How to Size a Servo Motor: Inertia Ratio, Torque and RMS

Picking a servo motor by its peak torque number alone is the fastest way to end up with a drive that trips on overload, a machine that hunts and oscillates, or a motor that quietly cooks itself over a long shift. Peak torque tells you what the motor can do for a fraction of a second. It says nothing about whether the motor can hold that pace all day, and nothing about whether the load will fight the motor’s ability to control it. Those two questions are answered by RMS torque and by the inertia ratio, and both are easy to skip when a catalog is waving a big peak number at you.

This guide walks through all three checks in the order a good sizing exercise follows. You will see what each number means, the formula behind it, and a full worked example for a ball-screw axis with real figures you can trace end to end. By the end you will know how to compute reflected inertia, why the gear ratio matters so much, how to build an RMS torque from a move profile, and the common mistakes that make a correctly rated motor still fail on the machine.

What servo sizing actually checks

A servo motor is not sized by a single rating. It is qualified against the load through three separate lenses, and a real motor has to clear all of them at once. The first lens is inertia matching, which asks whether the motor can control the load without instability. The second is peak torque, which asks whether the motor can produce enough instantaneous torque to accelerate the load in the time you demand. The third is continuous or thermal torque, which asks whether the motor can sustain the average effort of the full cycle without overheating.

These three tend to pull in different directions. A bigger motor has more peak and continuous torque, but it also has more rotor inertia, which can push the inertia ratio the wrong way if the load is light. A high gear ratio slashes reflected inertia and multiplies torque at the load, but it also raises the motor speed you need and eats into your torque budget through efficiency. Good sizing is the balance point where all three checks pass with a sensible margin, not the point where any single number looks impressive.

The inertia ratio

The inertia ratio is the reflected load inertia seen at the motor shaft divided by the motor’s own rotor inertia.

Inertia ratio = reflected load inertia / motor rotor inertia

It matters because the servo loop has to accelerate and decelerate both the motor’s own rotor and everything the mechanism connects to it. When the load inertia is many times larger than the rotor, the motor struggles to command the load precisely. The system becomes prone to overshoot, oscillation, and long settling times, and you have to detune the gains to keep it stable, which sacrifices the very responsiveness you bought a servo for.

The rule of thumb most drive makers publish is to aim for a ratio of 5 to 1 or less for high dynamic response and precise positioning, and to treat 10 to 1 as the usual upper limit for general motion. Some rigid, well-coupled systems tolerate higher ratios, but the stiffer and more backlash-free the coupling, the more you can get away with. If your ratio comes in above 10, the fixes are a larger motor, a higher gear ratio, or a lighter mechanism. You can test any combination fast with the Servo Motor Sizing Calculator.

Reflected inertia and the gear ratio

Reflected inertia is the load’s inertia as it appears back at the motor shaft, and it depends entirely on the mechanism between the motor and the load. A direct-coupled rotary load reflects its inertia one to one. A belt or rack turns linear mass into an equivalent rotary inertia through the pulley or pinion radius. A ball screw converts the moving mass into rotary inertia through the lead, and it usually adds the screw’s own inertia on top.

For a ball screw, the inertia of a linearly moving mass reflected to the screw is the mass multiplied by the lead divided by two pi, all squared. A short lead keeps reflected inertia small, which is one reason fine-pitch screws feel so controllable, while a coarse lead trades that control for speed.

The gear ratio is the strongest lever you have. Reflected inertia scales with the inverse square of the gear ratio, so a 3 to 1 reduction cuts reflected load inertia to one ninth of its direct value. That same reduction multiplies available torque at the load by three, though real gearboxes give back a little to efficiency and add their own inertia. Because the effect is squared, a modest reduction can rescue an inertia ratio that looks hopeless with a direct coupling.

Reflected inertia through a gearbox = load inertia at output / gear ratio squared

Peak torque and RMS torque

Peak torque is the largest instantaneous torque the motor must produce, which almost always happens during acceleration. It is the sum of the torque needed to accelerate the combined inertia and the torque needed to overcome the load itself.

