29 Figure D.1 – Examples of static pressure coefficient of LS1P and LS2P microphones relative to the low-frequency value as a function of relative frequency f/fo ...32 Figure D.2 – Gener
General principles
General
Reciprocity calibration of microphones can be performed using three microphones, with two being reciprocal, or by utilizing an auxiliary sound source along with two microphones, where one is reciprocal.
NOTE If one of the microphones is not reciprocal it can only be used as a sound receiver.
General principles using three microphones
By acoustically connecting two microphones with a coupler, one can measure the electrical transfer impedance using one microphone as a sound source and the other as a sound receiver Once the acoustic transfer impedance is established, the product of the pressure sensitivities of the coupled microphones can be calculated By utilizing pair-wise combinations of three microphones labeled (1), (2), and (3), three independent products are obtained, allowing for the derivation of an expression for the pressure sensitivity of each microphone.
General principles using two microphones and an auxiliary sound
To begin, connect the two microphones acoustically using a coupler and determine the product of their pressure sensitivities (refer to section 5.1.2) Subsequently, expose both microphones to the same sound pressure generated by the auxiliary sound source.
The ratio of the two output voltages will then equal the ratio of the two pressure sensitivities
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
The pressure sensitivity of each microphone can be derived from the product and the ratio of their respective pressure sensitivities.
To determine the ratio of pressure sensitivities, a direct comparison method can be employed This method may involve using an auxiliary sound source, such as a third microphone, which possesses mechanical or acoustical characteristics that differ from those of the microphones undergoing calibration.
Basic expressions
Laboratory standard microphones and similar microphones are considered reciprocal and thus the two-port equations of the microphones can be written as: p q z i z
11 (1) where p is the sound pressure, uniformly applied, at the acoustical terminals
(diaphragm) of the microphone in pascals (Pa);
U is the signal voltage at the electrical terminals of the microphone in volts
The volume velocity (\$q\$) through the microphone's diaphragm is measured in cubic metres per second (m³/s), while the current (\$i\$) flowing through its electrical terminals is in amperes (A) The electrical impedance of the microphone, denoted as \$z_{11} = Z_e\$, is expressed in ohms (Ω) when the diaphragm is blocked Conversely, the acoustic impedance, represented as \$z_{22} = Z_a\$, is measured in pascal-seconds per cubic metre (Pa⋅s⋅m⁻³) when the electrical terminals are unloaded The reverse and forward transfer impedances, \$z_{12} = z_{21} = M_p Z_a\$, are quantified in volt-seconds per cubic metre (V⋅s⋅m⁻³), with \$M_p\$ indicating the microphone's pressure sensitivity in volts per pascal (V⋅Pa⁻¹).
NOTE Underlined symbols represent complex quantities
Equations (1) may then be rewritten as: p q Z i Z M
+ a a p a p e (1a) which constitute the equations of reciprocity for the microphone
Microphones (1) and (2), with pressure sensitivities \(M_{p,1}\) and \(M_{p,2}\), are acoustically connected via a coupler According to Equations (1a), a current \(i_1\) flowing through the electrical terminals of microphone (1) generates a short-circuit volume velocity of \(M_{p,1} i_1\) when the diaphragm pressure is zero, resulting in a sound pressure of \(a_{12} p_{1}\).
2 Z M i p = at the acoustical terminals of microphone (2), where Z a,12 is the acoustic transfer impedance of the system
The open-circuit voltage of microphone (2) will then be: a,12 1 p,2
U = ⋅ LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Thus the product of the pressure sensitivities is given by:
Insert voltage technique
The insert voltage technique is used to determine the open-circuit voltage of a microphone when it is electrically loaded
To measure the open-circuit voltage of a microphone with a specific internal impedance, it is connected to a load impedance A small impedance, significantly lower than the load impedance, is placed in series with the microphone, and a calibrating voltage is applied across this setup.
Let a sound pressure and a calibrating voltage of the same frequency be applied alternately
When the calibrating voltage is set to match the voltage drop across the load impedance caused by the sound pressure on the microphone, the open-circuit voltage will equal the calibrating voltage in magnitude.
