--- title: "Reciprocity, losslessness and return loss from S-matrix" subject: "RF and Microwave Engineering" chapter: 3 key: "semester_7/rf_and_microwave_engineering:s-matrix-reciprocal-lossless-test" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/s-matrix-reciprocal-lossless-test site: BCT Engineering --- # Reciprocity, losslessness and return loss from S-matrix RF and Microwave Engineering · Chapter 3: RF and Microwave Network Theory and Analysis - Asked in: 2079 Bhadra, 3+1+1+1+2 marks - Video: [S-Parameters #5. Lossless & Reciprocal Networks Explained | Input Return Loss, S11 & Matching Basics](https://www.youtube.com/watch?v=_YmHF5COg_w) — Technologies Discussion - 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/s-matrix-reciprocal-lossless-test - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-3 ## Question Why S-parameters are used in high frequencies? The S-matrix of certain microwave network is given as S = [0.4 + j0.5 j0.6; j0.6 0.4 − j0.5]. a) Is the network reciprocal? b) Is the network lossless? c) What is the return loss at the input? d) If the input power to the network is 5 watts, what is the reflected power? ## Short answer S12=S21 so reciprocal. Column sums |S11|^2+|S21|^2=0.77≠1, so not lossless. Return loss = -20 log|S11| = 3.87 dB; reflected power = |S11|^2 × 5 = 2.05 W. ## Answer #### Why S-parameters are used at high frequencies At microwave frequencies voltages and currents are hard to measure (and not unique in waveguides), and the open/short terminations needed for Z, Y or h parameters are hard to realise and may make active devices oscillate. S-parameters use **incident and reflected waves** measured with **matched loads**, they are measured directly by a network analyzer, and they give reflection, transmission and power flow directly. #### Given ```text S11 = 0.4 + j0.5 S12 = j0.6 S21 = j0.6 S22 = 0.4 - j0.5 ``` #### a) Is the network reciprocal? A network is reciprocal if [S] is symmetric, S12 = S21. Here S12 = j0.6 = S21, so **the network is reciprocal**. #### b) Is the network lossless? A lossless network has a unitary S-matrix, so each column must satisfy |S11|² + |S21|² = 1. ```text |S11|² = 0.4² + 0.5² = 0.16 + 0.25 = 0.41 |S21|² = 0.6² = 0.36 |S11|² + |S21|² = 0.77 ≠ 1 ``` (The second column also gives 0.41 + 0.36 = 0.77. The orthogonality condition S11·S12* + S21·S22* = 0 happens to hold, but the column sums fail.) So **the network is not lossless**: 23% of the power fed at port 1 is absorbed in the network. #### c) Return loss at the input ```text |S11| = √0.41 = 0.6403 RL = -20 log|S11| = -20 log(0.6403) = 3.87 dB ``` **Answer: return loss at the input = 3.87 dB** #### d) Reflected power for 5 W input ```text Pref = |S11|² · Pin = 0.41 × 5 = 2.05 W ``` **Answer: reflected power = 2.05 W** (For interest: power transmitted to a matched port 2 = 0.36 × 5 = 1.80 W, and power lost in the network = 5 − 2.05 − 1.80 = 1.15 W.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Is S = [0.4+j0.5, j0.6; j0.6, 0.4-j0.5] reciprocal, and why? A. Yes, because S12 = S21 = j0.6 B. No, because S11 is not equal to S22 C. No, because the entries are complex D. Yes, because S11 = S22 Answer: A. Reciprocity needs only the matrix to be symmetric, S12 = S21, which holds. S11 = S22 is the symmetry (not reciprocity) condition. (Key point: Reciprocity test) 2. For this matrix, |S11|^2 + |S21|^2 equals A. 1.00, lossless B. 0.77, so the network is not lossless C. 0.41 only D. 1.77 Answer: B. |S11|^2 = 0.16 + 0.25 = 0.41 and |S21|^2 = 0.36, giving 0.77, not 1, so it is lossy. (Key point: Lossless test) 3. The input return loss is A. -20 log(0.41) = 7.7 dB B. -20 log(0.6403) = 3.87 dB C. -10 log(0.77) = 1.1 dB D. 20 log(0.6) = -4.4 dB Answer: B. |S11| = sqrt(0.41) = 0.6403, so RL = -20 log(0.6403) = 3.87 dB. (Key point: Return loss) 4. With 5 W input, the reflected power is A. 3.2 W B. 2.05 W C. 1.8 W D. 3.85 W Answer: B. Pref = |S11|^2 x Pin = 0.41 x 5 = 2.05 W. 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