Stepper Motor Torque-Speed Calculator: Driver Voltage & Resonance

Electrical torque envelope

This tool derives a sinusoidal two-phase model; it does not digitize or reproduce a measured pull-out curve. All angles are mechanical unless stated otherwise. With full-step angle $\theta_s$, the electrical cycle count is $p=\pi/(2\theta_s)$. A two-phase-on holding test gives peak-current torque constant $K_t=T_h/(\sqrt{2}I_r)$ and peak phase back-EMF constant $K_e=K_t$ numerically in SI. Trinamic QSH5718 v2.6, pp. 13–14, gives the corresponding RMS back-EMF and current conventions.

For sine peak current $I$, mechanical angular speed $\omega$ and effective supply $U=V_{bus}-V_{drop}$, the phase-voltage estimate at maximum-torque alignment (90° electrical load angle) is

The resistive voltage and motoring back EMF are in phase only under this alignment assumption; inductive voltage is in quadrature. Solve the quadratic for nonnegative current, cap it at the selected peak current, then use $T=K_tI$. Above $U/K_e$, this model provides zero positive motoring torque. The formula is derived here from the phase RL equation and electromechanical power conservation, assuming current aligned with back EMF and no field weakening. Explicit one-radian factors preserve the engine's angle units.

For a general electrical load angle $\delta$, the same ideal two-phase equations give $T=K_t I\sin\delta$ and, with $E=K_e\omega$ and $X=p\omega L$,
$U^2=(RI+E\sin\delta)^2+(XI+E\cos\delta)^2.$
This relation is derived from the phase-voltage equations documented by MathWorks. The calculator fixes $\delta=90°$ for its envelope; it does not solve the operating load angle. A smaller load angle can require more voltage at the same current. The supply and knee outputs therefore cannot establish full-current waveform feasibility at the entered load.

Load and speed

Inertia is referred to the motor shaft. The maximum design-load speed is the largest nonnegative intersection, found by bounded bisection; zero means the load already exceeds the low-speed torque estimate. The full-current knee solves the voltage equation at the selected current and fixed 90° alignment. The supply output uses the same assumed alignment. The speed intersection and torque margin are comparisons with that envelope, not waveform-feasibility or synchronism checks. The acceleration allowance can be negative if steady running already exceeds the design allowance.

Current rise and microsteps

For a standstill phase starting at zero, $\tau=L/R$ and $i(t)=\min[I_{pk},(U/R)(1-e^{-t/\tau})]$. Thus $t_{rise}=-\tau\ln(1-RI_{pk}/U)$. This diagnostic excludes back EMF and current reversal and is not used as an artificial per-microstep torque multiplier. The chart and animation show this same ideal current rise.

With $\mu$ microsteps per full step, $f_{STEP}=\mu\omega/\theta_s$, $\theta_\mu=\theta_s/\mu$, and the ideal single-command incremental torque is $T_{inc}=T_0\sin(90°/\mu)$. Analog Devices, Mastering Precision, p. 4, supplies this sine-law approximation. Its 1/8 row prints 17.508%; the equation instead gives 19.509%. The model retains the equation; the checked 1/16 row is unaffected.

• Two identical phases, sinusoidal permanent-magnet flux, negligible mutual inductance, constant winding resistance and inductance. Holding torque is a two-phase-on measurement at the entered rated current.
• Driver current is an ideal regulated sine wave. The RMS/peak selector describes the driver setting; it does not change the datasheet current convention. Rated RMS current gives equal total copper loss to two phases each carrying rated DC current.
• The voltage and torque envelope assumes current aligned with back EMF: a 90° electrical load angle. The entered load defines the load line only; it does not determine the operating current phase or load angle.
• Back EMF is inferred from holding torque rather than measured. Magnetic saturation, current phase error, PWM ripple, blanking time, decay mode, iron loss and detent torque are omitted.
• Load torque, viscous coefficient and positive acceleration are fixed during a sweep. The design factor multiplies their sum. For a constant-speed limit, set acceleration to zero.
• Motor defaults come from Trinamic QSH5718 v2.6 p. 4. Driver voltage, loss allowance, load and design factor are illustrative user-adjustable values.
• The animation is a standstill current-rise experiment. It neither simulates rotation nor changes the heavy torque-speed calculations. At standstill, interval outputs are zero because no STEP pulses are being commanded.

Use for preliminary comparisons of two-phase hybrid or PM motors with chopper current drivers. The smooth electrical envelope assumes maximum-torque alignment, a 90° electrical load angle; it is not a validated pull-out or pull-in characteristic. The voltage and knee outputs do not test full-current waveform feasibility at the entered load. A positive margin or a supply greater than the displayed alignment estimate cannot establish that feasibility. Full-step and half-step high-speed behavior may differ, even when microsteps per full step is set to 1 or 2.

Resonance: low-speed mechanical ringing and mid-band instability depend on the motor, load inertia, mounting, damping and driver control. No universal resonance band or invented torque notch is applied. A positive calculated margin does not establish stable operation. Verify the manufacturer's motor-driver torque curve and test the complete axis. Appropriate microstepping, damping, current-loop tuning and acceleration through an observed critical band can help; crossing faster still needs sufficient acceleration torque. Oriental Motor's Stepper Motor Basics and Trinamic AN-001 describe these dependencies.

Check phase-current waveforms at the maximum required speed. Keep the supply and transients within driver and motor limits. Copper loss is at the selected current with constant resistance, not a thermal prediction or a power-supply size. Instantaneous phase peaks and standstill duty must also satisfy the motor and driver ratings. This tool does not calculate regeneration, braking energy, stopping distance, temperature rise, shaft stress or transmission efficiency.