 0.09 0.09 0.09 0.08 0.09 0.08 0.09 0.10 0.09 0.09 0.09 0.09 0.08 0.09 0.10 0.10
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Table 6 The standard uncertainty of the reference microphone used in the full-automatic calibration of the SLM f (Hz)
vdm ( 102 V)
sm ( 102 V/Pa)
cm (Pa2 /V)
un ðsdB Þ (dB)
uðsm Þ ( 105 V/Pa)
jcm uðsm Þj ( 104 Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
1.62 1.62 1.62 1.61 1.61 1.62 1.61 1.62 1.61 1.61 1.61 1.60 1.59 1.57 1.54 1.50 1.44 1.26 1.20
4.50 4.50 4.50 4.50 4.50 4.50 4.49 4.49 4.48 4.47 4.46 4.44 4.41 4.36 4.28 4.16 3.99 3.52 3.33
)7.89 )7.91 )7.90 )7.88 )7.88 )7.93 )7.89 )7.95 )7.91 )7.99 )7.99 )8.02 )8.08 )8.18 )8.30 )8.57 )8.95 )10.12 )10.66
0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05 0.06
7.95 7.94 7.94 7.94 7.94 7.94 7.93 7.92 7.91 7.88 7.88 7.84 7.78 7.70 7.56 7.35 9.39 10.36 11.78
6.27 6.28 6.28 6.26 6.26 6.29 6.26 6.30 6.26 6.30 6.30 6.29 6.29 6.30 6.27 6.30 8.40 10.48 12.56
automatic calibration, Eq. (4) is used, and for the full-automatic one, Eq. (6) is used. The terms cR uðRÞ and cz uðzÞ, which are related to the SLM positioning, are common in Eq. (4) and Eq. (6). The sensitivity coefficients cR and cz can be obtained by fitting Table 7 The standard uncertainty for the sensitivity of the specified sound level meter (B&K 2231) and its comprising components un ðssdB Þ pffiffiffi Ss rs n ss un ðssdB Þ 1:96 uðss Þ cs uðss Þ f (Hz) rs ð10 20 1Þ cs 2 2 2 ( 10 V/Pa) ( 10 V/Pa) (V/Pa) (dB) ( 104 Pa2 /V) ( 102 V/Pa) ( 105 Pa) ( 10 V/Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
3.27 3.97 3.68 3.58 2.09 2.81 1.98 3.02 2.68 3.73 3.20 3.91 3.49 3.18 2.75 2.78 4.14 2.68 2.55
1.03 1.26 1.16 1.13 0.66 0.89 0.63 0.96 0.85 1.18 1.01 1.24 1.10 1.01 0.87 0.88 1.31 0.85 0.81
1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22
)7.77 )7.76 )7.77 )7.79 )7.78 )7.74 )7.78 )7.72 )7.75 )7.68 )7.68 )7.65 )7.59 )7.50 )7.39 )7.16 )6.86 )6.06 )5.76
3.38 3.46 3.42 3.41 3.29 3.34 3.28 3.36 3.33 3.43 3.38 3.45 3.40 3.37 3.34 3.34 3.48 3.33 3.32
)2.63 )2.68 )2.66 )2.66 )2.56 )2.58 )2.55 )2.59 )2.58 )2.64 )2.59 )2.64 )2.59 )2.53 )2.47 )2.39 )2.38 )2.02 )1.91
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the spatial distribution along the corresponding axis. The fitting parameters are listed in Tables 3 and 4 with their uncertainties. For the sound pressure level measurement at each frequency by SLM through a semi-automatic procedure, 12 data are acquired, and 10 of them are taken in the average excluding for the maximum and the minimum value. Hence, the standard deviations rp in Table 1 have to be obtained after reducing the value of the sound pressure level to the value of thepacoustic pressure. The standard uncertainty uðdp Þ by ffiffiffiffiffi the resolution of the SLM dp = 12 was also given in Table 1. The resolution of the SLM d in dB scale is 0.01 dB independent to the frequency, and it was transformed to the Pa scale dp through Eq. (5). For the full-automatic calibration, the expanded uncertainty is given in Table 5 with its components consisting the standard uncertainties given in Eq. (6) and Eq. (7). The standard uncertainty uðpm Þ for the referenced microphone through Eq. (5), uðzÞ for the positioning of SLM along the radiation axis, and uðRÞ along the radial axis are given in Table 5, and their elements are already given in Tables 2–4, respectively. The standard uncertainty of the reference microphone defined in Eq. (8) and used in the full-automatic calibration of the SLM is tabulated along with its associated elements in Table 6. The estimated values of the standard uncertainties for the SLM calibration according to Eq. (10) are given in Table 7. The standard uncertainty for the DVM measurements defined by Eq. (11), and its components are given in Table 8. Finally, in Table 9 the estimations of the uncertainty components of the amplifying network of the calibration system are listed.
