--- title: "Importance of S-parameters and two-port derivation" subject: "RF and Microwave Engineering" chapter: 3 key: "semester_7/rf_and_microwave_engineering:s-parameters-two-port-derivation" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/s-parameters-two-port-derivation site: BCT Engineering --- # Importance of S-parameters and two-port derivation RF and Microwave Engineering · Chapter 3: RF and Microwave Network Theory and Analysis - Asked in: 2080 Baisakh, 2+6 marks; 2071 Magh, 4+5 marks; 2069 Bhadra (old course), 4+6 marks (asked 3 times) - Video: [Scattering Parameters of Two Port Network in Microwave Engineering by Engineering Funda](https://www.youtube.com/watch?v=fF1I4_PQMXA) — 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/s-parameters-two-port-derivation - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-3 ## Question Why is S-parameter important in microwave network analysis? Using a two-port network, derive the S-parameters. ## Short answer S-parameters are crucial because they use matched loads, avoid open/short instability, and directly give gain and reflection. For a two-port, defining normalized waves a_n=V_n^+/√Z0, b_n=V_n^-/√Z0, we get b1=S11 a1+S12 a2, b2=S21 a1+S22 a2, measured by terminating unused ports in Z0. ## Answer #### Importance of S-parameters in microwave network analysis - At microwave frequencies, voltage and current are difficult to measure (and not uniquely defined in waveguides), but **power waves** travelling in and out of a port are easy to measure. - Z, Y, h parameters need open or short terminations; these are hard to realise at high frequencies and can make active devices oscillate. S-parameters are measured with **matched (Z0) loads**. - S-parameters directly give reflection coefficient, VSWR, return loss, insertion loss, gain and isolation. - They can be measured accurately with a **vector network analyzer** and cascaded (via T-parameters) for system design. #### Derivation of S-parameters for a two-port network ```text a1 -> -> b2 o-----------+-------+-----------o port 1 | [S] | port 2 o-----------+-------+-----------o <- b1 <- a2 ``` At each port *n* the total voltage and current are the sum of incident (+) and reflected (−) waves on a line of characteristic impedance Z0: ```text Vn = Vn+ + Vn- In = (Vn+ - Vn-)/Z0 ``` Define the **normalized incident and reflected waves** (their squares are power): ```text an = Vn+/√Z0 = (Vn + Z0·In)/(2√Z0) bn = Vn-/√Z0 = (Vn - Z0·In)/(2√Z0) Incident power = |an|²/2 Reflected power = |bn|²/2 ``` For a linear network the outgoing waves are linear combinations of the incoming waves: ```text b1 = S11·a1 + S12·a2 b2 = S21·a1 + S22·a2 | b1 | | S11 S12 | | a1 | | b2 | = | S21 S22 | | a2 | ``` Each parameter is found by making one incident wave zero, i.e. terminating that port in a **matched load Z0** (a matched load reflects nothing, so aₖ = 0): - **S11 = b1/a1 |a2=0**: input reflection coefficient with output matched. - **S21 = b2/a1 |a2=0**: forward transmission coefficient (gain or loss). - **S22 = b2/a2 |a1=0**: output reflection coefficient with input matched. - **S12 = b1/a2 |a1=0**: reverse transmission (isolation). #### Relation with impedance parameters Starting from V = Z·I and substituting V = √Z0(a + b) and I = (a − b)/√Z0 gives ```text √Z0 (a + b) = Z (a - b)/√Z0 (Z - Z0·U) a = (Z + Z0·U) b [S] = (Z - Z0·U)(Z + Z0·U)^-1 (U = unit matrix) ``` For a one-port this reduces to S11 = (ZL − Z0)/(ZL + Z0) = Γ, the familiar reflection coefficient. #### Useful results - Return loss = −20 log|S11| dB; insertion loss = −20 log|S21| dB. - Input VSWR = (1 + |S11|)/(1 − |S11|). - Reciprocal network: S12 = S21; lossless network: |S11|² + |S21|² = 1 and S11·S12* + S21·S22* = 0. - Example: an ideal matched attenuator of 3 dB has S11 = S22 = 0, S12 = S21 = 1/√2 ≈ 0.707. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How are the normalized incident and reflected waves defined at port n? A. an = Vn+ x sqrt(Z0), bn = Vn- x sqrt(Z0) B. an = Vn+/sqrt(Z0), bn = Vn-/sqrt(Z0) C. an = In+ / Z0, bn = In- / Z0 D. an = Vn, bn = In Answer: B. Dividing the wave voltages by sqrt(Z0) makes |an|^2 and |bn|^2 proportional to power. (Key point: Normalized waves) 2. To find S11 and S21, a2 is set to zero. How is this done physically? A. Open-circuit port 2 B. Short-circuit port 2 C. Terminate port 2 in a matched load Z0 so nothing is reflected back D. Remove port 2 from the network Answer: C. A matched load absorbs all power and reflects none, so the incident wave a2 at port 2 is zero. (Key point: Setting a2 = 0) 3. Which parameter is the reverse transmission (isolation), and how is it defined? A. S12 = b1/a2 with a1 = 0 B. S12 = b2/a1 with a2 = 0 C. S12 = b1/a1 with a2 = 0 D. S12 = b2/a2 with a1 = 0 Answer: A. S12 is the wave leaving port 1 due to excitation at port 2, measured with port 1 matched (a1 = 0). (Key point: Meaning of S12) 4. From the normalized relations, the S-matrix in terms of Z is A. [S] = (Z + Z0 U)(Z - Z0 U)^-1 B. [S] = (Z - Z0 U)(Z + Z0 U)^-1 C. [S] = Z / Z0 D. [S] = Z Z0 Answer: B. Substituting V = sqrt(Z0)(a+b) and I = (a-b)/sqrt(Z0) gives (Z - Z0 U)a = (Z + Z0 U)b, so S = (Z - Z0 U)(Z + Z0 U)^-1. 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