# Linkwitzâ€“Riley filter

A Linkwitz- Riley filters (LR ) filter is an electronic filter which is used in Linkwitz -Riley crossovers. It is named after its inventors Siegfried Linkwitz and Russ Riley. It is also known as a double Butterworth filter, since it is formed by cascading two Butterworth filter.

## General

A LR crossover consists of a parallel arrangement of a respective LR low-pass filter and a high pass LR whose outputs eg can be used for driving power amplifiers for operation of multi- way active speaker systems. The rated frequency of both filters is the same and is called at crossover arrangement takeover frequency.

LR- filters are realized by cascading two Butterworth filter, each having a gain of -3 dB at its rated frequency. The resulting Linkwitz -Riley filter now has a gain of -6 dB In measuring frequency.

In summation of the high-pass and low-pass output, the gain is 0 dB at the crossover frequency. The whole arrangement behaves like an all-pass with a flat amplitude and frequency of the slowly varying phase. The flat amplitude response is compared to Butterworth crossovers, the biggest advantage because they have a gain of 3 dB at the acquisition frequency for summing the outputs.

Cascading two Butterworth filter of order n give a Linkwitz- Riley filters 2n -th order. Theory can be built this way each LR crossover 2n -th order. Crossovers with higher than fourth order are hardly used because the group delay at the crossover frequency increases around too much. In addition, for larger orders than 4 the complexity of the electrical structure rises too much. This extra components makes the system error and interference-prone and too dependent on tolerances of the individual components.

## Frequently used types

### Second order ( LR2, LR -2)

Linkwitz -Riley second order crossovers have a slope of 12 dB / octave (40 dB / decade ). They can be of two filters liabilities build first-order or Sallen -Key topology with a quality factor of Q0 = 0.5. The phase difference of 180 ° between low-pass and high -pass output of the switch can be compensated by inverting a signal. In speakers, this is achieved by reversing the polarity of a chassis, if one uses passive crossovers. With active crossovers is added as a inverting amplifier unit wired operational amplifier ( op amp ) is added or leaves an OPV invert the signal. In this balanced cabling can both signal wires (the " cold" to the "hot " wire) are interchanged.

### Fourth -order ( LR4, LR -4)

Linkwitz -Riley crossovers fourth order are the most popular speaker crossovers. Two Butterworth filter can be cascaded to it. The slope is 24 dB / octave (80 dB / decade ). The phase difference between low-pass and high -pass output of the crossover is 360 degrees so they appear to be in phase with the low-pass lags a full period.

### Eighth order ( LR8, LR -8)

Linkwitz -Riley crossovers are eight order of 48 dB / octave (160 dB / decade ) is very steep. They are built of two cascaded Butterworth 4th order filter.