A Better SVF Preamp

In an­oth­er di­gres­sion from my run­ning se­ries of posts on us­ing Rust to pro­gram the Bela, I de­cid­ed to do some ana­log au­dio pro­cess­ing. Why ana­log in­stead of the more fa­mil­iar to me dig­i­tal sig­nal pro­cess­ing route? There are sev­er­al rea­sons, one even re­lat­ed to the afore­men­tioned se­ries!

First, the un­re­lat­ed rea­son: I was sort­ing through some old box­es while look­ing for some­thing and dis­cov­ered my long-un­used col­lec­tion of elec­tron­ics com­po­nents. In­clud­ing a set of com­po­nents that looked a lot like some kind of au­dio project, for­got­ten for more than a decade—yes, re­al­ly, the ship­ping dates were on the la­bels! The (ex­pen­sive) Lun­dahl LL1538 au­dio trans­former was a good hint, but the slight­ly yel­lowed schemat­ic for a Bo Han­sén 1975 DI made it clear what it was. So I or­dered a PCB from 51x­Au­dio and the hand­ful of miss­ing—prob­a­bly lost in trans­port over the course of two moves—com­po­nents and fi­nal­ly fin­ished the project:

Photo of a fully assembled Bo Hansén Active DI-Box 1975 PCB in a PCB holder, with a soldering station in the background

And it worked im­me­di­ate­ly! With the elec­tron­ics bug in me re­vived (the “en­thu­si­asm”-kind not the “pro­gram­ming er­ror”-kind), I got back to think­ing about the next project I plan with the Bela—which brings us to the sec­ond, Bela-re­lat­ed rea­son why I'm do­ing this. Yeah, yeah, “fin­ish the first one first,” I hear you say, but I an­nounced this one right in the sec­ond post al­ready! If you don't re­mem­ber and aren't in the mood to go back to the pre­vi­ous post, here's the TL;DR: I plan to build a poly­phon­ic (bass) gui­tar us­ing Cy­c­fi Nu mod­u­lar pick­ups, us­ing the Bela to per­form per-string au­dio pro­cess­ing.

But wait, isn't that just more dig­i­tal pro­cess­ing? It is, but a poly­phon­ic set­up is a com­plex beast, typ­i­cal­ly re­quir­ing ad­di­tion­al con­trollers, mul­ti­ple am­pli­fiers, etc.; and some­times you just want to plug and play. And to do that, you re­al­ly want some­thing that be­haves like an or­di­nary, mono­phon­ic pick­up, which the Nus are not, in mul­ti­ple ways:

  1. They are ac­tive, i.e., they re­quire a bat­tery.
  2. They are poly­phon­ic (duh) and there­fore out­put per-string sig­nals in­stead of a sin­gle sig­nal for all strings.
  3. In­ter­nal­ly, the cap­sules use low-Z, wide-band­width pick­ups.

The first dif­fer­ence isn't much of an is­sue, and is par for the course for bass gui­tars, where ac­tive pick­ups and/or ac­tive preamps are very com­mon—two of the three I own have a bat­tery com­part­ment. Gui­tar play­ers are of­ten a lit­tle more con­ser­va­tive, but even there, ac­tive pick­ups, such as EMGs or Fish­mans, are pop­u­lar in some gen­res.

The sec­ond dif­fer­ence is eas­i­ly re­solved us­ing a sum­ming am­pli­fi­er. A sum­ming am­pli­fi­er is the ba­sis of any ac­tive mix­er and con­sists of a sin­gle op-amp (op­er­a­tional am­pli­fi­er) and a hand­ful of re­sis­tors—and in most cas­es some de­cou­pling ca­pac­i­tors to stop any DC off­sets from prop­a­gat­ing. If you are in­ter­est­ed in an over­view of how sum­ming am­pli­fiers work and why they are typ­i­cal­ly bet­ter than pas­sive mix­ers—such as the av­er­age mul­ti-pick­up bass's bal­ance con­trol—check out El­liot Sound Prod­uct's ar­ti­cle on au­dio sig­nal mix­ing. They gen­er­al­ly have a lot of great ar­ti­cles on au­dio elec­tron­ics, so def­i­nite­ly worth check­ing out if you are in­ter­est­ed in these things—and I as­sume you are, hav­ing read this far.

