Modelling analog hardware from the papers

A generated plugin is only as good as what it is written against. If the library offers a filter and a tanh, a request for “an LA-2A” comes back as a filter and a tanh with a nice name. So over the past two months the DSP library under Sonorie grew models of the hardware people actually ask for, each one built from the papers that describe it and checked by measurement.

The rule is the one the library has always had: every claim is measured. Below is what each model follows, and the number that says it does.

Three compressors, three circuits

Optical (LA-2A style). What makes an optical compressor sound the way it does is its photocell, which recovers fast from a little reduction and slowly from a lot, and remembers what it has been through. The model has a fast state and a slow memory that holds about half the reduction, so the release is program-dependent by construction: a short burst lets go in a fraction of a second, a long loud passage takes seconds. Measured ratios: 3.0:1 in Compress, 15:1 in Limit.

FET (1176 style). The 1176 is a feedback compressor, which is awkward digitally: a feedback loop with a gain between 3 and 19 and a 20-microsecond attack oscillates if you run it one sample late, which is why software versions usually fudge the attack. This one solves the loop every sample by bisection, so 79% of the reduction lands on the first sample of a transient. The ratio buttons are loop gains, and they measure 3.9:1 and 20.0:1. The FET adds its own second harmonic as it works.

VCA (dbx and console bus style). An exponential gain law measured at 4.000:1, a true-RMS detector, and the two-speed auto release found on console bus compressors.

Tape

Magnetic tape saturates through hysteresis, and the standard model is Jiles–Atherton’s, in the form Jatin Chowdhury published for real time use in 2019. It is solved with a fourth-order Runge–Kutta step at twice the sample rate, with his implicit Newton solver available for heavier settings.

Around it sit the parts a tape machine actually has: record and playback equalisation to the NAB, IEC and AES standards, built as exact inverse pairs so the chain is flat to 0.05 dB when nothing saturates. NAB at 15 ips puts a 10 kHz tone 9.9 dB hotter on the tape, which is why highs saturate first. There is also playback head loss from the gap, spacing and thickness of the tape, and it obeys the speed law: halve the speed and the curve moves down an octave.

Tubes

A static curve is not a triode stage. What a driven stage does is governed by its two capacitors. The input coupling capacitor charges through grid current once the grid conducts and pushes the stage towards cut-off, and the cathode bypass shifts the bias. The model follows Koren’s triode with Dempwolf and Zölzer’s grid-current fit, solved with Newton’s method. It produces blocking distortion the way a real stage does: after a 5 V burst, 3.05 V sits on the coupling cap, and the quiet signal that follows reads 18 dB down until the cap drains.

The guitar-amp tone stack is Yeh and Smith’s exact model of the Bassman network. The Baxandall tone control is the 1952 feedback network solved symbolically, so a cut is exactly the mirror of the same boost.

A circuit solver, and two pedals solved from their schematics

For circuits too awkward to derive by hand, the library has a nodal DK solver. It takes a netlist of resistors, capacitors, diodes, transistors, JFETs and ideal op-amps, inverts the linear part once and runs Newton iteration only on the nonlinear parts each sample. We validated it the strict way: a diode clipper solved by the engine agrees with a separately hand-derived model of the same circuit to within a microvolt.

The two-transistor fuzz is its full schematic, solved whole. With nothing tuned, the second transistor’s collector biases to 4.52 V, against the 4.5 V builders aim for. Its well-known cleanup when you roll back the guitar volume falls out of the physics: from a 100 Ω source the third harmonic sits 13.8 dB below the fundamental, from a 50 kΩ source 84 dB below.

The green overdrive’s drive stage is its netlist too, op-amp and diodes included. Its gain measures 39.0 dB against 39.6 dB derived by hand, the bass stays 15 dB cleaner at 100 Hz (the voicing that makes it sit in a mix), and the clipped peaks still follow the input: that is the clean-on-top character of the pedal. Its tone stage is structural, and labelled so.

Filters

The Moog-style ladder is Zavalishin’s zero-delay-feedback form, solved in closed form, with Huovilainen’s per-stage saturation for drive. It self-oscillates at its cutoff (500.4 Hz for a 500 Hz setting) and stays bounded there. The diode ladder and the Sallen–Key filter follow the same ZDF approach in Pirkle’s form.

Reverbs

The spring is Abel and Parker’s dispersive model: high frequencies arrive before low ones, which is the chirp you hear when you kick a spring tank. It adds the transition-frequency resonance from Välimäki, Parker and Abel (JAES 2010). That resonance sits on the output; inside the loop it turned a two-second decay into an oscillator. The plate is Dattorro’s figure-of-eight from his 1997 paper, with his delay lengths scaled to the sample rate. Its decay time is checked against the figure’s own arithmetic.

What is labelled structural

Not everything is a component-level model, and the library says which ones are not. The passive tube EQ is Pultec-style by structure: a second-order boost and a first-order cut at different corners, which is what produces the famous dip, but not the EQP-1A’s component values. The overdrive’s tone stage is the same. Calling either of those a circuit model would be marketing, and the next person to build on it would trust it for the wrong reasons.

Graded while it builds

When a request names a unit the builder knows (the 1176, the LA-2A, a console bus compressor, a Pultec, a Moog ladder, a Tube Screamer), every compile is graded against that unit’s documented behaviour: compression slope, attack class, harmonic balance, the EQ dip, the filter slope, the mid hump. Each test point carries its source and a deliberately wide tolerance, because the grade is about shape, not clone precision. With fewer than two points that can be measured, there is no score at all: a fidelity percentage built on one number is marketing.

Product names are trademarks of their owners and are used only to identify the hardware a model refers to. Sonorie is not affiliated with or endorsed by any of them.