# Motor constants

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{{Short description|Constants describing characteristics of electrical motors}}
The '''motor size constant''' (<math>K_\text{M}</math>) and '''motor velocity constant''' (<math>K_\text{v}</math>, alternatively called the '''[back EMF](/source/Counter-electromotive_force) constant''') are values used to describe characteristics of electrical motors.

== Motor constant==

<math>K_\text{M}</math> is the ''motor constant''<ref>{{Cite web| url=http://www.motioncomp.com/pdfs/Motor_Constant_Great_Equalizer.pdf | title=MCG distance-based training - application note#4 | access-date=2014-01-04 | archive-date=2021-04-13 | archive-url=https://web.archive.org/web/20210413125847/http://www.motioncomp.com/pdfs/Motor_Constant_Great_Equalizer.pdf | url-status=dead}}</ref> (sometimes, ''motor size constant''). In [SI units](/source/International_System_of_Units), the motor constant is expressed in [newton metre](/source/newton_metre)s per square root [watt](/source/watt) (<math>\text{N}{}\cdot{}\text{m} / \sqrt{\text{W}}</math>):
: <math>K_\text{M} = \frac{\tau}{\sqrt{P}}</math>
 	
where
* <math>\scriptstyle \tau</math> is the motor [torque](/source/torque) ([SI unit](/source/International_System_of_Units): newton–metre)	
* <math>\scriptstyle P</math> is the [resistive power loss](/source/Joule_heating) ([SI unit](/source/International_System_of_Units): watt)
 	
The motor constant is winding independent (as long as the same conductive material is used for wires); e.g., winding a motor with 6 turns with 2 parallel wires instead of 12 turns single wire will double the velocity constant, <math>K_\text{v}</math>, but <math>K_\text{M}</math> remains unchanged. <math>K_\text{M}</math> can be used for selecting the size of a motor to use in an application. <math>K_\text{v}</math> can be used for selecting the winding to use in the motor.

Since the torque <math>\tau</math> is current <math>I</math> multiplied by <math>K_\text{T}</math> then <math>K_\text{M}</math> becomes 
: <math>K_\text{M} = \frac{K_\text{T} I}{\sqrt{P}} = \frac{K_\text{T} I }{\sqrt{I^2 R}} = \frac{K_\text{T}}{\sqrt{R}}</math>

where
* <math>I</math> is the [current](/source/Electric_current) ([SI unit](/source/International_System_of_Units), ampere)	
* <math>R</math> is the [resistance](/source/Electrical_resistance_and_conductance) ([SI unit](/source/International_System_of_Units), ohm)
* <math>K_\text{T}</math> is the motor torque constant ([SI unit](/source/International_System_of_Units), newton–metre per ampere, N·m/A), see below

If two motors with the same <math>K_\text{v}</math> and torque work in tandem, with rigidly connected shafts, the <math>K_\text{v}</math> of the system is still the same assuming a parallel electrical connection. The <math>K_\text{M}</math> of the combined system increased by <math>\sqrt{2}</math>, because both the torque and the losses double. Alternatively, the system could run at the same torque as before, with torque and current split equally across the two motors, which halves the resistive losses.