Evaluation of the acoustic transfer impedance
Z = can be evaluated from the equivalent circuit in Figure 1, where Z a,1 and Z a,2 are the acoustic impedances of microphones (1) and
Figure 1 – Equivalent circuit for evaluating the acoustic transfer impedance Z a,12
In certain scenarios, the theoretical evaluation of \$Z_{a,12}\$ is possible When the sound pressure remains uniform throughout the coupler, which occurs when the coupler's physical dimensions are significantly smaller than the wavelength, the gas within the coupler acts as a pure compliance Consequently, \$Z_{a,12}\$ can be expressed as \$Z'_{a,12}\$ based on the equivalent circuit depicted in Figure 2.
(assuming adiabatic compression and expansion of the gas):
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Figure 2 – Equivalent circuit for evaluating Z ’ a,12 when coupler dimensions are small compared with wavelength
V is the total geometrical volume of the coupler in cubic metres (m 3 );
V e,1 is the equivalent volume of microphone (1) in cubic metres (m 3 );
V e,2 is the equivalent volume of microphone (2) in cubic metres (m 3 );
The acoustic impedance (\$Z\$) of the gas within the coupler is measured in pascal-seconds per cubic meter (Pa⋅s/m³) It is influenced by several factors, including the angular frequency (\$\omega\$) in radians per second (rad/s), the static pressure (\$p_s\$) in pascals (Pa), and the static pressure at reference conditions (\$p_{s,r}\$) in pascals (Pa) Additionally, the ratio of specific heat capacities at measurement conditions is represented by \$\kappa\$, while \$\kappa_r\$ denotes this ratio at reference conditions.
Values for κ and κr in humid air can be derived from equations given in Annex F
At higher frequencies, evaluating \( Z_{a,12} \) becomes complex when dimensions are not small relative to the wavelength However, if the coupler is cylindrical with a diameter matching that of the microphone diaphragms, the system can be treated as a homogeneous transmission line at frequencies where plane-wave transmission is applicable.
Figure 3 – Equivalent circuit for evaluating Z ’ a,12 when plane wave transmission in the coupler can be assumed
Z a,12 is then given by Z' a,12 (assuming adiabatic compression and expansion of the gas):
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
The acoustic impedance of plane waves in the coupler, denoted as \$Z_{a,0}\$, can be expressed as \$Z_{a,0} = \frac{\rho c}{S}\$ when losses in the coupler are disregarded Here, \$\rho\$ represents the density of the gas in kilograms per cubic meter (kg⋅m⁻³), and \$c\$ is the speed of sound in the gas measured in meters per second (m⋅s⁻¹).
The cross-sectional area of the coupler is denoted as \$S_0\$ in square metres (m²), while \$l_0\$ represents the length of the coupler, defined as the distance between the two diaphragms in metres (m) The complex propagation coefficient is expressed as \$\gamma = \alpha + j\beta\$, measured in metres to the power of minus one (m⁻¹).
Values for ρ and c in humid air can be derived from equations given in Annex F
The real part of γ accounts for the viscous losses and heat conduction at the cylindrical surface and the imaginary part is the angular wave number
If losses are neglected, γ may be approximated by putting α equal to zero and β equal to ω/c in Equation (4)
Allowance shall be made for any air volume associated with the microphones that is not enclosed by the circumference of the coupler and the two diaphragms (see 7.3.3.1).
Heat-conduction correction
The assessment of Z' a,12 in the previous section is based on the assumption of adiabatic conditions within the coupler However, real-world scenarios reveal that heat conduction at the coupler's walls leads to deviations from these ideal adiabatic conditions, particularly in smaller couplers and at lower frequencies.
At low frequencies, assuming uniform sound pressure and constant wall temperature, the impact of heat conduction losses can be quantified using a complex correction factor ΔH applied to the geometric volume V, as outlined in Equation (3) Detailed expressions for the correction factor ΔH are provided in Annex A.
