# Motor constants

Motor constants are the proportionality factors that link a motor's electrical quantities to its mechanical ones: the torque constant relates current to torque, the speed and back-EMF constants relate speed to induced voltage, and the motor constant relates torque to the square root of electrical power loss. For a given motor they are fixed by design, though they carry manufacturing tolerances and shift with temperature.<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup>

| Key fact | Detail |
|---|---|
| Torque constant \( k_M \) | \( k_M = M/I \), the torque produced per unit current; a design parameter set by geometry, magnetic field density, and winding (Lorentz force)<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup> |
| Speed constant \( k_n \) | \( k_n = n/U_{\mathrm{ind}} \), used to compute ideal no-load speed for a given voltage, friction losses not considered<sup>[2](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)</sup> |
| Generator (back-EMF) constant \( k_G \) | \( k_G = 1/k_n = U_{\mathrm{ind}}/n \); identical to the torque constant, only in different units (Nm/A · π/30 = V/rpm)<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup> |
| SI equality | In SI units the torque constant and back-EMF constant are the same number: 1 N·m/A = 1 V·s<sup>[3](https://www.electronicsguide.net/foundations-and-theory/analog-electronics/feedback-and-control-systems/control-system-modeling-and-block-diagrams/)</sup> |
| Motor constant \( K \) | \( K = M/\sqrt{P_J} = k_M/\sqrt{R} = 1/\sqrt{\Delta n/\Delta M} \), torque per square root of electrical heat loss<sup>[2](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)</sup> |
| Speed/torque gradient \( \Delta n/\Delta M \) | \( \Delta n/\Delta M = (\pi/30000) \cdot R/k_M^2 \approx n_0/M_S \); a smaller value means a more powerful motor whose speed changes less with load<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup><sup> • </sup><sup>[2](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)</sup> |
| Tolerances and temperature | Constants have tolerances of up to about 10% and change with motor temperature; catalog values apply at 25 °C<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup> |

## Torque constant and back-EMF constant

The torque constant \( k_M \) gives the proportional relation between input current and output torque. It is a design parameter that includes the motor's geometry, the magnetic field density, and the winding; the underlying physics is the [Lorentz force](https://www.edgepedia.org/lorentz-force) on a current-carrying wire in a magnetic field. For coreless DC motors, torque and current are strictly proportional, which allows the motor to be used as a torque probe by simply measuring current. For motors with an iron core the proportionality still holds for realistic currents; only at extremely high currents, which can hardly ever be reached, would the produced torque be smaller due to saturation effects in the iron.<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup>

When the coil of a motor turns, the changing magnetic flux induces an emf that, by [Lenz's law](https://www.edgepedia.org/lenzs-law), opposes the driving voltage; this is the back emf. It is zero when the motor is not turning and increases proportionally to angular velocity, so a motor switched on draws its maximum current at standstill, and a loaded motor that slows down draws more current as its back emf falls.<sup>[4](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf)</sup> In a permanent-magnet [DC motor](https://www.edgepedia.org/dc-motor) the back emf is written \( E = K_E \omega \), where \( K_E \) is the back-EMF constant.<sup>[5](https://www.nidec.com/en/technology/motor/glossary/item/back-emf/)</sup>

**Why the two constants are equal.** In SI units \( K_t \) and \( K_e \) are the same number. The electrical power absorbed by the back emf is \( e_b i = K_e \omega i \), and the mechanical power developed is \( T_m \omega = K_t i \omega \); since the electromechanical coupling itself neither stores nor dissipates energy, the two powers are equal and \( K_t = K_e \). The units agree as well: 1 N·m/A = 1 J/A = 1 V·s, the radian being dimensionless.<sup>[3](https://www.electronicsguide.net/foundations-and-theory/analog-electronics/feedback-and-control-systems/control-system-modeling-and-block-diagrams/)</sup> A coil-based derivation reaches the same result: with \( N \) turns, \( e = 2RNBL\omega = K_E \omega \), and substituting \( K_T = 2RNBL \) gives \( K_T = K_E \). Nidec's motor text interprets this as the motor being a bidirectional energy converter between electricity and mechanics, with \( K_T \) the conversion coefficient in the electricity-to-machine direction and \( K_E \) in the machine-to-electricity direction.<sup>[6](https://www.nidec.com/en/technology/motor/basic/00013/)</sup>

