Kv, Kt and Ke: one motor constant in three units

Motor datasheets list a speed constant in rpm per volt, a back-EMF constant in volts per krpm, and a torque constant in mN·m or oz·in per amp. They look like three separate specifications. They are one number in different units, and the only real trap is which volts and which amps the number was measured in. This article shows why the constants are linked, how to convert them for brushed DC and BLDC motors, where the factors of √3, √2 and 1.5 come from, and how to use them to choose a winding for a robot joint. Two real datasheets are worked through along the way: the maxon RE 40 brushed DC motor and the ODrive D5065 270KV brushless motor.

One constant, three names

A motor converts electrical power into mechanical power. On the electrical side that power is the back-EMF E times the current I. On the mechanical side it is torque T times angular speed ω. Resistive heating, friction and iron loss are handled separately, so the conversion itself loses nothing:

Two proportionalities are enough. Back-EMF grows linearly with speed, E = K_eω, and torque grows linearly with current, T = K_tI. Substituting both gives K_eωI = K_tIω, so the speed and current cancel:

K_e and K_t are one constant seen from the electrical side and from the mechanical side. If a datasheet shows two numbers that disagree after unit conversion, the cause is a unit or a convention, not physics. The hobby unit for K_v is rpm per volt, which is the SI value multiplied by 60/(2π) = 9.549.

maxon calls these the speed constant k_n (rpm/V), the torque constant k_M (mN·m/A) and the generator constant k_G (V per 1000 rpm), and notes that k_n is essentially the inverse of k_M and k_G = 1/k_n [1]. The 9,549 product above is not a vendor number: it is 60/(2π) × 1000, the unit conversion that makes the SI equality hold.

Brushed DC motor: the simple case

A permanent-magnet brushed motor has one winding and one current, so there is no convention to choose. The armature voltage equation and the torque equation are

Eliminate the current and the motor becomes a straight line from no-load speed to stall torque:

This is where K_v gets its meaning: it is the no-load speed per volt, ignoring the small R·I₀ drop. The slope depends on R and K_t, not on the supply voltage, so raising the voltage slides the whole line to the right without changing its steepness. The quantity K_m = K_t/√R, the motor constant, sets that slope and is the number that says how much torque you get per square root of watt of copper loss.

A real datasheet: maxon RE 40 (part 148877)

The maxon RE 40 is a well-known 150 W graphite-brush motor of the kind used in robotics and laboratory equipment. Its datasheet [2] lists, at 48 V, a torque constant of 60.3 mN·m/A, a speed constant of 158 rpm/V, a terminal resistance of 1.13 Ω and a no-load current of 68.6 mA. Everything else in the table can be rebuilt from just those few numbers plus the supply voltage, which is a good way to check that you have understood them.

All eight rows agree to the rounding of the datasheet. Only three motor-specific numbers are doing the work: K_t (= K_e), R and I₀. The speed constant is not an independent specification; it is the torque constant read in other units, and the datasheet's 158 rpm/V and 60.3 mN·m/A multiply to 9,527, within 0.3 % of 9,549.

At the nominal point the motor draws 3.17 A and delivers 138 W from 152 W, an efficiency of 90.5 %. The best efficiency of 92 % occurs at only 1.71 A, or 99 mN·m, because the two losses are balanced when R·I² equals the no-load loss. A motor at its best efficiency is lightly loaded and fast, far from the 2.56 N·m stall point, which is why a small fast motor behind a gearbox often beats a large slow one.

Two caveats. First, the equality K_t = K_e describes a permanent-magnet motor; in a wound-field motor both constants scale with the field flux, so they change with field current. Second, brush voltage drop shifts the no-load speed slightly, so measure K_e by turning the motor as a generator on open circuit rather than from V/ω₀.

BLDC and PMSM: the same balance, with conventions

A three-phase machine obeys the same power balance. The difference is that there are three voltages and three currents, each of which can be quoted as peak or RMS, per phase or line-to-line. Every factor of √3, √2 or 1.5 in this section comes from that bookkeeping, not from new physics.

Let ψ be the peak permanent-magnet flux linkage of one phase, p the number of pole pairs and ω_m the mechanical speed. The phase back-EMF is e_a = −pω_mψ sin θ_e, the same expression used in the winding sequence article. Its peak is pψω_m, and the flux linkage itself comes from the winding: ψ = k_w·N·Φ₁, the winding factor times the series turns times the fundamental flux per pole.

