[Buddhaheart]

Here’s one way to analyze the sentence. First recognize this is a complex sentence. The main or principal (independent) clause (PCl) is " the senator was on his cellphone talking loudly about one of his recent successes." The subordinate (dependent) clause (SCl) is "when Rachel arrived at her father`s table."

The SCl is an adverb clause (AdvCl) of time, qualifying the verb phrase (VP) ‘was talking’ in the PCl.

Now you can look at each clause as a simple sentence and analyze it separately.

The subject (S) of the PCl is the noun phrase (NP) ‘the senator’ where ‘the’ is the definite article (DA) modifying the head noun (HN) ‘senator’. The predicate (Pred) is ‘was on his cellphone talking loudly about one of his recent successes.’ ‘Was ... talking’ is the main verb phrase (MVP) or predicator. The MVP indicates a past time and a progressive or continuous aspect. The Pred contains 2 prepositional phrases (PP) and an adverb (Adv): ‘on his cellphone’,‘about one of his recent successes’ and ‘loudly’. The 1st PP contains the preposition (P) ‘on’ and object of the preposition (OP) ‘his cellphone’. The P governs the OP and the PP functions as an adverbial modifier (AdvM) of ‘was calling’. The OP contains a possessive pronoun (PPN) ‘his’ that modifies the N ‘cellphone’. The 2nd PP contains the preposition (P) ‘about’ and ‘of’ and their respective OP ‘one’ and ‘his recent successes’. The 2nd OP contains a possessive pronoun (PPN) ‘his’, an adjective (Adj) ‘recent’ and a N ‘successes’ and is further modified by the Adv ‘Loudly’.

The time connective between the 2 clauses is ‘when’. The S of the SCl is the proper noun (PN) ‘Rachel’. The Pred is ‘arrived at her father`s table’. The main verb (MV) is ‘arrived’. It indicates past time. The PP is ‘at her father’s table’ where ‘at’ is the P and ‘her father’s table’ is the OP. The OP contains the N ‘table’ which is modified by the possessive ‘father’s’ which in turn is modified by ‘her’. The OP acts as an adverbial of place modifying the V ‘arrived’.

I might have a question regarding the phrase ‘at her father`s table’. I don’t know what exactly this means or better wording may be used to make this clear to the reader.