--- title: "Impedance matching techniques" subject: "Electromagnetism" chapter: 5 key: "semester_3/electromagnetism:impedance-matching-quarter-wave-stub" source: https://www.bctengineer.com/q/semester_3/electromagnetism/impedance-matching-quarter-wave-stub site: BCT Engineering --- # Impedance matching techniques Electromagnetism · Chapter 5: Transmission lines - Asked in: 2070 Chaitra, 2 marks; 2069 Chaitra, 2 marks; 2068 Chaitra, 3+3 marks (asked 3 times) - Video: [Electro Magnetics Theory - Impedance Matching](https://www.youtube.com/watch?v=7Xe2xmEAH7M) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/impedance-matching-quarter-wave-stub - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-5 ## Question What are the techniques that can be taken to match the transmission line with a mismatched load? Explain impedance matching using quarter wave transformer and single stub methods. ## Short answer Quarter-wave transformer: insert λ/4 line of Z₀′ = √(Z₀ Z_L) between line and load; single-stub: connect a shorted stub of length l in shunt at distance d from load, where d makes y = 1±jb and l gives susceptance ∓jb, canceling the imaginary part. ## Answer A mismatched load causes reflections, standing waves, power loss and possible damage to the source. Matching makes the line see a load equal to Z₀, so Γ = 0 and SWR = 1. #### Matching techniques 1. Quarter-wave transformer (λ/4 section). 2. Single-stub matching (short- or open-circuited stub in shunt). 3. Double-stub (and triple-stub) matching: fixed stub positions, adjustable lengths. 4. Lumped L-C matching networks at low frequencies. 5. Tapered lines (exponential, linear) for broadband matching. 6. Changing the load itself (antenna design, loading coils). #### Quarter-wave transformer A λ/4 line of characteristic impedance Z₀′ inserted between the line and the load gives Z_in = Z₀′²/Z_L. For a match we need Z_in = Z₀: **Z₀′ = √(Z₀ Z_L)** (for a real Z_L) ```text Z0 line ----[ Z0' , λ/4 ]---- Z_L ``` Example: to match a 100 Ω load to a 50 Ω line, Z₀′ = √(50×100) = 70.7 Ω. For a complex Z_L, move toward the generator to the nearest voltage maximum (or minimum). There the impedance is purely resistive (R_max = Z₀·S at a maximum, R_min = Z₀/S at a minimum), and the transformer is placed there with Z₀′ = √(Z₀ R_max) (or √(Z₀ R_min)). Limitation: it matches at one frequency only (narrow band). #### Single-stub matching A short-circuited stub is connected in parallel with the main line at a distance d from the load. ```text Generator ---------+--------- d ---------- Z_L (Z0) | (Z0) +-- stub (length l, Z0, shorted end) ``` Working with admittances (y = Y/Y₀): 1. Find y_L = 1/z_L. 2. Move toward the generator a distance d until the normalised admittance becomes y = 1 ± jb (it meets the g = 1 circle). 3. The stub, of length l, supplies a susceptance ∓jb that cancels the imaginary part. A shorted stub has y = −j cot βl, so l comes from cot βl = ±b. 4. The total admittance is now Y₀, so the line is matched from the stub position back to the generator. On the Smith chart: mark y_L, rotate it clockwise on its constant-|Γ| circle to the g = 1 circle (this gives d), read b, then move from the short-circuit point (y = ∞) toward the generator to the point −jb (this gives l). Only d and l have to be found, and no adjustable element is needed along the line, but the stub must be fixed at that particular point; this is why double-stub matching is used when the stub position cannot be changed. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does matching achieve? A. It makes the load impedance zero B. It maximises reflection C. The load appears as Z0, so Gamma = 0 and SWR = 1 D. It raises SWR to reduce power loss Answer: C. A matched line has no reflected wave. (Key point: Purpose) 2. What impedance should a lambda/4 section have to match a real load Z_L to a Z0 line? A. Z0 + Z_L B. sqrt(Z0 Z_L) C. (Z0 + Z_L)/2 D. Z0 Z_L Answer: B. Z_in = Z0'^2/Z_L = Z0 gives Z0' = sqrt(Z0 Z_L), e.g. 70.7 ohm for 50 and 100 ohm. (Key point: Quarter-wave transformer) 3. What is the main limitation of the quarter-wave transformer? A. It matches at one frequency only (narrow band) B. It cannot match real loads C. It needs an adjustable short D. It works only for Z_L < Z0 Answer: A. The section is lambda/4 only at the design frequency. (Key point: Limitation) 4. How does a single shunt stub give a match? A. It is placed at the load to double the resistance B. It is placed where y = 0 to add reactance C. It rotates the load toward the generator D. Placed where y = 1 +/- jb, it supplies -/+ jb to cancel the susceptance Answer: D. The stub position d brings the real part of y to 1, and the stub length l cancels the leftover susceptance. (Key point: Single stub) ## Related in Electromagnetism - [Derive input impedance of transmission line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aderive-input-impedance-line) — Electromagnetism, chapter 5 - [Line with 50+j50 load, first maximum, power](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-50-j50-load-first-maximum) — Electromagnetism, chapter 5 - [Smith chart, load 80+j100 on 50 Ω line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Asmith-chart-load-80-j100-50ohm) — Electromagnetism, chapter 5 - [Waveguide vs transmission line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-advantages-over-transmission-line) — Electromagnetism, chapter 6, asked 3× - [30 m line at 2 MHz, Γ SWR Zin](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Alossless-line-30m-2mhz-reflection-swr) — Electromagnetism, chapter 5 - [Smith chart, load 100+j150 on 75 Ω line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Asmith-chart-100-j150-load) — Electromagnetism, chapter 5 ## Related in other subjects - [Double-stub matching 110+j110 Ω on 100 Ω line](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-110-j110) — RF and Microwave Engineering, chapter 2, asked 2× - [Matching techniques and double-stub S-matrix](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Amatching-techniques-double-stub-smatrix) — RF and Microwave Engineering, chapter 2 - [Single stub matching for ΓL = 0.5∠51°](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Asingle-stub-reflection-0-5-51) — RF and Microwave Engineering, chapter 2 - [Double-stub for inductive antenna load on 78 Ω line](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-inductive-load-78-ohm) — RF and Microwave Engineering, chapter 2