I've been wondering how to calculate crossover frequency of the third-order crossover and its parameters from the schematic.
Let us consider second-order crossover, there are two formulas to calculate the parameters of the crossover which are crossover frequency and quality factor. The crossover frequency of the second-order filter is calculated from resonance formula: f = 1/(2 x pi x sqrt(L x C)) and the quality factor is calculated by Q = sqrt(CR^2/L).
But, how about third-order system? Are there formulas for calculating crossover point and Q-factor as the second-order system?
I know the simulator software can do it very well. But sometimes it should be more handy with hand calculation.
Assume we have a series 6.8uF capacitor and a shunt 0.2mH inductor, we can easily obtain crossover frequency and Q-factor of this second-order filter by using two formulas above. What if we add a second series capacitor of about 18uF to it, now the crossover is converted to be third-order system, can we find crossover frequency and Q-factor (if any) of this third-order filter?
Let us consider second-order crossover, there are two formulas to calculate the parameters of the crossover which are crossover frequency and quality factor. The crossover frequency of the second-order filter is calculated from resonance formula: f = 1/(2 x pi x sqrt(L x C)) and the quality factor is calculated by Q = sqrt(CR^2/L).
But, how about third-order system? Are there formulas for calculating crossover point and Q-factor as the second-order system?
I know the simulator software can do it very well. But sometimes it should be more handy with hand calculation.
Assume we have a series 6.8uF capacitor and a shunt 0.2mH inductor, we can easily obtain crossover frequency and Q-factor of this second-order filter by using two formulas above. What if we add a second series capacitor of about 18uF to it, now the crossover is converted to be third-order system, can we find crossover frequency and Q-factor (if any) of this third-order filter?
Last edited:
What schematic?
You can get the textbook 3rd order Butterworth formulas from any decent text on filter design. From LDC 7 (Vance Dickason) attached.
Only 2nd order filters have a Q value as such -higher orders have several. And you need to be careful exactly what it is in question: a 3rd order electrical filter is not necessarily going to provide 3rd order acoustical slopes, so textbook values are rarely much use anyway.
You can get the textbook 3rd order Butterworth formulas from any decent text on filter design. From LDC 7 (Vance Dickason) attached.
Only 2nd order filters have a Q value as such -higher orders have several. And you need to be careful exactly what it is in question: a 3rd order electrical filter is not necessarily going to provide 3rd order acoustical slopes, so textbook values are rarely much use anyway.
Attachments
Here is the schematic of example.
Thank you for the textbook's formulas. What was in the question is the electrical response of the filter, to my understanding. Those formulas you provided are useful when designing the filter. However, the objective of my question was about the inversion of the formulas.
For instance, regarding the schematics:
Second-order filter
Crossover frequency: f = 1/(2 x pi x sqrt(0.2 x 6.8)) = 4,316 Hz
Q value: Q = sqrt(6.8 x 3.2^2/0.2) = 0.59
Third-order filter
Crossover frequency: f = ?
Q value: Q = ?
If you read my post, you'll see I pointed out a 3rd order filter doesn't have a single Q value (only 2nd order filters have that) but several. You can back-calculate a crossover frequency assuming it conforms to a given electrical order and slope (most do not) by rearranging the individual formulas.
Just to clarify on your question. The crossover frequency is the frequency where the -3dB point is defined. That is a definition. If you design a filter and enter parameters like slope, crossover frequency, and Q, you get a filter which attenuates 3 dB at crossover frequency.
That does not necessarily mean that you want 3 dB attenuation at the point where you separate one speaker response to the other. That is a matter of how you want your response.
The exception is a 4th order Linkwitz-Riley filter. It has 6dB attenuation at the crossover point. And it is common to call this -6 dB point the crossover frequency. However in filter theory it is not.
That does not necessarily mean that you want 3 dB attenuation at the point where you separate one speaker response to the other. That is a matter of how you want your response.
The exception is a 4th order Linkwitz-Riley filter. It has 6dB attenuation at the crossover point. And it is common to call this -6 dB point the crossover frequency. However in filter theory it is not.
In the Laplace domain, an ideal inductor with inductance L has an impedance sL and an ideal capacitor with capacitance C has an impedance 1/(sC), where s is the Laplace variable. (Depending on the kind of calculation you want to do, you can also interpret it as j omega or as the Heaviside differentiation-to-time operator.)
Knowing this, you can use the equations for series and parallel connections and for voltage dividers to calculate the transfer function of the filter. For your high-pass, it will be some constant times s3 in the numerator and a third-order polynomial in s in the denominator.
You can now equate the denominator to zero and solve s to calculate the poles of the filter. That boils down to solving a cubic equation, how that can be done is explained on Wikipedia.
You will probably find a negative real pole and a complex pole pair with a negative real part, although you might also find three negative real poles. By definition, the natural frequency of the complex pole pair is
fn = √((Re(s))2 + (Im(s))2)/(2π)
and its quality factor is
Q = -√((Re(s))2 + (Im(s))2)/(2 Re(s))
That and the position of the real pole define the response shape.
It gets simpler when you just want to calculate the shape of the frequency response, rather than the pole positions. You don't need to solve any cubic equation then. You just substitute s = j2πf in the transfer function for each frequency of interest.
Knowing this, you can use the equations for series and parallel connections and for voltage dividers to calculate the transfer function of the filter. For your high-pass, it will be some constant times s3 in the numerator and a third-order polynomial in s in the denominator.
