Linkwitz-Riley Filter :
Here is a catalog of line-level circuits that I have found useful for building active loudspeakers. Many other topologies are possible, but one should always analyze a circuit's signal handling capability and its contribution to overall system noise before choosing it. A CAD software package such as CircuitMaker is most convenient for analyzing and designing active filters. LspCAD software allows you to see how an active filter changes the measured frequency response of a driver and lets you optimize it to a target response. All the line level filters below are included in LspCAD standard and professional versions. Component values for all the filters below and for a dual power supply can be determined from a circuit design spreadsheet contributed by Bernhard Faulhaber. It covers more cases than the earlier spreadsheet by Alister Sibbald.
1 - Buffer stage
1 - Buffer stage
A buffer as the first stage of an active crossover/equalizer provides the necessary low source impedance to the following filter networks. The buffer also provides a high impedance load to the preamplifier output circuit and the option of a highpass filter for dc blocking. (w-xo-lp2.gif, pmtm-eq1.gif, 38xo_eq.gif) Top
2 - 12 dB/oct Linkwitz-Riley crossover
The two outputs from the LR2 crossover filter are 180 degrees out of phase at all frequencies, which requires to use one of the drivers with reversed polarity, so that the two acoustic outputs add in phase. At the crossover frequency the filter outputs are 6 dB down.
The LR2 circuit uses the Sallen-Key active filter topology to implement the 2nd order transfer function. The response is defined by w0 and Q0 which sets the location of a pole pair in the complex frequency s-plane and by an additional two zeros at s = 0 for the highpass filter. In the case of the LR2 filters Q0 = 0.5, and Q0 = 0.71 for each of the two cascaded 2nd order filters that form the LR4 filter. The frequency response is obtained by setting s = jw and solving the transfer function for magnitude and phase. The formulas below can be used to design filters with different values for w0 or Q0, or to analyze a given circuit for its w0 and Q0 values.
Any order Linkwitz-Riley filters can be implemented by a cascade of 2nd order Sallen-Key filters. The Q0 values for each stage are listed in the table below. The component values of each stage for a given crossover frequency f0 can be calculated by using Q0 and selecting a convenient value for C2 or R2 in the formulas above.
Crossover filters of higher order than LR4 are probably not useful, because of an increasing peak in group delay around f0.
3 - 24 dB/oct Linkwitz-Riley crossover
The 24 dB/oct LR4 crossover filter provides outputs which are 360 degrees offset in phase at all frequencies. At the transition frequency Fp the response is 6 dB down. The electrical network will only give the targeted exact acoustic filter response, if the drivers are flat and have wide overlap. This is seldom the case. The steep filter slopes make the combined acoustic response less sensitive to magnitude errors in the driver responses, but phase shift errors usually have to be corrected with an additional allpass network. (xo12-24b.gif, 38xo_eq1.gif, models.htm#E) Top
4 - Delay correction
A first order allpass filter section with flat amplitude response but phase shift that changes from 0 degrees to -180 degrees, or -180 degrees to -360 degrees, is often used to correct phase response differences between drivers. Multiple sections may delay the tweeter output and compensate for the driver being mounted forward of the midrange. Active crossover circuits that do not include phase correction circuitry are only marginally useable. (allpass.gif, allpass2.gif, models.htm#E, 38xo_eq1.gif) Top
5 - Shelving lowpass
This type of circuit is useful to bring up the low frequency response in order to compensate for the high frequency boost from front panel edge diffraction. It can also serve to equalize the low frequency roll-off from an open baffle speaker. (shlv-lpf.gif, 38xo_eq1.gif) Top
A passive RC version of the shelving lowpass is shown below.
6 - Shelving highpass
A passive RC version of the shelving highpass is shown below.
7 - Notch filter
Notch filters are used to introduce dips in the frequency response in order to cancel driver or room resonances. The three circuits above have the same response. A) is difficult to realize because of the large inductor. B) is used to remove the peak in the 6 dB/oct dipole response. C) gives convenient component values for room EQ below 100 Hz. (room EQ, inductr1.gif, inductr2.gif, 38xo_eq1.gif ) Top
8 - 6 dB/oct dipole equalization
Equalization of the dipole frequency response roll-off usually requires not only a 6 dB/oct boost towards low frequencies, but also removal of a peak in the response. The three circuits differ in their ability to remove such peak.
A) The shelving lowpass filter cannot correct for a peak.
9 - 12 dB/oct highpass equalization ("Linkwitz Transform", Biquad)
A majority of drivers exhibit second order highpass behavior because they consist of mechanical mass-compliance-damping systems. They are described by a pair of zeroes at the s-plane origin and a pair of complex poles with a location defined by Fs and Qt. The circuit above allows to place a pair of complex zeroes (Fz, Qz) on top of the pole pair to exactly compensate their effect. A new pair of poles (Fp, Qp) can then be placed at a lower or a higher frequency to obtain a different, more desirable frequency response.
The associated phase and group delay responses are shown below.
Not only is the frequency response extended, but the time response is also improved, as indicated by the reduced overshoot and ringing of the lower cut-off highpass filter step response.
It can be seen from the s-plane description of the transfer functions that the complex poles of the driver in the box are canceled by a set of complex zeros in the equalizer. The specified real axis poles of the equalizer, together with the driver zeros at the s-plane origin, determine the overall loudspeaker response in frequency and time.
