How to convert lumped L/C/R to transmission-line equivalents on PCB

Answer: mapping L and C to transmission-line stubs

You convert a lumped L or C into PCB transmission-line form by sizing an open- or short-circuited stub (or a series stub) so its input reactance/susceptance at f0 matches the lumped value; a frequency‑independent resistor cannot be realized with uniform lossless lines.

Use these calculable relations (lossless line, θ = βl)

• Shunt capacitor C: B = ωC = (1/Z0) tan θ ⇒ θ = arctan(ω C Z0)

• Shunt inductor L: B = −1/(ωL) = −1/(Z0 tan θ) ⇒ tan θ = ω L / Z0 ⇒ θ = arctan(ω L / Z0)

• Series inductor L: X = ωL = Z0 tan θ ⇒ θ = arctan(ω L / Z0)

• Series capacitor C: X = −1/(ωC) = −Z0 cot θ ⇒ tan θ = ω C Z0 ⇒ θ = arctan(ω C Z0)

For electrically short stubs, tan θ ≈ θ (rad) and cot θ ≈ 1/θ; keeping θ ≤ 10–15° limits the tanθ ≈ θ error to about 3–4% at the band edge.

What does an equivalent TL form mean?

It is a distributed network (uniform lines plus open/short stubs and—if needed—impedance inverters) whose input impedance or 2‑port transfer equals the lumped design over the target band. At a single f0 you can match element‑by‑element. For bandwidth, synthesize the whole ladder: map L/C with Richards’ variable (s → j tan θ), insert unit elements to keep realizable impedances, then use Kuroda identities to convert unavoidable series stubs into shunt stubs and to swap extreme impedances. A quarter‑wave inverter with Zt transforms Zin = Zt^2/Zload at θ = 90°.

When do lumped models stop working?

Distributed effects matter once features approach λg/10. With microstrip on FR‑4 (εr ≈ 4.2), εeff is typically 2.8–3.0, so vp ≈ c/√εeff ≈ 0.58–0.60c. At 2 GHz, λ0 = 150 mm and λg ≈ 88–90 mm; any trace, gap, or component pad pattern longer than about 9 mm should be treated as distributed rather than lumped. The definitions are vp = 1/√(L′C′), β = ω√(L′C′), and θ = βl.

Worked examples at 2 GHz on FR‑4 microstrip

C → open shunt stub: Replace C = 0.5 pF at 2 GHz with an open stub, choose Z0 = 50 Ω. Compute ω C Z0 = 2π·2e9·0.5e−12·50 ≈ 0.314 ⇒ θ = arctan 0.314 ≈ 17.5° (0.305 rad). With εeff ≈ 2.9, vp ≈ 0.588c so λg ≈ 88.2 mm; length l = θ λg/(2π) ≈ 4.28 mm.

L → shorted shunt stub (corrected math): For L = 2 nH at 2 GHz with Z0 = 75 Ω, set −1/(Z0 tan θ) = −1/(ωL) ⇒ tan θ = ωL/Z0 = (2π·2e9·2e−9)/75 ≈ 0.335 ⇒ θ ≈ 18.5° (0.323 rad). Length l ≈ 0.323·88.2/2π ≈ 4.5 mm. This is electrically short; if you need a shunt inductor much larger than this, either increase Z0 to shorten θ further or use a lumped part.

Broadband synthesis with Richards and Kuroda

Normalize impedances and frequency to f0, substitute s = j tan θ to convert every lumped L/C into a TL element, insert unit elements to keep realizable Z0, and apply Kuroda identities to eliminate series stubs and to move extreme impedances into manufacturable ranges. Finally, denormalize θi(f) = 2π f li / vp and choose Z0i from the PCB geometry.

How do I pick trace widths for a target Z0?

For microstrip, IPC‑2141 formulas give first‑order width/height. Example: FR‑4 with h = 0.18 mm and εr ≈ 4.2 gives about 50 Ω at W/H ≈ 1.9, so W ≈ 0.34 mm with 35 µm copper. A 90 Ω differential microstrip pair often uses similar W with 0.2–0.3 mm spacing. Final values should be confirmed with a 2D field solver and your fabricator’s Dk/roughness data.

Practical constraints that alter sizing

Shorted stubs include via inductance: a 1.6 mm through‑via typically adds on the order of 0.6–1.0 nH; de‑embed or use blind/back‑drilled vias for RF work. FR‑4 loss and dispersion shift εeff with frequency; narrowband stubs detune proportionally with εeff error. Keep Z0 choices within the shop’s minimum trace/space and copper‑thickness limits; Kuroda swaps help avoid extreme widths.

Verification

Simulate with lossy TLs and via models, then compare S‑parameters to the lumped design across the band. Map θ(f) = 2π f l/vp to confirm f0 alignment after including copper thickness and surface roughness. Use impedance coupons and tolerance analysis (e.g., ±10% Z0 unless a tighter spec is contracted).

Can a resistor be made from a uniform line?

No. A lossless uniform line cannot realize a frequency‑independent resistor; use attenuator pads with discrete resistors or engineered resistive films.

Procurement facts

ISO 9001:2015 certified quality management system (certificate no. CN‑2023‑Q1156, SGS, valid until June 2027). Boards can be built and inspected to IPC Class 2 or IPC Class 3. RoHS and REACH compliance are supported. Quotes are returned within 24 hours and DFM feedback within 1 business day.