Peak torque = acceleration torque + load torque

Acceleration torque = (motor inertia + reflected inertia) x angular acceleration

Load torque is the steady effort the machine demands even at constant speed, from friction, gravity on a vertical axis, cutting forces, or process load. During acceleration the motor pays both bills at once, which is why peak torque is highest there. The motor’s peak rating, often two to three times its continuous rating for a short burst, has to sit above this number with margin.

RMS torque is the root-mean-square of the torque over the entire motion cycle, including the acceleration phase, the constant-speed run, the deceleration phase, and the dwell where the motor may hold position at low or zero torque. It represents the thermally equivalent steady torque, the single continuous value that would heat the motor the same amount as the varying real profile.

RMS torque = square root of ( sum of (torque squared x time) / total cycle time )

The motor’s continuous rated torque must be at or above the RMS torque. This is the check that peak-only sizing skips, and it is the one that governs whether the motor survives a long production shift.

How to size a servo step by step

A clean sizing exercise runs in a fixed order so nothing gets missed.

First, define the move. Write down the load mass, the mechanism and its lead or radius, the travel, the target speed, and the acceleration time. This is the motion profile, and every torque number flows from it.

Second, compute the reflected load inertia. Convert the moving mass to rotary inertia through the mechanism, add the screw or pulley inertia, and if there is a gearbox divide by the gear ratio squared.

Third, pick a candidate motor and read its rotor inertia and torque ratings from the catalog. Compute the inertia ratio and check it against your target of 10 or less, ideally near 5.

Fourth, compute angular acceleration from the target speed and the acceleration time, then the acceleration torque, then add load torque to get peak torque. Check it against the motor’s peak rating.

Fifth, build the RMS torque across the full cycle including dwell, and check it against the motor’s continuous rating. If any check fails, adjust the motor, the gear ratio, or the move profile, and run the loop again.

Worked example: a ball-screw axis

Take a horizontal ball-screw axis that moves a 20 kg load on a 10 mm lead screw, where the screw itself weighs about 1 kg. We want to accelerate to 3,000 rpm in 0.1 s, and the axis carries a small steady load torque from friction.

Start with reflected inertia. The 20 kg mass on the 10 mm lead, plus the screw’s own inertia, reflects to about 1.01 kg.cm2 at the motor shaft. This is a light, well-behaved load, which is exactly what a short lead buys you.

Now pick a motor. The candidate has a rotor inertia of 0.20 kg.cm2. The inertia ratio is 1.01 divided by 0.20, which is 5.0 to 1. That lands right on the target for high response, so the first check passes cleanly.

Add the steady load torque of 0.1 N.m and the peak torque during acceleration is 0.48 N.m. That is the number the motor’s peak rating has to clear, and a motor with a peak capability comfortably above 0.48 N.m will handle the acceleration burst.

Finally, build the RMS torque over the whole accelerate-run-decelerate-dwell cycle. During acceleration the motor pushes 0.48 N.m, during the constant-speed run it only fights the 0.1 N.m load, during deceleration the load inertia helps so the torque is low, and during the dwell it holds with almost nothing. Rooting the mean of the squared torques over the full cycle time gives an RMS torque of 0.18 N.m. So the right motor has a continuous rated torque of at least 0.18 N.m, a peak capability above 0.48 N.m, and rotor inertia that keeps the ratio at or under 10. Run your own axis through the Servo Motor Sizing Calculator to reproduce these figures and try variations.

How to read the result

The lesson from the numbers is that the three checks describe three different-sized motors, and you have to satisfy the largest demand from each. The RMS torque of 0.18 N.m sets the floor for the continuous rating. The peak torque of 0.48 N.m sets the floor for the short-burst rating. The inertia ratio sets an upper bound on how small a rotor you can use before control suffers, and a lower bound on how big a motor you can add before the load becomes too light to match.

Notice how far apart peak and RMS are. The peak of 0.48 N.m is more than two and a half times the RMS of 0.18 N.m, which is normal for a short, sharp move followed by a run and a dwell. If you had sized purely on peak, you might have reached for a much larger motor than the thermal duty needs, wasting money and pushing the inertia ratio around. If you had sized purely on RMS, you would have picked a motor that overheats never but stalls on acceleration. Reading all three together is what lands the right motor.

Common mistakes

The first mistake is sizing by peak torque alone. A motor that can hit the peak for a moment may still overheat if its continuous rating sits below the RMS torque of the cycle. Peak is a sprint rating, RMS is the marathon rating, and production runs marathons.

