Antimagic Field, PHB, page 213
Spells and other magical effects...are suppressed in the sphere and can't protrude into it...While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.
The question remaining for
Do any charm spells stop working when the caster is in an Antimagic Field?,
is whether a spell a wizard is concentrating on is suppressed by Antimagic Field. The answers to the other question rely on the specific text indicating targeted creatures, ("Any active spell or other magical effect on a creature or an object in the sphere is suppressed,") however the general text is not considered. The answers to the other question also assume a "link" for telepathic effects, but if that is so, isn't there also a "link" for all concentration spells? My concern is the rule makers did not consider this, so it is a hole in the rules.
Exact scenario: Caster is concentrating on any spell that has an effect outside of the Antimagic Field, and then the caster enters the Antimagic Field. Is the spell he is concentrating on suppressed?
Note: Other questions answer whether the wizard is aware or whether his concentration ends (Do you know that Detect Magic is being suppressed if it intersects with Antimagic Field?, Does a spellcaster know when concentration ends?)