Source: https://docs.jmrp.io/phonometry/signals/metrology/reciprocity-calibration/ # Microphone calibration by reciprocity (IEC 61094-2 and IEC 61094-3) A laboratory standard microphone is calibrated without a reference microphone. Two microphones are coupled acoustically, one drives and the other receives, and the electrical transfer impedance `Z_e,12 = U_2/i_1` that is measured is the product of their sensitivities times the acoustic transfer impedance of whatever couples them: `M_1 M_2 = Z_e,12 / Z_a,12`. In a coupler (IEC 61094-2:2009) `Z_a,12` is the acoustic impedance of a small enclosed volume of gas; in a free field (IEC 61094-3:2016, read with its corrigendum COR1:2016) it is that of a spherical wave between the two acoustic centres. Three microphones measured in pairs give three products, and each sensitivity follows from them alone. The modules are `metrology.reciprocity_coupler`, `metrology.reciprocity_free_field` and `metrology.reciprocity_calibration`. The readings on this page are synthetic, built from sensitivities the page knows, so every step can be checked against them. ## How the measurement goes Three microphones, two of them reciprocal, or two and an auxiliary sound source, one of them reciprocal (5.1.1). Each pair is measured in turn: one is driven by a current, measured across a calibrated impedance in series with it, and the other's open-circuit voltage is read by the insert voltage technique (5.3). In a coupler the two face each other across a cavity of known shape; in a free field, on their principal axes in an anechoic room. | Requirement | Value | Clause | | :--- | :--- | :--- | | Reference conditions | 23,0 °C, 101,325 kPa, 50 % relative humidity | 61094-2 and 61094-3, 4 | | Polarizing voltage | 200,0 V recommended, and the one used reported | 61094-2, 6.2; 61094-3, 6.2 | | Plane-wave coupler | Distance between the diaphragms 0,5 to 0,75 times the diameter | 61094-2, C.2 | | Large-volume coupler | Length to diameter about 0,3, with a wave-motion correction | 61094-2, C.3 | | Free-field distance | Greater than ten nominal diameters | 61094-3, 7.3 | | Free-field mounting | On a cylinder of the microphone's diameter at least twenty diameters long | 61094-3, 6.4 | | Reported uncertainty | Expanded, `k = 2` (61094-2) or for a 95 % coverage probability (61094-3) | 61094-2, 7.5; 61094-3, 7.8 | ## The microphones, the coupler and the transfer impedance Each microphone enters through its lumped acoustic impedance (Annex E: the equivalent volume, the resonance frequency and the loss factor) and its front cavity (E.2, E.3). Figure C.2 draws each microphone of a large-volume coupler with its face a distance F back from the cavity, through a bore of the front-cavity diameter B: `LargeVolumeCoupler` takes that length as `port_length_m`, and the two bores of the LS1P coupler of Table C.2 add 2,4 % to its volume, a tenth of a decibel on each sensitivity. `check_coupler` holds the coupler to what its formulas need: `omega rho a^2 > 100 eta` for Formulas (A.3) and (A.4), the domain `0,125 < R < 8`, `X > 5` of the approximation (A.2), and the Annex F conditions. The ratio C.2 recommends is reported, and so is a frequency below 20 Hz with the approximation, where A.2 asks for the full solution or a larger uncertainty component. ```python import numpy as np from phonometry import fluids, metrology mics = tuple( metrology.ReciprocityMicrophone( equivalent_volume_m3=v, resonance_frequency_hz=f0, loss_factor=d, front_cavity_volume_m3=0.534e-6, # measured, E.3 front_cavity_depth_m=1.95e-3, # D