What furnace do you have.....brand/model??
Furnace has fused DC power to it.
When t-stat contacts close calling for heat DC power goes t a timed delay relay.
This allows fan motor to start (DC Powered).
Fan comes up to speed and purges combustion chamber prior to gas valve opening.
As fan speed increases a 'sail switch' closes and DC power goes to circuit board. (Sail switch has an arm that sticks out into air stream...closes contact and dc goes to high limit switch which is always closed unless high temp in exchanger)
Circuit board now has DC power to send to gas valve and spark ignitor.
Main flame lights off.........flame is proven...circuit board keeps DC voltage on gas valve until t-stat contacts open.
DC power drops off of time delay relay....but DC power thru other side of relay keeps fan running to cool exchanger down. Coil inside relay cools off....shuts down power to fan.
Furnace works when AC Power is available.
Fan turns on when just using batteries...............BUT no heat.
Slow fan motor speed, sail switch not closing.
Dirty/loose contacts...high resistance