== Units ==
The motor constant may be provided in one of several units. The table below provides conversions between common SI units
{| class="wikitable sortable"
|+
| colspan="2" rowspan="2" |Conversion Factor
! colspan="6" |Ending Unit
|-
|<math>k_t, \frac{Nm}{A_{pk}}  </math>
|<math>k_t, \frac{Nm}{A_{RMS}} </math>
|<math>k_v, \frac{V_{LL,RMS}}{\frac{rad}{s}} </math>
|<math>k_v, \frac{V_{LL,pk}}{\frac{rad}{s}} </math>
|<math>k_v, \frac{V_{LL,pk}}{rpm}  </math>
|<math>k_v, \frac{V_{LL,RMS}}{rpm}  </math>
|-
! rowspan="6" |Starting Unit
|<math>k_t, \frac{Nm}{A_{pk}}  </math>
!'''<math>1 </math>'''
!<math>\sqrt2 </math>
!<math>\frac{1}{\sqrt2} </math>
!'''<math>1 </math>'''
!<math>\frac{\pi}{30} </math>
!<math>\frac{\pi}{30\sqrt2} </math>
|-
|<math>k_t, \frac{Nm}{A_{RMS}} </math>
!<math>\frac{1}{\sqrt2} </math>
!'''<math>1 </math>'''
!<math>\frac{1}{2} </math>
!<math>\frac{1}{\sqrt2} </math>
!<math>\frac{\pi}{30\sqrt2} </math>
!<math>\frac{\pi}{30} </math>
|-
|<math>k_v, \frac{V_{LL,RMS}}{\frac{rad}{s}} </math>
!<math>\sqrt2 </math>
!<math>2 </math>
!'''<math>1 </math>'''
!<math>\sqrt2 </math>
!<math>\frac{\pi\sqrt2}{30} </math>
!<math>\frac{\pi}{30} </math>
|-
|<math>k_v, \frac{V_{LL,pk}}{\frac{rad}{s}} </math>
!'''<math>1 </math>'''
!<math>\sqrt2 </math>
!<math>\frac{1}{\sqrt2} </math>
!'''<math>1 </math>'''
!<math>\frac{\pi}{30} </math>
!<math>\frac{\pi}{30\sqrt2} </math>
|-
|<math>k_v, \frac{V_{LL,pk}}{rpm}  </math>
!<math>\frac{30}{\pi} </math>
!<math>\frac{30\sqrt2}{\pi} </math>
!<math>\frac{30}{\pi\sqrt2} </math>
!<math>\frac{30}{\pi} </math>
!'''<math>1 </math>'''
!<math>\frac{1}{\sqrt2} </math>
|-
|<math>k_v, \frac{V_{LL,RMS}}{rpm}  </math>
!<math>\frac{30\sqrt2}{\pi} </math>
!<math>\frac{60}{\pi} </math>
!<math>\frac{30}{\pi} </math>
!<math>\frac{30\sqrt2}{\pi} </math>
!<math>\sqrt2 </math>
!'''<math>1 </math>'''
|}
<math>V_{Line-Neutral} = \frac{V_{LL}}{\sqrt3}</math><ref>{{Cite web |title=More Basics of Three-Phase AC Sinusoidal Voltages |url=https://blog.teledynelecroy.com/2018/03/more-basics-of-three-phase-ac.html |access-date=2025-12-02 |language=en}}</ref>

== Motor velocity constant, back EMF constant ==

<math>K_\text{v}</math> is the motor velocity, or motor speed,<ref name="kk"/> constant (not to be confused with kV, the symbol for ''kilovolt''), measured in [revolutions per minute](/source/revolutions_per_minute) (RPM) per volt or radians per volt second, rad/V·s:<ref>{{Cite web|url=http://learningrc.com/motor-kv/|title = Brushless Motor Kv Constant Explained • LearningRC|date = 29 July 2015}}</ref>
: <math>K_\text{v} = \frac{\omega_\text{no-load}}{V_\text{peak}}</math>

The <math>K_\text{v}</math> rating of a [brushless motor](/source/brushless_motor) is the ratio of the motor's unloaded [rotational speed](/source/rotational_speed) (measured in RPM) to the peak (not RMS) voltage on the wires connected to the coils (the ''[back EMF](/source/Counter-electromotive_force)''). For example, an unloaded motor of {{nowrap|<math>K_\text{v}</math> {{=}} 5,700 rpm/V}} supplied with 11.1&nbsp;V will run at a nominal speed of 63,270&nbsp;rpm (= 5,700&nbsp;rpm/V × 11.1&nbsp;V).

The motor may not reach this theoretical speed because there are non-linear mechanical losses. On the other hand, if the motor is driven as a generator, the no-load voltage between terminals is perfectly proportional to the RPM and true to the <math>K_\text{v}</math> of the motor/generator.

