electroacoustics.induction_loop_components

Components of an audio-frequency induction-loop system: the loop, its amplifier, the neck loop.

IEC 62489-1:2010, with its Amendment 1:2014 (read in the consolidated BS EN 62489-1:2010+A1:2015), says how the manufacturer of an induction-loop amplifier, of a loop and of a neck loop measures and states what the component does. IEC 60118-4:2014 then judges the installed system built from them (phonometry.electroacoustics.induction_loop). This module holds the parts of IEC 62489-1 that are arithmetic, and the loop formulas both standards lean on.

A current $I$ in a loop of $N$ turns produces a magnetic field strength $H$, in amperes per metre, that is proportional to $NI$ everywhere and falls with distance from the conductor. IEC 60118-4:2014 E.1 gives the one closed form it uses, the field at the centre of a single-turn square loop of side $d$, in its own plane,

and extends it to a rectangle of sides $d_1$ and $d_2$: “Provided that the sides d1, d2 of a rectangular loop are not extremely different”, $d$ is taken as $\sqrt{d_1{d}_2}$ (loop_centre_field). Everything else about the field is a figure in that standard. Here it is computed: rectangular_loop_field sums the Biot-Savart field of the four straight sides exactly, at any point, which is how the E.1 formula is checked (it is exact for the square and off by 0,35 dB for a rectangle of aspect ratio 1,5). Two of the figures are reproduced: the curves of Figure E.2 b), both components within about half a decibel, taken across the 10-unit width of its 15 by 10 loop, as E.1 describes them (the figure’s panel a) puts the vertical-field line along the length, see docs/ERRATA.md), and the currents of Figure H.1 through loop_current. Figures E.3 to E.5 and A.1 draw the same kind of pattern and are not checked point by point.

IEC 62489-1:2010 5.4.10.2 fixes where an amplifier’s field is measured: 1,4 m above the centre of a horizontal square loop whose resistance and inductance equal the rated load. loop_current gives the current that loop needs for a field strength, and loop_dimensions answers 5.4.11 the other way round, the largest square loop, or loop of aspect ratio 3:1, whose centre reaches 400 mA/m at that height with the amplifier’s current.

The loop is a resistance in series with an inductance (IEC 62489-1 B.2). The resistance is $R=\rho l/a$ for a conductor of length $l$ and area $a$ (loop_resistance), with the resistivity of standard annealed copper, 1/58 ohm square millimetre per metre at 20 degrees Celsius and a temperature coefficient of 0,00393 per degree (IEC 60028:1925, clause I). The inductance of a rectangle of round wire is Grover’s Formula (58) (rectangular_loop_inductance); for $N$ turns it is multiplied by $N^2$, the perfect coupling Table B.1 assumes. The impedance $|Z|=\sqrt{R^2+(2\pi fL{)}^2}$ rises above the frequency where the reactance equals the resistance, where it is $\sqrt{2}$ times the resistance (IEC 60118-4 E.3), and the voltage the amplifier has to deliver is $U=I|Z|$ (LoopImpedance).

Table B.1 of IEC 62489-1 is the oracle. Its six resistances are reproduced by the 20 degree resistivity with the perimeter of each loop, which for the 0,35 m by 0,45 m counter loop is 1,6 m and not the printed 1,5 m (see docs/ERRATA.md). Grover’s formula reproduces four of its six inductances to the printed microhenry once the term for the flux inside the wire is left out: 189, 22, 47 and 109 microhenries. Table B.1 does not say which formula it used, and the two it does not reproduce, the neck loop (79 against 85) and the 15 m by 40 m loop (206 against 218), are not reached by any reading of the wire size either, so they are not claimed. The internal term is kept by default here, because at audio frequencies the skin depth in copper (2 mm at 1 kHz) is larger than the radius of every conductor Table B.1 lists.

IEC 62489-1 clause 5 lists what an amplifier’s specification states, and which of it is measured. The measured characteristics that are computations are:

• the maximum (distortion-limited) output current, 5.4.7: the 1 kHz load current is raised until the total harmonic distortion across the resistive part of the load reaches the rated value, and the current is the total voltage across that resistance over its value (maximum_output_current);
• the compliance voltage, 5.4.8: the maximum positive-going and negative-going peak voltages across the rated load, over at least 60 s of the specified pink noise, averaged and divided by $\sqrt{2}$ (compliance_voltage);
• the noise, 5.4.9: the equivalent input noise voltage $U_{\mathrm{n}}=U{I}_{\mathrm{n}}/I$ and the signal-to-noise ratio $S=10\lg (I_{\mathrm{r}}^2/I_{\mathrm{n}}^2)$ dB (equivalent_input_noise_voltage, amplifier_signal_to_noise_ratio);
• the frequency response, 5.4.12: the load current at one-third-octave centres from at least 50 Hz to 8 kHz, with the response at 1 kHz taken as 0 dB (amplifier_frequency_response);
• the automatic gain control, 5.4.13: the steady-state output current against the source e.m.f., and the recommendation Amendment 1 added as 5.4.13.4, an input range of at least 32 dB for an output change of at most 3 dB (agc_characteristic);
• the phase error of the quadrature network of a phased loop array, 5.4.14: the maximum deviation from 90 degrees between the loop currents over 100 Hz to 5 kHz (quadrature_phase_error).

The rated conditions (5.4.1 to 5.4.6) are what the manufacturer states and are not measured; the standard measuring conditions of 5.2.2 put the output current 10 dB below the rated one. The test signal of 5.4.8.2 b) is the pink noise of IEC 60118-4 6.4, generated by loop_test_noise.

Clause 9, added by Amendment 1, specifies a neck loop by the input voltage that produces 400 mA/m at the telecoil position of the test jig of Annex E, the smallest magnitude of its input impedance over 100 Hz to 5 kHz rounded to the nearest ohm, and its frequency response, stated in text as the frequencies where it differs from the response at 1 kHz by 3 dB (neck_loop_characteristics). The draft Amendment 2 (prEN 62489-1:2010/prA2:2017, read in E DIN EN 62489-1/A2:2017-10) replaces Annex D with two example specifications: type 1, for audio sources powered by two primary 1,5 V cells and higher voltage supplies, by a DC resistance of 32 ohm plus or minus 5 %, and type 2, a neck loop with a transformer or an amplifier, by an input DC resistance of at least 32 ohm, both reaching 400 mA/m on the jig with at most 1,06 V at the input (NECK_LOOP_TYPES, verify_neck_loop). It is a draft, and this module cites it as one. Its Table D.1 (the field a type 1 loop reaches from a 3 V and a 9 V battery) rests on a source model the draft does not give and is not reproduced.

The loop listener and the assistive listening device of Annex F, and the monitoring devices of clause 10, are specified by values to be met (a sensitivity of 150 mV into 32 ohm for 400 mA/m, a volume range of 40 dB plus or minus 5 dB, a THD of 3 %) that are single comparisons with nothing to compute. The target frequency response of Annex F is stated by two corner frequencies and two final slopes, with its shape only in Figure F.1. The electromagnetic exposure of IEC 62489-2 is outside this library.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

agc_characteristic(
    source_emf_db: ArrayLike,
    output_level_db: ArrayLike,
) -> AgcCharacteristic

The output/input characteristic of an amplifier and its AGC range (5.4.13).

From standard measuring conditions the source e.m.f. is reduced to zero and raised in steps, the output current being read at each once it has settled (5.4.13.2), and the result is plotted with the e.m.f. on a logarithmic abscissa and the output current level in dB referred to the rated maximum output current (5.4.13.3). Amendment 1 adds the recommendation of 5.4.13.4: “an input level range of at least 32 dB for a maximum output level change of 3 dB”.

That range is read here as the widest span of input level over which the output varies by at most 3 dB, on the characteristic interpolated linearly between the measured steps, and found exactly: its ends are steps, or the inputs where the line crosses a step’s output level or that level 3 dB up or down. For a characteristic that rises and then holds, which is what an automatic gain control is (Annex A), it runs from the input where the output is 3 dB below its top to the highest input measured.

Parameters

Returns: An AgcCharacteristic.

AgcCharacteristic(
    source_emf_db: np.ndarray,
    output_level_db: np.ndarray,
    agc_range_db: float,
    agc_range_start_db: float,
    agc_range_end_db: float,
)

The steady-state output/input characteristic of an amplifier (5.4.13).

Attributes

AgcCharacteristic.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw output level against input level, as Figure A.1.

The span of agc_range_db is shaded, with the 3 dB output band it is taken over.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

property

The range 5.4.13.4 recommends, 32 dB, for comparison with agc_range_db.

amplifier_frequency_response(
    frequencies_hz: ArrayLike,
    output_current_a: ArrayLike,
) -> AmplifierFrequencyResponse

The frequency response of an amplifier into its rated load (5.4.12).