The second mistake is ignoring the inertia ratio. A motor with plenty of torque but a rotor far too small for the load will be unstable no matter how well the torque checks pass. The machine will overshoot and oscillate, and no amount of tuning fully rescues a ratio that is badly out of range.

The third mistake is forgetting the gear ratio’s squared effect on inertia. Teams sometimes add a gearbox for torque and are surprised the inertia ratio improves dramatically, or they remove one and cannot understand why the axis suddenly hunts. Reflected inertia scales with the inverse square of the ratio, so small ratio changes move the inertia number a lot. A fourth error is leaving the dwell out of the RMS calculation, which inflates the average torque and makes you oversize.

When simple sizing does not apply

The clean three-check method assumes a horizontal axis with a fixed, repeating move profile and a rigid coupling. Real machines bend those assumptions. A vertical axis carries a constant gravity load that adds to torque going up and subtracts going down, and it needs a holding brake and regeneration handling that a horizontal axis does not. A flexible coupling or a long belt introduces compliance that lowers the safe inertia ratio, so a ratio that is fine on a stiff screw may be unstable on a springy belt.

Continuous-motion applications such as flying shears, winders, and cyclic cams have torque profiles that never really dwell, so the RMS calculation has to integrate the full varying profile rather than lean on a simple accelerate-run-dwell shape. Regeneration during deceleration can also demand a braking resistor sized to dump the returned energy. In all of these cases the three checks still apply, but you feed them a more detailed profile and add the extra hardware the duty calls for. For the surrounding drivetrain math, the Gear Ratio and Gearmotor Torque Calculator and the Conveyor Belt Speed and Motor Power Calculator handle the mechanism side.

Three expert tips

Aim for an inertia ratio near 5, not right at 10

Ten to one is the ceiling for general motion, not the target. Sitting right at the limit leaves no room for a heavier fixture, a longer tool, or a coupling that turns out softer than the datasheet promised, any of which can push you over the edge into instability. Sizing for a ratio around 5 gives the servo loop the authority to control the load crisply and leaves headroom for the machine to change over its life without a motor swap.

Always include the dwell in the RMS torque

The dwell, where the motor holds position at low or zero torque between moves, is part of the thermal cycle and it lowers the RMS. Leaving it out treats the motor as if it accelerates forever, which inflates the average torque and makes you buy a bigger motor than the duty needs. Add the full cycle time including every dwell to the RMS denominator so the thermal picture matches how the machine actually runs.

Use the gear ratio to fix inertia, not just torque

People reach for a gearbox to get more torque at the load and forget it is the most powerful inertia-matching tool they have. Because reflected inertia falls with the square of the ratio, adding a 3 to 1 reduction cuts the load inertia the motor sees to a ninth while tripling torque. When an inertia ratio comes in too high, try a gear ratio change before you jump to a much larger, more expensive motor.

Free industrial automation calculators

You do not have to grind through this arithmetic by hand. These free tools cover servo sizing and the drivetrain and pneumatic math around it, so you can size a whole automated axis from one place.

• Servo Motor Sizing Calculator for inertia ratio, peak torque, and RMS torque in one pass.
• Gear Ratio and Gearmotor Torque Calculator to set the reduction that fixes both torque and inertia.
• Conveyor Belt Speed and Motor Power Calculator for belt-driven axes and transport.
• VFD Energy Savings Calculator to compare variable frequency drive savings on constant-speed motors.
• Pneumatic Cylinder Force and Air Consumption Calculator for the pneumatic actuators alongside your servos.
• Air Receiver Tank Size and Compressed-Air Demand Calculator for sizing the air supply that feeds them.
• The full Industrial Automation hub for every related tool in one place.

Frequently asked questions

What is a good inertia ratio for a servo motor?

Aim for 5 to 1 or less for high dynamic response and precise positioning, and treat 10 to 1 as the usual upper limit for general motion. Rigid, backlash-free couplings tolerate higher ratios than soft belts, but past 10 to 1 most systems start to overshoot and oscillate, and you have to detune the gains to keep them stable.

Why can’t I size a servo motor by peak torque alone?

