Motor Model¶

The motor model is a two-phase hybrid stepper with sinusoidal back-EMF, a detent torque term, viscous and dry friction and a rigid load. It is written once (model/stepper_motor_f.sci) and used by the prototype, the Xcos plant block (in C) and the Python twins. This chapter gives the equations, the sign conventions, the parameters of the 17HS19-2004S1 and the limits of the model.

State equations¶

The state is \(x = [\,i_A,\ i_B,\ \omega,\ \theta\,]\): the two phase currents (A), the mechanical speed (rad/s) and the mechanical rotor angle (rad). With \(p = N_s / 4\) electrical periods per mechanical turn (50 for a 200-step motor):

Electrical part

\[ \begin{aligned} L\,\frac{di_A}{dt} &= v_A - R\,i_A - e_A, \qquad e_A = -K_t\,\omega\,\sin(p\theta) \\ L\,\frac{di_B}{dt} &= v_B - R\,i_B - e_B, \qquad e_B = K_t\,\omega\,\cos(p\theta) \end{aligned} \]

Mechanical part

\[ \begin{aligned} J\,\frac{d\omega}{dt} &= T_e - T_L - T_f \\ \frac{d\theta}{dt} &= \omega \end{aligned} \]

Torques

\[ \begin{aligned} T_e &= K_t\,\left(-i_A \sin(p\theta) + i_B \cos(p\theta)\right) - T_d \sin(4p\theta) \\ T_f &= B\,\omega + T_c \tanh\left(\frac{\omega}{\omega_c}\right) \end{aligned} \]

Symbol	Meaning	Unit
\(v_A\), \(v_B\)	phase voltages applied by the power stage	V
\(R\), \(L\)	phase resistance and inductance (no mutual inductance between phases)	Ω, H
\(K_t\)	torque constant = back-EMF constant	N·m/A = V·s/rad
\(T_d\)	detent torque amplitude (period: one full step)	N·m
\(T_L\)	load torque	N·m
\(B\)	viscous friction	N·m·s/rad
\(T_c\), \(\omega_c\)	dry (Coulomb) friction and its smoothing speed	N·m, rad/s
\(J\)	rotor + load inertia	kg·m²

The back-EMF and the torque come from the same flux linkage, so the electrical power \(e_A i_A + e_B i_B\) equals the mechanical power \(T_e\,\omega\) (without the detent term, which does no net work over a step). Test T00 checks this balance on 1000 random states (largest error 1.1·10⁻¹³ W).

Sign convention¶

With the current vector

\[ i_A = I \cos\varphi, \qquad i_B = I \sin\varphi \]

the torque is \(T_e = K_t\,I \sin(\varphi - p\theta)\): it is zero and restoring at \(p\theta = \varphi\). The rotor therefore settles at \(p\theta = \varphi\), increasing \(\varphi\) turns it forward, and phase A leads phase B by 90°. The electrical angle between the field and the rotor, the load angle (electrical degrees),

\[ \delta = \varphi - p\theta \]

is the most useful quantity in every test: with a load \(T_L\) the rotor lags by \(\delta = \arcsin\left(T_L / (K_t I)\right)\), the largest static torque is \(K_t I\) at \(\delta = 90^\circ\), and beyond \(\delta \approx 180^\circ\) the rotor falls into the next pole — the motor has lost steps.

Note

An early project note used \(i_A = I \sin\varphi\), \(i_B = I \cos\varphi\). With the torque equation above that turns the rotor backwards; the library, the prototype and this manual all use the cosine/sine form.

Parameters of the 17HS19-2004S1¶

Parameter	Value	Source
\(R\)	1.60 Ω	datasheet (±10 %, 25 °C); older listings state 1.4 Ω — measure
\(L\)	3.0 mH	datasheet at 1 kHz (±20 %)
\(I_{\mathrm{rated}}\)	2.0 A	datasheet
\(K_t\)	0.2086 N·m/A	derived: holding torque 0.59 N·m with both phases at 2 A, \(K_t = 0.59 / (\sqrt{2} \cdot 2)\) — measure the back-EMF
\(N_s\)	200 (1.8°)	datasheet
\(J\)	82 g·cm² = 8.2·10⁻⁶ kg·m²	datasheet, rotor only
\(B\)	1·10⁻⁵ N·m·s/rad	assumption
\(T_d\)	0 (tests: 0.02 N·m)	not in the datasheet; 0.02 N·m ≈ 3 % of the holding torque is an assumption
\(T_c\)	0 (tests: 0.01 N·m)	assumption
\(\omega_c\)	0.05 rad/s	numerical smoothing of the dry friction

The parameters are a Scilab structure returned by motor_17hs19_2004s1() (params/motor_17hs19_2004s1.sci); a test or a scenario changes a field (mot.Td = 0.02;) without touching the equations.

Measuring K_t

Spin the unpowered shaft (a drill), measure the open-circuit phase voltage amplitude \(E\) and the speed \(\omega\): \(K_t = E / \omega\). If the datasheet holding torque is meant for one phase only, K_t would be 0.295 N·m/A — 41 % more, which changes every speed limit in this manual.

Characteristic values¶

Quantity	Formula	17HS19-2004S1
Electrical time constant	\(\tau = L / R\)	1.875 ms
Steady current at 3.2 V	\(V / R\)	2.0 A (T01)
Rotor natural frequency at current \(I\)	\(f_n = \frac{1}{2\pi}\sqrt{p K_t I / J}\)	254 Hz at 2 A, no load
Back-EMF amplitude	\(K_t\,\omega\)	1.31 V per rev/s
Holding torque	\(\sqrt{2}\,K_t I\) (both phases)	0.59 N·m at 2 A
Static torque of the current vector	\(K_t I\)	0.417 N·m at 2 A

The natural frequency is the stiffness of the "magnetic spring" \(p K_t I\) against the inertia; it decides the low-speed resonance (Results).

Numerical solution¶

Use	Solver
Open-loop studies (`stepper_simulate`)	Scilab `ode` (lsoda), rtol 10⁻⁸, atol 10⁻¹⁰
Closed loop, averaged power stage	fixed-step RK4, 4 steps per PWM period (12.5 µs)
Closed loop, switching power stage	RK4 between the switching instants, at least 16 steps per period
Python twin of the open-loop model	scipy `solve_ivp` (Radau), rtol 10⁻¹⁰

The dry friction \(T_c \tanh(\omega/\omega_c)\) is stiff near zero speed (time constant \(J\omega_c / T_c\)). RK4 stays stable while \(h\,T_c / (J\omega_c) < 2.8\): with \(\omega_c\) = 0.05 rad/s and the 12.5 µs step of the closed loop that is \(T_c\) up to about 0.09 N·m. Increase \(\omega_c\) for larger friction values.

Limits of the model¶

The model is meant for designing and testing the controller. It does not include:

magnetic saturation (inductance and K_t drop at high current);
harmonics of the back-EMF and of the torque other than the detent term;
mutual inductance between the phases;
temperature (R rises about 0.4 %/K in copper);
stiction (the dry friction is smoothed);
elasticity of the coupling and the load (one rigid inertia).

Effects that depend on these — the exact speed of the mid-band instability, the noise spectrum, the current at full saturation — must be checked on the hardware.