How tube saturation is modeled: the Koren equations explained

Two ways to fake a tube

The simplest way to add "tube sound" is a waveshaper: take every sample and pass it through a fixed curve, often a hyperbolic tangent, that rounds off the peaks. It's cheap and it adds harmonics, but it has nothing to do with how a tube behaves. Every "tube" sounds the same, and the curve doesn't react to the circuit around it.

The other way is physical modeling: describe the electrical behavior of the tube with equations, put it in a virtual circuit, and solve that circuit for every sample. Gain, headroom and harmonics are no longer designed by hand; they emerge from the tube's physics. That's the approach behind Triodia, and the starting point is a model published by Norman Koren.

The Koren triode model

In 1996, engineer Norman Koren published "Improved VT models for SPICE simulations", a set of equations that describe how much current flows through a tube for any combination of grid and plate voltage. It became the most widely used tube model in circuit simulators because it matches real tube curves closely with only a handful of parameters. For a triode:

Here V_p is the plate voltage, V_g the grid voltage and I_p the resulting plate current. Five parameters give each tube its personality:

• μ (mu): the amplification factor. About 100 for a 12AX7, about 20 for a 12AU7. This sets how much gain the tube can give.
• Ex: the exponent that shapes how the current curves as the tube turns on. Close to 1.5, as in the classic "three-halves power law" of vacuum tubes.
• K_g1: scales the overall current, related to the tube's transconductance.
• K_p: controls the shape of the curves near cutoff, where the tube is almost off. This is where much of the asymmetric, 2nd-harmonic distortion comes from.
• K_vb: fine-tunes how the curves bend at low plate voltage.

The ln(1 + exp(…)) part is a smooth version of "if the result is negative, use zero". It lets the current fade out gradually near cutoff instead of switching off abruptly, just like a real tube.

A tube is nothing without its circuit

The equation alone tells you current from voltages, but audio flows through a circuit. Triodia places the tube in the classic common-cathode gain stage used in countless preamps:

• A high-voltage supply (B+) feeds the plate through a plate resistor.
• As the grid voltage swings with the audio, the plate current changes, and so does the voltage drop across the resistor. That changing plate voltage is the amplified, inverted output.
• The bias, the grid's resting voltage, is computed automatically for each tube so it sits in a sensible operating point.
• When the signal swings the grid positive, the grid starts to conduct and clamps the peaks. This grid conduction is a big part of how real tube stages clip.

There's a catch: the plate voltage depends on the current, and the current depends on the plate voltage. There's no simple formula for the answer, so the plugin solves it numerically, iterating until both sides agree, for every signal level.

The details that make it sound real

Miller effect

Every triode has a tiny capacitance between grid and plate. Because the plate swings in the opposite direction to the grid, amplified by the stage gain, that capacitance looks far larger from the input. It forms a low-pass filter that gently rounds off the highs, and it's stronger in high-gain tubes. In a pentode like the EF86 the screen grid shields the plate, so the effect is practically zero, just like in the real circuit.

Output transformer

Many vintage units end in a transformer. Its core saturates with strong low frequencies, which adds weight and soft compression on the bass, and its resonance smooths the extreme top end.

Oversampling

Saturation creates harmonics above the original signal. If those harmonics go past half the sample rate, they fold back down as inharmonic "aliasing", the harsh, digital edge that gives cheap distortion away. Running the tube stage at 2× the sample rate and filtering before returning to the original rate keeps those harmonics clean.

What it means for your ears

Because the behavior comes from the physics, the differences between tubes are real rather than presets. A 12AX7 saturates earlier than a 12AU7 because of its μ, not because someone tuned a curve. Measured in Triodia, about 99% of the triodes' distortion is 2nd harmonic, the "warm" signature tube gear is known for. See the numbers for all nine tubes in 12AX7 vs 12AU7 on vocals, or put the theory to work with how to get warm tube vocals for free.