The third is­sue is where things get more com­pli­cat­ed. First off, what does low-Z and wide band­width even mean in this con­text? Low-Z is short for low im­ped­ance (Z be­ing the stan­dard sym­bol used for im­ped­ances in elec­tron­ics lit­er­a­ture). Es­sen­tial­ly, im­ped­ance is the fre­quen­cy-de­pen­dent ana­log to a re­sis­tance for AC sig­nals. Typ­i­cal­ly, a low-Z pick­up will have few­er wind­ings and there­fore low­er DC re­sis­tance, low­er in­duc­tance, and low­er wind­ing ca­pac­i­tance than a reg­u­lar pick­up. This leads to a wider band­width, i.e., they pick up high­er fre­quen­cies, typ­i­cal­ly with­out los­ing low-fre­quen­cy re­sponse, and a res­o­nance fre­quen­cy out­side the au­di­ble fre­quen­cy spec­trum; but al­so sig­nif­i­cant­ly re­duced out­put, lead­ing to the ne­ces­si­ty of ac­tive elec­tron­ics (or trans­form­ers in Lace Alu­mi­tones).

Wait. Res­o­nance fre­quen­cy? Yes, all coils, in­clud­ing trans­form­ers and pick­ups, have a res­o­nance fre­quen­cy, de­ter­mined by their in­duc­tance and wind­ing ca­pac­i­tance—as well as any ex­ter­nal loads, such as long ca­bles or a rolled down tone knob. These form a res­o­nant, sec­ond-or­der (12 dB/oc­tave) low-pass fil­ter. The oth­er par­a­sitics of the coil af­fect the so-called Q of this fil­ter. Q is of­ten de­scribed as band­width, but that on­ly re­al­ly ap­plies to sec­ond-or­der band-pass fil­ters. For low-pass fil­ters it makes more sense to think about Q as the gain at the res­o­nant fre­quen­cy.

Is this res­o­nance de­sir­able? Well, yes and no. In au­dio trans­form­ers for pro­fes­sion­al au­dio ap­pli­ca­tions, it is un­de­sir­able and a Zo­bel net­work is used to tame the res­o­nance and en­sure a flat re­sponse across the spec­trum (check out ESP for de­tails). Gui­tar pick­ups are a dif­fer­ent sto­ry. Here, the res­o­nance is usu­al­ly smack dab in the mid­dle of the au­dio spec­trum—with res­o­nance fre­quen­cies rang­ing from 800 Hz for hum­buck­ers to 8 kHz for sin­gle coils—and a large fac­tor in the per­ceived “tone” of the pick­up. In fact, an am­pli­fi­er with a low in­put im­ped­ance will lead to sig­nif­i­cant damp­ing of the res­o­nance lead­ing to a tone that is of­ten de­scribed as “dead.” By the way, a DI box as shown above is used con­vert a high-im­ped­ance sig­nal to a bal­anced low-im­ped­ance sig­nal for a mi­cro­phone preamp when record­ing. In any case, gi­tar­renelek­tron­ik.de has a num­ber of good ar­ti­cles on gui­tar pick­up res­o­nance and how to af­fect it (al­beit on­ly in Ger­man).

So, how can we em­u­late the re­sponse of clas­sic pas­sive pick­up with a wide band­width ac­tive pick­up? There are a cou­ple op­tions. One op­tion would be to add a pas­sive RLC-net­work, op­tion­al­ly with switch­able ca­pac­i­tors and re­sis­tors, al­though that would be af­fect­ed by ex­ter­nal loads too, un­less an ac­tive buf­fer is added. An­oth­er op­tion, the route I chose, is to use an ac­tive fil­ter. A great fil­ter topol­o­gy for this ef­fect is the state vari­able fil­ter:

