© The Author(s), Published by EDP Sciences, 2026

1 Introduction

The development of techniques aiming at controlling the directivity of electroacoustic transducers is conventionally driven by (spatial) audio motivations [1], such as improving sound coverage over a targeted zone [2, 3], spatially localizing sound sources [4, 5] or avoiding back reflections in sound reinforcement contexts [6]. Similar techniques are also deployed in active noise cancellation contexts, either to separate the upstream and downstream sound fields on the sensor’s side [7], or to directionally steer noise reduction performance on the actuator’s side [8–12]. Furthermore, controlling the directivity of sound sources is particularly timely in the context of active metasurfaces achieving anomalous refraction of sound [13, 14], active non-reciprocal metamaterials [15–17], and even at the heart of the Willis coupling which has attracted a significant attention in recent years [18–21]. Although all directivity control techniques generally apply to both sensors (microphones) and actuators (loudspeakers), the focus will be put on loudspeaker systems in the following.

Loudspeakers directivities are inherently dependent on frequency. Generally a loudspeaker system (driver + cabinet) is considered almost omnidirectional for wavelengths that are larger than the largest dimension of the loudspeaker system. But as frequency increases above the k d_max = 1 “diffraction limit” (where d_max is the largest dimension and k the wavenumber), diffraction enters into play and yields frequency (relative to size)-dependent directivity patterns, that are inherently uncontrollable (as fixed by design) [22]. That means a given construction inevitably yields a given directivity. Besides resorting to additional waveguides and horn radiators, likely to increase the loudspeaker systems dimensions [23], several techniques have been developed in a view to controlling the directivity of electroacoustic transducers, provided the wavelength of operation is larger than the individual transducer dimensions. The most widespread delay-and-sum beamforming (DSB) consists in feeding a line (or surface) array of N loudspeakers with time-delayed versions of a given audio signal in order to steer sound power towards a prescribed direction in space [23, 24]. By deploying sub-arrays, it is even possible to extend the bandwidth of operation, but still limited by the system dimensions [25]. Parametric arrays leverage ultrasound focusing, relying on tiny ultrasound speakers (the reduced size of which allows covering a rather large frequency bandwidth), mapping the audible range by modulating two ultrasound signals of slightly different frequencies [26]. Although performing in achieving highly directional sound beams, these techniques require a significant processing power. Therefore, much simpler directivity control configurations, such as gradient loudspeakers resorting to only a pair of sound sources, allow achieving unidirectional directivities [27] although still governed by the diffraction limit. The following will rely on the latter technique as a simple use-case of directivity control with Corona Discharge Transducers (CDTs).

The CDT reported in reference [28] consists of an ultra-thin layer of partially-ionized air particles that is put in motion within an intense, oscillating electric field, allowing moving the surrounding medium and generating sound without resorting to a physical mechanical radiator. This transduction principle is characterized by a combination of two sound sources: one monopolar source resulting from local heat release of the corona discharge, and one dipolar source linked to the back and forth electrostatic force. Although the two sources are interdependent (and proportional to the driving voltage, with fixed relative strengths), the very thin dimension of the CDT makes them almost coincident, making this type of loudspeaker an almost frequency-independent unidirectional sound source along the axis normal to the radiating surface (however, its lateral geometry and dimensions inevitably yield a prescribed directivity). We will show in the following how stacking two adjacent CDTs fed with different input voltages allows independently controlling the monopolar and dipolar sources. The latter configuration then allows achieving interesting directivity patterns, still outperforming conventional loudspeakers owing to their very thinness.

This paper reports a concept of dual CDTs achieving controllable unidirectional patterns over a broad frequency range. In a first section, the fundamental principle of the CDT is reminded, followed by the formulation of the sound field generated by a dual CDTs. Then, numerical simulations are provided, which are finally experimentally verified on a laboratory prototype. Concluding remarks lead to discussion on the applicability of the dual CDTs to directional audio, as well as active noise control [29] and as a unit-cell for active metamaterials and metasurfaces [30].

2 The corona discharge transducer

2.1 Description of the transduction principle

Corona discharge transducers (CDTs) allow for sound generation thanks to the ionization of molecules in the surrounding fluid, moved by an intense oscillating electric field. In the “wire-to-grid” configuration, the transducer is constituted by a pair of electrodes (forming two parallel planes, as illustrated in the sketch of Fig. 1): the first one is a perforated metallic grid electrically connected to the ground, and is designated the “collector electrode”; the second one, designated “corona electrode”, is made of an ultra thin wire (diameter of the order of 100 μm), arranged in a pattern of parallel lines along a same plane parallel and distant of d to the collector plane, put at a sufficiently high voltage so as to trigger the extraction of electrons from surrounding molecules. The breakdown voltage U0, above which ionization occurs, depends on the geometry of the CDT (inter-electrode distance, corona wires diameter) and on the medium properties (relative permittivity, density, temperature, etc.), and is of the order of a few kVs in our study. As an order of magnitude, for a square CDT with a width in the range of 100 mm, a corona wire diameter of 0.1 mm, a distance between two consecutive corona wires in the range of 10 mm, and the gap between the collector and the corona wires in the range of 5 mm, the breakdown voltage in the air is in the range of 6 kV.

Figure 1. Schematic cut view of the electrode pair: the corona electrode consisting of an array of ultra-thin emitter wires parallel to each others (thin dots in this cut view), and a collector electrode consisting of a grounded perforated grid (dashed thick line in this cut view).

When applying an offset voltage U_DC higher than U₀, the positive ions are accelerated from the ionization region (a few μm around the corona wires) towards the collector electrode. They then collide with the neutral particles present in the drift region (the inter-electrode space), which are then moved, giving rise to an “ionic wind” (constant flow). If an alternating voltage u_ac(t) is superimposed to U_DC (ensuring total voltage U_DC + u_ac remains higher than the breakdown voltage), the transducer makes the surrounding fluid medium oscillate, responsible for sound generation.

