SynthMaker How an OTA filter works: two integrators and a current
An OTA filter is two OTAs, each charging a capacitor, in a loop: a 12 dB per octave low-pass whose cutoff follows the OTAs' bias current, 6.75 Hz per µA here, so 120 µA gives 810 Hz. Feeding the middle back sets the resonance: Q = Rq / 100 kΩ.
circuit OtaFilter setting input: 6 V, from 0.6 V to 30 V setting output: 20 V, from 2 V to 100 V setting cutoff: 120k, from 1M to 33k setting resonance: 300k, from 47k to 1M V1 vcc gnd 15V R1 in p1 100k R2 p1 gnd 220 A1 p1 m1 c1 abc1 CA3080 Rlp out m1 100k Rq bp m1 resonance R4 m1 gnd 220 Ca c1 gnd 1n U3 c1 bp bp opamp R5 bp p2 100k R6 p2 gnd 220 A2 p2 gnd c2 abc2 CA3080 Cb c2 gnd 1n U4 c2 out out opamp Rk1 vcc abc1 cutoff Rk2 vcc abc2 cutoffendPress Play to see the wave after the circuit, at out.
What you're hearing
An OTA and a capacitor make an integrator whose speed is a current. An OTA's output is a current: the voltage between its inputs times gm = Iabc / 2Vt, where Iabc is the current into its bias pin (see how a VCA works). Into a capacitor, a current changes the voltage at I / C. At 120 µA, gm is 2.32 mS, so 1 mV between the inputs makes 2.32 µA, and that moves a 1 nF capacitor 2.3 V per ms. Double the bias current and it moves twice as fast. Feed the capacitor's voltage back to the OTA's minus input, and it chases the input but can't keep up with its fastest parts: a one-pole low-pass, 6 dB per octave, here with its corner at 810 Hz. The 100 kΩ and 220 Ω dividers keep the OTAs' inputs down at millivolts.
Two in a loop make a 12 dB low-pass. A1 charges Ca, buffered by U3: that's the band-pass. A2 charges Cb from it, buffered by U4: the low-pass, and the output. The output goes back to A1's minus input through 100 kΩ, closing the loop. Its cutoff is f0 = gm × 0.0022 / (2π × 1 nF) = 2.32 mS × 0.0022 / (2π × 1 nF) ≈ 810 Hz: 6.75 Hz for every µA of bias current. Above it the treble falls 12 dB per octave: at four times the cutoff it's 23.6 dB down. It's the loop of a state-variable filter, two integrators with the band-pass between them. The difference is what sets their speed: resistors there, a current here.
The cutoff knob sets the current. The bias pins sit about 0.6 V above ground, so the current is (15 V − 0.6 V) / Rk, and one knob turns Rk1 and Rk2 together, a dual pot:
| cutoff (Rk) | bias current | cutoff frequency |
|---|---|---|
| 1 MΩ | 14 µA | 98 Hz |
| 470 kΩ | 31 µA | 207 Hz |
| 220 kΩ | 66 µA | 442 Hz |
| 120 kΩ | 120 µA | 810 Hz |
| 68 kΩ | 212 µA | 1.43 kHz |
| 47 kΩ | 306 µA | 2.07 kHz |
| 33 kΩ | 435 µA | 2.94 kHz |
That's 4.9 octaves from end to end. The knob turns on a log taper, so it moves the cutoff evenly in octaves. The cutoff is proportional to the current, so a control voltage at the far end of Rk would move it in hertz, not octaves. A synth puts a transistor before the bias pin, an exponential converter, so that each volt moves the cutoff an octave. This one doesn't follow the keys: you tune it by hand, or wire something into cutoff.
Resonance: less damping. Rq feeds the band-pass back to A1's minus input, and that damps the loop. The bigger Rq, the less it damps and the more the filter rings: Q = Rq / 100 kΩ.
| resonance (Rq) | Q | at the cutoff | peak |
|---|---|---|---|
| 47 kΩ | 0.47 | −6.5 dB | none |
| 100 kΩ | 1.0 | 0 dB | +1.3 dB |
| 220 kΩ | 2.2 | +6.9 dB | +7.1 dB |
| 300 kΩ | 3.0 | +9.6 dB | +9.7 dB |
| 470 kΩ | 4.7 | +13.5 dB | +13.5 dB |
| 1 MΩ | 10 | +20 dB | +20 dB |
At 1 MΩ the peak is ten times the level below the cutoff: every change of cutoff rings like a struck bell. Q stays the same wherever the cutoff is, and at the knob's top it's 10, still short of oscillating on its own.
