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1074 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS<br />

+<br />

V i<br />

–<br />

0 dB<br />

–3 dB<br />

0 dB<br />

400 Hz<br />

Cmid = 47 �F<br />

0 dB<br />

400 Hz<br />

L low = 3.3 mH<br />

5 kHz<br />

Chigh = 3.9 �F<br />

–3 dB<br />

dB<br />

For the crossover network of Fig. 23.83 with three 8-� speakers, the<br />

resulting values are<br />

R 8 �<br />

Llow ����� �3.183 mH 3.3 mH<br />

2pf1 2p(400 Hz)<br />

(commercial value)<br />

R 8 �<br />

Lmid ����� �254.65 mH 270 mH<br />

2pf2 2p(5 kHz)<br />

(commercial value)<br />

1<br />

1<br />

Cmid ����� �49.736 mF 47 mF<br />

2pf1R 2p(400 Hz)(8 �)<br />

(commercial value)<br />

1<br />

1<br />

Chigh ����� �3.979 mF 3.9 mF<br />

2pf2R 2p(5 kHz)(8 �)<br />

(commercial value)<br />

as appearing on Fig. 23.83.<br />

For each filter, a rough sketch of the frequency response is included<br />

to show the crossover at the specific frequencies of interest. Because all<br />

three speakers are in parallel, the source voltage and impedance for<br />

each are the same. The total loading on the source is obviously a function<br />

of the frequency applied, but the total delivered is determined<br />

solely by the speakers since they are essentially resistive in nature.<br />

To test the system, let us apply a 4-V signal at a frequency of 1 kHz<br />

(a predominant frequency of the typical human auditory response<br />

curve) and see which speaker will have the highest power level.<br />

At f � 1 kHz,<br />

XLlow � 2pfLlow � 2p(1 kHz)(3.3 mH) � 20.74 �<br />

(ZR �0°)(Vi �0°) (8 ��0°)(4 V�0°)<br />

Vo ��� ���<br />

ZT<br />

8 ��j 20.74 �<br />

� 1.44 V��68.90°<br />

8 �<br />

5 kHz<br />

Lmid = 270 �H<br />

8 �<br />

–3 dB<br />

8 �<br />

–3 dB<br />

FIG. 23.83<br />

Three-way, 6-dB-per-octave, crossover network.

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