At high frequencies, wave motion within the coupler leads to variations in sound pressure at different points In right-cylindrical couplers, transmission line theory applies, allowing for the consideration of heat conduction and viscous losses along the cylindrical surface through the complex propagation coefficient and acoustic impedance Additionally, heat conduction at the coupler's end surfaces and microphone diaphragms can be incorporated by adding components to the microphones' acoustic impedances Detailed expressions for the complex propagation coefficient and acoustic impedance for plane-wave propagation are provided in Annex A.
Capillary tube correction
The coupler is equipped with capillary tubes to balance the static pressure both inside and outside Additionally, these two capillary tubes allow for the introduction of gases other than air.
The acoustic input impedance of an open capillary tube is given by:
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU. t C a, a,C Z tanh l
Z a,t is the complex acoustic wave impedance of an infinite tube in pascal-seconds per cubic metre (Pa⋅s⋅m –3 ); l C is the length of the tube in metres (m)
The shunting effect of the capillary tubes can be taken into account by introducing a complex correction factor Δ C to the acoustic transfer impedances given in Equations (3) and (4):
′′ٛ + Δ (6) where n is the number of identical capillary tubes used;
Z ” a,12 is the acoustic transfer impedance Z' a,12 corrected for heat conduction according to 5.5
An expression for the acoustic input impedance Z a,C of an open capillary tube is given in
Final expressions for the pressure sensitivity
Method using three microphones
Let the electrical transfer impedance U 2 /i 1 (see 5.2) be denoted by Z e,12 with similar expressions for other pairs of microphones
Taking into account the corrections given in 5.5 and 5.6, the final expression for the modulus of the pressure sensitivity of microphone (1) is:
Similar expressions apply for microphones (2) and (3)
The phase angle of the pressure sensitivity for each microphone is determined by a similar procedure from the phase angle of each term in the above expression
NOTE When complex quantities are expressed in terms of modulus and phase, the phase information should be referred to the full four-quadrant phase range, i.e 0 - 2 π rad or 0 – 360°.
Method using two microphones and an auxiliary sound source
If only two microphones and an auxiliary sound source are used, the final expression for the modulus of the pressure sensitivity is:
The pressure sensitivities are evaluated by comparing them to an auxiliary source, as outlined in section 5.1.3 This measurement is conducted exclusively for internal use at the MECON Limited locations in Ranchi and Bangalore, with materials supplied by the Book Supply Bureau.
6 Factors influencing the pressure sensitivity of microphones
General
The pressure sensitivity of a condenser microphone depends on polarizing voltage and environmental conditions
A polarized condenser microphone operates under the principle of maintaining a constant electrical charge across all frequencies However, this stability is compromised at very low frequencies, where the time constant for charging the microphone is influenced by the product of its capacitance and polarizing resistance Although the open-circuit sensitivity measured through the insert voltage technique remains accurate, the absolute output from the connected preamplifier diminishes at low frequencies due to this time constant effect.
Pressure sensitivity requires specific measurement conditions to be met During calibration, it is crucial to maintain control over these conditions to ensure that the resulting uncertainty components remain minimal.
Polarizing voltage
The sensitivity of a condenser microphone is approximately proportional to the polarizing voltage and thus the polarizing voltage actually used during the calibration shall be reported
To comply with IEC 61094-1 a polarizing voltage of 200,0 V is recommended.
Ground-shield reference configuration
As per section 3.3 of IEC 61094-1:2000, the open-circuit voltage must be measured at the microphone's electrical terminals while connected to a designated ground-shield configuration, utilizing the insert voltage technique outlined in section 5.3 Additionally, specifications for ground-shield configurations applicable to laboratory standard microphones are provided.
The appropriate ground-shield configuration shall apply to both transmitter and receiver microphones during the calibration, and the shield should be connected to ground potential
If any other arrangement is used, the results of a calibration shall be referred to the reference ground-shield configuration
If the manufacturer specifies a maximum mechanical force to be applied to the central electrical contact of the microphone, this limit shall not be exceeded.
