If modified, the specified method can be used to define the characteristics of finite antenna arrays taking into account edge effects and the interaction of the radiators.
Let us assume that a separate radiator in the array is an aggregate of metal constructive elements located above a plane or circular cylindrical screen and apertures in the screen connecting the radiator with the attenuator or feeder line to the reflective phase shifter. The structure of the radiating curtain may include dielectric substratum and a multilayer dielectric covering in the form of a sheet of the corresponding shape located parallel to the screen.
Within the induced EMF method, the currents in the radiators are defined by the following equation:
 |
(1) |
when 
are the matrices of internal and mutual impedances, conductivities, and transmission coefficients defined relative to basis functions in the expansions of electric and equivalent magnetic currents 
of the radiators
 |
(2) |
where р, t is the number of the radiator and the constructive elements in the radiator respectively; 
are basis functions in the expansions of electric and magnetic currents; 
is the coefficient vector in the abovementioned expansions; 
is the vector of EMF and MMF defined relative to the corresponding basis functions in expansions (2) expressed via the electric and magnetic field of the reflective array feed or the waves of the attenuator of a transmissive array. The dimension of basis functions in (2) can be chosen in such a way that the coefficients of the vector will have the dimensions of the electric current and voltage. So these coefficients will be called the currents and voltages of the radiators. Let us assume that the finite array is composed of exactly the same equally spaced radiators.
The matrix of internal and mutual conductivities is a sum of two summands
 |
(3) |
where 
is the matrix of the external, internal and mutual conductivities; 
is the matrix if internal conductivities characterizing the radiation of coupling apertures in the transmission line that connect radiators with reflective phase shifters or the attenuator. The coefficients of the matrix represent a sum of partial conductivities
 |
(4) |
where 
is the wave admittance of the q-th internal wave of the transmission line; 
is the reflection coefficient of this wave regarding the load in the plane of the coupling aperture; 
is the coefficient taking into account the "interaction" between the basis functions of the t-th aperture via the transmission line over the q-th natural wave.
The solution of equation (1) using the methods we know becomes difficult when there is a lot of radiators and there are dielectric coverings and substrata. We can make it considerably easier if we manage to express the solution of equation (1) via the solution of the problem regading the excitation of the corresponding regular structure. To do it, we expand the finite array to an infinite one and select loads for additional radiators so that we exclude the influence of these radiators on current distribution in the finite array while modelling. After that we use the successive reflections method. According to this method, currents in the elements of an infinite array are represented as a sum of currents excited during the successive reflection of waves between the apertures of the radiators and loads (the phase shifters of reflective arrays or the attenuator of transmissive arrays, as well as the loads of radiators expanding an finite array to an infinite one). We need to solve the well studied problem regarding the excitation of a regular radiating structure for each reflection.
As you will see later, the models of loads in the radiators of a finite antenna array and additional radiators are different from regular models of physical loads. That is why the method below can be called the generalized method of successive reflections.