of Table C.1 front_cavity_diameter_m=18.6e-3, # B of Table C.1 ) for v, f0, d in ((148e-9, 8200.0, 1.05), (141e-9, 8350.0, 1.00), (152e-9, 8050.0, 1.10)) ) coupler = metrology.PlaneWaveCoupler( length_m=7.5e-3, diameter_m=18.6e-3, capillary=metrology.CapillaryTube(length_m=0.05, radius_m=0.2e-3, count=2), ) conditions = {"temperature_c": 22.4, "static_pressure_pa": 100200.0, "relative_humidity_percent": 46.0} f = metrology.exact_frequencies(20, 10000, fraction=3) check = metrology.check_coupler(f, coupler, mics, **conditions) print(check.passes, round(check.length_to_diameter_ratio, 3), check.ratio_recommended) # True 0.613 True large = metrology.LargeVolumeCoupler(length_m=12.55e-3, diameter_m=42.88e-3, port_length_m=0.80e-3) # E, C, F of Table C.2 print(round(large.cavity_volume_m3 * 1e9), round(large.closed_volume_m3(mics[:2]) * 1e9)) # 18124 19626 (mm³) ``` A plane-wave coupler is the transmission line of Formula (4), with the losses of Formulas (A.3) to (A.5) and the excess volume of each front cavity (7.3.3.1); a large-volume coupler is the compliance of Formula (3), its volume multiplied by `Delta_H = kappa / (1 + (kappa - 1) E_V)` of Formula (A.1). Both are divided by `Delta_C = 1 + n Z''_a,12 / Z_a,C` of Formula (6) for the capillary tubes of Annex B. ```python pairs = ((0, 1), (1, 2), (2, 0)) za = [metrology.coupler_transfer_impedance(f, coupler, mics[i], mics[j], **conditions) for i, j in pairs] print(za[0].heat_conduction_correction_db[[1, 16, 25]].round(3)) # [-0.287 -0.051 0.004] print(za[0].capillary_correction_db[[1, 16, 25]].round(3)) # [-0.213 0.011 -0. ] ``` `temperature_transfer_function` sums Gerber's full solution, which A.2 requires below 20 Hz unless the uncertainty component is increased instead, and which gives every entry of Table A.1 to its stated 0,000 01; Formula (A.2) is the approximation. The capillary impedance gives Tables B.1 and B.2, whose "0,1667 mm" radius is 1/6 mm, to within 0,002 GPa·s/m³: the residue reaches 0,0011 in the real part and 0,0018 in the imaginary near the resonances of the tubes, and 0,0005 and 0,0007 from 20 Hz to 250 Hz, a little more than the printed half-unit (the 4,288 below against the printed 4,287). ```python print(metrology.temperature_transfer_function(1.0, 100.0).round(5)) # (0.97179+0.02758j) print(metrology.temperature_transfer_function(1.0, 100.0, method="approximation").round(5)) # (0.97179+0.02758j) air = fluids.air(temperature_c=23.0, static_pressure_pa=101325.0, relative_humidity_percent=50.0) tube = metrology.capillary_tube_impedance([20.0, 1000.0], length_m=0.05, radius_m=1 / 6000, gas=air) print((tube.impedance_pa_s_m3 / 1e9).round(3)) # [3.015+0.097j 8.331+4.288j] ``` Table C.3 is the wave-motion correction of the air-filled large-volume coupler for LS1P microphones: ```python print(metrology.large_volume_wave_motion_correction([1000.0, 2000.0]).correction_db) # [-0.002 -0.06 ] ``` ## The pressure sensitivities Formula (2) gives the product of each pair and Formula (7) the three complex sensitivities. A product does not change when every sensitivity changes sign together, so the sign of the square root is a convention: the library takes the branch continuous in frequency with a non-negative real part at the lowest frequency. With two microphones, the ratio measured against an auxiliary source replaces the third pair, Formula (8). ```python true = [m0 * m.complex_equivalent_volume_m3(f) / m.equivalent_volume_m3 for m0, m in zip((0.0496, 0.0473, 0.0512), mics)] ze = [true[i] * true[j] * z.transfer_impedance_pa_s_m3 for (i, j), z in zip(pairs, za)] # Formula (2) c = metrology.pressure_reciprocity(f, ze, za) print(c.sensitivity_level_db[:, 16].round(2)) # [-26.03 -26.44 -25.76] print(c.sensitivity_mv_per_pa[:, 16].round(2)) # [49.93 47.64 51.51] print(np.abs(c.sensitivity_v_per_pa - true).max() < 1e-15) # True ``` ```python pair = metrology.pressure_reciprocity_pair(f, ze[0], za[0], true[0] / true[1]) print(pair.sensitivity_level_db[:, 16].round(2)) # [-26.03 -26.44] ``` ## A free-field calibration Formula (7) of IEC 61094-3 carries the distance between the acoustic centres, the density of the air, the phase of the spherical wave and its attenuation over the distance between the diaphragms; the attenuation is the five steps of Annex B and the wave number takes the speed of sound with the dispersion of IEC 61094-2 F.3. Formulas (8) and (9) as printed omit the factor `-j` of Formula (7), which turns the phase by 45°; the library follows Formula (7). ```python ff = metrology.exact_frequencies(1000, 40000, fraction=3) ffc = {"temperature_c": 23.1, "static_pressure_pa": 101100.0, "relative_humidity_percent": 52.0} att = metrology.reciprocity_air_attenuation(ff, **ffc) print(att.attenuation_db_per_m[[0, 10, 13, 15]].round(4)) # [0.0053 0.1348 0.4713 0.9877] ``` ```python ratio = ff / 23000.0 true_ff = [m0 / (1.0 - ratio**2 + 1.3j * ratio) for m0 in (0.0128, 0.0124, 0.0131)] centre = 4.5e-3 / (1.0 + (ff / 30000.0) ** 2) ze_ff = [ metrology.free_field_transfer_impedance( ff, true_ff[i], true_ff[j], diaphragm_distance_m=0.25, acoustic_centres_m=(centre, centre), **ffc ) for i, j in pairs ] # Formula (D.1) cf = metrology.free_field_reciprocity( ff, ze_ff, diaphragm_distances_m=(0.25, 0.25, 0.25), acoustic_centres_m=(centre, centre, centre), **ffc ) print(cf.sensitivity_level_db[:, 10].round(2)) # [-37.76 -38.03 -37.55] print(np.abs(cf.sensitivity_v_per_pa - true_ff).max() < 1e-15) # True ``` The acoustic centre comes from the inverse-distance law (6.5), and `check_free_field_arrangement` holds the arrangement to the ten diameters 7.3 recommends between the microphones, the twenty 6.4 recommends along the supporting cylinder and the accuracy domain of B.2: ```python r = np.array([0.15, 0.20, 0.25, 0.30, 0.40, 0.50]) alpha = metrology.reciprocity_air_attenuation([10000.0], **ffc).attenuation_np_per_m[0] p = 0.85 * np.exp(-alpha * r) / (r - 2.4e-3) # your readings at one frequency print(round(1000 * metrology.acoustic_centre(r, p, attenuation_np_per_m=alpha).position_m, 2)) # 2.4 ``` ```python arrangement = metrology.check_free_field_arrangement( ff, diaphragm_distances_m=(0.25, 0.25, 0.25), microphone_diameter_m=13.2e-3, support_length_m=0.3, **ffc ) print(arrangement.passes, arrangement.annex_a_range) # True True ``` ## The uncertainty budgets Both parts list their components in Table 1 and find those of the acoustic transfer impedance by changing each parameter by its uncertainty one at a time; `coupler_parameter_uncertainty` and `free_field_parameter_uncertainty` do that, from the standard uncertainties gathered in a `CouplerInputUncertainties` or a `FreeFieldInputUncertainties` (one left at zero adds no component), and `reciprocity_uncertainty_budget` combines everything with `k = 2`. ```python pu = metrology.coupler_parameter_uncertainty( f, coupler, mics, metrology.CouplerInputUncertainties( u_coupler_length_m=3e-6, u_coupler_diameter_m=3e-6, u_capillary_radius_m=2e-6, u_static_pressure_pa=10.0, u_temperature_k=0.05, u_relative_humidity_percent=2.0, u_front_cavity_depth_m=3e-6, u_front_cavity_volume_m3=1e-9, u_equivalent_volume_m3=1e-9, u_resonance_frequency_hz=50.0, u_loss_factor=0.03, ), **conditions, ) print(pu.components_db["capillary_tube_dimensions"][[1, 