# Motor constants

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The constants KM (motor size constant) and Kv (motor velocity constant, or the back EMF constant) are values used to describe characteristics of electrical motors.

## Motor constant

KM is the motor constant (sometimes, motor size constant). In SI units, the motor constant is expressed in (Nm/sqrt(W)):

$K_{\mathrm {M} }={\frac {\tau }{\sqrt {P}}}$ where

• $\tau$ is the motor torque (SI units, N·m)
• $P$ is the resistive power loss (SI units, W)

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, Kv, but KM remains unchanged. KM can be used for selecting the size of a motor to use in an application. Kv can be used for selecting the winding to use in the motor.

Since the torque $\tau$ is current I multiplied by KT then Km becomes

$K_{M}={\frac {K_{T}\cdot I}{\sqrt {P}}}={\frac {K_{T}\cdot I}{\sqrt {I^{2}\cdot R}}}={\frac {K_{T}}{\sqrt {R}}}$ where

• $I$ is the current (SI units, A)
• $R$ is the resistance (SI units, Ohm)
• $K_{T}$ is the Motor Torque Constant (SI units, N·m/A), see below

If two motors with the same Kv and torque work in tandem, with rigidly connected shafts, the Kv of the system is still the same assuming a parallel electrical connection. The KM of the combined system increased by ${\sqrt {2}}$ , 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.

## Motor velocity constant, back EMF constant

Kv is the motor velocity constant (not to be confused with kV, the abbreviation for kilovolt), measured in RPM per volt or radians per volt-second (rad/V-s): $K_{\mathrm {v} }={\frac {\omega _{\mathrm {No-Load} }}{V_{\mathrm {Peak} }}}$ The Kv rating of a brushless motor is the ratio of the motor's unloaded rotational speed (measured in RPM) to the peak (not RMS) voltage on the wires connected to the coils (the back EMF). For example, an unloaded motor of Kv 5,700 rpm/V supplied with 11.1 V will run at a nominal speed of 63,270 rpm (5,700 rpm/V × 11.1 V).

Actually, 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 Kv of the motor/generator.

The terms Ke, Kb are also used, as are the terms back EMF constant. or the generic electrical constant. In contrast to KV the value Ke is often expressed in SI units ${\frac {V\cdot s}{rad}}$ , thus it is an inverse measure of KV. Sometimes it is expressed in non SI units V/krpm.

The field flux may also be integrated into the formula:

$E_{b}=K_{\omega }\phi \omega$ where $E_{b}$ is back EMF, $K_{\omega }$ is the constant, $\phi$ is the flux, and $\omega$ is the angular velocity

An inverse measure is also sometimes used, which may be referred to as the speed constant.

By 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

KT is the torque produced divided by armature current. It can be calculated from the motor velocity constant Kv.

$K_{\mathrm {T} }={\frac {\tau }{I_{A}}}={\frac {60}{2\pi \cdot K_{V}}}$ where $I_{A}$ is the armature current of the machine (SI units A). KT is primarily used to calculate the armature current for a given torque demand:

${I_{A}}={\frac {\tau }{K_{\mathrm {T} }}}$ The SI units for the torque constant are newton-metres per ampere (N·m/A). Since N·m = J, and A = C/s, then N·m/A = J·s/C = V·s (same units as back EMF constant).

The relationship between Kt and Kv is not intuitive, to the point that many people simply assert that torque and Kv are not related at all. An analogy with an hypothetical linear motor can help to convince that it is true. Suppose that a linear motor has a Kv of 2 m/s per volt, that is, the linear actuator generates one volt of back-EMF when moved (or driven) at a rate of 2 m/s. Conversely, s=V·Kv (s is speed of the linear motor, V is voltage).

The useful power of this linear motor is P=V·I, P being the power, V the useful voltage (applied voltage minus back-EMF voltage), and I the current. But, since power is also equal to force multiplied by speed, the force F of the linear motor is F=P/(V·Kv) or F=I/Kv. The inverse relationship between force-per-amp and Kv 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 meters and assume for simplicity that all force is applied at the outer perimeter of the rotor, giving 1 meter of leverage.

Now, supposing that Kv (angular speed per volt) of the motor is 3600, it can be translated to "linear" by multiplying by 2π meters (the perimeter of the rotor) and dividing by 60, since angular speed is per minute. This is about 377 Kv linear (m/s per volt).

Now, if this motor is fed with current of 2 A and assuming that back-EMF is exactly 2 V, it is rotating at 7200 RPM and the mechanical power is 4 W, and the force on rotor is 4/2.377 or 0.0053 N. The torque on shaft is 0.0053 N·m at 2 A because of the assumed radius of the rotor (exactly 1 meter). Assuming a different radius would change the linear Kv but would not change the final torque result. To check the result, remember that power = torque·2π·angular speed/60.

So, a motor with 3600 Kv will generate 0.00265 N·m of torque per ampere of current, regardless of its size or other characteristics. This is exactly the value estimated by the KT formula stated earlier.