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This would have to be something in the heater relay. This includes the thermostat W2 ( some label it Aux) terminal, the wire running from this W2 terminal to the air handler, and the heat relay that this wire hooks into at the air handler. This wire and it's relays (thermostat and heat relay in air handler) are the only thing that energize different in heat that do not energize in cooling that would cause the 5 amp fuse to blow. It could be this wire is bare somewhere from point A to point B and is touching something it shouldn't, could be the relay is shorted out or it could be a short in the thermostat.
The w2 circuit is energized in two instances during a heat call.
1. When you turn the demand temperature 3 or more degrees above the ambient temperature of the thermostat the w2 (electric heat kit) energizes to help the heat pump.
2. When the demand is less then 3* of the ambient but the system has run for around 8 mins without reaching the demand temperature.
I take that back there is a third way it is energized in a heat call.
When the outdoor unit goes into defrost mode the unit sends a signal to the heat relay in the air handler to bring on the heat kit. This almost never happens though as if it is cold enough out for the system to go into defrost then the heat kit is almost always already on because the system has been running for over 8 mins.
Unit has a new thermostat, doesn't explain why it would kick on in a rapid cycle, but not after sitting. Tried a 10 amp fuse and doesn't blow but still wont start the unit unless it is already running in A/C mode. Could it be a weak capacitor at the unit itself?