Under standard measuring conditions, with the automatic gain control and compression disabled or with the input the manufacturer states, the signal frequency is stepped through the one-third-octave centres “over the range of at least 50 Hz to 8 kHz” and the load current measured at each (5.4.12.2). The result is presented with the response at 1 kHz taken as 0 dB (5.4.12.3).

Parameters

Returns: An AmplifierFrequencyResponse.

amplifier_signal_to_noise_ratio(
    rated_current_a: float,
    noise_current_a: float,
) -> float

The signal-to-noise ratio of an amplifier (IEC 62489-1:2010 5.4.9.2).

the rated output current against the A-weighted output current with the source e.m.f. reduced to zero.

Parameters

Returns: The signal-to-noise ratio, in dB.

AmplifierFrequencyResponse(
    frequencies_hz: np.ndarray,
    output_current_a: np.ndarray,
    response_db: np.ndarray,
)

The frequency response of an amplifier into its load (IEC 62489-1:2010 5.4.12).

Attributes

AmplifierFrequencyResponse.at(frequency_hz: ArrayLike) -> np.ndarray

The response at any frequency inside the measured range, in dB.

Interpolated linearly against the logarithm of frequency, the way a response measured at one-third-octave centres is read between them.

Parameters

Returns: The response there, dB re 1 kHz.

Raises

AmplifierFrequencyResponse.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the response against frequency, 0 dB at 1 kHz (5.4.12.3).

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

compliance_voltage(voltage_v: ArrayLike) -> float

The compliance voltage of an amplifier (IEC 62489-1:2010 5.4.8).

The voltage across the rated load is recorded “over a period of at least 60 s” of the specified pink noise with the automatic gain control fully in operation, or with the RMS output current 6 dB below the maximum output current (5.4.8.2 c). The magnitudes of its most positive and most negative peaks are averaged and divided by $\sqrt{2}$ (5.4.8.2 d), which gives the RMS voltage of the sine with the same peaks.

Parameters

Returns: The compliance voltage, in volts.

Raises

equivalent_input_noise_voltage(
    source_emf_v: float,
    output_current_a: float,
    noise_current_a: float,
) -> float

The equivalent input noise voltage of an amplifier (IEC 62489-1:2010 5.4.9).

With the amplifier below the action of its automatic gain control, a 1 kHz source e.m.f. $U$ gives an output current $I$; with the e.m.f. reduced to zero the A-weighted output current is $I_{\mathrm{n}}$. The 1 kHz input that would give the same current as the noise is

Parameters

Returns: The equivalent input noise voltage, in volts.

loop_centre_field(
    current_a: float,
    length_m: float,
    *,
    width_m: float | None = None,
    turns: int = 1,
) -> float

The field at the centre of a loop, in its plane (IEC 60118-4:2014 E.1).

with $d$ the side of a square loop, or $\sqrt{d_1{d}_2}$ for a rectangle, which E.1 allows “Provided that the sides d1, d2 of a rectangular loop are not extremely different”. For the square it is exact; for a rectangle it is the approximation E.1 prints, and rectangular_loop_field gives the exact value, which is larger by $\sqrt{(d_1^2+d_2^2)/(2d_1{d}_2)}$: 0,35 dB at an aspect ratio of 1,5 and 3,3 dB at 4.

Parameters

Returns: The field strength at the centre, in A/m.

loop_current(
    length_m: float,
    width_m: float,
    *,
    height_m: float = 1.4,
    field_strength_a_per_m: float = 0.4,
    turns: int = 1,
) -> float

The current a loop needs for a field strength above its centre.

The exact on-axis field of the rectangle (see the module docstring), inverted for the current. The defaults are the conditions of IEC 62489-1:2010 5.4.10.2, 400 mA/m at 1,4 m above the centre of the loop plane, which is also where IEC 60118-4:2014 Figure H.1 plots the current required against the loop’s size and aspect ratio, and the curves of that figure are reproduced: 4,88 A for a 10 m square and 6,41 A for 10 m by 30 m, read as about 4,89 A and 6,41 A.

Parameters

Returns: The RMS current in each turn, in amperes.

loop_dimensions(
    current_a: float,
    *,
    aspect_ratio: float = 1.0,
    height_m: float = 1.4,
    field_strength_a_per_m: float = 0.4,
    turns: int = 1,
) -> tuple[float, float]

The largest loop an amplifier’s current drives to a field strength (5.4.11).