You can now equate the denominator to zero and solve s to calculate the poles of the filter. That boils down to solving a cubic equation, how that can be done is explained on Wikipedia.
You will probably find a negative real pole and a complex pole pair with a negative real part, although you might also find three negative real poles. By definition, the natural frequency of the complex pole pair is
fn = √((Re(s))2 + (Im(s))2)/(2π)
and its quality factor is
Q = -√((Re(s))2 + (Im(s))2)/(2 Re(s))
That and the position of the real pole define the response shape.
It gets simpler when you just want to calculate the shape of the frequency response, rather than the pole positions. You don't need to solve any cubic equation then. You just substitute s = j2πf in the transfer function for each frequency of interest.
Last edited:
I used to read on some tutorials. They suggest that the crossover frequency depends on type of crossover characteristics, which are as follows:
Chevbychev: -1.5dB point
Butterworth: -3dB
Bessel: -4.5dB
Linkwitz-Riley: -6dB
Furthermore, if we consider the graph of amplitude we obtain from measurements, how can we recognize the type/ characteristics of the slope so as to indicating the crossover point?
For example, consider the green curve on post #3, the third-order response, which type of slope; CC, BW, BS, or LR, to indicate the crossover point?
For Chebyshev, the cut-off frequency is often taken to be the frequency where the response starts to deviate more than allowed by the ripple specification. For example, where the response starts to deviate more than 0.1 dB for a 0.1 dB Chebyshev filter.
Regarding recognizing filter responses: pretty much in the same way as bird watchers recognize the species of bird they are watching: look at the filter magnitude and group delay responses very carefully and compare them with theoretical filter responses in a filter synthesis handbook. For example, a Chebyshev filter has passband ripple, so when your filter has a monotonic response, it cannot be a Chebyshev filter. Linkwitz-Riley filters are only defined for even order, so an odd-order filter cannot be Linkwitz-Riley. If the magnitude reponse is monotonic and quite flat, it could well be an optimally flat magnitude filter, a.k.a. Butterworth filter. The only problem is that there are infinitely many responses that don't quite fit with any of the ideal responses.
Regarding recognizing filter responses: pretty much in the same way as bird watchers recognize the species of bird they are watching: look at the filter magnitude and group delay responses very carefully and compare them with theoretical filter responses in a filter synthesis handbook. For example, a Chebyshev filter has passband ripple, so when your filter has a monotonic response, it cannot be a Chebyshev filter. Linkwitz-Riley filters are only defined for even order, so an odd-order filter cannot be Linkwitz-Riley. If the magnitude reponse is monotonic and quite flat, it could well be an optimally flat magnitude filter, a.k.a. Butterworth filter. The only problem is that there are infinitely many responses that don't quite fit with any of the ideal responses.
Last edited:
Which also brings us back to the point that much of this is relatively moot, since drive units are not flat resistive electrical loads unless you make them that way, are rarely flat in the frequency / amplitude domain either, and you have phase / delay on top of that, so idealised electrical filters almost never provide the stated response in practice for a sufficient BW (if at all). The above is a good illustration, since you even get to the point of asking: what actually is a Linkwitz-Riley filter? Siegfried didn't define them for anything other than even order due to the basis -but in many practical situations you can create, say, a 3rd order, a 5th order etc. with the important Linkwitz characteristics of complimentary phase & the filters down -6dB at the crossover frequency. In a similar way -what type of Chebyshev filter? If it's a Type II, you can use what could be loosely defined as that to ~mimic a different filter's characteristic for the initial octave of the rolloff, and a few of us do just that -it can work quite well, since it may allow you to save on some expensive components, reduce losses & also keep GD down compared to if you generated a full ladder of the target order. Ideal electrical filters are interesting, but in practice, neither they or the assumptions they're based on tend to be particularly accurate when working with actual drive units & non-flat frequency, impedance, phase etc. responses.
And that's why, for someone who likes to play with transfer functions, it is more fun to design RIAA corrected phono preamplifiers and DAC reconstruction filters than to design loudspeaker crossovers. When you design a RIAA corrected phono preamplifier that follows the ideal curve within +/- 0.1 dB, you almost never get remarks about it being useless because of the non-flat response of the cartridge and the loudspeakers and the acoustics of the listening room.
One of life's little ironies. At least you know who / where to blame... 😉Sorry but not true.
Like I said, the corner frequency by definition is the frequency where the filter output voltage is half of the pass-through frequency, either f=0 or f >> fc.
Changes between Chevbychev, Butterworth or Bessel are only changes in Q of the filter. In implementation for a certain Q, the -3 dB point might shift slightly depending on Q.
It is not so that a designer determines the Q of a filter and calculated fc differently depending on Q. You have a filter formula (like @MarcelvdG posted) and you plug in the desired fc and Q.
More information and background in the book Active Filter Cookbook by Don Lancaster. Despite of the funny name this is a serious book. Chapter 4 discusses the various orders and shapes.
I forgot to add this in my previous post. I am talking about the definition and calculation of filters. For the design of a passive cross-over totally different requirements might exist for the filter design. As @Scottmoose pointed out, a speaker is a complicated load and designing a filter with appropriate corrections and whatnot is not trivial.
But that does invalidate the theory for designing filters in any way.
But that does invalidate the theory for designing filters in any way.