  The objective of this thesis is to ascertain the dimensions of an RF 2.856GHz
photoinjector through a combination of analytical and computational approaches.
The phase velocity within a single cavity exceeds 'c', rendering it inadequate
for storing the requisite energy for beam acceleration. To surmount this
limitation, we aim to devise a multi-celled cavity design. However, the
alterations in electromagnetic fields and resonant frequency within the
multi-celled cavity are intricate and sensitive, presenting challenges in
obtaining precise dimensions solely via computer simulations. Prior to
numerical methods, it is essential to analyze the photoinjector using
theoretical frameworks. We employ perturbation theory and the construction of
an equivalent circuit to elucidate the underlying physics of the photoinjector
and the electrical oscillations within the cell structure. Detailed analytical
methods for the equivalent circuit are explored. Through theoretical analysis,
the dimensions and simulation outcomes can be determined quantitatively.