The preceding studies [28, 31, 32] have shown that this corona discharge transducer configuration could be modeled as a combination of two volumetric sound sources, one monopolar “heat” source H (due to the heat release around the corona electrodes), and another dipolar “force” source F (relative to the electrostatic force moving the surrounding fluid back and forth), the orientation of which depends on the sign of the ac voltage: a positive u_ac yields a force oriented towards the collector electrode and vice-versa.

An extremely simple electroacoustic model can be deduced from the coupling equations between the plasma generation, and the two sound sources strengths. Indeed, it has been shown that the offset voltage U_DC can be linked to the current I flowing through the corona wire according to the Townsend formula [32, 33]:

$$\begin{array}{c}\hfill I=C{U}_{\mathrm{DC}}(U_{\mathrm{DC}}-U_0)\end{array}$$(1)

where C is a constant that depends on the transducer geometry, and that can be experimentally determined. Note that equation (1) still holds for oscillating voltages, when substituting U_DC + u_ac for U_DC. It is then possible to express the two sources strengths H and F as a function of the voltage feeding the transducer [28, 30, 32]. Assuming the ac voltage is much lower than the DC one, it is possible to linearize the expression of equation (1) and express linear relationships between the heat source power H (in W) and bulk electrostatic force F (in N) as:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H& =C(3U_{\mathrm{DC}}^2-2U_{\mathrm{DC}}{U}_0)u_{\mathrm{ac}}\hfill \\ \hfill \overrightarrow{F}& =\frac{C\overrightarrow{d}}{{\mu }_{\text{ion}}}(2U_{\mathrm{DC}}-U_0)u_{\mathrm{ac}}\hfill \end{array}\end{array}$$(2)

where μ_ion designates the mobility of the ions, and $\overrightarrow{d}$ is the vector orthogonal to the electrode planes, defined from the corona electrode to the collector electrode ($d=\Vert \overrightarrow{d}\Vert$ the plasma thickness).

It can be observed that the two sound sources are interdependent. Their individual influence is independent on frequency but on the constant parameters U₀, U_DC, C, d and μ_ion, and indirectly on the transducer cross-section area S. Also, the lateral dimensions of the CDT are likely to change the current-voltage law (Eq. (1)), thus C.

For the numerical study of Section 4, we will define the heat source power density $h=\frac{H}{S.d}=h_0{u}_{\mathrm{ac}}$ (in W/m³), and the dipolar force source density $\varphi =\frac{\Vert \overrightarrow{F}\Vert }{S.d}=f_0{u}_{\mathrm{ac}}$ (in N/m³), where S is the transducer effective cross-section area.

2.2 Sound field generated by a single CDT

Let’s consider a square CDT of cross-section area S = w ² where w is the square width, and thickness d (here the square geometry has been chosen for simplifying the fabrication of the prototype, but the following could also apply to any shape, for example a circular one). The corona electrode is made of a pattern of parallel thin wires (colored in deep purple), while the collector electrode is made of a perforated grid, illustrated with the black dots pattern. We will consider a spherical coordinates system ($\overrightarrow{r},\theta ,\phi$), whose origin O is located at the CDT geometrical center, as illustrated in Figure 2. In this system, $r=\Vert \overrightarrow{r}\Vert$ is the distance to center O, angle θ designates the elevation with respect to z, and angle φ designates the azimuth over the CDT reference plane (parallel to the electrodes planes comprising O). In addition to this coordinates system, we will denote z the axis normal to the CDT reference plane, with unit vector ${\overrightarrow{e}}_z$. In this orientation, we can notice that $\overrightarrow{d}·{\overrightarrow{e}}_z<0$, thus a negative sign in the following expressions of the electrostatic force. We will assume d ≪ λ where λ denotes the wavelength, yielding the two sources centers (heat and force sources) are collocated at O. As d is of the order of 5 mm, this assumption holds up to 10 kHz. In this example, the height of the corona electrode is supposed much smaller than the lateral dimension w of the transducer: d ≪ w ≪ λ.

Figure 2. Sketch of a square CDT in free field: (a) 3D representation with cartesian and spherical coordinates definitions; (b) simplified cut view in the x O z plane. The blue circle represents the monopolar heat source, while the red circle pair highlights the dipolar force source, and their relative strength being qualitatively represented by the circles radii.

We can then derive the wave equation as [28]:

$$\begin{array}{c}\hfill {\mathrm{\nabla }}^{2}p-\frac{1}{c^2}\frac{{\partial }^{2}p}{\partial t^2}=-\frac{\gamma -1}{c^2}\frac{\partial h}{\partial t}+\overrightarrow{\mathrm{\nabla }}·\overrightarrow{\varphi }\end{array}$$(3)

where γ is the specific heat ratio (equal to 1.4 in the air in adiabatic transformation). This can be formulated in the frequency domain as:

$$\begin{array}{c}\hfill ({\mathrm{\nabla }}^2+k^2)p=-\frac{j\omega (\gamma -1)}{c^2}h+\overrightarrow{\mathrm{\nabla }}.\overrightarrow{\varphi }\end{array}$$(4)

where k is the wavenumber.

