Source: https://docs.jmrp.io/phonometry/signals/metrology/comparison-calibration/ # Microphone calibration by comparison (IEC 61094-5 and IEC 61094-8) A working standard microphone is not calibrated by reciprocity. It is put beside a reference microphone whose sensitivity is already known, or in its place, both are exposed to the same sound pressure, and the ratio of their output voltages carries the reference's sensitivity over to it. IEC 61094-5:2016 does this in a **pressure field**, in a coupler or a jig; IEC 61094-8:2012 does it in a **free field**, in an anechoic room or behind a time window. Annex D of the first part writes the model once for both, `M_test = M_ref · R_V / R_P`, and `metrology.comparison_calibration` computes it in sensitivity levels, `L_test = L_ref + 20 lg R_V - 20 lg R_P + sum(C_j)`, with `R_V` the ratio of the output voltages, `R_P` that of the effective sound pressures (reduced to unity by the procedure) and `C_j` the corrections each part asks for. This page runs a pressure calibration of a WS2P against an LS2P and a free-field calibration of a WS2F on synthetic readings, so every step can be checked against the sensitivities that built them, then the time-selective processing of IEC 61094-8 Annex B, the phase of the sensitivity, the effect of different acoustic impedances (IEC 61094-5 7.4 and 7.5) with the air between two microphones by the circuit the part refers to, the correction for microphones of different diameters by the model it refers to (6.5), and the validation of a jig or a coupler (6.7). ## How the measurement goes With **simultaneous excitation** both microphones sit in the field at once, each on its own measuring channel; in a coupler or a jig they face each other about 1 mm apart (IEC 61094-5 5.1.2). With **sequential excitation** they take the same place in turn: either the exchange does not change the sound pressure significantly, or any change is detected and corrected, for example with a monitor microphone near the source (IEC 61094-5 5.1.3, IEC 61094-8 5.2, 6.5 and A.2). | Requirement | Value | Clause | | :--- | :--- | :--- | | Reference conditions | 23,0 °C, 101,325 kPa, 50 % relative humidity | 61094-5 and 61094-8, 4 | | Simultaneous, in a coupler or a jig | Diaphragms about 1 mm apart; the microphones interchanged and the measurement repeated | 61094-5, 5.1.2 | | Sequential | The exchange does not change the sound pressure significantly, or any change is detected and corrected, for example with a monitor microphone; a monitor, when used, senses the changes at the test position | 61094-5, 5.1.3; 61094-8, 5.2, 6.5 | | Free-field mounting | On a semi-infinite rod of the microphone's diameter | 61094-8, 6.7 | | Reported uncertainty | Expanded, with `k = 2`, at each frequency | 61094-5, 7.9; 61094-8, 8.8 | ## A pressure calibration by simultaneous excitation Annex C removes the gains of the two channels and the asymmetry of the coupler: with the reference on channel 1 the level reading difference is `L_C12`, after the interchange it is `L_C21`, and Formula (C.3) gives `L_ref - L_test = (L_C12 - L_C21) / 2`. Here the channels differ by 0,55 dB, port A hears up to 0,13 dB more than port B, and each reading is repeated three times: ```python import numpy as np from phonometry import metrology f = metrology.exact_frequencies(250, 20000, fraction=3) # 20 exact one-third octaves x = f / 1000 l_ref = -38.0 + 0.04 * np.log10(x) - 0.25 * (x / 20) ** 2 # the LS2P's certificate, dB re 1 V/Pa l_true = -38.6 + 0.08 * np.log10(x) + 0.2 * (x / 12) ** 2 - 0.5 * (x / 20) ** 4 # the WS2P: unknown gain_1, gain_2 = 0.35, -0.20 # the two measuring channels, dB field_a = 0.03 * np.sqrt(x) # port A of the coupler hears 0.015 dB to 0.13 dB more rng = np.random.default_rng(61094) noise = rng.normal(0.0, 0.004, (2, 3, f.size)) # three repeats of each reading l_c12 = (l_ref + gain_1) - (l_true + gain_2) + field_a + noise[0] # reference on channel 1 l_c21 = (l_true + gain_1) - (l_ref + gain_2) + field_a + noise[1] # interchanged c = metrology.simultaneous_comparison(f, l_ref, l_c12, l_c21) print(c.determinations, c.sensitivity_level_db[[2, 8, 19]].round(2)) # 3 [-38.63 -38.58 -38.44] print(np.abs(c.sensitivity_level_db - l_true).max().round(3)) # 0.003 ``` A pressure calibration without the interchange is refused, since 5.1.2 requires it. The "mean of the two ratios" 5.1.2 asks for is taken in decibels, as Annex C does: the geometric mean, the only one in which the gains cancel exactly (the arithmetic mean of the two ratios here would sit 0,018 dB to 0,027 dB high). IEC 61094-5 6.6 corrects the reference's sensitivity to the conditions of the test when the two microphones are different models, and IEC 61094-8 7.6 does it whatever the models. `metrology.environmental_sensitivity_correction` corrects a sensitivity level for the static pressure, the temperature and the humidity to first order, from the microphone's own coefficients in the units IEC 61094-2 Annex D gives them: ```python env = metrology.environmental_sensitivity_correction( f, static_pressure_kpa=99.2, # the laboratory during the test temperature_c=21.5, relative_humidity_percent=45.0, static_pressure_coefficient_db_per_kpa=-0.005, # the LS2P's own coefficients temperature_coefficient_db_per_k=0.002, ) # from its