--- title: "Transmission line behavior at low and microwave bands" subject: "RF and Microwave Engineering" chapter: 1 key: "semester_7/rf_and_microwave_engineering:transmission-line-low-vs-microwave" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/transmission-line-low-vs-microwave site: BCT Engineering --- # Transmission line behavior at low and microwave bands RF and Microwave Engineering · Chapter 1: Introduction - Asked in: 2079 Chaitra, 10 marks - Video: [Transmission Lines: Part 1 An Introduction](https://www.youtube.com/watch?v=yezmCNGTVYU) — TheSiGuy - 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/transmission-line-low-vs-microwave - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-1 ## Question Differentiate the behavior of a transmission line at conventional low frequency and microwave bands. ## Short answer At low frequency the line is electrically short, so it is just a connection with no phase change and input impedance equals the load. At microwave the line is many wavelengths long, so it is distributed with characteristic impedance, phase delay, reflections, standing waves, and can act as stubs, transformers, or resonators. ## Answer At low frequency, a transmission line (for example a pair of wires) is only a connection. At microwave frequency the same line is several wavelengths long, so it becomes a **circuit element** whose behaviour depends on its length, its characteristic impedance and the load. #### Low-frequency behaviour - Length ≪ λ, so there is no phase change from end to end, and V and I are the same at both ends. - Modelled with lumped series R and L, and sometimes shunt C (power line models). - Load impedance is seen directly at the input: Zin ≈ ZL. - Losses are mainly I²R. Skin effect and radiation are negligible. - Mismatch does not matter much, and there are no standing waves. - Two-wire lines, twisted pair and ordinary cable are used. #### Microwave behaviour - Length is comparable to λ, so the line is treated as **distributed** with R′, L′, G′, C′ per metre. - Voltage and current are waves: V(z) = V⁺e^(-γz) + V⁻e^(γz), with γ = α + jβ. - Characteristic impedance Z0 = √((R′ + jωL′)/(G′ + jωC′)) ≈ √(L′/C′). - Input impedance changes with length: Zin = Z0 (ZL + jZ0 tan βl)/(Z0 + jZL tan βl). - A λ/4 line inverts the impedance: Zin = Z0²/ZL. - A λ/2 line repeats the load. - Shorted or open stubs act as pure L or C. - Mismatch gives reflection Γ = (ZL − Z0)/(ZL + Z0), standing waves and VSWR, so matching is required. - Skin effect (δ = 1/√(πfμσ)) and dielectric loss raise the attenuation. Open lines radiate, so coax, waveguide, microstrip or stripline are used. - Higher-order modes appear when the cross-section approaches λ/2. #### Summary table | Property | Low frequency | Microwave | |---|---|---| | Electrical length | Negligible | Several λ | | Model | Lumped | Distributed (telegrapher's equations) | | Zin | ≈ ZL | Depends on l, Z0, ZL | | Standing waves | None | Present if mismatched | | Loss mechanism | I²R | Skin, dielectric, radiation | | Line types | Two-wire, cable | Coax, waveguide, microstrip, stripline | | Phase delay | None | βl, important | | Use as component | No | Stubs, transformers, filters, resonators | | Analysis tools | Circuit laws | Smith chart, S-parameters | #### Example A 7.5 cm shorted line at 1 GHz (λ = 30 cm) is λ/4 long, so it looks like an **open circuit**. At 50 Hz the same piece of wire is a dead short. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. At low frequency, what is Zin of a short transmission line? A. Approximately ZL B. Always Z0 C. Always zero D. Always infinite Answer: A. With length much less than lambda there is no phase change, so Zin is about ZL. (Key point: Input impedance) 2. What does a lambda/4 line do to a load impedance? A. Repeats it B. Inverts it: Zin = Z0^2/ZL C. Doubles it D. Makes it zero Answer: B. Zin = Z0^2/ZL at lambda/4; lambda/2 repeats the load. (Key point: Quarter-wave) 3. A 7.5 cm shorted line at 1 GHz (lambda = 30 cm) looks like what? A. A short circuit B. A resistor C. An open circuit D. A matched load Answer: C. It is lambda/4 long, so the short is inverted into an open. (Key point: Example) 4. What is the reflection coefficient of a load ZL on a line Z0? A. (ZL + Z0)/(ZL - Z0) B. (ZL - Z0)/(ZL + Z0) C. ZL/Z0 D. Z0/ZL Answer: B. Gamma = (ZL - Z0)/(ZL + Z0). 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