Source: https://jmrplens.github.io/phonometry/devices/electroacoustics/loudspeakers/ # Loudspeaker Characterisation (IEC 60268-5) A loudspeaker datasheet is a bundle of conventions, and IEC 60268-5 is the standard that fixes them: what "sensitivity" refers to, over which band a frequency range is read, how directivity is plotted and at what drive the distortion is measured. This guide covers the chain phonometry implements around a measured on-axis response: the **sensitivity conventions** where cross-datasheet comparisons usually go wrong, the baffled **radiating piston** that is the canonical physical model behind a cone's radiation impedance and directivity, and the **rated-characteristics report** that renders the standard's data sheet, with every panel drawn to the IEC 60263 scale conventions. The distortion and frequency-response measurements that feed it live in [Electroacoustics](https://jmrplens.github.io/phonometry/devices/electroacoustics/electroacoustics/); the microphone counterpart has [its own guide](https://jmrplens.github.io/phonometry/devices/electroacoustics/microphones/). ## 1. Sensitivity conventions (IEC 60268-5) Distortion and response figures only compare across devices when the *sensitivity* conventions behind them match, and two conventions trip people constantly: the microphone reference level and the loudspeaker power-and-distance normalization. **Loudspeakers (IEC 60268-5 clause 20.3).** The characteristic sensitivity is the sound pressure produced at 1 m on the reference axis, in the free field, referred to an input of 1 W into the rated impedance; expressed as a level re 20 µPa (clause 20.4) it is the familiar "dB @ 1 W/1 m". Two normalizations hide in that sentence: - *The electrical one.* The test voltage is $U_p = \sqrt{R \cdot 1\,\text{W}}$, numerically $\sqrt{R}$: 2.83 V into a rated 8 Ω. A datasheet quoting "dB @ 2.83 V/1 m" for a 4 Ω loudspeaker is feeding it 2 W, which flatters the figure by 3 dB against a true 1 W/1 m rating; check the rated impedance before comparing. - *The geometric one.* 1 m is a **reference** distance, not necessarily the measurement distance. The standard has you measure in the far field (at 0.5 m or an integer number of metres, clause 7.1) and refer the result back with the inverse-distance law, $L_p(1\ \text{m}) = L_p(r) + 20\log_{10}(r / 1\ \text{m})$. That scaling only holds where the level actually falls 6 dB per doubling of distance, hence the free field of the diagram; at 1 m from a large multi-way cabinet the near field may not have ended yet, and the quoted "1 m" figure is then a referred quantity, not what a microphone placed at 1 m would read. The microphone counterpart, the sensitivity level re 1 V/Pa and its reference-value pitfall, is covered in [Microphone Characterisation (IEC 60268-4)](https://jmrplens.github.io/phonometry/devices/electroacoustics/microphones/). ## 2. Radiating piston: radiation impedance and directivity The rigid circular **piston in an infinite baffle** is the canonical radiator behind a loudspeaker cone, the open end of a duct and the radiation efficiency of any finite vibrating surface (Beranek & Mellow §4.19, §13.7). Its mechanical radiation impedance is $Z_\mathrm{r} = \rho c S\,(R_1 + jX_1)$ with $S = \pi a^2$ and the dimensionless resistance and reactance functions (Eqs. (13.117), (13.118)) $$ R_1(x) = 1 - \frac{2 J_1(x)}{x}, \qquad X_1(x) = \frac{2 H_1(x)}{x}, \qquad x = 2ka, $$ with $J_1$ the Bessel function and $H_1$ the Struve function of order one. At low frequency $R_1 \to (ka)^2/2$ and the reactance is mass-like with the radiation mass $M_\mathrm{r} = 8\rho a^3/3$ (Eq. (4.151)); at high frequency $R_1 \to 1$, $X_1 \to 0$ and the piston radiates as into an infinite tube. The far field follows the directivity $D(\theta) = 2 J_1(ka\sin\theta)/(ka\sin\theta)$, whose first null is at $ka\sin\theta = 3.8317$. ```python import numpy as np from phonometry import radiating_piston res = radiating_piston(radius=0.1, frequencies=np.geomspace(20, 20000, 200), angles=np.linspace(0.0, np.pi / 2, 91)) print(round(res.radiation_mass, 4)) # 8 rho a^3 / 3, kg print(round(float(res.directivity_index[0]), 2)) # 3.01 dB half-space limit res.plot() # R1 and X1 vs ka ``` *The `.plot()` of the result is the classic Beranek & Mellow impedance figure computed for this piston: below $ka \approx 1$ the load is almost purely mass-like ($X_1$ dominates and $R_1 \propto (ka)^2$, the regime where a loudspeaker's output is stiffness- and mass-limited), and above it the resistance settles on $\rho c S$ with interference ripples while the reactance dies away.