--- title: "Magic tee, degenerate modes, dominant mode for a = 3b" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:magic-tee-modes-a-3b" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/magic-tee-modes-a-3b site: BCT Engineering --- # Magic tee, degenerate modes, dominant mode for a = 3b RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2072 Magh, 6+2+2 marks - Video: [Modes in Rectangular Waveguide (TE Mode, TM Mode, Dominant Mode, Evan Scant Mode, Degenerative Mode)](https://www.youtube.com/watch?v=yW8QUzt78tI) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/magic-tee-modes-a-3b - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Describe magic Tee based on S-parameters. Differentiate between dominant and degenerate modes. Consider a rectangular waveguide having dimension of a = 3b and find the dominant mode among TM01, TM10, TM11, TM21, TM12, TM02 and TM20. ## Short answer In a magic tee, H-arm input splits equally in-phase and E-arm anti-phase (S13=S23=S14=S24=1/√2). Dominant mode has the lowest cutoff; degenerate modes share the same cutoff. For a=3b, TM10, TM01, TM02, TM20 vanish; of TM11, TM21, TM12, TM11 has the lowest cutoff fc=1.054c/(2b), making it dominant. ## Answer #### 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. ```text E-arm (4) | | -----------+ +----------- (1) ===========+ +=========== (2) -----------+ +----------- / / / / H-arm (3) ``` **S-matrix (derived from symmetry, reciprocity, E–H isolation and losslessness):** ```text | 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 | Point | Dominant mode | Degenerate modes | |---|---|---| | Meaning | Mode with the lowest cut-off frequency | Different modes with the same cut-off frequency | | Number | One per guide | Two or more modes sharing fc | | Rectangular example | TE10 (a > b) | TEmn and TMmn, e.g. TE11 and TM11 | | Circular example | TE11 | TE01 and TM11 (both use 3.832) | | Use | Chosen for single-mode operation | Avoided; cause mode conversion | #### 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: ```text 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). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For the magic tee, an input at the H-arm gives: A. Equal in-phase outputs at 1 and 2 and nothing at the E-arm B. Outputs 180° apart at 1 and 2 C. All power at port 4 D. All power reflected Answer: A. S13 = S23 = 1/√2 and S43 = 0. (Key point: Magic tee S-matrix behaviour) 2. What is a degenerate mode pair? A. Modes with the lowest cut-off frequency B. Different modes sharing the same cut-off frequency, e.g. TE11 and TM11 in a rectangular guide C. Modes that never propagate D. Modes with zero field Answer: B. Degenerate modes have the same fc, whereas the dominant mode is the single one with lowest fc. (Key point: Dominant versus degenerate) 3. Why do TM01, TM10, TM02 and TM20 not exist in a rectangular guide? A. Their cut-off is infinite B. Ez = E0 sin(mπx/a) sin(nπy/b) is zero everywhere if m = 0 or n = 0 C. They are TE modes D. They violate a = 3b Answer: B. With m or n zero, Ez vanishes and so do all fields derived from it. (Key point: Non-existent TM modes) 4. For a = 3b, with fc proportional to sqrt((m/3)^2 + n^2), which TM mode has the lowest cut-off? A. TM21 B. TM12 C. TM11 D. TM01 Answer: C. TM11 gives 1.054, TM21 gives 1.202, TM12 gives 2.028 (in units of c/2b), so TM11 is lowest. 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