Peak torque only tells you what the motor can produce for a short burst during acceleration. It says nothing about whether the motor can sustain the average effort of the full cycle without overheating, which is the RMS torque check, or whether it can control the load, which is the inertia ratio check. A motor that clears peak but not RMS will overheat on a long shift.

What is reflected inertia?

Reflected inertia is the load’s inertia as it appears back at the motor shaft, after passing through the mechanism. A ball screw converts the moving mass to rotary inertia through the lead, a belt or rack does it through the pulley radius, and a gearbox divides the result by the gear ratio squared. In the worked example, a 20 kg mass on a 10 mm lead plus the screw reflected to about 1.01 kg.cm2.

How does the gear ratio affect inertia?

Reflected inertia scales with the inverse square of the gear ratio. A 3 to 1 reduction cuts the load inertia the motor sees to one ninth while multiplying torque at the load by three. Because the effect is squared, a small gear ratio change can rescue an inertia ratio that looks impossible with a direct coupling, which makes the gearbox the strongest inertia-matching tool you have.

How do I calculate peak torque?

Peak torque is acceleration torque plus load torque. Acceleration torque is the sum of the motor rotor inertia and the reflected load inertia, multiplied by the angular acceleration. In the example, a combined inertia of 1.21 kg.cm2 at an angular acceleration of 3,142 rad/s2 gave about 0.38 N.m, and adding a 0.1 N.m load produced a peak torque of 0.48 N.m.

What is RMS torque and why does it matter?

RMS torque is the root-mean-square of the torque over the full motion cycle, including acceleration, constant-speed run, deceleration, and dwell. It is the thermally equivalent steady torque, the single value that would heat the motor the same as the varying real profile. The motor’s continuous rated torque must be at or above it, which is why RMS governs whether the motor survives a long production shift.

How did the example reach 3,142 rad/s2?

Accelerating to 3,000 rpm means reaching about 314 rad/s, since 3,000 rpm converts to roughly 314 radians per second. Reaching that speed from rest in 0.1 s gives an angular acceleration of 314 divided by 0.1, which is 3,142 rad/s2. That angular acceleration multiplied by the combined inertia is what produced the acceleration torque.

What continuous and peak torque did the example motor need?

The example needed a motor with a continuous rated torque of at least 0.18 N.m, which is the RMS torque of the cycle, and a peak capability above 0.48 N.m, which is the peak torque during acceleration. Its rotor inertia had to keep the inertia ratio at or under 10, and at 0.20 kg.cm2 against 1.01 kg.cm2 reflected, the ratio was 5.0 to 1.

Should I include the dwell time in the RMS calculation?

Yes. The dwell, where the motor holds position at low or zero torque between moves, is part of the thermal cycle and it lowers the RMS torque. Leaving it out inflates the average and makes you oversize the motor. Always add the full cycle time, including every dwell, to the RMS denominator so the thermal picture matches how the machine actually runs.

How does sizing change for a vertical axis?

A vertical axis carries a constant gravity load that adds to torque on the way up and subtracts on the way down, so the torque profile is no longer symmetric. It usually needs a holding brake to keep the load in place when powered off, and it may need a braking resistor to absorb the energy the load returns during downward moves. The three checks still apply, but you feed them the gravity-adjusted torque.

What if the inertia ratio is too high?

You have three levers. Fit a larger motor with more rotor inertia, add or increase a gear reduction, which cuts reflected inertia by the ratio squared, or lighten the mechanism itself. The gearbox is usually the most efficient fix because the squared effect means even a modest reduction moves the ratio a lot, and it multiplies torque at the same time.

Does a bigger motor always help sizing?

Not always. A bigger motor adds torque and thermal headroom, but it also adds rotor inertia, which can push the inertia ratio too low if the load is light, and it costs more and draws more from the drive. The goal is the balance point where the inertia ratio, peak torque, and RMS torque all pass with sensible margin, not the largest motor that fits.

Servo sizing is really three questions asked together: can the motor control the load, can it hit the acceleration burst, and can it sustain the thermal duty. The inertia ratio, peak torque, and RMS torque each answer one, and the right motor clears all three with margin rather than acing one and failing the rest. Run your own axis through the calculators above, watch the gear ratio do double duty on torque and inertia, and never let a big peak number talk you out of checking the RMS.