Electric-Drive Models, Transfer Functions & Transients

The useful drive model is the smallest model that preserves the state and limit causing the behavior under study. A rigid-inertia model may size an acceleration; a resonance problem needs at least two inertias and shaft stiffness; inverter current saturation must be represented to predict recovery from a large command.

Mechanical balance #

Reduced to one shaft,

$$J\dot\omega=T_e-T_L-B\omega-T_c\mathop{\mathrm{sgn}}(\omega),$$

where $J$ is total reflected inertia, $T_e$ electromagnetic torque, $T_L$ load torque, $B$ viscous friction, and $T_c$ Coulomb friction. Every torque must use the same shaft and sign convention.

For constant net torque, acceleration time is

$$t=\frac{J(\omega_2-\omega_1)}{T_e-T_L}.$$

A 0.08 kg·m² system accelerating 0→150 rad/s with 18 N·m motor torque against 6 N·m load needs 1.0 s ideally. If the current limit permits only 10 N·m, it needs 3.0 s; a one-second speed ramp will saturate and lag.

Reflecting gears #

For speed ratio $n=\omega_m/\omega_L$, load inertia reflected to the motor is $J_L/n^2$ while load torque referred to the motor is approximately $T_L/n$ before efficiency. Mixing reflected inertia with unreferred torque is a common sizing error.

Electrical and converter states #

The DC motor model adds $L\dot i=V-Ri-k_e\omega$ and $T_e=k_ti$. AC-vector-control models add flux and rotating-frame current states. Converter delay, sampling, voltage saturation, current limit, dead time, and DC-bus dynamics can dominate a loop even when the motor equations are accurate.

Elastic mechanics #

Two inertias joined by stiffness $K_s$ and damping $C_s$ introduce torsional resonance. A speed sensor on the motor can report calm motion while the load oscillates. Backlash adds a state-dependent dead zone and impact when teeth re-engage; it cannot be represented by a constant gain.

Validation sequence #

Identify resistance and current-loop response with the shaft controlled; inertia/friction from torque and acceleration/coast-down; load torque across operating points; then resonances from frequency response or bounded excitation. Validate on data not used for fitting and include saturation tests. A model that fits one gentle speed step may still be useless for reversal, field weakening, or impact load.