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import radiating_piston res = radiating_piston(radius=0.075, frequencies=np.geomspace(20, 20000, 400)) res.plot() # normalized R1 and X1 against ka plt.show() ```

`radiating_piston` returns a `RadiatingPistonResult` with the normalized `resistance`/`reactance`, the mechanical `radiation_resistance`/`radiation_reactance`, the `radiation_mass`, the `directivity_index`, the far-field `directivity` pattern (when `angles` are given) and `.plot()`. The building blocks `piston_resistance`, `piston_reactance` and `piston_directivity` are also callable directly. The piston is the companion radiator of the [reactive silencers](https://jmrplens.github.io/phonometry/devices/noise-control/silencers/). The far-field **directivity pattern** is a plottable result in its own right: `piston_directivity_pattern(ka)` samples $D(\theta)$ at one or more $ka$ values and returns a `PistonDirectivity` (the polar angle grid, the linear `directivity` and its dB form `directivity_db`, and the `ka` values) whose `.plot()` draws the classic polar beam pattern. Several $ka$ are shown as one family, so the main lobe narrowing and the side lobes appearing read at a glance: ```python from phonometry import piston_directivity_pattern pattern = piston_directivity_pattern([3.0, 8.0, 16.0]) print(pattern.directivity_db.shape) # (3, 361): one row per ka pattern.plot() # polar beam pattern in dB (needs matplotlib) ```

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```python import matplotlib.pyplot as plt from phonometry import piston_directivity_pattern piston_directivity_pattern([3.0, 8.0, 16.0]).plot() plt.show() ```

Behind both curves sits the same physical object, and `.plot_geometry()` on the piston result draws it: the plate in its rigid baffle to scale, with the far-field lobe of the highest computed frequency overlaid on the radiation side. *The piston the impedance and directivity curves describe, to scale: at 4 kHz this 10 cm radius plate has $ka \approx 7.3$, so the overlaid lobe is already several times narrower than the low-frequency hemisphere and the first side lobes have appeared.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import radiating_piston res = radiating_piston(0.1, np.array([500.0, 2000.0, 4000.0]), angles=np.linspace(-np.pi / 2, np.pi / 2, 181)) # One line: the piston in its baffle with the lobe of the highest frequency. res.plot_geometry() plt.show() ```

## 3. Loudspeaker characteristics report (IEC 60268-5) The rated characteristics IEC 60268-5 defines around a measured on-axis response gather into a single **loudspeaker characteristics** result that renders the standard's rated-characteristics data sheet. Two of the numbers are computed from the response rather than merely repeated: - **Characteristic sensitivity level** (20.3/20.4). The on-axis level averaged over a stated band, referred to 1 W into the rated impedance $R$ at 1 m: $L_M = \overline{L} + 20\log_{10}(d/d_0) + 20\log_{10}(U_p/U)$ with $U_p = \sqrt{R\cdot 1\,\text{W}} = \sqrt R$. The default drive is that $\sqrt R$ (2.83 V into 8 Ω), so with a 1 m response the sensitivity level is the band mean. - **Effective frequency range** (21.2). The band over which the response stays within 10 dB of the level averaged over the one-octave band in the region of maximum sensitivity; troughs narrower than 1/9 octave are neglected. ```python import numpy as np from phonometry import ( LoudspeakerDirectivity, ReportMetadata, loudspeaker_characteristics, radiating_piston, ) freqs = np.geomspace(30, 24000, 320) spl = 87.0 + 1.2 * np.sin(2 * np.log2(freqs / 900.0)) spl -= 10 * np.log10(1 + (50.0 / freqs) ** 6) # low-frequency roll-off spl -= 10 * np.log10(1 + (freqs / 16000.0) ** 7) # high-frequency roll-off