As we all know, the resonance formula for second-order filter is given by f = 1/(2 x pi x sqrt(LC)). As can be seen, there are two variables involved which are the product of inductor and capacitor values used in the filter.
However, in third-order, there are three components used which should mean the formula would have been changed to have three variables involved, and four variables for fourth-order. Do I understand correctly? If so, what are resonance formulas for third- and fourth-order filters, respectively.
Finally, the first-order is the only case that has no resonance formula because there’s no two elements to interact with each other as the higher-orders, do I understand it correctly?
However, in third-order, there are three components used which should mean the formula would have been changed to have three variables involved, and four variables for fourth-order. Do I understand correctly? If so, what are resonance formulas for third- and fourth-order filters, respectively.
Finally, the first-order is the only case that has no resonance formula because there’s no two elements to interact with each other as the higher-orders, do I understand it correctly?
Well, let me explain the reason why I asked this question.
In second-order filter, with resonance formula, we can calculate the alternative options for L and C values at the same frequency. However, Q is changed instead.
For instance, let's have a standard second-order high-pass filter with C = 4.7uF and L = 0.3mH. By replacing L = 0.3mH with a 0.2mH, to maintain the same cut-off frequency, we need to alter C as well. And using the resonance formula, we can calculate the new C by using the product of the original L and C divided by new C, that is (4.7 x 0.3) / 0.2 = 6.8uF.
As can be seen, the resonance formula is useful for designing second-order filters. But, how about third-order or fourth-order? Can I perform similarly with the second-order?
This is my reason for asking this question.
In second-order filter, with resonance formula, we can calculate the alternative options for L and C values at the same frequency. However, Q is changed instead.
For instance, let's have a standard second-order high-pass filter with C = 4.7uF and L = 0.3mH. By replacing L = 0.3mH with a 0.2mH, to maintain the same cut-off frequency, we need to alter C as well. And using the resonance formula, we can calculate the new C by using the product of the original L and C divided by new C, that is (4.7 x 0.3) / 0.2 = 6.8uF.
As can be seen, the resonance formula is useful for designing second-order filters. But, how about third-order or fourth-order? Can I perform similarly with the second-order?
This is my reason for asking this question.
I meant - your concept of altering different values of L and C but keeping the same LxC product (to keep the same resonant frequency) when designing filter is wrong, because there is only one optimal value (in fact - a narrow range) of L and C for a given driver, to be able to get flat frequency response of the whole loudspeaker system. Do not overthink with math - just change L and C values in a crossover simulator until you get flat flat frequency response and good phase response. While doing that, be careful - there should be no peaking (i.e. resonance) in the frequency response of the filter (that is what I meant by "filters should not resonate").
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It's called resonance whenever Q > 1/2, resulting in complex poles and in damped oscillatory terms in the impulse and step responses. Most filter types have those. It does not necessarily mean that you have peaking in the transfer.
Anyway, regarding the original question: it is rather complicated. It involves these steps:
Write the impedance of each inductor as sL and the impedance of each capacitor as 1/(sC), where s is the Laplace variable. It's a complex variable.
Calculate the transfer of the filter. For a ladder filter, you can do that by repeatedly applying the formula for a voltage divider, for example. Instead of calculating the transfer from input voltage to output voltage, you can also calculate the transfer from input voltage to input current (input admittance, reciprocal of the input impedance) as that normally has the same denominator and it's the denominator that matters. The input admittance is usually easier to calculate than the transfer from input to output. In either case, don't forget to include the load resistance.
The transfer or input admittance should now be rewritten as the ratio of two polynomials. The order of the denominator polynomial should be equal to the filter order.
Now one of the root finding methods for polynomials can be used to find the values of s for which the denominator is zero. These values are known as the poles of the filter. For third- and fourth-order polynomial equations, general analytic root finding methods are known, you can read all about it on Wikipedia. I never used them, I chicken out when it comes to this. For fifth- and higher orders, such methods do not exist, but there are numerical methods to approximate the solutions.
When there are complex-conjugate pairs of poles in the solution, their natural frequency is by definition fn = |s|/(2π), where |s| = √(Re(s) • Re(s) + Im(s) • Im(s)).
Anyway, regarding the original question: it is rather complicated. It involves these steps:
Write the impedance of each inductor as sL and the impedance of each capacitor as 1/(sC), where s is the Laplace variable. It's a complex variable.
Calculate the transfer of the filter. For a ladder filter, you can do that by repeatedly applying the formula for a voltage divider, for example. Instead of calculating the transfer from input voltage to output voltage, you can also calculate the transfer from input voltage to input current (input admittance, reciprocal of the input impedance) as that normally has the same denominator and it's the denominator that matters. The input admittance is usually easier to calculate than the transfer from input to output. In either case, don't forget to include the load resistance.
The transfer or input admittance should now be rewritten as the ratio of two polynomials. The order of the denominator polynomial should be equal to the filter order.
Now one of the root finding methods for polynomials can be used to find the values of s for which the denominator is zero. These values are known as the poles of the filter. For third- and fourth-order polynomial equations, general analytic root finding methods are known, you can read all about it on Wikipedia. I never used them, I chicken out when it comes to this. For fifth- and higher orders, such methods do not exist, but there are numerical methods to approximate the solutions.
When there are complex-conjugate pairs of poles in the solution, their natural frequency is by definition fn = |s|/(2π), where |s| = √(Re(s) • Re(s) + Im(s) • Im(s)).
By the way, for anyone interested in filter synthesis, I highly recommend DeVerl S. Humpherys, The Analysis, Design, and Synthesis of Electrical Filters, Prentice-Hall, 1970. I borrow it from the Delft university library when I really have to go into the details of filter synthesis.
A 4th order linkwitz riley is 2x 2nd order butterworth cascaded. That doesn't help with designing passive filters because the sections interact. and generally you are trying to attain some target curve rather than an actual textbook crossover. If you want to understand the theory better, I think you are going to have to dig into laplace transforms and circuits and solve them. You might develop an intuition with Phasor math as well.
The final coefficient in the denominator of a typical filter function will be (2*pi*Fo)^n, =1/ (the product of all of the component values) where n is order and Fo is "resonance" (not really) and for a 4th order the term is 1/LCLC ; and the same (except 3 terms and LCL or CLC) for a 3rd order, but that isn't going to tell you much because a filter can be detuned and not have a typical slope....
The final coefficient in the denominator of a typical filter function will be (2*pi*Fo)^n, =1/ (the product of all of the component values) where n is order and Fo is "resonance" (not really) and for a 4th order the term is 1/LCLC ; and the same (except 3 terms and LCL or CLC) for a 3rd order, but that isn't going to tell you much because a filter can be detuned and not have a typical slope....