Math Bridge: Servo Timing, Resolution, and Deadband

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Math BridgeActuatorsStruggle-friendly runway

Which servo limit comes from 50 Hz, one microsecond, and ±5 microseconds?

One thread that keeps sampling, timer steps, and anti-hunting deadband separate.

Max, the actuators guideMax guides
The one targetTranslate three timing limits into angle.
The chapter case50 Hz; 500–2,400 µs; ±5 µs.
What it buys youFind what really limits fine positioning.

A technician must decide whether angular deadband is safe before changing servo deadband on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is servo deadband. The middle card applies this page's rule. The green card is angular deadband. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only servo deadband, so the numeric fixture does not switch without explanation.

Servo deadband changes angular deadband An input card leads through the rule angle = deadband x 180 degrees / 1,900 us to the angular deadband result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A wider timing deadband creates a wider angle that the servo cannot resolve.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 5 us.

  2. 2

    Name the relationship. angle = deadband x 180 degrees / 1,900 us

  3. 3

    Substitute with units. 5 x 180 / 1,900 = 0.474 degrees

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change servo deadband

Try Predict the direction of angle = deadband x 180 degrees / 1,900 us. Test another servo deadband, then compare angular deadband.

5 us
Chapter baseline
Angular deadband

Observe A wider timing deadband creates a wider angle that the servo cannot resolve. Reset servo deadband to 5 and compare angular deadband.

Explain A wider timing deadband creates a wider angle that the servo cannot resolve.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only servo deadband moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only servo deadband. It is not a field prediction.

1. A servo receives timed commands

The frame repeats at 50 Hz. Pulse width chooses angle. A timer rounds pulse width to fixed ticks. The servo also ignores a small deadband to avoid hunting.

Max: These are three limits. Do not call all of them “resolution.”

2. Map pulse width to angle

1

Find the pulse span2,400−500=1,900 µs.

2

Spread 180 degreesq=180°/1,900 per 1 µs.

3

Find RMS roundingσq=q/√12.

3. Add sampling and deadband

fmax=fs/2; q=180°Δt/1,900 µs; σq=q/√12; θdead=deadband×180°/1,900 µs

Nyquist limits motion content. Timer ticks limit requested angle spacing. Deadband is a larger physical no-response zone.

4. Try the servo deadband

fNyquist=fs/2; qθ=180°Δt/Δtpulse; σ=qθ/√12; θdead=deadband×180°/Δtpulse

TryChange the deadband while the chapter's timer and 50 Hz stream stay fixed.

Nyquist ceiling
0.5 Hz sweep margin
1 µs angle step
Timer RMS angle
Deadband angle
Deadband / RMS

ObserveAt ±5 µs, the 50 Hz stream gives 25 Hz, the chapter's sweep has 50× sampling margin, one timer tick is 0.0947°, RMS rounding is 0.0273°, and deadband is ±0.474° or 17.3× the RMS floor.

ExplainThe physical deadband, not the timer's tiny rounding error, sets the larger fine-position limit in this example. The 25 Hz value answers a separate motion-speed question.

Technical boundaries.

Linear pulse mapping is a calibration model.

servo-specific endpoints
Needs separate evidence
potentiometer noise
Needs separate evidence
gear backlash
Needs separate evidence
load
Needs separate evidence
torque-speed behaviour
Needs separate evidence
supply sag
Needs separate evidence
control-loop tuning
Needs separate evidence
horn geometry
Needs separate evidence
saturation
Needs separate evidence
jitter
Needs separate evidence
mechanical stops
Needs separate evidence

Use field evidence or a deeper model before release.

5. Read the one-second sweep

A 0°→180°→0° cycle taking about two seconds has a fundamental near 0.5 Hz. The 25 Hz Nyquist ceiling is 50 times higher, so command sampling is comfortable even though mechanics may still lag.

6. Verify position safely

Calibrate real endpoints under load. Record supply current, pulse width, measured angle, deadband, repeatability, backlash, temperature, stall behaviour, and the safe parked state.

7. Check yourself

What angle does one microsecond represent here?
Answer: 180°/1,900=0.0947°.
Why is ±0.474° larger than 0.0273°?
Answer: Deadband is a deliberate no-response range; 0.0273° is only ideal timer-rounding RMS.
Does a 25 Hz command limit mean the horn can move at 25 Hz?
Answer: No. Mechanical bandwidth, load, torque, and the inner controller can be much slower.
Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

50 Hz
Frequency, sample rate, or event rate
20 ms
Time, interval, or service-life value
500–2,400 µs
Time, interval, or service-life value
1,900 µs
Time, interval, or service-life value
180°
Temperature or angle value
25 Hz
Frequency, sample rate, or event rate
0.5 Hz
Frequency, sample rate, or event rate
50×
Percentage, ratio, or gain
1 µs
Time, interval, or service-life value
0.0947°
Temperature or angle value
0.0273°
Temperature or angle value
±5 µs
Time, interval, or service-life value
±0.474°
Temperature or angle value
17.3×
Percentage, ratio, or gain
around-1° whole-servo
Charge or energy value

They do not certify loaded positioning.

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