certificate's 101.325 kPa, 23.0 °C and 50 % print(env.correction_db[0].round(4)) # 0.0076 ``` ## The uncertainty budget of Annex D Table D.1 is a worked budget for a set-up like this one at 2 kHz, in the coupler of Figure A.1, which is for frequencies up to 10 kHz (A.1; above, the jig of A.2 reaches 20 kHz). `metrology.IEC61094_5_TABLE_D1` holds each of its eight rows with its printed standard uncertainty, and the stated value and divisor of the seven that state one; the repeatability row states no value and prints 0,025 dB as a standard uncertainty. `metrology.comparison_uncertainty_budget` combines the components on `metrology.combine_uncertainty` and multiplies by `k = 2`: ```python d1 = { "reference": 0.025, "capacitance": 0.006, "non_linearity": 0.017, "impedance": 0.003, "polarizing_voltage": 0.005, "repeatability": 0.025, "drift": 0.017, "rounding": 0.003, } # Table D.1, standard uncertainties at 2 kHz b = metrology.comparison_uncertainty_budget(d1, frequency_hz=2000) print(round(b.combined_uncertainty_db, 4), round(b.expanded_uncertainty_db, 3)) # 0.0437 0.087 print(round(b.linear_combined_uncertainty_db, 4)) # 0.0436 ``` D.3 prints 0,040 dB and 0,08 dB, which are not the root-sum-square of its own eight components (an [erratum](https://docs.jmrp.io/phonometry/reference/errata/)). The strict calculation in linear form D.3 mentions, and does not print, gives 0,043 614 dB; the Spanish UNE-EN 61094-5:2017 leaves six of the eight values of Table D.1 blank and prints 0,004 dB in D.3. With one budget per frequency, the expanded uncertainty goes to the calibration, which draws it as a band: ```python budgets = [ metrology.comparison_uncertainty_budget( {**d1, "impedance": 0.003 + 0.09 * (fx / 20000) ** 2}, frequency_hz=fx ) for fx in f ] u = [bb.expanded_uncertainty_db for bb in budgets] c = metrology.simultaneous_comparison( f, l_ref, l_c12, l_c21, reference_environment=env, expanded_uncertainty_db=u ) print(np.round(u, 3)[[0, 16, 19]]) # [0.087 0.101 0.205] print(c.sensitivity_mv_per_pa[[2, 8, 19]].round(2)) # [11.72 11.78 11.98] ``` ## A WS3 microphone in the jig Table A.1 gives the corrections to add to the sensitivity level of a WS3 microphone calibrated against an LS2aP in the jig of Figure A.4, with an expanded uncertainty of a tenth of each: ```python j = metrology.jig_diameter_correction() print(j.correction_db[[0, 9, 13]]) # [-0.004 -0.235 -1.443] print(j.standard_uncertainty_db[13].round(4)) # 0.0722 ``` ### Beyond Table A.1: microphones of different diameters "The effect of a non-uniform pressure distribution over the surface of the diaphragm will be significantly greater if the test and reference microphones are of different diameters. A theoretical model which can be used to apply corrections and assess the uncertainties in this case is given in the literature (for example [1])" (6.5). Its reference [1], Barham, Barrera-Figueroa and Avison, Metrologia 51 (2014) 129, is where Table A.1 comes from. The gap between the diaphragms is a cylinder of the reference's front-cavity radius `a` and length `L`, closed by the reference and driven, in a radially symmetrical field, through the annulus round the smaller microphone. Its pressure is a series of the radial modes of the cavity, `p(r, z) = sum C_n cosh(mu_n z) J0(k_n r/a)` with `J1(k_n) = 0`, and each microphone reads it averaged over its diaphragm with its radial sensitivity, the deflection of a membrane under a uniform pressure, `J0(j01 (f/f0) r/a)/J0(j01 f/f0) - 1`, `f0` being the diaphragm's resonance frequency and `b` taking the place of `a` for the test microphone. The correction is the ratio of the two averages, `-20 lg|R_P|`. `metrology.diameter_sound_field_correction` computes it from the geometry and the two resonance frequencies, at the speed of sound of the IEC 61094-2 Annex F air: ```python ws3 = { "reference_radius_m": 4.650e-3, # an LS2: its front cavity and diaphragm "test_diaphragm_radius_m": 2.065e-3, # a WS3: its diaphragm "test_outer_radius_m": 2.975e-3, # and its outline "separation_m": 0.5e-3, # Figure A.4 "reference_resonance_frequency_hz": 22000, "test_resonance_frequency_hz": 100000, } # the inputs of Table 1 of [1] model = metrology.diameter_sound_field_correction(j.frequencies_hz, **ws3) print(round(model.speed_of_sound, 1), model.correction_db[[0, 9, 13]].round(3)) # 345.9 [-0.004 -0.233 -1.434] print(np.abs(model.correction_db - j.correction_db).max().round(4)) # 0.0093 print((model.separation_change_db / np.abs(model.correction_db))[[0, 13]].round(3)) # [0.063 0.054] ``` At the reference air of clause 4 the model is within 0,0093 dB of every row of Table A.1, inside the 0,144 dB the NOTE gives the largest correction as its expanded uncertainty; the paper prints no speed of sound, and at 344,8 m/s the model gives every row to the thousandth printed. As printed, the paper's formulas with its own inputs do not give its own table: the library reads the radial sensitivity of its Formula (4) less 1 and the annulus of its Formula (2) from the WS3's overall radius, the two readings that do (an [erratum](https://docs.jmrp.io/phonometry/reference/errata/)). A.2 sets the expanded uncertainty of Table A.1 at 10 % of each correction, "which is approximately the change observed by doubling the distance between the