Magic tee, degenerate modes, dominant mode for a = 3b

Magic tee based on S-parameters

A magic tee joins an E-plane tee and an H-plane tee at one point: ports 1 and 2 collinear, port 3 the H-arm, port 4 the E-arm.

              E-arm (4)
                | |
     -----------+ +-----------
 (1) ===========+   +=========== (2)
     -----------+ +-----------
               / /
              / /  H-arm (3)

S-matrix (derived from symmetry, reciprocity, E–H isolation and losslessness):

              | 0   0   1   1 |
 [S] = (1/√2) | 0   0   1  -1 |
              | 1   1   0   0 |
              | 1  -1   0   0 |

• H-arm input → equal in-phase outputs at 1 and 2, nothing at E-arm.
• E-arm input → equal 180° out-of-phase outputs at 1 and 2, nothing at H-arm.
• Input at 1 → half to E-arm and half to H-arm; port 2 isolated.
• Equal in-phase inputs at 1 and 2 add at the H-arm (sum); anti-phase inputs add at the E-arm (difference).
• All ports matched, lossless, reciprocal. Uses: balanced mixers, duplexers, impedance bridges, monopulse radar.

Dominant vs degenerate modes

Dominant mode among the given TM modes for a = 3b

For TM modes the field Ez ∝ sin(mπx/a)·sin(nπy/b), so m = 0 or n = 0 gives zero field. Therefore TM01, TM10, TM02 and TM20 do not exist. The remaining candidates are TM11, TM21 and TM12:

 fc,mn = (c/2) √((m/a)² + (n/b)²),   a = 3b
       = (c/2b) √((m/3)² + n²)

 TM11: √(1/9 + 1)  = 1.054   -> fc = 1.054 · c/(2b)
 TM21: √(4/9 + 1)  = 1.202   -> fc = 1.202 · c/(2b)
 TM12: √(1/9 + 4)  = 2.028   -> fc = 2.028 · c/(2b)

Answer: TM11 has the lowest cut-off frequency, so it is the dominant mode among the listed TM modes (order: TM11 < TM21 < TM12; TM10, TM01, TM20, TM02 do not exist).