Table 8 The standard uncertainty for the measurement of the DVM indication and its components pffiffiffi pffiffiffi f rv rv = n vd r vd r= 3 cv uðvs Þ cv uðvs Þ 4 4 3 4 (Hz) ( 10 V) ( 10 V) ( 10 V) ( 10 ) ( 106 V) ( 101 Pa/V) ( 104 V) ( 105 Pa) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
3.26 2.61 2.28 2.92 3.26 2.06 3.26 3.61 3.26 3.26 5.67 5.16 3.95 4.73 6.70 4.81 4.81 5.67 2.92
1.03 0.82 0.72 0.92 1.03 0.65 1.03 1.14 1.03 1.03 1.79 1.63 1.25 1.49 2.12 1.52 1.52 1.79 0.92
2.53 2.53 2.53 2.54 2.54 2.52 2.53 2.52 2.53 2.50 2.50 2.49 2.48 2.44 2.41 2.33 2.24 1.98 1.88
8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00
1.17 1.17 1.17 1.17 1.17 1.16 1.17 1.16 1.17 1.16 1.16 1.15 1.14 1.13 1.11 1.08 1.03 0.91 0.87
5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51
1.03 0.82 0.72 0.92 1.03 0.65 1.03 1.14 1.03 1.03 1.79 1.63 1.25 1.49 2.12 1.52 1.52 1.79 0.92
5.68 4.55 3.98 5.09 5.69 3.59 5.69 6.29 5.69 5.69 9.89 8.99 6.89 8.24 11.68 8.39 8.39 9.89 5.09
Y.T. Kim et al. / Applied Acoustics 65 (2004) 967–984 Table 9 Standard uncertainity for the effective gain of the amplifying network pffiffiffi f (Hz) rg ( 103 ) rg = n ( 103 ) ge cg ( 103 Pa) uðge Þ ( 103 ) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
9.19 6.33 0.61 7.68 5.19 0.34 0.56 4.38 0.39 6.59 3.96 2.04 3.97 9.27 7.95 1.92 7.87 6.81 3.61
2.90 2.00 0.19 2.43 1.64 0.11 0.18 1.38 0.12 2.08 1.25 0.64 1.26 2.93 2.52 0.61 2.49 2.15 1.14
1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01
)1.38 )1.38 )1.38 )1.38 )1.38 )1.38 )1.38 )1.37 )1.38 )1.37 )1.37 )1.36 )1.35 )1.33 )1.32 )1.27 )1.22 )1.08 )1.02
2.90 2.00 0.19 2.43 1.64 0.11 0.18 1.38 0.12 2.08 1.25 0.64 1.26 2.93 2.52 0.61 2.49 2.15 1.14
983
cg uðge Þ ( 106 Pa) )4.02 )2.76 )0.26 )3.36 )2.27 )0.15 )0.24 )1.90 )0.17 )2.85 )1.71 )0.88 )1.70 )3.91 )3.31 )0.77 )3.04 )2.32 )1.17
Summaries of the estimation results of the standard uncertainty for the semiautomatic calibration of the SLM are listed in Table 1, and that for the full-automatic calibration of SLM are listed in Table 5. As seen in Tables 1 and 5, the expanded uncertainty for the semi-automatic calibration is greater than the full-automatic calibration. 5. Conclusions In this paper, the theoretical model for estimating the uncertainty of the SLM calibration is developed and tested for two cases, such as a semi-automatic calibration and a full-automatic calibration. The method of using the curve fitting parameters is adopted to evaluate the uncertainty in positioning the SLM to the original position of the reference microphone. The measurement of time delay, between the input electric signal to the sound source and the arrival of the sound signal to the receiver, enables us to obtain the positional uncertainty along the radiation axis. Our finding that the fitting parameter B0 ¼ B=2A in Eq. (21) turned out to be used to obtain the positional uncertainty on the radial axis is useful to get the overall uncertainty of the SLM calibration in a free-field condition by the comparison method. Acknowledgements This work was financially supported by the Maintenance and Improvement of National Measurement Standard in KRISS.
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References [1] International Organization for Standardization, Guide to the expression of uncertainty in measurement, 1993. [2] Kim YT, Lee YB, Jho MJ. A fast automatic calibration system for a sound level meter in an anechoic room. Applied Acoustics 2003;64:459–70. [3] International Electrotechnical Commission, Sound level meters, IEC standard 651, 1979. [4] International Electrotechnical Commission, Integrating-averaging sound level meters, IEC standard 804, 1985. [5] Korea Association of standards and Testing Organizations. Standard calibration procedure of sound level meter, KASTO 96-16-011-098, 1996. [6] International Organization for Standard, Acoustics Preferred frequencies, ISO266, 2nd ed., 1997. [7] Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of acoustics. 3rd ed. New York: John Wiley and Sons; 1982. [8] Kim YT, Jho MJ, Eun HJ, Kim MG. A matrix model for ultrasonic source calibration and radiation field prediction using a pulsed planar scanning system. J Korean Phys Soc 2000;37(3):221–31.
A theoretical model for the evaluation of measurement uncertainty of a sound level meter calibration by comparison method in an anechoic room Y.T. Kim *, Y.B. Lee, M.J. Jho, S.J. Suh Acoustics and Vibration Group, Division of Physical Metrology, Korea Research Institute of Standards and Science, P.O. Box 102 Yusong, Daejeon 305-600, Republic of Korea Received 17 September 2001; accepted 19 March 2004 Available online 19 June 2004
Abstract A theoretical model for the evaluation of measurement uncertainty of a sound level meter (hereafter as ‘SLM’) calibration by comparison method in an anechoic room was developed. Through this model, the uncertainties in the semi-automatic calibration and that in the fullautomatic calibration were estimated for the recently developed SLM calibration system. In order to estimate the standard uncertainty against the SLM positioning, which is a significant uncertainty component, the sound field curve-fitting formulae were adopted. The validity of the curve-fitting method was proven by the similarity of the spatial distributions of radiation sound field produced by the plane circular piston source and that by the cone shape source. A linear equation was used to fit the measurements of the sound field distribution along the radiation axis. A quadratic equation was used to fit the measurements along the radial axis normal to the radiation axis. The fitting parameters gave us the sensitivity coefficients of the propagation of the uncertainty. In addition, one of the quadratic fitting parameters was found to be a positional uncertainty itself. Using this model, the expanded uncertainties were evaluated for the semi-automatic and full-automatic calibration of SLM. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Uncertainty; Calibration; Sound level meter; Field distribution; Curve-fitting; Sensitivity coefficient
*
Corresponding author. Tel.: +82-42-868-5301; fax: +82-42-868-5643. E-mail address: [email protected] (Y.T. Kim).
0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.03.007
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1. Introduction Whilst there is already a published guide for the expression of uncertainty in measurement [1], it contains only general descriptions on the measurement uncertainty. A theoretical model to estimate the uncertainty for the calibration of the SLM, however, is not yet published. The SLM is the most widely used device to measure noise levels in the industrial field because of the low cost, the compactness of its size, and the relative ease to handle. Demands for the SLM calibration are on the increase, and an automated calibration system [2] was recently developed to reduce the calibration time, and is presently used in our laboratory. Besides developing a theoretical model to estimate the uncertainty of the automated calibration system of the SLM, this paper also estimated the expanded uncertainties of the calibration system for the ‘semi-automatic’ and the ‘full-automatic’ calibration procedures. In order to include all the uncertainty components for the development of this theoretical model, a careful analysis of the measurement system and procedure has been made. The theoretical models for the estimation of the expanded uncertainties corresponding to each calibration procedure are similar, but not identical. Among the uncertainty components in each calibration procedure, the standard uncertainties in SLM positioning are common and significant. In order to include them to the theoretical model, the spatial distribution of the sound field radiated from a cone type speaker must be known, which was calculated and measured in the present work. Sound field curve-fittings are employed to get the positional uncertainties. Due to the axial symmetry of the sound field, there is no azimuth-angle dependency on the positional uncertainties. One of the three parameters for the quadratic fitting of the radial distribution of the sound field is found to be the positional uncertainty for the radial orientation of the SLM. Based on the calibration system construction described in Section 2, the details of the theoretical model for the evaluation of the SLM-calibration are described in Section 3, and the uncertainty estimation results are given in Section 4.
