|The Audio Pages|
|Elliott Sound Products||Design of Passive Crossovers|
Page Updated 09 Aug 2004
- 1.0 Introduction
- 2.0 Filter Types
- 3.0 Speaker Effects on Filter Response
- 4.0 Selecting the Filter Slope, Alignment & Components
- 5.0 The Maths Behind the Filters
- 6.0 Attenuation Networks
- 7.0 Power Losses / Resistor Power
- 8.0 Winding the Coils
- 9.0 Conclusion
Figure 1.1 - Speaker Test Setup
|Remember that once a Zobel or notch filter has been determined for a driver, that becomes part of the driver. The network and driver must be treated as one, since the network's purpose is to remove some objectionable characteristic of the attached driver - most commonly unwanted impedance variations.|
- Disconnect the speaker, and set the voltage to a convenient level (say 1V)
- Connect the speaker, and sweep the frequency to locate the centre of the "flat" range, where the voltage does not change appreciably
- Measure the voltage
Vr = Vin - VsWhere Vr is voltage across resistor, Vin is unloaded voltage, Vs is voltage across speaker
I = Vr / RWhere I is current, and R is the value of the resistor (10 ohms is suggested)
Z = Vs / I
Vr = Vin - Vs = 1 - 0.4 = 0.6 V
I = Vr / R = 0.6 / 10 = 0.06 A
Z = Vs / I = 0.4 / 0.06 = 6.67 Ohms
2.0 Filter Types
|Filter||Main Characteristic||Other Characteristics||Q|
|Butterworth||Maximally flat amplitude||-||0.707|
|Bessel||Maximally flat phase||Fastest settling time||0.5 to 0.7 (typ)*|
|Chebyshev **||Fastest rolloff||Slight peaks / dips||0.8 to 1.2 (typ)|
** also spelled Tchebychev in some texts
3.0 Speaker Effects on Filter Response
One area where measurement is essential when designing passive crossovers, is the loudspeaker driver itself. There is usually very little information in the makers' data that will prepare you for the behaviour of a loudspeaker / crossover network combination, and these data are usually derived empirically. In some cases the voice coil inductance will be quoted, and if so, this may be a bonus, as will be shown shortly.
In almost every case, the crossover frequency selected for the woofer and midrange driver will be at a frequency where the voice coil inductance is significant. As frequency increases, the effect of the voice coil inductance is to increase the driver's impedance, and this plays havoc with the crossover network's performance.
Figure 3.1 - Equivalent Circuit of a Loudspeaker
Figure 3.2 - Impedance Curve of "Simulated" Loudspeaker
Figure 3.3 - Addition of an Impedance Correction Zobel Network
f = Rvc / ( 2 π Lvc )Where ...
f = frequencyOnce this figure is found, it is a simple matter to calculate the capacitance for the Zobel network ...
Rvc = Resistance of voice coil
Lvc = Inductance of voice coil
C = 1 / ( 2 π f Rvc )Using the simulated speaker above as an example, we already know that Rvc is 6.2 ohms, so ...
f = 6.2 / ( 2 * π * 1.5E-3 ) = 658HzIt should come as no surprise that this is almost exactly the value found by simulation, so we can safely assume that the formula works, and is easy enough to use. The resistance will nearly always be approximately equal to the voice coil resistance - in some cases it may be found that a small variation is needed, but this is unlikely to be significant.
C = 1 / ( 2 * π * 658 * 6.2 ) = 39E-6F = 39uF
Figure 3.4 - Resulting Loudspeaker / Zobel Impedance Curve
Most tweeters and midrange drivers can benefit from using a compensation circuit at their resonant frequency when a passive crossover is used. This is especially true if you are using a crossover network with a slow rolloff, or the frequency is too close to the resonant frequency of the driver. With a 6dB/octave filter, I suggest an absolute minimum of about 1.5 octaves between the driver resonance and crossover frequency. A tweeter with a 900Hz resonance should therefore be crossed over at a minimum of 2,500Hz, but preferably higher. If you use the minimum possible frequency separation, there will be a small peak at tweeter resonance - this is a combination of the tweeter's resonance itself, and the fact that the crossover cannot maintain the correct rolloff if the load impedance changes.
Figure 3.5 - Tweeter Resonance
Figure 3.6 - Compensation Circuit, and Equivalent Circuit of Tweeter
C = 1 / ( 2 * π * Rvc * F3 )The details of the simulated tweeter are ...
