# i2t protection

The instantaneous energy dissipated by a motor is proportional to the square of the current circulating through it and to the time this current lasts circulating through it. The nominal current is the current that a motor can stand in a continuous manner without exceeding its thermal limits. Therefore, any current above the nominal one creates an accumulation of thermal energy in the motor's surroundings that the cooling systems will have to dissipate. If this process of accumulating thermal energy exceeds the cooling system's ability to dissipate it, the system is bound to reach its thermal limits, and permanent damage to the motor or the surrounding elements can be inflicted. The so-called *i2t *is an indirect magnitude proportional to the energy dissipated by the motor, and the *i2t* protection is a control mechanism aimed to ensure that the integral of the power dissipated by the motor in the form of thermal energy does not exceed its thermal limits.

The energy dissipated by a motor is defined as:

\[ E_{system}=P· t=i_{RMS}^2· R· t\] |

Where \( i_{RMS}\) is the RMS current flowing through the motor and R is its resistance.

The nominal current that can flow through the motor is determined by the power it can dissipate continuously without exceeding its thermal limits.

\[ E_{system-nom}=P· t=i_{RMS-nom}^2· R· t\] |

In a transient peak, the motor could tolerate an excess of energy with respect to the continuous limit.

The excess of energy could be expressed as follows:

\[ E_{trans}=i_{RMS-peak}^2· R· t-i_{RMS-nom}^2· R· t\] |

Most times we do know the current rating system, the peak current and the maximum duration of the peak. Therefore, the equation could be simplified as follows:

\[ E_{excess}=\frac{E_{trans}}{R}=(i_{RMS-peak}^2-i_{RMS-nom}^2)· t\] |

This excess of energy is called *I-squared-t* (\(
i^2t\)), and it is expressed in amper^{2} per second.

\[ i^2t=(i_{RMS-peak}^2-i_{RMS-nom}^2)· t_{peak}\] |

The following picture shows a graphical representation of the \(
i^2t\) limit algorithm implemented in the controller. In the left side of the graph, the system is working with its nominal current. Under this situation, the system could be working infinite time. Once the actual current crosses the motor nominal current, the algorithm starts to integrate the excess of energy (red zone). If the excess of energy reaches the prefixed value, current will be decreased to its nominal value, and an interrupt will be generated.

Once the system has started to limit the current, it will not allow new overcurrent peaks until the \( i^2t\) accumulated goes below half of its maximum allowed value.

For example, if the system is configured with the following parameters:

\[ i_{RMS-nom} = 1 A\] |

\[ i_{RMS-peak} = 2 A\] |

\[ t_{peak} = 1 s\] |

The i^{2}t variable or energy excess will have the following value:

\[ i^2t=(i_{RMS-peak}^2-i_{RMS-nom}^2)· t_{peak}=3 A- s\] |

It means that the system will tolerate a peak of 2A during 1s, but also some other combinations, like for example:

\[ i_{RMS-nom} = 1 A\] |

\[ i_{RMS-peak} = 1.5 A\] |

\[ i^2t=(i_{RMS-peak}^2-i_{RMS-nom}^2)· t_{peak}= 3 A- s- t_{peak} = 2.4 s\] |

The system will not allow new overcurrent peaks until the energy excess goes below \( \frac{i^2t}{2}=1.5 A·s\)