16]].round(4)) # [0.0086 0.0002] x = f / 10000 b = metrology.reciprocity_uncertainty_budget( f, { **pu.components_db, "series_impedance": 0.002, "voltage_ratio": 0.003, "cross_talk": 0.001 + 0.004 * x**2, "polarizing_voltage": 0.0025, "radial_wave_motion": 0.03 * x**2, "heat_conduction_theory": 0.002, "repeatability": 0.004, "rounding": 0.0006, }, ) print(b.expanded_uncertainty_db[[1, 16, 25]].round(3)) # [0.023 0.015 0.041] pf = metrology.free_field_parameter_uncertainty( ff, metrology.FreeFieldInputUncertainties( u_distance_m=0.1e-3, u_acoustic_centre_m=0.5e-3, u_static_pressure_pa=20.0, u_temperature_k=0.1, u_relative_humidity_percent=2.0, u_air_attenuation_ratio=0.1 / np.sqrt(3.0), ), diaphragm_distances_m=(0.25, 0.25, 0.25), acoustic_centres_m=(centre, centre, centre), **ffc, ) xf = ff / 40000 bf = metrology.reciprocity_uncertainty_budget( ff, { **pf.components_db, "series_impedance": 0.002, "voltage_ratio": 0.003, "cross_talk": 0.002 + 0.01 * xf, "noise": 0.005, "reflections": 0.01 + 0.02 * xf, "polarizing_voltage": 0.0025, "repeatability": 0.01, }, field="free_field", ) print(bf.expanded_uncertainty_db[[0, 10, 15]].round(3)) # [0.049 0.054 0.072] ``` ## What this guide covers Implemented: Formulas (2) to (8) of IEC 61094-2 with Annexes A (Gerber's full solution and the approximation (A.2), Formulas (A.3) to (A.5)), B, C (Tables C.1, C.2 and C.3, with the frequency scaling of C.3 for hydrogen) and E (Formula (E.1)); Formulas (6) to (9) and (D.1) of IEC 61094-3 with COR1, the attenuation of Annex B, the acoustic centre of 6.5 and the arrangement of 6.4 and 7.3; Table 1 of both parts and their one-at-a-time recalculation. Table A.1 of IEC 61094-2 is reproduced to its stated 0,000 01 and Table B.1 of IEC 61094-3 to its last figure; Tables B.1 and B.2 of IEC 61094-2 to within 0,002 GPa·s/m³. Read from the text: dimension F of Table C.2 is the length of the bore Figure C.2 draws between each microphone face and the cavity. Not implemented: the measurements themselves; the radial wave-motion corrections of plane-wave couplers; the environmental corrections, which are the first-order correction of the comparison guide; the determination of the microphone parameters of E.3 and E.4; Formula (4) of IEC 61094-3, which needs the radiation impedance and the scattering factor; the time-selective processing of Annex D; the acoustic centres of Annex A, printed only as a figure. ## See also - [Microphone calibration by comparison (IEC 61094-5 and IEC 61094-8)](https://docs.jmrp.io/phonometry/signals/metrology/comparison-calibration/): carrying a reciprocity-calibrated reference over to a working standard microphone. - [Humid air](https://docs.jmrp.io/phonometry/fluids/humid-air/): the IEC 61094-2 Annex F air. - [Measurement uncertainty](https://docs.jmrp.io/phonometry/signals/metrology/gum-uncertainty/): the GUM the budgets are combined by. - [Errata in published sources](https://docs.jmrp.io/phonometry/reference/errata/): IEC 61094-2 Tables B.1 and B.2, IEC 61094-3 Formulas (8) and (9), and B.2 Step 1. - API reference: [`metrology.reciprocity_calibration`](https://docs.jmrp.io/phonometry/reference/api/metrology/reciprocity-calibration/), [`metrology.reciprocity_coupler`](https://docs.jmrp.io/phonometry/reference/api/metrology/reciprocity-coupler/), [`metrology.reciprocity_free_field`](https://docs.jmrp.io/phonometry/reference/api/metrology/reciprocity-free-field/).