microphones"; `separation_change_db` is that change by the model, of the order A.2 states and below it. Table A.1 holds for one pair of microphones at the one separation of Figure A.4 and at fourteen frequencies; the model holds for any test microphone that fits inside its reference's front cavity, any separation, and any frequency below the two resonance frequencies: ```python f12 = metrology.exact_frequencies(1000, 20000, fraction=12) # every twelfth of an octave fine = metrology.diameter_sound_field_correction(f12, **ws3) print(f12.size, round(f12[50]), fine.correction_db[50].round(3)) # 53 17783 -1.141 ``` `fine.correction_db` goes to a calibration through `corrections_db=` as Table A.1 does. The model assumes a radially symmetrical field, which is why 6.5 and A.2 ask for a coaxial source in the far field or a diffuse field. ## A free-field calibration by substitution By IEC 61094-8 A.2 each microphone is read against the monitor, and the quotient of the two ratios is the output ratio corrected for any change of the source: `20 lg R_V = 20 lg(V_test / V_mon,2) - 20 lg(V_ref / V_mon,1)`, with `V_ref` and `V_mon,1` the output voltages of the reference and the monitor in the first reading and `V_test` and `V_mon,2` those of the second. The monitor's two readings go together, as a `metrology.MonitorReadings`. The reference here is an LS2P calibrated in a pressure field, so it takes the free-field to pressure difference of IEC/TS 61094-7 (illustrative values below), and the source drifts by 0,25 dB between the readings: ```python ff = metrology.exact_frequencies(500, 20000, fraction=3) xf = ff / 1000 c_ff = 0.05 * xf**1.3 # the LS2P's free-field to pressure difference: illustrative l_ref_p = -38.0 + 0.04 * np.log10(xf) - 0.25 * (xf / 20) ** 2 # its pressure calibration l_ws2f = -38.3 + 0.1 * np.log10(xf) - 0.3 * (xf / 20) ** 3 # the WS2F: unknown field_1 = 74.0 + 0.05 * np.sin(xf) # the free field while the reference is in place field_2 = field_1 + 0.25 # the source has drifted by the time the WS2F is monitor = -40.0 # the monitor microphone near the source, dB re 1 V/Pa def free_field_budget(fx): x20 = fx / 20000 return metrology.comparison_uncertainty_budget( { "reference": 0.06 + 0.14 * x20, # pressure calibration and IEC/TS 61094-7 "source_stability": 0.01, "positioning": 0.02, "alignment": 0.01 + 0.04 * x20**2, "free_field": 0.03 + 0.12 * x20**2, "non_linearity": 0.017, "rounding": 0.003, "repeatability": 0.02, }, frequency_hz=fx, field="free_field", ) free = metrology.sequential_comparison( ff, l_ref_p, l_ref_p + c_ff + field_1, # the reference's output, dB re 1 V l_ws2f + field_2, # the WS2F's output in its place monitor=metrology.MonitorReadings( reference_level_db=monitor + field_1, test_level_db=monitor + field_2 ), field="free_field", reference_free_field_difference_db=c_ff, expanded_uncertainty_db=[free_field_budget(fx).expanded_uncertainty_db for fx in ff], ) print(bool(np.abs(free.sensitivity_level_db - l_ws2f).max() < 1e-9)) # True print(free.expanded_uncertainty_db[[0, 12, 16]].round(2)) # [0.16 0.26 0.51] ``` Table 2 of IEC 61094-8 lists the components and prints no values; the ones above are illustrative. `metrology.IEC61094_8_TABLE_1` holds the typical expanded uncertainty of each way of calibrating the reference: ```python row = metrology.IEC61094_8_TABLE_1["primary_pressure"] print(row.expanded_uncertainty_1khz_db, row.expanded_uncertainty_10khz_db) # 0.12 0.4 ``` ## The effective free-field region of a time window A time window of length `tau` simulates a free field inside a prolate spheroid with the source and the microphone at its foci, `d` apart, and the major diameter `A = d + tau c` (Formula (B.1)), at the speed of sound of the IEC 61094-2 Annex F air: ```python r = metrology.free_field_region(1.0, 0.005) # 1 m, a 5 ms window, at 23.0 °C print(round(r.speed_of_sound, 1), round(r.major_axis_m, 2)) # 345.9 2.73 print(round(r.semi_minor_axis_m, 2), round(r.rod_clearance_m, 2)) # 1.27 0.86 ``` ## Time-selective processing (Annex B) Annex B turns a room that is not anechoic into a free field: an impulse response separates the direct sound from the reflections in time, a window keeps the first, and a Fourier transform gives the frequency response of the direct sound alone (B.1.1). The impulse response can come from a stepped sine taken to the time domain by Formula (B.3) (`metrology.stepped_sine_impulse_response`), from a sweep or a maximum length sequence (`room.impulse_response`, `room.mls_impulse_response`), or from a direct impulse. `metrology.time_selective_response` weights it with a time window and evaluates Formula (B.2) at any frequency. The window "normally has 'tapered' edges" (B.1.2) and a rectangular one "is not recommended because it usually leads to spectral leakage" (B.1.3), so the default is a Tukey window with an eighth of it tapered at each edge, a proportion the library chose. Here an LS2P reference and a WS2F stand in turn 1 m from the source, and a reflection off the rig arrives along a path of 1,48 m. The WS2F hears it off axis, so it does not cancel in the ratio of the two outputs as the direct sound does (8.5). Each is measured by a stepped sine from 0 Hz to 100 kHz in 50 Hz steps; the microphones are illustrative: ```python c0 = metrology.free_field_region(1.0, 