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It's known (and it's the basis of many designs of microwave devices) that electromagnetic wave propagation in a medium occurs in a good way (i.e. without reflection and higher order modes excited), if there are no discontinuities, or at least they are spatially slow compared to the wavelength.

There are lots of examples of this:

1) Horn Antenna: Why is a horn antenna (which as a slow and sweet enlargement of its transverse section) better than an abrupt interruption of its supplying rectangular waveguide? Because the last will have an abrupt discontinuity, the antenna won't be matched well to free space impedance and so its radiation will be mostly reflected back to the waveguide.

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2) Transitions between transmission lines: for instance this is a slow transition from a microstrip line to a couple of symmetric strips. Why can't this transition be fast (less than a wavelength)? There would be a huge mismatch.

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3)Polarizers: I've seen that many antennas have not a single polarizer layer, but many layers in which there is a slow rotation of the strips in order not to abruptly rotate the electric field to get the desired polarization. A single layer is often referred not to work at multiple frequencies, but only inside a narrow band. enter image description here

My question is: why a slow transition may determine an almost perfect match, without reflection? A slow transition still is a succession of discontinuities. Each of them will cause reflection, and the starting and ending points of the transition will be mismatched as if the transition were fast. I can't imagine why slow discontinuities are so well seen by electromagnetic waves.

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In the abstract, a horn antenna is no better than a simple waveguide end. In the concrete though one selects the size of the end aperture to control antenna gain (directivity). Directivity is directly proportional to the size of the aperture while beamwidth being proportional to the reciprocal of directivity is proportional to the reciprocal of the aperture area.

On the other hand the standing wave on a uniform transmission line depends on the load impedance at the end of the line, here at the throat of the pyramid. Note though that the coaxial to waveguide transition is an antenna whose length is adjusted to minimize reflection from it and act as a match to the aperture. The larger the aperture the larger the ratio of the electric to magnetic field is at the end but perfect match to free-space would require a ratio of $377\Omega$, a figure that can only be approached but not reached, especially starting with the $50\Omega$ coaxial line.

Because of their low directivity untapered waveguide end even when properly matched as a radiator is rarely used but large arrays of such waveguides are used in high power electrically steered phased arrays.

To see how tapering helps think of the problem to match between $50$ to $377\Omega$. The wider taper increases the wave impedance of the line, i.e., the ratio of $E/H$, and helps matching the coax to free-space by reducing the frequency sensitivity of it for the wave impedance is a purely geometric property. The frequency width of the $50\Omega$ coax probe is limited by the its stray capacitive and inductive couplings to the waveguide walls. This frequency dependence is avoided by the smooth taper at the cost of a much larger size.

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