Motor Protection Functions: Thermal, Stall and Unbalance Protection Explained

Motor protection functions are the set of relay algorithms — thermal overload, stall/locked-rotor, and current unbalance (negative-sequence) protection — that detect abnormal operating conditions in an induction or synchronous motor and issue a trip before winding insulation, rotor bars, or bearings sustain permanent damage. Unlike simple overcurrent protection on a feeder, motor protection must track the motor’s own thermal behaviour, starting dynamics, and sensitivity to supply asymmetry, because a motor’s failure modes are driven by heat accumulated in the stator and rotor rather than by fault current alone. This guide walks through the underlying formulas for each function, provides a fully worked numerical example for each, and summarizes selection criteria that engineers specifying protection relays for medium-voltage motors across the Middle East and Gulf region (including Egypt and Saudi Arabia), North Africa, CIS countries, and Sub-Saharan Africa should apply when coordinating relay settings with motor thermal withstand curves.

Motor protection relays typically bundle these three functions together with earth fault, differential and undervoltage elements, but the thermal, stall, and unbalance functions are the ones that require an engineering calculation tied to the motor’s own nameplate and thermal-withstand data — they cannot be set from generic feeder overcurrent rules of thumb. For background on the switchgear that houses motor starters and circuit-breakers, see our medium-voltage switchgear hub; for the relays themselves, see our protection relay hub; and for upstream transformer sizing that determines available fault current at the motor terminals, see our distribution transformer guide.

The method: three formulas, three failure modes

1. Thermal overload (ANSI 49). The relay runs a thermal replica of the motor, integrating I²t against a cooling/heating time constant τ. The classic IEC-type formula for trip time is:

t = τ × ln[(I/I_B)² − (I_p/I_B)²] / [(I/I_B)² − 1]

where I is the actual current, I_B is the thermal reference (pickup) current, I_p is the pre-load current before the overload started, and τ is the motor’s thermal time constant. This is the standard “thermal replica” characteristic referenced by IEC 60255-8 / IEC 60255-149 thermal electrical relay standards.

2. Stall / locked-rotor protection (ANSI 48/51LR). A motor’s rotor and stator can withstand locked-rotor current only for a limited time, defined by the manufacturer’s hot and cold locked-rotor withstand curves. Stall protection compares actual current against a pickup set above normal running current but below the full starting inrush, combined with a timer shorter than the locked-rotor withstand time:

t_trip < t_withstand(I_actual), with pickup set between normal full-load current and starting current

3. Current unbalance / negative-sequence protection (ANSI 46). Any voltage asymmetry on the supply produces a negative-sequence current component I2 in the motor. Because the rotor’s impedance to negative-sequence current (which appears to the rotor as double supply frequency) is far lower than its positive-sequence impedance, a given percentage of negative-sequence current causes disproportionately more heating than the same percentage of positive-sequence overload. The negative-sequence current is obtained from the classic symmetrical-component transform:

I2 = (Ia + a²Ib + aIc) / 3, where a = 1∠120°

and a simplified NEMA-style unbalance index commonly used for field checks is:

% Unbalance = (max deviation from average phase current / average phase current) × 100

Worked example 1: Thermal overload trip time

Consider a medium-voltage induction motor rated 315 kW, 6.6 kV, with efficiency 0.95 and power factor 0.87. The rated full-load current is:

I_rated = P / (√3 × V × cosφ × η) = 315,000 / (1.732 × 6,600 × 0.87 × 0.95) ≈ 33.3 A

The thermal element is set with pickup I_th (I_B) = 1.05 × I_rated ≈ 35 A, and the motor’s heating time constant τ = 25 minutes. Assume the motor was cold (I_p = 0) when an overload of 50 A (≈1.5× rated) develops.

(I/I_B)² = (50/35)² = 2.041

t = 25 × ln(2.041 − 0) / (2.041 − 1) = 25 × ln(2.041)/1.041 = 25 × 0.713/1.041 ≈ 17.1 minutes

So the relay allows about 17 minutes at 1.5× rated current before tripping on the thermal replica — a result any protection engineer can verify by re-running the formula with the site’s actual τ and pickup values from the motor datasheet.

