Let us present the matrix of equations (1) as a formal sum

(5)

where matrix is defined by expression (4). Matrix is diagonal; let us define the coefficients of this matrix using the following expression

(6)

where

(7)

in fact we have

(8)

However, now every distribution of the electric current corresponding to the basis functions in expansions (2) is considered as a radiator connected with the load of the transmission line having wave impedance . The coefficients of the reflection of loads are defined by expression (7). Obviously, short circuits will be such loads, since the length of transmission lines connecting these loads with radiators must equal zero.

The distributions of magnetic current in (2) can also be considered as radiators loaded to a multiport transducer with the matrix of conductivities . Let us assume that radiators and the multiport transducer are connected between each other with zeo-length lines with wave conductivities .

The distributions of electric and magnetic current specified above will be called electric and magnetic partial radiators or simply partial radiators further on.

Let us replace the initial matrix in equation (1) with the matrix of an infinite equally spaced array. Taking into account (5), equation (1) can be presented as

(9)

The coordinates of the vector in the right part (9) corresponding to radiators expanding the finite array to an infinite one equal zero.

Equations (1) and (9) are equivalent if we assign for additional radiators in (4), (6) the following

(10)

It follows from (9) that the currents and voltages of additional radiators equal zero if conditions (10) are met, therefore these radiators are in fact not there.

Equation (9) describes the excitation of an infinite array, but this array is irregular since the loads of its radiators are not the same. To switch to the equation describing the excitation of a regular structure, let us picture the excitation of the array as excitation by generators with EMF and MMF and waves reflected from the loads of partial radiators.

Let us assume that

(11)

Then the partial radiators with transmission lines form a regular structure.

Let us introduce these designations

(12)

where are the vectors of voltages and currents of loads of electric and magnetic partial radiators respectively. Let us present (12) as a sum of voltages and currents of falling and reflected waves in the corresponding transmission lines

(13)

where the upper +, - indices indicate falling and reflected from loads waves respectively; are the diagonal matrices of the wave impedances and conductivities of transmission lines connecting loads with electric and magnetic partial radiators.