The Complex Poynting Theorem Reactive Power, Radiative Q, and Limitations on Electrically Small Ante - Electromagnetic Compatibility, 1995. Symposium Record., 1995 IEEE International Symposium on

The complex Poynting theorem does not calculate the correct reactive power for all possible field conditions. Upon subdividing fields into modal constituents, the theorem is valid for all TE or TM field combinations, but it is not valid for all TE and TM combinations. The method needed to find reactive power under all modal field combinations is reviewed. Proofs that there are severe size limitations on electrically small antennas are based upon reactive power calculated using the complex Poynting theorem. This commonly believed and significant antenna result is neither proven nor true for fields consisting of mixed modal types. An example is given where it breaks down. The result emphasizes a research direction in which to seek an efficient, electrically small antenna. INTRODUCTION The complete time-dependent expression for power in ac fields is inconvenient to use; the detailed information is just too extensive and elaborate. Instead power is described by a complex number: the real part is the magnitude of unidirectional power pulses and the imaginary part is the magnitude of directionally oscillating power pulses. Real energy leaves the source and never retums, reactive energy oscillates about a fixed position in space; that within a half wavelength of the source retums to it twice each cycle, directly affecting and influencing its operational characteristics. Real and reactive powers are 90" out of time phase [l]. Real power from multiple sources is directly additive under all circumstances, reactive power is not. For example, let a single power source (engine) turn a shaft driving identical electric generators with independent but equal magnitude loads: one has an inductive reactance and the other an equal capacitive reactance. If the real powers are in phase or 180" out of phase, the reactive powers cancel. If the real powers are 90" out of phase the reactive powers algebraically add. In these cases, after initial buildup the source is not and is, respectively, affected by reactive power: The relative phases of the real powers is important in determining the net reactive power. Similar conclusions apply to radiation fields. With superimposed, coherent radiation fields the net complex power depends upon the magnitudes and phase differences between the additive real and reactive powers. As an example, consider the powers on the surface of a sphere that contains more than one radiator, and fields whose description requires both field polarizations. The net reactive power is equal to the sum of the reactive power per polarization only if the real powers per polarization are in or 180" out of phase. This point was missed in previous discussions of power in radiation fields. The result is that the surface integral of the commonly accepted complex Poynting vector is equal to the net complex power only in simplified cases. Although most antennas satisfy the simpler conditions, some do not. Proofs [2,3,4] that the radiation Q increases precipitously for antennas whose maximum length-towavelength ratio is small and decreasing are based upon the reactive power being additive in all cases [5]. The proof, therefore, does not apply to all possible antennas and it is, at least in principle, possible to build an electrically small, efficient antenna that exceeds the accepted limits; detailed analyses are presented elsewhere [5,6]. The purpose of this paper is to detail why the common integral expression for complex power is not correct for all cases. TIME 'VARYING AND COMPLEX POWER IN CIRCUITS Inspection of several textbooks covering power systems, circuits, and power radiation shows that all add complex powers using simple ad'dition. For example, if real and reactive powers are designated by P and R, respectively, and subscript "i" indicates values at a particular circuit node, the power equations are written as