The Servo That Overshot Every Time
We put a 400 W servo on a rotary indexer. It had plenty of torque on paper — the catalog said it would handle the inertia. On the floor, it overshot the target position by 3 degrees every cycle, oscillated twice, and settled in 800 ms. The cycle time target was 400 ms. We tuned the gains up, tuned them down, changed the gear ratio, nothing worked. Finally we checked the inertia ratio: the indexer’s inertia was 22 times the motor’s rotor inertia. The servo couldn’t control that load. It wasn’t under-powered — it was mismatched.
Servo motor sizing for custom machines is where engineers who can size a cylinder or a bearing still go wrong. Torque is easy. Speed is easy. Inertia matching is the one that nobody calculates until the motor is on the machine and oscillating. This article is the sizing process I use now, after that indexer incident and every servo tuning nightmare since.
The Three Numbers That Determine Servo Sizing
A servo motor that works needs three things: enough torque to accelerate the load, enough speed to reach the required RPM, and an inertia ratio that the drive can actually control. Miss any of the three and you’re tuning on the bench instead of running production.
1. Torque: Not Just Peak, but RMS
The peak torque rating on the motor datasheet is the maximum the motor can produce for a short time (usually a few seconds). It’s not the torque you design around. The number that decides whether the motor overheats is the RMS (root mean square) torque over the motion profile.
For a typical move — accelerate, run at constant speed, decelerate — the RMS torque is higher than you think. The acceleration torque and deceleration torque are the peaks, and the RMS is the square-root-of-the-summed-squares average. If your cycle has high acceleration followed by a long dwell, the RMS might be close to the continuous rating. If it’s rapid start-stop-start-stop, the RMS is higher and you need a bigger motor.
The acceleration torque itself is:
T_accel = (J_load + J_motor) × α + T_friction + T_external
Where α is the angular acceleration (rad/s²). This is why the load inertia matters so much — it adds directly to the torque required to accelerate.
2. Speed: Top RPM and the Gear Ratio
The motor’s rated speed (typically 3,000 RPM or 5,000 RPM for a standard servo) needs to be above the required output speed after the gearbox. If you need 300 RPM at the load and use a 10:1 gearbox, the motor needs to spin at 3,000 RPM — right at the rated speed. That’s fine, but you’ve used all your speed margin. Add 10–20% margin for voltage drops and friction. If you need 3,000 RPM at the load with a 10:1 ratio, you need 30,000 RPM at the motor — which means you need a different gearbox or a higher-speed motor.
3. Inertia Ratio: The Number Nobody Checks
This is the one. The inertia ratio is the load inertia reflected to the motor shaft divided by the motor’s rotor inertia. For example, if the load inertia is 1.2 × 10⁻⁴ kg·m² and the motor rotor inertia is 0.05 × 10⁻⁴ kg·m², the ratio is 24:1.
Different drives tolerate different ratios:
| Inertia Ratio | What Happens |
|---|---|
| < 5:1 | Easy to tune, stiff response, fast settling |
| 5:1 to 10:1 | Standard range; most drives handle it with tuning |
| 10:1 to 20:1 | Marginal; tuning is harder, settling time increases |
| 20:1 to 30:1 | Difficult; expect oscillation and poor settling |
| > 30:1 | Most drives can’t control it. Use a gearbox to reduce the reflected inertia. |
The fix for a high inertia ratio is a gearbox. A 5:1 gearbox reduces the reflected load inertia by 25× (inertia scales with the square of the ratio). That indexer that was 22:1? With a 5:1 gearbox, it becomes 22/25 = 0.88:1. The motor controls it easily. The tradeoff is that the gearbox also reduces speed by 5×, so you need a motor that spins faster — or you accept the slower output speed.
The 10:1 rule: If your inertia ratio is above 10:1, add a gearbox. If it’s above 20:1, you must. Don’t try to tune your way out of an inertia mismatch. The servo drive isn’t lying when it oscillates — it physically can’t control that load.
Motion Profile: The Input Most People Skip
Servo sizing isn’t just about static numbers. It’s about the actual motion you’re asking the motor to do. A move that accelerates fast, moves at constant speed, and decelerates fast has a different RMS torque than a move that eases in and eases out.
