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9.4. Frequency response

The frequency response of a complete horn loudspeaker, in the range where the throat impedance of the horn is a resistance as given by Eq. (9.1), is determined by solution of the circuit of Fig. 9.3. A horn-loaded drive unit behaves very differently from a direct radiator. The diaphragm of a direct-radiator loudspeaker is mass controlled because a flat on-axis response is given at frequencies where the acceleration of the diaphragm is constant. Because the velocity decreases with frequency, so does the radiated power for ω > c/a, but this is compensated for by an increasingly narrow directivity pattern, which is how the flat on-axis response is maintained. By contrast, a horn has a fairly constant directivity pattern over its operating frequency range. Hence, for a flat frequency response, the radiated power must also be constant, which can only be achieved if the velocity is constant. Hence the diaphragm of horn-loaded drive unit is resistance controlled. For purposes of analysis, we shall divide the frequency range into three parts, A, B, and C, as shown in Fig. 9.5.

Figure 9.5 Normalized frequency response of the mechanical force $\tilde{f}$ $\tilde{f}$ _T or velocity $\tilde{u}$ $\tilde{u}$ _T at the throat of a horn drive unit, in the frequency region where the mechanical impedance at the throat is a pure resistance ρ ₀ cS _T. The ordinate is a logarithmic scale, proportional to decibels.

Midfrequency range

In the midfrequency range, designated as B in Fig. 9.5, the response is equal to the reference efficiency given by Eq. (9.8). Here, the response is “flat” with frequency, and, for the usual high-frequency units used in auditoriums with 300-Hz cutoff frequencies, the flat region extends from a little above 500 to a little below 3000 Hz. In this region the velocity of the diaphragm is constant with frequency, rather than decreasing in inverse proportion to frequency as was the case for a direct-radiator loudspeaker.

Resonance frequency

It is apparent from Fig. 9.3 that because ωC _M1 is small, zero reactance will occur at the frequency where

f_{0} = \frac{1}{2 π \sqrt{M_{M D} (C_{M S} C_{M B} / (C_{M S} + C_{M B}))}} .

$f_{0} = \frac{1}{2 π \sqrt{M_{M D} (C_{M S} C_{M B} / (C_{M S} + C_{M B}))}} .$

(9.12)

In practice, this resonance usually is located in the middle of region B of Fig. 9.5 and is heavily damped by the conductance G _MT, so that the velocity of the diaphragm is resistance controlled.

Low frequencies

At frequencies well below the resonance frequency, the response will drop off 6 dB for each octave decrease in frequency if the throat impedance is a resistance as given by Eq. (9.1). This case is shown as region A in Fig. 9.5.

Figure 9.6 Analogous circuit for a horn drive unit in the region where the diaphragm would be stiffness controlled if the horn admittance were infinite. The actual value of the mechanical admittance of the horn at the throat is z _MT.

Let us simplify Fig. 9.3 so that it is valid only for the low-frequency region, well below the resonance of the diaphragm. Then the inductance L _E, the mass M _MD, the compliance C _M1, and the conductances G _MS and G _MB may all be dropped from the circuit, giving us Fig. 9.6.

Assuming the throat admittance of the horn is a pure conductance as given by Eq. (9.3), the frequency at which the frequency response is 3 dB down is given in terms of the Thiele–Small parameters by

ω_{L} = (1 + \frac{V_{A S}}{V_{B}}) \frac{ω_{S} Q_{E S} S_{D} c}{S_{D} c + (1 + S_{D} / S_{T}) ω_{S} Q_{E S} V_{A S}} .

$ω_{L} = (1 + \frac{V_{A S}}{V_{B}}) \frac{ω_{S} Q_{E S} S_{D} c}{S_{D} c + (1 + S_{D} / S_{T}) ω_{S} Q_{E S} V_{A S}} .$

(9.13)

where V _B is the volume of the back cavity. In practice, however, the throat impedance Z _MT of the horn near the lowest frequency at which one wishes to radiate sound is not a pure resistance. Hence, region A needs more careful study. Solving for the mechanical admittance at the diaphragm of the drive unit yields

Y_{M c} = \frac{{\tilde{u}}_{c}}{{\tilde{f}}_{c}} = \frac{j ω C_{M 2} {(S_{T} / S_{D})}^{2} Y_{M T}}{j ω C_{M 2} + {(S_{T} / S_{D})}^{2} Y_{M T}}

$Y_{M c} = \frac{{\tilde{u}}_{c}}{{\tilde{f}}_{c}} = \frac{j ω C_{M 2} {(S_{T} / S_{D})}^{2} Y_{M T}}{j ω C_{M 2} + {(S_{T} / S_{D})}^{2} Y_{M T}}$

(9.14)

where

C_{M 2} = \frac{C_{M S} C_{M B}}{C_{M S} + C_{M B}}

$C_{M 2} = \frac{C_{M S} C_{M B}}{C_{M S} + C_{M B}}$

(9.15)

and Y _MT is the mechanical admittance at the throat of the horn with area S _T. The mechanical impedance at the diaphragm of the drive unit is the reciprocal of Y _Mc,

Z_{M c} = \frac{{\tilde{f}}_{c}}{{\tilde{u}}_{c}} = {(\frac{S_{D}}{S_{T}})}^{2} Z_{M T} - j \frac{1}{ω C_{M 2}}

$Z_{M c} = \frac{{\tilde{f}}_{c}}{{\tilde{u}}_{c}} = {(\frac{S_{D}}{S_{T}})}^{2} Z_{M T} - j \frac{1}{ω C_{M 2}}$

(9.16)

where Z _MT = 1/Y _MT is the mechanical impedance at the throat of the horn with area S _T.