result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), impedance=(np.geomspace(20, 20000, 260), 6.6 + 24 * np.exp(-(np.log2(np.geomspace(20, 20000, 260) / 52.0) ** 2) / 0.12)), distortion=(np.geomspace(50, 5000, 140), 0.3 + 2.6 * np.exp(-(np.log2(np.geomspace(50, 5000, 140) / 70.0) ** 2) / 0.45)), directivity=LoudspeakerDirectivity( piston=radiating_piston(0.075, np.array([1000.0, 2000.0, 4000.0]), angles=np.radians(np.linspace(0, 90, 46))), frequency=2000.0, ), ) print(round(result.sensitivity_level_db, 1)) # dB, 1 W / 1 m print(tuple(round(x) for x in result.effective_range)) # Hz result.report("loudspeaker.pdf", metadata=ReportMetadata(measurement_standard="IEC 60268-5")) ``` `loudspeaker_characteristics` returns a `LoudspeakerCharacteristics` with the computed `sensitivity_level_db`, `effective_range`, `reference_level_db`, `characteristic_sensitivity_pa` and `minimum_impedance`, and a `.report()` that writes the fiche. The on-axis response is drawn with its tolerance band and the effective-range markers to the IEC 60263 proportion (one frequency decade equal to 25 dB), and the polar directivity on the IEC 60263 25 dB reference circle. The impedance modulus (with the 80 %-of-rated line), the total-harmonic-distortion curve and the directivity feed the secondary panels, reusing the radiating piston directivity of section 2 and a [swept-sine THD](https://jmrplens.github.io/phonometry/devices/electroacoustics/swept-sine-distortion/) result for the distortion curve. None of the standard's measuring conditions is checked here: the functions reduce and report the curves they are handed, so the mounting, the excitation, the gate and the field condition are the operator's to keep and to state with the result. Thiele-Small parameter extraction from the impedance curve is not implemented either: the impedance panel reports the measured modulus against the rated-impedance lines. The rated characteristics are also available interactively through `.plot()`, which draws **one concept per figure** with the same panel code the report composes, selected by `quantity`. Passing an axes draws on it: ```python result.plot() # on-axis response (default) result.plot(quantity="impedance") # |Z| modulus with the rated / 80 % lines result.plot(quantity="thd") # total harmonic distortion vs frequency result.plot(quantity="directivity") # polar response on the 25 dB circle ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics freqs = np.geomspace(30, 24000, 320) spl = 87.0 + 1.2 * np.sin(2 * np.log2(freqs / 900.0)) spl -= 10 * np.log10(1 + (50.0 / freqs) ** 6) # low-frequency roll-off spl -= 10 * np.log10(1 + (freqs / 16000.0) ** 7) # high-frequency roll-off result = loudspeaker_characteristics(freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0)) result.plot() # quantity="response" (the default) plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics freqs = np.geomspace(30, 24000, 320) spl = 87.0 - 10 * np.log10(1 + (50.0 / freqs) ** 6) fz = np.geomspace(20, 20000, 260) result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), impedance=(fz, 6.6 + 24 * np.exp(-(np.log2(fz / 52.0) ** 2) / 0.12)), ) result.plot(quantity="impedance") plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics freqs = np.geomspace(30, 24000, 320) spl = 87.0 - 10 * np.log10(1 + (50.0 / freqs) ** 6) thd_f = np.geomspace(50, 5000, 140) result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), distortion=(thd_f, 0.3 + 2.6 * np.exp(-(np.log2(thd_f / 70.0) ** 2) / 0.45)), ) result.plot(quantity="thd") plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import ( LoudspeakerDirectivity, loudspeaker_characteristics, radiating_piston, ) freqs = np.geomspace(30, 24000, 320) spl = 87.0 - 10 * np.log10(1 + (50.0 / freqs) ** 6) result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), directivity=LoudspeakerDirectivity( piston=radiating_piston(0.075, np.array([1000.0, 2000.0, 4000.0]), angles=np.radians(np.linspace(0, 90, 46))), frequency=2000.0, ), ) result.plot(quantity="directivity") plt.show() ```

The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iec60268_5_loudspeaker_example.pdf) *Loudspeaker characteristics fiche (`LoudspeakerCharacteristics.report`), the IEC 60268-5 rated-characteristics table beside the on-axis response, impedance, THD and polar panels drawn to the IEC 60263 25 dB-per-decade and 25 dB reference-circle conventions.