Hand-drawn schematic of a three-op-amp state variable filter. Frequency is set by the resistors RF and the capacitors C, resonance is set by the resistors RQ1 and RQ2, where RQ2 is the resistance to ground

While oth­er sec­ond-or­der fil­ter topolo­gies re­quire few­er than three op-amps, the SVF topol­o­gy al­lows in­de­pen­dent con­trol of fre­quen­cy and Q. Ad­di­tion­al­ly, both Alem­bic and Wal, two of the most pre­mi­um/lux­u­ry-lev­el bass builders al­ready use ac­tive low-pass fil­ters in their preamps! Pre­sum­ably, they use SVFs, al­though it isn't re­al­ly doc­u­ment­ed, since their elec­tron­ics are pro­pri­etary. For an over­view of ac­tive fil­ters for au­dio, you can, again, check out ESP.

Now let's look in­to the cir­cuit in more de­tail! In the schemat­ic above, fre­quen­cy and Q are set by

f=12πRFCf = \frac{1}{2 \pi R_F C}

and

Q=13g,Q = \frac{1}{3 g},

where gg is the volt­age di­vider gain

g=RQ2RQ1+RQ2.g = \frac{R_{Q2}}{R_{Q1} + R_{Q2}}.

In the lit­er­a­ture, vari­able SVFs are typ­i­cal­ly di­men­sioned by re­plac­ing the re­sis­tors RFR_F with a du­al-gang rheo­stat (vari­able re­sis­tor) ad­justable be­tween 0 Ω and RVF\mathit{RV}_F in se­ries with fixed re­sis­tors RF0R_{F0} each. Along with the ca­pac­i­tors CC this sets the fre­quen­cy range to

fmin=12π(RVF+RF0)Cf_\mathrm{min} = \frac{1}{2 \pi (\mathit{RV}_F + R_{F0}) C} fmax=12πRF0Cf_\mathrm{max} = \frac{1}{2 \pi R_{F0} C}

Sim­i­lar­ly, Q can be con­trolled by re­plac­ing ei­ther RQ1R_{Q1} or RQ2R_{Q2} by a rheo­stat RVQ\mathit{RV}_Q in se­ries with a fixed re­sis­tor RQ0R_{Q0} . If RQ1R_{Q1} is kept fixed and RQ2R_{Q2} is var­ied, this re­sults in a Q be­tween

Qmin=1+RQ1RVQ+RQ03Q_\mathrm{min} = \frac{1 + \frac{R_{Q1}}{\mathit{RV}_Q + R_{Q0}}}{3}

and

Qmax=1+RQ1RQ03.Q_\mathrm{max} = \frac{1 + \frac{R_{Q1}}{R_{Q0}}}{3}.

I plugged in the num­bers for my de­sired fre­quen­cy ad­just­ment range of 500 Hz to 10 kHz and Q ad­just­ment range of 0.5 (-6.02 dB) to 5.62 (+15 dB), and the 50 kΩ of the po­ten­tiome­ters—vari­able volt­age di­viders that can be wired as rheostats by short­ing the wiper to one of the oth­er two pins—re­sult­ing in RF02.61kΩR_{F0} \approx 2.61\mathrm{k\Omega} , C6.2nFC \approx 6.2\mathrm{nF} , RQ01.62kΩR_{Q0} \approx 1.62\mathrm{k\Omega} , and RQ125.5kΩR_{Q1} \approx 25.5\mathrm{k\Omega} . Then I got all the nec­es­sary parts and set them up on a bread­board.

Breadboard with circuit with two DIP8 ICs, a dual and a single gang potentiometer, two quarter inch jack connectors and a 9V battery

And it worked. Kind of. While it did what it was sup­posed to, the ad­just­ment range of the po­ten­tiome­ters was down­right ter­ri­ble, with more than 80% of the ad­just­ment hap­pen­ing in the last 20% or so of the ro­ta­tion range. Not ex­act­ly us­able on stage. One method to solve this is to use po­ten­tiome­ters with a dif­fer­ent ta­per. In this case, due to the re­cip­ro­cal na­ture of our ranges, it would have to be a re­verse au­dio/in­verse log­a­rith­mic ta­per. Hard to source, es­pe­cial­ly as an in­di­vid­u­al, and the con­for­mance—how close the re­sis­tances of a du­al-gang po­ten­tiome­ter are to one an­oth­er—is gen­er­al­ly much worse than for lin­ear po­ten­tiome­ters. And you want them to be close, oth­er­wise the Q will fluc­tu­ate while ad­just­ing the fre­quen­cy.

