--- title: "Double-stub tuner for 75+j75 Ω on 50 Ω line" subject: "RF and Microwave Engineering" chapter: 2 key: "semester_7/rf_and_microwave_engineering:double-stub-tuner-75-j75" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/double-stub-tuner-75-j75 site: BCT Engineering --- # Double-stub tuner for 75+j75 Ω on 50 Ω line RF and Microwave Engineering · Chapter 2: RF and Microwave Transmission Lines - Asked in: 2069 Bhadra (old course), 3+15 marks - 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/double-stub-tuner-75-j75 - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-2 ## Question What is double-stub tuner? Assuming a load of 75 + j75 ohm is connected to a 50-ohm transmission line, find the lengths and spacing for a two-stub impedance matching system with three-eighths wavelength separation between the stubs. ## Short answer A double-stub tuner uses two fixed shunt stubs separated by a fixed distance; only stub lengths are adjusted. For 75+j75 Ω on 50 Ω line with 3λ/8 spacing, place stub 1 at the load and stub 2 0.375λ away. Short-stub lengths: solution A 0.0981λ and 0.0477λ; solution B 0.2625λ and 0.3917λ. ## Answer #### Double-stub tuner A **double-stub tuner** is an impedance-matching network with two shunt stubs (usually short-circuited) at **fixed positions** on the line, a fixed distance apart (λ/8, 3λ/8 or λ/4). Only the stub lengths are adjusted. Stub 1 moves the admittance onto the rotated g = 1 circle. After travelling the spacing d this point lands on the g = 1 circle, and stub 2 cancels the remaining susceptance. It suits adjustable coaxial or waveguide tuners because nothing slides along the main line. Its limit is the forbidden region g > 1/sin²βd, which cannot be matched. #### Design **Given:** ZL = 75 + j75 Ω, Z0 = 50 Ω, spacing 3λ/8. **Assumptions:** short-circuited stubs, with the first stub at the load. #### Step 1: Normalise and convert to admittance ```text zL = 1.500 + j1.500 yL = 1/zL = 0.333 − j0.333 ΓL = 0.542∠40.6°, VSWR = 3.370 ``` On the chart, zL is at 0.1936λ WTG and yL is diametrically opposite, at 0.4436λ WTG. #### Step 2: First stub at the load The first stub is placed at the load, so y1 = yL = 0.333 − j0.333. #### Step 3: Forbidden-region check and rotated g = 1 circle ```text βd = 360° × 0.375 = 135° g must be ≤ 1/sin²βd = 2.000 g1 = 0.3333 → matchable ``` Rotate the g = 1 circle by 0.375λ (270° on the chart) toward the load. The point just after stub 1 must lie on this rotated circle, so move from y1 along the constant g = 0.333 circle until it meets the rotated circle. #### Step 4: Susceptance of stub 1 Analytically (Pozar), with t = tan βd: ```text t = tan 135° = −1.0000 b1 = −B + [1 ± √((1+t²)g − g²t²)]/t (1+t²)g − g²t² = 0.5556, √ = 0.7454 Sol A: b1 = −1.4120 Sol B: b1 = +0.0787 ``` #### Step 5: Move 0.375λ to stub 2 and find b2 ```text Sol A: y after stub 1 = 0.333 − j1.745 (0.3309λ) → after 0.375λ: y2 = 1.000 + j3.236 b2 = −3.2361 Sol B: y after stub 1 = 0.333 − j0.255 (0.4559λ) → after 0.375λ: y2 = 1.000 − j1.236 b2 = +1.2361 ``` Both y2 values lie on the g = 1 circle, as required. #### Step 6: Stub lengths - For a short-circuited stub: y = −j cot βl, so l = (1/2π)·cot⁻¹(−b); measure from the short-circuit point (y = ∞, right end of the chart) toward the generator. ```text Sol A short: l1 = 0.0981λ, l2 = 0.0477λ Sol B short: l1 = 0.2625λ, l2 = 0.3917λ ``` | Quantity | Solution A | Solution B | |---|---|---| | Stub 1 from load | at load | at load | | Stub spacing | 0.375λ | 0.375λ | | b1, b2 | -1.412, -3.236 | 0.079, 1.236 | | Short l1, l2 | 0.0981λ, 0.0477λ | 0.2625λ, 0.3917λ | **Answer:** stub 1 at the load, stub 2 at 3λ/8 = 0.375λ from it. Solution A: l1 = 0.0981λ, l2 = 0.0477λ (shortest). Solution B: l1 = 0.2625λ, l2 = 0.3917λ. ```text Z0=50Ω 3λ/8 o------+---------------+--------[75+j75Ω] | | stub 2 stub 1 l2=0.0477λ l1=0.0981λ SC SC ``` **Check:** y at stub 2 = 1.000 + j0.000, so VSWR drops from 3.37 to 1. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What defines a double-stub tuner? A. One stub whose position slides along the line B. Two shunt stubs at fixed positions a fixed distance apart, with only the stub lengths adjusted C. Two series stubs whose spacing is varied D. Two stubs of fixed length whose spacing is adjusted Answer: B. The stubs sit at fixed points (λ/8, 3λ/8 or λ/4 apart) and only their lengths are tuned, so nothing slides on the main line. (Key point: Definition) 2. What does stub 1 do? A. Cancels the load reactance directly B. Doubles the load conductance C. Moves the admittance onto the real axis of the chart D. Moves the admittance onto the rotated g = 1 circle so that it reaches g = 1 after the spacing Answer: D. After the spacing d, a point on the rotated g = 1 circle lands on g = 1, where stub 2 cancels its susceptance. (Key point: Role of stub 1) 3. What is the limitation of a double-stub tuner? A. Loads with g > 1/sin^2(βd) lie in the forbidden region and cannot be matched B. Loads with g < 1 cannot be matched C. It only works for purely resistive loads D. It cannot match above 1 GHz Answer: A. Admittances in the forbidden region cannot be brought onto the rotated g = 1 circle by stub 1. (Key point: Limit) 4. For ZL = 75 + j75 Ω on 50 Ω with 3λ/8 spacing, which stub lengths give the shortest solution? A. l1 = 0.2625λ, l2 = 0.3917λ (solution A) B. l1 = 0.0981λ, l2 = 0.3917λ C. l1 = 0.0981λ, l2 = 0.0477λ (solution A) D. l1 = 0.375λ, l2 = 0.375λ Answer: C. Solution A has b1 = -1.412, b2 = -3.236, giving short stubs of 0.0981λ and 0.0477λ; B is longer. (Key point: Solutions) ## Close variants set in other exams - [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× ## Related in RF and Microwave Engineering - [Double-stub with assumed load on 50 Ω line](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-arbitrary-load-50-ohm) — RF and Microwave Engineering, chapter 2 - [Double-stub shunt tuner for admittance load](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-admittance-load) — RF and Microwave Engineering, chapter 2 - [Double-stub for 80+j180 Ω at 3 GHz](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-80-j180-3ghz) — RF and Microwave Engineering, chapter 2 - [Double-stub on 75 Ω coax, load 0.4+j0.85](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-coax-normalized-load) — 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 - [Double-stub 3λ/8 for 300+j300 Ω antenna](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Adouble-stub-300-j300) — RF and Microwave Engineering, chapter 2 ## Related in other subjects - [Impedance matching techniques](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aimpedance-matching-quarter-wave-stub) — Electromagnetism, chapter 5, asked 3×