Planetary Gearbox Inertia Matching —
Gear Ratio Selection for Servo Performance

T_motor = (J_motor + J_load/i²) × α_motor + T_friction

Define: J_total = J_motor + J_load/i²
J_ratio = J_load_reflected / J_motor = (J_load/i²) / J_motor

Servo bandwidth ω_c (rad/s) — simplified open-loop:
ω_c ∝ K_p / J_total = K_p / [J_motor × (1 + J_ratio)]

→ Achievable bandwidth decreases as J_ratio increases
→ At J_ratio = 1: bandwidth = K_p / (2 × J_motor) — 50% of ideal
→ At J_ratio = 5: bandwidth = K_p / (6 × J_motor) — 17% of ideal
→ At J_ratio = 10: bandwidth = K_p / (11 × J_motor) — 9% of ideal

This bandwidth reduction limits how fast the servo can:
• Respond to position commands (limits acceleration profile steepness)
• Reject disturbances (limits stiffness against cutting/impact forces)
• Settle to target position (limits positioning time)

The optimal ratio formula has a beautifully simple physical interpretation: the gear ratio that maximises load acceleration is the one that makes the reflected load inertia equal to the motor inertia. At this ratio, exactly half the motor torque accelerates the motor itself and half accelerates the load — a 50/50 split that is thermodynamically efficient and mechanically balanced.

In practice, i_optimal often falls between standard catalogue ratio steps. A Korean servo engineer who calculates i_optimal = 17.3 must choose between i = 15 and i = 20 from the catalogue. Both are acceptable — the inertia ratio varies only modestly across this range. The engineer should also verify that the chosen ratio delivers the required output speed at the motor’s rated RPM.

Problem: Film reel drive axis is unstable during deceleration at high speed. Motor: J_motor = 0.8 kg·cm². Full reel: Ø600mm, 25 kg (J_reel = 2,812 kg·cm²). Drive pulley r = 50 mm (J_pulley = 0.15 kg·cm²). Linear film mass 2 kg on belt. Current ratio: i = 5.

Solution: Upgrade from i=5 to i=25 (EP-AF090 P1 two-stage). J_ratio drops from 390:1 to 7.7:1 — within acceptable range for HFFS speed control. This result matches the Art10 recommendation and now shows the mathematical basis for that choice.

Problem: Select optimal ratio for B-axis rotary table. Motor: J_motor = 4.2 kg·cm². Table + fixture: J_table = 380 kg·cm² (varies 200–500 with workpiece). Target: J_ratio ≤ 5:1 at maximum load.

Solution: EP-AFH i=50 two-stage. At this ratio the rotary table is completely motor-inertia-dominated — the load contribution is negligible — and the servo loop is controlled almost entirely by motor properties. This is why high-ratio CNC rotary tables are inherently easy to tune regardless of workpiece weight variation.

Problem: 500 kg payload AMR, wheel radius r=0.10m, J_motor = 0.35 kg·cm². Effective rotary inertia of vehicle+payload at wheel: J_vehicle = m × r² = 700 × 100² = 7,000,000 kg·cm².

Insight: AGV drive wheels are fundamentally inertia-mismatched in the gearbox sense — the vehicle mass is so large that matching it to a compact motor would require impractically high ratios. The correct architecture is velocity control with outer-loop position correction from navigation sensors, not tight inertia matching. This is why AGV drives are specified on torque, noise, and speed sync (Art12) — not on inertia ratio.

When J_load/i² is comparable to J_motor (i.e. the system is near the optimal ratio), the gearbox input inertia may push the total past the J_ratio target. Always include J_gearbox_input when J_ratio is calculated to be below 3:1 — the correction may push it above the target.

At high input speeds, the gearbox input stage generates bearing centrifugal loads and churning losses that are themselves speed-dependent. For ratios above 3,000 rpm input, include J_gearbox_input and verify against the maximum input speed specification for the selected EP series frame and ratio.

AGV drives (Case 3 above), conveyor head drums, and screw rotation axes all operate at J_ratio far above the standard guideline — but they use velocity control with outer-loop position correction from encoders, sensors, or navigation systems. In these cases the servo loop does not need tight inertia matching; it needs reliable speed control, which works acceptably at J_ratio up to 30:1 or more with well-tuned velocity PI gains.

If the required output speed range dictates a low ratio (e.g. i=3 for a high-speed indexer that must reach 500 rpm output from a 1,500 rpm motor), and this produces J_ratio = 15:1, the engineer should accept the mismatch and compensate through motor sizing: specifying a motor with higher rotor inertia (typically a larger frame motor in the same power class) to reduce J_ratio without changing the gear ratio.

CNC rotary table with variable workpiece weight (empty 50 kg to full 500 kg) has a 10:1 inertia variation that no fixed gear ratio can simultaneously optimise for both extremes. The standard approach is to select the ratio that keeps J_ratio ≤ 5:1 at the maximum load condition — accepting that at minimum load the system is over-reduced and slightly less efficient, but stable at both extremes.

n_out_max = maximum axis speed (rpm) from machine spec. n_out_min (if applicable). This gives the speed-based ratio limit: i_speed = n_motor_rated / n_out_max. This is the minimum allowable ratio — ratios below this exceed motor rated speed.

