Choosing equalizer bandwidth

[Fancy Lime]

Hi there!

I'm working on an EQ for a distortion pedal. It is probably either going to be a three band with parametric mids or a four band with all frequencies fixed. Gyrators are probably the way to go here, I'm starting off with the basic topology used in many boss pedals, like the Metal Zone or the EQ-7. Trying to figure out how to choose the Q factors for each band, I plugged the values of the EQ-7 (https://www.hobby-hour.com/electronics/s/schematics/boss-ge7-equalizer-schematic.png) into Jack Orman's Gyrator calculator (http://www.muzique.com/lab/gyrator.htm). The Q values are between 3.11 and 4.07. Now the bands are centered at (very approximately) 100, 200, 400, 800, 1600, and 3200 Hz. (The highest of the 7 bands is a shelving filter with a supposed frequency of 6400Hz that actually has it's -3db cut-off at 4100Hz.) Since the bands are an octave apart, I was expecting them to be an octave wide, no? Half an octave to the left, half an octave to the right. Instead they are around 0.4 octaves wide according to Jack's calculator.

I doubt that they made the bands too narrow by accident or on purpose, meaning I am not understanding something right. Can someone help me clear this up. Understanding this seems sort of crucial for building a useful EQ.

Thanks,
Andy

[Mark Hammer]

Two years ago, I had dinner with a bunch of pedal guys I admire, one of them being Joel Korte of Chase Bliss.  I told Joel that I felt the "ideal" guitar tone shaper had 6 knobs and a maybe a couple of toggles.  The 6 knobs were:
- bass shelving EQ
- two bands of semi-parametric with boost/cut and sweepable centre-frequency
- a sweepable 2-pole lowpass filter.

Toggles to adjust Q would be optional, but being able to control the bandwidth and add a couple of peaks (or dips or one of each) wuld get you what you need 99% of the time.

Joel cogitated and came up with his Condor pedal.  It's a smidgen different than what I envisioned, but not much.  And it is remarkable flexible as a tone shaper.  If you wish to build it into the distortion pedal, I will suggest  one sweepable resonant cut/boost before your clipping stage/s, one after, and either a sweepable, or at least switchable, 2-pole lowpass before the output.  Will it get you everything?  No, but it should get you a universe of tones.

[Rob Strand]

The point often missed about those Gyrator equalizers is the Q is *not* constant.   The Q and the amount of boost/cut are linked.   At low boost/cut the Q is very low.    You only get the design Q at full boost.    The Q is roughly proportional to the boost.

As the circuit stands it's not so easy to see.  However, imagine designing a boost only equalizer except you make the boost variable by adjusting the series resistor.    To first order approximation that's what is going on.  Now look at the design equations.  The Q depends on the total series resistance.

Suppose you design an equalizer with 12dB boost and Q=3 then design another with 15dB boost and Q=3.  When you set each equalizer to say 6dB boost  the actual Q produced on the 12dB design will be about 1 but the Q on the 15dB design is 0.65.    To get 6dB boost the 15dB design need to backed-off the boost more relative to 15dB that means the Q gets backed-off more than the 12dB design.

When set to 6dB the *total series resistance* must be set to the same resistance in both designs.  The L and C values (and minimum series R value) for the 12dB and 15dB cases are not the same so the final Q is different for any given resistance.

My advice for these things is to tweak the Q by ear *at the settings you are going to use*.    Typically you might only use 6 to 10dB  and a Q of around 1.0 usually sounds pretty good.     That just happens to match-up with a Q=3 design for 12dB or 15dB equalizer.

One more point is calculator only calculates maximum Q.   The series R value is linked to the L in the gyrator design.  When you add a boost pot there is a second series R which adds to the gyrator series R, it doesn't affect the L but it does affect the amount of boost and the Q.   So what I'm saying is you won't be able to use the calculator to see the Q variations at different boosts because the calculator only has one series R, the one that affects L.

I was expecting them to be an octave wide, no? Half an octave to the left, half an octave to the right. Instead they are around 0.4 octaves wide according to Jack's calculator.

When you have a numbers of band those equalizers perform much differently to a single band.  The centre frequency is about as expected but the Q's and maximum boost get screwed up because the bands interact.
The best way to convince yourself is to compare spice with the design equations.  You will see it's almost pointless using a calculator for the graphic equalizer case.  For widely split bands the calculator is OK but you will still find it doesn't agree with what you expected from the design equations.

