Update Content - 2022-08-24

content/zettels/stepper_motor.md

@@ -9,18 +9,91 @@ Tags
 :
 
 
-## Errors between steps (micro-stepping) {#errors-between-steps--micro-stepping}
+## Types of Stepper motors {#types-of-stepper-motors}
 
-For a two phase stepper motor, there are (typically) **200 steps per revolution** (i.e. 1.8% per step).
+<https://blog.orientalmotor.com/stepper-motor-basics-pm-vs-vr-vs-hybrid>
+
+-   Permanent Magnet
+-   Variable Reluctance
+-   Hybrid
+
+<a id="figure--fig:stepper-two-phase-hybrid-stepper"></a>
+
+{{< figure src="/ox-hugo/stepper_two_phase_hybrid_stepper.png" caption="<span class=\"figure-number\">Figure 1: </span>Interior of a two phase hybrid stepper motor. This motor has eight windings and 50 roto teeth" >}}
+
+<a id="figure--fig:stepper-hybrid-schematic"></a>
+
+{{< figure src="/ox-hugo/stepper_hybrid_schematic.png" caption="<span class=\"figure-number\">Figure 2: </span>Schematic of a two phase hybrid stepper motor. This motor has four windings and 15 pole pairs" >}}
+
+
+## Micro Stepping {#micro-stepping}
+
+From (<a href="#citeproc_bib_item_2">Ronquist and Winroth 2016</a>):
+
+> By varying the magnitude and direction of the winding currents, the rotor is continuously attracted in the desired direction.
+> A "step" occurs whenever a rotor tooth moves slightly to align itself to an electromagnet tooth.
+>
+> It is possible to decrease the step size of the hybrid stepper motor by using a control logic called **microstepping**.
+> As opposed to fully exciting each phase in turn, as described previously, microstepping involves transitioning between each phase shift.
+> That is, the current references are defined by sinusoidal signals displaced 90 electrical degrees from each other.
+> For most time instances, then, both phases are excited to a certain degree.
+> The result is that the electric position vector can be placed between two teeth.
+> The resolution of the motor has therefore been increased.
+
+From (<a href="#citeproc_bib_item_1">Condit 2004</a>):
+
+> There are several factors that affect the linearity of microstepping in real motors.
+> The first limitation is static friction in the system.
+>
+> [...]
+>
+> Another limitation is the fact that the torque versus position curve is not perfectly sinusoidal.
+> The toothed shape of the motor and other physical characteristics of the motor contribute to this.
+> Figure [3](#figure--fig:stepper-real-pos-vs-actual-pos) shows a plot of actual position vs expected position for a typical motor.
+
+<a id="figure--fig:stepper-real-pos-vs-actual-pos"></a>
+
+{{< figure src="/ox-hugo/stepper_real_pos_vs_actual_pos.png" caption="<span class=\"figure-number\">Figure 3: </span>Real vs actuator rotor position" >}}
+
+
+## Open Loop errors {#open-loop-errors}
+
+Nice references:
+
+-   (<a href="#citeproc_bib_item_3">Vyas, Patel, and Shah 2015</a>)
+-   (<a href="#citeproc_bib_item_2">Ronquist and Winroth 2016</a>)
+
+<div class="seealso">
+
+References about these errors can be search for using "torque ripple", "Cogging torque" and "load dependent error" keywords.
+
+</div>
+
+
+### Error with period equal to one **turn** {#error-with-period-equal-to-one-turn}
+
+A stepper motor has a position error with a period equal to a full turn.
+
+An example is shown in Figure [4](#figure--fig:stepper-error-one-turn-period) (from (<a href="#citeproc_bib_item_2">Ronquist and Winroth 2016</a>)).
+The high frequency errors that can be observed have a period of one step (i.e. 200 periods each turn).
+
+<a id="figure--fig:stepper-error-one-turn-period"></a>
+
+{{< figure src="/ox-hugo/stepper_error_one_turn_period.png" caption="<span class=\"figure-number\">Figure 4: </span>Angle error of the stepper motor during a 100rpm (i.e. 0.6s per turn)" >}}
+
+
+### Error with period equal to one **step** {#error-with-period-equal-to-one-step}
+
+For a two phase stepper motor, there are (typically) **200 steps per revolution** (i.e. 1.8 degrees per step).
 
 Between each step, even when using some micro-stepping, there are some position errors that are due to non-perfect magnetic and electromagnetic fields.
 
 The period of this error is corresponding to 200 period/revolution.
 
-Then scanning, this can lead to high frequency vibrations.
+Then scanning, this can lead to **high frequency vibrations**.
 
 This is what is typically limiting the accuracy of the stepper motor (usually specified in between 3% and 5% of the step increment).
-This is approximately corresponding to **1mrad**.
+This is approximately corresponding to **1mrad** and can be around 0.1mrad for best stepper motors.
 
 <div class="exampl">
 
@@ -31,12 +104,23 @@ A rotation of 1 turn per second will induce vibrations at 200Hz with an amplitud
 
 Note that this error is not a pure sine, it also has some harmonics with corresponding periods of 1/100 revolution and 1/50 revolution.
 
-This error should repeat every turn and can be calibrated provided it is repeatable over time.
-
 One way to reduce these errors is to use a ball-screw mechanism with a smaller pitch.
 The price to pay is smaller velocity.
 
 
+### Load Dependent Error {#load-dependent-error}
+
+If the electromagnetic torque would be the only torque acting on the system, the electrical angle generated by the control system would correspond directly to the reference angle.
+
+The position error is to a large degree due to the so called load angle when the motor is positioned by an open-loop controller.
+The load angle results from applying an external torque to the stepper motor, **causing the magnetic rotor to be out of phase with the electrical field**.
+
+The most common way to limit these errors is to always operate the motor with its rated winding currents.
+This results in significant energy losses and heating of the motor which deprive the motor of its efficiency.
+
+Another option is to use a position sensor such as an encoder with a feedback controller.
+
+
 ## Manufacturers {#manufacturers}
 
 | Manufacturers                                                            | Country        |
@@ -53,4 +137,7 @@ The price to pay is smaller velocity.
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Condit, Reston. 2004. “Stepping Motors Fundamentals.” Microchip Technology.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Ronquist, Anton, and Birger Winroth. 2016. “Estimation and Compensation of Load-Dependent Position Error in a Hybrid Stepper Motor.” Linköping University, Automatic Control; Linköping University, Automatic Control.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Vyas, Darshit C, Jinesh G Patel, and Mrs Heli A Shah. 2015. “Microstepping of Stepper Motor and Sources of Errors in Microstepping System.” <i>Int. Journal of Engineering Research and General Science</i> 3 (2).</div>
 </div>