# Influence of Microstepping Signal Shape on Shaft Movement Precision and Torque Variation of the Stepper Motor

^{*}

## Abstract

**:**

## 1. Introduction

#### Stepper Motor Principle of Operation

## 2. Derivation of $\mathbf{p}$-Circle Signals

## 3. Materials and Methods

^{®}incremental optical encoder [17], which provides a resolution of $180e3$ pulses per revolution (physical line count 9000 with x20 interpolation provided by the readhead). For $1.9$ deg/step mechanical resolution of motor with x32 microstepping, this yields exactly $28.125$ pulses/microstep, which is sufficient to measure shaft oscillations at the sub-microstep scale for microstepping resolutions up to and including x32. Encoder data are processed by the TMS320F28335 eQEP module into the angular position. The PC is used to acquire, save and analyse data obtained by the microcontroller.

^{®}environment, in which phase-space equations presented in Section 2 are implemented and cast to the time domain. This ensures that for every p-norm signal, the same set of phasor angles is achieved, and in case of an ideal motor geometry, no load applied to the motor and ideal currents control, no difference in shaft position should be observed between all analysed signals. Figure 13, Figure 14, Figure 15 and Figure 16 exemplify currents generated in motor coils on the test setup.

## 4. Results

`tfest()`function of the Matlab

^{®}environment, in order to estimate the parameters of model (7). For every signal and microstepping resolution, both the mean and the median values were calculated from datasets of size specified in Table 5.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of a bipolar stepper motor. Letters A and B denote separate electrical coils. The coil letter color denotes the winding direction of the stator pole (either clockwise or counterclockwise).

**Figure 2.**Analysis of the vector difference ${i}_{\mathrm{TR}}$ between two consecutive current phasors ${i}_{\mathrm{I}}$ and ${i}_{\mathrm{II}}$. Incremental torque corresponds with rejection of ${i}_{\mathrm{TR}}$ onto ${i}_{\mathrm{I}}$ denoted by ${i}_{\mathrm{ROT}}$. Signal curvature is then defined by projection ${i}_{\mathrm{MOD}}$.

**Figure 6.**Torque variation and increment relative to a sine-cosine microstepping of equal resolution; $p=3$, microstepping resolution x8.

**Figure 7.**Torque variation and increment relative to a sine-cosine microstepping of equal resolution; $p=3$, microstepping resolution x16.

**Figure 8.**Torque variation and increment relative to a sine-cosine microstepping of equal resolution. For $p=10$ resolution, x8 becomes too low to guarantee the drop in torque variation at ${\varphi}_{\mathrm{e}}=\frac{\pi +k\pi}{4}$.

**Figure 9.**Torque variation and increment relative to a sine-cosine microstepping of equal resolution; $p=10$ microstepping resolution x16.

**Figure 10.**Maximum and minimum incremental torque for different signal shapes and microstepping resolutions. Quadrature microstepping with resolution x2 is equivalent to half step movement.

**Figure 12.**Photo of the test setup. The acquired shaft position dataset can be seen on the PC screen.

**Figure 17.**Presentation of a full collection of 12,000 data points sampled at 1 kHz. Odometry count is the raw position data obtained from the incremental encoder. The motor was working under the load in the quadrature x16 mode.

**Figure 18.**Zoom on the region of rapid change in amplitude of shaft oscillation with related A (red) B (blue) coil currents reference shown.

**Figure 19.**An example of input-output signals used for system identification. Output signal obtained from raw position data by defining steady initial condition and debiasing.

**Figure 20.**Medium and median values of estimated parameters for selected microstepping signals at x8 resolution. Horizontal axis is not of uniform scale, since quadrature signal corresponds with p = ∞. Axis label s denotes sine-cosine microstepping.

**Figure 21.**Medium and median values of estimated parameters for selected microsptepping signals at x16 resolution.

**Figure 22.**Medium and median values of estimated parameters for selected microsptepping signals at x32 resolution.

**Figure 26.**Influence of microstepping resolution for the selected p-circle signal on shaft position error. Time range is chosen so as to include one full step of recorded data with lowest angular velocity.

**Figure 27.**Positioning error against signal shape for different microstepping resolutions. Error calculated only from experiments in which load was applied to the motor shaft.

A | B |
---|---|

$+{i}_{\mathrm{n}}$ | $-{i}_{\mathrm{n}}$ |

$+{i}_{\mathrm{n}}$ | $+{i}_{\mathrm{n}}$ |

$-{i}_{\mathrm{n}}$ | $+{i}_{\mathrm{n}}$ |

$-{i}_{\mathrm{n}}$ | $-{i}_{\mathrm{n}}$ |

${\mathbf{J}}_{\mathbf{pulley}}[\mathbf{kg}\xb7{\mathbf{cm}}^{2}]$ | ${\mathbf{m}}_{\mathbf{pulley}}\left[\mathbf{kg}\right]$ | ${\mathbf{T}}_{\mathbf{load}}[\mathbf{kg}\xb7\mathbf{cm}]$ | ${\mathbf{m}}_{\mathbf{load}}\left[\mathbf{kg}\right]$ | $\mathbf{r}\left[\mathbf{cm}\right]$ | ${\mathbf{f}}_{\mathbf{PWM}}\left[\mathbf{kHz}\right]$ |
---|---|---|---|---|---|

0.114 | 0.0581 | 6.67 | 3.09 | 2.16 | 50 |

**Table 3.**Nominal parameters of the SM57HT76-2804B bipolar stepper motor. “spr” means “steps per revolution”.

${\mathbf{I}}_{\mathbf{phase}}\left[\mathbf{A}\right]$ | ${\mathbf{R}}_{\mathbf{phase}}\left[\mathbf{\Omega}\right]$ | ${\mathbf{L}}_{\mathbf{phase}}\left[\mathbf{mH}\right]$ | ${\mathbf{T}}_{\mathbf{n}}[\mathbf{kg}\xb7\mathbf{cm}]$ | ${\mathbf{J}}_{\mathbf{rotor}}[\mathbf{kg}\xb7{\mathbf{cm}}^{2}]$ | spr |
---|---|---|---|---|---|

2.8 | 1.13 | 3.6 | 18.9 | 0.48 | 200 |

x8 | x16 | x32 |
---|---|---|

$0.5$ rpm | $0.25$ rpm | $0.176$ rpm |

x8 | x16 | x32 |
---|---|---|

552 | 552 | 783 |

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**MDPI and ACS Style**

Bednarski, B.; Jackiewicz, K.; Gałecki, A.
Influence of Microstepping Signal Shape on Shaft Movement Precision and Torque Variation of the Stepper Motor. *Energies* **2021**, *14*, 6107.
https://doi.org/10.3390/en14196107

**AMA Style**

Bednarski B, Jackiewicz K, Gałecki A.
Influence of Microstepping Signal Shape on Shaft Movement Precision and Torque Variation of the Stepper Motor. *Energies*. 2021; 14(19):6107.
https://doi.org/10.3390/en14196107

**Chicago/Turabian Style**

Bednarski, Bogdan, Krzysztof Jackiewicz, and Andrzej Gałecki.
2021. "Influence of Microstepping Signal Shape on Shaft Movement Precision and Torque Variation of the Stepper Motor" *Energies* 14, no. 19: 6107.
https://doi.org/10.3390/en14196107