--- title: "Three-port directional coupler using S-parameters" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:directional-coupler-three-port" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/directional-coupler-three-port site: BCT Engineering --- # Three-port directional coupler using S-parameters RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2075 Bhadra, 4 marks - Video: [Directional Coupler (Basics, Working, Structure, Scattering Parameters & Applications) Explained](https://www.youtube.com/watch?v=4poo1q3GvTg) — 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/directional-coupler-three-port - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Analyze a three-port directional coupler using S-parameters. ## Short answer A lossless, reciprocal, matched three-port coupler is impossible. A practical three-port is a four-port coupler with the isolated port terminated internally; its S-matrix has through coefficient alpha and coupled coefficient j beta, with coupling C = -20 log beta dB. ## Answer A three-port directional coupler is analysed with the 3 × 3 S-matrix, using port 1 = input, 2 = through, 3 = coupled. **Step 1 – can a lossless, reciprocal, fully matched 3-port exist?** Suppose S11 = S22 = S33 = 0 and Sᵢⱼ = Sⱼᵢ: ```text | 0 S12 S13 | [S] = | S12 0 S23 | | S13 S23 0 | ``` Unitary conditions: ```text |S12|² + |S13|² = 1 |S12|² + |S23|² = 1 |S13|² + |S23|² = 1 S13*·S23 = 0, S23*·S12 = 0, S12*·S13 = 0 ``` The last three need at least two of S12, S13, S23 to be zero, which then breaks one of the first three. So **no lossless, reciprocal, matched 3-port coupler exists**. (If reciprocity is dropped, a lossless matched 3-port is possible: that is the circulator.) **Step 2 – practical 3-port coupler.** It is a 4-port coupler whose isolated port is terminated internally in a matched load. With the ideal 4-port coupler (S12 = α, S13 = jβ, α² + β² = 1), deleting port 4: ```text | 0 α jβ | [S] = | α 0 0 | | jβ 0 0 | ``` It is matched and reciprocal but **lossy** (power entering port 2 or 3 is partly absorbed in the internal load). Coupling C = −20 log β dB; e.g. β = 0.316 gives a 10 dB coupler. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A lossless, reciprocal, matched 3-port is impossible because A. it would violate Maxwell's equations at DC B. matching always creates loss C. reciprocity requires S11 ≠ 0 D. the unitary conditions force at least two of S12, S13, S23 to be zero, breaking the power conditions Answer: D. Orthogonality of columns requires two zero entries which contradicts unit row norms. (Key point: Impossibility) 2. A practical 3-port coupler is A. a 4-port coupler with its isolated port terminated in a matched load B. a lossless Y-junction C. a circulator D. a short-circuited tee Answer: A. The internal load makes it matched and reciprocal but lossy. (Key point: Practical coupler) 3. With β = 0.316, the coupling C = -20 log β is about A. 3 dB B. 10 dB C. 20 dB D. 0.316 dB Answer: B. -20 log(0.316) = 10 dB. 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