2. Construction of the calibration system Fig. 1 shows the calibration system of the SLM automated in the present work. The sound-generating part was composed of a signal generator (B&K 1049), a digital voltmeter (hereafter as ‘DVM’) I (Keithley 2000) with a frequency counting unit, a power amplifier (B&K 2706), and a speaker (Forstex FE208) which cabinet measures 440(L) 390(W) 490(H) in millimeter. The sound-detection part is composed of a microphone (B&K 4165) coupled with a preamplifier (B&K 2639), a measuring amplifier (B&K 2636), a DVM II (Keithley 2000), and an oscilloscope (HP 54601A). The signal generator, the DVM I in the generating part and the DVM II in the receiving part, were connected with a console PC (Pentium II, 200 MHz) through the GPIB interfaces. The spatial distance between the location of the sound source and
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Fig. 1. Schematic diagram of the automated calibration system of sound level meters.
the location of the microphone or the SLM was maintained at a 1-m separation from each other.
3. Theoretical background A free-filed calibration of SLM by comparison method is made by comparing the sound pressure levels detected by a standard reference microphone to that by a target SLM, in the same acoustic pressure field by a single-tone sinusoidal wave at the frequencies prescribed in the standards [3–6]. The direct output of the comparison of the two sound pressure levels is the difference designated by D, as defined in the following equation: D ¼ SPLs SPLm ;
ð1Þ
where SPLs and SPLm are the sound pressure levels measured by a calibration system connecting the SLM and the microphone, respectively. Since the sound pressure level is defined as SPL 20 logðp=pref Þ, the direct output of the comparison of the two sound pressure levels in Eq. (1) can be written as: ps d¼ ¼ 10D=20 ; ð2Þ pm
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where ps and pm are the acoustic pressures corresponding to SPLs and SPLm , respectively. Hence, the estimation of the expanded uncertainty of Eq. (1) is equivalent to that of Eq. (2). In this paper, the measurement uncertainty of Eq. (2) is mainly discussed, and is used to reduce in dB scale, which is equivalent to the estimation of the expanded uncertainty in Eq. (1). It is practically impossible to measure under the same environmental conditions (temperature, humidity, and ambient pressure), but with the assumption that the quantities ps and pm are measured under similar conditions, the uncertainty components related to these environmental conditions are neglected here. The combined uncertainty for the uncorrelated input quantities is defined as the sum of the standard uncertainties corresponding to each input quantity by the law of the propagation of uncertainty [1]. The ratio of acoustic pressures d is determined by measuring ps (using the SLM) and pm (using the reference microphone). These measurements are uncorrelated with each other. Hence, 2 the combined uncertainty of d can now be written as u2c ðdÞ ¼ ðod=ops Þ 2 u2 ðps Þ þ ðod=opm Þ u2 ðpm Þ, where the partial derivatives ðod=ops Þ and ðod=opm Þ are called the ‘‘sensitivity coefficients’’. The combined relative standard uncertainty of the calibration system for the SLM in an anechoic room (or free field) can be written, as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uc ðd Þ u2 ðps Þ u2 ðpm Þ ¼ : ð3Þ þ 2 d ps2 pm Since the measurements of ps and pm are not made by direct reading of pressures, thus the uncertainties uðps Þ and uðpm Þ are also the combined uncertainties of other input quantities. For the semi-automatic measurement, the direct reading quantity for the measurement of ps is the sound pressure level, and that for the measurement of pm is the electric voltage. While in the full automatic measurement, the direct reading quantities for the measurement of ps and pm are the electric voltages. For the semi-automatic method, the direct reading of sound pressure level indication is converted to the acoustic pressure by: ps ¼ pref 10ðSPL=20Þ , where pref is 20 lPa. In this case, the standard uncertainty uðps Þ is composed of three uncertainties: one in repeated measurements, in the resolution of the SLM, and in the SLM positioning. The position of the SLM ½Rslm ; zslm is assumed to locate at the original position ½R; z of the reference microphone with uncertainties ½uðRÞ; uðzÞ. In here, Rslm and zslm are the radial normal distance from the radiation axis to the acoustical center of the SLM and the axial distance from the acoustical center of the source speaker to that of the SLM, respectively. Hence, the standard uncertainty uðps Þ can be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 rp dp pffiffiffi þ pffiffiffiffiffi þ c2R u2 ð RÞ þ c2z u2 ð zÞ; uðps Þ ¼ ð4Þ n 12 where cR ops =oðoRÞ, and cz ops =oz are the sensitivity coefficients, which will be discussed later. The uðRÞ, and uðzÞ are the standard uncertainties in the SLM positioning. The rp is the standard deviations of the acoustic pressure measurements and
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dp is the acoustic pressure resolution corresponding to the resolution of sound pressure level d. The relationship between d and dp can easily be derived from ps dp ¼ pref 10ðSPLdÞ=20 dp ¼ ps 10d=20 1 : ð5Þ For the full automatic method, the electric voltage of the measuring amplifier output signal vds , whose input is then the output of the SLM vs , is measured by a DVM, so that ps ¼ vs =ss , where ss is the sensitivity of the SLM in mV/Pa scale and vs ¼ vds =ges is the output signal voltage of the SLM. The ges and vds are the effective gain of the amplifying network and the voltage-indication of the DVM II, respectively. The sensitivity of the SLM can be determined by comparing the DVM readout with the acoustic pressure of the sound field applied to the SLM in an anechoic room. The standard uncertainty for the SLM can then be written including the terms relating to the SLM orientation, as: u2c ðps Þ ¼ c2s u2 ðss Þ þ c2v u2 ðvds Þ þ c2g u2 ðges Þ þ c2R u2 ð RÞ þ c2z u2 ð zÞ; vds =s2s ges ,
ð6Þ
2 vds =ss ges ,
where cs ops =oss ¼ cg ops =oges ¼ cv ops =ovds ¼ 1=ss ges , cR ops =oðdRÞ, and cz ops =oz are the sensitivity coefficients. The standard uncertainties, such as uðss Þ, uðvds Þ, uðges Þ, uðRÞ, and uðzÞ, and the sensitivity coefficients cR and cz will be discussed after the standard uncertainty of the reference microphone. To determine the acoustic pressure incident to the reference microphone pm , the electric voltage of the measuring amplifier output signal, whose input is the reference microphone output, is measured by DVM II, so that pm ¼ vm =sm , where sm is the sensitivity of the microphone and vm ¼ vdm =gem the signal voltage of the reference microphone output. gem and vdm are the gain of the amplifier and the indication of DVM II, respectively. The gain of the amplifier can be measured by connecting the signal generator output directly to the preamplifier instead of applying an acoustic field to the microphone, as well as by comparing readout of the DVM with the electric input voltage. The reference microphone sensitivity can be measured by the reciprocity calibration method in the coupler [7], and the nominal uncertainty given in the calibration certificate can then be used to estimate the standard uncertainty of the acoustic pressure. The orientational effect of the reference microphone in the sound field on the uncertainty is associated with only the distance between the microphone position and the SLM position. Hence, the consideration of the reference microphone orientation will be redundant, and the estimation of uncertainty caused by the reference microphone alignment is not necessary. Therefore, the standard uncertainty for the reference microphone can be written as: u2 ðpm Þ ¼ c2m u2 ðsm Þ þ c2g u2 ðgem Þ þ c2v u2 ðvdm Þ; vdm =s2m gm ,
