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

The figure reads from left to right. The blue card is pwm resolution. The middle card applies this page's rule. The green card is ideal drive step. 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 pwm resolution, so the numeric fixture does not switch without explanation.

PWM resolution changes ideal drive step An input card leads through the rule PWM step = 12,000 mV / 2^bits to the ideal drive step result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. More PWM bits shrink voltage steps; motor mechanics are a separate fixture.

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 More PWM bits shrink voltage steps; motor mechanics are a separate fixture.

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 uniform-error model assumes targets are spread within each step.

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.

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

12 V
Voltage or voltage-step value
8-bit
Digital resolution or converter setting
46.875 mV
Voltage or voltage-step value
13.53 mV
Voltage or voltage-step value
49.9 dB
Gain, loss, margin, or level ratio
1.8°
Temperature or angle value
0.520°
Temperature or angle value
2,048-step
Cycle, step, or position count
0.1758°
Temperature or angle value
0.0507°
Temperature or angle value
roughly 10×
Percentage, ratio, or gain
50 Hz
Frequency, sample rate, or event rate
25 Hz
Frequency, sample rate, or event rate

They describe command grids, not guaranteed actuator accuracy.

Return to the assessment with the maths unpacked →