Active Filters




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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

2 – 12 dB/oct Linkwitz-Riley crossover

3 – 24 dB/oct Linkwitz-Riley crossover

4 – Delay correction

5 – Shelving lowpass & passive circuit

6 – Shelving highpass & passive circuit

7 – Notch filter

8 – 6 dB/oct dipole equalization

9 – 12 dB/oct highpass equalization
(“Linkwitz Transform”, Biquad)

10 –
Variable gain & fixed attenuation

11 –
Line driver

12 –
Power supply
13 – Printed circuit boards

14 –
Literature


—————————————————————–


1 – Buffer stage

buffer - Active Filters

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

LR2 - Active Filters

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 acoustic frequency and polar response is controlled by the electrical
filters and the response of the mounted drivers. The electrical filter will not
give the desired results, if there is insufficient overlap and flatness of the
driver frequency response and when they are offset from each other. This can be corrected in
many cases with the addition of a phase shift correcting network. I consider the
crossover marginally useful, because the 12 dB/oct roll-off of the
highpass filter below the crossover frequency does not reduce the excursions of
a driver’s cone when flat frequency response is obtained. My earlier assumption
that the group delay of a 4th order LR4 crossover
at low frequencies would introduce audible distortion was not correct. Therefore
I recommend not to use the LR2 crossover. (38xo_eq1.gif,
FAQ19, xo12-24b.gif)

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.

lp 2 - Active Filters

hp 2 - Active Filters

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.








LR2 LR4 LR6 LR8 LR10
Q0 of stage 1 0.5 0.71 0.5 0.54 0.5
Q0 of stage 2 0.71 1.0 1.34 0.62
Q0 of stage 3 1.0 0.54 1.62
Q0 of stage 4 1.34 0.62
Q0 of stage 5 1.62
dB/octave slope 12 24 36 48 60

Crossover filters of higher order than LR4 are probably
not useful, because of an increasing peak in group delay around f0.

Top


3 – 24 dB/oct Linkwitz-Riley crossover

LR4 - Active Filters

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



IMG 2344c3as - Active Filters

Russ Riley and Siegfried Linkwitz, September 2006, Douglas City, CA
In the
sixties,
early seventies, I
worked with Russ Riley at Hewlett-Packard’s Palo Alto R&D laboratory
for the development of RF and Microwave test equipment. Like many other
engineers we had “G-Jobs”, building such things as
electronic ignitions for our VW bugs and vans, FM receivers, phase-locked
pulse width FM demodulators, short-wave receivers, audio pre- and power
amplifiers, third octave audio analyzers, headphone equalizers, and of
course, loudspeakers. After measuring the acoustic and electrical
responses of commercial speakers we equalized them and tried to understand
why they were designed with strange looking driver layouts, used large
baffles, were stuffed with a variety of internal damping materials and
used various box stiffening and damping techniques. Eventually we
completely redesigned them and built our own speakers. Russ and his wife,
Vicky, an accomplished organist, always had the most critical and reliable ears. He
was an ingenious design engineer, a strong contributor, who inspired and
challenged many of us on our HP and unofficial design projects.

Russ retired after over 40 years
in R&D for HP/Agilent and now lives
with his wife in a remote mountain valley, in a genuine log cabin, amongst
pear, plum and walnut trees, berry bushes, chicken and deer, the sounds of a large creek, and
the pine and fir trees that climb up the slopes. He died peacefully
in his log cabin on December 6, 2010.

4 – Delay correction

delay - Active Filters

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

shlv lp - Active Filters

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.

passive down shelf - Active Filters


6 –
Shelving highpass

shlv hp - Active Filters

A circuit used to boost high frequencies or to smooth the
transition between a floor mounted woofer and a free standing midrange. (shlv-hpf.gif,
38xo_eq1.gif, models.htm#F) Top

A passive RC version of the shelving highpass is shown
below.

passive up shelf - Active Filters


7 – Notch filter

notch2 - Active Filters

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

notch filter - Active Filters



8 – 6 dB/oct dipole equalization

dpl eq2 - Active Filters

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. (Models A2)
The three circuits differ in their ability to
remove such peak.

dpl eq3 - Active Filters

A) The shelving lowpass filter cannot correct for a peak.

B) The bridged-T based circuit is limited in the shape
of curves that can be realized. It has also higher gain for opamp noise
than signal at high frequencies.

C) The shelving lowpass with added notch filter is the most flexible circuit. (models.htm#D)
Top



9 – 12 dB/oct highpass equalization
(“Linkwitz Transform”, Biquad)

pz eq2 - Active Filters

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.