2 vdm =sm gm ,
ð7Þ
where cm ¼ cg ¼ and cv ¼ 1=sm gm . For the semi-automatic calibration, the direct reading of the SLM indicator is made instead of the DVM readout through the amplifying network in the calibration system. Hence,
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Eq. (4) together with Eq. (7) is used to estimate the uncertainty of the SLM calibration. For the full-automatic calibration, as seen in Eq. (6) and Eq. (7), the uncertainty components uðsÞ , uðvd Þ , and uðge Þ are common. The sensitivities ss and sm are the quantities of different devices, such as the SLM and the reference microphone. However, the gains ges and gem are of the same amplifying network, and vds and vdm are the readout voltages of the same DVM. Hence, the uncertainties uðvd Þ and uðge Þ appeared in Eq. (6) are not needed to be in Eq. (7) for the full automatic calibration [2], and the last two terms on the right hand side of the Eq. (7) can be eliminated, uðpm Þ ¼ jcm uðsm Þj:
ð8Þ
Hence, Eq. (6) through Eq. (8) is used in the estimation of the uncertainty of the full-automatic SLM calibration. The sensitivities of the SLM or the reference microphone given in the dB scale are commonly defined as sdB 20 logðs=sref Þ, where the reference sensitivity sref is 1 V/Pa and s is the sensitivity in the V/Pa scale. The standard uncertainty for the reference microphone sensitivity can be transformed from the nominal value un , given in the dB (re 20 lPa) scale by a reciprocity calibration method with a 95% confidence level, to an expression of un ðsmdB Þ s 10 20 1 : uðsm Þ ¼ ð9Þ 1:96 Since the SLM sensitivity was determined by comparing the DVM readout voltage of the direct output of SLM to the acoustic pressure amplitude corresponding to the sound pressure level readout of the SLM, the standard uncertainty for the SLM sensitivity can be written with its nominal uncertainty component, as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 h i2 r s un ðssdB Þ pffiffiffi þ 10 20 1 ; uðss Þ ¼ ð10Þ 1:96 n where a 95% confidence level is assumed for the nominal uncertainty. The standard uncertainty of the measurement by DVM uðvds Þ appeared in Eq. (6) is composed of an uncertainty by the repeated measurement of voltages and the nominal uncertainty in the certificate of the DVM. Assuming that the nominal uncertainty in the certificate had a uniform or rectangular distribution of possible values, the standard uncertainty of the DVM measurement can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 rv vd r pffiffiffi þ pffiffiffi : ð11Þ uðvs Þ ¼ n 3 Here, r is the nominal relative uncertainty of the DVM in the certificate, rv is the standard deviation pffiffiffi of the DVM measurement, n is the number of measurement repetition, and 3 is due to the assumption of a uniform or rectangular distribution of possible values. The gain of the amplifying network ges in Eq. (6) can be changed by different settings of the level range attenuator of the measuring amplifier and is an effective
Y.T. Kim et al. / Applied Acoustics 65 (2004) 967–984
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value that should be measured. This gain is not equal to the nominal value of the measuring amplifier gain. The standard uncertainty of the effective gain of amplifying network can then be written as: rg uðge Þ ¼ pffiffiffi ; ð12Þ n where rg and n are the standard deviation and the number of measurements, respectively. To determine the sensitivity coefficients of cR and cz , and the standard uncertainties uðRÞ and uðzÞ in Eq. (6), the spatial distribution of the radiation field produced not by an ideal piston source, but by a real speaker of practical use must be known. The field distribution can be given by the Rayleigh–Sommerfeld equation [7], either by a direct numerical double-integration or by an exact analytical solution of it, with the proper geometrical boundary of the real speaker source used. The exact analytical solution is somewhat difficult to obtain. It can, however, easily be estimated and verified that the radiation field from the real speaker is similar to that from the ideal piston source. For the verification of the similarity of the radiation fields, a simple geometry of a cone is assumed as shown in Fig. 2. To this geometry, the original form of the Rayleigh–Sommerfeld equation is applicable Z Z jkr0 q0 cu0 k e pðr; hÞ ¼ j dS 0 ; ð13Þ 2p r0 s where q0 , c, u0 , and k are the density of air, the sound speed in air, the membrane excitation velocity, and the wave number. The distance r0 from the surface element dS 0 to the field point is altered from the original form for the plane piston to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 ¼ r2 þ rs2 2rrs cos /s ; ð14Þ where r, rs , and /s are the distance from the origin to the field point, the distance from the origin to the surface element, and the angle between r and rs . The projection x
x
dS cosϕ
φs θ
rs zs
ϕ
p ( r ,θ )
r′ r
dS
Rs z
p ( R, Θ
R′
Φs
(
dS ′ =
R Θ
y
z
y (a)
(b)
Fig. 2. Configuration of coordinate system of the cone-shape radiator for the numerical prediction of radiation sound field.
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dS on the xy-plane of the surface element dS 0 on the cone is equal to that of the plane circular piston [7], as dS ¼ Rs dRs dUs , where Rs and Us are the radial distance on the xy-plane from the origin to dS and the angle between Rs and the projection R on the xy-plane of r. As seen in the Fig. 2, the cone-angle is designated as u, so the surface element can be written as: dS 0 ¼
1 Rs dRs dUs : cos u
ð15Þ
To use this projected coordinate system in a numerical calculation of Eq. (13), r0 must be rewritten as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 ¼ R02 þ z02 ; ð16Þ where R0 is the projection of r0 on the x–y plane and z0 ¼ z zs is the axial distance corresponding to r0 . Fig. 3 shows the radiation field spatial distributions of sound at 1.0 kHz and that at 6.3 kHz of frequency from the speaker modeled in a cone shape together with that
Fig. 3. Examples of the prediction results of the sound pressure level field distribution for the 1 and 6.3 kHz frequency with cone-shape and plane-shape models.