Something I forgot to mention is the value of the boost/cut pot affects things as well.

You have to use graphic equalizer pot tapers for those equalizers otherwise all the boost/cut is bunched-up in the last 10% of the pot sweep.   You can reduce that effect by using lower value pots but you then have to watch out for the increase in noise.

Single-band circuit vs multi-band circuit with same gyrator:



Here the Q's are close because the Q is high but it doesn't always work out like that.

[Fancy Lime]

Hi Mark,
I agree that what you describe is pretty much the ideal guitar EQ. Although I would reserve this type of design for 19" units and make the mid bands fully parametric. The challenge I am facing here is the flexibility-simplicity trade-off. Which to me is the most difficult part of pedal design. I am trying to make something that is as flexible as possible, while being easy and intuitive to use and not impossibly difficult to build. As usual, pick any two of those three and you're fine; but striking a good balance between all three? Much harder. There is, however definitely going to be a Sallen-Key low pass after the main EQ, with a bit of resonance around 3kHz and a cut-off around 5-6kHz, and probably a 3-way switch for some control of the cut-off frequency. This may end up being made variable and replacing the Hi control of the main EQ. Having a shelving characteristic instead of a band for bass would be preferable but I have not yet figured out if that can be done with the basic design I am looking at. I have not found any examples for it. So right now the idea is to use two overlapping low-Q bands at around 30Hz and 100Hz, controlled by the same pot. That trick is brought to you by the Hi-EQ of the Boss HM-2.


Hi Rob,
well, yes: if the "virtual order" of the filter (in this case defined by the properties of the gyrator) remains the same, then Q and boost/cut are coupled. But that is the same for all EQ designs that I can think of at the moment, no? I'm not sure if it would be very musical if that were not the case. You are certainly right about the "design by ear". Designing an EQ, or anything that is supposed to make a "sound", based on numbers on paper alone is not likely going to be all that great. The matter is just too complex and hast too much to do with taste and feelings to make it possible to predict what EQ-curve we will find sonically pleasing, at least beyond a certain level of complexity. Depending on experience, of course. So my usual design approach is 1) design on paper what I *think* will do what I want; 2) breadboard that; 3) modify until I am satisfied by ear; 4) trace the design back from the bredboard. What I end up with is sometimes close and sometimes very different from what I started. Right now I am trying to understand some theoretical aspects so I can do the "modify by ear" bit a little less blindly.

As for the pot values: I have never heard of "graphic taper" or anything like it. What would that be? I was going to try linear and W-taper (the tube screamer 20k variety because they are the ones readily available) pots. The pot values you see seems to differ a lot between designs. From about 0.1 to 10 times the value of the "to V-ref" resistor in the gyrator. Can you share any insights into the pros and cons of those choices?

Thanks and Cheers,
Andy

[Rob Strand]

But that is the same for all EQ designs that I can think of at the moment, no?

Most do it for sure.  IIRC the gyrator type EQ does that a little more than other types.  The Rane company developed a constant-Q equalizer.

So my usual design approach is

You should get there that way, no problems.  It's a bit fiddly with those gyrators so I have had a couple gyrators on the breadboard to switch between so you can A/B things.  Maybe tweaking along the way when one obvously isn't what you want.    Other way is to use a separate parametric EQ, tweak the knobs then measure the frequency response and match it with a gyrator design.

As for the pot values: I have never heard of "graphic taper" or anything like it. What would that be? I was going to try linear and W-taper (the tube screamer 20k variety because they are the ones readily available) pots.

I use the term graphic taper since the Japanese code is "G".  They don't have much use outside of graphic equalizers.  The US name for the taper is "W" but I have seen W used for other tapers by some manufacturers so watch out.   The Japanese codes are quite consistent.   

The pot values you see seems to differ a lot between designs. From about 0.1 to 10 times the value of the "to V-ref" resistor in the gyrator. Can you share any insights into the pros and cons of those choices?

For for W-taper/G-taper pots:  Ideally when you set the pot to half way up the boost direction you get half the maximum boost.  So from that graph at 75% rotation is about 0.92 the way through the pot; some pots might differ.  I would simulate the the design at that setting and tweak the pot to get a half the maximum boost.  It's not that straight forward.  You can only buy pots in large steps of resistance so what you would really do is pick a pot resistance then adjust the feedback/input resistors and gyrator to get the half boost at half the pot position.