Luck­i­ly, I was re­cent­ly in con­tact with Joel de Guz­man, prin­ci­pal ar­chi­tect and en­gi­neer at Cy­c­fi Re­search who is de­sign­ing a cus­tom Nu Mul­ti for me. And he al­so hap­pens to be a C++ de­vel­op­er like I am—Rust 🦀 is still most­ly re­strict­ed to hob­by work. He sug­gest­ed us­ing the po­ten­tiome­ters as ac­tu­al three-ter­mi­nal po­ten­tiome­ters in­stead of two-ter­mi­nal rheostats. So how does that work? Let's fig­ure it out!

To do that, we'll have a look at each of the fil­ter's stages in­di­vid­u­al­ly. So let's be­gin with the in­put / high-pass stage. Here, both the in­put and the out­puts of the oth­er two stages come to­geth­er as feed­back:

Hand-drawn schematic of the input stage of the state variable filter, including the input and feedback resistors, as well as RQ1 and RQ2

So can we just use our po­ten­tiome­ter to com­plete­ly re­place RQ1R_{Q1} and RQ2R_{Q2} , since they al­ready form a volt­age di­vider? Well, not re­al­ly. While the one end of the ad­just­ment range would re­sult in a Q of ⅓ (-9.54 dB), which is far too high­ly damped to be us­able but could po­ten­tial­ly work, the oth­er end of the range would re­sult in a Q of… ∞. Not good. So we def­i­nite­ly need re­sis­tors at both ends of the po­ten­tiome­ter. But just adding those would re­sult in a sim­i­lar­ly ill-suit­ed ad­just­ment range as be­fore, so what can we do? We can fake a non-lin­ear ta­per by adding a “shap­ing” re­sis­tor in par­al­lel to one of the sides, i.e., be­tween the wiper and one of the two ter­mi­nals, of the po­ten­tiome­ter as de­scribed in an old Ge­oFex ar­ti­cle. In this case, be­tween the wiper and ter­mi­nal three (the clock­wise ter­mi­nal) of the po­ten­tiome­ter:

Hand-drawn schematic of the potentiometer-based voltage divider network used to set Q. The band-pass output is connected to RQCCW; pin 1 of the potentiometer RVQ is connected to the other side of RQCCW; pin 2 (the wiper) of RVQ is connected to the non-inverting input of the input stage as well as RQS; the other side of RQS is connected to the final pin of RVQ and RQCW; finally, the other side of RQCW connects to ground

To sim­pli­fy the cal­cu­la­tions, let's use re­sis­tances rel­a­tive to RVQ\mathit{RV}_Q , the to­tal re­sis­tance of the po­ten­tiome­ter, des­ig­nat­ed by low­er­case rr . This re­sults in

g=rQcw+((1t)rQS)rQcw+((1t)rQS)+t+rQccw,g = \frac{r_{Q\mathrm{cw}} + ((1-t) \parallel r_{QS})}{r_{Q\mathrm{cw}} + ((1-t) \parallel r_{QS}) + t + r_{Q\mathrm{ccw}}},

where t[0,1]t \in [0, 1] is the trav­el of the po­ten­tiome­ter. By plug­ging in t=0t = 0 and QminQ_\mathrm{min} as well as t=1t = 1 and QmaxQ_\mathrm{max} in­to our volt­age di­vider and Q-equa­tion, we ar­rive at

rQcw=3Qmin11+1rQS+13(QmaxQmin)r_{Q\mathrm{cw}} = \frac{\frac{3 Q_\mathrm{min} - 1}{1 + \frac{1}{r_{QS}}} + 1}{3 (Q_\mathrm{max} - Q_\mathrm{min})}

and

rQccw=(3Qmax1)rQcw1.r_{Q\mathrm{ccw}} = (3 Q_\mathrm{max} - 1) r_{Q\mathrm{cw}} - 1.