Use the appropriate formula from Module 2 for your drive topology. For variable-load applications (workpiece changes, film reel depleting), calculate at the worst case (maximum inertia condition). Include all rotating and translating masses connected to the output shaft.

i_optimal = √(J_load / J_motor). Calculate J_ratio at catalogue ratios near i_optimal. Select the ratio that meets the J_ratio target for your application type (from Module 3 table) while also satisfying i ≥ i_speed from Step 1.

T_output = T_motor_rated × i × η. Confirm T_output ≥ required load torque × service factor. This step may override the inertia-optimal ratio if the torque requirement dictates a different frame size or series.

For belt, chain, or gear-loaded output shafts: apply the overhang multiplier from Art16 and confirm the effective bearing load is within EP-AB or EP-AF permissible values. Inertia matching and radial load capacity are independent checks — both must pass.

Select backlash grade from Art8 based on positioning accuracy. If inertia-optimal ratio falls between standard EP-AB steps, check EP-ADS non-standard ratios for a closer match. Korea Ever-Power application team confirms all six steps for any specific Korean machine specification — same business day.

Q
Our Korean packaging machine servo is oscillating and hunting at low speeds. The control engineer has increased damping but the axis is still slow to settle. Is this an inertia ratio problem?

Oscillation and hunting at low speeds that does not respond well to increased damping is a classic high-J_ratio symptom. When J_ratio is large, the servo loop must use high proportional gain to achieve adequate stiffness, but the high gain combined with the large load inertia produces the oscillatory response. Increasing damping (derivative gain) helps but has diminishing returns at high J_ratio. The correct fix is to increase the gear ratio — even one step up in the EP-AB catalogue (e.g. from i=10 to i=15) reduces J_ratio by 2.25× and may immediately resolve the hunting without any control gain changes. Measure J_load at the output shaft, calculate J_ratio at your current ratio, and compare to the targets in Module 3. If J_ratio exceeds 10:1 on a precision positioning axis, increasing the ratio is the correct mechanical solution.

Q
Can I use a worm gear reducer for inertia-critical applications where I need a very high ratio?

Μειωτήρες με ατέρμονα κοχλία can provide high ratios (up to 100:1 in single stage) with inherent self-locking — useful for applications requiring position holding on power-off. However, their efficiency of 40–70% means that a significant fraction of motor torque is lost to friction, and their typical backlash of 15–30 arcmin makes them unsuitable for closed-loop precision servo positioning. For inertia matching purposes: the worm’s high ratio does reduce reflected load inertia by i² — so a worm at i=60 reduces load inertia by 3,600× — but the friction torque loss and high backlash prevent the high-bandwidth servo response that the inertia matching is supposed to enable. Use planetary gearboxes for inertia-critical servo axes; consider worm reducers only where self-locking, very high ratio, and open-loop speed control are the primary requirements rather than servo bandwidth.

Q
How does the inertia matching calculation change for a two-stage gearbox compared to a single-stage?

For a two-stage gearbox with total ratio i = i₁ × i₂, the reflected load inertia formula is unchanged: J_load_reflected = J_load / i². The total ratio i is used directly — there is no need to calculate stage by stage. The only two-stage-specific addition is the intermediate stage inertia: the first stage planet carrier and output shaft rotate at n_motor / i₁, contributing an intermediate inertia term of J_intermediate / i₁². For standard Korea Ever-Power EP two-stage units, this intermediate inertia is included in the published J_gearbox_input value — it already accounts for both stages’ internal rotating components at their respective speeds. When Korea Ever-Power provides J_gearbox_input for a two-stage EP-AB unit, that value is ready to use directly in J_total = J_motor + J_gearbox_input + J_load/i² without further stage-by-stage decomposition.

Q
My Korean CNC machine builder specifies a J_ratio ≤ 3:1 for all servo axes. Is this unnecessarily conservative for axes like the tool changer or chip conveyor?

Yes — a blanket J_ratio ≤ 3:1 for all axes is unnecessarily conservative for non-precision servo axes and will increase gearbox BOM cost without performance benefit. The J_ratio ≤ 3:1 guideline is appropriate for high-dynamic precision axes (spindle, feed axis, rotary table B/C axis) where servo bandwidth directly affects machining accuracy and cycle time. For the tool changer (moderate speed, position control to pocket location), J_ratio ≤ 10:1 is adequate. For the chip conveyor (speed control only, no precision positioning), J_ratio ≤ 30:1 is fully acceptable. The axis-by-axis differentiation approach used in Art19 (injection molding five-axis BOM) applies equally here: specifying by actual axis requirements rather than applying the most conservative specification to all axes reduces BOM cost by ₩300,000–800,000 per machine at equivalent or better performance.