L = 1 / ( 4 * π² * Fo² * C )
Rvc = 5.8 OhmsSubstituting our tweeter, we will obtain the following ...
Fo = 907 Hz
F3 = 635 Hz
C = 1 / ( 2 * π * 5.8 * 635) = 43uF
L = 1 / ( 4 * π2 * 9072 * 43E-6 ) = 716 uH
Figure 3.7 - Tweeter Impedance With Correction Circuit
Figure 3.8 - 12dB Filter Response With (a) and Without (b) Compensation
Figure 3.9 - 6dB Filter Response With (a) and Without (b) Compensation
|A word of warning is worthwhile here. Never operate an amplifier into a crossover network with the drivers disconnected. It may be tempting to look at the response, but at a frequency equal to the series resonant frequency of the inductor and capacitor, the network may present almost a dead short circuit to the amplifier (depending on the filter type - second order filters are the greatest risk).|
Current is limited only by series resistance, and dangerous voltages can be developed across the capacitor and inductor. These can be sufficient to damage the capacitor (due to over-voltage), and can give you a very nasty electric shock. The amplifier may not survive this abuse either, so it could be a very expensive temptation indeed.
With only 10V RMS applied at the resonant frequency of the 3kHz filter shown below (and assuming a total series resistance of 1 ohm), the amplifier will be supplying 8.3A RMS, and there will be 98V RMS across the inductor and capacitor. Provided the Zobel network is left in place, the resonance is damped so heavily that any risk is eliminated.
In case you were wondering, the voice coil temperature used in the examples below (150°C) is not as outrageous as it may seem. Since loudspeakers have an efficiency of typically 1% or less, this means that 99% of all the power going to the speaker must be dissipated as heat. Although there is some air movement through the voice coil gap, it cannot keep the temperature down low enough to ensure that the effects described will not disturb the behaviour of the crossover network. An efficiency of 1% indicates just over 92dB/m/W, which is quite a respectable figure in the world of loudspeakers!
The question remains - what can be done about it?
The answer (regrettably) remains - virtually nothing!
Figure 3.10 - Filter Performance at Ambient Temperature (Z = 8 Ohms)
Figure 3.11 - Filter Performance at Elevated Temperature (Z = 11 Ohms)
Figure 3.12 - Combined Crossover Response at Elevated Temperature
¹ I have since been advised that B&O do use this exact technique with the Dalek (aka BeoLab 5). I don't know how effective it is, but at least someone has thought of it. In reality it's probably not needed for a home system sub because few people will push the system hard enough to invoke thermal compression.
The loading on a loudspeaker cone, and therefore its Thiele/ Small parameters, will also vary with changes in the atmospheric conditions. High humidity, altitude or temperature make air less dense - such variations will cause changes to the loading on the cone, and thus the speaker's parameters.
4.0 Selecting the Filter Slope, Alignment & Components
Selecting the best slope is important, both to protect the tweeter (in particular), and to ensure that the drivers are all operated within their optimum frequency and power handling ranges. A first order (6dB/octave) filter has the most predictable response, and is affected less by impedance variations than higher orders. On the negative side, the loudspeaker drivers will be producing sound at frequencies that are very likely outside their upper or lower limits. At low powers (less than 10W or so), this is usually not a major issue, but it becomes much more important when amplifier powers of 50W or more are considered.
The traditional passive crossover is (and for the most part always has been) the Butterworth - at least for second order filters and above. Although this may seem the ideal, it is not, since there is a 3dB peak at the crossover frequency when the outputs are summed. It is now commonly accepted that this peak is also present in almost all speaker systems when the loudspeaker outputs are summed acoustically - i.e. in normal operation. Many (reputable and/or higher priced 'audiophile') loudspeaker manufacturers will modify the crossover to compensate for this effect, but most of the formulae you find on the Net (and even in books and magazines) will simply use the nominal impedance of the drivers, and work out a pair of conventional Butterworth filters. IMO, this is not usable as a crossover for high quality reproduction, but it remains firmly entrenched regardless.
4.3 World's Worst Passive Component
It is worth pointing out that inductors are, in general, the worst passive component imaginable. This is particularly true for use in crossover networks. Because it is not possible to use a core material without introducing audible distortion, the coils used for crossovers are nearly always air-cored. This means that many more turns than might otherwise be needed must be used to get the needed inductance, and that means either a very large, heavy and expensive coil, or a smaller and lighter coil with significant resistance.
