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
and the electrical
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).
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 00-axes. Summing as a scalar means generally summation of the 00 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 -300 and +300.
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.
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 = (a0)2, and the quality factor Q = 1/a1. 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 2e-order LP filter is:
The phase needs an absolute correction to obtain the usual graphs that go from 00 to -1800 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 = (QBW)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 2e- order low pass systems as defined above.
Higher order filters
Just like 2e-order systems we can derive transfer functions for 4e-order systems. In fact these consist of a cascade of two 2e-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 4e-order filter is given by:
For a BW filter the two stages have different parameters. Q1 = 0.5412, Q2 = 1.3065 (both FSF=1). LR crossovers have two identical stages with Q1,2 = 0.707 (FSF=1). This makes the calculation for 4e -LR crossovers easier whereas the calculation for BW needs to be done with two different stages. For 4e -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 2e-order high pass we find:
and for 4e-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.