Acoustics

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15.8. Neutralization of stray capacitances

Inevitably, in the real world, we encounter stray capacitances between adjacent rings. This may be in the form of capacitance between the stator rings themselves (anything up to 10% of the capacitance of the air gap), in the wiring and most significantly within the windings of the inductors, which need many turns of wire to produce the high inductance values needed. The capacitances between adjacent rings and within the wiring can be measured directly, but the winding capacitances of the inductors are best found indirectly by measuring their self-resonance frequencies. The effect of the stray capacitance is to bypass the inductors at high frequencies and thus reduce the amount of delay in the delay line. This in turn narrows the directivity pattern and thus lifts the on-axis response. Although we will describe a method for neutralizing the stray capacitances, it is best not to neutralize all of them in practice because some lift in the on-axis response is usually needed to compensate for the high-frequency roll-off due to the inertia of the membrane.

Figure 15.13 Directivity patterns at various frequencies of a 280 mm diameter membrane discretized into six equal rings, each having a width of 20 mm, and a center disk with a radius of 20 mm, using the delay line of Fig. 15.12 with six sections.

Figure 15.14 On-axis responses of a 280 mm diameter membrane with the delay configured to simulate an oscillating sphere where the delay is continuous (gray) and discretized into six equal rings and center disk (black) using the delay line of Fig. 15.12.

Table 15.2

Quantities for idealized electrostatic loudspeaker and delay line.
Parameters	Resistor and constants	Inductors (H)	Capacitors (pF)	Turnover frequencies (kHz)
a = 14 cm	R _T = 282 kΩ	L ₁ = 2.66	C ₁ = 0	f ₁ = 16.9
d = 1 mm	ρ ₀ = 1.18 kg/m³	L ₂ = 4.90	C ₂ = 5.75	f ₂ = 9.18
r = 1 m	c = 345 m/s	L ₃ = 7.80	C ₃ = 20.0	f ₃ = 5.77
e _in = 2 $\sqrt{2}$ $\sqrt{2}$ kV_rms	ε ₀ = 8.85 pF/m	L ₄ = 11.5	C ₄ = 43.6	f ₄ = 3.92
E _P = 2 kV		L ₅ = 16.9	C ₅ = 89.4	f ₅ = 2.66
N = 6		L ₆ = 28.4	C ₆ = 211	f ₆ = 1.59

A scheme for neutralizing the stray capacitances C _Sn is shown in Fig. 15.15 in the form of the cross-coupled capacitors C _Xn. For effective neutralization, we set

C_{X n} = C_{S n} .

$C_{X n} = C_{S n} .$

(15.55)

Although the effect of these neutralizing capacitors is to make the stray capacitances between adjacent rings vanish, the capacitances between opposite rings are effectively increased to include C _Sn. Hence, we modify Eq. (15.34) for the inductors to

L_{n} = 2 R_{T}^{2} (C_{T n} + C_{S n}),

$L_{n} = 2 R_{T}^{2} (C_{T n} + C_{S n}),$

(15.56)

and Eq. (15.35) for the turnover frequency to

Figure 15.15 Constant impedance delay line with compensation for stray capacitance C _Sn in the form of C _Xn.

ω_{n} = \frac{1}{2 R_{T} (C_{T n} + C_{S n})},

$ω_{n} = \frac{1}{2 R_{T} (C_{T n} + C_{S n})},$

(15.57)

where the voltage transfer function of each delay section is still given by Eq. (15.33). The time delay T _n per section is defined by

T_{n} = \frac{z_{n}}{c} = \frac{1}{ω_{n}} = 2 R_{T} (C_{T n} + C_{S n}) = \frac{L_{n}}{R_{T}} .

$T_{n} = \frac{z_{n}}{c} = \frac{1}{ω_{n}} = 2 R_{T} (C_{T n} + C_{S n}) = \frac{L_{n}}{R_{T}} .$

(15.58)

so that the total capacitance per section is given by

C_{T n} = \frac{z_{n}}{2 c R_{T}} - C_{S n} .

$C_{T n} = \frac{z_{n}}{2 c R_{T}} - C_{S n} .$

(15.59)

We can now furnish each section of the delay with its respective component values

C_{n} = 2 (C_{T n} - C_{R n})

$C_{n} = 2 (C_{T n} - C_{R n})$

(15.60)

where C _Tn is given by Eq. (15.59) and C _Rn by Eq. (15.30). From Eq. (15.58) we have

L_{n} = \frac{z_{n} R_{T}}{c} .

$L_{n} = \frac{z_{n} R_{T}}{c} .$

(15.61)

Each delay section is defined by the transmission matrix

[\begin{matrix} {\tilde{e}}_{n - 1} \\ {\tilde{i}}_{n - 1} \end{matrix}] = A_{n} \cdot [\begin{matrix} {\tilde{e}}_{n} \\ {\tilde{i}}_{n} \end{matrix}]

$[\begin{matrix} {\tilde{e}}_{n - 1} \\ {\tilde{i}}_{n - 1} \end{matrix}] = A_{n} \cdot [\begin{matrix} {\tilde{e}}_{n} \\ {\tilde{i}}_{n} \end{matrix}]$

(15.62)

where each element of A _n is given by

a_{11} (n) = \frac{R_{T} L_{n} (2 C_{T n} + C_{S n} + C_{X n}) s^{2} + L_{n} s + R_{T}}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},

$a_{11} (n) = \frac{R_{T} L_{n} (2 C_{T n} + C_{S n} + C_{X n}) s^{2} + L_{n} s + R_{T}}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},$

(15.63)

a_{12} (n) = \frac{2 R_{T} L_{n} s}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},

$a_{12} (n) = \frac{2 R_{T} L_{n} s}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},$

(15.64)

a_{21} (n) = \frac{(C_{T n} + 2 C_{X n}) s (R_{T} L_{n} (C_{S n} + \frac{C_{T n} C_{X n}}{C_{T n} + 2 C_{X n}}) s^{2} + L_{n} s + R_{T})}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},

$a_{21} (n) = \frac{(C_{T n} + 2 C_{X n}) s (R_{T} L_{n} (C_{S n} + \frac{C_{T n} C_{X n}}{C_{T n} + 2 C_{X n}}) s^{2} + L_{n} s + R_{T})}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},$

(15.65)

a_{22} (n) = \frac{R_{T} L_{n} (C_{S n} + C_{X n}) s^{2} + L_{n} s + R_{T}}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},

$a_{22} (n) = \frac{R_{T} L_{n} (C_{S n} + C_{X n}) s^{2} + L_{n} s + R_{T}}{R_{T} L_{n} (C_{S n} - C_{X n}) s^{2} + L_{n} s + R_{T}},$

(15.66)

The remaining calculations proceed as per the previous section from Eq. (15.46) onwards.

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15.8. Neutralization of stray capacitances

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Acoustics