By comparison to inductors, capacitors are positively benign. Their series (internal) resistance is usually extremely low, and self-resonance is influenced more by lead length than by the component itself. Even high loss capacitors will dissipate far less power than the best low loss inductors. There are some capacitors that should be avoided though, most notably bipolar (non-polarised) electrolytics. When they are new, they work very well, but if they handle appreciable current they will lose capacitance (and gain distortion) over time.
|Capacitor Type||Typical Temperature Coefficient|
|Polyester (Mylar®)||600 to 900 ppm/°C|
Resistors are again benign, although they will always contribute heat if dissipating any power. While non-inductive resistors are available and are recommended, the error introduced by a normal (slightly inductive) resistor will typically be far smaller than the normal production differences between supposedly identical loudspeaker drivers. Any errors introduced will generally not be apparent within the audio band. The inductance of most power resistors is such that the wiring may introduce greater errors than the resistors themselves, given that each 10mm of (straight) wire adds about 5nH of inductance to the circuit.
5.0 And Now ... Some (more) Maths
Figure 5.1 - 6dB/Octave 2-Way Passive Crossover
C = 1 / ( 2 π Z f )Where:
L = Z / ( 2 π f )
C = capacitance in faradsThese can be "simplified", and reduce to the following ...
L = inductance of the coil in Henrys
f = frequency in hertz
Z = (actual) impedance of the speaker in ohms
C = 0.159 / ( Z f )Thus, for a crossover frequency of 3,000Hz at 6 ohms (a standard I shall use throughout these examples) ...
L = ( 0.159 Z ) / f
C = 0.159 / ( 6 * 3,000 ) = 8.83 uF
L = ( 0.159 * 6 ) / 3,000 = 318 uH
Figure 5.2 - 12dB/Octave 2-Way Passive Crossover
C = 0.0796 / ( Z * f )The derivation of these is marginally interesting, and will help you to understand the Butterworth and (sub) Bessel alignments a little better. The full original formulae are ...
L = ( 0.3183 * Z ) / f
C = 1 / ( 2 * π * Z * d * f )Where ...
L = ( Z * d ) / ( 2 * π * f )
d = 1/ Q = 1 / 0.707 = 1.414 (Butterworth)
d = 1 / Q = 1 / 0.5 = 2 (sub-Bessel)
d = 1 / Q = 1 / 0.57 = 1.75 (Bessel)
6.0 Attenuation Networks
dB = 20 log ((Rl / Z) + 1)For our example, this gives ...
dB = 20 log ((0.53 / 6) +1) = 20 log (1.088) = 0.73 dB
Figure 6.1 - 2dB L-Pad Attenuator
dB = 20 log (V1/V2) = 20 log (Vr), so reversing the formula we get ...For this example, we have 3.5dB, so substituting ...
Vr = 1 / (antilog (dB / 20))
Vr = 1 / (antilog (3.5 / 20)) = 1 / (antilog 0.175) = 1 / 1.496 = 0.668
I = V / Z = 1 / ZNow, find the voltage drop across the series resistor Rs, then the value of Rs ...
Vs = 1 - VrThe process is quite simple so far. Now we need to determine the value of the parallel resistor, Rp. We already know that the voltage across the parallel combination of Z and Rp - it is equal to Vr (I told you that an input voltage of 1V was convenient, didn't I? . The value if I (current) does not change, so we can determine the current through Z and Rp easily, and then Rp itself ...
Rs = Vs / I
Iz = Vr / ZNow, let's substitute all the values for the example into the formulae. As I said, this is a little tedious, but easily remembered. Recall does not come from rote learning, it comes fromunderstanding, and this is just simple arithmetic and Ohm's law.
Ip = I - Iz
Rp = Vr / Ip
I = 1 / Z = 1 / 6 = 0.1667 AmpsThat was easy enough, so now for Rp ...
Vs = 1 - Vr = 1 - 0.668 = 0.332 Volts
Rs = Vs / I = 0.332 / 0.1667 = 1.99 (2.0) Ohms
Iz = Vr / Z = 0.668 / 6 = 0.111 AmpsNow we might want to check that the values really will give us what we wanted - I recommend this final check, because there are resistor values that are easily created (or are standard), and we want to use these if possible. As a result, we will substitute 2R (2 x 1R in series) for Rs, and 12R for Rp, as these are standard values. The first thing we need, is to determine Rt - the total parallel combination of Z and Rp (Z || Rp). We could do that from the current calculated earlier, but that may re-introduce any error made earlier.
