Single Order SVF

[atavacron]

I came across the single order state variable filter on Rod Elliott’s site. It’s implemented in his bass amp as a unity gain crossover. I have a parametric EQ application for it that would require variable Q and static gain from both outputs. I could accommodate that gain being not 1, as long as both outputs were the same distance from 1. Here’s the basic circuit, with my best shot at a Q adjust.



I made the frequency variable with a linear pot. I’ve seen a rev log specified in Q adjust circuits that tie inverting and non-inverting inputs of the mix amp to ground, and Rqmax is often placed between the wiper and ground.

Rod describes R1 = R2 = R3 = R4, as the mix amp in normal non-inverting 3-amp SVFs is typically described - except typically R4 is positive feedback from bandpass and R1 is negative feedback from lowpass. This is different, so I’m wondering:

1) If the network to the left would do the variable Q job without changing gain over the RVq travel

2) If a variable resistance from the R3/R4 junction to ground would do the same, albeit only raising the Q

3) If R1 = R4 = R, and R2 = R3 = 2R, is the gain at both outputs still unity, but the Q (without an adjust) has changed

4) If you invert the LP output and sum it with the HP output, is the result a regular notch

FWIW the low shelf button on the Calrec PQ1549 turns a typical 3-amp SVF into the functional equivalent of this circuit, albeit with fixed Q and negative input. I assume the designer chose this method over simply grabbing the LP output because the result is a first-order response. Screencap is @gyraf ‘s redraw.

 

[gyraf]

I may have misunderstood something fundamental, but I don't think you have any option to change Q on first-order filters (iirc they're always ca. 0.7) - once you get q higher, it's no longer first-order..?

 

[atavacron]

I may have misunderstood something fundamental, but I don't think you have any option to change Q on first-order filters (iirc they're always ca. 0.7) - once you get q higher, it's no longer first-order..?

Ah. Well, the goal is the notch sum, which is not drawn. That notch sum gets subtracted from the original signal - also not drawn. The result (hopefully) is a 6dB/oct bandpass, with adjustable Q, and frequency controlled by just the one pole as drawn.

But i suppose if the mix amp isn’t adjusting Q for either HP or LP, the sum wouldn’t adjust either.

 

[user 37518]

I may have misunderstood something fundamental, but I don't think you have any option to change Q on first-order filters (iirc they're always ca. 0.7) - once you get q higher, it's no longer first-order..?

With first order filters, we could say that, properly speaking, there is no such thing as Q. Q may be defined as the ratio of energy stored vs energy dissipated during one period of signal. But Q is also related to the damping factor as Q=1/(2D), where I am using D to represent the damping factor. The damping factor is non-existent in a first-order filter, since the damping factor appears in the solution of a 2nd order differential equation as the coefficient of an exponential function, so Q really makes no sense with first-order filters, which produce first-order differential equations. The consequence of this is that the canonical transfer function of a first-order filter does not include Q anywhere in it, whereas in a second-order canonical transfer function, Q always appears in the coefficient of the second term in the denominator as w0/Q (omega zero/Q), regardless of whether it is a low-, high-, or band-pass filter, or even all-pass and band-reject filters. The term w0 is defined as the undamped natural frequency, or simply the natural frequency.

In band-pass filters, the ratio w0/Q may also be the Bandwidth of the filter, but it is important to note that, technically speaking, there is no such thing as a first-order band-pass filter. Actually, there is no such thing as an odd-order band-pass filter, since every band-pass filter transfer function will have a characteristic polynomial of even order. For this reason, it is possible to factor a high-order band-pass filter transfer function (for instance a 4th order band-pass filter) into a product of second-order band-pass transfer functions with the same Q in each characteristic polynomial but with different natural frequencies. Even though the Q of the total filter (that is the ratio of the center frequency to the frequency interval between the -3dB cut-off frequencies, aka bandwidth) might not correspond to the Q of each individual transfer function.