In case where the heat and force sources were considered as point sources and collocated (same position $\overrightarrow{r_0}$), the integration of equation (4) using the Green’s function $g(\overrightarrow{r}|\overrightarrow{r_0})$ would result in [22, 31, 32]:

$$\begin{array}{cc}\hfill p(\overrightarrow{r})& =\int \int {\int }_V[-j\omega \frac{\gamma -1}{c^2}h+\overrightarrow{\mathrm{\nabla }}.\overrightarrow{\varphi }]g\left(\overrightarrow{r}|\overrightarrow{r_0}\right)\mathrm{d}{V}_0\hfill \\ \hfill & =jk[\frac{\gamma -1}{c}\underset{H}{\underbrace{\int \int {\int }_{V}h\mathrm{d}{V}_0}}+\text{sgn}(\overrightarrow{d}·{\overrightarrow{e}}_z)(\frac{j}{\mathit{kr}}+1)\hfill \\ \hfill & \times cos\theta \underset{F=||\overrightarrow{F}||}{\underbrace{\int \int {\int }_V\varphi \mathrm{d}{V}_0}}]\frac{e^{-jkr}}{4\pi r}\hfill \end{array}$$(5)

where the sign of the force source depends on the orientation of the electrode pair with respect to the axis z, and H and $F=\Vert \overrightarrow{F}\Vert$ are derived by linearization and expressed in equation (2). As the ratio between H and F is assumed constant for a given transducer and given DC voltage U_DC, the individual contributions of the heat and force sources on the total sound pressure, thus directivity D_CDT(θ) resulting the heat and force point sources pair, are fixed by construction.

However, in the present case, the spatial extent of the sources is far from a point, and one needs to account for the directivity of the corresponding radiating surface. If we consider the radiation in the far field, the total radiated pressure can be derived as the product of the expression of equation (5) and the “geometric” directivity D_piston(θ, φ) of the uniform piston of surface S.

In the case of a square piston of width w, the “geometric” directivity function reads [22]:

$$\begin{array}{cc}\hfill D_{\text{piston}}(\theta ,\phi )& =\text{sinc}(\frac{\mathit{kw}}{2}sin\theta cos\phi )\hfill \\ \hfill & \times \text{sinc}(\frac{\mathit{kw}}{2}sin\theta sin\phi )\hfill \end{array}$$(6)

where sinc designates the sine cardinal function. We will intentionally disregard the radiation contribution of the small piston of cross-section area S to focus only on the heat and force source contributions D_CDT(θ) to the overall directivity. However, one should bear in mind that the directivities will be affected by the geometry of the radiating square piston at high frequencies, as soon as $\frac{\mathit{kw}}{2}>1$.

Let’s assume now that the transducer dimensions are much smaller than the distance of observation (r ≫ w ≫ d), so that the problem can be simplified as the one of point sources in a free spherical domain. Also, discarding the directivity of the piston, we can simplify the problem to a 2D axisymmetric problem, so that the spherical coordinates can be limited to only r and θ (φ may only appear in D_piston(θ, φ)).

In the far field (k r ≫ 1), the sound pressure can be approximated as:

$$\begin{array}{c}\hfill p(r,\theta )=jk\frac{e^{-jkr}}{4\pi r}\underset{F_{\text{CDT}}}{\underbrace{[\frac{(\gamma -1)}{c}H-Fcos\theta ]}}.\end{array}$$(7)

The term inside the brackets of equation (7) corresponds to the pressure forces exerted by the combination of the monopolar and dipolar sources at any angles θ over the boundaries of the dual CDTs, that can be rewritten to take the form of a unidirectional function:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{\text{CDT}}(\theta )& =F_t\underset{D_{\text{CDT}}(\theta )-\text{unidirectional}}{\underbrace{[(1-\mu )+\mu cos(\theta +\pi )]}}\hfill \end{array}\end{array}$$(8)

where $F_t=\frac{(\gamma -1)H+cF}{c}$ is a total force (in N, independent on frequency) and $\mu =\frac{\mathit{cF}}{(\gamma -1)H+cF}\in [0,1]$ (dimensionless, independent on frequency) is the ratio of the bidirectional source to the overall directivity.

Then, with one single CDT, it is possible to achieve a given directivity D_CDT(θ), prescribed by the ratio between the Heat “weight” (1 − μ) and the Force “weight” (μ), for any frequency. The following section will show how to decouple the monopole and dipole sources, and then fully control the directivity when two CDTs are stacked back-to-back.

3 The dual corona discharge transducer

3.1 General description

Now we stack two CDTs with same shape and dimensions back-to-back, namely sharing a common collector electrode, as illustrated in Figure 3. For the sake of generality, we can consider that the two CDTs may present slight differences in terms of construction, thus giving rise to different electroacoustic parameters such as their coupling factors (C₁ and C₂), offset voltages (U_DC1 and U_DC2) or even breakdown voltages (U₀₁ and U₀₂). If they are individually driven by two different voltages u_ac1 and u_ac2, it will yield two different sets of monopolar sources (H₁, H₂) and dipolar sources (F₁, F₂). As in the preceding section, we will consider that the CDTs’ thicknesses are still much smaller than the wavelength of interest (for example if d = 5 mm, the total thickness is 1 cm, thus subwavelength up to the range of 5 kHz). Then, following the same assumptions as in Section 2.2, the total sound pressure in the far field becomes:

Figure 3. Simplified sketch of a dual Corona Discharge Transducer – cut view along the x O z plane. The observation point is assumed in the far field (r1 ≈ r2 ≈ r ≫ d) where r1 and r2 are the distance to the centers of each individual CDT and r denotes the distance to the center of the dual CDTs.