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from the plane circular piston model [8]. As seen in Fig. 3, the spatial distribution of the sound field pattern radiated from the speaker is similar to that from the plane piston. Hence, one may take an approximated form the radiated field from the speaker as that from the plane circular piston. The acoustic pressure along the radiation axis by the plane circular piston source is inversely proportional to the axial distance z in the free-field (far-field) region [7]. Hence, the inverse of the acoustic pressure along the radiation axis can now be fitted by the linear equation: 1 ¼ az þ b ¼ a z b0 ; ð17Þ p where a and b are the fitting parameters, and especially the fitting parameter b0 indicates an effective acoustic center of the sound source. The sensitivity coefficients in Eq. (6) can be written as: cð zÞ ¼
op a ¼ : oz ð az þ bÞ2
ð18Þ
For the precise determination of the axial distance z, the time delay dt between the moment of electric signal applying to the speaker and that of electrical signal received by the SLM can be used. Although small, the possibility of SLM missorientation along the radiation axis still exists, so that the axial distance can be written as: z cdt, where c is the speed of sound. Hence, the standard uncertainty for the SLM positioning can be written as: r uð zÞ ¼ cuðdtÞ ¼ c pffiffiffi : ð19Þ n The acoustic pressure distribution along the radial axis normal to the radiation axis can be approximated as a Bessel function [7]. Moreover, in a small range around the axial point in the free-field region, a quadratic approximation can be appropriate, p ¼ AR2 þ BR þ C;
ð20Þ
where A, B, and C are the fitting parameters. Eq. (20) can be rewritten as: p ¼ AðR B0 Þ2 þ E, where B0 ¼ B=2A and E ¼ C B2 =4A. Since the fitting result B0 is equal to the distance from the center of the spatial distribution along the radial axis to the microphone center, and the average value of B0 turned out to be the positional standard uncertainty uðRÞ, i.e. n P B0 i¼1 u ð RÞ ¼ : ð21Þ n Now, the sensitivity coefficient cR in Eq. (6) is deduced from Eq. (20), cR ¼ 2AR þ B:
ð22Þ
The expanded uncertainty is determined by the relation: U ¼ kuc ðd Þ;
ð23Þ
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where k is the coverage factor. For normal distribution, the coverage factor is 2, which gives a 95% confidence level. The expanded uncertainty conversion from the ratio d to that of the difference D can be obtained by: d þU UdB ¼ 20 log : ð24Þ d Consequently, the expanded uncertainty in the dB scale for the semi-automatic calibration can be obtained from Eq. (24) by substitution of Eqs. (3)–(5), (7) and (18)–(23). The expanded uncertainty in the dB scale for the full-automatic calibration can be obtained from Eq. (24) by substituting the Eqs. (3), (6), (8)–(12), and (18)–(23).
4. Results The SLM calibration by a comparison method requires the placement of the reference microphone at certain position and exchange of it to the SLM at the same position in an anechoic room. The distance between the position of the previously placed reference microphone and that of the lately exchanged SLM is found to be an error of the SLM position. For the uniformly and equally distributed sound field, there is no propagation of the positional uncertainty to the uncertainty of the acoustic pressure measurement. Generally, the sound field radiated from the speaker is spatially distributed with various magnitudes, so that the reference microphone and the SLM measure the different acoustic pressures according to their positions. This means that the uncertainty of the SLM position propagates to the uncertainty of the measurement values of the acoustic pressure. Although the spatial distribution of the sound field radiated from the speaker is similar to that from the plane piston, those are not same. Because the prediction through the Rayleigh–Sommerfeld equation gives only the relative values of the acoustic pressure amplitude in the sound field, the predicted results cannot be used as a real field to reduce the positional uncertainty to the effect on the acoustic pressure measurements. The best way is to use the equation determined through the curve-fitting method of the measured spatial distribution of the sound field radiated from the real speaker source. Fig. 4 is composed of the result measuring the sound pressure level by using the reference microphone with the variation of the frequency and the axial distance (Fig. 4(a)) and its comparison results with numerical predictions and linear curve-fitting values. Where, Fig. 4(b), (c) and (d) are the examples at 1.0 kHz frequency results represented as an inverse of acoustic pressure, an acoustic pressure, and a sound pressure level, respectively. As seen in Fig. 4(a), the sound fields radiated from the speaker and predicted by numerical method are commonly normalized to 85 dB at the axially 1-m position from the speaker. As seen in figures Fig. 4(b), (c) and (d), the measurement result agrees well with the linearly fitted curve, and differs infinitesimal but distinct from the prediction result. This difference reveals an increasing tendency as far off from the normalized point, and its degree depends on the frequency.
Y.T. Kim et al. / Applied Acoustics 65 (2004) 967–984
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110 25 cm 75 cm
50 cm 100 cm
100
-1
95 90
Experimental Numerical method Linear fit
3 1/p [Pa ]
SPL [dB]
105
2 1
85 80
100
(a)
2.0
0.5
(c)
20
40 60 80 100 Axial distance [cm]
0
20
(b)
SPL [dB]
1.0
0.0
0
Experimental Numerical method Linear fit
1.5
p [Pa]
1000 10000 Frequency [Hz]
100 98 96 94 92 90 88 86 84
Experimental Numerical method Linear fit
20 (d)
40 60 80 100 120 Axial distance [cm]
40 60 80 100 Axial distance [cm]
Fig. 4. Selected examples of the measurement results of the radiation field distribution along the radiation axis and three-type of the fitting result expression of the 1 kHz sound field.
