# DC Motor Back-EMF & Torque Calculator (E=kω, T=kI)

Compute the back-EMF and electromagnetic torque of a DC motor from its machine constant: E=k·ω, T=k·I. In SI units the back-EMF constant equals the torque constant (k_e=k_t=k, V·s/rad=N·m/A). Also returns the mechanical (air-gap) power P=T·ω=E·I. Speed accepted in rad/s or RPM.

> Canonical page: https://elysiatools.com/en/tools/dc-motor-back-emf

- **Category:** Math & Numbers

- **Keywords:** DC motor, back-EMF, back EMF, counter-EMF, torque constant, k_e, k_t, machine constant, armature current, angular velocity, RPM, electromechanical, motor

## Overview

Calculate a DC motor's back-EMF, electromagnetic torque, and mechanical power from the motor constant, angular speed, and armature current. Enter speed in rad/s or RPM and receive the results in JSON format.

## Inputs

- **Motor Constant k (V·s/rad = N·m/A)** (number): Machine constant k. In SI units k_e (V·s/rad) = k_t (N·m/A) = k.
- **Angular Speed ω** (number): Rotor angular speed in the selected unit (rad/s or RPM).
- **Speed Unit** (select)
- **Armature Current I (A)** (number): Armature current I (A). May be 0 (then torque and power are 0).
- **Decimal Places** (number)

## When to use

- Estimate back-EMF from a motor's machine constant and angular speed.
- Calculate electromagnetic torque from the motor constant and armature current.
- Check mechanical power using the relationship P = T·ω = E·I.

## How it works

- Enter the motor constant k in V·s/rad or N·m/A.
- Enter angular speed in rad/s or select RPM and enter the rotational speed.
- Enter armature current in amperes and choose the number of decimal places.
- The calculator converts RPM to rad/s when needed, then computes E = k·ω, T = k·I, and P = T·ω.

## Use cases

- Motor design checks involving back-EMF and torque constants.
- Electrical and mechanical calculations for DC motor experiments.
- Quick verification of electromechanical power relationships.

## Frequently asked questions

### What does the motor constant k represent?

In SI units, k is both the back-EMF constant and torque constant: k_e = k_t = k.

### Can I enter speed in RPM?

Yes. Select RPM, and the calculator converts the value to rad/s for the calculations.

### What formulas does the calculator use?

It uses E = k·ω for back-EMF, T = k·I for torque, and P = T·ω = E·I for mechanical power.

### What units are used for the results?

Back-EMF is returned in volts, torque in N·m, speed in rad/s and RPM, and mechanical power in watts.

### Can the armature current be zero?

Yes. A zero current produces zero torque and zero mechanical power, while back-EMF still depends on speed and k.

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