$$\begin{array}{cc}\hfill p_t(r,\theta )& =jk[(\frac{\gamma -1}{c}{H}_1-F_{1}cos\theta )\frac{e^{-jk{r}_1}}{4\pi r_1}\hfill \\ \hfill & +(\frac{\gamma -1}{c}{H}_2+F_{2}cos\theta )\frac{e^{-jk{r}_2}}{4\pi r_2}]\hfill \end{array}$$(9)

where $r_1=|\overrightarrow{r}-{\overrightarrow{r}}_{01}|$ and $r_2=|\overrightarrow{r}-{\overrightarrow{r}}_{02}|$ are the distances of observations relative to the centers of each CDT ${\overrightarrow{r}}_{01}$ and ${\overrightarrow{r}}_{02}$. Without loss of generality, we can observe that the distance d between the two centers is much smaller than the distance of observation $r_1=r-\frac{d}{2}cos\theta \approx r_2=r+\frac{d}{2}cos\theta \approx r\gg d$ (see Fig. 3) and then:

$$\begin{array}{cc}\hfill p_t(r,\theta )& =jk(\frac{\gamma -1}{c}(H_1+H_2)+(-F_1+F_2)cos\theta )\hfill \\ \hfill & \times \frac{e^{-jkr}}{4\pi r}\hfill \\ \hfill & =jk{F}_t(1-{\mu }_t+{\mu }_{t}cos\theta )\frac{e^{-jkr}}{4\pi r}\hfill \end{array}$$(10)

where $F_t=\frac{\gamma -1}{c}(H_1+H_2)+(F_2-F_1)$ is a global force and ${\mu }_t=\frac{c(F_2-F_1)}{(\gamma -1)(H_1+H_2)+c(F_2-F_1)}$ is the unidirectional factor of the dual CDTs arrangement, that both depend on u_ac1 and u_ac2 (and the parameters of the heat and force sources U _{D c i} , U_0i , C _i ).

In the following, the directivity resulting from the monopole/dipole pair of the dual CDTs (discarding D_piston) will be denoted:

$$\begin{array}{c}\hfill D_t(\theta )=(1-{\mu }_t)+{\mu }_{t}cos\theta .\end{array}$$(11)

3.2 Case of ideally identical CDTs

In the ideal case where both CDTs share exactly the same physical properties, then feeding each with a same ac voltage u_ac1 = u_ac2 will cancel the dipolar terms (as the symmetry of the transducer yields opposite electrostatic force orientation on both sides of the collector electrode), thus achieving a fully monopolar source of amplitude $\frac{\gamma -1}{c}2H$ with H = H₁ = H₂:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p_t{(r,\theta )|}_{u_{\mathrm{ac}1}=u_{\mathrm{ac}2}}& =jk\frac{\gamma -1}{c}2H\frac{e^{-jkr}}{4\pi r}·\hfill \end{array}\end{array}$$(12)

On the other hand, if the two CDTs are fed with opposite ac voltages (u_ac1 = −u_ac2), this will cancel the monopolar contributions and thus yields a fully bi-directional source of amplitude 2F with F = F₂ = −F₁:

$$\begin{array}{c}\hfill p_t{(r,\theta )|}_{u_{\mathrm{ac}1}=-u_{\mathrm{ac}2}}=jk2Fcos\theta \frac{e^{-jkr}}{4\pi r}·\end{array}$$(13)

Finally, it is always possible to identify a combination of voltages u_ac1 and u_ac2 that allows achieving a targeted force F _t and directivity D _t (θ)=(1 − μ _t )+μ _t cosθ after:

$$\begin{array}{c}\hfill \{\begin{array}{c}{F}_t(1-{\mu }_t)=C(3U_{\mathrm{DC}}^2-2U_{\mathrm{DC}}{U}_0)\hfill \\ \left(\frac{\gamma -1}{c}\right)(u_{\mathrm{ac}1}+u_{\mathrm{ac}2})\hfill \\ F_t{\mu }_t=\frac{\mathit{Cd}}{{\mu }_{\text{ion}}}(2U_{\mathrm{DC}}-U_0)(u_{\mathrm{ac}2}-u_{\mathrm{ac}1}).\hfill \end{array}\end{array}$$(14)

By denoting ${\alpha }_h=C\frac{\gamma -1}{c}(3U_{\mathrm{DC}}^2-2U_{\mathrm{DC}}{U}_0)$ and ${\alpha }_f=\frac{\mathit{Cd}}{{\mu }_{\text{ion}}}(2U_{\mathrm{DC}}-U_0)$, the two sources coefficients, it finally yields:

$$\begin{array}{c}\hfill \{\begin{array}{c}{u}_{\mathrm{ac}1}=F_t\frac{{\alpha }_f-({\alpha }_h+{\alpha }_f){\mu }_t}{2{\alpha }_h{\alpha }_f}\hfill \\ u_{\mathrm{ac}2}=F_t\frac{{\alpha }_f+({\alpha }_h-{\alpha }_f){\mu }_t}{2{\alpha }_h{\alpha }_f}\hfill \end{array}\end{array}$$(15)

thus u_ac1 and u_ac2 can be directly derived from the settings of F _t and μ _t and vice-versa. A more practical protocol consists in setting one voltage (e.g. u_ac1) and prescribing a target unidirectional factor μ _t , then it yields:

$$\begin{array}{c}\hfill \{\begin{array}{c}{F}_t=2{\alpha }_h{\alpha }_f\frac{1+{\mu }_t}{{\alpha }_f-({\alpha }_h+{\alpha }_f){\mu }_t}{u}_{\mathrm{ac}1}\hfill \\ u_{\mathrm{ac}2}=\frac{{\alpha }_f+({\alpha }_h-{\alpha }_f){\mu }_t}{{\alpha }_f-({\alpha }_h+{\alpha }_f){\mu }_t}{u}_{\mathrm{ac}1}.\hfill \end{array}\end{array}$$(16)

3.3 Case of CDTs with different characteristics

In the most general case where the electroacoustic properties of the two CDTs are unknown, or if they happen to present slight geometrical and electrical differences, we can generalize the preceding protocol to derive the voltages u_ac1 and u_ac2 allowing achieving a targeted directivity D _t (θ) (and specifying for example one of the two ac voltages u_aci ).