Keep the input small, or the OTAs bend. An OTA's current is a tanh of its input, a straight line only for a few millivolts: at 30 mV it falls 10% short. For slow changes the loop holds A1's two inputs together: a 110 Hz sine at 6 V puts only 0.2 mV across them. A saw's sudden drop arrives whole before the capacitors can move. At 6 V that's 27 mV across A1's inputs, where tanh is 8% short: only just bending. Press "Input 30 V" and it's 138 mV. The OTA then gives nearly all of its bias current and no more, so its capacitor can't change faster than Iabc / C = 120 µA / 1 nF, 0.12 V per µs. The saw's edges turn into ramps, the ringing roughens, and the sound gets gritty. Circuits 6 turns the input up to 16 V for a little grit.
What the model covers, and what it leaves out. SynthMaker's OTA is the core's law: the tanh, the bias pin's junction and its current setting gm. It has no input bias current, no limit on its output swing and no temperature. Its two OTAs are identical, so their cutoffs move exactly together; real chips differ a little from each other and drift as they warm up. The op-amp buffers are ideal. The circuit runs at twice the sample rate, where even the 2.94 kHz top of the knob plays only a fraction of a percent low.
Try this
- Press "Resonance 1M: it rings", then turn cutoff from 1 MΩ to 33 kΩ. The ringing peak sweeps from 98 Hz to 2.9 kHz, nearly five octaves.
- Press "Resonance 47k: no peak", then "Resonance 300k: Q 3". The same cutoff, from soft and dull to a vocal bump 9.7 dB high.
- Press "Input 30 V: gritty", then turn input back down to 6 V. The edges go from rough to smooth. Output is set higher on that button so the levels match: it's the shape that changes.
- Open the example sound "Random Filter". Circuits 3's sample and hold turns cutoff to a new place on every eighth note. To change the parts, open it in SynthMaker.
The parts list
circuit OtaFilter setting input: 6 V, from 0.6 V to 30 V setting output: 20 V, from 2 V to 100 V setting cutoff: 120k, from 1M to 33k setting resonance: 300k, from 47k to 1M V1 vcc gnd 15V R1 in p1 100k R2 p1 gnd 220 A1 p1 m1 c1 abc1 CA3080 Rlp out m1 100k Rq bp m1 resonance R4 m1 gnd 220 Ca c1 gnd 1n U3 c1 bp bp opamp R5 bp p2 100k R6 p2 gnd 220 A2 p2 gnd c2 abc2 CA3080 Cb c2 gnd 1n U4 c2 out out opamp Rk1 vcc abc1 cutoff Rk2 vcc abc2 cutoffendcutoff is written from 1M to 33k, so turning it up makes the resistors smaller, the current bigger and the cutoff higher. One setting turning two resistors is a dual pot.
How we checked it
Checked against ngspice, a standard circuit simulator: the whole filter at Q 3 and Q 10, driven by a 500 Hz sine, is within 0.01% of ngspice across the audible band at the 2× rate it runs at. Measured in SynthMaker, the cutoff follows the bias current at 6.75 Hz per µA, from 97.6 Hz at 14 µA to 2.93 kHz at 435 µA.
Build it step by step
- The lesson: A filter from two OTA chips (opens in a new tab). Ned builds it one chip at a time: an OTA, a capacitor for a gentle slope, a second chip for a steeper one, then the resonance. Then he plugs the sample and hold into its cutoff.
- The video: Two chips make this filter bubble: an OTA filter, built from parts. It loads only when tapped.
- The same filter, wider and hotter: An acid line from two OTA chips (opens in a new tab), with an Envelope on the cutoff. Its video: One cable makes this acid line squelch: two OTA chips, built from parts.
- The manual: OTAs and pots.
Further reading
- What an OTA is, and its use in filters: Wikipedia: Operational transconductance amplifier.
- The LM13700 data sheet: its transconductance, 19.2 × Iabc, and voltage-controlled filters built from it, a two-pole low-pass and a state-variable among them. PDF
- What an integrator does: Wikipedia: Integrator.
- Filters whose cutoff a voltage sets: Wikipedia: Voltage-controlled filter.
- What Q measures: Wikipedia: Q factor.
- Why a capped current makes ramps: Wikipedia: Slew rate.
- How SynthMaker simulates a circuit: M. Holters and U. Zölzer, "A Generalized Method for the Derivation of Non-Linear State-Space Models from Circuit Schematics", EUSIPCO 2015. PDF