The terms <math>K_\text{e}</math>,<ref name="kk">{{citation| url = http://hades.mech.northwestern.edu/images/6/61/Asst7.pdf| title = Mystery Motor Data Sheet| work = hades.mech.northwest.edu}}</ref> <math>K_\text{b}</math> are also used,<ref>{{citation| url =http://www.smma.org/pdf/SMMA_motor_glossary.pdf|title = GENERAL MOTOR TERMINOLOGY| work = www.smma.org}}</ref> as are the terms ''back EMF constant'',<ref>{{citation| url = http://www.mathworks.co.uk/help/toolbox/physmod/elec/ref/dcmotor.html|title =DC motor model with electrical and torque characteristics - Simulink| work =www.mathworks.co.uk}}</ref><ref>{{citation| url = http://www.micro-drives.com/motor-calculations.aspx| title = Technical Library > DC Motors Tutorials > Motor Calculations| work = www.micro-drives.com| url-status = dead| archiveurl = https://web.archive.org/web/20120404160332/http://www.micro-drives.com/motor-calculations.aspx| archivedate = 2012-04-04}}</ref> or the generic ''electrical constant''.<ref name="kk"/> In contrast to <math>K_\text{v}</math> the value <math>K_\text{e}</math> is often expressed in SI units volt–seconds per [radian](/source/radian) (V⋅s/rad), thus it is an inverse measure of <math>K_v</math>.<ref>{{cite web |url=http://www.precisionmicrodrives.com/tech-blog/2014/02/02/reading-the-motor-constants-from-typical-performance-characteristics |title=Home |website=www.precisionmicrodrives.com |url-status=dead |archive-url=https://web.archive.org/web/20141028075543/http://www.precisionmicrodrives.com/tech-blog/2014/02/02/reading-the-motor-constants-from-typical-performance-characteristics |archive-date=2014-10-28}} </ref> Sometimes it is expressed in non SI units volts per kilorevolution per minute (V/krpm).<ref>{{Cite web| title=General motor terminology | url=http://www.smma.org/pdf/SMMA_motor_glossary.pdf | archive-url=https://web.archive.org/web/20140502084952/http://www.smma.org/pdf/SMMA_motor_glossary.pdf | archive-date=2014-05-02}}</ref>
: <math>K_\text{e} = K_\text{b} = \frac{V_\text{peak}}{\omega_\text{no-load}} = \frac{1}{K_\text{v}}</math>

The field flux may also be integrated into the formula:<ref>{{citation |title=DC motor starting and braking |url=http://iitd.vlab.co.in/?sub=67&brch=185&sim=470&cnt=1 |work=iitd.vlab.co.in |archive-url=https://web.archive.org/web/20121113123938/http://iitd.vlab.co.in/?sub=67&brch=185&sim=470&cnt=1 |archive-date=2012-11-13 |access-date=2012-03-23 |url-status=live }}</ref>
: <math>K_\omega = \frac{E_\text{b}}{\phi\omega}</math>

where <math>E_\text{b}</math> is back EMF, <math>K_\omega</math> is the constant, <math>\phi</math> is the [flux](/source/magnetic_flux), and <math>\omega</math> is the [angular velocity](/source/angular_velocity).

By [Lenz's law](/source/Lenz's_law), a running motor generates a back-EMF proportional to the speed. Once the motor's rotational velocity is such that the back-EMF is equal to the battery voltage (also called DC line voltage), the motor reaches its limit speed.

== Motor torque constant ==
<math>K_\text{T}</math> is the torque produced divided by armature current.<ref>{{citation| url = http://electronics.stackexchange.com/questions/33315/understanding-motor-constants-kt-and-kemf-for-comparing-brushless-dc-motors| title = Understanding motor constants Kt and Kemf for comparing brushless DC motors }}</ref> It can be calculated from the motor velocity constant <math>K_\text{v}</math>. For a single coil the relationship is:
: <math>
K_\text{T} = \frac{\tau}{I_\text{a}} = \frac{60}{2\pi K_\text{v(RPM)}} = \frac{1}{K_\text{v(SI)}} 
</math>

where <math>I_\text{a}</math> is the [armature](/source/Armature_(electrical_engineering)) current of the machine (SI unit: [ampere](/source/ampere)). <math>K_\text{T}</math> is primarily used to calculate the armature current for a given torque demand: 
: <math>
I_\text{a}  = \frac{\tau}{K_\text{T}} 
</math>

The SI units for the torque constant are newton meters per ampere (N·m/A). Since 1&nbsp;N·m = 1&nbsp;J, and 1&nbsp;A = 1&nbsp;C/s, then 1&nbsp;N·m/A = 1&nbsp;J·s/C = 1&nbsp;V·s (same units as back EMF constant).