Pressure distribution over the diaphragm
Pressure sensitivity in microphones refers to the assumption that sound pressure is applied uniformly across the diaphragm When a microphone experiences a non-uniform pressure distribution, its output voltage will differ from that produced under uniform pressure conditions, even if the mean values are the same This is primarily because microphones tend to be more sensitive to sound pressure at the center of the diaphragm The variation in output voltage due to non-uniform tension distribution on the diaphragm can differ among various microphone models.
For cylindrical couplers, as described in Annex C, both longitudinal and radial wave motions
(symmetric as well as asymmetric) will be present The radial wave motion will result in a
The diaphragm experiences non-uniform pressure distribution due to variations from an ideal piston source that should cover the entire end surface of the coupler This issue arises when the combined microphone and coupler geometry deviates from a perfect right angle cylinder Additionally, imperfections in the backplate or diaphragm geometry, as well as inconsistencies in diaphragm tension and homogeneity, contribute to asymmetric radial wave motion generated by the transmitter microphone.
For optimal calibration, it is essential to achieve a uniform sound pressure distribution within ± 0.1 dB across the diaphragm's surface However, maintaining this uniformity is challenging due to the inherent geometrical imperfections of microphones and couplers While radial wave motion is unavoidable, using couplers that match the microphone diaphragm's diameter minimizes radial wave motion and reduces sensitivity to these geometrical flaws.
For high-frequency calibrations requiring high accuracy, it is advisable to utilize multiple couplers of varying dimensions This approach helps in accurately assessing the sensitivity of microphones and allows for the application of a theoretically grounded correction for radial wave-motion effects.
Dependence on environmental conditions
Static pressure
The acoustic resistance and mass of the gas between the diaphragm and backplate, along with the compliance of the cavity behind the diaphragm, influence the microphone's pressure sensitivity, which varies with static pressure and frequency This relationship can be assessed through reciprocity calibrations conducted at various static pressures for the microphone being tested.
Annex D contains information on the influence of static pressure on the pressure sensitivity of laboratory standard condenser microphones.
Temperature
The pressure sensitivity of a microphone is influenced by the acoustic resistance and mass of the gas between the diaphragm and backplate, which vary with temperature Additionally, the mechanical dimensions of the microphone and its sensitivity are affected by temperature, mechanical tension in the diaphragm, and the spacing between the diaphragm and backplate These interdependencies collectively impact the microphone's performance as a function of frequency.
The combined dependence can be determined for a microphone under test by making reciprocity calibrations at different temperatures
Annex D contains information on the influence of temperature on the pressure sensitivity of laboratory standard condenser microphones
NOTE If a microphone is exposed to excessive temperature variations a permanent change in sensitivity may result.
Humidity
The thermodynamic state of the air in the cavity behind a microphone's diaphragm is minimally affected by humidity, and laboratory standard microphones show no sensitivity changes as long as condensation is avoided.
Certain conditions can affect the stability of polarizing voltage and backplate charge, thereby impacting sensitivity For instance, the insulation material's surface resistance between the microphone's backplate and housing may degrade in high humidity, especially if contaminated This deterioration significantly influences the microphone's sensitivity at low frequencies, particularly affecting the phase response.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Transformation to reference environmental conditions
When reporting the results of a calibration, the pressure sensitivity should be referred to the reference environmental conditions if reliable correction data are available
The actual conditions during the calibration should be reported
NOTE During a calibration, the temperature of the microphone can be different from the ambient air temperature
General
Calibration accuracy is influenced not only by the factors outlined in Clause 6 but also by the method, equipment, and care taken during the process To reduce the impact of these uncertainties, it is essential to measure or calculate known influencing factors with the highest possible precision.
Electrical transfer impedance
Various methods are used for measuring the electrical transfer impedance with the necessary accuracy, and no preference is given
To accurately measure the current through a transmitter, it is essential to assess the voltage across a calibrated impedance connected in series with the transmitter microphone The ground shield reference configuration must be attached to the microphone to ensure precise current determination Additionally, the calibration of the series impedance should account for any cable capacitance and other load impedances present during the voltage measurement This approach enables the calculation of electrical transfer impedance using a voltage ratio and the calibrated series impedance.
