They are separate structures with one defensive purpose.
The integrated Walls surround the city. Lovrijenac stands outside that circuit on the western rock, positioned to protect the sea approach and reinforce the vulnerable Pile side.
Bokar is the best place to understand the pairing.
From the Walls near Fort Bokar, look across the bay toward Lovrijenac. The two fortifications frame the western approach and make it obvious why Dubrovnik did not rely on one continuous wall alone.
The ticket detail is worth knowing before you buy twice.
The official FAQ currently states that a City Walls ticket includes Lovrijenac. If you start with a Lovrijenac ticket and later decide to visit the Walls, an additional payment is required. Always confirm current terms before purchase because ticket policies can change.
Allow separate time for the fortress.
Lovrijenac is not simply another tower on the circuit. Reaching it means leaving the Old Town / Pile area and climbing separately, so treat it as a second visit rather than part of your 1.5–2 hour Walls timing.
Quick answers
Is Lovrijenac included with a Dubrovnik Walls ticket?
The official operator’s current FAQ says yes: a ticket purchased for the City Walls can also be used to visit Lovrijenac.
Is Lovrijenac physically part of the City Walls?
No. It is a separate fortress west of the integrated Walls.
How much extra time should I allow?
Treat Lovrijenac as a separate stop and give yourself additional time for the walk and climb rather than including it in the normal Walls circuit estimate.
References: Official City Walls FAQ · Dubrovnik Tourist Board: City Walls and Fortresses