0.005).speed_of_sound # 345.9 m/s at 23.0 °C f_step = np.arange(0.0, 100000.0 + 1.0, 50.0) # 0 Hz to 100 kHz in 50 Hz steps def mic(fx, level_db, f0, d): # a microphone's sensitivity, V/Pa: illustrative return 10 ** (level_db / 20) / (1 - (fx / f0) ** 2 + 1j * d * fx / f0) direct, rig = 1.0 / c0, 1.48 / c0 # the direct sound over 1 m, the rig's reflection over 1.48 m def heard(m, off_axis): # each step: the direct sound and the rig's reflection return m * ( np.exp(-2j * np.pi * f_step * direct) + 0.25 * off_axis * np.exp(-2j * np.pi * f_step * rig) ) ir_ref = metrology.stepped_sine_impulse_response(f_step, heard(mic(f_step, -38.0, 22000, 1.1), 1.0)) ir_ws2f = metrology.stepped_sine_impulse_response( f_step, heard(mic(f_step, -26.4, 18000, 0.9), 1 / (1 + 1j * f_step / 6000)) ) # the WS2F hears the reflection off axis print(ir_ref.duration_s, round(ir_ref.sample_rate_hz)) # 0.02 200050 ``` The impulse response lasts the inverse of the frequency step, 20 ms (B.2.2: 120 Hz is enough in a small anechoic room "because the primary reflections all occur before 8 ms", about 30 Hz in a large one). B.2.1 says the increment "will determine the time domain resolution"; it is the length it sets, as B.2.2 says (an [erratum](https://docs.jmrp.io/phonometry/reference/errata/)). `ir_ws2f.plot()` draws the whole record: The window has to end before the reflection arrives. A point on the boundary of the region of Formula (B.1) reflects along a path of `A = d + tau c`, so a reflection whose path is `L` stays outside while `tau <= (L - d) / c`: ```python tau = metrology.reflection_free_window_s(1.0, 1.48) window = {"window_start_s": direct - 0.0005, "window_end_s": direct + tau} ref = metrology.time_selective_response(ir_ref.impulse_response, ir_ref.sample_rate_hz, **window) ws2f = metrology.time_selective_response(ir_ws2f.impulse_response, ir_ws2f.sample_rate_hz, **window) print(round(1000 * tau, 2), round(ref.frequency_resolution_hz)) # 1.39 530 ``` A window of 1,89 ms tells frequencies apart no closer than 530 Hz, the low-frequency limit 6.2.3 warns time-selective techniques have. The levels and phases through the window are the two outputs of a substitution: ```python fw = metrology.exact_frequencies(1900, 20000, fraction=3) # 2 kHz to 20 kHz m_ref = mic(fw, -38.0, 22000, 1.1) # the reference's free-field sensitivity, from its certificate tsel = metrology.sequential_comparison( fw, 20 * np.log10(np.abs(m_ref)), ref.level_db_at(fw), ws2f.level_db_at(fw), field="free_field", phase=metrology.SequentialComparisonPhase( reference_sensitivity_phase_deg=np.degrees(np.angle(m_ref)), reference_output_phase_deg=ref.phase_deg_at(fw), test_output_phase_deg=ws2f.phase_deg_at(fw), ), ) m_ws2f = mic(fw, -26.4, 18000, 0.9) # what the page knows the WS2F to be print(tsel.sensitivity_level_db[[0, 6, 10]].round(2)) # [-26.34 -25.46 -26.6 ] print(bool(np.abs(tsel.sensitivity_level_db - 20 * np.log10(np.abs(m_ws2f))).max() < 1e-4)) # True record = 20 * np.log10(np.abs(ws2f.record_response_at(fw) / ref.record_response_at(fw))) print(np.abs(record - 20 * np.log10(np.abs(m_ws2f / m_ref))).max().round(2)) # 1.91 ``` Through the window the WS2F's sensitivity comes back to within 0,0001 dB; from the whole record, the reflection included, it would be up to 1,91 dB off. **The direct impulse method** (B.6) drives the source with a short rectangular pulse, whose spectrum is Formula (B.10); its first zero, `1/(2b)`, "must be approximately an order of magnitude higher than the upper limit of the frequency range of interest". The formula is that of a pulse lasting `2b`, although the text calls `b` the duration (an [erratum](https://docs.jmrp.io/phonometry/reference/errata/)), and `metrology.rectangular_pulse` takes the whole duration: ```python T = metrology.rectangular_pulse_duration_s(20000) # a first zero ten times 20 kHz pulse = metrology.rectangular_pulse(T, amplitude_v=10.0) print(T, pulse.half_duration_s, round(pulse.first_zero_hz)) # 5e-06 2.5e-06 200000 print(pulse.level_db_at([20000.0]).round(2)) # [-0.14] ``` The responses to several pulses, averaged synchronously (`signals.time_synchronous_average`; B.6.2 counts on `10 lg n` dB of signal-to-noise ratio from `n` of them), go to `time_selective_response` with `excitation=pulse`, which divides the pulse's spectrum out. ## The phase of the sensitivity Both parts carry the phase over with the modulus: IEC 61094-5 5.1.1 calculates "the sensitivity (both modulus and phase) of the test microphone", and IEC 61094-8 5.1 "both the modulus and phase of the free-field sensitivity of the microphone under test". D.2 holds for the complex ratios, so the phase follows the level term by term, `phi_test = phi_ref + arg R_V - arg R_P`, and the interchange of Annex C cancels the phase shifts of the two channels and of the coupler as (C.3) cancels their gains: `phi_ref - phi_test = (Phi_C12 - Phi_C21) / 2`, with `Phi_C12` the phase of channel 1's reading re channel 2's. Neither part prints this form; it is (C.3) written for the complex ratio. The phase inputs go together, as a `metrology.SimultaneousComparisonPhase` here and a `metrology.SequentialComparisonPhase` in a sequential calibration, whose monitor's phases go with its levels in its `metrology.MonitorReadings`. The pressure calibration