And why are there no odd-order band-pass filters? simple, you need at least two poles and one zero to make a band-pass filter, one pole for the low-cut frequency and the other pole for the high-cut frequency, which results in a transfer function with a second-order characteristic polynomial. So poles always have to be added in pairs (which produces even-order transfer functions) to keep the frequency response symmetric, otherwise, the filter might have a band-pass-ish response, but it is no longer considered a band-pass filter in the proper sense of the word.

Usually, when people speak of a first-order band-pass filter, they are actually referring to a second-order band-pass filter transfer function with a single pole in the low and high cut-off frequencies, because these filters produce a 6dB/octave attenuation slope on each side of the center frequency. The problem is that people have been erroneously taught that filter order refers to the attenuation slope, so most people associate the term 'first-order' with a 6dB/octave attenuation slope, 'second-order' with a 12dB/oct slope, and so on. But the truth is that the word 'order' refers to the order of the transfer function's characteristic polynomial, not to the attenuation slope; it is a mathematical term, not an electrical or audio term. Some people try to solve this ambiguity by making a compromise and calling band-pass filters with 6dB/oct slopes as first-order band-pass filters with second degree transfer functions, or band-pass filters with 12dB/oct slopes as second-order band-pass filters with fourth degree transfer functions, and so on.... but I don't like this 'solution', it is a contradiction in terms if you ask me, it would be like calling cows two-legged animals with 4 extremities.

So, to sum up. If you guys are talking about parametric equalizers with a 'bell' shape, those enter into the category of band-pass filters, and there are no first-order (or odd-order, e.g., no third- or fifth- order) band-pass filters. Mathematically you could select any Q value, practically it is another story, because the circuit might not actually achieve the required Q. Q has no meaning with first-order filters, although I guess it is possible to make a first-order circuit that emulates the effect of what a higher/lower Q would do in a second-order circuit (e.g., more or less ringing near the cut-off frequency), but it is not properly named Q (a proper term might be 'resonance' as with synthesizer filters).

P.S. A purely passive low-pass second-order filter which uses only 2 capacitors and 2 resistors, cannot have a Q higher than 0.5, in fact, the best you can do is go near a Q of 0.5 without actually reaching it. If you want to have a Q higher than 0.5 you must have a mix of inductors and capacitors, not simply capacitors. This is the reason why amplifiers such as op-amps are used, which can increase the value of Q by means of different techniques, for instance by adding positive feedback (as with the case of Sallen-Key circuits) without having to use inductors. At RF frequencies it is a different story, since inductors are relatively easier to realize than at audio frequencies (and much smaller) and, since most op-amps do not work at RF frequencies, the passive LCR filter becomes the norm, but I digress.

 

[atavacron]

Usually, when people speak of a first-order band-pass filter, they are actually referring to a second-order band-pass filter transfer function with a single pole in the low and high cut-off frequencies, because these filters produce a 6dB/octave attenuation slope on each side of the center frequency.

I am posting about a second order notch, built from the sum of the two outputs in the OP. That notch would be subtracted from the original signal to produce a second order bandpass. I didn't coin the term "Single order state variable filter," I'm just trying to figure out if its resultant summed notch can have variable bandwidth, using a conventional positive-feedback approach. I figured it would be simpler to just draw the thing above, because if this subcircuit doesn't work, then the point is moot.

 

[user 37518]

I am posting about a second order notch, built from the sum of the two outputs in the OP. That notch would be subtracted from the original signal to produce a second order bandpass. I didn't coin the term "Single order state variable filter," I'm just trying to figure out if its resultant summed notch can have variable bandwidth, using a conventional positive-feedback approach. I figured it would be simpler to just draw the thing above, because if this subcircuit doesn't work, then the point is moot.

I was addressing gyraf's comment.

 

[atavacron]

I was addressing gyraf's comment.

Okee. FWIW, you have good points.

 

[gyraf]

Thanks, user 37518 - haven't seen this described in-depth before..

/Jakob E.