The voltage applied to the transmitter microphone must be carefully selected to minimize the impact of harmonics—either from the source or produced by the microphone—on the uncertainty of pressure sensitivity measurements, ensuring that this effect remains significantly lower than the random uncertainty.
Noise or other interference such as cross-talk, whether of acoustical or other origin, shall not unduly affect the determination of the pressure sensitivity
NOTE 1 Frequency selective techniques can be used to improve the signal-to-noise ratio
Cross-talk measurement can be achieved by replacing the receiver microphone with a dummy microphone that matches its capacitance and external geometry, allowing for the assessment of the difference in electric transfer impedance It is essential to position the coupler and microphones as they would be during calibration Another method involves setting the polarizing voltage to zero volts during calibration For both approaches, employing frequency selective techniques is advisable.
Acoustic transfer impedance
General
Several factors influence the acoustic transfer impedance but the major source of uncertainty in its determination is often the microphone parameters, especially for small couplers.
Coupler properties
The coupler cavity's shape and dimensions must be designed to meet the requirements of 6.4 When the coupler's largest dimension is significantly smaller than the wavelength of sound in the gas, the sound pressure will remain largely uniform throughout.
MECON Limited is licensed for internal use in Ranchi and Bangalore, with materials supplied by the Book Supply Bureau To ensure effective coupling that is independent of shape, especially at high frequencies and with larger couplers, it may be necessary to fill the cavity with helium or hydrogen.
The uncertainty on coupler dimensions affects the acoustic transfer impedance by different amounts that vary with frequency It also influences the heat conduction and capillary tube corrections
Examples of couplers are given in Annex C
NOTE 1 Cylindrical couplers used in a frequency range where the dimensions are not small compared to the wavelength should be manufactured with the utmost care so that asymmetric sound fields are not excited
NOTE 2 The influence on a microphone of an asymmetric sound pressure distribution in the coupler may be ascertained by changing the relative position of the coupler and microphones, for instance by incrementally rotating each microphone about its axis If such a change affects the electrical transfer impedance, this effect should be taken into account when estimating the uncertainty
NOTE 3 If the coupler is filled with a gas other than air, care should be taken to avoid leakage of the gas to the cavity behind the diaphragm of the microphone, by sealing the contacting surface with a thin layer of grease If diffusion of the gas into the back cavity takes place, through the diaphragm or by other means, the microphone cannot be calibrated in this way as the microphone sensitivity is altered unpredictably
7.3.2.2 Heat conduction and viscous losses
The correction for heat conduction and viscous losses shall be calculated from the equations given in Annex A for cylindrical couplers within the range of dimensions as described in
Annex C In the calculations the total coupler volume is understood as the sum of the geometrical volume of the coupler and the front cavity volumes of the coupled microphones
Similarly the total surface area is understood as the sum of the surface area of the coupler and the surface areas of the front cavities of the coupled microphones
When using capillary tubes, it is essential to calculate the acoustic impedance according to the equations provided in Annex B To reduce uncertainty in tube dimensions, long and narrow capillary tubes are recommended Additionally, the correction factor for these tubes can be determined using Equation (6) in section 5.6.
The acoustic transfer impedance is influenced by specific physical properties of the gas within the coupler, which are affected by environmental factors like static pressure, temperature, and humidity For detailed information on these properties and their relationship with environmental conditions, refer to Annex F, which focuses on humid air.
The resulting uncertainty on the quantities is a combination of the uncertainty on the equations in Annex F and the uncertainty on the measurement of the environmental conditions.
Microphone parameters
A laboratory standard microphone has a recessed cavity in front of the diaphragm
The volume of the front cavity contributes to the total geometrical volume \( V \) of the coupler, as indicated in Equation (3) Additionally, the depths of the front cavities affect the length \( l_0 \) of the coupler, as shown in Equation (4) Due to production tolerances, it is essential to determine the volume and depth of the front cavity individually for each microphone during calibration in plane-wave couplers (refer to Annex E).