above, with the phase of each reading: ```python phi_ref = np.degrees(np.angle(1 / (1 - (f / 22000) ** 2 + 1.1j * f / 22000))) # the LS2P's certificate: illustrative phi_true = np.degrees(np.angle(1 / (1 - (f / 18000) ** 2 + 0.9j * f / 18000))) # the WS2P: unknown shift_1, shift_2 = 1.5 * x, -0.8 * x # the two channels' phase shifts, degrees late_a = 0.4 * x # port A hears the field later, degrees jitter = np.random.default_rng(5).normal(0.0, 0.02, (2, 3, f.size)) p12 = (phi_ref + shift_1) - (phi_true + shift_2) + late_a + jitter[0] # reference on channel 1 p21 = (phi_true + shift_1) - (phi_ref + shift_2) + late_a + jitter[1] # interchanged cp = metrology.simultaneous_comparison( f, l_ref, l_c12, l_c21, phase=metrology.SimultaneousComparisonPhase( reference_sensitivity_phase_deg=phi_ref, channel_phase_difference_deg=p12, interchanged_channel_phase_difference_deg=p21, ), ) print(cp.sensitivity_phase_deg[[2, 8, 16, 19]].round(1)) # [ -1.1 -4.6 -35.9 -102.9] print(np.abs(cp.sensitivity_phase_deg - phi_true).max().round(3)) # 0.016 ``` The difference of the two readings is twice the phase difference of the microphones, so it is taken into one turn before it is halved: the result is unique while they differ by less than 90°. In a free field the phases are those at the acoustic centres (IEC 61094-8 5.1), which 7.3 positions "at the measurement points"; in a sequential substitution each goes in turn to the same point, where `arg R_P` is 0, and a delay common to the two readings, like the travel time from the source in the time-selective example, cancels in their quotient. ## Microphones of different acoustic impedance "Differences in the acoustic impedance between the test and reference microphones can cause the sound pressure at the test and reference microphones to differ" (7.4), most where a pressure and a free-field microphone meet above 10 kHz (7.5). Neither clause prints a model: 7.4 sends the reader to the literature and 7.5 asks for the error to be estimated and added to the budget. `metrology.impedance_pressure_ratio` writes the effect for two circuits with each microphone's equivalent volume, `V_e = kappa_r p_s,r / (j omega Z_a)` (IEC 61094-1 6.2.2, `kappa_r` = 1,40): `R_P = (V_x + V_e,ref) / (V_x + V_e,test)`. In a closed coupler small against the wavelength a sequential substitution changes the pressure by the ratio of the two admittance sums of IEC 61094-2 Formula (3), `V_x` being the rest of the load (the gas, the source, a monitor); for a simultaneous excitation Table D.1 puts "the acoustical impedance of the microphone [...] in series with that of the air in the space between the two microphones", so that the two "see slightly different pressures [...] (see 7.4 and [2])": each diaphragm takes the pressure of a common source behind the same series impedance, and `V_x` is the equivalent volume of that impedance, which the circuit of [2] gives (below). `metrology.ReciprocityMicrophone`, the microphone of [the reciprocity calibration](https://docs.jmrp.io/phonometry/signals/metrology/reciprocity-calibration/), holds the lumped parameters of IEC 61094-2 E.4, the low-frequency volume, the resonance frequency and the loss factor, and its `complex_equivalent_volume_m3` gives `V_e` from them. Its ratio to the low-frequency volume holds no `kappa_r`, so with the 10 mm³ that Table 3 of IEC 61094-1 gives an LS2P, `V_e` is the volume of 6.2.2 with its 1,40. E.4 finds the representation "of sufficient accuracy" up to "about 1,3 times the resonance frequency of the microphone", 20,8 kHz for the WS2F below. The front cavity the microphone also holds enters the transfer impedance of a coupler, not `V_e`: ```python fi = metrology.exact_frequencies(1000, 20000, fraction=3) front = {"front_cavity_volume_m3": 34e-9, "front_cavity_depth_m": 0.5e-3, "front_cavity_diameter_m": 9.3e-3} # an LS2aP's (Table C.1 of IEC 61094-2) lumped_ls2p = metrology.ReciprocityMicrophone( equivalent_volume_m3=10e-9, resonance_frequency_hz=22000, loss_factor=1.1, **front ) # an LS2P: 10 mm³ (IEC 61094-1 Table 3); the resonance and loss: illustrative lumped_ws2f = metrology.ReciprocityMicrophone( equivalent_volume_m3=30e-9, resonance_frequency_hz=16000, loss_factor=0.5, **front ) # a more compliant WS2F: illustrative rp = metrology.impedance_pressure_ratio( fi, reference_equivalent_volume_m3=lumped_ls2p.complex_equivalent_volume_m3(fi), test_equivalent_volume_m3=lumped_ws2f.complex_equivalent_volume_m3(fi), coupling_equivalent_volume_m3=600e-9, # the coupler's gas, source and monitor: illustrative ) print(rp.level_difference_db[[0, 10, 13]].round(3)) # [-0.281 -0.421 0.37 ] print(rp.standard_uncertainty_db[[0, 10, 13]].round(3)) # [0.163 0.243 0.213] ``` A monitor that senses "changes in the sound pressure at the test/reference microphone position" (5.1.3) takes this ratio out. Without one, `rp.level_difference_db` and `rp.phase_difference_deg` go to the comparison as `pressure_level_difference_db` and as the `pressure_phase_difference_deg` of its phase inputs. Where the ratio is estimated rather than corrected for, `rp.standard_uncertainty_db` is the `"impedance"` component of the budget: the level difference read as the semi-range of a rectangular distribution, as Table D.1 reads its 0,005 dB at 2 kHz. The same row warns about exactly