Servo Sizing Basics: Inertia Ratio, Torque and Choosing the Gearbox
The short answer
Sizing a servo axis means checking four things: the load inertia reflected to the motor shaft against the motor's own inertia, the peak torque needed for the fastest acceleration in the move profile, the continuous or RMS torque over the whole duty cycle, and the maximum speed at that torque. Most axes that never tune properly failed on the first check, not the others: the inertia ratio is too high and no amount of gain adjustment will fix it.
Inertia ratio, and why it dominates
Reflected inertia is the load's inertia as the motor experiences it through the transmission. For a gearbox of ratio N, the load inertia divides by N squared. A load of 0.4 kg·m² behind a 10:1 gearbox reflects as 0.004 kg·m².
That squared term is the whole reason gearboxes exist on servo axes, and it is why changing a 5:1 for a 10:1 has four times the effect on the ratio that people expect.
Rules of thumb, and they are rules of thumb rather than limits:
| Ratio of reflected load inertia to motor inertia | Behaviour |
|---|---|
| Below 3:1 | Easy to tune, stiff, high bandwidth |
| 3:1 to 10:1 | Normal industrial territory, tunes fine |
| 10:1 to 20:1 | Needs care, lower bandwidth, may need notch filters |
| Above 20:1 | Difficult. Expect poor settling and low gains |
High-precision or high-bandwidth applications aim for the top row. A conveyor or a simple index can live comfortably in the third. If a machine builder tells you an axis "just never tuned well", ask for the inertia ratio before touching the gains.
Torque, in two numbers
Peak torque is what the fastest acceleration in the profile requires: the sum of the inertial torque, which is total inertia times angular acceleration, plus friction, plus any gravity or process load. It must sit inside the motor's peak curve at the speed where it occurs, and it must be inside the drive's peak current rating.
RMS torque is the root mean square over the whole cycle including dwell time. It is what heats the motor, and it must sit inside the continuous curve at the average speed. An axis sized only on peak torque will pass a demonstration and overheat in production.
Do not forget the transmission's own inertia. Couplings, pulleys and especially ballscrews contribute, and on a long screw the screw itself can dominate everything else on the axis.
What the gearbox costs you
Reducing reflected inertia by N squared is not free.
- Speed. The motor must turn N times faster for the same load speed, and the motor's rated speed becomes the limit.
- Backlash, which for a positioning axis is directly an accuracy loss. Precision axes use planetary gearboxes with backlash specified in arcminutes, and pay for it.
- Efficiency and torsional stiffness. A compliant gearbox introduces a resonance between motor and load, and that resonance is what your notch filter is fighting.
The checks people skip
Duty cycle including dwell. A move profile is not the whole story; the time between moves is what allows cooling.
Speed at peak torque. Motor curves fall off at speed. Peak torque available at 1,000 rpm may not exist at 4,000.
Drive current, not just motor torque. The drive has its own peak and continuous limits and its own derating with ambient temperature and switching frequency.
Cable length and voltage. Long motor cables and a low DC bus limit achievable speed under load.
Vertical axes. Gravity is a continuous torque even at standstill, holding brakes are not a substitute for a correctly sized motor, and a counterbalance changes the sums entirely.
Where to learn it
Servo and motion control covers sizing, tuning and motion profiles, and Allen-Bradley Kinetix servo motion covers a specific platform end to end. For the drive side more generally, see VFD and variable frequency drives. Related reading: servo and motion control basics in a PLC.
Frequently asked questions
Can I fix a high inertia ratio in software? No. You can lower the gains until the axis is stable, which means slower and softer, and that is a compromise rather than a fix. Change the ratio.
Is direct drive better? For some applications, yes, because it removes backlash and compliance entirely. It also means the load inertia is the reflected inertia, so it demands a much larger motor and is only sensible where the ratio works out.
Vendor sizing software or hand calculation? Use the vendor tool, because it holds real motor and drive curves and derating data. Do the hand calculation as well, at least for inertia and peak torque, so you can tell when the tool has been given the wrong input.