How do we choose rQSr_{QS} ? I just put tried to plug in a few val­ues, and the op­ti­mum ap­pears to be some­where be­tween 114\frac{1}{14} and 112\frac{1}{12} for my de­sired ad­just­ment range. For oth­er ranges it will be dif­fer­ent. If you felt like it, you could even run an op­ti­miz­er to find the low­est pos­si­ble er­ror across the en­tire range. But with RQS=3571ΩR_{QS} = 3571\mathrm{\Omega} , RQcw=3361ΩR_{Q\mathrm{cw}} = 3361\mathrm{\Omega} , and RQccw=3347ΩR_{Q\mathrm{ccw}} = 3347\mathrm{\Omega} for RVQ=50kΩ\mathit{RV}_Q = 50\mathrm{k\Omega} . Ob­vi­ous­ly, we still have to round these to ac­tu­al­ly avail­able E se­ries val­ues, but they re­sult in very good ad­just­ment be­hav­ior:

Diagram showing the relationship between the rotation of the potentiometer RVQ and the resonance gain in decibels, as well as the idealized response

The fre­quen­cy is con­trolled by the two es­sen­tial­ly iden­ti­cal in­te­gra­tor stages. So let's just look at one of them:

Hand-drawn schematic of one of the integrator stages including RF and the feedback capacitor C. A dashed line points out that the inverting input is a virtual ground

What RFR_F does here is a lit­tle less ob­vi­ous. In the pre­vi­ous case, we had a volt­age di­vider con­nect­ed to the high-im­ped­ance non-in­vert­ing in­put of an op-amp. Here, what we have is just a sin­gle re­sis­tor con­nect­ed to the in­vert­ing in­put, which I la­beled as vir­tu­al ground. What does that mean? Well, op-amps have a very high gain and the neg­a­tive feed­back will try to push the node at the in­vert­ing in­put to as close as the same volt­age as is present at the non-in­vert­ing in­put of the op-amp. Which in this case is ground. So in­stead of a very high-im­ped­ance volt­age in­put, we es­sen­tial­ly have a very low-im­ped­ance cur­rent in­put and RFR_F con­trols the cur­rent flow­ing in­to the node at the in­vert­ing in­put! So what can we do with a po­ten­tiome­ter in this po­si­tion? We can wire it up as a “cur­rent split­ter” that con­trols both how much cur­rent flows from the pre­vi­ous stages out­put and how much of it flows to ground or to vir­tu­al ground:

Hand-drawn schematic of the potentiometer-based frequency-setting network. Pin 3 of the potentiometer RVF is connected to the output of the previous stage; pin 2 is connected to RFS, which in turn is connected to the inverting input of the integrator; finally, pin 1 of RVF connects to ground via an additional resistor RFCCW. The current flowing in from the previous stage is indicated as i0 and the current flowing into the virtual ground at the inverting input is called iin

Here i0i_0 is the cur­rent flow­ing out of the pre­vi­ous stage's out­put and iini_\mathrm{in} is the cur­rent flow­ing in­to the vir­tu­al ground node. To sim­pli­fy cal­cu­la­tions, let's de­scribe the above cir­cuit us­ing fixed re­sis­tors:

Simplified version of the previous schematic, where the resistance between pins 3 and 2 of RVF is called RA; the sum of the resistances RFCCW and the resistance between pins 2 and 1 of RVF is called RB; and RFS is abbreviated as R

