| Electronics > Projects, Designs, and Technical Stuff |
| Question: Cascading two Sallen-Key low pass filter elements? |
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| Benta:
Yansi, I'm perfectly aware of what a Linkwitz-Riley filter is, and what the rationale behind it is. I've designed several myself. I challenged your assertion that it needs to be Butterworth filters. Bessel, Chebyshev and other responses work just as well. The main point is, that at crossover there is not N x 90 degrees phase shift, but N x 180 degrees, so that a simple amplitude addition doesn't cause peaking. This can be done in many ways: LP/HP, LP/AP+subtraction etc. For a subwoofer LP, there's no reason at all to go for a second-order squared, a full 4th order filter is the better solution. |
| T3sl4co1l:
A trivial proof that Chebyschev cannot be used: The passband ripple in one channel must be balanced by stopband ripple in the other channel. However, [type 1] Chebyschev does not exhibit stopband ripple. Therefore a summed response cannot be flat. Possibly an elliptical type can be designed in this way, but I wonder if the poles and zeroes can actually be placed in such a way that both amplitude and phase are correct for a flat overall response. I don't have the tools to prove that, unfortunately... Now, if you allow that there is ripple in the overall response, you can do whatever you please, and the phase constraint also goes away. The easiest type to prove is the Butterworth, because it is self complementary. This is also relevant to diplexers: a highpass and lowpass can be connected in parallel (using the respective singly-terminated prototypes) to give flat (constant Z) response. No other type can be used in this way, because the complement of a lower-Q (Bessel-like) filter must have higher-Q poles (like a Chebyschev), and therefore zeroes in the impedance response, and therefore you can't simply connect another network in series or parallel to extract what energy would otherwise be reflected. The Butterworth has an all-pole response in both transfer and impedance, and can therefore be connected in this way. On a related note, I wonder what response cellphone base station diplexers have. They use many poles and several zeroes, so they can't be constant-Z; but the antenna doesn't need to see constant Z, it can have power reflected at any bandstop frequency. So long as the antenna still matches with the filter, of course. Tim |
| Benta:
It's true that it won't work with standard LP/HP filters. LP/AP + subtraction does work. |
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