Electronic Cam & Electronic Gearing Calculator — Tracking Lag, Max Speed & Servo Bandwidth
Estimate master–slave position tracking lag for electronic gearing or camming, then determine allowable master speed or minimum equivalent servo bandwidth from angular-error tolerance and residual delay.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Calculation task
Synchronization relationship
Operating speed
Equivalent tracking model
Position tracking tolerance
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Electronic cam and electronic gearing synchronization model
This calculator estimates the angular position tracking lag between a master axis and a slave axis using an equivalent first-order closed-loop tracking model with additional uncompensated transport delay. It supports both constant-ratio electronic gearing and electronic cam motion represented by the maximum absolute cam slope.
Master-to-slave motion factor
For electronic gearing:
k = |i|
where:
- i = signed electronic slave/master gearing ratio [-]
- k = absolute master-to-slave speed multiplier [-]
The sign of the gearing ratio defines direction, but the tracking-lag magnitude is calculated from its absolute value.
For electronic cam synchronization:
k = max |dθs/dθm|
where:
- θs = slave-axis reference position
- θm = master-axis position
- k = maximum absolute slope of the electronic cam relationship [-]
The peak slave reference speed used by the calculation is:
ωs,peak = k · ωm
where:
- ωm = peak absolute master speed [rad/s]
- ωs,peak = peak slave reference speed [rad/s]
Equivalent servo tracking lag
The equivalent time constant associated with the specified closed-loop tracking bandwidth is:
τbw = 1 / (2πfbw)
and the combined equivalent lag time is:
teq = Td + τbw
where:
- fbw = equivalent closed-loop position tracking bandwidth [Hz]
- Td = additional residual uncompensated delay [s]
The estimated angular tracking-lag bound is:
Δθest = ωs,peak · teq · 180/π
where Δθest is expressed in degrees.
The calculator also separates this value into two components:
Δθdelay = ωs,peak · Td · 180/π
Δθdynamic = ωs,peak / (2πfbw) · 180/π
so that:
Δθest = Δθdelay + Δθdynamic
Lag-to-tolerance utilization
For the tracking-lag check:
U = Δθest / Δθallow
R = Δθallow − Δθest
where:
- U = lag/tolerance utilization [-]
- R = remaining angular error allowance [deg]
- Δθallow = maximum permitted absolute slave position error [deg]
A utilization below 1 means the calculated steady-speed tracking-lag bound is below the entered tolerance. This quantity is not a servo stability ratio or phase margin.
Maximum allowable master speed
For a known tracking bandwidth and residual delay, the maximum master speed allowed by the angular-lag criterion is:
ωm,max = Δθallow,rad / (k · teq)
with:
Δθallow,rad = Δθallow · π/180
Minimum equivalent servo bandwidth
At a specified master speed, the delay consumes part of the permitted position-error budget. The remaining allowable first-order lag time is:
tremaining = Δθallow,rad / ωs,peak − Td
If tremaining > 0, the minimum equivalent tracking bandwidth is:
fbw,min = 1 / (2πtremaining)
If tremaining ≤ 0, no finite bandwidth can satisfy the selected angular-error tolerance in this model because the residual-delay error alone already meets or exceeds the allowed error.
When to use this calculator
When to use this electronic cam and gearing calculator
Use this calculator for preliminary synchronization analysis of a motion-control system in which a servo-driven slave axis follows a master axis through either electronic gearing or an electronic cam relationship.
It provides three engineering calculation modes:
- Evaluate tracking lag at known speed — estimate the master/slave position tracking lag for a specified master speed, electronic gearing ratio or cam slope, servo bandwidth, residual delay, and permitted angular error.
- Determine allowable master speed — calculate the maximum master-axis speed that keeps the modeled tracking lag within the selected position tolerance.
- Size minimum equivalent servo bandwidth — determine the minimum equivalent closed-loop tracking bandwidth required at a specified master speed.
In electronic gearing mode, enter the signed slave/master ratio. In electronic cam mode, enter the maximum absolute derivative of slave position with respect to master position over the relevant cam profile.
The calculator is useful when evaluating whether available tracking performance is compatible with a required synchronization tolerance, or when establishing an initial bandwidth or speed requirement before detailed servo-system verification.
It is not a mechanical gear ratio calculator and does not determine gear tooth counts, gearbox reduction ratios, or transmission geometry. The gearing ratio entered here describes an electronic master/slave motion relationship.
It also does not generate an electronic cam profile or cam table. A cam profile must already be defined well enough to establish its maximum absolute master-to-slave slope.
Detailed servo tuning, stability analysis, acceleration and jerk feasibility, torque sizing, mechanical resonance, drive saturation, and machine-specific transient behavior require separate analysis.
How to interpret the result
How to interpret the synchronization results
Estimated position tracking lag
This is the calculated angular slave-position lag bound at the specified peak master speed. It combines the equivalent first-order servo tracking component with the entered residual transport delay.
In the tracking-lag check, the result is compared directly with the maximum permitted absolute slave position error.
Lag/tolerance utilization
- U < 1 — calculated lag is below the entered tolerance.
- U = 1 — calculated lag is at the entered tolerance.
- U > 1 — calculated lag exceeds the entered tolerance.