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
The measured volume of the front cavity often differs from the volume calculated using the coupler's cross-sectional area \( S_0 \) and cavity depth This discrepancy arises from variations in the cavity diameter compared to the coupler, the presence of screw threads on the inner wall, and an additional annular air space around the microphone diaphragm The excess volume, defined as the difference between the actual front volume and the calculated volume, should be treated as an additional terminating impedance when applying Equation (4) This can be achieved by setting \( Z_{a,1} \) and \( Z_{a,2} \) to represent the impedance of the microphone in parallel with the impedance from the excess volume.
NOTE 1 This excess volume can in some instances be negative
NOTE 2 For front cavities with an inner thread, the larger surface of the thread results in increased heat conduction that affects the acoustic transfer impedance If this effect is neglected when calculating the acoustic transfer impedance, the corresponding uncertainty component should be increased accordingly
The acoustic impedance of a microphone varies with frequency and is primarily influenced by the characteristics of the stretched diaphragm, the air in the cavity behind it, and the backplate's geometry It can be approximated using a series network of compliance, mass, and resistance, which can also be represented by compliance, resonance frequency, and loss factor Compliance is typically expressed as the low-frequency value of the real part of the microphone's equivalent volume, as outlined in IEC 61094-1:2000.
At low frequencies, heat conduction in the cavity behind the diaphragm increases the equivalent volume of LS1 microphones by up to 5%.
The acoustic impedance \( Z_a \) of each microphone is crucial to the overall acoustic transfer impedance \( Z_{a,12} \) of the system Errors in measuring \( Z_a \) can significantly affect calibration accuracy, especially at high frequencies.
Methods for determining the acoustic impedance are described in Annex E
NOTE The accuracy to which the microphone parameters need to be measured in order to obtain a certain overall accuracy is related to the coupler used and the frequency
To accurately determine the polarizing voltage, it is essential to measure this voltage directly at the microphone terminals, especially when sourced from a high-impedance supply due to the microphone's finite insulation resistance Alternatively, verifying that the microphone's insulation resistance is sufficiently high allows for valid measurements of the polarizing voltage supply with the microphone removed or at a low impedance port of the supply.
Imperfection of theory
The reciprocity theorem and the derivation of acoustic transfer impedance rely on idealized assumptions regarding microphones, the sound field in couplers, diaphragm movement, and coupler geometry when closed with microphones However, there are instances where these assumptions may not hold true.
− Small scale imperfections in the transmitter microphone may lead to asymmetric wave-motion which cannot be accounted for;
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
− Microphones may not be reciprocal The effect of this can be minimized by combining only microphones of the same model;
− Radial wave-motion corrections, if applied, are based on idealized movements of the microphone diaphragms or on empirical data;
− The excess volume of the microphone front cavity, see 7.3.3.1, may not be dealt with correctly;
− A lumped parameter representation of the microphone acoustic impedance is only an approximation to the true impedance;
− Viscous losses along the coupler surface have been estimated by an approximate theory
Viscous losses from the inner thread in the front cavity and surface roughness are not considered, which impacts the acoustic transfer impedance at high frequencies.
Uncertainty on pressure sensitivity level
The uncertainty on the pressure sensitivity level should be determined in accordance with
ISO/IEC Guide 98-3 When reporting the results of a calibration the uncertainty, as function of frequency, shall be stated as the expanded uncertainty of measurement using a coverage factor of k = 2
The uncertainty analysis of acoustic transfer impedance, as indicated in Equation (7), is typically conducted by recalculating while varying each component individually according to its associated uncertainty The resulting differences from the unchanged components are utilized to assess the standard uncertainty linked to each component.
Table 1 outlines several components that influence calibration uncertainty However, not all components may apply to every calibration setup, as different methods are utilized for measuring electrical transfer impedance, determining microphone parameters, and coupling microphones.
The uncertainty components presented in Table 1 are primarily frequency-dependent and should be calculated as standard uncertainty While these components are typically expressed in a linear format, a logarithmic representation is also permissible due to the small magnitude of the values, resulting in a final expanded uncertainty of measurement that remains largely unchanged.
