Servo Motor Torque-Speed Curve Calculator

The fundamental linear torque-speed relationship comes from balancing the motor's voltage equation. The supply voltage (V) is split between overcoming the back-EMF ($K_e \omega$) and the resistive drop in the armature ($I_a R_a$).

Since motor torque is proportional to armature current ($T = K_t I_a$), we can substitute to get the core equation plotted by the simulator:

Where:
$T$ = Output Torque (Nm)
$\omega$ = Angular Speed (rad/s)
$K_t$ = Torque Constant (Nm/A)
$K_e$ = Back-EMF Constant (Vs/rad)
$R_a$ = Armature Resistance (Ω)
$V$ = Supply Voltage (V)
Note: In SI units, $K_t$ and $K_e$ are numerically equal for an ideal motor.

From torque and speed, we calculate mechanical power and efficiency. Peak mechanical power occurs at half the no-load speed and half the stall torque.

Where:
$P_{mech}$ = Mechanical Output Power (W)
$\eta$ = Efficiency (0 to 1)
$P_{elec}$ = Electrical Input Power = $V I_a$
The efficiency curve shows that motors are most efficient in the mid-speed, mid-torque region, not at the extremes.

Robotic Arm Actuation: Engineers use these curves to select a servo motor strong enough to lift a payload (high stall torque) and fast enough for the required cycle time (high no-load speed). The "continuous operating region" is kept within the curve to prevent overheating.

CNC Machine Tool Drives: In a milling machine, the servo must maintain constant torque under varying cutting loads to ensure a smooth surface finish. A low Ra (stiff curve) is critical here, which is why high-performance servos use low-resistance windings and cooling.

Conveyor Belt Systems: The motor must accelerate the belt and load from rest (requiring high starting torque) and then run at a constant speed. The torque-speed curve helps size the motor to handle the inertial load (modeled by the Rotor Inertia, J, parameter) without stalling.

Electric Vehicle Window Lift: This is a classic case of a dynamic load. The motor must overcome static friction to start moving the window (stall torque), then run at speed. The curve predicts if the motor will stall when the window is ice-sealed, a key safety and reliability consideration.

When starting with this simulator, there are several pitfalls that beginners to CAE often encounter. First and foremost is the assumption that "the torque constant Kt and the back EMF constant Ke are the same value, just with different units". While they are numerically equal in an ideal motor, real-world motors can show differences due to design or magnetic saturation effects. If both values are listed on the datasheet, use them. If only Ke is available, a practical approach is to start by assuming "Ke ≒ Kt".

Next is the tendency to underestimate the armature resistance Ra. This value fluctuates significantly with temperature. The datasheet value is typically for room temperature (25°C). During actual continuous operation, the coil heats up, and it's not uncommon for the resistance to increase by a factor of 1.5. For example, if you input Ra=1Ω to find the peak efficiency, in a real motor the efficiency peak may shift once it heats up, potentially reducing output. When considering thermal management, also run simulations using the resistance value at your expected maximum operating temperature.

Finally, a fundamental understanding: this tool shows the characteristics of "the motor alone in a steady state". In an actual system, factors like gearbox efficiency, moment of inertia, and the driver's current limit can drastically alter the curve. For instance, attaching a 10:1 gearbox to a motor with a stall torque of 1Nm will yield only 8Nm at the output shaft if the gearbox efficiency is 80%. To evaluate this "system-level characteristic", the next step is modeling that includes the gearbox and load inertia.