This allows to extend the response of a closed box woofer to lower frequencies,
in the above circuit example from 55 Hz to 19 Hz, provided the driver has adequate
volume displacement capability and power handling.
The equalizer frequency response is shown below, correcting for a woofer with
peaked response (Qp = 1.21) and early roll-off (Fp = 55 Hz), to obtain a
response that is 6 dB down at 19 Hz and with Q = 0.5 .

pz frsp - Active Filters

The associated phase and group delay responses are shown
below.

pz frsp2 - Active Filters

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.

pz step - Active Filters

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.

pz plane - Active Filters

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.

pz impulse - Active Filters

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.

pz time - Active Filters

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.

Of course, none of the driver mechanical
parameters like mass, compliance and damping have been changed in the process of
equalization, only the input signal to the driver has been modified.

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)

The ‘CFL
Linkwitz Transform Designer with Monte Carlo Sensitivity Ananlysis
‘ by
Charlie Laub makes component value selection easy and shows the effect of
component tolerances upon the frequency response. Keep in mind that the LT is
based on a measurement of driver parameters Fs and Qt. Only the small signal
parameters are easy to define. Fs and Qt change with increasing signal level and
to varying degree for different drivers. This makes the equalization imprecise,
but it remains effective in practice.

Top




10
– Variable gain & fixed attenuation

gainadj3 - Active Filters

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.

passive atten - Active Filters

For designing an attenuator with specified output
impedance Rout see: attnrout.gif




11
– Line driver

linedrv1 - Active Filters

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.

wm1 - Active Filters

The WM1 board can be used for:

  • Equalization of an existing speaker with passive
    crossovers, baffle step correction and extension of the low frequency
    response.
  • Pole-zero equalization of a closed box woofer and a LR2
    crossover lowpass filter. Variable gain.
  • Pole-zero equalization of a midrange and a LR2 crossover
    highpass filter.
  • Dipole woofer equalization with notch and variable
    gain. LR2 crossover lowpass.
  • Dipole woofer equalization for low Qts drivers.
  • Low frequency, individual channel and overall response
    equalization of multi-way speakers, so long as elements of this topology
    allow you to generate the response you need.
  • Equalization of add-on
    woofer
    , FAQ10, FAQ15

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.

mt1 - Active Filters

The MT1 board can be used to construct:

  • A 2-way speaker with crossovers of order 1, 2, 3, or 4.
    The tweeter channel has variable gain and delay circuitry to align the
    tweeter’s acoustic center with the woofer. The input buffer stage can
    provide 4p to 2p
    polar response (baffle step) correction.
  • The tweeter and midrange channels of a 3-way system.
    The midrange highpass filter of the woofer to mid crossover would have to be
    provided by the WM1 board.
  • The tweeter and upper midrange or upper midrange and
    lower midrange channels of a 4-way system.
  • A great variety of active multi-channel line level
    filters in combination with the WM1 board.
  • Crossover for add-on
    woofer
    , FAQ10, FAQ15

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.

See the Circuit Board page for
ordering information. Top

.

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:

[1] 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.

[2] Herman J. Blinchikoff & Anatol I. Zverev,
Filtering in the Time and Frequency Domains
, John Wiley, 1976. A broad and
fundamental look at filters.

[3] Arthur B. Williams & Fred J. Taylor, Electronic Filter Design
Handbook
, McGraw-Hill, 1995. Design and analysis formulas for all types of
filters.

[4] Jasper J. Goedbloed, Electromagnetic Compatibility, Prentice
Hall,1990. Fundamental concepts and practices for dealing with radio frequency
interference.

[5] Henry W. Ott, Noise Reduction Techniques in Electronic Systems, John
Wiley, 1976. Practical steps to combat RFI.

[6] Manfred Zollner & Eberhard Zwicker, Elektroakustik, Springer,
1998. The most comprehensive and solid engineering
level presentation of electro-acoustic transducers and related subjects.

In
German, no comparable English language text available, to my knowledge.

[7] Walter G. Jung, editor, Op Amp Applications, Analog
Devices
, 2002. Everything you ever wanted to know about using operational
amplifiers, and not just at audio frequencies.
Top

——————————————————————



| Build-Your-Own
| Main Panel | Dipole Woofer | Crossover/EQ
| Supplies |

| System Test | Design Models | Prototypes
| Active Filters | Surround
| FAQ |





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