Ip = I - Iz = 0.1667 - 0.111 = 0.0557 Amps
Rp = Vr / Ip = 0.668 / 0.0557 = 11.99 (12.0) Ohms
Rt = 1 / (1 / Rp + 1 / Z) = 1 / (1 / 12 + 1 / 6) = 1 / (0.0833 + 0.1667) = 4 OhmsDamn! That was close
Vd = (Rs / Rt) + 1 = (2 / 4) + 1 = 1.5
dB = 20 log(Vd) = 20 log (1.5) = 3.52 dB
7.0 Determining Power Losses
Figure 7.1 - Power Distribution Chart
I would suggest a power rating of 10W for the Zobel resistor - this provides a very large safety margin.
The tweeter power at resonance is more than 20dB down from the maximum level using a 12dB crossover. Since this is the case, a 5W resistor is more than adequate.
The L-Pad will be subjected to a maximum of 15% of the power, but will dissipate very little of this. 5W resistors are again more than enough to handle the power.
Since we determined that the resistance will be about 0.53 ohms, so at full power it will dissipate less than 10% of the 100W input, which is about 10W. (0.53 ohms is 8.8% of 6 ohms) The average will be much less than this, and heating will not be a major problem.
8.0 Winding The Coils
Figure 8.1 - Typical Coil Former and Dimensions
L = 0.8 * a² * N² / (6a + 9l + 10c) uHWhere
N is the number of turns.All dimensions are in inches. This is easily converted in a spreadsheet or program, but modifying the formula itself is too tedious. For a real example of a coil wound with 0.83 mm (20 AWG) wire, having a design inductance of 637 uH and a resistance of 0.53 ohms has the following dimensions ...
a is the average radius.
c is the height of the windings
l is the length of the coil.
N = 99 turns
l = 11.2 mm (0.44")
Id = 44.8 mm (1.76")
Od = 58 mm (2.28")
c = 6.64 mm (0.26")
Figure 8.2 - Mounting Inductors to Minimise Coupling
- Sub-Bessel filters are to be preferred for flattest overall response (Similar response to Linkwitz-Riley alignment and easier to design)
- There is a vast amount to be gained by using a biamped system to cover the bass and mid+high crossover. Keep passives for higher frequencies, where their bulk, cost, power loss, and other flaws are minimised.
- A really well designed crossover is of no use if the box is not designed correctly, is inadequately braced, or has drivers mounted equidistant from two or more edges - these cause high frequency refractions that "smear" the stereo image.
- Likewise, voice coil (time) alignment can make a huge difference to the linearity of the system as a whole.
- The crossover will never compensate for a poor selection of drivers, regardless of the work you put into it.
- Some very simple crossovers may appear to give a more "musical" sound reproduction, but are not accurate - in the long term, they cause listener (and driver) fatigue, and are adding things to the music that was never there in the first place.
- Your listening room has more effect on the sound than any of the other points made above! However, a good system has a much better chance of sounding acceptable in a bad room than a bad system (which will sound bad everywhere!).
- A fully active system (using electronic crossovers and separate amps for all drivers) will almost certainly give a better result than the most carefully designed passive system, and may even work out cheaper ... Some passive crossover networks can become very complex and expensive indeed.
The "simple" passive crossover is actually vastly more complex than is commonly believed. The "new" Diaural™ "inductor only" crossovers are not a panacea for the ills of the world (despitemassive marketing hype to the contrary), but fall into the 'overly simple and grossly coloured' category. There is nothing (repeat - nothing!) about the Diaural system that is new, or will benefit the vast majority of systems. They will probably sound "lively" and perhaps "musical" by initial direct comparison to a conventional crossover, but require very careful driver selection indeed if gross response and phase errors are to be avoided. These are even patented - how in God's name those ratbags got a patent on something that has been done by others for years, we will never know. I have a copy of the patent, and all the variations are shown - so much for the "cone of silence" that was placed on anyone who saw them in the early days. There is absolutely nothing remarkable about the principle, other than the complete and total neglect of almost everything I covered in this article. Especially noteworthy is the fact that they use ... inductors, which as discussed above have more compromises than any other passive component.
Lynn Olson ...
- Looking Over My Shoulder, Parts I and II
- The Soul of Sound
Page created and copyright © 20 May 2001 Updated 09 Aug 2004 - added material, and corrected some small errors./ 04 Nov 2003 - corrected attenuator calculation./ 17 Jun 2002 - added spreadsheet and impedance info./ 08 Oct 05 - Included component selection info, table 2, etc.