Motor Torque and Back-EMF Calculator
Find a DC motor's back-EMF, speed, torque, and efficiency from supply voltage, armature current, armature resistance, and the torque constant.
⚙️ What is Motor Torque and Back-EMF?
Back-EMF (counter-electromotive force) is the voltage a spinning DC motor generates internally, opposing the supply voltage, given by the armature voltage equation E = V - I*Ra. Torque is the motor's rotational output, T = Kt*I, directly proportional to armature current through the motor's torque constant Kt.
Motor and drive engineers use these two equations together constantly, to predict a DC motor's operating speed and torque for a given supply and load, size motor drivers and current-limiting circuitry against the stall current, estimate efficiency across the operating range, and understand why motor current spikes at startup (zero back-EMF, maximum current draw) and falls as the motor spins up.
A common point of confusion is treating the torque constant Kt and the back-EMF constant Ke as separate quantities that need separate lookup. In consistent SI units they are numerically identical, a direct consequence of energy conservation, so this calculator uses one Kt input to compute both torque and speed.
This calculator computes back-EMF, rotational speed, torque, and efficiency from your supply voltage, armature current, armature resistance, and torque constant, and plots the classic straight-line torque-versus-speed characteristic that defines a permanent-magnet DC motor's operating range.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — 24 V Motor Under Moderate Load
V = 24 V, I = 5 A, Ra = 0.8 Ω, Kt = 0.1 N·m/A
Example 2 — Small 12 V Motor, Light Load
V = 12 V, I = 2 A, Ra = 0.5 Ω, Kt = 0.05 N·m/A
Example 3 — 48 V Industrial Motor, Heavier Load
V = 48 V, I = 10 A, Ra = 0.3 Ω, Kt = 0.2 N·m/A
❓ Frequently Asked Questions
🔗 Related Calculators
What is back-EMF in a DC motor?
Back-EMF (counter-electromotive force) is the voltage a spinning DC motor generates internally, opposing the supply voltage, exactly analogous to how a generator produces voltage from mechanical rotation. It grows with speed and is what naturally limits current draw once a motor is running.
What is the formula for back-EMF and torque?
The armature voltage equation gives back-EMF as E = V - I*Ra, where V is supply voltage, I is armature current, and Ra is armature resistance. Torque is T = Kt*I, where Kt is the motor's torque constant in newton-meters per amp.
Why are the torque constant and back-EMF constant the same number?
In SI (consistent) units, Kt (N*m/A) and Ke (V per rad/s) are numerically identical for the same motor, a direct consequence of energy conservation: electrical power converted to torque times speed must equal back-EMF times current. This calculator uses a single Kt value for both.
How do I find motor speed from back-EMF?
Speed in radians per second is omega = E / Kt, where E is the back-EMF and Kt is the motor constant. Multiply by 60/(2*pi) to convert to RPM, the more commonly quoted unit on motor datasheets.
What is the torque-speed curve of a DC motor?
For a fixed supply voltage, a permanent-magnet DC motor's torque-speed relationship is a straight line: maximum torque occurs at zero speed (stall, where all voltage drops across armature resistance), and torque falls linearly to zero at the no-load speed, where back-EMF nearly equals the supply voltage.
How is motor efficiency calculated here?
Efficiency is mechanical output power (back-EMF times current) divided by electrical input power (supply voltage times current), expressed as a percentage. This captures armature I-squared-R resistive losses but not additional losses like friction, windage, or core losses that a real motor also has.
What happens at the stall condition?
At stall, speed is zero, so back-EMF is zero, and the entire supply voltage drops across the armature resistance: the stall current is V/Ra, the maximum current (and torque) the motor can draw. This calculator flags an input error if you enter a current that would require exceeding this physical limit.
Does this calculator account for load torque directly?
No, it works from armature current, which is set by whatever load torque the motor is driving at steady state. If you know the load torque instead, divide it by Kt to get the corresponding armature current, then use that as this calculator's current input.
What is a typical torque constant value?
Small hobby DC motors often have Kt in the range of 0.01 to 0.05 N*m/A, while larger industrial DC motors can range from 0.1 up to several N*m/A depending on size and winding design. Always use the manufacturer's datasheet value for accurate results.
What units does this calculator use?
Supply voltage in volts, armature current in amps, armature resistance in ohms, and the torque constant in newton-meters per amp (numerically the same as volts per radian-per-second). Results are given in volts, RPM, and newton-meters.