this pairing: "When microphones have significantly differing impedances (for example WS2F microphone compared against LS2P at frequencies above 10 kHz), the measurement uncertainty can be considerably larger and should be established experimentally." Above 10 kHz the value here is the model's estimate, not that experiment; where the ratio crosses 0 dB, near 16 kHz, it gives almost nothing at all. ### The air between the two microphones The "Microphone impedance" row of Table D.1 refers the simultaneous excitation to its reference [2], Jarvis, NPL Report CIRA(EXT) 010 (1996). Its Appendix B models a symmetric coupler as a ladder from a source in its middle: on each side a mass `Z_L`, the compliance `Z_c` of that side's half of the volume, a second `Z_L` and the microphone, with `Z_c = kappa p0 / (j omega V/2)`, `Z_L = j omega rho (L/4) / (3 pi r^2)` and `V = pi r^2 L` for a tube of length `L` and radius `r`. Seen from each diaphragm, the middle of the ladder is a pressure common to both behind one impedance, `Z_x = Z_L + Z_L Z_c/(Z_L + Z_c)`, so each diaphragm takes `p Z_a/(Z_a + Z_x)`, the divider above, and the ratio of the two is the one Appendix B prints. `metrology.air_gap_series_impedance_pa_s_m3` gives `Z_x` for a coupler of known length and radius. Appendix B's example is an LS2P against a high sensitivity WS2P in a tube 2 mm long and 12,7 mm across, each microphone a resistance, a mass and a compliance in series: ```python fj = np.arange(100.0, 20000.0 + 1.0, 100.0) # Appendix B's frequencies w = 2 * np.pi * fj def z_mic(r, c, m): # Z_m = r + j w m + 1/(j w c), Appendix B return r + 1j * w * m + 1 / (1j * w * c) def v_e(z): # the equivalent volume of IEC 61094-1 6.2.2 return 1.40 * 101325 / (1j * w * z) z_x = metrology.air_gap_series_impedance_pa_s_m3(fj, gap_length_m=2e-3, gap_radius_m=6.35e-3) gap = metrology.impedance_pressure_ratio( fj, reference_equivalent_volume_m3=v_e(z_mic(330e6, 48e-15, 1210)), # the LS2P of Appendix B test_equivalent_volume_m3=v_e(z_mic(7e7, 2.6e-13, 820)), # its high sensitivity WS2P coupling_impedance_pa_s_m3=z_x, ) print(gap.level_difference_db[[69, 199]].round(4)) # [ 0.0064 -0.025 ] print(gap.phase_difference_deg[124].round(3)) # -0.158 ``` The WS2P hears 0,0064 dB more than the LS2P at 7 kHz and 0,025 dB less at 20 kHz, and lags it by up to 0,158° at 12,5 kHz. Appendix B draws the same curves for `P1/P2`, the LS2P re the WS2P, so the other way up, with its 1,21 kg/m³ for the density of the air; with that density the conformance report reproduces them. In the published scan its impedance of the LS2P shows no imaginary unit on the mass term, where blank space stands; the second microphone's has it, and the graph needs it (an [erratum](https://docs.jmrp.io/phonometry/reference/errata/)). ## The diffuse-field comparison of IEC 61183 The diffuse-field method of IEC 61183 clause 5 is a sequential comparison without a monitor, and the library computes it through the same model: ```python d = metrology.diffuse_field_sensitivity( [1000.0, 2000.0], [94.3, 94.6], [94.0, 94.1], reference_random_incidence_level_db=[-26.0, -26.2], ) s = metrology.sequential_comparison([1000.0, 2000.0], [-26.0, -26.2], [94.0, 94.1], [94.3, 94.6]) print(d.diffuse_field_level_db, s.sensitivity_level_db) # [-25.7 -25.7] [-25.7 -25.7] ``` ## Validating a jig or a coupler "Calibrations performed in any particular jig or coupler shall be validated by comparison with calibrations performed in other jigs and couplers and alternative sound sources. A separate validation is necessary for each different type of microphone. If the test microphone is a laboratory standard microphone, then the jig or coupler can be validated by comparing a comparison calibration with a reciprocity calibration. For some microphones, it can be necessary to use more than one jig and/or coupler to cover a full frequency range with low uncertainty" (6.7). `metrology.verify_jig_or_coupler` sets a calibration made in the jig or coupler beside another calibration of the same microphone, a comparison made elsewhere or, for a laboratory standard microphone, the `ReciprocityCalibration` of [the reciprocity calibration](https://docs.jmrp.io/phonometry/signals/metrology/reciprocity-calibration/) with the index of that microphone, and compares them at the frequencies they share. The clause prints no criterion. The library reads "validated" as the two agreeing within the expanded uncertainty of their difference, `|L_cal - L_val| <= U_Δ`, which for two independent calibrations is the root-sum-square `U_Δ = sqrt(U_cal^2 + U_val^2)`; two calibrations that share a component are correlated, and the part they share comes out of it (below). Here the jig of A.2 is validated against the coupler of A.1 for the WS2P of the first example, the coupler used up to the 10 kHz it is for. Near 6,3 kHz the jig has a standing wave close to the microphones, the anomaly Barham et al. found in a test box at that frequency, and the field moves between the reading and the interchange, so (C.3) no longer cancels it: ```python fv = f[6:] # the jig of A.2, 1 kHz to 20 kHz mode = 0.5 * np.exp(-(((fv / 1000 - 6.3) / 0.6) ** 2)) # the field moves near 6.3 kHz jitter = np.random.default_rng(42).normal(0.0, 0.01, (2, 3, fv.size)) j12 = (l_ref[6:] + gain_1) - (l_true[6:] + gain_2) + field_a[6:] + mode + jitter[0] j21 = (l_true[6:] + gain_1) - (l_ref[6:] + gain_2) + field_a[6:] + jitter[1] jig = metrology.simultaneous_comparison(fv, l_ref[6:], j12, j21, expanded_uncertainty_db=u[6:]) coupler = metrology.simultaneous_comparison( f[:17], l_ref[:17], l_c12[:, :17], l_c21[:, :17], expanded_uncertainty_db=u[:17] ) # the coupler of A.1, up to 10 kHz check = metrology.verify_jig_or_coupler(jig, coupler) print(check.passes, check.failing_frequencies_hz.round()) # False [6310.] print(check.unvalidated_frequencies_hz.round()) # [12589. 