Sgt Kups, who was 38 and from Nuneaton, Warwickshire, was an electronics technician. He leaves his wife and three children.
He was one of two soldiers who died in separate incidents on Friday.
Neither of the deaths was thought to be the result of hostile action.
Sgt Kups was based in Tidworth, Wiltshire and has been described by his commanding officer as "a man of real substance."
His family has issued a statement paying tribute to him.
It said: "Jonathan was a wonderful husband and loving father to three children. He was a loyal man with a wide circle of friends, a devoted son, son-in-law, grandson and brother."
His children said: "You're the best Dad; always in our hearts - our hero."
Lieutenant Colonel Mark Smyth, commanding officer of 104 Force Support Battalion Reme, said: "Sergeant Jonathan Kups, or 'Koops' as he was known, arrived at the battalion in 2011.
"A real character, his dry and quick-witted sense of humour made him a pleasure to work with and he was extremely popular across all the ranks."
Lance Corporal James Sowersby, of the Electronic Counter Measure Section, General Platoon, said: "Sergeant Kups was a family man who was always talking to the section about his three children and his wife.
"He received many letters every day. He loved the gym and he could be there up to three times a day to maintain his fitness.
"He was good at his job and will leave a large gap in our small section."
The identity of the other soldier has not yet been released; he was from 28 Engineer Regiment, attached to 21 Engineer Regiment, and died at Forward Operating Base Shawqat, in Helmand.
The number of British military deaths since operations began in Afghanistan in 2001 now stands at 432.
A serviceman who died at Camp Bastion in Helmand Province, southern Afghanistan, has been named as Sergeant Jonathan Eric Kups, of the Royal Electrical and Mechanical Engineers.