3D Printing Calculators

FDM calibration & firmware

PID Tuning (Ziegler–Nichols)

When a heater sits at the edge of oscillation, Ku is that gain and Tu is the period in seconds. Classic Ziegler–Nichols turns those two numbers into Kp, Ki and Kd. Firmware choice here is documentation — the three gains are the same math.

Inputs

Units

Proportional gain where the temperature just sustains oscillation, on the firmware’s 0–255 scale (the Kp you set).

s

Seconds between oscillation peaks at that Ku.

Does not change Kp, Ki or Kd. Only reminds you which commands exist.

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Results

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Calculation method

Classic Ziegler–Nichols PID from the ultimate-sensitivity experiment: Kp = 0.6 × Ku, Ki = 2 × Kp / Tu, Kd = Kp × Tu / 8, with Tu in seconds. The firmware dropdown does not alter the numbers; Marlin and Klipper simply expect different G-code or config keys. Invalid if Ku or Tu is not positive. There is no unit conversion: Marlin (M301/M304) and Klipper (pid_Kp/pid_Ki/pid_Kd) both express the gains on a 0–255 heater-PWM scale per °C, with Ki per second and Kd in seconds, so enter as Ku the Kp value your firmware was running.

Exact formula

Kp = 0.6 × Ku. Ki = 2 × Kp / Tu. Kd = Kp × Tu / 8. Marlin M301/M304 and Klipper pid_Kp/pid_Ki/pid_Kd take the same numbers on a 0–255 PWM scale. Limits: Classic Ziegler–Nichols is aggressive and can overshoot. Prefer firmware PID_CALIBRATE / M303; never leave a heater unattended hunting Ku.

Where to paste the numbers

Marlin: M301 for hotend, M304 for bed, then M500. Klipper: pid_Kp, pid_Ki, pid_Kd under the heater section, then SAVE_CONFIG after a real PID_CALIBRATE if you can.

Frequently asked questions

How do I find Ku and Tu?

Raise Kp (others zero) until the temperature oscillates at constant amplitude. Ku is that Kp. Tu is the time between peaks. Then come here. Or skip this and run M303 / PID_CALIBRATE.

Marlin vs Klipper values differ on forums.

Both autotunes (M303 and PID_CALIBRATE) run a relay test and apply these same classic formulas, but they estimate Ku and Tu differently, at different temperatures and fan settings. The math on this page is identical for both.

Bed Tu is huge compared with the hotend.

A bed is a slow plant: hotends typically oscillate with Tu of about 20–40 s, beds often 100 s or more. Do not paste hotend PID onto the bed.

My bed overshoots 10 °C.

ZN classic is snappy. Lower Kp a bit or use the firmware autotune, which often damps more.

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