--- title: "S-matrices of E-plane tee, H-plane tee and duplexer" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:eplane-hplane-tee-duplexer-smatrix" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/eplane-hplane-tee-duplexer-smatrix site: BCT Engineering --- # S-matrices of E-plane tee, H-plane tee and duplexer RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2081 Baisakh, 8 marks - 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/eplane-hplane-tee-duplexer-smatrix - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Prepare S-Matrices of a perfectly working E-Plane Tee, H-Plane Tee and a Duplexer, and explore their uses. ## Short answer E-plane tee has S13=1/√2, S23=-1/√2, splitting equally 180 degrees out of phase, useful in balanced mixers. H-plane tee has S13=S23=1/√2, splitting in phase for power dividers. A duplexer is an ideal circulator with S=[[0,0,1],[1,0,0],[0,1,0]], directing TX to antenna and antenna to RX while isolating TX from RX. ## Answer For each junction, port numbering is: ports 1 and 2 are the collinear arms and port 3 is the side arm. A "perfectly working" device means a lossless junction with its side arm matched (for tees) or all ports matched (for the duplexer). #### 1. E-plane tee (series tee) The side arm is in the plane of the E-field (on the broad wall). A wave fed into port 3 splits equally into ports 1 and 2 with **180° phase difference**. ```text | 1/2 1/2 1/√2 | [S] = | 1/2 1/2 -1/√2 | | 1/√2 -1/√2 0 | ``` Check: each row has |S|² sum = 1/4 + 1/4 + 1/2 = 1 and rows are orthogonal, so it is lossless; S33 = 0 (matched side arm); S13 = −S23 (anti-phase split). **Uses:** power division with phase reversal, balanced mixers, part of the magic tee, impedance tuners (with a shorted side arm). #### 2. H-plane tee (shunt tee) The side arm is in the plane of the H-field (on the narrow wall). A wave fed into port 3 splits equally into ports 1 and 2 **in phase**. ```text | 1/2 -1/2 1/√2 | [S] = | -1/2 1/2 1/√2 | | 1/√2 1/√2 0 | ``` Again lossless and reciprocal, S33 = 0, and S13 = S23 (in-phase split). **Uses:** in-phase power dividers and combiners, feeding antenna arrays, part of the magic tee. #### 3. Duplexer (3-port circulator type) A duplexer lets one antenna be shared by a transmitter (port 1), antenna (port 2) and receiver (port 3): transmitter power goes only to the antenna, and the received echo goes only to the receiver. A perfect duplexer is an ideal circulator: ```text | 0 0 1 | [S] = | 1 0 0 | TX(1) -> ANT(2) -> RX(3) | 0 1 0 | ``` - Matched at all ports, lossless (unitary), and **non-reciprocal** (S21 = 1 but S12 = 0), made with a magnetized ferrite. - Transmitter → antenna: b2 = a1. Antenna → receiver: b3 = a2. Transmitter → receiver: S31 = 0 (isolation protects the receiver). ```text TX (1) ---->+-----+----> ANT (2) | (→) | +--+--+ | v RX (3) ``` **Uses:** radar and communication transceivers using a single antenna, isolators (with port 3 terminated), reflection amplifiers. Note: the two tees are reciprocal and cannot have all three ports matched; only the non-reciprocal circulator can be lossless and matched at all ports. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A signal fed to the side arm of an E-plane tee divides into ports 1 and 2 A. equally and in phase B. unequally and in phase C. only into port 1 D. equally with 180° phase difference Answer: D. E-field lines bend in opposite directions, giving S13 = -S23. (Key point: E-plane tee phase) 2. In the H-plane tee, the split from port 3 to ports 1 and 2 is A. equal and in phase (S13 = S23) B. equal and anti-phase C. all to port 1 D. zero because of isolation Answer: A. The H-plane tee is a shunt junction with in-phase split. (Key point: H-plane tee phase) 3. The ideal circulator-type duplexer [[0,0,1],[1,0,0],[0,1,0]] sends A. TX to RX directly B. TX to antenna, antenna to RX, and gives S31 = 0 (TX to RX isolation) C. antenna to TX D. RX to antenna Answer: B. S21 = 1 means port 1 to 2 (TX to antenna) and S32 = 1 means antenna to RX. (Key point: Duplexer matrix) 4. Which device can be lossless and matched at all three ports? A. An E-plane tee B. An H-plane tee C. A non-reciprocal circulator (ferrite based) D. Any reciprocal 3-port Answer: C. Reciprocal 3-ports cannot be matched at all ports and lossless; non-reciprocal ones can. (Key point: Reciprocity) ## Related in RF and Microwave Engineering - [Short note: E-plane tee](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aeplane-tee-note) — RF and Microwave Engineering, chapter 4 - [Short note: E-plane tee against H-plane tee](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aeplane-vs-hplane-tee-note) — RF and Microwave Engineering, chapter 4 - [Duplexer for half-power airport surveillance radar](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aradar-duplexer-smatrix) — RF and Microwave Engineering, chapter 4 - [Hybrid tee with terminated arms](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Ahybrid-tee-terminated-arms) — RF and Microwave Engineering, chapter 4 - [Three-port model and magic tee with shorted arms](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Athree-port-magic-tee-shorted-arms) — RF and Microwave Engineering, chapter 4 ## Related in other subjects - [Working principle of plane table survey](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aworking-principle-plane-table) — Surveying I, chapter 6 - [Truss element stiffness matrices](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Atruss-element-stiffness-matrices) — Computational Techniques in Civil Engineering, chapter 4 - [Total station principle, importance and uses](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atotal-station-principle-importance-uses) — Surveying II, chapter 9 - [Masonry pier under axial load and in-plane moment](https://www.bctengineer.com/question.md?key=semester_5%2Fdesign_of_timber_and_masonry_structures%3Amasonry-pier-axial-moment-check) — Design of Timber and Masonry Structures, chapter 7