## Speed constant and generator constant

The speed constant \( k_n \) [rpm/V] and the generator constant \( k_G \) [V/rpm] both describe the proportionality between motor speed and induced voltage (back EMF); the speed constant is the inverse of the generator constant. Its principal application is determining the ideal no-load speed produced by a particular applied voltage, with friction losses left out of the calculation. In maxon units the two constants are tied to the torque constant by \( k_n \cdot k_M = (30000/\pi) \cdot [\mathrm{min}^{-1}/\mathrm{V} \cdot \mathrm{mNm}/\mathrm{A}] = 1 \), and the back-EMF constant is identical to the torque constant given only in different units (Nm/A · π/30 = V/rpm).<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup><sup> • </sup><sup>[2](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)</sup>

## Motor constant and speed/torque gradient

The speed/torque gradient \( \Delta n/\Delta M \) [rpm/mNm] indicates how much speed is lost with increasing torque. A smaller value means a more powerful motor whose speed changes less under load variations. For most motors the gradient is constant and can be calculated as the quotient of ideal no-load speed and ideal stall torque; a hotter motor is weaker and the value increases. In maxon's formulation \( \Delta n/\Delta M = (\pi/30000) \cdot R/k_M^2 \approx n_0/M_S \).<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup><sup> • </sup><sup>[2](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)</sup>

The motor constant \( K \) [NmW<sup>-1/2</sup>] is the form more often found in the literature. It gives the amount of torque per square root of power loss, \( K = M/\sqrt{P_J} = k_M/\sqrt{R} = 1/\sqrt{\Delta n/\Delta M} \), so a high motor constant means a motor converts current to torque with little resistive heating. Ansys Motor-CAD defines it the same way, as the ratio between average torque and the square root of the conductor losses.<sup>[2](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)</sup><sup> • </sup><sup>[7](https://ansyshelp.ansys.com/public/Views/Secured/MotorCAD/v252/en/Motor-CAD_UG/MotorCAD/topics/drive_pmdc_output_data.html)</sup>

**Exceptions to the straight line.** For motors with cored windings (maxon flat, EC-i, frameless, and ECX TORQUE), the speed-torque line is not straight: the gradient is not constant and depends on speed, and in the continuous operating range it must be approximated by a formula rather than read off as a single number.<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup>

## Temperature and tolerances

Motor constants describe general behavior but are not fixed laboratory truths: they carry tolerances of up to about 10% and change with motor temperature, and catalog values apply at maxon standard conditions of 25 °C.<sup>[1](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)</sup> The temperature dependence is quantified for torque motors by Schaeffler: at a winding temperature of +130 °C the motor constant decreases to 0.84 times its normal value, and the constant is stated in the performance data for +20 °C and is accurate for the linear control range in a stationary state at low speeds.<sup>[8](https://www.schaeffler-industrial-drives.com/remotemedien/media/_shared_media/08_media_library/01_publications/schaeffler_2/catalogue_1/downloads_6/re1_de_en.pdf)</sup>

## Brushless motor conventions

For brushed DC motors the constants translate directly: the torque constant is \( 9.549/K_v \) N·m/A when \( K_v \) is given in rpm/V. For sinusoidally driven BLDC or PMSM motors the numbers depend on the current and voltage convention: \( K_t \) is 8.27/\( K_v \) N·m per peak phase ampere and 11.70/\( K_v \) N·m per RMS phase ampere, and the line-to-line peak back-EMF constant is \( 9.549/K_v \) V·s/rad. A 270 \( K_v \) motor turned at 1,000 rpm, for example, shows a line-to-line peak of about 3.70 V.<sup>[9](https://emdesignlabs.com/learn/kv-kt-ke-motor-constants/)</sup>

