Closed-form solution to electromagnetic fields in far zone radiated by toroidal antennas

In this paper, a single-wound toroidal helix antenna (SWTHA) and a contra-wound toroidal helix antenna (CWTHA) are analyzed in an analytical way. In this connection, the total winding length of a SHTA is estimated and a fairly accurate and closed-form expression of the length is obtained. Meanwhile, a closed form solution of the electromagnetic fields in far zone or their alternative vector potential components is obtained. The formulation is analytical, based on the assumption of uniform SWTHA and CWTHA currents and the large ratio a/b of the toroidal antenna radius a over the turn radius b. The solution is general enough for any number N of wound turns. The results are finally reduced to those existing results available elsewhere in literature by letting the radius b be zero (for infinitely thin wire and circular loop antennas) and the winding number N be zero (for the normal circular loop antennas). The reduced results of the potential components are the same as those in the textbooks. The analytical results are obtained for the first time, useful for antenna designers and also antenna engineering education.