Magic Tees

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A magic tee is a four-port, 180 degree hybrid power divider, 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  (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, because they are the only two that are in line with each other. Port 4 is the Δ (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.

Here's another take on the port nomenclature, from Ryan:

A magic tee has four ports, differentiated accordingly:

1. Sum, P-port
2. Co-linear
3. Co-linear
4. Difference, S-port

I believe the pair of identifying letters (S,P) are more logically correlated to the principle of polarization in optics, relative to a plane of incidence.  The P-port is titled parallel because the three ports (1,2,3) define a plane which with propagation is parallel with and the S-port is labelled with an S due to its normal orientation to that plane and the German ‘senkrecht’, meaning perpendicular, or vertical.

To play devil’s advocate, the following three ports (2,3,4) can also define a plane to which port 4 is parallel and port 1 is perpendicular.  The functional response will remain unchanged, namely the summation, difference and relative phasing.  In this view, proof by disproof, only the S & P titles would change therefore there is no logical relation between the letter (S or P) and the mathematical function described by ‘sum’ or ‘difference’.

To summarily discredit the challenge above and reaffirm that port 1 is titled P-port for a reason is that ports 1,2,3 define a plane in which the E-field  vectors are all parallel in all three ports.  This is most likely a pedantic rehash of arcane waveguide principles but I would offer the semantical tie to perpendicular in German ‘senkrecht’ as the source for the "S" in S-port as it equivalently applies to S-pol linear polarization in the optical plane of incidence.

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 four-port 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 Σ and Δ 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.