A Stepper Motor. Some Measurements and a Model for LTSpice

As P2 is well suited for control jobs with step motors, this might perhaps be interesting for others here?
Idea was and is to learn more about step motors and their behaviour. I have been using them for many years but still there is lot to learn. If possible, I especially want to learn more about "Midband Instability".

So let's begin with a few links:
https://en.wikipedia.org/wiki/Stepper_motor explains some basics and makes clear, that we are here speaking of Hybrid Stepper Motors.
https://homepage.divms.uiowa.edu/~jones/step/ is also a good source of general information about step motors and how to drive them.

A calculation model is for me some form of "quantitative knowledge", so I was interested to have a model for the step motor.
https://www.mdpi.com/1996-1073/15/17/6159 describes a model, unfortunately not as SPICE text.
In Aichers thesis (German), there is an advanced model: https://mediatum.ub.tum.de/doc/601890/601890.pdf

This model is about a "Brushless DC Motor" and somewhat simpler: https://resources.pcb.cadence.com/i/1480186-pspice-app-note-brushless-dc-motor-model/1?
I decided to redo this model and start with it.
In this model each of the two windings is on the electrical side characterized by a resistance (constant) RW1, a (constant) inductance L1 and the Back-EMF-Voltage. This voltage is generated by the spinning motor.

In LTSpice torque is modelt as current 1A=1Nm and speed of the shaft is a voltage 1V= 1rev/sec.
The mechanical side of the winding model is the generation of a torque BtorqueW1.

The torques ( as currents ) come into the part of the model, which deals with motor mechanics:

Here we see the inertia as a capacitor a resistor for friction, that is speed dependant, a current source for friction, which is dependant from the direction of speed and the detent torque.
To get the angle of the shaft a current source and a capacitor do the integration.

As test-item I decided to use a motor "17HS19-2004S1", which is a Hybrid-Motor Nema17 with length 48mm, 1.8Degrees step angle (50 pole pairs), 0.59Nm holding torque at 2.0Amps, Rotor moment of inertia 82gr*cm², 1.4Ohms resistance and 3.0mH inductance per phase. https://www.omc-stepperonline.com/de/nema-17-bipolar-0-9-grad-44ncm-62-3oz-in-1-68a-2-8v-42x42x47mm-4-draehte-17hm19-1684s

Unfortunately some more data is needed, so I tried to do some simple measurements on my lathe. The setup shall give information about drag torque, detent torque and also about the generated voltage. To measure (average) torques the adhesive tape will lift a weight and the difference will show on the scale below. As the radius is well known, torques can be calculated. To get the speed, the frequency of the Back-EMF-Voltage is documented.

Here I am not sure, about the impact of the simplification of the model. In this measurement only the magnetic field of the permanent magnet is relevant. When the motor is driven by external voltage, the current through the coils will strengthen the field, which should lead to somewhat higher voltage??? ("Reluctance"?) This measurement was used to get the constant Cb=1,23V/(rev/sec)

Last not least the motor acting as a generator can be loaded with a resistor. So a current (RMS) will be generated which gives a mean torque. The constant for torque CT from this measurement but multiplied by 2 for both phases is similar to 0,59/2.0A.

Detent torque was "measured" by manually turning the chuck and trying to get the maximum torque. This only gives a rough idea.

The following picture shows the model together with it's parameters for the chosen motor 17HS19-2004S1:

To be continued....

Cheers Christof