M. V. Indenbom – Ph.D. (Eng.), Senior Research Scientist, Head of Sector of Antenna Department, JSC «All-Russian Scientific and Research Institure of Radiotechnics» (Moscow)
Convex phased array antennas (PAA) allow wide angles (hemisphere or more) electric scanning, conformal placement of PAA elements on convex surface of aircraft.
In common case the periodic coordinate grids with constant sizes of unit cell is not available on convex surfaces. Periodic grids are available on developed surface only, for example on cylindrical surface with arbitrary guiding curve. On cone surface the periodic grids are available only in special cases. Usually convex arrays are quasi-periodic with slowly changing unit cell sizes.
The adequate description of convex PAA characteristics is available only if mutual coupling effects are taken into account. For this purpose it is necessary to calculate electromagnetic fields near convex PAA surface. Asymptotic differential equation method was early developed for electromagnetic field calculation of periodic cylindrical array with arbitrary guiding curve and large sizes in wavelength. At present paper this method is applied for analysis arbitrary quasi-periodic convex antenna arrays.
Any vector of incidence waves in antenna elements inputs may be presented by linear superposition of partial excitation vectors with constant amplitude and linear phase distribution of its components. If reflected wave amplitudes in elements inputs or another PAA characteristic calculated for partial incidence wave vectors it may be founded for available incidence wave distribution is a matter of linear superposition principle.
The neared surface electromagnetic field of slot antenna array by partial excitation vector is presented as expansion of a spatial har-monics series analogy with infinite planar array electromagnetic field Floque\\\'s expansion. The spatial harmonics in the method of asymptotic differential equation are expressed by Airy functions.
The task to find the series coefficients of periodic planar or convex cylindrical array is reduced to solve a system of linear algebraic equations. This system is obtained as result using any calculating method in one of array unit cells. The equation systems order is not depends from number of PAA elements. It depends only from number of modes in aperture of one radiation element. For resonances slots only one linear equation is needed. For quasi-periodic convex antenna array the linear algebraic equation exchanges by the linear integral equation. The integral variance is a coordinate alone certain curve on antenna surface.
It appears what that equation is a second kind integral equation of Volterra with polar kernel what have and only one solution. That solution may be obtained by iteration method. The first approximation is obtained as a linear combination of special functions introduced, analyses and calculated in the paper.
On base of established expressions the numerical results calculated for input admittances and realized reflection coefficients on inputs of slot array elements located in cone surface coordinate triangular net with spacing between elements depending at forms coordinate. Calculated results show what although partial incidence waves have equal amplitudes maximal voltage complex amplitude in slots, admittance and reflection coefficient are changing along antenna surface. They may to have oscillator character in case if arias present were any spatial harmonics transition from propagation to damping conditions. Presenting of such arias depend from spacing and differential phase shift between array elements. This is difference between convex and planar antenna array. For planar array voltage in slots, admittance and reflection coefficient in element inputs for partial incidence wave vector are not depending from element placement in array (if edge effect is neglect).
Obtained expressions had common character and may be used for analysis an arbitrary convex arrays with quasi-periodic slot radiator placement in case large surface sizes relative a wavelength. They also may be expanded for convex PAA with multimode aperture radiators.