Servo Motor Torque & Speed

This calculator finds the operating point of a brushed permanent-magnet DC servo motor on the linear motor model. From the supply voltage $V$, the datasheet no-load speed $n_{nl}$ and the winding resistance $R_a$ it derives the torque constant and the stall torque, then solves two operating points: one at the operating current $I$ you measure (torque, speed, back-EMF, input and output power, efficiency) and one at the load torque $T_L$ your mechanism demands (the current, speed and output power the motor settles at). The same model describes the small DC motor inside a hobby RC servo and a brushed industrial servo motor. The servo's position loop, gearbox and PWM drive are outside it, so enter values at the motor shaft.

The supply voltage divides between the winding resistance and the back-EMF, $V = I R_a + K_e \omega$, and the torque is proportional to the current, $T = K_t I$. In SI units the torque constant in N·m/A equals the back-EMF constant in V·s/rad. With the no-load current neglected, the back-EMF equals the supply voltage at no load, so $K_t = K_e = V/\omega_{nl}$, where $\omega_{nl} = 2\pi n_{nl}/60$. At current $I$ the speed is $\omega = (V - I R_a)/K_e$ and the output power is the back-EMF times the current, $P_{out} = (V - I R_a) I = T\omega$; the rest of $P_{in} = VI$ is the copper loss $I^2 R_a$. The efficiency, $1 - I R_a/V$, is therefore the operating speed as a fraction of the no-load speed. At zero speed there is no back-EMF, the current is $V/R_a$ and the torque is the stall torque $T_{stall} = K_t V/R_a$. Every operating point lies on the straight torque-speed line $\omega = \omega_{nl}(1 - T/T_{stall})$, so under a load torque $T_L$ the motor draws $I_L = T_L/K_t$ and runs at $n_L = n_{nl}(1 - T_L/T_{stall})$. A load torque above the stall torque is reported as a stall: zero speed at the stall current.

The model's main simplification is the no-load current $I_0$, the current the motor draws with nothing on its shaft to supply its friction, brush and iron losses. The calculator neglects it, so the torque it reports is the electromagnetic torque and the shaft delivers about $K_t (I - I_0)$. Efficiency is overstated most at light load, where $I_0$ is a large share of $I$, and $K_e = V/\omega_{nl}$ comes out slightly high, because some of the supply voltage drops across $R_a$ even at no load. For a motor whose no-load current is a few percent of its operating current, these errors are a few percent. The winding resistance rises about 0.39% per °C, so a hot motor has less stall torque and loses more speed under load than a cold one. Brush voltage drop, winding inductance, magnetic saturation at high current and the driver's current limit are not modelled.