Now for non-critical designs with a few bands you can get reasonable behaviour (ie. not cramped at the ends) using linear pots provided you choose a largish feedback/input resistor (RF) relative to the pot resistance (RV).  It helps not to target large amounts of boost/cut as well.    So RV = 2*RF is pretty good maybe upto 3*RF.  For RV=2*RF and 12dB boost you get just under 4dB at 75% of the pot rotation whereas ideally that should be 6dB.   4*RF to 5*RF is OK but can become too cramped.   In general the smaller RV compared to the RF gives less cramping of the control but the penalty is noise.  I'm not saying it's noisy I'm just saying you should check how much noisier it is compared to using a W-taper pot.   The W-taper pot is always better from a noise perspective but there's cases where it makes little difference and it's much more convenient using linear pots.

EDIT: fixed the ratios.

[ElectricDruid]

Since the bands are an octave apart, I was expecting them to be an octave wide, no? Half an octave to the left, half an octave to the right. Instead they are around 0.4 octaves wide according to Jack's calculator.

I doubt that they made the bands too narrow by accident or on purpose, meaning I am not understanding something right. Can someone help me clear this up. Understanding this seems sort of crucial for building a useful EQ.

I suspect this is to do with how Q is defined in bandpass filters. It's the centre frequency, divided by the bandwidth between the lower -3dB point and the upper -3dB point. Imagine what the frequency response looks like - typical bell curve. The part that is regarded as the "passband width" is only the bit between -3dB points at the top of the bell. Since the "skirts" of the bell actually go out quite a bit wider than that (especially for lower-order filters) you might well want to set the Q a bit higher than the raw numbers suggest to stop one band interfering with another quite so much. As usual, there's a trade-off there between having bands that don't overlap too much and having a sound which is unnaturally "peaky".

if the "virtual order" of the filter (in this case defined by the properties of the gyrator) remains the same, then Q and boost/cut are coupled. But that is the same for all EQ designs that I can think of at the moment, no?

Aside from the Rane "Constant Q" design (and probably subsequent copies) mentioned earlier, yes, that's true.

https://www.rane.com/note101.html

I think this is a bit over-played though. Saying that the Q varies with cut/boost makes it sound like the curve gets wider or narrower as you range the cut or boost. I don't think it does, really. The skirt goes out just as wide at high boost as at low boost. What happens is that because of the way Q is defined, the Q works out much lower for a low, rounded hill than a tall mountain. Doesn't mean you don't reach the foothills for both at the same place:

(or see the MT-2 curves below)

There's a picture on that Rane page that explains what a "Constant Q" EQ looks like.

Instead of "squashing" the curve as you turn down the boost, it "shrinks" it.

There's one other issue, which is the coupling between frequency and resonance. The gyrator-based EQ design suffers very badly from this, but not all EQ sections do.

For example, state-variable-filter-based EQs don't have that problem. In the SVF, resonance and frequency can be completely separate. They use 3 op-amps per section rather than the one you need for a gyrator though, so cheaper designs settle for the gyrator.

The Wein-bridge EQ used for mids in the MT-2 also provides a very constant Q as frequency is varied:


Hope these thoughts give you a few more ideas to play with.

T.

[Fancy Lime]

Rob, Tom, thanks!

Damn, I'm gonna have to learn some kind of SPICE simulation, don't I? Any suggestions that work on Linux? KiCad any good?

The Wien bridge: I think that probably only makes sense for a semi parametric and at the moment I lean more toward a non-parametric 4 band. Semi-parametric with gyrators only makes sense when the center frequency shift is small. Even then it's not ideal but should be workable. Switching frequencies is probably the better idea for that.

I thought a bit about implementation and realized that the gyrators can easily go on a separate board, meaning that designing the layout of the main board is pretty much independent of the EQ section. So the EQ can easily be swapped, which is great for a DIY project. If someone thinks that 2 bands are plenty, they can build it with 2. If someone thinks they absolutely must have 12 bands, no problem either. Or install a switch to toggle between several differently tuned EQ's.

About the changing Q: I tend to like slight boosts or cuts with low Q, and more intense ones with a higher Q. So I think the behavior of the gyrators s really more feature than bug for me.

Cheers,
Andy

[Rob Strand]

Damn, I'm gonna have to learn some kind of SPICE simulation, don't I? Any suggestions that work on Linux?

Spice is extremely useful for things like those graphic equalizers.   You pretty much know in advance you have to do more than simple hand calculations.  I haven't tried KiCad.   LT spice used to run under Wine.   Some of the PC spice programs didn't  run under Wine.