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Oh ho ho, it's Magic by Pilot, a Scottish group, 1975
A magic tee is a four-port, 180 degree hybrid splitter, realized in waveguide. Originally developed in World-War II, and first published by W. A. Tyrell in a 1947 IRE paper, it has very similar properties to the rat-race coupler, which is usually realized in microstrip or stripline.
Like all of the coupler and splitter structures, the magic tee can be used as a power combiner, or a divider. It is ideally lossless, so that all power into one port can be assumed to exit the remaining ports.
The convention used in Pozar's book "Microwave Engineering" is shown on the following figure, though not all waveguide vendors adhere to it. Port 1 is the (sum) port, and is sometimes called the H-plane port, and sometimes called the P-port for "parallel". A signal incident on port 1 equally splits between ports 2 and 3, and the resulting signals are in phase. Ports 2 and 3 are sometimes called the co-linear ports (thanks Bill!), because they are the only two that are in line with each other. Port 4 is the (difference or delta) port, and is sometimes called the E-plane port, or the S-port for "series". A signal incident on the difference port splits equally between ports 2 and 3, but the resulting signals are 180 degrees out of phase (thanks Harald!)
The math behind the magic tee is too much for us to present here for now. Maybe it's just better to leave it as "magic" and not try to analyze it.
We used HFSS v10 to model a magic tee, using an example right out of the Ansoft HFSS book. This exercise will help you visualize how the E-field of a signal entering the sum port remains in the same up-and-down direction and polarity as it splits to ports 2 and 3, while the E-field of a signal entering the delta port wraps around into two opposing polarities as it splits between ports 2 and 3. The interior dimensions of the waveguide are 50 mm by 20 mm. This is not a standard waveguide size, the broad wall is approximately two inches, which puts it close to WR187. You can tell that Ansoft is run by mathematicians, not microwave engineers, or they would have picked a "real" waveguide band. Below is the model:
The next picture shows how it was meshed:
The next two pictures show the E-field vectors for signals entering the sum port, then the delta port. Now you can see how the delta port excites opposing phases in the CO-linear arms.
Cool stuff! The next plot shows the phase of the transmission coefficients out the CO-linear ports, when driven by the delta port. Note the 180 degree difference.
Last, here are some of the S-parameters of the fourport network, including the transmission coefficient between sum and delta ports (red trace), which is better than -50 dB. The input match S11 (blue trace) could be better, which would require some tuning. Guess that's why you'd never buy a magic tee from Ansoft! If you go back to Tyrell's paper, he suggests adding tuning to the and arms. This can be done with tuning screws, rods, or plates. We're not going to get into this right now.
The magic tee in the photo below is WR-62. If you cut it open you could see how it was tuned. We like it just fine in one piece.