9.5. Examples of horn calculations

Example 9.1. Find the maximum efficiency of a 2-inch theater horn drive unit designed to operate in the frequency range above 500   Hz with the following Thiele–Small parameters:
  • R E   =   6.4   Ω
  • Q ES   =   0.8
  • f S   =   250   Hz
  • S D   =   13.2   cm2
  • V AS   =   0.1   L.
Solution. From Eq. (9.9) the optimum throat area is
ST=13.2×10−4×2×3.14×250×0.8×0.1×10−313.2×10−4×348.8+2×3.14×250×0.8×0.1×10−3=6.11cm2,
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which from Eq. (9.11) gives a maximum efficiency of
Eff(max)=100SDc(13.2×10−4×348.8+2×3.14×250×0.8×0.1×10−3+2×3.14×250×0.8×0.1×10−3)2=36.7%.
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Example 9.2. To extend the high-frequency response of the previous example, reduce the throat area to S T   =   0.66   cm2 (S T /S D   =   0.05) and recalculate the efficiency. Also determine the upper resonance frequency and the Q of the resonance, as well as the theoretical lower cutoff frequency that would be obtained with an infinite horn of low cutoff frequency and a 0.1   L back cavity (i.e., V B   =   V AS ).
Solution. Then we obtain the efficiency from Eq. (9.8), which gives
Eff=100×0.05×13.2×10−1×348.8(1+0.05)((13.2×10−1×348.8+2×3.14×250×0.8×0.1)×0.05+2×3.14×250×0.8×0.1)=14.2%,
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which is still quite good. If we set the depth d of the front cavity to d   =   1   mm so that V F   ≈   dS D   =   1.32   cm3, we obtain the high-frequency resonance from Eq. (9.21), which gives
fU=2500.1×10−31×10−3×13.2×10−4{1+0.05×(1+13.2×10−4×348.82×3.14×250×0.8×0.1×10−3)}=2.42kHz.
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Using Eq. (9.22) we obtain a Q value of
QU=2×3.14×2420×{0.66×10−4×348.810−3×13.2×10−4+2×3.14×250(10.8+2×3.14×250×0.1×10−313.2×10−4×348.8)}−1=0.762,
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which is fairly optimal for a flat frequency-response with a smooth roll-off. Finally, from Eq. (9.13) we obtain the following lower cutoff frequency with an ideal infinite horn:
fL=12×3.14(1+1)2×3.14×250×0.8×13.2×10−4×348.813.2×10−4×348.8+(1+20)×2×3.14×250×0.8×0.1×10−3=59.4Hz.
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Of course this is unrealistic in practice as the lower cutoff frequency is more likely to be determined by the characteristics of the horn, which we shall examine in the next section. The full frequency response is plotted in Fig. 9.5.
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