* ## 4. Gain before feedback of a reinforcement system Everything above characterises a loudspeaker on its own. Put it in a room with an open microphone and the interesting quantity is no longer a datasheet number but a **closed loop**: the loudspeaker feeds the audience, and it also feeds the microphone that drives it. Long (*Architectural Acoustics* 2e, Chapter 18, Equations (18.13) to (18.24)) writes that loop with two decibel gains: $$ \begin{aligned} Z_\mathrm{S} &= L_{\mathrm{H}\text{-}\mathrm{L}} - L_{\mathrm{T}\text{-}\mathrm{M}} && \text{(Eq. (18.17)) open-loop system gain} \\ G_\mathrm{S} &= L_{\mathrm{H}\text{-}\mathrm{M}} - L_{\mathrm{H}\text{-}\mathrm{L}} + D_\mathrm{M}(\theta) && \text{(Eq. (18.18)) feedback-loop gain} \end{aligned} $$ $Z_\mathrm{S}$ is the level the loudspeaker produces at an average listener minus the level the talker produces at the microphone; $Z_\mathrm{S} = -6$ dB, typical of an auditorium or a church, means a comfortable conversational level at twice the talker-to-microphone distance. $D_\mathrm{M}(\theta)$ is the directivity index of the microphone toward the loudspeaker *relative to* the talker: zero for an omnidirectional microphone, about −2 to −3 dB for a cardioid pointed at the talker. Summing the infinite series of round trips, the system oscillates when the loop gain reaches unity, that is $Z_\mathrm{S} + G_\mathrm{S} = 0$ (Equation (18.16)). *The four points of the loop. Only the sound field the loudspeaker produces at two of them, and the type and orientation of the microphone, enter the criterion: the loudspeaker's own type, number and power do not appear anywhere in it.* A working system needs margin below that threshold. Long takes **10 dB** for an equalised system (other authors quote 12 dB unequalised and 6 dB carefully equalised): 6 dB of it covers a sustained tone adding in phase with a reflection off a hard surface, and the remaining 4 dB is safety. With several microphones open at once the returned signals add at the mixer, which the number-of-open-microphones correction $\Delta L_\text{nom} = 10\log_{10} N_\mathrm{m}$ accounts for (Equation (18.23)). The criterion is then $$ Z_\mathrm{S} + L_{\mathrm{H}\text{-}\mathrm{M}} + \Delta L_\text{nom} \le L_{\mathrm{H}\text{-}\mathrm{L}} - D_\mathrm{M}(\theta) - 10. $$ ```python from phonometry import electroacoustics as ea # An auditorium at Zs = -6 dB, omnidirectional microphone, one mic open. res = ea.feedback_stability(-6.0, 76.0, 80.0) print(res.is_stable, round(res.headroom, 1)) # True 0.0 dB print(round(res.maximum_level_at_microphone, 1)) # 76.0 dB = L(H-L) - 4 # A cardioid buys back exactly its relative directivity index. card = ea.feedback_stability(-6.0, 78.0, 80.0, microphone_directivity=-2.0) print(round(card.maximum_level_at_microphone, 1)) # 78.0 dB = L(H-L) - 2 # Four open microphones cost 10 lg 4 = 6 dB of that headroom. print(round(ea.open_microphone_correction(4), 1)) # 6.0 dB res.plot() # the gain structure against the oscillation and margin lines ``` Those two special cases are Long's own (Equations (18.21) and (18.22)): with $Z_\mathrm{S} = -6$ dB the criterion collapses to $L_{\mathrm{H}\text{-}\mathrm{M}} \le L_{\mathrm{H}\text{-}\mathrm{L}} - D_\mathrm{M}(\theta) - 4$, so an omnidirectional microphone must see the loudspeaker 4 dB below the average level in the audience, while a cardioid may let it rise to 2 dB below. > **A sign slip in the printed equation.