IEC 62489-1:2010 5.4.11.1 states “the linear dimensions of a square loop, and a loop of aspect ratio 3:1, for which the magnetic field strength, measured as specified in 5.4.10.1, is 400 mA/m”. Above the centre, the field of a given current first grows with the loop and then falls, so two sizes reach any field below the peak. The characteristic is the larger one: the biggest loop the amplifier can serve, which is the size a specification is read for.

Parameters

Returns: (short_side_m, long_side_m), in metres.

Raises

loop_impedance(
    resistance_ohm: float,
    inductance_h: float,
    *,
    frequencies_hz: ArrayLike | None = None,
) -> LoopImpedance

The impedance of a loop from its resistance and inductance.

Parameters

Returns: A LoopImpedance.

loop_resistance(
    perimeter_m: float,
    conductor_area_mm2: float,
    *,
    turns: int = 1,
    conductor_temperature_c: float = 20.0,
) -> float

The resistance of a copper loop (IEC 62489-1:2010 B.2).

$R=\rho Nl/a$, with the resistivity of standard annealed copper, 1/58 ohm square millimetre per metre at 20 degrees Celsius, carried to the conductor’s temperature by the coefficient 0,00393 per degree (IEC 60028:1925, clause I). B.2 quotes “approximately 0,017 ohm/m” for 1 mm² at 25 degrees Celsius; Table B.1 is computed at 20 degrees, where its six rounded resistances hold the resistivity between 1,716 and 1,738 times $10^{-8}$ ohm metres, and 1/58 is inside that range.

Parameters

Returns: The DC resistance of the loop, in ohms.

LoopField(
    current_a: float,
    length_m: float,
    width_m: float,
    turns: int,
    x_m: np.ndarray,
    y_m: np.ndarray,
    z_m: np.ndarray,
    h_x_a_per_m: np.ndarray,
    h_y_a_per_m: np.ndarray,
    h_z_a_per_m: np.ndarray,
)

The magnetic field of a rectangular induction loop at a set of points.

The loop lies in the plane $z=0$, centred on the origin, with its sides of length length_m along $x$ and width_m along $y$, and the current runs counterclockwise seen from $+z$, so the field inside the loop points to $+z$. For a horizontal loop on the floor $z$ is the height and h_z_a_per_m the vertical component a standing listener’s telecoil picks up; for a vertical loop on a counter the same arrays describe it turned on its side.

Attributes

LoopField.level_db(component: str = 'z') -> np.ndarray

The field strength level of one component, dB re 400 mA/m.

Parameters

Returns: $20\lg (|H|/0,4A/m)$; minus infinity where the component vanishes, which is the null line outside a loop.

property

The magnitude of the field strength vector, in A/m.

LoopField.plot(
    ax: Axes | None = None,
    *,
    component: str = 'z',
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the field strength level along the points, as Figures E.2 and E.3.

The abscissa is the coordinate that varies most among the points, and the ordinate the level of one component in dB re 400 mA/m, with the 0 dB reference and the plus or minus 3 dB band of IEC 60118-4:2014 8.4.3 around it.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

LoopImpedance(
    resistance_ohm: float,
    inductance_h: float,
    frequencies_hz: np.ndarray,
    impedance_ohm: np.ndarray,
)

The impedance of a loop, a resistance in series with an inductance.

IEC 62489-1:2010 5.4.3.2 states the rated load “as a series combination of resistance and inductance”, and B.2 represents the loop the same way. The magnitude $|Z|=\sqrt{R^2+(2\pi fL{)}^2}$ is $\sqrt{2}$ times the resistance at the corner frequency $f_{\mathrm{c}}=R/(2\pi L)$, “1,4 times” in IEC 60118-4:2014 E.3, and the voltage an amplifier has to deliver for a current $I$ is $U=I|Z|$, the $U_{\mathrm{h}}$ of E.3.

Attributes

LoopImpedance.at(frequency_hz: ArrayLike) -> np.ndarray

The magnitude of the impedance at any frequency, in ohms.

Parameters

Returns: $|Z|$ there, in ohms.

property

The frequency where the reactance equals the resistance, in hertz.

LoopImpedance.drive_voltage(
    current_a: float,
    frequency_hz: ArrayLike,
) -> np.ndarray

The RMS voltage that drives a current through the loop (IEC 60118-4 E.3).

Parameters

Returns: $U=I|Z|$, in volts.

LoopImpedance.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the magnitude of the impedance against frequency.

The resistance is drawn as a floor and the corner frequency marked, where the magnitude has risen to $\sqrt{2}$ times the resistance.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

maximum_output_current(
    resistor_voltage_v: ArrayLike,
    thd_percent: ArrayLike,
    *,
    load_resistance_ohm: float,
    rated_thd_percent: float,
) -> MaximumOutputCurrent

The maximum (distortion-limited) output current of an amplifier (5.4.7).