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Measured quantity Relevant subclause no
Cross-talk 7.2 Inherent and ambient noise 7.2
Distortion 7.2 Frequency 7.2 Receiver ground shield 6.3
Coupler volume 7.3.2.1; 7.3.2.2 Coupler surface area 7.3.2.1; 7.3.2.2
Unintentional coupler/microphone leakage Capillary tube dimensions 7.3.2.3
Front cavity depth 7.3.3.1 Front cavity volume 7.3.3.1
Additional heat conduction caused by front cavity thread
Heat conduction theory Annex A Adding of excess volume 7.3.3.1; 7.4
Rounding error Repeatability of measurements Static pressure corrections 6.5; Annex D Temperature corrections 6.5; Annex D
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Heat conduction and viscous losses in a closed cavity
In a closed coupler, heat conduction between the air and walls leads to a gradual transition from adiabatic to isothermal conditions, influenced by the calibration frequency and coupler dimensions Additionally, sound particle velocity along the coupler surfaces causes viscous losses, affecting the sound pressure generated by the transmitter microphone, which acts as a constant volume displacement source Two methods for calculating the resulting sound pressure are presented.
– A low frequency solution based on heat conduction only and applicable to large-volume couplers and plane-wave couplers in the frequency range where wave-motion can be neglected
– A broad-band solution applicable to plane-wave couplers only, including both heat conduction and viscous losses
Plane-wave and large-volume couplers are described in Annex C
At low frequencies, the sound pressure in the coupler is uniform, allowing heat conduction effects to be interpreted as an apparent increase in the coupler's volume This is represented by a complex correction factor, ΔH, which modifies the geometrical volume, V.
The correction factor is given by:
The equation \( H = 1 + - E \Delta \kappa \kappa \) (A.1) describes the complex temperature transfer function \( E_V \), which represents the ratio of the spatial average of the sinusoidal temperature variation due to sound pressure to the variation that would occur with perfectly non-conducting walls in the coupler Tabulated values for \( E_V \) are available in [A.1] 2, depending on the parameters \( R \) and \( X \).
R is the length to diameter ratio of the coupler;
The equation \( X = f \cdot l \cdot \alpha_t \) relates the frequency \( f \) in hertz (Hz) to the volume-to-surface ratio \( l \) of the coupler in meters (m) and the thermal diffusivity \( \alpha_t \) of the enclosed gas in square meters per second (m²·s⁻¹).
Tabulated values of E V for some values of R and X are given in Table A.1 The figures given are considered accurate to 0,000 01
2 Figures in square brackets refer to Clause A.4
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Finite cylindrical couplers, as detailed in Annex C, exhibit an approximation of the complex quantity E V that yields errors of less than 0.01 dB at frequencies exceeding 20 Hz.
The modulus of E V , as calculated from Equation (A.2), is accurate to 0,01 % within the range
Real part of E V Imaginary part of E V
The first two terms in Equation (A.2) constitute an approximation that may be used for couplers that are not right circular cylinders
When calibrations are performed at frequencies below 20 Hz using the couplers described in
Annex C, the full frequency domain solution given in [A.1] shall be used, or the corresponding uncertainty component shall be increased accordingly
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
At high frequencies, viscous losses, alongside thermal losses, diminish the effective cross-sectional area of the coupler due to the boundary layer effect, while simultaneously increasing the effective length of the coupler because of the reduced speed of sound Conversely, at low frequencies, the couplers outlined in Annex C experience a balance between these two effects, although heat conduction continues to play a role The interplay of heat conduction and viscous losses in sound propagation within cylindrical tubes has been analyzed based on Kirchhoff’s theory.
The complex expressions for the propagation coefficient and the acoustic impedance of the coupler to be used in Equation (4) are:
= ⎜⎜⎝ + ⎜⎜⎝ − − ⎟⎟⎠⎟⎟⎠ (A.4) where η is the viscosity of the gas in pascal-seconds (Paãs); a is the radius of the coupler in metres (m)
Values for c, η, ρ and α t in humid air can be derived from equations given in Annex F
Heat conduction losses also occur at the end surfaces, in addition to the losses at the cylindrical surface These losses can be addressed by incorporating an admittance of \( \frac{1}{Z_{a,h}} \) into each microphone admittance as shown in Equation (4), refer to [A.3].