These conventions are a documented source of error. Microchip's motor-control guidelines note that datasheets rarely state whether the back-EMF constant is expressed line-to-line or line-to-neutral, or in RMS or zero-peak, and that the published torque constant usually omits whether it refers to six-step or sine-drive operation or how current is measured, so use of the published torque constant should in general be avoided without clarification from the manufacturer.<sup>[10](https://ww1.microchip.com/downloads/en/DeviceDoc/motorBench-mclv2-modifications-guidelines.2.285.pdf)</sup> In a brushless motor the back emf is an alternating voltage rather than a DC value, and it can be measured by back-driving the motor.<sup>[5](https://www.nidec.com/en/technology/motor/glossary/item/back-emf/)</sup>

## Measuring the constants

Standard identification tests separate the constants from the other motor parameters. A locked-rotor test, applying a voltage step with the rotor held, yields the stator resistance and inductance. A fixed-speed no-load test, spinning the motor on a dynamometer with no electrical load, permits estimation of the back-EMF constant, which in SI units equals the torque constant. A final controlled-motor test converts stator currents to torque using the identified torque constant and fits friction torque from the zero-speed intercept and the viscous damping coefficient from the slope; a speed run-down test then gives the motor inertia.<sup>[11](https://www.mathworks.com/help/simscape-electrical/ug/parameterize-a-pmsm-from-measurements.html)</sup> For a simple DC check, spinning the shaft at a known speed and measuring the open-circuit voltage gives the back-EMF constant directly as voltage divided by speed.<sup>[5](https://www.nidec.com/en/technology/motor/glossary/item/back-emf/)</sup>

## References

1. [Motor data and simulation, maxon Support](https://support.maxongroup.com/hc/en-us/articles/360013761160-Motor-data-and-simulation)
2. [Motor constants, maxon Knowledge Support Academy (PDF)](https://www.maxongroup.com/assets/public/caas/v1/media/219142/data/4cbfb8e3f8b30686be19b6672f127c4a/maxon-knowlege-support-academy-pdf-en-download.pdf)
3. [Control System Modeling and Block Diagrams, Electronics Guide](https://www.electronicsguide.net/foundations-and-theory/analog-electronics/feedback-and-control-systems/control-system-modeling-and-block-diagrams/)
4. [13.6 Electric Generators and Back Emf, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf)
5. [Back-emf, Counter electromotive force, Back-emf constant, Nidec Glossary of Motor Terms](https://www.nidec.com/en/technology/motor/glossary/item/back-emf/)
6. [2-1-3. Rotating Speed of DC Motor and Counter-electromotive Force, Nidec Basic Motor Information](https://www.nidec.com/en/technology/motor/basic/00013/)
7. [Drive [PMDC Output Data], Ansys Motor-CAD User Guide](https://ansyshelp.ansys.com/public/Views/Secured/MotorCAD/v252/en/Motor-CAD_UG/MotorCAD/topics/drive_pmdc_output_data.html)
8. [Torque motors catalog, Schaeffler Industrial Drives (PDF)](https://www.schaeffler-industrial-drives.com/remotemedien/media/_shared_media/08_media_library/01_publications/schaeffler_2/catalogue_1/downloads_6/re1_de_en.pdf)
9. [Kv, Kt and Ke Explained for DC and BLDC Motors, EM Design Labs](https://emdesignlabs.com/learn/kv-kt-ke-motor-constants/)
10. [MCLV-2 alternative I/V ratings with motorBench Development Suite, Microchip (PDF)](https://ww1.microchip.com/downloads/en/DeviceDoc/motorBench-mclv2-modifications-guidelines.2.285.pdf)
11. [PMSM Parameterization from Measurements, MATLAB & Simulink, MathWorks](https://www.mathworks.com/help/simscape-electrical/ug/parameterize-a-pmsm-from-measurements.html)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electromagnetic quantities and history › Electromagnetic quantities*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Sep 17, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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