Bandwidth and Q of antennas radiating TE and TM modes

The time-domain Poynting theorem is used to develop a general expression for the complex Poynting vector applicable to any single-frequency electromagnetic radiation field. It is found that the traditional complex Poynting vector applies to TE or TM fields, which we call simple fields, but that it does not apply to TE and TM fields, which we call compound fields. Either TE or TM fields are generated by most antennas. We show that previously imposed theoretical minimum size-to-wavelength ratios for useful antenna operation apply to simple fields but not always to compound ones. We conclude that electrically small, efficient compound antennas may be possible. As an example, the general form of the Poynting vector is used to analyze a compound source consisting of four antenna elements; idealized, superimposed, properly phased and oriented, coherent, electric and magnetic, dipole and quadrupole radiators. When properly driven, the antenna supports zero reactance on a circumscribing virtual surface of radius a, even in the limit as the radius-to-wavelength ratio of that surface goes to zero. The directivity pattern has a fixed 9 dB gain; the radiative Q of the surface is less and the bandwidth more by a factor of (ka)², where k is the wave number, than for similarly sized radiators of simple fields