Let’s consider the two CDTs (denoted with subscript i) characterized by the following parameters:

• offset voltage U_DCi ;

• breakdown voltage U_0i ;

• cross section area S _i (affecting the current-voltage constant C _i );

• inter-electrode gap d _i (affecting the current-voltage constant C _i ).

In this case, the heat and force sources coefficients read:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\alpha }_{h1}=\frac{\gamma -1}{c}{C}_1(3U_{\mathrm{DC}1}^2-2U_{\mathrm{DC}1}{U}_{01}),\hfill \\ {\alpha }_{f1}=\frac{C_1{d}_1}{{\mu }_{\text{ion}}}(2U_{\mathrm{DC}1}-U_{01})\hfill \\ {\alpha }_{h2}=\frac{\gamma -1}{c}{C}_2(3U_{\mathrm{DC}2}^2-2U_{\mathrm{DC}2}{U}_{02}),\hfill \\ {\alpha }_{f2}=\frac{C_2{d}_2}{{\mu }_{\text{ion}}}(2U_{\mathrm{DC}2}-U_{02}).\hfill \end{array}\end{array}$$(17)

There exist pairs of voltages (u_ac1, m , u_ac2, m ) for which the dual CDTs arrangement yields an omnidirectional sound source (μ _t  = 0), and another pair (u_ac1, d , u_ac2, d ) for which the dual CDTs yields a bi-directional source (μ _t  = 1). Experimentally, it is possible to identify such two pairs of voltages (see Sect. 5.1) and deduce the relationships between α _f1 and α _f2 on the one hand, and between α _h1 and α _h2 on the other hand:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\alpha }_{f2}={\alpha }_{f1}\frac{u_{\mathrm{ac}1,m}}{u_{\mathrm{ac}2,m}}\hfill \\ {\alpha }_{h2}=-{\alpha }_{h1}\frac{u_{\mathrm{ac}1,d}}{u_{\mathrm{ac}2,d}}.\hfill \end{array}\end{array}$$(18)

Also, computing the ratio between the sound pressures measured at a same distance r and at angle θ = 0 rad, for both cases (p _m (r, 0)=p _t (r, 0)| _{μ t  = 0} for the monopole setting and p _d (r, 0)=p _t (r, 0)| _{μ t  = 1} for the dipole setting), yields the ratio between α _h1 and α _f1 as:

$$\begin{array}{c}\hfill \frac{{\alpha }_{h1}}{{\alpha }_{f1}}=\frac{u_{\mathrm{ac}2,d}}{u_{\mathrm{ac}2,m}}\frac{p_m(r,0)}{p_d(r,0)}·\end{array}$$(19)

This ratio can finally be used to extract the voltage pair (u_{ac1, μ t} , u_{ac2, μ t} ) achieving the target directivity μ _t after equation (14):

$$\begin{array}{cc}\hfill \frac{1-{\mu }_t}{{\mu }_t}& =\frac{{\alpha }_{h1}}{{\alpha }_{f1}}\frac{u_{\mathrm{ac}2,m}}{u_{\mathrm{ac}2,d}}\hfill \\ \hfill & \times \frac{u_{\mathrm{ac}1,{\mu }_t}{u}_{\mathrm{ac}2,d}-u_{\mathrm{ac}1,d}{u}_{\mathrm{ac}2,{\mu }_t}}{u_{\mathrm{ac}1,m}·u_{\mathrm{ac}2,{\mu }_t}-u_{\mathrm{ac}1,{\mu }_t}·u_{\mathrm{ac}2,m}}\hfill \end{array}$$(20)

which finally yields:

$$\begin{array}{c}\hfill \frac{u_{\mathrm{ac}2,{\mu }_t}}{u_{\mathrm{ac}1,{\mu }_t}}=\frac{{\mu }_t\frac{{\alpha }_{h1}}{{\alpha }_{f1}}+(1-{\mu }_t)}{{\mu }_t\frac{{\alpha }_{h1}}{{\alpha }_{f1}}\frac{u_{\mathrm{ac}1,d}}{u_{\mathrm{ac}2,d}}+(1-{\mu }_t)\frac{u_{\mathrm{ac}1,m}}{u_{\mathrm{ac}2,m}}}·\end{array}$$(21)

Note that this result shows a constant ratio between the two voltages feeding the dual CDTs, independent of frequency. Indeed, unidirectional factor μ _t is a real gain ranging from 0 to 1, coefficients α _{h i} and α _{f i} are the real, frequency-independent, electroacoustic factors for the heat and force sources within each CDT i (similarly to the force factor for an electrodynamic transducer), and ratios $\frac{u_{\mathrm{ac}2,m}}{u_{\mathrm{ac}1,m}}$ and $\frac{u_{\mathrm{ac}2,d}}{u_{\mathrm{ac}1,d}}$ don’t depend on frequency neither. Then, the voltage ratio expressed in equation (21) represents a real gain between the two input voltages feeding the dual CDTs in a view to achieving a target directivity, independent of frequency, applying to any kind of signal, including audio signals.

In practice, we can fix one of the two CDT voltages (for example u_ac1) and keep the same value for all settings, in this case, u_ac1, m  = u_ac1, d  = u_{ac1, μ t}  = u_ac1, ref, to simply derive u_{ac2, μ t} .

In the following, we will illustrate some examples of directivities achievable with the dual CDTs assembly.

4 Numerical simulations

In this section, we will give some examples of directivities simulated with COMSOL Multiphysics on a dual CDTs arrangement, as described in Section 3. In these simulation examples, we will consider two ideally identical CDTs, with constitutive parameters of Table 1, and follow the methodology described in Section 3.2. Note that the electrical parameters (C, U₀, U_DC) provided in this table are estimated beforehand by measuring the voltage-current curve on a prototype of same dimensions and geometry (here the square transducer prototype presented in Sect. 5). The interested readers can gain more insight on this calibration procedure in reference [32], page 49 and following.

