### Filters: the acoustical and the electrical outcome

### Theory of filtering

Just like passive crossovers, active crossovers can be designed as Butterworth (BW), Bessel, Chebychev or Linkwitz-Riley (LR). The most commonly used for speaker design are BW and LR (and sometimes Bessel).

What follows is a mathematical analysis of fundamental filter theory.

For active crossovers two aspects are considered; the acoustical
response
and the electrical
response.

The acoustical response is related directly to the physics of speakers
and the radiating pattern of sound waves from a speaker (modeled as a
point source or a piston radiator).

### Acoustical response

Key for the acoustical response is that two speakers
mounted on a baffle are generally non-coincident. Simply stated, the
voice coils of the speakers (for example the low and high speaker) are
not in the same vertical plane. As a rule of thumb, the voice coil can
be taken as the acoustical centre of radiation. As a result, the
speakers are spatially separated.

Especially the vertical plane of radiation will lobed
because of interference patterns between the two speakers. Placing the
two speakers less than one wavelength (at the corner frequency) from
each other can minimize this lobing effect (distance d1).

The acoustical horizontal displacement (distance d2) results in a phase
delay between the two speakers. The signals of each speaker at the
corner frequency (Fc) will be summed out-of-phase. If not accounted for
in the filter design, the filter will not work properly especially when
lambda at Fc is comparable to the horizontal displacement. The group
delay time for the horizontal displacement can be easily calculated
with the basic formula:

in which c=340m/s

Besides, the axes of directivity will be tilted because of the spatial separation. This tilt can be calculated with:

It must be noticed that the acoustical centre changes with the natural phase characteristic of the speaker. The general rule of thumb is an approximation for d2 in the zero delay plane (ZDP), the plane in which d2 is not a function of frequency. Especially at the resonance frequency of the speaker the general rule of thumb is not very accurate (this is where the group delay of the speaker peaks, or where the phase changes much). The acoustical point is at this frequency often behind the speaker and so diverts from the ZDP.

To correct for the acoustical effects, the speaker design on this web uses a backward mounting of the tweeter on the baffle for the horizontal displacement. Vertical separation is much less than lambda at Fc reducing the lobing.

The crossover for the woofer is not critical for this acoustical displacement since the wavelength is here much larger than the acoustical separation.

Besides these acoustical displacements two speakers will
sum their radiated responses in the free field at the crossover
frequency. This is according to fundamental physics. For this
acoustical summation, the acoustical responses are summed as scalar
quantities i.e. not as a vector. In reality this is a wrong approach
since the power distribution of a speaker will divert from its 0^{0}-axes.
Summing as a scalar means generally summation of the 0^{0}
responses as approximation. Nevertheless, this theoretical approach
will suffice as a model for the direct sound waves towards the
listener.

It must be noticed that it is also very difficult to
account crossover design for the power distribution of speakers. Not
even mentioning the effects of room acoustics. Modeling with these
parameters makes development of filters very complex. Besides high end
speakers have a remarkable equal power distribution between -30^{0}
and +30^{0}.

### Summing acoustical sine waves

The mathematical review for summing acoustical responses is shown below.

First, the acoustical response from a speaker can be written as a standard wave equation or in the complex format:

in which A is the amplitude of the wave and F is the phase.

Summing the power of two waves (or two speakers) x(t) and y(t), we will get:

The time component is not of interest for the power calculation and can be discarded.

The sum can be written as:

From this point basic complex calculation determines the amplitude and phase of the total summed response.

Of course, these two last equations can be simplified but leaving it as is makes it easy to substitute a third response.

### Electrical
response: a mathematical review for low-pass filters**.**

**.**

The electrical response or electrical transfer function of a filter is the most important task in speaker design. This part takes care that each speaker will be driven with that bandwidth that suite its electrical and acoustical parameters best.

A short fundamental approach that is used in the active speakers on this web will be shown here.

Assume a second order low pass filter. For this filter a second order polynomial describes the transfer function, that is:

It is common to normalize the polynomial with two
scaling factors: the frequency scaling factor: FSF = (a_{0})^{2},
and the quality factor Q = 1/a_{1}. Also s =j*2(pi)f in
which j is the complex variable. By making these substitutions the
equation becomes:

in which fc is the cut-off frequency

The actual transfer function for low pass filtering is H(s) = A/P(s) in which A is a predefined gain for example from an amplifier. For simplicity A=1. The transfer function can now be written as:

This transfer function is a complex function and is
commonly used as a standard form for calculations. For audio crossover
designs it is common to use an expression in dB for the amplitude and
an expression in degrees for the phase. These can be obtained from
standard complex mathematics. So the transfer function for a **2 ^{e}-order
LP filter** is:

The phase needs an absolute correction to obtain the
usual graphs that go from 0^{0} to -180^{0}
instead of the jumps. FSF and Q are parameters that determine the
behavior of the transfer function (either a Butterworth, Bessel or
Linkwitz-Riley (LR)). These can parameters can be found in table books.

For BW, FSF=1 and Q=0.707. For LR, FSF=1 and Q = 0.5.
Actually LR = (BW)^{2} so Q = (Q_{BW})^{2}
=0.5. Making use of these substitutions, the equations can be
simplified. However, Q is an important design parameter and will be
used as variable. Further, we will focus on BW and LR only.

Summing two electrical filter responses at the corner
frequency is an important aspect in crossover design to study the
effect on the total amplitude response curve. For this summation the
equations for summing sound waves can be used for which H(f) relates to
the amplitude and Arg(H(f)) to the phase of the specific response.

It is interesting to notice that closed box designs can be accurately
modeled as 2^{e}- order low pass systems as defined
above.

### Higher order filters

Just like 2^{e}-order systems we can
derive transfer functions for 4^{e}-order systems.
In fact these consist of a cascade of two 2^{e}-order
stages to combine to higher even order filters. The transfer functions
are derived from multiplying two of these stages. This results from the
original polynomial:

Then the transfer function for a 4^{e}-order
filter is given by:

For a BW filter the two stages have different
parameters. Q_{1 }= 0.5412, Q_{2}
= 1.3065 (both FSF=1). LR crossovers have two identical stages with Q_{1,2
}= 0.707 (FSF=1). This makes the calculation for 4^{e}
-LR crossovers easier whereas the calculation for BW needs to be done
with two different stages. For **4 ^{e}
-order low pass LR** we get:

### High pass filters

Analogous to low pass filters, high pass filters can be
defined with transfer functions. For this the same parameters FSF=1 and
Q accounts as stated before for BW and LR. For **2 ^{e}-order
high pass **we find:

and for **4 ^{e}-order high
pass LR **we find:

The equations for the phase are equal to the ones shown for low pass filters. The usual corrections to the phase can be applied to avoid the "jumping" in the phase diagrams.