# Input Impedance and Directivity of Large Circular Loops

Assuming a uniform current distribution along a **small circular loop >** has allowed us to obtain closed-form expressions for the radiation resistance and directivity. When the loop circumference is comparable to the wavelength, the current distribution cannot longer be assumed uniform but a Fourier series is applied to approximate it. Also, a delta gap voltage source is used as the excitation in the theory of large loops, so the drawbacks regarding lack of convergence of the input impedance will appear here as in the case of the cylindrical antenna with delta-gap source. Nevertheless, the theoretical results are still useful as a reference frame.

In the theory of loops, the circumference C is measured in wavelengths, C/λ, and the wire thickness is taken into account via the Ω parameter,

where *r* is the loop radius, so C = 2π*r*, and *a* is the wire radius. Fig. 1 shows the computed radiation pattern in decibels for a loop with C/λ = 1 and Ω = 10. This pattern is quite similar to that shown in Fig. 5.12 of “Antenna Theory Analysis and Design” by Constantine A. Balanis, 4th Edition, Wiley 2016, for the same loop dimensions.

To establish a numerical comparison between AN-SOF and theory, and to demonstrate that the radiation pattern in Fig. 1 is actually that predicted by theory, the following table shows the directivity in dBi of the circular loop antenna for various electrical sizes (C/λ) and thicknesses (Ω):

Ω | C/λ | D [dBi] Theory | D [dBi] AN-SOF | Error % |

8 | 1 | 3.344 | 3.36 | 0.478 |

10 | 1 | 3.412 | 3.411 | -0.0293 |

12 | 1 | 3.442 | 3.439 | -0.0872 |

20 | 1 | 3.476 | 3.473 | -0.0863 |

Ω | C/λ | D [dBi] Theory | D [dBi] AN-SOF | Error % |

8 | 1.48 | 4.626 | 4.684 | 1.25 |

10 | 1.45 | 4.592 | 4.615 | 0.501 |

12 | 1.43 | 4.523 | 4.54 | 0.376 |

20 | 1.39 | 4.354 | 4.368 | 0.322 |

*Directivity of large circular loops predicted by theory and calculated by AN-SOF.*

We can see that the percentage error between the AN-SOF and theoretical results is negligible from a practical point of view. The largest discrepancies, around 1%, appear when the loop is made of a thicker wire (Ω = 8) as we could have expected since AN-SOF uses the exact kernel and the theoretical loop a thin-wire approximation.

Regarding the loop input impedance, Fig. 2 shows the comparison between theory and AN-SOF of the input resistance and reactance as a function of the loop circumference normalized to the wavelength and for a wire thickness of Ω = 10. A constant segmentation of 35 segments has been used in the AN-SOF model, so each segment is about 7% of a wavelength long for the largest loop, C/λ = 2.5 (2.5/35 ~ 0.07).

It is remarkable that the agreement between theory and the computed results is quite satisfactory over such a large range of variation.