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I tried to set the antenna in the dielectric, thinking that the minimum frequency of the S-parameter would move, but it began to behave very strangely. What shoudl happens to the operating frequency of the antenna in the dielectric? Should it move, or does the surrounding dielectric change the resistance of the antenna, making it unworkable?

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The dimensions of the elements of an antenna are dependent on the wavelength (frequency) of the incident waves to be received.
The wavelength of the waves will depend on the refractive index of the medium though which they are travelling.
Thus if the wave is travelling in a dielectric that might mean that the wavelength for which the antenna was designed is not the wavelength of the incoming wave.

You may also have a problem if the dielectric is "lossy" - absorbs the incoming waves.

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In a dielectric the propagating wavelength gets shorter in ratio with the refractive index so you would expect that the formerly otherwise properly tuned antenna in air would be electrically large and now would tune to a lower frequency but if you did nothing else but placed the antenna in some dielectric then the antenna S parameters will surely behave funny ways because your external matching circuits will not be affected by the dielectric.

Reality is even more complicated because the radiation behavior will also depend on the size of the dielectric block relative to the antenna dimensions and also on the refractive index. Unless the linear dimension of the dielectric in the direction of the main lobe is at least as large as the Fraunhofer (Rayleigh) distance the reflections from the boundary will interact with the antenna itself distorting the beam shape.

Note too that the presence of the dielectric reduces the bandwidth of the antenna (ie., increases its "Q") as well.

Another funny thing is that if you place a phased array in a dielectric then grating lobes may show up because for the same frequency the element spacing is too large now electrically.

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