The relationship between <math>K_\text{T}</math> and <math>K_\text{v}</math> is not intuitive, to the point that many people simply assert that torque and <math>K_\text{v}</math> are not related at all. An analogy with a hypothetical [linear motor](/source/linear_motor) can help to convince that it is true. Suppose that a linear motor has a <math>K_\text{v}</math> of 2&nbsp;(m/s)/V, that is, the [linear actuator](/source/linear_actuator) generates one volt of back-EMF when moved (or driven) at a rate of 2&nbsp;m/s. Conversely, <math>s = VK_\text{v}</math> (<math>s</math> is speed of the linear motor, <math>V</math> is voltage).

The useful power of this linear motor is <math>P = VI</math>, <math>P</math> being the power, <math>V</math> the useful voltage (applied voltage minus back-EMF voltage), and <math>I</math> the current. But, since power is also equal to force multiplied by speed, the force <math>F</math> of the linear motor is <math>F = P/(VK_\text{v})</math> or <math>F = I/K_\text{v}</math>. The inverse relationship between force per unit current and <math>K_\text{v}</math> of a linear motor has been demonstrated.

To translate this model to a rotating motor, one can simply attribute an arbitrary diameter to the motor armature e.g. 2&nbsp;m and assume for simplicity that all force is applied at the outer perimeter of the rotor, giving 1&nbsp;m of leverage.

Now, supposing that <math>K_\text{v}</math> (angular speed per unit voltage) of the motor is 3600&nbsp;rpm/V, it can be translated to "linear" by multiplying by 2π&nbsp;m (the perimeter of the rotor) and dividing by 60, since angular speed is per minute. This is linear <math>K_\text{v} \approx 377\ (\text{m} / \text{s}) / \text{V}</math>.

Now, if this motor is fed with current of 2&nbsp;A and assuming that back-EMF is exactly 2&nbsp;V, it is rotating at 7200&nbsp;rpm and the mechanical power is 4&nbsp;W, and the force on rotor is <small><math>
\frac{P}{V * K_\text{v(SI)}}=\frac{4}{2 * 377} 
</math></small>&nbsp;N or 0.0053&nbsp;N. The torque on shaft is 0.0053&nbsp;N⋅m at 2&nbsp;A because of the assumed radius of the rotor (exactly 1&nbsp;m). Assuming a different radius would change the linear <math>K_\text{v}</math> but would not change the final torque result. To check the result, remember that <math>P = \tau\, 2\pi\, \omega / 60</math>.

So, a motor with <math>K_\text{v} = 3600\text{ rpm} / \text{V} = 377\text{ rad} / \text{V·s}</math> will generate 0.00265&nbsp;N⋅m of torque per ampere of current, regardless of its size or other characteristics. This is exactly the value estimated by the <math>K_\text{T}</math> formula stated earlier.
{| class="wikitable"
|+<big>EXAMPLE: Torque applied at different diameters</big>, <small><math>K_\text{v (rpm/V)}</math>= 3600 rpm/V ≈ 377 rad/s/V , <math>K_\text{T}</math> ≈ 0.00265 N.m/A (each calculatable if one is known)</small>,

