# What is meant by Butterworth filter

As of June 17, 2017

### Crossovers

Introduction: Since I cannot take over the entire explanation of all the basics, I am only trying to give a few practical explanations on the surface. If you are really interested in this, you should, if you have the appropriate training, look into a specialist book for better or worse. Especially since you should have knowledge and some experience to create the circuit board afterwards. Nevertheless, one should have the basics of electronics and simple mathematics ...
As the time for answering e-mails is tight, please refrain from making precise inquiries about the calculation of individual turnouts.

Since I can miscalculate or make mistakes with some things, I do not give any guarantee for all content on this page (like all other individual pages on www.selfmadehifi.de also )

I'm only going into analog filters here, digital ones work completely differently.

The task of crossovers in acoustics is to divide the entire audible range into individual areas.

Types: You mainly use low, band and high passes in acoustics. A low-pass filter (integrating element) allows low frequencies to pass through and attenuates high frequencies; with a high-pass filter (differentiating element), the opposite is true. A band pass lets through mid frequencies (a frequency band) and attenuates frequencies below and above. It is often created by simply (but decoupled!) Connecting a high and a low pass one after the other.
Bandstop filters attenuate one frequency band, but let the rest of the range through. This can be further increased with a notch filter (also known as a notch filter), which attenuates only one frequency from a range extremely strongly. All-passes are rather rare, they show a phase shift depending on the frequency, the amplitude curve remains constant.
Anything else:
A crossover branch is characterized (in addition to the type) by the cutoff frequency, slope and quality.
The cut-off frequency fg (or fc cut frequency) indicates the frequency of the filter at which the voltage has dropped to approx. 70.7%, which is around 3 dB and is the point at which the power is halved.

The slope shows how strong the filter is in the frequency range to be attenuated. For high and low pass filters of the first order (only one passive component for the filter) it is 6dB / octave, which corresponds to 20dB / decade. This means, for example, with a low-pass above the cut-off frequency, the level decreases by 20dB with every tenfold increase in frequency, which is 6dB when the frequency is doubled. 2nd order filter attenuates with 12dB / octave, 3rd order with 18dB etc. Passive filters usually work from the 1st to a maximum of 4th order. Above that, only active filters are usually used that allow extremely steep edges by being connected one after the other.

In the case of bandpasses, the definitions sometimes diverge somewhat: The electrical engineer speaks of bandpasses of the 2nd order, although they only go up and down with 6dB / oct. dampen. Reason: What is meant is the mathematical order: the degree of the polynomial in the denominator of the transfer function G (f). This means that bandpass filters that work at high and low frequencies with the same slope (standard) always have an even order.

Passive switches divide the frequency range directly in front of the loudspeaker. Passive because they do not contain any active components (transistors, operational amplifiers). A pass. The crossover generally only consists of capacitors (capacitances C), coils (inductivities L) and ohmic resistors (R). In principle, complex components (C, L) represent a frequency-dependent resistance. With coils it increases with increasing frequency, with capacitors it decreases, with resistors it remains constant. Various filters with different behaviors can be set up by arranging them accordingly.
How passive frequency branches are calculated can be viewed and simulated extremely easily in BassCADe (can be downloaded here).
In practice, e.g. for loudspeakers, 2nd order filters are mostly used, but 1st or 3rd order high and low pass filters are also used here, but are less common.
The following figure shows the structure of passive low-pass filters; in the case of high-pass filters, with the same arrangement, only the coils (L) with the capacitors (C) are exchanged. A loudspeaker serves as a conclusion. This is a kind of resistance, but it is not frequency-neutral and therefore (unfortunately) also influences the characteristic.
I will not go into more detail on passive turnouts here, as the software mentioned above offers sufficient explanations, calculations and simulation options for individual turnout branches. It can also be used to correct the effects of a midrange or bass loudspeaker or to adjust the level of the tweeter.

For first-order low-pass passes:
L = Z / (2 * pi * fc)

or equivalent for RC low-pass filters:
C = 1 / (2 * pi * fc * R)

For high passes:
C = 1 / (2 * pi * fc * Z)
equivalent for RC elements: C = 1 / (2 * pi * fc * R)

Structure of passive low-pass filters:

At active points all this happens before the final stage. An active switch works with at least one active BE (component) that is supposed to compensate for the attenuation. The RC orders of magnitude differ here, since work is carried out with high and medium resistance. The reasons for this are lower currents and smaller components (Rs and Cs). In this case, however, in addition to the active BE (generally today operational amplifiers OP / OPV) only capacitors and resistors are used. (In the LF range, coils are too large, lossy and impractical.) By arranging the components accordingly, e.g. in feedback branches, different frequency curves are generated here. For this purpose, the OP is often operated in an inverting manner (because of the simple gain calculation V = Z2 / Z1). These circuits all apply without offset, which means a symmetrical power supply (plus minus xVolt with center tap 0V to ground) for the OP.
Here are the basic circuits of an active low-pass and high-pass filter of the 1st order with an OP in an inverting design. The output signal has a phase shift of 180 ° compared to the input signal. That means inverted here, i.e. a positive input voltage results in a negative output voltage and vice versa.

