Phasor Alternatives to Friis’ Transmission Equation

Two alternatives to Friis' transmission equation in terms of phasor voltage waves are presented. In one formulation, antennas are characterized by the complex effective length vectors. An additional form introducing field gain, which serves effectively as a phasor counterpart to the power gain, is proposed. Both forms show the same degree of symmetry and modularity as the original Friis' equation, but thanks to using phasors instead of power quantities, they allow for superposition of fields or voltages. Although the new transmission equations are formulated in a frequency domain, they also constitute a simple and intuitive way to calculate time-domain responses of antennas, such as in ultrawideband systems. Application of the equations is demonstrated on a simple example of coupling between two nonaligned dipoles in the vicinity of a conductive reflector.