4.1 Model geometry

The numerical model is designed using COMSOL Multiphysics Acoustic Module in 3D space dimension. The fluid domain is defined as a sphere of radius R = 3 m, with a Perfectly Matched Layer (PML) of 2 m, making the simulation extent of 5 m. Each of the individual CDT is modeled as identical parallelepipeds (Blocks) of square cross-section on the x O y plane, with width w and height d, sharing a common boundary at z = 0. We assume here the φ = 0 (x O z) and $\phi =\frac{\pi }{2}$ (y O z), as well as the $\phi =\frac{\pi }{4}$ planes, are symmetry planes, therefore simplifying the 3D domain comprised between planes φ = 0 and $\phi =\frac{\pi }{4}$. This space domain simplification is highlighted in Figure 4a as a deep purple triangular surface over the reference plane z = 0 (corresponding to the shared collector electrode). The corresponding simplified COMSOL 3D model is illustrated on Figure 4b.

Figure 4. Model geometry in COMSOL Multiphysics: (a) Sketch of the dual CDTs reference plane (z = 0) highlighting symmetries, and definition of spherical coordinates angles θ and φ; (b) snapshot of the COMSOL 3D model simplified geometry.

4.2 Model physics

The propagating domain, including the CDTs volume, is considered filled with air with mass density ρ and speed of sound c. Each CDT is modeled as the combination of a “Heat Source” of individual power density Q_heat, i  = h₀ u_aci where h₀ is defined in Section 2.1 (also specifying the ratio of specific heat capacities γ and heat capacity at constant pressure C _P ), and a “Dipolar Domain Source” of individual volumetric force $q_d={(-1)}^i{f}_0{u}_{\mathrm{ac}i}·\overrightarrow{e_z}$, where i ∈ {1, 2} designates the CDT number 1 or 2 (as in Fig. 4). In this definition, the electrodes of the upper transducer CDT1 (resp. lower transducer CDT2) are arranged so that the projection of the electrostatic force ${\overrightarrow{F}}_1$ (resp. ${\overrightarrow{F}}_2$) over z is negative (resp. positive) for a positive u_ac1 (resp. u_ac2), as in the geometry presented in Section 3.

4.3 Other settings and processing methodology

The effect of the CDTs shape and dimensions on directivity start to enter into play for $\frac{\mathit{kw}}{2}>1$ where w = 13 cm, yielding a diffraction limit of approximately f _d  = 1 kHz. The study is performed in the frequency domain, for octave band frequencies spanning from f_min = 125 Hz to f_max = 1 kHz. The mesh inside the propagating domain and the CDTs is “Free Triangular” with maximal mesh size $\delta =\frac{c}{6.f_{\text{max}}}$, and the PML layer is meshed with a “Swept” mesh of maximum element size 10 cm.

The processing starts by specifying voltage u_ac1 as well as the target directivity parameter μ _t . From these values, u_ac2 is set after equation (16), depending on the coefficients ${\alpha }_{h1}={\alpha }_{h2}=\frac{\gamma -1}{c}C(3U_{\mathrm{DC}}^2-2U_{\mathrm{DC}}{U}_0)$ and ${\alpha }_{f1}={\alpha }_{f2}=\frac{\mathit{Cd}}{{\mu }_{\text{ion}}}(2U_{\mathrm{DC}}-U_0)$. The results are presented in the following subsection.

4.4 Numerical results

4.4.1 Directivity

Figure 5 show the achieved directivities, along the plane $\phi =\frac{\pi }{4}$ for different parameter settings μ _t  ∈ {0, 0.3, 0.5, 0.63, 0.75, 1}. The first series of curves (plain lines) represent the directivities per octave band computed as the sound pressure magnitude over the boundary of the propagating medium (i.e. the interior boundary of the PML) divided by the maximum of sound pressure magnitude over this boundary:

Figure 5. Computed directivity $D(\theta ,\frac{\pi }{4})$ in the $\phi =\frac{\pi }{4}$ plane for each octave band between 125 Hz and 1 kHz simulated with COMSOL Multiphysics (plain lines) and compared to the theoretical expression of equation (10) (plus markers), for different values of μ t : (a) omnidirectional (μ t  = 0); (b) subcardioid (μ t  = 0.3); (c) cardioid (μ t  = 0.5); (d) supercardioid (m u t  = 0.63); (e) hypercardioid (μ t  = 0.75); (f) bidirectional (μ t  = 1).

$$\begin{array}{cc}\hfill D_{\text{COMSOL}}(\theta )& =\frac{\left|p(r=R,\theta ,\phi =\pi /4)\right|}{{max}_{\theta \in [0,\pi ]}\left(\left|p(r=R,\theta ,\phi =\pi /4)\right|\right)},\hfill \\ \hfill & \text{for}\theta \in [0,\pi ].\hfill \end{array}$$(22)

The second series (plus markers) represents the same quantity defined from the theoretical value D _t (θ) as in equation (11), multiplied by the square piston directivity D_square(θ, φ) of equation (6).

We can observe that the dual CDTs arrangement allows achieving the targeted directivites in each case (COMSOL and theoretical pressure fields perfectly match for each settings and each frequency). Also, we can see a deviation from the targeted directivity D _t (θ) of equation (11) for frequencies f >  500 Hz, highlighting the directional characteristics of the square piston. This is particularly obvious for the omnidirectional directivity, where D_COMSOL(π/2)_|500 Hz = 0.957 and D_COMSOL(π/2)_|1000 Hz = 0.780. However, it can still be considered omnidirectional (for this specific case) up to 1 kHz as those directivity minima always exceed $\frac{1}{\sqrt{2}}$.