Worked example 2: Stall/locked-rotor timer setting

Using the same 33.3 A motor, assume a locked-rotor (starting) current multiplier of 6× rated, so I_start ≈ 6 × 33.3 ≈ 200 A, and a normal accelerating (start) time of 8 seconds to reach running speed. The motor’s hot locked-rotor thermal withstand time at 6× current is 12 seconds (from the manufacturer’s thermal curve).

The stall element pickup is set at 0.9 × I_start ≈ 180 A (roughly 5.4× rated current) so that it stays below normal running current after start-up but remains active during a genuine stall. The associated timer is set to:

t_stall = t_start + margin = 8 s + 2 s = 10 s

Since 10 s < 12 s (the hot withstand time), the relay will always trip before the rotor reaches its thermal limit if the motor fails to accelerate and stays at locked-rotor current beyond the expected start duration — while still allowing a normal 8-second start to complete without a nuisance trip.

Worked example 3: Current unbalance and derating

Field current readings on the same motor show Ia = 36 A, Ib = 33 A, Ic = 30 A. The average is 33 A, and the maximum deviation from average is 3 A, giving:

% Unbalance = (3/33) × 100 ≈ 9.1%

Because negative-sequence heating effects are widely approximated (as an illustrative engineering rule, not a fixed standard constant) as roughly double the equivalent positive-sequence heating for the same per-unit current, the effective additional thermal loading factor can be estimated as:

Additional heating factor ≈ 2 × (I2/I1)² ≈ 2 × (0.091)² ≈ 0.0166, or roughly a 1.7% increase in effective thermal loading

While this specific 9.1% unbalance looks moderate, it illustrates why unbalance protection (ANSI 46) is set on a percentage-current-unbalance basis with an inverse-time characteristic, tripping faster as unbalance increases, rather than on a single instantaneous threshold — a small unbalance sustained continuously can still shorten motor life through cumulative rotor heating.

Comparison table: motor protection functions at a glance

Selection criteria and common pitfalls

Selecting and commissioning these functions correctly depends on matching relay settings to the specific motor, not to generic defaults:

• Use the motor’s own thermal time constant and locked-rotor withstand curve. These values differ across motor designs and frame sizes; copying settings from a similar-looking motor is a frequent source of nuisance trips or, worse, under-protection.
• Distinguish “hot” and “cold” locked-rotor withstand times. A motor that has just tripped on overload and is restarted has less thermal margin than one starting from ambient — stall protection timers should respect the hot curve, which is always shorter.
• Coordinate the thermal element with the starting current profile. If the thermal pickup is set too close to rated current without accounting for the normal start transient, the relay may trip during a legitimate start rather than only during an actual stall or overload.
• Set unbalance protection on percentage current unbalance, not on absolute amperes, since motors of different ratings tolerate different absolute unbalance currents but similar percentage unbalance for a given design class.
• Check upstream supply quality. Voltage unbalance from an unevenly loaded distribution transformer or a single-phase-heavy feeder can trigger the 46 function even when the motor itself is healthy — always verify supply voltage balance before assuming a motor or relay fault.
• Re-verify settings after motor rewinds or re-ratings. A rewound motor may have a different thermal time constant than the original nameplate assumed by the relay.

Standards

The thermal replica characteristic used in ANSI 49 elements follows the IEC 60255-8 (superseded by IEC 60255-149) thermal electrical relay standards. Effects of supply voltage unbalance on induction motor performance and derating are addressed in IEC 60034-26, while general motor rating and performance definitions follow IEC 60034-1. IEEE C37.96, the Guide for AC Motor Protection, provides complementary application guidance widely referenced alongside these IEC documents, particularly for stall and unbalance coordination practices.

Related guides

See also our guides on gas-insulated switchgear, ring main units and metal-clad switchgear.