In this study, the expanded uncertainties are estimated for two cases of the SLM calibrations, one for semi-automatic calibration, and the other for full automatic calibration. For the semi-automatic calibration, the expanded uncertainty in the dB scale is given by Eq. (24) with the comprising standard uncertainty components listed in Table 1. The first two terms in the right-hand side of Eq. (4) are also given in Table 1. The standard uncertainty elements for the calculation of the reference microphone uðpm Þ, the standard uncertainty elements used in the calculation for the positioning of the SLM along the radiation axis uðzÞ, and that along the radial axis uðRÞ in Table 1 are separately presented in Tables 2–4, respectively. Table 2 is a representation of the results estimated through Eq. (7), which corresponds to the reference microphone standard uncertainty for the semi-automatic calibration. The standard uncertainty of the source sound field calibration by the reference microphone with comprising uncertainty components and sensitivity coefficients are given. As seen in the Table 2, the absolute values of the products jcm uðsm Þj, cg uðgem Þ, and jcv uðvdm Þj are found to have values of the order of 109 , 104 , and 103 , respectively. Since jcv uðvdm Þj is the greatest among them, the uncertainty relating to the measurement by a DVM is the most significant for the source sound field calibration measurement. The uncertainties of the reference microphone sensitivities uðsm Þ with the frequency variation listed in Table 2 is the reduction of the nominal uncertainties given in the dB re 1 V/Pa scale by a reciprocity
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Table 1 Uncertainty estimation result for the semi-automatic calibration of the sound level meter pffiffiffiffiffi pffiffiffi f (Hz) uðpm Þ cc uðzÞ cR uðRÞ rp n dp 12 uðps Þ uðdÞ UdB 3 3 6 3 ( 10 Pa) ( F 10 Pa) ( 10 Pa) ( 10 Pa) ( 104 Pa) ( 103 Pa) ( 103 ) (dB) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
2.57 2.05 1.72 2.30 2.43 1.58 2.37 2.64 2.37 2.49 4.06 3.70 2.92 3.61 5.02 3.68 3.97 5.22 3.05
)3.62 )3.60 )3.63 )3.55 )3.31 )3.44 )3.15 )3.31 )3.08 )3.50 )3.81 )3.70 )3.31 )3.33 )3.56 )2.89 )3.50 )3.63 )3.12
1.58 0.55 1.23 2.36 )0.26 0.23 0.74 0.54 0.49 0.28 )0.66 0.15 )0.54 )0.09 )1.49 )0.81 0.24 1.33 )0.10
2.59 2.07 1.81 2.32 2.59 1.64 2.59 2.87 2.59 2.59 4.51 4.10 3.14 3.75 5.33 3.82 3.82 4.51 2.32
1.20 1.18 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.18 1.20 1.18
4.45 4.15 4.06 4.24 4.20 3.81 4.08 4.38 4.02 4.36 5.90 5.52 4.57 5.02 6.41 4.80 5.18 5.79 3.89
1.45 1.31 1.24 1.36 1.37 1.16 1.33 1.44 1.32 1.41 2.01 1.87 1.52 1.74 2.29 1.70 1.85 2.20 1.40
0.25 0.22 0.21 0.23 0.23 0.20 0.23 0.24 0.22 0.24 0.34 0.32 0.26 0.29 0.39 0.29 0.31 0.37 0.24
calibration method to the mV/Pa scale by Eq. (9). The nominal sensitivity with its uncertainty in the calibration certificate of the reference microphone is shown in Fig. 5. Although the sensitivity coefficient cm ¼ vdm =s2m gm involves the effective gain of Table 2 The standard uncertainty of the reference microphone used in the semi-automatic calibration of SLM f (Hz) cm cg cv uðsm Þ uðgem Þ uðvdm Þ uðpm Þ ( 104 Pa2 /V) ( 101 Pa) ( 10þ1 Pa/V) ( 105 V/Pa) ( 103 ) ( 104 V) ( 103 Pa) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
)7.89 )7.91 )7.90 )7.88 )7.88 )7.93 )7.89 )7.95 )7.91 )7.99 )7.99 )8.02 )8.08 )8.18 )8.30 )8.57 )8.95 )10.12 )10.66
)3.52 )3.52 )3.52 )3.51 )3.51 )3.53 )3.51 )3.53 )3.51 )3.53 )3.53 )3.53 )3.53 )3.53 )3.52 )3.53 )3.53 )3.52 )3.52
2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.21 2.21 2.22 2.22 2.23 2.25 2.27 2.31 2.38 2.48 2.82 2.97
7.95 7.94 7.94 7.94 7.94 7.94 7.93 7.92 7.91 7.88 7.88 7.84 7.78 7.70 7.56 7.35 9.39 10.36 11.78
2.90 2.00 0.19 2.43 1.64 0.11 0.18 1.38 0.12 2.08 1.25 0.64 1.26 2.93 2.52 0.61 2.49 2.15 1.14
1.03 0.83 0.73 0.93 1.03 0.66 1.03 1.14 1.03 1.03 1.80 1.63 1.25 1.50 2.12 1.52 1.52 1.79 0.93
2.57 2.05 1.72 2.30 2.43 1.58 2.37 2.64 2.37 2.49 4.06 3.70 2.92 3.61 5.02 3.68 3.97 5.22 3.05
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Table 3 The fitting parameter in Eq. (17), the standard uncertainty for the positioning of the SLM on the radiation axis, and the sensitivity coefficient f (Hz)
a (Pa/m)
b ( 102 Pa)
cz ( 101 Pa/m)
uðzÞ (cm)
cz uðzÞ ( 103 Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
2.89 2.87 2.87 2.84 2.60 2.73 2.49 2.65 2.45 2.79 3.02 2.94 2.65 2.65 2.82 2.30 2.78 2.90 2.47
)6.33 )0.04 )0.06 )0.01 0.20 0.09 0.32 0.18 0.37 0.03 )0.21 )0.12 0.18 0.17 )0.01 0.52 0.04 )0.08 0.34
)3.62 )3.60 )3.63 )3.55 )3.31 )3.44 )3.15 )3.31 )3.08 )3.50 )3.81 )3.70 )3.31 )3.33 )3.56 )2.89 )3.50 )3.63 )3.12
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
)3.62 )3.60 )3.63 )3.55 )3.31 )3.44 )3.15 )3.31 )3.08 )3.50 )3.81 )3.70 )3.31 )3.33 )3.56 )2.89 )3.50 )3.63 )3.12
Table 4 The fitting parameters in Eq. (20), the standard uncertainty for the positioning of the SLM on the radial axis, and the sensitivity coefficient f (Hz)
A B C cR jB=2Aj ( 101 Pa/m2 ) ( 103 Pa/m) ( 101 Pa) ( 103 Pa/m) (mm)
uðRÞ (mm)
cR uðRÞ ( 106 Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
)0.17 )0.17 )0.17 )0.17 )0.17 )0.18 )0.19 )0.19 )0.22 )0.25 )0.30 )0.38 )0.51 )0.72 )1.13 )1.87 )2.75 )3.62 )5.46
1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21
1.61 0.56 1.26 2.41 )0.27 0.24 0.76 0.55 0.50 0.29 )0.67 0.15 )0.55 )0.09 )1.52 )0.83 0.24 1.36 )0.10
1.33 0.46 1.04 1.99 )0.22 0.20 0.62 0.45 0.42 0.24 )0.56 0.12 )0.45 )0.08 )1.60 )1.42 0.43 3.43 0.42
0.355 0.354 0.355 0.354 0.355 0.355 0.355 0.354 0.355 0.355 0.354 0.353 0.353 0.351 0.351 0.349 0.349 0.350 0.351
1.33 0.46 1.04 1.99 )0.22 0.20 0.62 0.45 0.42 0.24 )0.56 0.12 )0.45 )0.08 )1.26 )0.69 0.20 1.12 )0.08
3.90 1.39 3.13 5.91 0.65 0.54 1.63 1.16 0.93 0.48 0.93 0.16 0.44 0.05 0.71 0.38 0.08 0.47 0.04
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0.06
Sensitivity [V/Pa]
0.05 0.04 0.03 0.02 0.01 0.00 100
1000 Frequency [Hz]
Fig. 5. Calibration chart of the reference microphone sensitivity with nominal uncertainties represented as a function of frequency.