As we shall show in the next part, the mechanical impedance at the throat of ordinary types of horn at the lower end of the useful frequency range is equal to a mechanical resistance in series with a negative compliance. That is to say,

Z_{M T} \equiv R_{M T} + j \frac{1}{ω C_{M T}} .

$Z_{M T} \equiv R_{M T} + j \frac{1}{ω C_{M T}} .$

(9.17)

The bold R _MT indicates that this resistance varies with frequency. Usually, its variation is between zero at very low frequencies and ρ ₀ cS _T (as given by Eq. 9.1) at some frequency in region A of Fig. 9.5. Hence, the admittance Y _MT = l/Z _MT is a resistance in series with a negative mass reactance. In the frequency range where this is true, therefore, the reactive part of the impedance Z _Mc can be canceled out by letting (see Eqs. 9.16 and 9.17)

\frac{S_{D}^{2}}{S_{T}^{2}} \frac{1}{C_{M T}} = \frac{1}{C_{M 2}} = (\frac{1}{C_{M B}} + \frac{1}{C_{M S}}) .

$\frac{S_{D}^{2}}{S_{T}^{2}} \frac{1}{C_{M T}} = \frac{1}{C_{M 2}} = (\frac{1}{C_{M B}} + \frac{1}{C_{M S}}) .$

(9.18)

Then,

Z_{M c} = R_{M T} {(\frac{S_{D}}{S_{T}})}^{2} \equiv \frac{1}{G_{M c}}

$Z_{M c} = R_{M T} {(\frac{S_{D}}{S_{T}})}^{2} \equiv \frac{1}{G_{M c}}$

(9.19)

where G _Mc is the acoustic conductance of the throat of the horn at low frequencies transformed to the diaphragm.

The efficiency for frequencies where the approximate circuit of Fig. 9.6 holds, and where the conditions of Eq. (9.18) are met, is

E_{f f} = \frac{100 B^{2} l^{2} G_{M c}}{R_{E} + B^{2} l^{2} G_{M c}}

$E_{f f} = \frac{100 B^{2} l^{2} G_{M c}}{R_{E} + B^{2} l^{2} G_{M c}}$

(9.20)

assuming R _g >> R _E. The conductance G _Mc usually varies from “infinity” at very low frequencies down to

S_{T} / (S_{D}^{2} ρ_{0} c)

$S_{T} / (S_{D}^{2} ρ_{0} c)$

at some frequency in region A of Fig. 9.5.

High frequencies

At very high frequencies, the response is limited principally by the combined mass of the diaphragm and the voice coil M _MD. If the compliance C _M1 of the front cavity were zero, the response would drop off at the rate of 6 dB per octave (see region C of Fig. 9.5). It is possible to choose C _M1 to resonate with M _MD at a frequency that extends the response upward beyond where it would extend if it were limited by M _MD alone. We can understand this situation by deriving a circuit valid for the higher frequencies as shown in Fig. 9.7 . It is seen that a damped antiresonance occurs at a selected high frequency ω _U, which is given in terms of the Thiele–Small parameters of the drive unit by

ω_{U} = ω_{S} \sqrt{\frac{V_{A S}}{V_{F}} {1 + \frac{S_{T}}{S_{D}} (1 + \frac{S_{D} c}{ω_{S} Q_{E S} V_{A S}})}}

$ω_{U} = ω_{S} \sqrt{\frac{V_{A S}}{V_{F}} {1 + \frac{S_{T}}{S_{D}} (1 + \frac{S_{D} c}{ω_{S} Q_{E S} V_{A S}})}}$

(9.21)

with a Q_U value of

Q_{U} = ω_{U} {\frac{S_{T} c}{V_{F}} + ω_{S} (\frac{1}{Q_{E S}} + \frac{ω_{S} V_{A S}}{S_{D} c})}^{- 1}

$Q_{U} = ω_{U} {\frac{S_{T} c}{V_{F}} + ω_{S} (\frac{1}{Q_{E S}} + \frac{ω_{S} V_{A S}}{S_{D} c})}^{- 1}$

(9.22)

where V _F is the volume of the front cavity. Above this resonance frequency, the response drops off 12 dB for each octave increase in frequency (see region C of Fig. 9.5).

Because the principal diaphragm resonance (Eq. 9.12) is highly damped by the throat resistance of the horn, it is possible to extend the region of flat response of a drive unit over a range of four octaves by proper choice of C _M1 at higher frequencies and by meeting the conditions of Eq. (9.18) at lower frequencies.

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Table of Contents for Acoustics

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9.4. Frequency response

Midfrequency range

Resonance frequency

Low frequencies

High frequencies

Table of Contents for
Acoustics