Since both the in­vert­ing in­put and ground are at 0 V, RR and RbR_b are es­sen­tial­ly par­al­lel re­sis­tors for the pur­pose of com­put­ing i0i_0 :

i0=uprevRa+(RbR)i_0 = \frac{u_\mathrm{prev}}{R_a + (R_b \parallel R)}

iini_\mathrm{in} can now be com­put­ed us­ing stan­dard cur­rent di­vider rules:

iin=i0RbRR=uprev(RaRb+1)R+Ra\begin{aligned} i_\mathrm{in} &= i_0 \frac{R_b \parallel R}{R} \\ &= \frac{u_\mathrm{prev}}{\left(\frac{R_a}{R_b} + 1\right)R + R_a} \end{aligned}

Ef­fec­tive­ly, at least for the pur­pose of com­put­ing iini_\mathrm{in} , our cur­rent split­ter acts like a re­sis­tor

RF(RaRb+1)R+Ra,R_F \equiv \left(\frac{R_a}{R_b} + 1\right)R + R_a,

or in terms of re­sis­tances rel­a­tive to RVF\mathit{RV}_F

rF(1tt+rFccw+1)rFS+1t.r_F \equiv \left(\frac{1-t}{t + r_{F\mathrm{ccw}}} + 1\right)r_{FS} + 1-t.

As be­fore, we can plug in t=0t = 0 and t=1t = 1 and ar­bi­trar­i­ly choose rFSr_{FS} to op­ti­mize the ro­ta­tion­al be­hav­ior of our con­trol. rFccwr_{F\mathrm{ccw}} is then de­ter­mined by

rFccw=1fmaxfmin11rFSr_{F\mathrm{ccw}} = \frac{1}{\frac{f_\mathrm{max}}{f_\mathrm{min}} - 1 - \frac{1}{r_{FS}}}

For my de­sired fre­quen­cy range of 500 Hz to 10 kHz, rFS=29r_{FS} = \frac{2}{9} re­sults in a very good per­for­mance. Us­ing that val­ue and RVF=50kΩ\mathit{RV}_F = 50\mathrm{k\Omega} re­sults in RFS=9091ΩR_{FS} = 9091\mathrm{\Omega} , RFccw=3704ΩR_{F\mathrm{ccw}} = 3704\mathrm{\Omega} , and C=1.8nFC = 1.8\mathrm{nF} . These re­sult in the fol­low­ing ad­just­ment curve:

Diagram showing the relationship between the rotation of the dual-gang potentiometer RVF and the resonance frequency in Hertz, as well as the idealized response. The frequency axis is plotted logarithmically

Are we done? Well, for this ar­ti­cle, pret­ty much, as build­ing the fil­ter is still a to-do. How­ev­er, I did plug it all in­to LT­Spice and sim­u­lat­ed it with re­al op-amp mod­els and an op-amp volt­age fol­low­er driv­en ground po­ten­tial. The lat­ter is nec­es­sary—and in­clud­ed in the bread­board cir­cuit above as well—since a bat­tery on­ly pro­vides a sin­gle volt­age, not a pos­i­tive and a neg­a­tive volt­age, so we have to fake it. In any case this is the re­sult­ing small-sig­nal gain:

Bode plot of the simulated small-signal gain. The resonant peaks and frequency vary as expected

So the sim­u­la­tion agrees with all those cal­cu­la­tions, yay! Re­al­i­ty may turn out to be a lit­tle more com­plex, but I need to wait for some parts be­fore I can try it in prac­tice. Ad­di­tion­al­ly, the mod­i­fied cir­cuit in­cludes some fair­ly low im­ped­ances, which is good for noise per­for­mance, but bad for bat­tery life, so I may have to move to high­er-val­ue po­ten­tiome­ters as well—which have be­come rar­er as well for some rea­son. Maybe some­thing in the 200 to 500 kΩ range? Once I have bread­board­ed ev­ery­thing and de­cid­ed on com­po­nent val­ues, I'll ei­ther have to con­vert it to a perf­board lay­out or try to de­sign a PCB with Ki­CAD—more ex­pen­sive but far more com­pact, es­pe­cial­ly with SMD com­po­nents. In any case, that's it for to­day, and as usu­al, feel free to fol­low me and send me a DM on Twit­ter if you have any ques­tions or com­ments.