The minimum quality factor of a rectangular antenna aperture

The functions which minimize the quality factor, or ratio of total reactive power, are found and shown to satisfy four different groups of symmetry. A notable feature of these minimum solutions is that the f(x) component always possesses a symmetry which is opposite to that of f(y). The analysis also demonstrates that Group II and Group III solutions in general possess eigenvalues that are distinct even in the case of square apertures, and that this feature may be interpreted as a rotational effect. It is noted that the analysis applies not only to quality factor functionals, but any positive, definite quadratic functional of the variables F(x) and F(y).