Math Bridge: One Rounding Law, Three Actuators

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Math BridgeActuatorsStruggle-friendly runway

What does an actuator lose when a smooth command becomes steps?

One thread from rounding error to PWM, stepper, and servo limits.

Max, the actuators guideMax guides
The one targetKeep quantisation separate from sampling.
The chapter case12 V, 8-bit; 1.8° and 0.1758°; 50 Hz.
What it buys youName the real resolution limit.

See the relationship before changing it

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 pwm resolution, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 8 bits.

  2. 2

    Name the relationship. PWM step = 12,000 mV / 2^bits

  3. 3

    Substitute with units. 12,000 / 2^8 = 46.875 mV

  4. 4

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

Predict, then change pwm resolution

Try Predict the direction of PWM step = 12,000 mV / 2^bits. Test another pwm resolution, then compare ideal drive step.

8 bits
Chapter baseline
Ideal drive step

Observe More PWM bits shrink voltage steps; motor mechanics are a separate fixture. Reset pwm resolution to 8 and compare ideal drive step.

Explain Each extra pulse-width modulation bit doubles the available duty settings, halving the ideal average-voltage increment across the same drive range.

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 pwm resolution moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only pwm resolution. It is not a field prediction.

1. A smooth target meets a grid

A controller can ask for any voltage or angle. Hardware often offers only fixed steps. Rounding chooses the nearest step and leaves a small signed error.

Max: Voltage steps and angle steps share the maths, but they do not share units.

2. Measure one rounding error

1

Name the stepCall the spacing between neighbouring choices q.

2

Bound the errorNearest-step rounding lies from −q/2 to +q/2.

3

Take its RMSA uniform error has σq=q/√12.

3. Add the separate time limit

qPWM=Vs/2^N; σq=q/√12; SNR≈6.02N+1.76 dB; fmax=fs/2

Bit depth sets command spacing. Refresh rate sets the fastest command motion that can be represented. One cannot repair the other.

4. Try the PWM bit depth

qPWM=Vs/2^N; σ=q/√12; qstep=360/steps; fNyquist=fs/2

TryMove the bit-depth control. The stepper and servo values stay fixed so you can see which limit changes.

PWM levels
12 V PWM step
PWM RMS error
Ideal PWM SNR
NEMA 17 RMS angle
28BYJ-48 step
28BYJ-48 RMS angle
Stepper refinement
Servo Nyquist limit

ObserveAt 8-bit, 12 V becomes 46.875 mV steps with 13.53 mV RMS error and 49.9 dB ideal SNR. The two steppers stay at 0.520° and 0.0507° RMS, while the servo stays at 25 Hz.

ExplainThe same rounding law compares PWM and stepper resolution. Nyquist belongs to a different question: how quickly the commanded angle may change.

Technical boundaries.

The voltage grid keeps the supply at 12 V while pulse-width modulation bit depth changes; its rounding-error estimate assumes targets are evenly spread within each step. Backlash, missed steps, torque, timer jitter, driver nonlinearity, servo mechanics, load, and control-loop behaviour are outside this grid calculation.

backlash
Needs separate evidence
missed steps
Needs separate evidence
torque
Needs separate evidence
timer jitter
Needs separate evidence
driver nonlinearity
Needs separate evidence
servo mechanics
Needs separate evidence
load
Needs separate evidence
control-loop dynamics
Needs separate evidence

Use field evidence or a deeper model before release.

5. Reproduce the chapter values

The NEMA 17 gives 1.8°/√12=0.520°. The 2,048-step motor gives q=360°/2,048=0.1758° and 0.0507° RMS, about 10× finer. A 50 Hz command stream gives 25 Hz by Nyquist.

6. Verify the physical result

Record timer resolution, driver mode, commanded and measured angle, load, backlash, missed steps, refresh rate, and the sensor used to prove motion.

7. Check yourself

Why divide q by √12?
Answer: It is the RMS value of an evenly distributed nearest-step error from −q/2 to +q/2.
Does more PWM bit depth raise a servo's 25 Hz limit?
Answer: No. Bit depth changes spacing; refresh rate sets the sampling limit.
Does 0.0507° RMS guarantee that motion accuracy?
Answer: No. Backlash, load, missed steps, and mechanics can dominate.
Honesty boundary.

The voltage grid keeps the supply at 12 V while pulse-width modulation bit depth changes; its rounding-error estimate assumes targets are evenly spread within each step. Backlash, missed steps, torque, timer jitter, driver nonlinearity, servo mechanics, load, and control-loop behaviour are outside this grid calculation.

12 V
Drive supply voltage
8-bit
Pulse-width modulation command resolution
46.875 mV
Drive voltage step at eight bits
13.53 mV
Root mean square drive rounding error
49.9 dB
Ideal signal-to-noise ratio at eight bits
1.8°
NEMA 17 full-step angle
0.520°
NEMA 17 root mean square angle-rounding error
2,048-step
28BYJ-48 steps per revolution
0.1758°
28BYJ-48 angle per step
0.0507°
28BYJ-48 root mean square angle-rounding error
roughly 10×
Stepper angle-resolution improvement
50 Hz
Servo command refresh rate
25 Hz
Servo command Nyquist frequency

They describe command grids, not guaranteed actuator accuracy.

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