Would a compact welder use MOSFETs or IGBTs
It's silicon carbide mosfets that have made high amperage/high voltage circuits possible. Now they don't have to use (what I think is, I never tore one open ) transformers and capacitors. Carbide is near indestructible. I had a wielding class a few years back and the high amperage welders ranged from the size of the night stand to a large kitchen desk. Now they can fit the same functionality into something the size of a microwave oven, not only can they do that, but the machines have more switching functionality and they have so much room left over, they combine the same funtionality of 3 welders (tig,mig and stick) into one. It is increadible. 


  Silicon Carbide (SiC) Power MOSFET devices are rapidly being
  introduced in different applications due their advantages over their
  silicon (Si) counterparts. Higher switching frequency (ƒsw) coupled
  with planar magnetics opens new possibilities in terms of increasing
  the power density of converter, while also improving their efficiency
  and lowering costs. In this paper, SiC MOSFETS and planar magnetics
  are employed to demonstrate new possibilities in arc welding machines
  technology. Design details of high power density power source for a
  multi-process welding machine are presented. Test results for
  3.9kW/150A and 200 kHz switching frequency working at rated load and under over load condition are included.


Source: https://www.researchgate.net/publication/300408761_Next_generation_arc_welding_machines_based_on_Silicon_Carbide_MOSFETS_and_high_frequency_planar_magnetics