Which method is more preferred?

Suppose a second-order low-pass filter has been being tweaked for smoothing amplitude of the woofer’s output at its cut-off region. As fas as I can think, there are two ways to achieve the goal. One is to swap the inductor, Dr1, for a larger value i.e. 3mH to 3.5mH and a little tweak on the capacitor, C1, may be required. The other way, which I have seen in many designs but I am unsure if I understand it correctly, is to add a small resistor, R1, in series with the original shunt capacitor, C1. As I tried playing with simulation software, both methods can lead the result to the goal. But I am still curious which method is actually more accurate/ preferred, please?

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Yes, that will pull the beginning of the cross lower, but it will also tend to reduce the slope. If you want to damp the existing filter at the knee without changing the out of band behaviour, you could put a larger resistor in parallel with the driver.
 
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