The independent clause is 'When the doorbell rang.' It is an adverbial clause, modifying the verb 'was sleeping'.
"You rang the doorbell" is an indepent clause, because it can exist on its own. So is "but no one answered," because it is using a coordinating conjunction, which acts as a independent marker for an independent sentence. See https://owl.english.purdue.edu/owl/resource/598/01/.
rang the doorbell is a predicate
The Doorbell Rang has 186 pages.
rang the doorbell is a predicate
The Doorbell Rang was created on 1965-10-08.
The past tense of "to ring" is "rang," so you say "I rang the doorbell" or "I have rung the doorbell many times."
Had is a past tense, which means it happened before (eg Sally had rung the doorbell) rang counts as a present tense (eg Sally rang the doorbell) The appropriate word for "had" would be "had rung" which is past tense.
inferance
I rang the doorbell repeatedly, but no one answered.
The past tense of 'ring' is 'rang', whilst the present perfect tense is 'have rung'.
Letter to Loretta - 1953 The Night the Doorbell Rang 8-11 was released on: USA: 27 November 1960