Madsen began his career as an on-field official before moving to the replay booth in 2009.
In a statement, NFL senior vice president Walt Anderson called Madsen a "terrific friend and colleague."
Anderson added: "Carl's love of football and dedication to officiating was ever-present, as he generously shared his time to mentor young officials at clinics across the country. A veteran of the Air Force, Carl had a tremendous spirit and will be greatly missed."
Scott Green, the executive director of the NFL Referees Association, said: "Carl will be missed by those who worked with him on the field and in replay. He had a nickname among his fellow officials of 'Big Country,' which was not only related to his size but to his big personality as a warm and generous man."
Madsen spent the first seven weeks of this season working on referee Brad Allen's crew.