15849. 19953.] print(check.difference_db[8].round(3), check.expanded_uncertainty_db[8].round(3)) # -0.247 0.128 ``` The jig agrees with the coupler at every frequency they share but 6,3 kHz, where they differ by 0,247 dB against 0,128 dB, and above 10 kHz it is not validated at all, the coupler not reaching there: a second set-up has to validate it there, as 6.7 allows. `check.normalised_difference` is the difference in units of its expanded uncertainty, -1,93 at 6,3 kHz. ### Calibrations that share a reference Both calibrations here take the same LS2P as their reference, so the first row of Table D.1, the calibration of that reference (0,025 dB, the "±0,05 dB with a coverage factor of k = 2" of its certificate), is in both budgets. Whatever error that calibration carries moves the two sensitivity levels together and cancels in their difference, while the root-sum-square counts it twice. In the terms of the GUM, two levels that each depend on a common quantity with a sensitivity of 1 have its variance as their covariance (JCGM 100:2008 F.1.2.3, Formula (F.2), whose sensitivity coefficients are then both 1), and the law of propagation for correlated inputs (5.2.2, Formula (13)) gives for the difference `u_Δ^2 = u_cal^2 + u_val^2 - 2 u_sh^2` with `u_sh` the standard uncertainty of what the two share. `shared_standard_uncertainty_db` takes it, one value or one per frequency of the calibration, and the band becomes `U_Δ = sqrt(U_cal^2 + U_val^2 - 2 (k u_sh)^2)` with the `k = 2` of 7.9: ```python shared = metrology.verify_jig_or_coupler(jig, coupler, shared_standard_uncertainty_db=0.025) print(shared.passes, shared.failing_frequencies_hz.round()) # False [6310.] print(shared.expanded_uncertainty_db[8].round(3), shared.normalised_difference[8].round(2)) # 0.106 -2.32 print(shared.correlation_coefficient[8].round(2)) # 0.31 ``` The band narrows from 0,124 dB to 0,101 dB at the low frequencies and from 0,128 dB to 0,106 dB at 6,3 kHz, where the two calibrations are correlated with a coefficient of 0,31, the reference's variance over the product of their standard uncertainties (Formula (14)). The verdict does not change here: the jig still fails at 6,3 kHz alone, now by 2,32 times the uncertainty of the difference. A pair that agrees within the root-sum-square can fail once the shared part is out of it. Which components the two share is the laboratory's to say: the calibration of a reference both use, its drift since then (the row "Drift in reference microphone sensitivity since last calibration", 0,017 dB) for two calibrations made close together, a measuring chain they both go through. Only a component that enters both levels with the same sign and size belongs here. A reciprocity calibration made in the same set of three microphones as the reference's is not such a case: in IEC 61094-2:2009 Formula (7) (folio 12) the electrical transfer impedances of the two pairs that go through the third microphone are in the numerator of one microphone's sensitivity and in the denominator of the other's, so those measurements enter the two levels with opposite signs. Their covariance is negative and widens the uncertainty of the difference beyond the root-sum-square, which `shared_standard_uncertainty_db` cannot express; only the factors common to both levels with the same sign may be given there. A shared part larger than either calibration's standard uncertainty is refused, since nothing can be shared beyond the whole of it. The clause prints neither a criterion nor an example, so the conformance report checks the verdict against its own definition. Jarvis and Watkins (1997), NPL Report CIRA(EXT) 022, who compared jig calibrations of working standard microphones at NPL with reciprocity calibrations of the same microphones, print no numeric criterion either: their Figures 6 and 7 (folio 4) plot the differences, and the text judges them against "the typical repeatability" of the measurements (folio 2), one standard deviation drawn as a band in Figure 7 and never printed as a number, with no number a criterion could be checked against. The residual differences "of less than 0.1 dB" of folio 5 report a result rather than set a test, and the report says nothing about what the two calibrations share. The shared part has printed numbers, in the GUM: two items calibrated against the same standard, of relative standard uncertainty $10^{-4}$, are correlated with coefficients of about 0,5, 0,990 and 1,000 for comparisons of $100 \times 10^{-6}$, $10 \times 10^{-6}$ and $1 \times 10^{-6}$ (F.1.2.3, Example 2). The conformance report checks `correlation_coefficient` against them, and the difference of two calibrations with the budget of Table D.1 against the