Why is the primary winding wire of a voltage step-up transformer usually a larger size than that of the secondary winding?
Why An ideal transformer conserves power, so primary volts times primary amps equals secondary volts times secondary amps. In a step-up transformer the secondary voltage is higher, which means its current is proportionally lower, and the low-voltage primary must carry the larger current. Wire size is chosen by current-carrying capacity, so the high-current primary needs heavier gauge wire to keep I-squared-R heating and voltage drop down.
Watch out Coupling between windings is set by the core and winding geometry, not by wire gauge, so the choice about improving coupling is wrong; wire thickness only affects copper loss.
Step up the volts, step down the amps. Big current needs big wire, so the low-voltage side gets the fat wire.