IEC 62489-1:2010 5.4.7.1 defines it as “The maximum current, produced by a sinusoidal input signal at 1 kHz, deliverable for at least 10 s into the rated load without exceeding the rated total harmonic distortion (THD)”. 5.4.7.2 measures it from standard measuring conditions: the load current is increased until the total harmonic distortion across the resistive part of the load equals the rated value, and “The current is then calculated from the total voltage across the resistive part of the load and the resistance value”. The 10 s the current has to be held for is a condition on the measurement, not a computation.

The steps of the level are given as the total RMS voltage across the resistance and the distortion read there, in ascending order of voltage; the current where the distortion first reaches the rated value is interpolated linearly between the two steps either side of it.

Parameters

Returns: A MaximumOutputCurrent.

Raises

MaximumOutputCurrent(
    resistor_voltage_v: np.ndarray,
    load_current_a: np.ndarray,
    thd_percent: np.ndarray,
    load_resistance_ohm: float,
    rated_thd_percent: float,
    maximum_current_a: float,
)

The maximum (distortion-limited) output current of an amplifier (5.4.7).

Attributes

MaximumOutputCurrent.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the distortion against the load current, with the rated THD.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

neck_loop_characteristics(
    frequencies_hz: ArrayLike,
    field_strength_level_db: ArrayLike,
    impedance_ohm: ArrayLike,
    *,
    input_voltage_v: float,
) -> NeckLoopCharacteristics

The input voltage, input impedance and frequency response of a neck loop.

The loop is arranged on the non-metallic jig of Annex E and the field measured at the telecoil position with a small inductor, about 3 mm by 10 mm, the size of a behind-the-ear telecoil (9.1.2). The rated input voltage is applied from a source of less than 0,4 ohm (9.3.2).

A passive loop is linear, so the input voltage for 400 mA/m at 1 kHz (9.1) follows from the level measured at any voltage: $U_{400}=U10^{-L/20}$, with $L$ the level at 1 kHz in dB re 400 mA/m. For an active loop that is not linear, measure at the voltage that gives 0 dB, where the scaling does nothing.

Parameters

Returns: A NeckLoopCharacteristics.

Constant (mapping).

NECK_LOOP_TYPES = {1: NeckLoopType(description='for audio sources powered by two primary 1.5 V cells and higher voltage supplies', min_dc_resistance_ohm=30.4, max_dc_resistance_ohm=33.6, max_input_voltage_v=1.06), 2: NeckLoopType(description='neck loop with a transformer or an amplifier', min_dc_resistance_ohm=32.0, max_dc_resistance_ohm=inf, max_input_voltage_v=1.06)}

NeckLoopCharacteristics(
    frequencies_hz: np.ndarray,
    field_strength_level_db: np.ndarray,
    impedance_ohm: np.ndarray,
    input_voltage_v: float,
    reference_input_voltage_v: float,
    minimum_impedance_ohm: float,
    response_db: np.ndarray,
    frequencies_3db_hz: tuple[float, ...],
)

What IEC 62489-1:2010+A1:2014 clause 9 states for a neck loop.

Attributes

NeckLoopCharacteristics.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the frequency response on the jig, 0 dB at 1 kHz.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

NeckLoopType(
    description: str,
    min_dc_resistance_ohm: float,
    max_dc_resistance_ohm: float,
    max_input_voltage_v: float,
)

One neck-loop type of the draft Annex D (prEN 62489-1:2010/prA2:2017).

Attributes

NeckLoopVerification(
    neck_loop_type: int,
    dc_resistance_ohm: float,
    input_voltage_v: float,
)

A neck loop judged against one type of the draft Annex D (prA2:2017).

Attributes

The limits of the type (limits) are read from NECK_LOOP_TYPES, so a verification cannot be built against other limits.

property

The NeckLoopType judged against.

property

Whether the loop meets both limits of its type.

NeckLoopVerification.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw each measured value as a share of its limit.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

property

Whether the DC resistance is inside the type’s range.

property

Whether 400 mA/m is reached within the type’s input voltage.

quadrature_phase_error(
    frequencies_hz: ArrayLike,
    phase_difference_deg: ArrayLike,
) -> QuadraturePhaseError

The maximum deviation from 90 degrees between two loop currents (5.4.14).