seven rows left once the reference's cancels. ## What this guide covers Implemented: the level model of IEC 61094-5 D.2; the simultaneous excitation with the interchange of Annex C (Formulas (C.1) to (C.3)), required for a pressure calibration; the sequential excitation against a monitor microphone; the average over the determinations; the phase of the sensitivity (61094-5 5.1.1, 61094-8 5.1), the phase form of (C.3) and of the monitor ratios being this library's reading of the complex ratio, as neither part prints it; the environmental correction to first order (this library's reading of 6.6 and 7.6, which print no formula); the reference's free-field to pressure difference in a free-field calibration; the WS3 corrections of Table A.1, whose 10 % expanded uncertainty, printed without a coverage factor, is read with the `k = 2` of 7.9 and D.2; the pressure ratio of microphones of different acoustic impedance (7.4, 7.5) in a closed coupler by the printed IEC 61094-2 Formula (3) and, for a simultaneous excitation, in series with the air between them, by the circuit of Appendix B of Jarvis (1996), the part's reference [2], whose series impedance the library gives for a coupler of known length and radius (read with the imaginary unit its first microphone's mass term has lost in the published scan), with the lumped equivalent volume of IEC 61094-2 E.4 and IEC 61094-1 6.2.2, and its standard uncertainty as Table D.1 reads a semi-range (which, above 10 kHz for a WS2F against an LS2P, the row asks to be established experimentally); the correction for a test microphone smaller than its reference (6.5) by the model of Barham et al. (2014), the part's reference [1], for any geometry and separation, read with the radial sensitivity of its Formula (4) less 1 and the annulus of its Formula (2) from the overall radius, the readings that give its Table 2, which is Table A.1, with the change on doubling the separation that A.2 names, its uncertainty left to the caller; the validation of a jig or a coupler against another calibration of the same microphone (6.7), whose criterion, agreement within the expanded uncertainty of the difference, the root-sum-square of the two less what they share by the GUM law of propagation for correlated inputs (JCGM 100:2008 5.2.2 and F.1.2.3), is this library's reading, the clause printing none and Jarvis and Watkins (1997) no numeric one, only a component that enters both levels with the same sign and size counting as shared; the budgets of Table D.1 and IEC 61094-8 Table 2 with `k = 2`; Tables A.1, D.1, 1 and 2 as published data; the effective free-field region of Formula (B.1) and the longest window it allows for a reflection; the time window of B.1.3 with the tapered edges of B.1.2, Formula (B.2) through it, the stepped sine of B.2 by Formula (B.3) and the rectangular pulse of B.6 (Formula (B.10)). Not implemented: the measurements themselves, and the choice of the set-ups a jig or a coupler is validated against, for each type of microphone (6.7); the sweeps, the maximum length sequences and the random noise of B.3 to B.5, which are `room.impulse_response`, `room.mls_impulse_response` and `electroacoustics.transfer_function` elsewhere in the library, and the synchronous averaging of B.6.2, which is `signals.time_synchronous_average`; the qualification of the free field by ISO 26101, on [its own page](https://docs.jmrp.io/phonometry/devices/emission/free-field-qualification/); the values of IEC/TS 61094-7, which are inputs, and the reciprocity calibration of the reference, which is in [Microphone calibration by reciprocity](https://docs.jmrp.io/phonometry/signals/metrology/reciprocity-calibration/); the second special case of Table D.1, a microphone calibrated as a system with its preamplifier (0,002 dB, and under 0,02 dB above 200 Hz), which a budget takes through `additional_components`; a covariance that widens the uncertainty of the difference, as between two microphones of the same reciprocity set through the pairs with the third (IEC 61094-2:2009 Formula (7)), which `verify_jig_or_coupler` does not take. ## See also - [Microphone calibration by reciprocity (IEC 61094-2 and IEC 61094-3)](https://docs.jmrp.io/phonometry/signals/metrology/reciprocity-calibration/): the primary calibration of the laboratory standard microphone a comparison takes as its reference. - [Free-field corrections of a sound level meter (IEC 62585)](https://docs.jmrp.io/phonometry/signals/metrology/free-field-corrections/): a meter compared with an LS2P reference in the same way. - [Random-incidence and diffuse-field response (IEC 61183)](https://docs.jmrp.io/phonometry/signals/metrology/random-incidence/): the diffuse-field comparison that computes through the same model. - [Measurement uncertainty](https://docs.jmrp.io/phonometry/signals/metrology/gum-uncertainty/): the GUM law of propagation the budgets are combined by. - [Errata in published sources](https://docs.jmrp.io/phonometry/reference/errata/): IEC 61094-5 D.3, UNE-EN 61094-5:2017 Table D.1, IEC 61094-8 8.4, B.2.1 and Formula (B.10), Barham et al. (2014) Formulas (1), (2) and (4), and Jarvis (1996) Appendix B. - API reference: [`metrology.comparison_calibration`](https://docs.jmrp.io/phonometry/reference/api/metrology/comparison-calibration/).