Since every filter represents a damped oscillating system, the Q factor indicates the suppression of this oscillation at the resonance frequency of the filter fr. The more the oscillation is suppressed, the better the impulse and step responses. When listening, this is shown by a significantly more precise and impulsive reproduction. Standard filters, which most sources recommend according to formulas, have a quality factor of 0.707, so that the resonance and cutoff frequencies match here. They are therefore also easier to calculate, since the values ​​remain the same if the (decoupled) components are exchanged and a high pass, e.g. a low pass, is made. In practice, especially with expensive components, somewhat lower quality values ​​between 0.5 and 0.6 are chosen because of the better impulsivity.

• Q <0.5 overdamped (no tendency to oscillate)
• Q = 0.5 (Linkwitz vote, "filter of critical damping") Excellent precision, level at the built-in resonance fc -6dB, no overshoot in the step response
• Bessel tuning: constant group delay very good precision, with 2nd order Q = 0.577, level with fc is -4.8dB
• Q = 0.707 (Butterworth vote), still good precision, level at fc (here is always the lower limit frequency) -3dB
• Q = 1 (> 0.707 Chebycheff vote), poor precision, level is (Qtc = 1) at fc 0dB, therefore overshoot
With qualities of 0.7 and above, it should be noted that the overshoot in the amplitude curve and the step response become increasingly stronger with a higher order.
Bessel filters (aka Thomson filters) have an excellent square wave behavior because they have a frequency-proportional phase shift. Due to the strong damping, they vibrate extremely little in the step response and higher-order filters also fade away quickly.
These filter types or characteristics are characterized by certain pole and zero positions of the transfer function.

A second order filter can be set up relatively easily with an OP stage. This can also be expanded to the 3rd order by resizing. (as with passive switches)
However, higher order filters have another disadvantage: the higher the order, the higher the group delay.

Normally, the calculation of filters based on the coefficients is time-consuming, which is why I skipped the whole further theoretical part in order to simplify everything. Only a few options that are often used are given.

The following examples include:
A relatively widespread standard 3rd order crossover:

As can be seen from the structure, the high pass is at the top and the low pass at the bottom.
The values ​​for the Butterworth characteristic:
Input resistance Rin
R = Rin / 2.
 Low pass High pass C2 = 1 / (2.5590 * fo * R) R2 = 1 / (15.4226 * fu * C) C3 = 1 / (32.5385 * fo * R) R3 = 1 / (1.21377 * fu * C) C4 = 1 / (2.9794 * fo * R) R4 = 1 / (13.2501 * fu * C) R1 = 2 * R C1 = C / 2 fo = 1 / (2 * pi * R * cubrt (C2 * C3 * C4)) fu = 1 / (2 * pi * C * cubrt (R2 * R3 * R4))
cubrt (cubicroot) means a cube root, i.e. the value in brackets to the power of a third. In general, one takes C twice as large as C2 if fu = fo, i.e. C1 = C2

Usually, however, OPV stages are only set up as 2nd order filters and these are then switched one after the other. This has the advantage, among other things, that the cut-off frequency can be changed simply by changing the resistance while maintaining a constant quality. This is very popular with subwoofer crossovers. In a non-inverting way it looks something like this:
Low pass:

For these 2nd order filters, choose R1 = R2 and C1 = C2. The respective frequency is determined by the RC element (T = R * C) and is approximately f = 1 / (6.3 * RC)
If you switch 2 of these stages one after the other, you get a 4th order low pass with 24dB / octave, which can be used very well as a subwoofer crossover (mono). The complete feedback without resistance results in a total gain of 1.
Unfortunately it is necessary to change 4 resistors due to the 4th order, so 2 tandem potentiometers (with mono!) Have to be coupled mechanically or, better still, a step switch has to be used.

The following then applies to the Bessel characteristic:
(Normally you set the C first and calculate the resistances from it ... But I do it the other way around here.)
R1 = R2 = R11 = R12 (sum of potentiometer and resistor) roughly the size of the input resistance, i.e. about 1k ... 100k
When setting, the following applies: the larger R, the lower the frequency, i.e. define the range (ratio Rmax to Rmin) with the respective fixed resistor and the associated potentiometer.
Calculate the associated capacitors from the specified resistance values ​​(4 fixed resistors and 2 tandem potentiometers).

 1st stage: 2nd stage: C1 = 1 / (8.47 * fc * R) C11 = 1 / (6.16 * fc * R) C2 = 1 / (9.24 * fc * R) C12 = 1 / (15.8 * fc * R)

However, since you need a mono signal at the input and should not only use one side channel, a mixer must be connected upstream, i.e. either one transistor stage each, preferably in emitter circuit, or also 2 OPVs, it is also possible with only one OPV.
The high-pass filter is constructed in the same way, except that all resistors and capacitors are interchanged in the arrangement. The equations then no longer apply (otherwise the quality would increase to over 0.7).

Here is an active 2nd order high-pass filter with an adjustable cut-off frequency as an example.

An adjustable crossover frequency in the range between 20 ... 200Hz is required for active subwoofer decoupling. This then results in the following values ​​(Butterworth) C1 = C2 = 150nF; R1 = R2 = 4.7k; R3 = R4 = 10k; P: tandem potentiometer with 2 times 47kOhm / lin.

If you switch n low-pass filters with the same cut-off frequency, quality and slope one after the other, the following approximation applies:

 for low passes (number n, cutoff frequency of a TP fgi): fg = fgi / sqrt (n) for high passes (number n, cutoff frequency of an HP fgi): fg = fgi * sqrt (n)