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A third- or fourth-order acoustic crossover often has just a second-order electrical filter. This requires that speaker drivers be well behaved a considerable way from the nominal crossover frequency, and further that the high-frequency driver be able to survive a considerable input in a frequency range below its crossover point.
Second-order Linkwitz–Riley crossovers (LR2) have a 12 dB/octave (40 dB/decade) slope. They can be realized by cascading two one-pole filters or using a Sallen Key filter topology with a Q 0 value of 0.5. There is a 180° phase difference between the low-pass and high-pass output of the filter, which can be corrected by inverting one signal.
The midwoofer-tweeter-midwoofer loudspeaker configuration (called MTM, for short) was a design arrangement from the late 1960s that suffered from serious lobing issues that prevented its popularity until it was perfected by Joseph D'Appolito as a way of correcting the inherent lobe tilting of a typical mid-tweeter (MT) configuration, at the crossover frequency, unless time-aligned. [1]
A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with = 4/3 F, = 1 Ω, = 3/2 H, and = 1/2 H. [3] Taking the impedance of the capacitors to be / and the impedance of the inductors to be , where = + is the complex frequency, the circuit equations yield the transfer function for this device:
Compared to finite-order approximations of the Gaussian filter, the Bessel filter has a slightly better shaping factor (i.e., how well a particular filter approximates the ideal lowpass response), flatter phase delay, and flatter group delay than a Gaussian filter of the same order, although the Gaussian has lower time delay and zero overshoot. [8]
Higher-order passive filters can also be constructed (see diagram for a third-order example). A third-order low-pass filter ( Cauer topology ). The filter becomes a Butterworth filter with cutoff frequency ω c =1 when (for example) C 2 =4/3 farad, R 4 =1 ohm, L 1 =3/2 henry and L 3 =1/2 henry.