Math Bridge: Microstep Current and Coil Lag

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Why are equal microstep angles not equal current steps?

One thread from a sine table and sense resistor to winding lag at speed.

Max, the actuators guideMax guides
The one targetCompute a microstep's current jump and lag.
The chapter case1/16, 1.5 A, 0.1 Ω, 800 steps/s.
What it buys youExplain smoothness and torque loss honestly.

A technician must decide whether microstep angle is safe before changing microsteps per quarter-cycle 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 microsteps per quarter-cycle. The middle card applies this page's rule. The green card is microstep angle. 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 microsteps per quarter-cycle, so the numeric fixture does not switch without explanation.

Microsteps per quarter-cycle changes microstep angle An input card leads through the rule angle = 90 degrees / microsteps to the microstep angle result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. More microsteps divide the same quarter-cycle into smaller command angles.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 16 steps.

  2. 2

    Name the relationship. angle = 90 degrees / microsteps

  3. 3

    Substitute with units. 90 / 16 = 5.625 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 microsteps per quarter-cycle

Try Predict the direction of angle = 90 degrees / microsteps. Test another microsteps per quarter-cycle, then compare microstep angle.

16 steps
Chapter baseline
Microstep angle

Observe More microsteps divide the same quarter-cycle into smaller command angles. Reset microsteps per quarter-cycle to 16 and compare microstep angle.

Explain More microsteps divide the same quarter-cycle into smaller command angles.

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 microsteps per quarter-cycle moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only microsteps per quarter-cycle. It is not a field prediction.

1. The driver commands current

A stepper driver compares the voltage across a small sense resistor with a reference. Microstepping changes that current target through sampled sine values.

Max: The table requests current; the inductive winding still needs time to reach it.

2. Build the quarter-sine table

1

Split 90 degreesFor N=16, each electrical angle step is 90°/16=5.625°.

2

Sample the sineIk=Itrip sin(kπ/(2N)).

3

Subtract neighboursThe first and last current jumps are unequal.

3. Add the electrical lag

Vsense=IcoilRsense; τ=L/R; i/Itarget=1−e^(−t/τ)

The chapter's 2.5 mH and 1.5 Ω winding has τ=1.67 ms, longer than the 1/800=1.25 ms command period.

4. Try the microstep count

Ik=Itrip sin(kπ/2N); Vsense=IRsense; τ=L/R; reached=1−e^(−t/τ)

TryChange the number of microsteps per quarter-cycle while the motor and 800 steps/s timing stay fixed.

Sense voltage at 1.5 A
Electrical angle step
First current jump
Last current jump
First / last
Winding τ
Command period
Target current reached
Steps per revolution

ObserveAt 1/16 stepping, the angle is 5.625°, the first jump is 0.147 A, the last is about 0.007 A, and the first is about 20.4× larger. The winding reaches only 52.8% in 1.25 ms.

ExplainMicrostep count changes the sine-table increments and steps/revolution. The fixed L/R and command period separately bound how closely real current follows those targets.

Technical boundaries.

This first-order single-target calculation

bipolar phase interaction
Needs separate evidence
chopper decay mode
Needs separate evidence
supply voltage headroom
Needs separate evidence
back-EMF
Needs separate evidence
detent torque
Needs separate evidence
load angle
Needs separate evidence
resonance
Needs separate evidence
driver nonlinearities
Needs separate evidence
heating
Needs separate evidence
cumulative motion error
Needs separate evidence

Use field evidence or a deeper model before release.

5. Read the non-uniform steps

Sine is steep near zero and flat near its peak. Equal angle intervals therefore cannot produce equal changes in commanded current.

6. Verify motion, not just the table

Record sense resistance, current limit, supply, winding R/L, decay mode, step rate, load, temperature, missed steps, resonance, and measured position.

7. Check yourself

What voltage represents 1.5 A through 0.1 Ω?
Answer: V=IR=1.5×0.1=0.150 V.
Why is the first sine step larger than the last?
Answer: Sine's slope is largest near zero and approaches zero near its peak.
Does 3,200 commands/rev guarantee 3,200 positions?
Answer: No. Current lag, torque, load, resonance, and missed steps can prevent commanded positions.
Honesty boundary.

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

1/16
Chapter input or worked result
200→3,200 steps/rev
Cycle, step, or position count
5.625°
Temperature or angle value
1.5 A
Current or responsivity value
0.1 Ω
Resistance or impedance value
0.150 V
Voltage or voltage-step value
0.147 A
Current or responsivity value
about 0.007 A
Current or responsivity value
20.4×
Percentage, ratio, or gain
1.67 ms
Time, interval, or service-life value
800 steps/s
Time, interval, or service-life value
1.25 ms
Time, interval, or service-life value
52.8%
Percentage, ratio, or gain

They do not certify position or torque; Under the Hood keeps those limits.

Return to the chapter with the maths unpacked →