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Unread 11-21-2017, 11:50 AM
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Re: Breaker Modeling

Let's put some physics to this.

Breakers trip because of heat. So we should do an energy balance and consider this an open system with no work being done. Heat energy enters the system because the breaker / wire assembly has a small resistance. Heat energy leaves the system via conduction or convection.

Q_breaker = Q_resistance - Q_cooling

All power dissipated in a resistor is converted to heat energy. This is expressed as P = I^2 * R. This gives you a result in Watts. Current can be measured with acceptable accuracy through the PDP.

Modelling the heat loss via conduction or convection gets a bit more complicated but can be assumed to be linear over time based on the governing equations.

To test the validity of this method let's generate a curve that hopefully matches reasonably well with the data sheet using these equations.

Using arbitrary numbers let's say we believe the breaker trips after applying 150 amps for 500 seconds (based roughly on the data sheet, in reality you'll want to determine this experimentally). We'll assign a resistance value of 0.001 Ohms (it's arbitrary and doesn't really impact the calculations). We'll also arbitrarily say that the breaker has 2 W of cooling via conduction.

Q_resistance = [(150 amps)^2 * 0.001 Ohms * 500 seconds] = 11250 Joules
Q_cooling = - (2 J/s * 500 seconds) = 1000 Joules

So we can say that the breaker will trip when it reaches 10250 Joules of heat (this corresponds to a temperature via a specific heat value) Note that I didn't look up specific heat values or resistances for anything for the sake of quick math so my Joules are probably way off from realistic numbers.

Now to solve for time to breaker trip at a different current we can do some algebra.

Q_breakerpop = I^2*R*t - P_cooling*t

t = Q_breakerpop / (I^2*R - P_cooling)

This reasonably creates a similar arc as the data sheet generates, but likely needs a lot of tuning via experimental values.

Since I is obviously not constant in a match a model could be generated tracking heat generated using the PDP data.
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Mike Schreiber

Kettering University ('09-'13) University of Michigan ('14-'18)
FLL ('01-'02), FRC Team 27 ('06-'09), Team 397 ('10), Team 3450/314 ('11), Team 67 ('14-'??)
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