4.4.2 Pressure along z axis

To further confirm the validity of the analytical model of the dual CDTs arrangement, the pressure as a function of distance along the z axis is also computed. The real and imaginary parts of the total sound pressure computed with COMSOL over the z axis at each frequency is then compared to the theoretical expression given in equation (10), also multiplied by the piston directivity of equation (6). The results obtained at f = 1 kHz are illustrated on Figure 6 for all cases. Although the far field condition is not fulfilled in the close vicinity of the dual CDT (distances shorter than λ_min = 34 cm), we observe a good agreement between the COMSOL full wave simulations and the simple theoretical model elaborated in Section 3.2 in the far field (kr ≫ 1), validating the analytical model.

Figure 6. Sound pressure (plain lines: real part; dashed lines: imaginary part) computed along the z axis with COMSOL Multiphysics (blue) and compared with the theoretical formulation of equation (10) at f = 1 kHz and for different values of μ t : (a) omnidirectional (μ t  = 0); (b) subcardioid (μ t  = 0.3); (c) cardioid (μ t  = 0.5); (d) supercardioid (m u t  = 0.63); (e) hypercardioid (μ t  = 0.75); (f) bidirectional (μ t  = 1).

The next section will present experimental results obtained on a square dual CDTs prototype.

5 Experimental validation

The square dual CDTs is assembled on a common frame, constructed through additive manufacturing. A perforated grid made of Aluminium of thickness t = 1 mm is assembled in the median plane of the frame, and two networks of thin Tungsten wires of diameter d_wire = 100 μm with inter-wire distance of 11 mm are assembled symmetrically on both sides, at a distance d = 5 mm to the metallic grid. The dual CDTs transducer is fed with DC+ac voltages from a custom dual-channel, High Voltage valve-based power amplifier, described in reference [32]. Figure 7 shows the construction principle and a picture of the experimental prototype taken from both sides.

Figure 7. Dual CDT prototype. (a) Construction principle. (b) Front view. (c) Rear view.

The dual CDTs is mounted on a microphone stand fixed to a Bruel & Kjaer Type 9640 Turntable System, and a PCB 378B02 half-inch microphone is aligned at 1 m from the dual CDTs center. The signal acquisition and processing is performed with a Bruel & Kjaer Type 3160 Pulse Multichannel Sound and Vibration Analyzer, each CDT channel on the power amplifier being excited with a synchronous swept sine ranging from 60 Hz to 2060 Hz with a sweep speed of 10 mdec/s, and individual voltages u_ac1 and u_ac2. The analysis consists in processing the transfer functions between the sound pressure sensed by the microphone and the output signal u_ac1 delivered to CDT1 amplification channel (that we set to a fixed value for all directivities). Figure 8 illustrates the experimental setup (the power amplifier and cabling are not visible here).

Figure 8. Experimental setup.

In this experimental evaluation, we will consider unknown CDTs parameters, following the methodology of Section 3.3.

5.1 Calibrating the directional corona discharge loudspeaker: Omnidirectional and bidirectional directivities

The first step consists in identifying the combination of voltages (u_ac1, m , u_ac2, m ) that gives rise to a pure omnidirectional directivity, and the pair (u_ac1, d , u_ac2, d ) that gives rise to a pure bidirectional directivity. To ensure the target directivity is achieved, two measurements of frequency responses are made for each settings:

• Measurement at angles θ = 0 and $\theta =\frac{\pi }{2}$ for the omnidirectional target, and identification of the value for which both curves are optimally aligned.

• Minimization of the frequency response at angle $\theta =\frac{\pi }{2}$ for the bi-directional target.

Once the two voltage pairs are identified, the frequency responses (FRF) are measured for each angle $\theta =n\frac{\pi }{36}$ with integer n ∈ [0 : 36]. The measured FRFs for the omnidirectional setting are illustrated in Figure 9a every $\frac{\pi }{6}$ and for the bidirectional setting they are shown in Figure 9b every $\frac{\pi }{12}$. We can observe that the frequency responses for both cases present the same high-pass behavior of 6 dB per octave, and that the response measured at θ = 0[π] rad in the bidirectional setting is about 12 dB higher than for the monopole setting, corresponding to the ratio of force to heat strengths $20{log}_{10}(\frac{F}{H})$. We can also note a glitch occurring in the measurements around 250 Hz, more prominent when signal to noise ratio is low (for example in the bidirectional setting, at angles around $\frac{\pi }{2}$), that might be the consequence of a notch filter effect due to a reflection on the metallic frame of the anechoic room (although the facility should be immune to such reflections).

Figure 9. Frequency responses $20{log}_{10}(\frac{p(1\mathrm{m},\theta )}{u_{\mathrm{ac}1}}.\frac{1\mathrm{V}}{20\mu \mathrm{Pa}})$ measured at different angles for (a) the monopolar and (b) the dipolar settings.

The corresponding directivity sampled at each octave central frequencies are illustrated in Figures 10a and 10f. The results reflect the calibration procedure, which allows determining the reference voltages pairs (u_ac1, m , u_ac2, m ) and (u_ac1, d , u_ac2, d ). The relative good agreement with the theoretical directivities (black plus markers) demonstrates the validity of the procedure up to 1 kHz. We can observe a slight misalignment at θ = 0[π] rad compared to the sensitivity at $\theta =\frac{\pi }{2}$ for f = 125 Hz and 500 Hz in the omnidirectional setting (Fig. 10a), but the mismatch represents around 0.5 dB which can be considered satisfactory.

Figure 10. Directivity measured at different angles for: (a): omnidirectional (μ t  = 0); (b) subcardioid (μ t  = 0.3); (c) cardioid (μ t  = 0.5); (d) supercardioid (μ t  = 0.63); (e) hypercardioid (μ t  = 0.75) and (f) bidirectional (μ t  = 1) settings. For each case, the theoretical directivity D t (θ)=1 − μ t  + μ t cosθ is presented with black plus markers.