amplifying network, it looks somewhat independent to the frequency, and is approximated 1.01. Whether the calibration method is automatic or semi-automatic, there may be two different ways to estimate the standard uncertainty uðps Þ in Eq. (3). For the semi-
Table 5 Summary of the uncertainty estimation results for the full-automatic calibration of the SLM f uðpm Þ cz uðzÞ cR uðRÞ cs uðss Þ cv uðvds Þ cg uðges Þ uðps Þ uðdÞ UdB (Hz) ( 104 Pa) ( 103 Pa) ( 105 Pa) ( 105 Pa) ( 105 Pa) ( 106 Pa) ( 103 Pa) ( 102 ) (dB) 125 6.27 160 6.28 200 6.28 250 6.26 315 6.26 400 6.29 500 6.26 630 6.30 800 6.26 1000 6.30 1250 6.30 1600 6.29 2000 6.29 2500 6.30 3150 6.27 4000 6.30 5000 8.40 6300 10.48 8000 12.56
)1.81 )1.80 )1.81 )1.77 )1.65 )1.72 )1.58 )1.65 )1.54 )1.75 )1.90 )1.85 )1.66 )1.67 )1.78 )1.45 )1.75 )1.82 )1.56
6.65 2.31 5.20 9.95 )1.12 0.98 3.12 2.27 2.08 1.20 )2.78 0.62 )2.27 )0.38 )6.30 )3.43 1.01 5.60 )0.42
)8.05 )6.44 )5.64 )7.24 )8.09 )5.08 )8.10 )8.90 )8.11 )8.10 )14.08 )12.87 )9.95 )12.00 )17.40 )12.80 )13.36 )17.92 )9.76
2.64 2.61 2.60 2.63 2.64 2.59 2.64 2.64 2.64 2.63 2.75 2.72 2.66 2.70 2.85 2.72 2.73 2.87 2.67
)1.43 )1.14 )1.00 )1.28 )1.43 )0.90 )1.43 )1.57 )1.43 )1.43 )2.48 )2.25 )1.73 )2.06 )2.94 )2.10 )2.10 )2.48 )1.28
1.81 1.80 1.81 1.77 1.65 1.72 1.58 1.65 1.54 1.75 1.90 1.85 1.66 1.67 1.78 1.45 1.75 1.82 1.56
5.38 5.40 5.39 5.29 4.97 5.14 4.76 4.97 4.67 5.23 5.63 5.49 4.97 5.00 5.30 4.43 5.50 5.90 5.68
0.09 0.09 0.09 0.09 0.09 0.09 0.08 0.09 0.08 0.09 0.10 0.09 0.09 0.09 0.09 0.08 0.09 0.10 0.10
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Table 6 The standard uncertainty of the reference microphone used in the full-automatic calibration of the SLM f (Hz)
vdm ( 102 V)
sm ( 102 V/Pa)
cm (Pa2 /V)
un ðsdB Þ (dB)
uðsm Þ ( 105 V/Pa)
jcm uðsm Þj ( 104 Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
1.62 1.62 1.62 1.61 1.61 1.62 1.61 1.62 1.61 1.61 1.61 1.60 1.59 1.57 1.54 1.50 1.44 1.26 1.20
4.50 4.50 4.50 4.50 4.50 4.50 4.49 4.49 4.48 4.47 4.46 4.44 4.41 4.36 4.28 4.16 3.99 3.52 3.33
)7.89 )7.91 )7.90 )7.88 )7.88 )7.93 )7.89 )7.95 )7.91 )7.99 )7.99 )8.02 )8.08 )8.18 )8.30 )8.57 )8.95 )10.12 )10.66
0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05 0.06
7.95 7.94 7.94 7.94 7.94 7.94 7.93 7.92 7.91 7.88 7.88 7.84 7.78 7.70 7.56 7.35 9.39 10.36 11.78
6.27 6.28 6.28 6.26 6.26 6.29 6.26 6.30 6.26 6.30 6.30 6.29 6.29 6.30 6.27 6.30 8.40 10.48 12.56
automatic calibration, Eq. (4) is used, and for the full-automatic one, Eq. (6) is used. The terms cR uðRÞ and cz uðzÞ, which are related to the SLM positioning, are common in Eq. (4) and Eq. (6). The sensitivity coefficients cR and cz can be obtained by fitting Table 7 The standard uncertainty for the sensitivity of the specified sound level meter (B&K 2231) and its comprising components un ðssdB Þ pffiffiffi Ss rs n ss un ðssdB Þ 1:96 uðss Þ cs uðss Þ f (Hz) rs ð10 20 1Þ cs 2 2 2 ( 10 V/Pa) ( 10 V/Pa) (V/Pa) (dB) ( 104 Pa2 /V) ( 102 V/Pa) ( 105 Pa) ( 10 V/Pa)
125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
3.27 3.97 3.68 3.58 2.09 2.81 1.98 3.02 2.68 3.73 3.20 3.91 3.49 3.18 2.75 2.78 4.14 2.68 2.55
1.03 1.26 1.16 1.13 0.66 0.89 0.63 0.96 0.85 1.18 1.01 1.24 1.10 1.01 0.87 0.88 1.31 0.85 0.81
1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22 3.22
)7.77 )7.76 )7.77 )7.79 )7.78 )7.74 )7.78 )7.72 )7.75 )7.68 )7.68 )7.65 )7.59 )7.50 )7.39 )7.16 )6.86 )6.06 )5.76
3.38 3.46 3.42 3.41 3.29 3.34 3.28 3.36 3.33 3.43 3.38 3.45 3.40 3.37 3.34 3.34 3.48 3.33 3.32
)2.63 )2.68 )2.66 )2.66 )2.56 )2.58 )2.55 )2.59 )2.58 )2.64 )2.59 )2.64 )2.59 )2.53 )2.47 )2.39 )2.38 )2.02 )1.91
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the spatial distribution along the corresponding axis. The fitting parameters are listed in Tables 3 and 4 with their uncertainties. For the sound pressure level measurement at each frequency by SLM through a semi-automatic procedure, 12 data are acquired, and 10 of them are taken in the average excluding for the maximum and the minimum value. Hence, the standard deviations rp in Table 1 have to be obtained after reducing the value of the sound pressure level to the value of thepacoustic pressure. The standard uncertainty uðdp Þ by ffiffiffiffiffi the resolution of the SLM dp = 12 was also given in Table 1. The resolution of the SLM d in dB scale is 0.01 dB independent to the frequency, and it was transformed to the Pa scale dp through Eq. (5). For the full-automatic calibration, the expanded uncertainty is given in Table 5 with its components consisting the standard uncertainties given in Eq. (6) and Eq. (7). The standard uncertainty uðpm Þ for the referenced microphone through Eq. (5), uðzÞ for the positioning of SLM along the radiation axis, and uðRÞ along the radial axis are given in Table 5, and their elements are already given in Tables 2–4, respectively. The standard uncertainty of the reference microphone defined in Eq. (8) and used in the full-automatic calibration of the SLM is tabulated along with its associated elements in Table 6. The estimated values of the standard uncertainties for the SLM calibration according to Eq. (10) are given in Table 7. The standard uncertainty for the DVM measurements defined by Eq. (11), and its components are given in Table 8. Finally, in Table 9 the estimations of the uncertainty components of the amplifying network of the calibration system are listed.
