Robotics, UAV Propulsion & Electric Drivetrain Engineering, BLDC Motor Kv & Kt Physics
BLDC Motor Kv Constant, Torque Kt & Shaft Power Formula
Calculate Exact SI Torque Constant Kt = 60 / (2*PI()*Kv) (N·m/A), Electromagnetic Torque (N·m), Resistive Voltage-Drop Loaded Speed (RPM), and Mechanical Shaft Power (kW).
Higher Kv NEVER means "more powerful"—by Lorentz-Faraday reciprocity, doubling BLDC Kv cuts torque per amp Kt in HALF!
Drone builders, e-foil designers, and robotics engineers select **Permanent-Magnet Brushless DC (`BLDC`) Motors** by **Velocity Constant `K_v`** (`RPM / Volt` unloaded), yet routinely guess torque! By exact SI conservation of electrical and mechanical power (`E_{bemf} * I = tau * omega`), **Torque Constant `K_t`** (`N·m/A`) is strictly the reciprocal of `K_v` expressed in `rad/(s·V)`: **`K_t = 60 / (2 * PI() * K_v) ~= 9.5493 / K_v`** (`N·m/A`)! Driven from DC Bus Voltage `V_{bus}` (`V`) across Winding Resistance `R_{ll}` (`Ohm`) at Motor Current `I` (`A`), **Electromagnetic Torque `tau = K_t * I`** (`N·m`) comes at the expense of Ohmic `I * R_{ll}` voltage drop, pulling **Loaded Shaft Speed down linearly along the torque-speed line `N = K_v * (V_{bus} - I * R_{ll})`** (`RPM`) and delivering **Mechanical Shaft Power `P_{mech} = (tau * N * 2 * PI() / 60) / 1,000`** (`kW`)!
Verify on a `420` Kv `48` V Heavy-Lift Cargo Drone / Robotics Actuator Outrunner—Motor_Velocity_Constant `K_v` = 420 RPM/V (B2 -> unloaded `20,160` RPM), DC_Bus_Voltage `V_{bus}` = 48.0 V (B3), Terminal_Resistance `R_{ll}` = 0.065 Ohm (B4), and Cruise_Current `I` = 55.0 A (B5): Exact SI Torque Constant `K_t` equals `0.02274` N·m/A (`22.74` mN·m/A, E2), Cruise Torque equals `1.25` N·m (E3), Loaded Speed drops to `18,658.5` RPM (`3.575` V `I*R` drop, E4), delivering `2.44` kW mechanical shaft power (E5)!
| BLDC Drivetrain KPI (420 Kv, 48 V Bus, R=0.065 Ohm: 55 A Cruise vs 160 A Climb Burst) | Formula | Verified Output |
|---|---|---|
| SI Torque Constant Kt (N·m/A) | 60 / (2 * PI() * Motor_Kv_RPM_V) | 0.02274 N·m/A (exact 9.5493 / Kv!) |
| Electromagnetic Torque (N·m) | Kt_Nm_A * Motor_Current_A | 1.25 N·m (3.64 N·m @ 160 A burst) |
| Loaded Shaft Speed N (RPM) | Motor_Kv * (Bus_V - Current_A * Resist_Ohm) | 18,658.5 RPM (15,792.0 RPM @ 160 A droop!) |
| Mechanical Shaft Power (kW) | (Torque_Nm * Loaded_RPM * 2*PI()/60) / 1000 | 2.44 kW (6.02 kW + 1.66 kW I²R copper heat!) |
Lorentz-Faraday BLDC Kv-to-Kt Reciprocity, Speed Droop & Shaft Power Formula
Validate `Motor K_v` in `[10, 15,000]` RPM/V, `Bus_Voltage V_{bus}` in `[3, 1,000]` V, `Resistance R_{ll} > 0` Ohm, and `Current I >= 0` below locked-rotor stall (`I * R_{ll} < V_{bus}`)!
=LET(kv_r, B2, vb_v, B3, rl_o, B4, im_a, B5, kt_n, 60 / (2 * PI() * kv_r), tq_n, kt_n * im_a, nr_p, kv_r * (vb_v - im_a * rl_o), IF(OR(kv_r < 10, kv_r > 15000, vb_v < 3, vb_v > 1000, rl_o <= 0, im_a < 0, im_a * rl_o >= vb_v), "Invalid BLDC Input", ROUND((tq_n * nr_p * 2 * PI() / 60) / 1000, 2)))160 A Full-Throttle Climb Burst Copper Droop Check (Set Current B5 = 160 A)
Command `160` A peak ESC phase current (`B5 = 160`): Torque E3 triples to `3.64` N·m and Mechanical Output E5 hits `6.02` kW, YET resistive drop (`160 * 0.065 = 10.4` V!) drags Loaded Speed E4 down `21.7%` below unloaded RPM to **`15,792.0` RPM** while dumping `I^2 * R = 1,664` W of copper heat straight into the stator windings!
=IF(ISTEXT(E5), E5, ROUND(B2 * (B3 - B5 * B4), 1))Try the example
Use the original values or change them to test your understanding.