T = KI
R
2
(1 – e
-t/τ
)
Where;
I
R
= Current, which when left to flow continuously, would produce a temperature T
max
, in the motor.
For an overload current 'I', the temperature is given by;
T = KI
2
(1 – e
-t/τ
)
For a motor not to exceed the rated temperature, the time 't' for which the motor can withstand the
current 'I' can be shown to be given by;
t = τLn [1/{1-(I
R
/I)
2
}]
An overload protection element should therefore satisfy the above relationship. The value of IR may
be the full load motor current or a percentage of it, depending on the motor design.
It is an oversimplification to regard a motor as a homogeneous body. The temperature rise of different
parts, or even of various points in the same part, can be very uneven. However, it is reasonable to
consider the current-time relationship follows an inverse fashion.
Thermal overload protection is designed to prevent the damage in the motors when the operating
temperature exceeds the maximum designed temperature.
2.3.1 Thermal Replica
Both the RMS and negative sequence currents are analysed to monitor the thermal state accounting
for any phase unbalance present. This thermal model takes in account the overheating, generated by
the negative phase sequence current in the rotor.
The equivalent motor heating current is calculated by:
I
eq
= √ (Irms² + (K Coefficient* I
2
²)
Where
I
rms: RMS current corresponding to the largest phase current.
I
2
: Negative phase sequence current.
K Coefficient is a constant proportional to the thermal capacity of the motor.
2.3.2 Thermal Trip
A multiple time constant thermal replica is used to account for different operating conditions of the
motor overload, starting or cooling conditions.
The equation used to calculate the trip time at 100% of thermal state is:
t = τ ln ((K²- A²)/ (K²- 1))
Where:
t: Time to trip (in seconds)
τ: Thermal time constant depending on the current value absorbed by the motor:
• Overload time constant τ= T
1 if IFL < Ieq ≤ 2xIFL
• Start-up time constant τ= T
2 if Ieq> 2xIFL
• Cooling time constant τ= T
r if interrupting device opened
K: Thermal overload equal to (Irms / k* IFL)
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