--- title: "Single-section series arm BPF and insertion loss" subject: "RF and Microwave Engineering" chapter: 6 key: "semester_7/rf_and_microwave_engineering:bpf-microstrip-insertion-loss" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/bpf-microstrip-insertion-loss site: BCT Engineering --- # Single-section series arm BPF and insertion loss RF and Microwave Engineering · Chapter 6: RF Design Practices - Asked in: 2081 Bhadra, 2+8 marks - Video: [RF Filter Design Part 5 Insertion Loss Method Introduction](https://www.youtube.com/watch?v=pHojDXV-L58) — RF MICROWAVE CIRCUIT - 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/bpf-microstrip-insertion-loss - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-6 ## Question Illustrate an example of a single section series arm passive BPF using microstrip. How are microwave filters designed using Insertion Loss Method? ## Short answer A single-section series-arm BPF is realized in microstrip as a λg/2 resonator coupled by input and output gaps; gap widths set coupling and bandwidth. Insertion-loss design: choose response and order from PLR, read g-values, scale and LP-to-BP transform, realize with coupled resonators or inverters, then layout and tune. ## Answer #### Single-section series-arm passive BPF in microstrip A single-section band-pass filter has a **series LC resonator in the series arm**. It passes signals near its resonant frequency f₀ = 1/(2π√(LC)) and blocks others. In microstrip, the series resonator is made as a **half-wavelength (λg/2) line** coupled to the input and output lines through **gaps**. The gaps act as series capacitors, and the λ/2 line behaves like a series-resonant circuit near f₀. ```text Lumped: in --| |--UUU-- out C L (series arm) Microstrip (end-coupled): 50 Ω λg/2 resonator 50 Ω =========| |=========================| |========= gap1 gap2 ``` The gap widths set the coupling and so the bandwidth. A smaller gap gives more coupling and a wider band. Several such resonators in cascade give a higher-order end-coupled BPF. A parallel-coupled-line layout is a more compact alternative. #### Filter design by the insertion loss method The **insertion loss method** starts from the wanted power loss ratio and synthesises a network that gives it: ```text PLR = 1 / (1 − |Γ(ω)|²) = 1 + M(ω²)/N(ω²) IL (dB) = 10 log10 PLR ``` **Steps** 1. **Specifications:** centre frequency f₀, fractional bandwidth Δ = (ω₂ − ω₁)/ω₀, passband ripple, stop-band attenuation, and R₀ = 50 Ω. 2. **Response and order:** Butterworth, PLR = 1 + k²(ω/ωc)²ᴺ (maximally flat), or Chebyshev, PLR = 1 + k²T_N²(ω/ωc) (equal ripple, sharper). Find N from attenuation charts. 3. **Low-pass prototype:** read the g-values from tables (R₀ = 1, ωc = 1). For example, Butterworth N = 3 gives 1, 2, 1. 4. **Impedance scaling and LP → BP transformation**, with ω → (1/Δ)(ω/ω₀ − ω₀/ω): - series L_k becomes a series LC: L' = R₀g_k/(ω₀Δ), C' = Δ/(ω₀R₀g_k) - shunt C_k becomes a parallel LC: L' = R₀Δ/(ω₀g_k), C' = g_k/(ω₀R₀Δ) 5. **Microwave realisation:** use admittance or impedance inverters with λ/2 resonators (end-coupled), λ/4 coupled lines (parallel-coupled), or stub resonators. Find the gap or coupling values from the inverter constants. 6. **Layout and tuning:** include gap, open-end and step effects, then simulate and tune the lengths and gaps. The method gives a predictable, exactly specified response, which is why it is the standard method for microwave filters. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is a single-section series-arm BPF made in microstrip? A. A λg/2 line coupled to the input and output lines by gaps B. A λ/4 short-circuited stub C. A wide low-impedance pad D. A via to ground Answer: A. The gaps act as series capacitors and the λ/2 line acts as a series resonant circuit near f0. (Key point: Series resonator in microstrip) 2. What effect does a smaller gap have in the end-coupled BPF? A. More coupling and a wider band B. Less coupling and a narrower band C. No change D. Lower resonant frequency only Answer: A. Gap widths set the coupling, and so the bandwidth. (Key point: Gap width) 3. Which substitution does the LP to BP transformation use? A. ω → (1/Δ)(ω/ω0 - ω0/ω) B. ω → ω0/ω C. ω → ω/ωc D. ω → 1/ω Answer: A. Δ is the fractional bandwidth. Series L becomes a series LC and shunt C becomes a parallel LC. (Key point: LP to BP transformation) 4. How does the insertion loss method define the filter response? A. Through the power loss ratio PLR = 1/(1 - |Γ|^2) B. Through the image impedance C. Through the line length only D. Through the substrate thickness Answer: A. A network is synthesised to give a chosen PLR, e.g. Butterworth or Chebyshev. 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