--- title: "Three-port model and magic tee with shorted arms" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:three-port-magic-tee-shorted-arms" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/three-port-magic-tee-shorted-arms site: BCT Engineering --- # Three-port model and magic tee with shorted arms RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2082 Bhadra, 4+4 marks - Video: [Magic Tee Junction (Basics, Working, Structure & Scattering Parameters) Explained](https://www.youtube.com/watch?v=0Krr-FoWLeU) — 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/three-port-magic-tee-shorted-arms - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Design a model of a three-port network and define its characteristic parameters. Prepare the S-Matrix of a rectangular magic tee having shorted both E and H-arms and explain how it behaves. ## Short answer A three-port network is characterized by S parameters: Sii is reflection, Sij transmission, reciprocity gives Sij=Sji, and losslessness gives unitarity. With both E and H arms shorted at their reference planes, the reduced two-port S matrix is [[-1,0],[0,-1]], so each collinear port reflects fully with 180 degree phase shift; moving the shorts can make it a switch or phase shifter. ## Answer #### Three-port network model and its parameters A three-port network (e.g. an E-plane or H-plane tee, circulator or power divider) is described by its scattering matrix, relating outgoing waves *b* to incident waves *a*: ```text port 3 | +---+---+ port 1 --| [S] |-- port 2 +-------+ | b1 | | S11 S12 S13 | | a1 | | b2 | = | S21 S22 S23 | | a2 | | b3 | | S31 S32 S33 | | a3 | ``` **Characteristic parameters:** - Sᵢᵢ: reflection coefficient at port *i* (return loss = −20 log|Sᵢᵢ|). - Sᵢⱼ: transmission from port *j* to *i* (insertion loss = −20 log|Sᵢⱼ|). - Reciprocal: Sᵢⱼ = Sⱼᵢ. Lossless: [S]*ᵀ[S] = I. Matched: Sᵢᵢ = 0. - A 3-port cannot be lossless, reciprocal and fully matched at the same time. Example, H-plane tee: S = [[1/2, −1/2, 1/√2], [−1/2, 1/2, 1/√2], [1/√2, 1/√2, 0]]. #### Magic tee with both E and H arms shorted Ideal magic tee (1, 2 collinear; 3 = H-arm; 4 = E-arm): ```text b1 = (a3 + a4)/√2 b3 = (a1 + a2)/√2 b2 = (a3 - a4)/√2 b4 = (a1 - a2)/√2 ``` A short at the reference plane of an arm reflects the wave with Γ = −1, so a3 = −b3 and a4 = −b4: ```text a3 = -(a1 + a2)/√2, a4 = -(a1 - a2)/√2 b1 = (a3 + a4)/√2 = -(2a1)/2 = -a1 b2 = (a3 - a4)/√2 = -(2a2)/2 = -a2 ``` So the reduced two-port (ports 1 and 2) has ```text | -1 0 | [S] = | | | 0 -1 | ``` **Behaviour:** each collinear arm behaves like a **short circuit**: all power entering port 1 (or 2) is reflected back with 180° phase, and nothing is transmitted to the other arm. The power entering port 1 splits into the E and H arms, is reflected by the shorts, and recombines entirely at port 1. **General case (shorts at distances l3 and l4):** Γ3 = −e^(−j2βl3), Γ4 = −e^(−j2βl4). Then ```text b1 = ((Γ3 + Γ4)/2)·a1 + ((Γ3 - Γ4)/2)·a2 b2 = ((Γ3 - Γ4)/2)·a1 + ((Γ3 + Γ4)/2)·a2 ``` If the shorts differ in position by λg/4 (Γ4 = −Γ3), then S11 = 0 and |S21| = 1: all power passes from port 1 to port 2. By moving the shorts, the tee can act as a variable reflector, switch or phase shifter; this property is used in magic-tee phase shifters and duplexers. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is impossible for a three-port network? A. Being lossless, reciprocal and matched at all ports at once B. Being reciprocal C. Having an S-matrix D. Being matched at one port Answer: A. Unitarity conditions cannot be satisfied simultaneously for a matched reciprocal lossless 3-port. (Key point: 3-port limitation) 2. A short at the reference plane of an arm reflects with Γ equal to A. +1 B. -1, so a3 = -b3 and a4 = -b4 C. 0 D. 1/sqrt2 Answer: B. A short circuit reverses the voltage, giving Γ = -1. (Key point: Short reflection) 3. With both E and H arms shorted, the reduced S-matrix of ports 1 and 2 is A. [[0, 1],[1, 0]]: full transmission B. [[1, 0],[0, 1]]: reflection in phase C. [[-1, 0],[0, -1]]: each collinear port reflects with 180°, no transmission D. [[0, 0],[0, 0]]: absorption Answer: C. b1 = -a1 and b2 = -a2, so each collinear arm looks like a short. (Key point: Resulting S-matrix) 4. If the shorts differ in position by λg/4 (Γ4 = -Γ3), then A. S11 = 1 and S21 = 0 B. S11 = S21 = 1/sqrt2 C. S21 = 0 and S11 = -1 D. S11 = 0 and |S21| = 1, so all power passes from port 1 to port 2 Answer: D. With Γ4 = -Γ3 the reflection terms cancel in b1 and add in b2. 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