Driver matching

Horn system response

Driver, chambers and finite horn in one lumped model: sound pressure, electrical impedance and cone excursion.

Sheet: 17 / 25
Units: Metric
Air: 20 °C

Fig. 1Sound pressure at 1 m, 2.83 V

SPLdrag on plot to read

Mean level: 106.6dB
Flare cut-off $f_c$: 45.5Hz
Peak excursion: 0.599mm
Minimum impedance: 5.64Ω
Compression ratio: 2.14: 1
Mouth $ka=1$: 108Hz

Fig. 2Electrical impedance magnitude

|Z|drag on plot to read

Fig. 3Diaphragm excursion, peak, at 2.83 V

x peakdrag on plot to read

Fig. 4Equivalent round horn

How it works

This sheet joins the pieces: a moving-coil driver described by its Thiele/Small parameters, a sealed chamber behind it, a small air cavity in front, and a finite horn ending in a mouth. It predicts sound pressure, the impedance the amplifier sees and how far the diaphragm moves. Mean level is averaged from 1.5 to 4 times the cut-off, and peak excursion is the largest value above cut-off.

The model

The Thiele/Small data are first converted to moving mass $M_{MS}$ , compliance $C_{MS}$ , loss $R_{MS}$ and force factor $Bl$ . The voice coil obeys $E_g = Z_E\,i + Bl\,v$ and the cone $Bl\,i = Z_M\,v$ , with

Z_E = R_E + j\omega L_E, \qquad Z_M = R_{MS} + j\omega M_{MS} + \frac{1}{j\omega C_{MS}} + S_d^{2}\left(Z_F + Z_R\right)

Solving the pair gives cone velocity and input impedance:

v = \frac{Bl\,E_g}{Z_E Z_M + (Bl)^{2}}, \qquad Z_{in} = Z_E + \frac{(Bl)^{2}}{Z_M}

Acoustic loads

Behind the cone the sealed chamber is a pure compliance. In front, the cavity compliance $C_F = V_{tc}/\rho c^{2}$ sits in parallel with the throat impedance $Z_t$ of the horn, so only part of the cone's volume velocity enters the throat:

Z_R = \frac{\rho c^{2}}{j\omega V_{rc}}, \qquad Z_F = \frac{Z_t}{1 + j\omega C_F Z_t}, \qquad U_t = \frac{S_d\,v}{1 + j\omega C_F Z_t}

$Z_t$ comes from the same conical-element transfer matrices as the throat impedance sheet, including the mirror images of the mouth when you select π or π/2 loading.

Output

The horn is treated as lossless, so the power entering the throat is the power radiated:

P = |U_t|^{2}\,\mathrm{Re}(Z_t), \qquad L_p = 10\log_{10}\frac{P\,\rho c}{\Omega\,r^{2}\,p_0^{2}}

with $r = 1$ m, $p_0 = 20\ \mu\text{Pa}$ and $\Omega$ the solid angle the mouth radiates into. One acoustic watt into half space is 112 dB at one metre.

What it leaves out

Directivity. The plot is power response; a real horn beams at high frequencies and measures hotter on axis.
Losses in the horn walls, leakage, chamber damping and cone break-up.
Semi-inductance and eddy losses of the voice coil, so the top octaves are approximate.
Higher-order modes. One-dimensional theory is trustworthy while the horn cross-section is small compared with the wavelength.

Use it to compare design choices quickly, then confirm with a full simulator and, above all, with measurement.

Loading sheet…

How it works

The model

Z_E = R_E + j\omega L_E, \qquad Z_M = R_{MS} + j\omega M_{MS} + \frac{1}{j\omega C_{MS}} + S_d^{2}\left(Z_F + Z_R\right)

Solving the pair gives cone velocity and input impedance:

v = \frac{Bl\,E_g}{Z_E Z_M + (Bl)^{2}}, \qquad Z_{in} = Z_E + \frac{(Bl)^{2}}{Z_M}

Acoustic loads

Z_R = \frac{\rho c^{2}}{j\omega V_{rc}}, \qquad Z_F = \frac{Z_t}{1 + j\omega C_F Z_t}, \qquad U_t = \frac{S_d\,v}{1 + j\omega C_F Z_t}

$Z_t$ comes from the same conical-element transfer matrices as the throat impedance sheet, including the mirror images of the mouth when you select π or π/2 loading.

Output

The horn is treated as lossless, so the power entering the throat is the power radiated:

P = |U_t|^{2}\,\mathrm{Re}(Z_t), \qquad L_p = 10\log_{10}\frac{P\,\rho c}{\Omega\,r^{2}\,p_0^{2}}

with $r = 1$ m, $p_0 = 20\ \mu\text{Pa}$ and $\Omega$ the solid angle the mouth radiates into. One acoustic watt into half space is 112 dB at one metre.

What it leaves out

Directivity. The plot is power response; a real horn beams at high frequencies and measures hotter on axis.
Losses in the horn walls, leakage, chamber damping and cone break-up.
Semi-inductance and eddy losses of the voice coil, so the top octaves are approximate.
Higher-order modes. One-dimensional theory is trustworthy while the horn cross-section is small compared with the wavelength.

Use it to compare design choices quickly, then confirm with a full simulator and, above all, with measurement.

Horn system response

Inputs

Driver

Horn

Chambers and drive

How it works

The model

Acoustic loads

Output

What it leaves out

Inputs

Driver

Horn

Chambers and drive

How it works

The model

Acoustic loads

Output

What it leaves out

Driver

Horn

Chambers and drive

§ How it works

The model

Acoustic loads

Output

What it leaves out

Driver

Horn

Chambers and drive

§ How it works

The model

Acoustic loads

Output

What it leaves out

How it works

How it works