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This problem focuses on the (greatly simplified) design of cross- over networks for audio speaker systems driven by a single amplifier. The purpose of these networks is to direct low-frequency signals to a low-frequency (LF) speaker, mid-range signals to a mid-range (MR) speaker, and high-frequency signals to a high-frequency (HF) speaker. The interconnection of the amplifier, modeled here as a cosinusoidal voltage source, the three speakers, each modeled here as a resistor, and the three cross-over networks is shown below.

The three cross-over networks N_{LF}, N_{MR}, and N_{HF} may each be a single capacitor, a single inductor, a series capacitor and inductor, or a parallel capacitor and inductor. The topology of each network, and the values of the component(s) in each network, are to be designed to provide the three voltage responses shown below as functions of frequency.

Note that \omega_L and \omega_H are the frequencies at which |v_{LF}|, |v_{MR}| and |v_{HF}| approximately fall to the value of V_a/\sqrt{2}. In this problem, we'll assume that \omega_L << \omega_H.

For each of the netowrks N_{\rm LF}, N_{\rm MR}, and N_{\rm HF}, what type of network should be used: a single capacitor; a single inductor; a series capacitor and inductor; or a parallel capacitor and inductor? For each, specify the necessary capacitances/inductances in terms of R_{\rm LF}, R_{\rm MR}, and R_{\rm HF}.

 
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