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15.6. A practical delay line

Delay Path Length. We saw in the last section that discretization of the delay into rings of finite width produces irregularities in the response at higher frequencies. However, the delay was an ideal delay like that produced by a DSP, whereas in practice, the delay is more likely to take the form of an analog delay line on the high-voltage side of the transformer, such as that shown in Fig. 15.12. Otherwise, a separate stepping up transformer would be needed to feed each ring. Unless complicated inductors with center taps or a very large number of inductors are used, analog delay lines tend to introduce a degree of attenuation at the higher frequencies. In this case, this turns out to have a smoothing effect on the response.

Figure 15.10 Cross section of a stator divided into concentric rings of equal width, where purely for illustration each ring has been shifted to the left by the distance that a wave would have traveled during the time delay applied to that ring.

Figure 15.11 Effects of discretization on a 280 mm diameter membrane where the stator is divided into six rings and a center disk of various widths with the delay configured to simulate an oscillating sphere.

Table 15.1

Dimensions of rings.
Ring	Equal delay (mm)	Equal area (mm)	Equal width (mm)
a ₀	72	53	20
a ₁ − a ₀	26	22	20
a ₂ − a ₁	17	17	20
a ₃ − a ₂	12	14	20
a ₄ − a ₃	7.7	12	20
a ₅ − a ₄	4.4	11	20
a ₆ − a ₅	1.4	10	20

Figure 15.12 Constant impedance delay line ignoring stray capacitance.

From Fig. 15.5, we see that the total delay path length z _T at each point along the radial ordinate w is given by

z_{T} = a - \sqrt{a^{2} - w^{2}} = a (1 - \sqrt{1 - \frac{w^{2}}{a^{2}}}) .

$z_{T} = a - \sqrt{a^{2} - w^{2}} = a (1 - \sqrt{1 - \frac{w^{2}}{a^{2}}}) .$

(15.26)

For discretized rings, the delay section z _n required for the nth ring is the difference between the total delay z _Tn at that ring and the sum of all the previous delay sections

z_{n} = z_{T n} - \sum_{m = 1}^{n - 1} z_{m},

$z_{n} = z_{T n} - \sum_{m = 1}^{n - 1} z_{m},$

(15.27)

where

z_{T n} = a (1 - \sqrt{1 - {(\frac{a_{n - 1} + a_{n}}{2 a})}^{2}}) .

$z_{T n} = a (1 - \sqrt{1 - {(\frac{a_{n - 1} + a_{n}}{2 a})}^{2}}) .$

(15.28)

Notice that we have taken the delay path to the mid-point (a _n
−1 + a _n)/2 of each ring. The nth path length z _n is related to the time delay T _n of the nth section by

z_{n} = c T_{n},

$z_{n} = c T_{n},$

(15.29)

where c is the speed of sound.

Delay Line ignoring Stray Capacitance. In Fig. 15.12, C _Rn are the ring capacitances that are given by

C_{R n} = \frac{ε_{0} π (a_{n}^{2} - a_{n - 1}^{2})}{2 d}

$C_{R n} = \frac{ε_{0} π (a_{n}^{2} - a_{n - 1}^{2})}{2 d}$

(15.30)

C_{R 0} = ε_{0} π a_{0}^{2} / (2 d)

$C_{R 0} = ε_{0} π a_{0}^{2} / (2 d)$

(15.31)

while C _n are shunt capacitors used to make up the required capacitance for the correct delay and impedance. The delay line comprises inductors L _n together with the total capacitances of each section

C_{T n} = C_{R n} + C_{n} / 2

$C_{T n} = C_{R n} + C_{n} / 2$

(15.32)

where R _T is the termination resistance on the far right-hand side of Fig. 15.12. The same resistance R _T is also connected across each inductor to create a series of Zobel networks such that the impedance presented to the preceding section is always 2R _T. The voltage transfer function of each delay section is given by

\frac{{\tilde{e}}_{n}}{{\tilde{e}}_{n - 1}} = \frac{ω_{n}}{s + ω_{n}}

$\frac{{\tilde{e}}_{n}}{{\tilde{e}}_{n - 1}} = \frac{ω_{n}}{s + ω_{n}}$

(15.33)

provided that the inductor values are set to

L_{n} = 2 R_{T}^{2} C_{T n},

$L_{n} = 2 R_{T}^{2} C_{T n},$

(15.34)

where s = jω and the angular turnover frequency is given by

ω_{n} = \frac{1}{2 R_{T} C_{T n}} .

$ω_{n} = \frac{1}{2 R_{T} C_{T n}} .$

(15.35)

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

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

(15.36)

so that the total capacitance per section is given by

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

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

(15.37)

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

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

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

(15.38)

where C _Tn is given by Eq. (15.37) and C _Rn by Eq. (15.30). From Eq. (15.36) we have

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

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

(15.39)

We wish to minimize the capacitor values so that most of the signal current flows through the rings. If we set C ₁ = 0 so that C _T1 = C _R1, then

R_{T} = \frac{z_{1}}{2 c C_{R 1}} .