Datasheets and ESCs quote the voltage between two terminals. For a star-connected winding the line-to-line peak back-EMF is √3 times the phase value:

The torque constant follows from the balanced three-phase power at the terminals. With current in phase with the back-EMF (the q-axis, i_d = 0), the power is √3 × V_LL,rms × I_rms. In peak quantities that is (√3/2) × E_LL,pk × I_pk. Setting it equal to Tω gives

Because the derivation uses terminal quantities, it holds for a delta-connected winding as well as a star. Measure at the terminals and the star/delta question disappears.

The last row applies to ideal six-step drive of a motor with flat-topped (trapezoidal) back-EMF: only two phases conduct, the voltage across them is the flat line-to-line value, and E_LL·I_dc = Tω gives K_t = K_e,LL just as in the DC motor. A motor with sinusoidal back-EMF driven by ideal 120° current blocks gets (3/π) × K_e,LL = 0.955 × K_e,LL per ampere of block amplitude, and torque ripple at six times the electrical frequency.

The ratio between the two most common quotes is 9.549/8.270 = 2/√3 = 1.155. A torque estimate made with 9.549/K_v for a controller that regulates peak phase current is therefore 15 % too high; one made with peak where RMS was meant is off by √2. FOC firmware that regulates i_q in amplitude-invariant form uses peak phase current, and a multimeter in AC mode reads RMS, so check the convention in your controller's documentation before trusting a number.

A real datasheet: ODrive D5065 270KV

ODrive's motor page [3] lists the D5065 270KV, a 14-pole (7 pole-pair) outrunner popular in hobby and research robot actuators. Its entries are: Kv 270 rpm/V, torque constant 0.031 N·m/A, phase resistance 39 mΩ and phase inductance 16 µH (both phase-to-neutral), 45 A continuous in free air and 85 A for 3 seconds. The footnotes matter as much as the numbers: "All voltage units are line-line amplitude" and "All current units are phase amplitude." That is exactly the line-to-line peak K_e and peak-phase-current K_t of the table above, so the published torque constant should equal 8.27/K_v. The page lists two sibling motors in the same format, which gives three checks:

The 8.27/K_v column matches all three published values to within their two-figure rounding. The 9.549/K_v column, the DC-motor rule applied to a brushless motor, overstates torque per amp by 14–16 %. This is the 2/√3 factor from the previous section appearing in a real datasheet.

Starting from just Kv = 270, p = 7, R_ph = 39 mΩ and the 45 A rating, you can derive the rest:

• K_e,LL = 9.549 / 270 = 0.0354 V·s/rad, which is 3.70 V line-to-line peak per 1,000 rpm.
• ψ = 0.0354 / (√3 × 7) = 2.92 mWb of flux linkage per phase.
• No-load speed is 6,480 rpm on a 24 V bus and 12,960 rpm on 48 V (before any load or field weakening).
• At 45 A continuous, torque is 1.40 N·m and copper loss is 1.5 × 45² × 0.039 = 118 W. At the 85 A peak the same constant would give 2.63 N·m and 423 W, which is consistent with a 3-second rating, and saturation, which lowers K_t at high current, would make the real figure lower.

The page itself cautions that all performance figures are approximate and depend on cooling, so treat these as first estimates.

Measuring K_e on the bench

The back-EMF test from the winding sequence article also gives K_v. You do not need a tachometer, because the electrical frequency on the scope already encodes the speed: f_e = p·n/60.

1. Disconnect the drive and turn the shaft at a steady speed, for example with a drill.
2. Read the electrical frequency f_e from the scope and compute n = 60·f_e/p, then ω_m = 2πn/60.
3. Read the line-to-line peak voltage between two terminals. A true-RMS meter gives the RMS value, which is the peak divided by √2 for a sine wave.
4. K_e,LL = V_pk/ω_m, then K_v = 9.549/K_e,LL and K_t,pk = 0.866 × K_e,LL.

This measures K_e cold and at open circuit. The torque constant under load is usually a little lower, for the reasons in the last section.

K_v is a winding choice, K_m is the motor

A common belief is that a high-K_v motor is the more powerful one. It is not. Put N turns in series per phase and everything scales with N: the flux linkage ψ, and therefore K_e and K_t, grow in proportion to N; the wire cross-section shrinks as 1/N for the same slot, so resistance grows as N². K_v falls as 1/N and resistance rises as 1/K_v². The combination that does not move is the motor constant,

K_v therefore says nothing about how good a motor is. The three ODrive motors above show it:

The highest-K_v motor has the lowest K_m; the lowest-K_v motor has the highest and is the larger machine. (These are three different motors, not rewinds of one, so their K_m values differ. Within one motor, K_m stays fixed as the turns change.) For the D5065, with K_t = 8.27/270 = 0.0306 N·m/A and R_ph = 39 mΩ (terminal-to-terminal 78 mΩ), K_m = 0.127 N·m/√W. The copper loss for any torque is the same whichever winding it carries. What changes is how that torque is paid for: with a low K_v you pay in volts, with a high K_v you pay in amps.

