Discussion
A patch on a substrate of effective permittivity 4 resonates at half a wavelength. How long is it at 3 GHz?
Answer: B
The free space wavelength at 3 GHz is 100 mm, and the substrate shortens it by the square root of the permittivity:
\[ L = \tfrac{100}{2\sqrt{4}} = 25 \]
millimetres, that is about 25 mm before the small fringing correction is subtracted. The square root dependence means a permittivity of nine halves the size again, which is why high permittivity substrates are chosen when compactness matters more than bandwidth or efficiency.
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