The phase angle between the currents of the two loops, measured across a low-value series resistor in each (5.4.14.3), is judged over 100 Hz to 5 kHz (5.4.14.2). Either loop may lead: an angle is folded into 0 to 180 degrees before its distance from 90 degrees is taken, so -90 and 270 degrees are as much in quadrature as 90.

Parameters

Returns: A QuadraturePhaseError.

QuadraturePhaseError(
    frequencies_hz: np.ndarray,
    phase_difference_deg: np.ndarray,
    deviation_deg: np.ndarray,
    max_deviation_deg: float,
    max_deviation_frequency_hz: float,
)

The phase error of a quadrature network for a phased loop array (5.4.14).

Two adjacent loops fed with currents about 90 degrees apart make a field whose direction rotates, instead of one that cancels at the nulls. A deviation $\delta$ from 90 degrees leaves an in-phase component $\cos (90^{\circ }-\delta )=\sin \delta$ of one field against the other, which raises the level where the two add and lowers it where they subtract, by $20\lg (1\pm \sin \delta )$ dB (5.4.14.1: at 85 degrees the in-phase part is $\cos 85^{\circ }=0,087$).

Attributes

property

Where they subtract, the level falls by this, in dB (a negative number).

$20\lg (1-\sin \delta )$, computed as $20\lg \left(2{\sin }^2(45^{\circ }-\delta /2)\right)$, which is the same number without the cancellation near 90 degrees. At a deviation of 90 degrees, currents in phase or in antiphase, the two fields cancel where they subtract and the fall is minus infinity.

property

Where the in-phase parts add, the level rises by this, in dB.

QuadraturePhaseError.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the deviation from 90 degrees against frequency.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

rectangular_loop_field(
    current_a: float,
    length_m: float,
    width_m: float,
    x_m: ArrayLike,
    y_m: ArrayLike,
    z_m: ArrayLike,
    *,
    turns: int = 1,
) -> LoopField

The magnetic field of a rectangular loop, by Biot and Savart.

Each of the four sides is a straight filament whose field closes exactly (see the module docstring), so the result carries no discretisation. It reproduces the curves of Figure E.2 b) of IEC 60118-4:2014 across the width of that figure’s loop, it is the kind of pattern Figures E.3 to E.5 draw, and at the centre of a square it is the $2\sqrt{2}I/(\pi d)$ of E.1. The conductor is a filament, so the field is that of a thin wire everywhere except within a few wire radii of it. Metal in the building, which Annex F describes, is not modelled.

Parameters

Returns: A LoopField.

Raises

rectangular_loop_inductance(
    length_m: float,
    width_m: float,
    conductor_area_mm2: float,
    *,
    turns: int = 1,
    internal_inductance: bool = True,
) -> float

The inductance of a rectangular loop of round wire (Grover, Formula (58)).

for sides $a$ and $b$ and a wire of radius $\rho$ (Grover, Inductance Calculations, 1946, p. 60; the factor 0,004 microhenry per centimetre there is ${\mu }_0/\pi$). The last term is the flux inside the wire, $\mu =1$ for copper. For $N$ turns the single-turn value is multiplied by $N^2$, the perfect coupling IEC 62489-1:2010 Table B.1 assumes.

Table B.1 is reproduced without the internal term (internal_inductance=False): the counter loop, the home loop, the small room and the typical place of worship come out at 189, 22, 47 and 109 microhenries, as printed. With the term they are about 3 % higher, from 2,7 % for the place of worship to 4,2 % for the counter loop. At audio frequencies the current fills the wire (the skin depth in copper is 2 mm at 1 kHz), which is why the term is included by default.

Parameters

Returns: The inductance, in henries.

Raises

verify_neck_loop(
    dc_resistance_ohm: float,
    input_voltage_v: float,
    *,
    neck_loop_type: int = 1,
) -> NeckLoopVerification

Does a neck loop meet one type of the draft Annex D (prA2:2017)?

The draft Amendment 2 to IEC 62489-1 (prEN 62489-1:2010/prA2:2017, read in E DIN EN 62489-1/A2:2017-10) specifies a type 1 neck loop, for audio sources powered by two primary 1,5 V cells and higher voltage supplies, by a DC resistance of 32 ohm plus or minus 5 % and a type 2 neck loop with a transformer or an amplifier by an input DC resistance of at least 32 ohm, both reaching 400 mA/m on the jig of Annex E with at most 1,06 V at the input (NECK_LOOP_TYPES). It is a draft, cited as one.

Parameters

Returns: A NeckLoopVerification.