The equalizer action is difficult to visualize in the time domain, because the driver output waveform is the convolution of the input signal s(t) with the impulse response of the equalizer h1(t), which in turn must be convolved with the impulse response h2(t) of the driver. Convolution is a process whereby the current value of the time response is determined by the time weighted integral over past behavior. Below are the responses of driver, equalizer and driver-equalizer combination, if the input signal s(t) is an impulse.
More illustrative are the responses to a 4-cycle, rectangular envelope 70 Hz toneburst s(t). For example, the driver output is the convolution of the burst s(t) with the driver's impulse response h2(t). Note that the driver phase leads the input signal, as would be expected for a highpass response. Upon turn-off of the input burst at 57.14 ms the driver response rings towards zero, governed by Fp = 55 Hz and Qp = 1.21.
The equalizer output response lags its burst input. This signal will force upon the driver a response correction so that it is no longer dominated by Fp = 55 Hz and Qp = 1.21. The equalizer output signal is convolved with the impulse response h2(t) of the driver to obtain the desired equalized driver output. Now, the decay of the driver output follows the 2nd order highpass filter response determined by Qp = 0.5 and Fp = 19 Hz of the equalizer, after the excitation has stopped.
The above circuit can also be used to correct the low frequency roll-off of a tweeter so that the equalized tweeter becomes a filter section in an exact LR4 acoustic highpass. (f0Q0fpQp.gif, pz-eql.xls, f0Q0.gif, FAQ15, sb80-3wy.htm, sb186-48.gif , sb186-50.gif) Top
10 - Variable gain & fixed attenuation
A major advantage of line-level active crossovers is the efficiency with which drivers of different sensitivity can be combined in a speaker system. The three circuits use linear taper potentiometers but obtain a gain variation that is approximately linear in dB. Circuits B and C assume a 10k ohm load such as the input impedance of the power amplifier. Circuit A is optimal between filter stages because of its low output impedance. The placement of the variable gain stage in the filter chain must be carefully considered, because it affects noise performance and signal handling. (gain-adj.gif, attnrout.gif, 38xo_eq1.gif) Top
Occasionally a fixed attenuation of A dB or a is needed for the input voltage V2 of a circuit stage with input impedance R3 when driven from an operational amplifier with output voltage V1. In the example below a 3 dB (a=1.41) attenuation is desired. The load Rin that is seen by the opamp should be about 2000 ohm. The following amplifier stage has an input impedance of 10k ohm.
For designing an attenuator with specified output impedance Rout see: attnrout.gif
11 - Line driver
The output stage of the filter must be capable of driving cables, which typically have a capacitances in the order of 150 pF per meter length, without going into oscillation. A 196 ohm resistor maintains a resistive load component and tying output to negative input for out-of-band frequencies (>100 kHz) reduces loop gain. All of the above circuits can drive cables if operational amplifiers such as the OPA2134 or OPA2604 are used. In most cases it is not necessary to have a separate line driver.
Performance of active circuits should always be checked for inter-stage clipping, and for oscillation with a wideband (>10 MHz) oscilloscope. Top
12 - Power supply
I recommend to leave the effort of building a regulated power supply to one of the many vendors that offer wallplug and tabletop models. An output specification of +/-12 V to +/-15 V DC at >250 mA and with <1% ripple and noise will suffice. Often such supplies can be found at electronic surplus stores. Top
13 - Printed circuit boards WM1 and MT1
To simplify the construction of active line-level equalizers and crossovers I offer three printed circuit boards, ORION/ASP, WM1 and MT1. The circuit traces are laid out to allow for a variety of filter designs. It is up to the user to determine the actual circuit configuration and component values. Then the necessary components and jumpers are loaded at the appropriate locations on the board to obtain the desired filter response. I will provide specific information for assembling the PHOENIX crossover/equalizer on the ORION/ASP pcb and a Linkwitz Transform on the WM1 pcb.
WM1 is designed to implement the functionality of circuits 1, 5, 6, 7, 8, 9 or 10 and various combinations of these. The circuit board provides two of the topologies below for two channels of equalization or for a more elaborate single channel response correction.
The WM1 board can be used for:
MT1 is designed to implement the functionality of circuits 1, 2, 3, 4, 5, 10 or 11 and various combinations of these. On the circuit board are two of the topologies below.
The MT1 board can be used to construct:
The circuit boards are practical tools to experiment with and to learn about active electronics. You will find that active loudspeaker systems give you the freedom to match drivers of greatly different sensitivities, are easier to design, and can give greater accuracy of sound reproduction, than is possible with passive, high-level crossovers and filters.
14 - Literature
Much useful information can be obtained from application notes of the various opamp manufacturers. If you need a refresher or an introduction to circuits, then read:
 Martin Hartley Jones, A practical introduction to electronic circuits, Cambridge University Press, 1995. It is a well illustrated, easy to read, yet technically solid text. It covers a broad range of devices - from tubes to ICs - and many basic circuit functions.
The following books cover a range of concepts and go into depth on specific, relevant topics to strengthen understanding of electronic circuits and electro-acoustic models.
 Herman J. Blinchikoff & Anatol I. Zverev, Filtering in the Time and Frequency Domains, John Wiley, 1976. A broad and fundamental look at filters.