** Long prints Equation (18.24) with > `+ D_M(theta)` on the right-hand side, which contradicts the Equations (18.20) to > (18.22) it generalises and would turn the benefit of a directional microphone > into a penalty. `feedback_stability` implements the sign of Equation (18.20), > so that $N_\mathrm{m} = 1$ reproduces Long's own special cases exactly. The defect is > recorded in the [errata registry](https://jmrplens.github.io/phonometry/reference/errata/). ### `feedback_stability()` parameters | Parameter | Type | Units | Range / default | Notes | | :--- | :--- | :--- | :--- | :--- | | `open_loop_gain` | float | dB | finite | $Z_\mathrm{S} = L_{\mathrm{H}\text{-}\mathrm{L}} - L_{\mathrm{T}\text{-}\mathrm{M}}$; about −6 dB in an auditorium | | `level_loudspeaker_at_microphone` | float | dB | finite | $L_{\mathrm{H}\text{-}\mathrm{M}}$ | | `level_loudspeaker_at_listener` | float | dB | finite | $L_{\mathrm{H}\text{-}\mathrm{L}}$ | | `microphone_directivity` | float | dB | default 0 | $D_\mathrm{M}(\theta)$, 0 omni, about −2 cardioid | | `open_microphones` | int | — | ≥ 1, default 1 | $N_\mathrm{m}$ | | `stability_margin` | float | dB | ≥ 0, default 10 | Long's equalised-system margin | Returns a `FeedbackStabilityResult` (`open_loop_gain`, `feedback_loop_gain`, `nom_correction`, `loop_gain`, `margin`, `headroom`, `is_stable`, `maximum_open_loop_gain`, `maximum_level_at_microphone`) with `.plot()`. `feedback_loop_gain` and `open_microphone_correction` are callable directly, and `plot_sound_reinforcement_geometry` draws the loop above. The criterion is a level bookkeeping exercise, not an acoustic model: it consumes the two direct-field levels the loudspeaker produces and does not compute them from a coverage pattern, nor does it predict the ring frequency or the effect of an equaliser or frequency shifter. Long excludes the reverberant field deliberately, because a uniform field cannot depend on where the microphone is or where it points; the direct-to-reverberant ratio enters the design separately. ## See also - [Electroacoustics](https://jmrplens.github.io/phonometry/devices/electroacoustics/electroacoustics/): the IEC 60268-3 distortion set, THD+N and SINAD, intermodulation and the $H_1$/$H_2$ frequency-response estimators that produce the curves this guide reports. - [Microphone Characterisation (IEC 60268-4)](https://jmrplens.github.io/phonometry/devices/electroacoustics/microphones/): the companion rated-characteristics report for the measuring side of the chain. - [Swept-sine distortion](https://jmrplens.github.io/phonometry/devices/electroacoustics/swept-sine-distortion/): the Farina/Novak harmonic separation whose THD(f) feeds the distortion panel. - [Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/): the radiated-power descriptor, for when the question is how much acoustic power the source emits rather than how it responds on axis. - API reference: [`electroacoustics.piston`](https://jmrplens.github.io/phonometry/reference/api/electroacoustics/piston/) and [`electroacoustics.loudspeaker`](https://jmrplens.github.io/phonometry/reference/api/electroacoustics/loudspeaker/). ## References - Beranek, L. L., & Mellow, T. J. (2012). *Acoustics: Sound fields and transducers*. Academic Press. ISBN 978-0-12-391421-7. [doi:10.1016/C2011-0-05897-0](https://doi.org/10.1016/C2011-0-05897-0). The transducer physics behind sections 1 and 2: loudspeaker radiation, the baffled-piston impedance and directivity (§4.19, §13.7), and the near-field to far-field transition. - Long, M. (2014). *Architectural acoustics* (2nd ed.). Academic Press. [doi:10.1016/C2012-0-03257-5](https://doi.org/10.1016/C2012-0-03257-5). Chapter 18 (the gain-before-feedback criterion of section 4: the open-loop and feedback-loop gains, the stability margin and the number-of-open-microphones correction, Equations (18.13) to (18.24)). ## Standards IEC 60268-5:2003+A1:2007, *Sound system equipment – Part 5: Loudspeakers*: the rated loudspeaker characteristics of section 3, namely the rated impedance (16), the rated frequency range (19.1), the characteristic sensitivity and its level referred to 1 W at 1 m (20.3/20.4), the effective frequency range against the −10 dB band (21.2), the directivity index (23.3) and the total harmonic distortion against frequency (24.1). IEC 60263:1982, *Scales and sizes for plotting frequency characteristics and polar diagrams*: the scale proportions of the characteristic graphs, one frequency decade equal to 25 dB on the ordinate (clause 2) and the polar diagram on a 25 dB reference-circle radius (clause 3). The piston quantities are verified against exact analytic oracles (the low- and high-frequency limits of the radiation impedance and the half-space directivity index).