Equivalent circuit model of microstrip patch antenna is well documented, and readily available even in antenna books.
I will suggest to refer the following for the same:
1. R. Bancroft, Microstrip and printed antenna design 2nd edition. Raleigh, NC, USA: Scitech Publishing, 2009. (for lumped circuit model) and
2. H. Pues and A. Van de Capelle, "Accurate transmission-line model for the rectangular microstrip antenna," in IEE Proc. H-Microw., Optics and Antennas, Dec. 1984, pp. 334-340. (for transmission line model).
Lumped Elements for RF and Microwave Circuits by Inder J. Bahl.
Check this amazing book, First chapter will give you clear idea about equivalent circuits for different microstrip configurations, and what happens with equivalent circuit when u make changes in microstrip.
An equivalent circuit of the patch antenna can be realized by a Parallel RLC circuit.
For different structure of the patch you have to change the Lumped elements of the circuit.
For this regards I can suggest you 2 books and few papers.
Book:
1. C.A. Balanis : Antenna Theory
2. D.M.Pozar : Microwave Engineering
Papers:
1. Luther JJ, Ebadi S, Gong X. Extraction of equivalent circuit model parameters of the feedless rectangular microstrip patch. In2013 IEEE Antennas and Propagation Society International Symposium (APSURSI) 2013 Jul 7 (pp. 302-303). IEEE.
2. Ping L. Equivalent circuit model of octagonal ultra wideband (UWB) antenna. Journal of Engineering Science and Technology. 2017 Feb 1;12(2):439-50.
3. Ansarizadeh M, Ghorbani A, Abd-Alhameed RA. An approach to equivalent circuit modeling of rectangular microstrip antennas. Progress In Electromagnetics Research. 2008;8:77-86.
Equivalent circuit model of microstrip patch antenna is well documented, and readily available even in various antenna books.
I will suggest to refer the following for the same:
1. R. Bancroft, Microstrip and printed antenna design 2nd edition. Raleigh, NC, USA: Scitech Publishing, 2009. (for lumped circuit model) and
2. H. Pues and A. Van de Capelle, "Accurate transmission-line model for the rectangular microstrip antenna," in IEE Proc. H-Microw., Optics and Antennas, Dec. 1984, pp. 334-340. (for transmission line model).