Understanding Torque Calculation in Electric Motors: A Technical Deep Dive

Torque calculation in electric motors is essential for performance and design optimization. It quantifies rotational force generated by the motor.

This article explores detailed formulas, variable definitions, common values, and real-world application cases for torque calculation.

• Calculate torque for a 3-phase induction motor with given power and speed.
• Determine torque output of a DC motor at specific current and magnetic flux.
• Find starting torque of a synchronous motor with known voltage and rotor parameters.
• Estimate torque in a brushless DC motor using back-EMF constant and current.

Comprehensive Tables of Common Torque Values and Parameters in Electric Motors

Fundamental Formulas for Torque Calculation in Electric Motors

Torque (T) in electric motors is the rotational force generated by the motor shaft, typically measured in Newton-meters (Nm). The calculation depends on motor type and available parameters.

1. Torque from Power and Angular Speed

The most general formula relates mechanical power output (P) and angular velocity (ω):

Where:

• T = Torque (Nm)
• P = Power output (Watts, W)
• ω = Angular velocity (radians per second, rad/s)

Angular velocity ω is related to rotational speed n (rpm) by:

Typical values:

• Power (P): ranges from a few watts in small motors to several megawatts in industrial motors.
• Speed (n): commonly 1500 or 3000 rpm for 50 Hz motors, 1800 or 3600 rpm for 60 Hz motors.

2. Torque in DC Motors Using Electromagnetic Parameters

For DC motors, torque can be calculated from the motor constant and armature current:

Where:

• T = Torque (Nm)
• k_t = Torque constant (Nm/A)
• I_a = Armature current (A)

Torque constant k_t depends on motor construction and magnetic flux. Typical values range from 0.05 to 5 Nm/A depending on motor size and type.

3. Torque in Synchronous Motors

For synchronous motors, torque can be derived from the power angle δ and motor parameters:

Where:

• T = Torque (Nm)
• V = Stator voltage (V)
• E = Internal generated voltage (V)
• ω = Angular velocity (rad/s)
• X_s = Synchronous reactance (Ω)
• δ = Power angle (radians)

This formula is critical for stability and control analysis in synchronous machines.

4. Torque in Induction Motors Using Slip and Rotor Parameters

Torque in an induction motor can be expressed as:

Where:

• T = Torque (Nm)
• V₁ = Stator voltage (V)
• R₂ = Rotor resistance (Ω)
• X₂ = Rotor reactance (Ω)
• s = Slip (unitless, between 0 and 1)
• ω = Angular velocity (rad/s)

Slip s is defined as:

Where:

• n_s = Synchronous speed (rpm)
• n = Rotor speed (rpm)

5. Torque in Brushless DC Motors (BLDC)

Torque in BLDC motors is often calculated using the back-EMF constant and current:

Where:

• T = Torque (Nm)
• k_t = Torque constant (Nm/A), often equal to back-EMF constant (V/(rad/s))
• I = Phase current (A)

Typical k_t values for BLDC motors range from 0.01 to 0.2 Nm/A depending on motor size and design.

Detailed Explanation of Variables and Their Typical Ranges

• Power (P): Mechanical output power, usually rated in kilowatts (kW). Small motors may have power as low as a few watts, while industrial motors can exceed several megawatts.
• Angular velocity (ω): Measured in radians per second, it is the rotational speed converted from rpm. For example, 1500 rpm corresponds to approximately 157 rad/s.
• Torque constant (k_t): Represents the relationship between current and torque in DC and BLDC motors. It depends on motor geometry and magnetic flux density.
• Armature current (I_a): The current flowing through the motor winding, directly influencing torque in DC motors.
• Slip (s): A dimensionless quantity representing the difference between synchronous speed and rotor speed in induction motors. Typical slip values range from 0.01 to 0.05 under normal operation.
• Rotor resistance (R₂) and reactance (X₂): Electrical parameters of the rotor circuit affecting torque production and efficiency.
• Power angle (δ): The angle between the stator voltage and rotor magnetic field in synchronous motors, critical for torque and stability.

Real-World Application Examples of Torque Calculation

Example 1: Calculating Torque of a 3-Phase Induction Motor

A 7.5 kW, 3-phase induction motor operates at 1450 rpm. Calculate the torque output.

Step 1: Convert speed to angular velocity:

Step 2: Use the power-torque relationship:

This matches typical rated torque values for this motor size, confirming the calculation.

Example 2: Torque Calculation in a DC Motor Using Current and Torque Constant

A DC motor has a torque constant k_t = 1.27 Nm/A. If the armature current is 10 A, find the torque output.

Step 1: Apply the torque formula:

This torque value can be used to assess motor capability under load conditions.

Additional Considerations for Accurate Torque Calculation

Several factors influence torque calculation accuracy in electric motors:

• Temperature Effects: Resistance values change with temperature, affecting torque in induction and DC motors.
• Magnetic Saturation: At high currents, magnetic materials saturate, reducing effective torque constant.
• Mechanical Losses: Friction and windage reduce net torque available at the shaft.
• Measurement Precision: Accurate current, voltage, and speed measurements are essential for reliable torque estimation.

References and Further Reading

• IEEE Transactions on Industrial Electronics – Torque Control in Electric Motors
• Engineering Toolbox – Electric Motor Torque Calculations
• Electrical4U – Torque in Electric Motors
• NEMA Standards for Electric Motors