Electrical – use a separate 24V transformer with an Ecobee 3 and an Aube relay

You guys down voted the only answer that is correct and up voted one that is completely wrong!!! You CANNOT use just one wire from a transformer and expect to get it to do a thing for you. The simple answer to your question is that if you add a transformer you want to tie the lines from it to Rh and C. Most modern thermostats I know of expect to pull power between Rh and C. You do NOT want to tie Rh and Rc together since your heating and cooling are on separate transformers and you don't know if they are truly floating.

The complex answer is that you should check to see if the control board of your heating system doesn't, in fact, have a common terminal and nobody bothered to run the wires for it. If it does, it might be simpler to just pull 18/3 wire from the heater to the thermostat. Depending on the run, you can use the old two conductor wire to pull the three conductor wire through.

My plan worked perfectly and I called Emerson Sensi customer service to check that it will not cause any issues and the confirmed. The 2 wires to the 24VAC relay that control the baseboard get connected to the Rh and W terminals on the thermostat. I cut the jumper between Rh and Rc to ensure I wouldn't cause a short by adding another power source I used an external 24VAC transformer, connected to Rc and C to provide power to the thermostat. In essence this is not a parallel circuit situation, but 2 independent circuits. One circuit from the external transformer provides power to the thermostat. The other circuit controls heat activation.