<small><u>V = 2 v, <math>I_\text{a}</math>= 2 A, P = 4 W , (any 2 makes the 3rd, <math>
P = VI 
</math></u>)</small>
!diameter = 2r
!r = 0.5 m
!r = 1 m
!r = 2 m
!Formula (<math>K_\text{v(rpm/V)}</math>)
!Formula (<math>K_\text{v(rad/s/V)}</math>)
!Formula (<math>K_\text{T}</math>)
!shorthand
|-
!<math>\tau</math> = motor torque (N.m/s)
|0.005305 N·m
|0.005305 N·m
|0.005305 N·m
|<math>
\frac{30I }{\pi K_\text{v(rpm/V)}} 
</math>
|<math>
\frac{I}{K_\text{v(rad/s/V)}} 
</math>
|<math>K_\text{T(N.m/A)}*I</math>
|<math>
K_\text{T} * I  
</math>
|-
!linear <math>K_\text{v}</math> (m/s/V) @ diameter
|188.5 (m/s)/V
|377.0 (m/s)/V
|754.0 (m/s)/V
|<math>
\frac{\pi r K_\text{v(rpm/V)}}{30} 
</math>
|<math>
r K_\text{v(rad/s/V)} 
</math>
|<math>
\frac{r}{K_\text{T(N.m/A)}} 
</math>
|<math>K_\text{v} * r </math>
|-
!linear <math>K_\text{T}</math> (N.m/A) @ diameter
|0.005305 N·m/A
|0.002653 N·m/A
|0.001326 N·m/A
|<math>
\frac{30}{\pi r K_\text{v(rpm/V)}} 
</math>
|<math>
\frac{ 1 }{ r K_\text{v(rad/s/V)}} 
</math>
|<math>
\frac{K_\text{T(N.m/A)}}{ r } 
</math>
|<math>K_\text{t} / r</math>
|-
!speed m/s @ diameter
(linear speed)
|377.0&nbsp;m/s
|754.0&nbsp;m/s
|1508.0&nbsp;m/s
|<math>
\frac{ \pi r V K_\text {v(rpm/V)}}{30} 
</math>
|<math>
V r K_\text{v(rad/s/V)} 
</math>
|<math>
\frac{rV}{K_\text{T(N.m/A)}} 
</math>
|linear <math>K_\text{v} * V = K_\text{v} * V r </math>
|-
!speed km/h @ diameter
(linear speed)
|1357&nbsp;km/h
|2714&nbsp;km/h
|5429&nbsp;km/h
|<math>
\frac{3\pi r V K_\text {v(rpm/V)}}{25} 
</math>
|<math>
3.6 Vr K_\text{v(rad/s/V)} 
</math>
|<math>
\frac{3.6 r V}{K_\text{T(N.m/A)}} 
</math>
|linear <math>K_\text{v} * V * \frac{3600}{1000}</math> 
|-
!torque (N.m) @ diameter
(linear torque)
|0.01061 N·m
|0.005305 N·m
|0.002653 N·m
|<math>
 \frac{30I }{\pi r K_\text{v(rpm/V)}}  
</math>
|<math>
\frac{I}{rK_\text{v(rad/s/V)} } 
</math>
|<math>\frac{K_\text{T} * I }{ r } </math>
|<math>\frac{\tau}{r} </math>
|-
!shorthand
|half diameter = half speed

<nowiki>*</nowiki> double torque
|full diameter = full speed

<nowiki>*</nowiki> full torque
|double diameter = double speed

<nowiki>*</nowiki> half torque
|<math>
K_\text{v(rad/s/V)} = \frac{2 \pi K_\text{v(rpm/V)} }{60} 
</math><math>
K_\text{T(N.m/A)} = \frac{60}{2 \pi K_\text{v(rpm/V)}} 
</math>
|<math>
K_\text{v(rpm/V)} = \frac{60 K_\text{v(rad/s/V)}}{2 \pi} 
</math><math>
K_\text{T(N.m/A)} = \frac{ 1 }{ K_\text{v(rad/s/V)}} 
</math>
|<math>
K_\text{v(rpm/V)} = \frac{60}{ 2 \pi K_\text{T(N.m/A)}} 
</math><math>
K_\text{v(rad/s/V)} = \frac{1}{K_\text{T(N.m/A)}} 
</math>
|<math>1 = 
{\displaystyle K_{\text{v (rad/s/V)}}}
 * 
{\displaystyle K_{\text{T (N.m/A)}}}</math><math>1 = linear 
{\displaystyle K_{\text{v(m/s/V)}}}
 * linear 
{\displaystyle K_{\text{T (N.m/A)}}}</math>
|}

==References==
{{reflist}}

==External links==
*{{citation| url = http://biosystems.okstate.edu/home/mstone/4353/downloads/Developement%20of%20Electromotive%20Force.pdf| title = Development of Electromotive Force| work = biosystems.okstate.edu| url-status = dead| archiveurl = https://web.archive.org/web/20100604201111/http://biosystems.okstate.edu/Home/mstone/4353/downloads/Developement%20of%20Electromotive%20Force.pdf| archivedate = 2010-06-04}}

Category:Electric motors

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Adapted from the Wikipedia article [Motor constants](https://en.wikipedia.org/wiki/Motor_constants) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Motor_constants?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.