FANUC electronic gearbox: why the synchronization coefficient is not simply hob starts divided by tooth count
The 0i-F Plus electronic gearbox definition separates mechanical speed ratio from feedback-pulse conversion. An original example with different detection resolutions shows why a tooth-count ratio alone cannot be entered as the synchronization coefficient.
Contents
For a two-start hob cutting a 45-tooth workpiece, the mechanical speed ratio is 2/45, yet the electronic gearbox synchronization coefficient may exceed 1. There is no contradiction: one compares mechanical revolutions, the other pulse counts measured on different scales.
Scope and original definition
The FANUC 0i-F Plus descriptions manual B-64692EN/01, §10.1, page 252 (public mirror of the original manual) defines the electronic gearbox EGB synchronization coefficient as K1=(L/T)×(β/α). L is the number of hob starts, T the tooth count, and α and β the detected pulses per revolution of the master and slave axes respectively. This concerns the digital-servo-side EGB structure in that section; the following section's Spindle EGB between two spindles is not assumed to share its wiring or parameter arrangement.
The following assumes spur gears with no additional differential component and considers only positive ratio magnitudes. Rotation direction and mechanical transmission relationships depend on the actual mechanism. Matching the system model does not establish that the function is configured.
Convert pulses to revolutions before comparing ratios
An original example uses L=2, T=45, master detection α=10000 pulse/rev and slave detection β=360000 pulse/rev. Then K1=(2/45)×36=1.6. One master revolution supplies 10000 input pulses, corresponding to 16000 slave pulses. Dividing by the slave's 360000 pulses per revolution gives 2/45 revolution, or 16°.
| Master detection scale α | Slave detection scale β | K1 | Slave angle for one master revolution |
|---|---|---|---|
| 10000 pulse/rev | 360000 pulse/rev | 1.6 | 16° |
| 20000 pulse/rev | 360000 pulse/rev | 0.8 | 16° |
The second row only doubles the master measurement scale. The same mechanical movement then produces twice as many master pulses, so the coefficient must halve. These values illustrate dimensions mathematically; they are neither recommended encoder specifications nor directly enterable parameters.
How large is the error if the detection-scale ratio is omitted?
If K1 in the first row is incorrectly set equal to 2/45, the slave angle per master revolution calculates as 360°×(2/45)×10000/360000: only about 0.444°, or 1/36 of the required 16°. Reducing master speed cannot correct a ratio error.
Conversely, the correct steady mechanical relationship is n_slave/n_master=K1α/β=L/T. At master speed 900 r/min, slave speed should be 40 r/min. Speed readings can help check the order of magnitude of the ratio, but matching average speeds does not verify phase tracking throughout the process.
Why encoder bit count is insufficient
Each “per revolution” in the formula must refer to the defined master or slave axis. If a detector is on the other side of a reduction gear, incorporate the transmission ratio into the effective detection scale first. Counts per motor revolution cannot simply stand in for counts per workpiece revolution. For offline checking, state fully “how many effective pulses per revolution of which axis,” then check cancellation of units.
This article explains only the coefficient's physical meaning. It supplies no synchronization-start procedure, parameter changes or actual hobbing program. Other advanced-function resources are in the OPTION section.