Table 8 The standard uncertainty for the measurement of the DVM indication and its components pffiffiffi pffiffiffi f rv rv = n vd r vd r= 3 cv uðvs Þ cv uðvs Þ 4 4 3 4 (Hz) ( 10 V) ( 10 V) ( 10 V) ( 10 ) ( 106 V) ( 101 Pa/V) ( 104 V) ( 105 Pa) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
3.26 2.61 2.28 2.92 3.26 2.06 3.26 3.61 3.26 3.26 5.67 5.16 3.95 4.73 6.70 4.81 4.81 5.67 2.92
1.03 0.82 0.72 0.92 1.03 0.65 1.03 1.14 1.03 1.03 1.79 1.63 1.25 1.49 2.12 1.52 1.52 1.79 0.92
2.53 2.53 2.53 2.54 2.54 2.52 2.53 2.52 2.53 2.50 2.50 2.49 2.48 2.44 2.41 2.33 2.24 1.98 1.88
8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00
1.17 1.17 1.17 1.17 1.17 1.16 1.17 1.16 1.17 1.16 1.16 1.15 1.14 1.13 1.11 1.08 1.03 0.91 0.87
5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51 5.51
1.03 0.82 0.72 0.92 1.03 0.65 1.03 1.14 1.03 1.03 1.79 1.63 1.25 1.49 2.12 1.52 1.52 1.79 0.92
5.68 4.55 3.98 5.09 5.69 3.59 5.69 6.29 5.69 5.69 9.89 8.99 6.89 8.24 11.68 8.39 8.39 9.89 5.09
Y.T. Kim et al. / Applied Acoustics 65 (2004) 967–984 Table 9 Standard uncertainity for the effective gain of the amplifying network pffiffiffi f (Hz) rg ( 103 ) rg = n ( 103 ) ge cg ( 103 Pa) uðge Þ ( 103 ) 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000
9.19 6.33 0.61 7.68 5.19 0.34 0.56 4.38 0.39 6.59 3.96 2.04 3.97 9.27 7.95 1.92 7.87 6.81 3.61
2.90 2.00 0.19 2.43 1.64 0.11 0.18 1.38 0.12 2.08 1.25 0.64 1.26 2.93 2.52 0.61 2.49 2.15 1.14
1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01
)1.38 )1.38 )1.38 )1.38 )1.38 )1.38 )1.38 )1.37 )1.38 )1.37 )1.37 )1.36 )1.35 )1.33 )1.32 )1.27 )1.22 )1.08 )1.02
2.90 2.00 0.19 2.43 1.64 0.11 0.18 1.38 0.12 2.08 1.25 0.64 1.26 2.93 2.52 0.61 2.49 2.15 1.14
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cg uðge Þ ( 106 Pa) )4.02 )2.76 )0.26 )3.36 )2.27 )0.15 )0.24 )1.90 )0.17 )2.85 )1.71 )0.88 )1.70 )3.91 )3.31 )0.77 )3.04 )2.32 )1.17
Summaries of the estimation results of the standard uncertainty for the semiautomatic calibration of the SLM are listed in Table 1, and that for the full-automatic calibration of SLM are listed in Table 5. As seen in Tables 1 and 5, the expanded uncertainty for the semi-automatic calibration is greater than the full-automatic calibration. 5. Conclusions In this paper, the theoretical model for estimating the uncertainty of the SLM calibration is developed and tested for two cases, such as a semi-automatic calibration and a full-automatic calibration. The method of using the curve fitting parameters is adopted to evaluate the uncertainty in positioning the SLM to the original position of the reference microphone. The measurement of time delay, between the input electric signal to the sound source and the arrival of the sound signal to the receiver, enables us to obtain the positional uncertainty along the radiation axis. Our finding that the fitting parameter B0 ¼ B=2A in Eq. (21) turned out to be used to obtain the positional uncertainty on the radial axis is useful to get the overall uncertainty of the SLM calibration in a free-field condition by the comparison method. Acknowledgements This work was financially supported by the Maintenance and Improvement of National Measurement Standard in KRISS.
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References [1] International Organization for Standardization, Guide to the expression of uncertainty in measurement, 1993. [2] Kim YT, Lee YB, Jho MJ. A fast automatic calibration system for a sound level meter in an anechoic room. Applied Acoustics 2003;64:459–70. [3] International Electrotechnical Commission, Sound level meters, IEC standard 651, 1979. [4] International Electrotechnical Commission, Integrating-averaging sound level meters, IEC standard 804, 1985. [5] Korea Association of standards and Testing Organizations. Standard calibration procedure of sound level meter, KASTO 96-16-011-098, 1996. [6] International Organization for Standard, Acoustics Preferred frequencies, ISO266, 2nd ed., 1997. [7] Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of acoustics. 3rd ed. New York: John Wiley and Sons; 1982. [8] Kim YT, Jho MJ, Eun HJ, Kim MG. A matrix model for ultrasonic source calibration and radiation field prediction using a pulsed planar scanning system. J Korean Phys Soc 2000;37(3):221–31.