## EQUIVALENCE PRINCIPLE

The ability to determine electromagnetic waves radiated fields via field-equivalence principles is a useful concept and the development can be traced back to Schelkunoff (2). The equivalence principle often makes an exact solution easier to obtain or suggests approximate methods that are of value in simplifying antenna problems. Field-equivalence principles are treated at length in the literature and we will not consider the many variants here. The book by Collin and Zucker (12) is a useful source of references in this respect. The basic concept is illustrated in Fig. 5. The electromagnetic source region is enclosed by a surface S that is sometimes referred to as Huygens’s surface.

<1 |

(a) Horn |

(c) Lens |

JS = n x (H1 — H) MS = —n x (E1 — E) |

(4) (5) |

E1, H1 |

/Js = n X (Hi – H) |

n |

V1 |

Si / / |

E, H |

г MS = – n x (E1 – E) (b) Equivalent problem |

(a) Original problem |

Figure 5. Equivalence principle with a closed Huygens surface S enclosing sources. |

In essence, Huygens’s principle and the equivalence theorem shows how to replace actual sources by a set of equivalent sources spread over the surface S (13). The equivalence principle is developed by considering a radiating source, electrically represented by current densities J1 and M1. Assume that the source radiates fields E1 and H1 everywhere. We would like to develop a method that will yield the fields outside the closed surface. To accomplish this, a closed surface S is shown by the dashed lines that enclose the current densities J1 and M1. The volume inside S is denoted by V. The primary task is to replace the original problem [Fig. 5(a)] by an equivalent one that will yield the same fields E1 and H1 [Fig. 5(b)]. The formulation of the problem can be greatly aided if the closed surface is judiciously chosen so that the fields over most of the surface, if not the entire surface, are known a priori. The original sources J1 and M1 are removed, and we assume that there exists a field E and H inside V. For this field to exist within V, it must satisfy the boundary conditions on the tangential electrical and magnetic field components on surface S. Thus on the imaginary surface S, there must exist equivalent sources (14): |

These equivalent sources radiate into an unbounded space. The current densities are said to be equivalent only outside region V, because they produce the original field (E1, H1). A field E or H, different from the original, may result within V. The sources for electromagnetic fields are always, apparently, electrical currents. However, the electrical current distribution is often unknown. In certain structures, it may be a complicated function, particularly for slots, horns, reflectors, and lenses. With these types of radiators, the theoretical work is usually not based on the primary current distributions. Rather, the results are obtained with the aid of what is known as aperture theory (15). This simple, sound theory is based upon the fact that an electromagnetic field in a source – free, closed region is completely determined by the values of tangential E or tangential H on the surface of the closed re- |

E1, H1 |

gion. For exterior regions, the boundary condition at infinity may be employed, in effect, to close the region. This is exemplified by the following case. Without changing the E and H fields external to S, the electromagnetic source region can be replaced by a zero-field region with appropriate distributions of electrical and magnetic currents (Js and Ms) on the Huygens surface. This example is overly restrictive and we could specify any field within S with a suitable adjustment. However, the zero internal field approach is particularly useful when the tangential electrical fields over a surface enclosing the antenna are known or can be approximated. In this case, the surface currents can be obtained directly from the tangential fields, and the external field can be determined. Assuming zero internal field, we can consider the electromagnetic sources inside S to be removed, and the radiated fields outside S are then determined from the electrical and magnetic surface current distributions alone. This offers significant advantages when the closed surface is defined as a two-hemisphere region, with all sources contained on only one side of the plane. If either the electrical or magnetic fields arising from these sources can be determined over the planar Huygens surface S, then the radiated fields on the far side of the plane can be calculated. The introduction of an infinite conducting sheet just inside the Huygens surface here will not complicate the calculations of the radiated fields in the other half-space (16). This infinite-plane model is useful for antennas the radiation of which is directed into the right hemisphere (Fig. 6), and has found wide application in dealing with aperture antennas. For instance, if the antenna is a rectangular horn, it is assumed the horn transitions into a infinite flange. All tangential fields outside the rectangular boundary along the infinite Huygens surface are taken to be zero. When the limitations of the half-space model are acceptable, it offers the important advantage that either the electrical or magnetic currents need to be specified. However, knowledge of both is not required. It must be emphasized that any of the methods described before will produce exact results over the Huygens surface. In the analysis of electromagnetic problems, often it is easier to form equivalent problems that will yield the same solution only within a region of interest. This is the case for aperture antenna problems. |

Figure 6. Some apertures yielding the same electromagnetic fields to the right side of the Huygens surface S. |

(b) Parabola |

W |

/IN |

(6) |

/ |

E (x) = E (sin 0)e-j |

(7) |

(8) |

(9) |

Lw/2 |

E(0) = -!-[ eij2nxlx)sine dx Lw J- |

‘w J-Lw/2 |

■ (nLw. „ sin sin 0 к_______ nLw sin 0 |

(11) |

(12) |

The steps that must be used to form an equivalent problem and solve an aperture antenna problem are as follows:

1. Select an imaginary surface that encloses the actual sources (the aperture). The surface must be judiciously chosen so that the tangential components of the electrical field and/or the magnetic field are known, exactly or approximately, over its entire span. Ideally, this surface is a flat plane extending to infinity.

2. Over the imaginary surface, form equivalent current densities JS and MS over S, assuming that the E and H fields within S are not zero.

3. Lastly, solve the equivalent-aperture problem.

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