Transmission Lines

At high frequencies, interconnects behave as transmission lines. Learn distributed L and C, characteristic impedance, propagation velocity and velocity factor, coax, twisted pair, microstrip and stripline, reflections and standing waves, input impedance of loaded lines, quarter-wave transformers, stubs, and signal-integrity rules.

Coaxial cables, PCB traces carrying DDR memory or USB 3, and antenna feeds are all transmission lines. Controlling their impedance prevents reflections that distort signals or waste RF power.

An analogy

A transmission line is like a long corridor for a wave: if the corridor suddenly narrows or ends (impedance change), part of the wave bounces back.

Distributed model

Each tiny segment has series inductance and shunt capacitance (plus loss R and G). A wave travels at v_p; a matched load absorbs it completely.

Line types

Coax (shielded, 50/75 Ω), twisted pair (differential, 100 Ω — Ethernet), microstrip (trace over ground plane on PCBs), stripline (trace between planes — better isolation), waveguides at microwave frequencies.

Reflections and termination

Mismatched ends create standing waves and ringing on digital edges. Series termination at the driver or parallel termination at the receiver (e.g. DDR ODT) fixes this.

Impedance transformation

A λ/4 line transforms Z_L to Z₀²/Z_L; a λ/2 line repeats the load; short-/open-circuited stubs act as reactances for matching and filters. Signal-integrity engineers also manage loss, crosstalk and via discontinuities.

Formulas

• Lossless line: Z₀ = √(L′/C′), v_p = 1/√(L′C′) = c/√ε_r
• Coax: Z₀ = (60/√ε_r) ln(D/d)
• Input impedance: Z_in = Z₀ (Z_L + jZ₀ tan βl)/(Z₀ + jZ_L tan βl)
• Quarter-wave transformer: Z₀ = √(Z_in Z_L)

Worked example: Coax impedance

Given: D/d = 3.5, ε_r = 2.25

Find: Z₀ and velocity factor.

• Z₀ = (60/1.5) × ln 3.5 = 50.1 Ω
• VF = 1/√2.25 = 0.667

Answer: 50 Ω, signals travel at 2 × 10⁸ m/s

Worked example: Quarter-wave match

Given: Match 100 Ω antenna to 50 Ω line

Find: Transformer impedance.

• Z = √(50 × 100) = 70.7 Ω

Answer: 70.7 Ω λ/4 section (≈ 75 Ω coax in practice)

Common mistakes

• Wrong: Ignoring line effects on fast digital edges. — Right: Rise time, not clock frequency, sets the need for termination.
• Wrong: Using free-space wavelength for line lengths. — Right: Divide by √ε_r (velocity factor).

Applications

• High-speed PCBs: USB, PCIe, DDR impedance control.
• RF feeds: Coax and microstrip to antennas.
• Networking: 100 Ω twisted pair in Ethernet.

Key takeaways

• Z₀ = √(L′/C′); v = c/√ε_r.
• Mismatches reflect; terminate lines.
• λ/4 transforms impedance.
• Control PCB impedance at high speed.

Going further

• TDR: Time-domain reflectometry to find faults.
• Differential pairs: Odd/even-mode impedance.
• Skin effect & dielectric loss: Frequency-dependent attenuation.