Radio Engineer Useful Formula
Babinet Principle can be used to find complementary impedances. Babinet's Principle states (in optics) that when a field behind a screen with an opening is added to the field of a complementary structure (that is a shape covering the screen hole), then the sum is equal to the field where there is no screen. Demonstration can be found in any good optic or antenna book (such as Balanis, Krauss, Stuzman). The end result (corollary as a matter of fact) of practical interest for antenna engineers is the following formula:
.
Where Zmetal and Zslot are input impedances of the metal and slot radiating pieces, and is the intrinsic impedance of the media in which the structure is immersed. In addition, Zslot is not only the impedance of the slot, but can be viewed as the complementary structure impedance (a dipole or loop in many cases). In addition, Zmetal is often referred to as Zscreen were the screen comes from the optical definition. It is noteworthy that the thin sheet or screen does not have to be metal, but rather any material that supports a (current density vector) leading to a magnetic potential . One issue with this equation, is that the screen must be relatively thin to the given wavelength (or range thereof). If it is not, modes can begin to form or fringing fields may no longer be negligible.
For a more general definition of Eta or intrinsic impedance, .
Please note that Babinet's principle does not account for polarization. In 1946, H.G. Booker published Slot Aerials and Their Relation to Complementary Wire Aerials to extend Babinet's principle to account for polarization (otherwise known as Booker's Extension). This information is drawn from, as stated above, Balanis's third edition Antenna Theory textbook.
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