Trapezoidal vs. S-Curve Motion
Trapezoidal velocity profiles (sudden acceleration steps) create high jerk and high peak torque. S-curve profiles (smooth acceleration ramps) reduce jerk but extend the move time. For pick-and-place where cycle time matters, trapezoidal is common. For precision applications where vibration matters, S-curve is better. The motor sizing changes with the profile — an S-curve move has lower peak torque than a trapezoidal move over the same distance and time.
Duty Cycle: How Long Is the Move vs. the Dwell?
If your machine moves for 0.5 seconds and then dwells for 5 seconds, the motor has time to cool between moves. The RMS torque is lower because the dwell (zero torque) pulls the average down. If your machine moves continuously (a conveyor, a flying cutoff), the motor is always under load and the RMS equals the continuous torque. Size for the worst-case continuous load, not the peak.
Gearboxes: When to Use One (and Which Type)
A gearbox isn’t just a speed reducer. It’s an inertia matching device. But choosing the wrong gearbox creates more problems than it solves.
Planetary vs. Timing Belt
| Gearbox Type | Backlash | Efficiency | Best For |
|---|---|---|---|
| Planetary (precision) | < 5 arcmin (can be <1 arcmin with preload) | 90–95% | Precision positioning, rotary indexers, servo axes |
| Timing belt | 10–30 arcmin (depends on belt tension) | 95–98% | Linear axes, long travel, cost-sensitive |
| Worm gear | 1–3 degrees (high) | 50–80% (low) | Vertical axes (self-locking), not for servo |
For servo-driven positioning, use a planetary gearbox. The backlash is low enough that the servo’s encoder on the motor side reflects the actual output position. A timing belt works for linear axes where the belt itself is the transmission — but you’re adding backlash in the belt drive on top of any gearbox.
Don’t Forget Gearbox Efficiency
A gearbox isn’t 100% efficient. The lost power becomes heat. A planetary gearbox at 92% efficiency means 8% of the motor’s torque becomes heat in the gearbox. For continuous-duty applications, this heat matters — the gearbox needs to be sized for continuous torque, not just peak.
Sizing for a Vertical Axis: The Holding Torque Problem
A vertical axis is different from a horizontal axis in one critical way: when the motor is off, gravity is still pulling the load down. The motor (or brake) has to hold the load continuously.
Two options:
- Motor with holding brake: Most servos offer a 24 V holding brake option. The brake engages when power is off. It holds the load. Size the brake for the full weight of the carriage and tool.
- Self-locking mechanism: A lead screw or worm gear is self-locking — the load can’t back-drive it. But these are inefficient (50% or less) and not suitable for high-cycle servo applications.
For vertical servo axes, always specify a motor with an integrated holding brake. Don’t rely on the gearbox or screw to hold the load. When the e-stop hits, the brake engages and the axis doesn’t drop.
A Servo Sizing Checklist
- What is the total moving mass? (Carriage + tool + payload)
- What is the required move distance and time? (Gives acceleration and speed)
- What is the load inertia reflected to the motor shaft? (Include the screw, belt, or coupling)
- What is the inertia ratio? (Load inertia / motor rotor inertia)
- If ratio > 10:1, what gearbox ratio brings it under 10:1?
- Does the motor’s rated speed exceed the required speed after gearing? (Add 10% margin)
- What is the RMS torque over the motion profile? (Compare to motor continuous rating)
- What is the peak torque? (Compare to motor peak rating — usually 2–3× continuous)
- For vertical axes: does the holding brake support the full load weight?
- Is the duty cycle continuous or intermittent? (Continuous needs more margin)
Tuning: Don’t Size, Then Ignore the Drive
Even a correctly sized servo needs tuning. Start with the drive’s auto-tune. Let it identify the load inertia and set the gains. Then watch the settling time. If the axis overshoots, lower the stiffness gain. If it’s too slow, raise it. Most modern servo drives have auto-tune that gets you 80% there in 30 seconds. The last 20% is manual — but it’s tuning a well-matched system, not fighting a mismatched one.
The Bottom Line
Custom machine servo selection isn’t about picking the motor with the most torque. It’s about matching the motor to the load: torque for acceleration, speed for the cycle time, and inertia ratio the drive can control. The motor that overshoots and oscillates wasn’t under-powered — it was the wrong size for the inertia. Calculate the ratio, add a gearbox when needed, and size the RMS torque for the real motion profile. The servo that settles in 200 ms on day one is the one the customer doesn’t call you about on day 90.