When a microphone features an inner thread in its front cavity, the additional heat conduction from the thread surface can be incorporated by augmenting the cross-sectional area \( S_0 \) in Equation (A.5) with the increased surface area of the thread, as referenced in [A.4].
Equations (A.3) – (A.4) are valid for the frequency range given by ω ρa 2 >100η This corresponds to frequencies higher than 3 Hz and 12 Hz for plane-wave couplers as given in
Table C.1 for type LS1P and LS2aP microphones respectively
[A.1] GERBER, H Acoustic properties of fluid-filled chambers at infrasonic frequencies in the absence of convection, Journal of Acoustical Society of America 36, 1964, pp 1427-1434
[A.2] ZWIKKER, C and KOSTEN, C.W Sound Absorbing Materials, 1949 Elsevier,
[A.3] MORSE, P.M and INGARD, K.U Theoretical Acoustics, 1968 McGraw-Hill, New York
[A.4] FREDERIKSEN, E Reduction of Heat Conduction Error in Microphone Pressure
Reciprocity Calibration Brỹel & Kjổr Technical Review, 1, 2001 pp14-23
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Acoustic impedance of a capillary tube
The acoustic input impedance Z a,C of an open capillary tube is determined by means of the transmission line theory, see 5.6: a ,C a ,t tan h C
The relationship between Z a,t and γ is given by (see [B.1] 3 ):
J o ( ), J 1 ( ) are the cylindrical Bessel functions of first kind, zero and first order respectively of complex argument; a t is the radius of the tube in metres (m);
(− ωρη k is the complex wavenumber in metres to the power minus one (m –1 ),
The viscosity of the gas is measured in pascal-seconds (Pa⋅s), while its density is expressed in kilograms per cubic meter (kg⋅m⁻³) Additionally, the thermal diffusivity of the gas is represented in square meters per second (m²⋅s⁻¹).
The equations above shall be used to calculate the correction factor ΔC given in Equation (6)
Values for c, η , ρ and αt in humid air can be derived from equations given in Annex F
Alternatively, the capillary tube may be blocked along its full length by a suitable wire after assembling the coupler and microphones In this case the correction factor ΔC equals 1
The formulas presented are based on the assumption of an ideal circular tube, which makes them highly sensitive to the tube's radius raised to the fourth power In real-world applications, the inner geometry of capillary tubes often deviates from a perfect circle, necessitating a flow calibration to accurately assess the effective radius.
Tables B.1 and B.2 present the tabulated values of the real and imaginary components of \( Z_{a,C} \) under standard environmental conditions, covering a typical range of parameters and frequencies These tables serve as a reference for validating calculation programs that utilize Equations B.1 to B.3.
3 Figures in square brackets refer to Clause B.2
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
In an actual calibration the equations given above should be used and the actual values of temperature, static pressure and relative humidity be applied
Table B.1 – Real part of Z a,C in gigapascal-seconds per cubic metre (GPa⋅s/m 3 )
NOTE The values given in this table are valid at reference environmental conditions only (see Clause 4 and Table F.2)
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Table B.2 – Imaginary part of Z a,C in gigapascal-seconds per cubic metre (GPa⋅s/m 3 )
NOTE The values given in this table are valid at reference environmental conditions only (see Clause 4 and Table F.2)
[B.1] ZWIKKER, C and KOSTEN, C.W Sound Absorbing Materials, 1949 Elsevier,
LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU.
Examples of cylindrical couplers for calibration of microphones
A coupler used in reciprocity calibration must ensure a uniform sound pressure distribution across the diaphragms of both transmitter and receiver microphones Achieving a consistent pressure distribution on the receiver microphone's diaphragm is crucial for accurate pressure sensitivity, as outlined in IEC 61094-1:2000 However, due to radial wave motion and diaphragm motion asymmetry, this ideal condition can only be approximated To enhance the frequency range of the coupler, it is beneficial to have a high radial resonance frequency, which necessitates a smaller coupler diameter Nonetheless, for practical purposes, the coupler's diameter should not be less than that of the diaphragms.