--- title: "S-parameters and three-port network S-matrix" subject: "RF and Microwave Engineering" chapter: 3 key: "semester_7/rf_and_microwave_engineering:s-matrix-three-port-network" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/s-matrix-three-port-network site: BCT Engineering --- # S-parameters and three-port network S-matrix RF and Microwave Engineering · Chapter 3: RF and Microwave Network Theory and Analysis - Asked in: 2080 Bhadra, 4+4 marks; 2074 Magh, 2+6 marks; 2073 Bhadra, 4+4 marks; 2073 Magh, 4+4 marks (asked 4 times) - Video: [Understanding S Parameters](https://www.youtube.com/watch?v=-Pi0UbErHTY) — Rohde & Schwarz - 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-three-port-network - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-3 ## Question Why do we use S-parameters for microwave analysis? Define the S-matrix for a 3-port network with appropriate example (properties of a 3-port network). ## Short answer S-parameters are used at microwave frequencies because they rely on matched terminations instead of shorts/opens, relate incident and reflected power waves measurable in waveguides, and are stable for active devices. The 3-port S-matrix is [b]=[S][a]; a key property is that it cannot be lossless, reciprocal and matched at all ports simultaneously, leading to designs like circulators. ## Answer #### Why S-parameters are used at microwave frequencies At microwave frequencies the Z, Y, h and ABCD parameters become hard to use because: - They need ideal **open or short circuits** at the ports, which are hard to make at high frequency (stray capacitance and inductance, radiation from an open end). - Active devices (transistors) often **oscillate** or are damaged when terminated in an open or short. - Voltage and current are **not uniquely defined** in a waveguide, but incident and reflected power waves are. S-parameters are measured with all ports terminated in a **matched load (Z0)**, they relate incident and reflected waves directly, and they are measured easily with a vector network analyzer. Their magnitudes give power flow (|Sij|² = fraction of power), so return loss, insertion loss and isolation are read off directly. #### S-matrix of a 3-port network For a 3-port network, let *aᵢ* be the normalized incident wave and *bᵢ* the reflected (outgoing) wave at port *i*. Then ```text | b1 | | S11 S12 S13 | | a1 | | b2 | = | S21 S22 S23 | | a2 | | b3 | | S31 S32 S33 | | a3 | ``` - Sᵢᵢ = bᵢ/aᵢ (all other ports matched): reflection coefficient at port *i*. - Sᵢⱼ = bᵢ/aⱼ (all ports except *j* matched): transmission coefficient from port *j* to port *i*. ```text port 3 | +---+---+ port 1 --| [S] |-- port 2 +-------+ ``` #### Properties of the S-matrix of a 3-port network 1. **Reciprocity (symmetry):** for a passive network without ferrites or plasma, Sᵢⱼ = Sⱼᵢ, i.e. [S] = [S]ᵀ. 2. **Lossless (unitary) property:** [S]*ᵀ[S] = [I], so for each column Σₖ|Sₖᵢ|² = 1, and for two different columns Σₖ SₖᵢSₖⱼ* = 0. 3. **Matched ports:** if all ports are matched, S11 = S22 = S33 = 0. 4. **Phase shift property:** if the reference plane at port *n* is moved outward (away from the junction) by a length lₙ, Sᵢⱼ is multiplied by e^(−jβ(lᵢ + lⱼ)). 5. **Key theorem:** a 3-port network **cannot be lossless, reciprocal and matched at all ports at the same time**. If it is lossless and matched, it must be non-reciprocal (a circulator); if it is reciprocal and lossless, at least one port is mismatched (e.g. a tee junction). #### Example 1: H-plane tee (reciprocal, lossless, port 3 matched) ```text | 1/2 -1/2 1/√2 | [S] = | -1/2 1/2 1/√2 | | 1/√2 1/√2 0 | ``` Power fed at port 3 divides equally and in phase between ports 1 and 2; ports 1 and 2 are not matched (S11 = 1/2). #### Example 2: ideal circulator (lossless, matched, non-reciprocal) ```text | 0 0 1 | [S] = | 1 0 0 | (1 -> 2 -> 3 -> 1) | 0 1 0 | ``` All ports are matched and the matrix is unitary but not symmetric, as the theorem requires. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why are Z, Y and h parameters hard to use at microwave frequencies? A. They give complex numbers B. They need ideal open or short circuits at the ports, which are hard to make and may make active devices oscillate C. They can only be used with DC D. They need matched loads at every port Answer: B. Open and short terminations suffer from stray reactances and radiation, and can destabilize transistors. S-parameters use matched loads instead. (Key point: Why S-parameters) 2. How is Sij (i not equal to j) measured? A. bi/aj with all ports except j terminated in matched loads B. bi/aj with all ports except j open-circuited C. ai/bj with all ports shorted D. bj/ai with port j shorted Answer: A. Sij = bi/aj is the transmission from port j to port i, measured with every other port terminated in a matched load so their incident waves are zero. (Key point: Definition of Sij) 3. Which statement about a 3-port network is correct? A. It can be lossless, reciprocal and matched at all ports simultaneously B. It cannot be lossless, reciprocal and matched at all ports at the same time C. Every 3-port must be reciprocal D. A matched 3-port must always have Sii not equal to 0 Answer: B. The three conditions cannot all hold together. A lossless matched 3-port must be non-reciprocal (circulator), and a reciprocal lossless one has a mismatched port (tee). (Key point: Key theorem for 3-ports) 4. The ideal circulator has S21 = S32 = S13 = 1 and others zero. What does this mean? A. Power flows 1 to 2 to 3 to 1, all ports matched, non-symmetric matrix B. Power flows both ways between all ports C. All ports reflect fully D. The matrix is symmetric and so reciprocal Answer: A. The ones give the rotation 1->2->3->1. Diagonal zeros mean matched, and the matrix is unitary but not symmetric, so it is non-reciprocal. 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