$R_{T} = \frac{z_{1}}{2 c C_{R 1}} .$

(15.40)

Each delay section is represented 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}]

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

(15.41)

where each element of A _n is given by

a_{11} (n) = {\frac{{\tilde{e}}_{n - 1}}{{\tilde{e}}_{n}} |}_{{\tilde{i}}_{n} = 0} = 1 + 2 \frac{s L_{n} R_{T}}{s L_{n} + R_{T}} s C_{T n},

$a_{11} (n) = {\frac{{\tilde{e}}_{n - 1}}{{\tilde{e}}_{n}} |}_{{\tilde{i}}_{n} = 0} = 1 + 2 \frac{s L_{n} R_{T}}{s L_{n} + R_{T}} s C_{T n},$

(15.42)

a_{12} (n) = {\frac{{\tilde{e}}_{n - 1}}{{\tilde{i}}_{n}} |}_{{\tilde{e}}_{n} = 0} = 2 \frac{s L_{n} R_{T}}{s L_{n} + R_{T}},

$a_{12} (n) = {\frac{{\tilde{e}}_{n - 1}}{{\tilde{i}}_{n}} |}_{{\tilde{e}}_{n} = 0} = 2 \frac{s L_{n} R_{T}}{s L_{n} + R_{T}},$

(15.43)

a_{21} (n) = {\frac{{\tilde{i}}_{n - 1}}{{\tilde{e}}_{n}} |}_{{\tilde{i}}_{n} = 0} = s C_{T n},

$a_{21} (n) = {\frac{{\tilde{i}}_{n - 1}}{{\tilde{e}}_{n}} |}_{{\tilde{i}}_{n} = 0} = s C_{T n},$

(15.44)

a_{22} (n) = {\frac{{\tilde{i}}_{n - 1}}{{\tilde{i}}_{n}} |}_{{\tilde{e}}_{n} = 0} = 1.

$a_{22} (n) = {\frac{{\tilde{i}}_{n - 1}}{{\tilde{i}}_{n}} |}_{{\tilde{e}}_{n} = 0} = 1.$

(15.45)

However, the first section contains no inductor, only the capacitance of the center disk

A_{0} = [\begin{matrix} 1 & 0 \\ s C_{R 0} & 1 \end{matrix}] .

$A_{0} = [\begin{matrix} 1 & 0 \\ s C_{R 0} & 1 \end{matrix}] .$

(15.46)

Hence, we can describe the whole delay line of Fig. 15.12 by multiplying together the chain matrices for all the sections

\begin{matrix} [\begin{matrix} {\tilde{e}}_{in} \\ {\tilde{i}}_{in} \end{matrix}] = A_{0} \cdot A_{1} \dots A_{N} \cdot [\begin{matrix} 1 & 0 \\ {(2 R_{T})}^{- 1} & 1 \end{matrix}] \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] \\ = A \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] \end{matrix}

$\begin{matrix} [\begin{matrix} {\tilde{e}}_{in} \\ {\tilde{i}}_{in} \end{matrix}] = A_{0} \cdot A_{1} \dots A_{N} \cdot [\begin{matrix} 1 & 0 \\ {(2 R_{T})}^{- 1} & 1 \end{matrix}] \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] \\ = A \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] \end{matrix}$

(15.47)

where

{\tilde{e}}_{in}

${\tilde{e}}_{in}$

and

{\tilde{i}}_{in}

${\tilde{i}}_{in}$

are the input voltage and current, respectively, and

{\tilde{e}}_{N}

${\tilde{e}}_{N}$

is the voltage across the termination impedance 2R _T. We evaluate

{\tilde{e}}_{N}

${\tilde{e}}_{N}$

from

{\tilde{e}}_{N} = {\tilde{e}}_{in} / a_{11} .

${\tilde{e}}_{N} = {\tilde{e}}_{in} / a_{11} .$

(15.48)

The then voltage and current at the junction of each section may be calculated by working back from the termination resistor R _T as follows

[\begin{matrix} {\tilde{e}}_{n} \\ {\tilde{i}}_{n} \end{matrix}] = A_{n + 1} \cdot A_{n + 2} \dots A_{N} \cdot [\begin{matrix} 1 & 0 \\ \frac{1}{(2 R_{T})} & 1 \end{matrix}] \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] .

$[\begin{matrix} {\tilde{e}}_{n} \\ {\tilde{i}}_{n} \end{matrix}] = A_{n + 1} \cdot A_{n + 2} \dots A_{N} \cdot [\begin{matrix} 1 & 0 \\ \frac{1}{(2 R_{T})} & 1 \end{matrix}] \cdot [\begin{matrix} {\tilde{e}}_{N} \\ 0 \end{matrix}] .$

(15.49)

Hence the driving pressure produced by each ring is

{\tilde{p}}_{n} = \frac{ε_{0} E_{P}}{d^{2}} {\tilde{e}}_{n}

${\tilde{p}}_{n} = \frac{ε_{0} E_{P}}{d^{2}} {\tilde{e}}_{n}$

(15.50)

{\tilde{p}}_{0} = \frac{ε_{0} E_{P}}{d^{2}} {\tilde{e}}_{in}

${\tilde{p}}_{0} = \frac{ε_{0} E_{P}}{d^{2}} {\tilde{e}}_{in}$

(15.51)

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15.6. A practical delay line

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Acoustics