The remaining angular error allowance is the difference between the permitted error and the estimated lag. A small positive reserve means the operating point is close to the selected criterion even though the calculated check still passes.
Maximum master speed
The maximum master speed is the speed at which the modeled lag reaches the selected angular-error limit for the entered electronic gearing ratio or cam slope, tracking bandwidth, and residual delay.
Increasing the master-to-slave ratio, increasing the peak electronic cam slope, reducing bandwidth, or increasing residual delay decreases this allowable speed.
Minimum equivalent bandwidth
The minimum equivalent bandwidth is the first-order tracking bandwidth required to keep the calculated position lag within the selected tolerance at the specified master speed.
This value should be interpreted as a tracking-performance requirement for the simplified model, not as a complete servo tuning specification.
Residual-delay error floor
The delay-floor result is especially important during bandwidth sizing. It represents the position error caused solely by the residual uncompensated delay at the specified slave reference speed.
If this delay floor reaches the permitted error, increasing bandwidth cannot make the modeled synchronization requirement feasible.
Zero-speed condition
At zero master speed the steady-speed reference demand is zero, so the bandwidth-sizing calculation can return a zero bandwidth requirement. This does not remove the need to verify transient motion when the axis accelerates, decelerates, engages a cam, or responds to disturbances.
Calculation example
Example: electronic gearing synchronization lag
Consider a slave servo axis electronically geared to a master axis. The design allows a maximum absolute slave position tracking error of 4°, and the objective is to check the expected steady-speed tracking lag.
Inputs
- Synchronization mode: electronic gearing
- Slave/master gearing ratio: i = 1.5
- Peak master speed: ωm = 4 rad/s
- Equivalent closed-loop tracking bandwidth: fbw = 20 Hz
- Residual uncompensated delay: Td = 0.002 s
- Maximum permitted slave position error: Δθallow = 4°
1. Peak slave reference speed
k = |1.5| = 1.5
ωs,peak = 1.5 × 4 = 6 rad/s
2. First-order tracking time constant
τbw = 1 / (2π × 20) = 0.007958 s
Therefore:
teq = 0.002 + 0.007958 = 0.009958 s
3. Residual-delay lag
Δθdelay = 6 × 0.002 × 180/π = 0.688°
4. First-order tracking lag
Δθdynamic = 6 / (2π × 20) × 180/π = 2.736°
5. Total estimated tracking lag
Δθest = 0.688 + 2.736 = 3.423°
Lag/tolerance utilization:
U = 3.423 / 4 = 0.8558
Remaining angular error allowance:
R = 4 − 3.423 = 0.577°
The modeled tracking lag is therefore below the specified 4° limit, using approximately 85.6% of the available angular-error allowance.
6. Speed and bandwidth limits
At the same 20 Hz bandwidth and 2 ms residual delay, the maximum master speed from the lag criterion is approximately:
ωm,max = 4.674 rad/s
Conversely, at the specified 4 rad/s master speed, the minimum equivalent tracking bandwidth required to meet the 4° tolerance is approximately:
fbw,min = 16.52 Hz
This result verifies only the stated first-order-plus-delay tracking-lag criterion. It does not establish servo stability, transient following error during acceleration, available motor torque, or mechanical suitability of the synchronized axes.
Assumptions and limitations
Assumptions and limitations
- The slave-axis tracking response is represented by an equivalent first-order closed-loop position response plus a residual uncompensated transport delay.
- The equivalent first-order time constant is derived from tracking bandwidth as τ = 1/(2πfbw).
- The error model uses a bounded slave reference speed. At constant velocity, the first-order-plus-delay model represents the asymptotic position lag; for a changing reference, the calculator uses peak speed as a conservative speed-based criterion.
- The model assumes consistent initialization and does not calculate synchronization error caused by an initial master/slave position mismatch.
- Electronic gearing is represented by the absolute value of the slave/master ratio. Ratio sign changes motion direction but not the calculated error magnitude.
- Electronic cam behavior is reduced to the maximum absolute slope |dθslave/dθmaster|. The detailed cam profile, cam table interpolation, local curvature, acceleration, jerk, and engagement or disengagement transients are not evaluated.
- The entered servo bandwidth must represent equivalent closed-loop position tracking behavior. It is not the controller sampling frequency, motor natural frequency, or mechanical resonance frequency.
- The residual-delay input is treated as a fixed uncompensated delay. Time-varying communication jitter and other stochastic timing effects are not modeled separately.
- The calculation does not verify actual frequency-domain phase margin, gain margin, closed-loop stability, resonance separation, or controller robustness.
- Motor torque, drive current, inertia, acceleration capacity, velocity limits, saturation, friction, backlash, structural compliance, and external disturbances are outside the calculation.
- The position-error tolerance is supplied by the user. The calculator does not determine whether that tolerance is appropriate for the machine, process, product, or applicable design requirements.
- Input and output units are fixed by the calculation: master speed in rad/s, servo bandwidth in Hz, residual delay in seconds, and angular position error in degrees.
Final motion-system design should verify synchronization over the actual trajectory and include the relevant servo dynamics, mechanical limits, transient loads, safety functions, and machine-specific acceptance criteria.