Industrial Electric Drives & Rotating Machinery, Kloss Asynchronous Motor Torque-Slip Law
Kloss Induction Motor Torque-Slip & Breakdown Formula
Calculate Synchronous Rotating-Field Speed Ns (RPM), Rotor Slip s (%), Kloss Electromagnetic Shaft Torque T(s) (N·m), and Direct-On-Line (s = 1) Locked-Rotor Starting Torque (N·m).
Push a squirrel-cage induction motor past its 14% breakdown slip knee and rising slip REDUCES torque—snapping the shaft straight to locked-rotor stall!
In a **Three-Phase Squirrel-Cage Asynchronous Induction Motor** (`Grid Frequency f` Hz, `Stator Pole Count P` -> **Synchronous Rotating Field Speed `N_s = (120 * f) / P`** RPM), rotor bars must turn slightly slower (`Shaft Speed N` RPM) at **Fractional Slip `s = (N_s - N) / N_s`** to cut stator flux lines and induce rotor current! Neglecting minor stator resistance, **Max Kloss’s (`1916`) Canonical Torque-Slip Equation** expresses electromagnetic torque at *any* slip `s` purely from **Pull-Out / Breakdown Torque `T_{max}`** (`N·m`) and **Breakdown Slip `s_m = R_{2,ref} / X_{eq}`**: **`T(s) = (2 * T_{max}) / ((s / s_m) + (s_m / s))`** (`N·m`)! For small operating slips (`s << s_m`), `T ~= 2 * T_{max} * (s / s_m)` rises linearly like a stiff DC shunt motor; at `s = s_m`, torque peaks at `T_{max}`; beyond `s > s_m` (the unstable front slope!), rotor leakage reactance chokes power factor so `T(s)` falls hyperbolically down to **Locked-Rotor Starting Torque (`s = 1`): `T_{start} = (2 * T_{max}) / ((1 / s_m) + s_m)`**!
Verify on a `4`-Pole, `60` Hz NEMA Design B Industrial Pump Motor—Grid_Frequency `f` = 60.0 Hz (B2), Stator_Poles `P` = 4 (B3 -> `N_s = 1,800` RPM in E2), Operating_Speed `N` = 1,755 RPM (B4), Breakdown_Slip `s_m` = 0.14 (`14%`, B5), and Pull-Out_Torque `T_{max}` = 310.0 N·m (B6): Operating Slip equals `2.50%` (E3), Operating Shaft Torque equals `107.3` N·m (`19.8` kW! E4), and Direct-On-Line (`s = 1`) Locked-Rotor Starting Torque equals `85.1` N·m (E5, showing why NEMA deep-bar rotors use skin effect at `60` Hz rotor frequency to boost starting `R_2`!)!
| Kloss Induction Motor KPI (60 Hz, 4-Pole -> Ns=1,800 RPM, sm=0.14, Tmax=310 N·m) | Formula | Verified Output |
|---|---|---|
| Synchronous Field Speed Ns (RPM) | (120 * Grid_Freq_Hz) / Stator_Poles_P | 1,800 RPM |
| Operating Rotor Slip s (%) | ((Ns_RPM - Shaft_N_RPM) / Ns_RPM) * 100 | 2.50% rated (17.00% > 14% UNSTABLE!) |
| Kloss Shaft Torque T(s) (N·m) | (2 * Tmax_Nm) / ((s / sm) + (sm / s)) | 107.3 N·m (304.2 N·m collapsing past knee!) |
| Locked-Rotor Torque @ s = 1 (N·m) | (2 * Tmax_Nm) / ((1 / sm) + sm) | 85.1 N·m (classical constant-R2 floor) |
Max Kloss (1916) Dimensionless Symmetrical Slip-Ratio Torque Formula
Validate `Frequency f` in `[10, 400]` Hz, even integer `Poles P` in `[2, 24]`, motoring speed `0 <= N < N_s`, `Breakdown_Slip s_m` in `[0.03, 0.80]`, and `Pull-Out Torque T_{max} > 1` N·m!
=LET(fg_h, B2, pc_n, B3, nr_p, B4, sm_b, B5, tm_n, B6, ns_r, (120 * fg_h) / pc_n, IF(OR(fg_h < 10, fg_h > 400, pc_n < 2, pc_n > 24, MOD(pc_n, 2) <> 0, nr_p < 0, nr_p >= ns_r, sm_b < 0.03, sm_b > 0.8, tm_n <= 1), "Invalid Kloss Input", ROUND((2 * tm_n) / ((1 / sm_b) + sm_b), 1)))Mechanical Jam / Voltage Sag Past Breakdown Knee (`N = 1,494` RPM -> `s = 17% > 14%`) Stall Check
Drag shaft speed down to `B4 = 1,494` RPM (`Slip s = 17.00%` in E3, past `s_m = 14%`!): Torque E4 has ALREADY peaked at `310` N·m (`1,548` RPM) and is now **falling** (`304.2` N·m)—any constant-torque crusher/conveyor load above `304.2` N·m decelerates uncontrollably to `0` RPM locked-rotor stall within fractions of a second!
=IF(ISTEXT(E5), E5, LET(ns_r, (120 * B2) / B3, sl_f, (ns_r - B4) / ns_r, ROUND((2 * B6) / ((sl_f / B5) + (B5 / sl_f)), 1)))Try the example
Use the original values or change them to test your understanding.