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Motor Characteristics Simulator

Servo Motor Torque-Speed Curve Calculator

Adjust Kt, Ke, Ra, and supply voltage V with sliders to compute torque-speed curves, output power, and efficiency in real time. Instantly find stall torque, no-load speed, and peak efficiency point.

Motor Parameters
Kt — Torque constant (Nm/A) 0.10
Ke — Back-EMF constant (Vs/rad) 0.10
Ra — Armature resistance (Ω) 1.0
V — Supply voltage (V) 24
J — Rotor inertia (kg·m²) 0.010
Results
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Stall Torque (Nm)
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No-load Speed (rpm)
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Peak Power (W)
—
Peak Efficiency (%)

Motor Equations

$T = \dfrac{K_t}{R_a}(V - K_e\omega)$

$P_{mech}= T\omega,\quad \eta = \dfrac{T\omega}{VI_a}$

Engineering note: Servo drives use PWM to vary effective voltage and shift the torque-speed characteristic. Lower Ra makes the curve steeper (stiffer motor).
Torque-Speed / Power / Efficiency
Motor Cross-Section (Animation)

What is a Servo Motor Torque-Speed Curve?

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What exactly is a torque-speed curve, and why is it so important for servo motors?
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Basically, it's the motor's performance map. It shows the maximum torque the motor can produce at any given speed before it stalls. For a servo motor, this curve defines its working envelope—how fast it can move a load and how much force it can exert. In this simulator, the curve is drawn in real-time based on the motor's electrical parameters.
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Wait, really? So the curve is just a straight line? What do the sliders for Kt, Ke, and Ra actually do to it?
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In an ideal DC motor model, yes, it's a straight line! Try moving the **Armature Resistance (Ra)** slider. You'll see a higher Ra makes the line slope down more steeply—the motor loses torque quickly as speed increases. A lower Ra gives a "stiffer" motor that holds torque better. The **Torque Constant (Kt)** slider scales the whole curve up or down.
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That makes sense. But what about the "no-load speed" and "stall torque" points on the graph? How are they calculated?
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Great observation! Those are the key boundaries. The stall torque (left y-intercept) is when speed is zero: $T_{stall}= (K_t V)/R_a$. The no-load speed (right x-intercept) is when torque is zero: $\omega_0 = V / K_e$. In the simulator, adjust the **Supply Voltage (V)**—you'll see both points shift, scaling the entire curve. This is how real servo drives use PWM to control speed!

Physical Model & Key Equations

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 ($I_a R_a$).

$$V = K_e \omega + I_a R_a$$

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:

$$T = \frac{K_t}{R_a}(V - K_e \omega)$$

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.

$$P_{mech}= T \omega$$ $$\eta = \frac{P_{mech}}{P_{elec}}= \frac{T \omega}{V I_a}= \frac{T \omega}{V (T/K_t)}$$

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.

Real-World Applications

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.

Common Misconceptions and Points to Note

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.

Related Engineering Fields

Calculating this torque-speed curve is merely the entry point to understanding DC servo motors. Unraveling this further reveals direct connections to crucial engineering fields like control engineering, thermal-fluid analysis (thermal CAE), and mechanical dynamics.

First, control engineering. This linear characteristic is essentially the "plant model" for speed or position control. For example, when designing a proportional-integral (PI) controller, the slope of this curve (the sensitivity of torque to speed change) directly influences the system's gain design. Furthermore, the current limit value defines the motor's output and is incorporated into the model as a saturation nonlinearity in the control system.

Next, thermal-fluid analysis (thermal CAE). The loss $P_{loss} = I_a^2 R_a$ all converts to heat. Based on this heat generation, you can simulate the thermal resistance of the motor casing or the performance of heat sinks to check if permissible temperatures are exceeded. Often, it is this thermal limit that actually determines the continuous operation region.

Furthermore, mechanical dynamics (multibody dynamics). Motors are used as "joints" in robot arms or conveyor systems. Here, the load's moment of inertia J demands acceleration torque $T_{acc} = J \frac{d\omega}{dt}$, moving the operating point along the torque-speed curve instant by instant. To evaluate dynamic motion, transient response analysis combining this inertia with the curve is essential.

For Further Learning

Once you've gained an intuitive understanding with this tool, the next step is to "get your hands on the equations". First, try recreating the derived torque-speed equation $$T = \frac{K_t}{R_a}(V - K_e \omega)$$ in Excel or Python (NumPy). By comparing how the curve changes when you modify parameters with the simulator's results, the meaning of the equation will become second nature.

The next recommended step is modeling the "transient state". Add the armature inductance La and the load's moment of inertia J, which were absent in the steady-state equation. This allows you to set up a system of differential equations representing the "time series" from when voltage is applied until speed rises. For example: $$\begin{cases} V = R_a I_a + L_a \frac{dI_a}{dt} + K_e \omega \\ T = K_t I_a = J \frac{d\omega}{dt} + T_{load} \end{cases}$$ Solving these simultaneous equations enables you to evaluate acceleration time and current surge.

Ultimately, integrate this knowledge and challenge yourself with system simulation that unifies the motor, driver, controller, and mechanical load. Powerful tools for this include MATLAB/Simulink, Python's SimPy, or physical modeling languages like Modelica. Mastering this allows you to move beyond simple component selection into the realm of "optimal design" that considers performance, thermal management, and responsiveness altogether.