This paper investigates optimization of embedded liquid crystal cavity in the patch antenna ground plane at Ka-band to get the largest frequency tuning across the Ka-band. Proposed antenna comprises of a patch radiator with microstrip feed line and liquid crystal (LC) cavity embedded thick ground plane. A dumbbell-shaped LC cavity can be achieved the largest frequency tuning of around 1.2 GHz in the ka-band compared to the pillarshaped and rectangular-shaped LC cavities.