5.2 Achieving unidirectional directivities

Once the voltages pairs corresponding to μ _t  = 0 and μ _t  = 1 have been identified, the other pairs have been derived corresponding to any of the targeted μ _t , following the methodology detailed in Section 3.3. The directivities achieved with the corresponding pairs of voltages (u_{ac1, μ t} , u_{ac2, μ t} ) for μ _t  ∈ {0; 0.3; 0.5; 0.63; 0.75; 1} are illustrated on Figure 10. Although some slight discrepancies might appear in the rear direction at certain frequencies for all of the measured unidirectional directivities, we can see a good agreement between the achieved directivities and the theoretical ones. These differences are mostly due to the rather short distance of measurement at 125 Hz and to the directivity of the square radiator at higher frequencies.

6 Conclusion

The control over directivity on conventional loudspeakers is known to be limited to low-frequencies due to their bulkiness. Given the compactness of the CDTs, it is possible to achieve very precise directivities in theory up to the range of 1 kHz following the proposed dual CDTs concept. In practice, however, the dimensions of the radiator induce a frequency limit above which the impact of planar geometry enters into play. But, in terms of directivity control, our results show that the introduced mechanism remains an advantageous asset, at least up to that frequency threshold, significantly outperforming loudspeakers of equal surface in realizing the targeted directivities. Though, it remains noticeably less performing in terms of overall sound power compared to conventional loudspeakers of same sizes, with a lower overall sensitivity and the typical high-pass behavior of its frequency response. This can be overcome by augmenting the radiating surface (though impacting the directivity performance), or further optimizing the electrodes arrangement (thinner corona wires, smaller air gap, etc.).

The dual CDTs concept has direct applications to directivity control on loudspeakers, owing to its direct conversion of electric voltage to acoustic flow velocity through the (geometry-dependent) radiation impedance. Besides, the dual CDTs will be an important asset as secondary source in ducted (1D) active noise reduction systems (e.g. active exhaust noise reduction). Indeed the capability to separate the net sound sources on each sides allows for controlling independently the up- and down-stream of the active control source, and thus have a full control over both sound fields individually. By extension, the dual CDTs will have direct application to active acoustic metamaterials and metasurfaces, as the control of monopolar/dipolar elements is at the heart of non-reciprocal concepts such as the Willis coupling effect.

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

Data are available on request from the authors.

References

All Tables

All Figures

	Figure 1. Schematic cut view of the electrode pair: the corona electrode consisting of an array of ultra-thin emitter wires parallel to each others (thin dots in this cut view), and a collector electrode consisting of a grounded perforated grid (dashed thick line in this cut view).
In the text

	Figure 2. Sketch of a square CDT in free field: (a) 3D representation with cartesian and spherical coordinates definitions; (b) simplified cut view in the x O z plane. The blue circle represents the monopolar heat source, while the red circle pair highlights the dipolar force source, and their relative strength being qualitatively represented by the circles radii.
In the text

	Figure 3. Simplified sketch of a dual Corona Discharge Transducer – cut view along the x O z plane. The observation point is assumed in the far field (r1 ≈ r2 ≈ r ≫ d) where r1 and r2 are the distance to the centers of each individual CDT and r denotes the distance to the center of the dual CDTs.
In the text

	Figure 4. Model geometry in COMSOL Multiphysics: (a) Sketch of the dual CDTs reference plane (z = 0) highlighting symmetries, and definition of spherical coordinates angles θ and φ; (b) snapshot of the COMSOL 3D model simplified geometry.
In the text

Figure 5. Computed directivity $D(\theta ,\frac{\pi }{4})$ in the $\phi =\frac{\pi }{4}$ plane for each octave band between 125 Hz and 1 kHz simulated with COMSOL Multiphysics (plain lines) and compared to the theoretical expression of equation (10) (plus markers), for different values of μ t : (a) omnidirectional (μ t  = 0); (b) subcardioid (μ t  = 0.3); (c) cardioid (μ t  = 0.5); (d) supercardioid (m u t  = 0.63); (e) hypercardioid (μ t  = 0.75); (f) bidirectional (μ t  = 1).

In the text

Figure 6. Sound pressure (plain lines: real part; dashed lines: imaginary part) computed along the z axis with COMSOL Multiphysics (blue) and compared with the theoretical formulation of equation (10) at f = 1 kHz and for different values of μ t : (a) omnidirectional (μ t  = 0); (b) subcardioid (μ t  = 0.3); (c) cardioid (μ t  = 0.5); (d) supercardioid (m u t  = 0.63); (e) hypercardioid (μ t  = 0.75); (f) bidirectional (μ t  = 1).

In the text

	Figure 7. Dual CDT prototype. (a) Construction principle. (b) Front view. (c) Rear view.
In the text

	Figure 8. Experimental setup.
In the text

	Figure 9. Frequency responses $20{log}_{10}(\frac{p(1\mathrm{m},\theta )}{u_{\mathrm{ac}1}}.\frac{1\mathrm{V}}{20\mu \mathrm{Pa}})$ measured at different angles for (a) the monopolar and (b) the dipolar settings.
In the text

	Figure 10. Directivity measured at different angles for: (a): omnidirectional (μ t  = 0); (b) subcardioid (μ t  = 0.3); (c) cardioid (μ t  = 0.5); (d) supercardioid (μ t  = 0.63); (e) hypercardioid (μ t  = 0.75) and (f) bidirectional (μ t  = 1) settings. For each case, the theoretical directivity D t (θ)=1 − μ t  + μ t cosθ is presented with black plus markers.
In the text