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once the wire heat's up it provides more resistance in the wire and causes the load to exceed the 15a breaker.
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That is not quite correct. Current flow increases with temperature. The problem is that breakers react to overcurrent, and overload, and you are asking too much from it. Breakers also have about a 20% tolerance. In some cases just changing out the breaker will help, but you will probably have to divide the load among circuits, or if possible supply a dedicated circuit..
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