The same logic applies to a gearbox. A ratio N multiplies the torque at the joint and divides the speed, so a high-K_v motor behind a larger ratio can deliver the same joint torque and speed as a low-K_v motor behind a smaller one. Choosing the gear ratio and K_v together is how you move the current/voltage balance to what your battery and controller handle best. Gearbox efficiency is separate and is not included in K_t.

Why a measured K_t comes out lower than the datasheet

• Mechanical and drag losses. K_t defines the electromagnetic torque. A torque sensor on the shaft reads that torque minus bearing and brush friction, windage and the iron-loss drag, which together behave like a roughly constant loss torque. This is what the no-load current I₀ represents in the DC motor equations: T_shaft = K_t(I − I₀). The apparent constant is therefore K_t(1 − I₀/I), which is worst at low current. For the maxon RE 40 (I₀ = 68.6 mA) the apparent K_t is 2 % low at the 3.17 A nominal current and 23 % low at 0.3 A. If you measure at the output of a gearbox, its efficiency is a further loss, and cogging adds torque ripple around the average.
• Magnet temperature. NdFeB remanence falls by roughly 0.1 % per °C, so a motor 80 °C above its measurement temperature loses of the order of 10 % of K_e and K_t. A hot motor spins faster per volt and makes less torque per amp. See the NdFeB grade article for the grade-dependent behaviour.
• Saturation. At high current the stator teeth saturate and K_t drops, while open-circuit K_e is unaffected.
• Back-EMF shape. The constants above assume a sine (or ideal trapezoid). Harmonics in the back-EMF add ripple and change average torque per amp, and the same MMF harmonics that cause cogging and rotor loss are involved.
• Reluctance torque. In an interior-magnet machine, torque has a second term from L_d≠L_q, so a single K_t is no longer the whole story.
• Current definition. Peak, RMS and DC-link amps differ by √2 and √3 factors, as the table shows.

Checklist before you trust a number

• Is K_e line-to-line or phase, peak or RMS? Is K_t per peak phase, RMS phase or DC-link ampere?
• Is K_v in rpm/V or rad/s per V, and does it refer to the DC bus voltage?
• Do 9.549/K_v and K_e,LL agree? Do K_t,pk and 0.866 × K_e,LL agree? If not, find the convention that explains the gap.
• At what magnet temperature and current was K_t measured?
• Is the winding compared on K_m, not on K_v?

Frequently asked questions

Is the torque constant Kt equal to the back-EMF constant Ke?

In SI units, yes for a brushed DC motor: Kt in N·m/A equals Ke in V·s/rad because electrical power E·I equals mechanical power T·ω. For a three-phase brushless motor the equality holds only when volts and amps are counted consistently. With line-to-line peak Ke and peak phase current, Kt = (√3/2) × Ke, about 0.866 × Ke.

How do I convert Kv to Kt?

With Kv in rpm/V, the line-to-line peak back-EMF constant is 9.549/Kv in V·s/rad. The torque constant is 9.549/Kv N·m/A for a brushed DC motor, 8.27/Kv N·m per peak phase ampere for a sinusoidally driven BLDC or PMSM, and 11.70/Kv N·m per RMS phase ampere.

What does Kv mean for a BLDC motor?

Kv is the no-load speed in rpm per volt of DC bus voltage. It is the inverse of the line-to-line peak back-EMF constant, so a 270 Kv motor turned at 1,000 rpm shows a line-to-line peak of about 3.70 V.

Does a higher Kv motor have more power?

No. Kv is set by the number of turns in the winding. Rewinding with fewer turns raises Kv and lowers Kt and resistance, but leaves the motor constant Km unchanged, so the torque available for a given copper loss does not change. A higher Kv motor reaches a higher speed per volt and needs more current for the same torque.

Why does my measured Kt not match the datasheet?

Datasheet Kt is the cold, low-current, ideal electromagnetic value. Friction and drag losses (which a shaft torque sensor includes), magnet temperature, core saturation at high current, back-EMF harmonics and the current definition (peak, RMS or DC link) can each move it by several percent.

References

1. maxon group, Motor constants (support article defining speed, torque and generator constants).
2. maxon group, RE 40 Ø40 mm, graphite brushes, 150 watt, part 148877. Values quoted at 48 V nominal voltage, read 10 October 2026.
3. ODrive Robotics, ODrive Motors (D5065 270KV, D6374 150KV and D5312s 330KV tables and footnotes), read 10 October 2026.
4. D. C. Hanselman, Brushless Permanent-Magnet Motor Design, 2nd ed., The Writing Business, 2003.