--- title: "Identify device from numeric 4×4 S-matrix" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:identify-device-s-matrix-numeric" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/identify-device-s-matrix-numeric site: BCT Engineering --- # Identify device from numeric 4×4 S-matrix RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2078 Chaitra, 10 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/identify-device-s-matrix-numeric - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Identify and explain the properties of a microwave passive device having following S-Matrix with necessary diagram: [S] = [0 0 1 1; 0 0 −1 1; 1 −1 0 0; 1 1 0 0]. ## Short answer It is a magic tee (hybrid E-H tee) with ports 1 and 2 collinear, port 3 E-arm, port 4 H-arm, normalized by 1/sqrt(2). H-arm input splits equally in phase to ports 1 and 2; E-arm input splits equally anti-phase; collinear arms are isolated. ## Answer #### Identification ```text | 0 0 1 1 | [S] = | 0 0 -1 1 | | 1 -1 0 0 | | 1 1 0 0 | ``` - Ports 1 and 2 are isolated (S12 = 0) and ports 3 and 4 are isolated (S34 = 0); all ports are matched (Sᵢᵢ = 0); the matrix is symmetric (reciprocal). - Port 4 receives **equal, in-phase** signals from ports 1 and 2 (S41 = S42 = 1): port 4 is the **H-arm (sum arm)**. - Port 3 receives **equal, anti-phase** signals from ports 1 and 2 (S31 = 1, S32 = −1): port 3 is the **E-arm (difference arm)**. These are the properties of a **magic tee (hybrid E-H tee, 180° hybrid)** with ports 1 and 2 as collinear arms. **Normalization:** as written, each row has |S|² sum = 2. A passive lossless device needs a sum of 1, so the matrix is the magic-tee matrix with the common factor 1/√2 left out: ```text | 0 0 1 1 | [S] = (1/√2) | 0 0 -1 1 | | 1 -1 0 0 | | 1 1 0 0 | ``` #### Diagram ```text E-arm (3) | | -----------+ +----------- (1) ===========+ +=========== (2) -----------+ +----------- / / / / H-arm (4) ``` #### Properties (with the 1/√2 factor) 1. **Input at H-arm (port 4):** b1 = b2 = a4/√2, b3 = 0. Power divides equally and in phase between the collinear arms; the E-arm gets nothing. 2. **Input at E-arm (port 3):** b1 = a3/√2, b2 = −a3/√2, b4 = 0. Equal split with 180° phase difference; the H-arm gets nothing. 3. **Input at port 1:** b3 = b4 = a1/√2, b2 = 0. Half the power goes to each of the E and H arms; the opposite collinear arm is isolated. 4. **Equal in-phase inputs at 1 and 2** (a1 = a2 = a): b4 = √2·a (all the input power comes out of the H-arm), b3 = 0. 5. **Equal anti-phase inputs at 1 and 2** (a1 = −a2 = a): b3 = √2·a (all power at E-arm), b4 = 0. 6. **Matched and lossless:** with E and H arms matched, all four ports are matched; [S]*ᵀ[S] = I (unitary). 7. **Reciprocal:** Sᵢⱼ = Sⱼᵢ. #### Applications - **Balanced mixer:** RF and LO fed to the E and H arms, mixer diodes in arms 1 and 2; LO noise cancels and the LO is isolated from the RF port. - **Duplexer:** transmitter and receiver at the isolated E and H arms, antenna at a collinear arm. - **Power combiner/divider:** combining two coherent sources (sum at H-arm). - **Impedance bridge:** with a known load on one collinear arm and an unknown on the other, the E-arm output is zero only when the loads are equal. - **Monopulse radar:** sum (Σ) and difference (Δ) signals from two antenna feeds. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The given S-matrix describes A. a magic tee (hybrid E-H tee) once the 1/sqrt2 factor is included B. a circulator C. an E-plane tee D. a directional coupler with 20 dB coupling Answer: A. Isolated pairs (1,2) and (3,4), matched ports, sum/difference arms identify a magic tee. (Key point: Identification) 2. Port 4 receives equal in-phase signals from ports 1 and 2 (S41 = S42 = 1), so port 4 is A. the E-arm (difference arm) B. the H-arm (sum arm) C. a collinear arm D. a matched load Answer: B. In-phase addition marks the H-arm. (Key point: Arm identification) 3. Why must the factor 1/sqrt2 be included? A. To make the device non-reciprocal B. To remove isolation C. Each row has |S|^2 sum of 2, but a lossless device needs 1 D. To make S11 nonzero Answer: C. Rows with two entries of magnitude 1 sum to 2, so scaling by 1/sqrt2 gives unity. (Key point: Normalization) 4. Equal anti-phase inputs a1 = -a2 = a at ports 1 and 2 give A. all power at the H-arm B. half at each of E and H arms C. no output anywhere D. all power at the E-arm (port 3) and none at the H-arm Answer: D. Difference signals go to the E-arm. 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