Transmission line behavior at low and microwave bands

At low frequency, a transmission line (for example a pair of wires) is only a connection. At microwave frequency the same line is several wavelengths long, so it becomes a circuit element whose behaviour depends on its length, its characteristic impedance and the load.

Low-frequency behaviour

• Length ≪ λ, so there is no phase change from end to end, and V and I are the same at both ends.
• Modelled with lumped series R and L, and sometimes shunt C (power line models).
• Load impedance is seen directly at the input: Zin ≈ ZL.
• Losses are mainly I²R. Skin effect and radiation are negligible.
• Mismatch does not matter much, and there are no standing waves.
• Two-wire lines, twisted pair and ordinary cable are used.

Microwave behaviour

• Length is comparable to λ, so the line is treated as distributed with R′, L′, G′, C′ per metre.
• Voltage and current are waves: V(z) = V⁺e^(-γz) + V⁻e^(γz), with γ = α + jβ.
• Characteristic impedance Z0 = √((R′ + jωL′)/(G′ + jωC′)) ≈ √(L′/C′).
• Input impedance changes with length: Zin = Z0 (ZL + jZ0 tan βl)/(Z0 + jZL tan βl).
  • A λ/4 line inverts the impedance: Zin = Z0²/ZL.
  • A λ/2 line repeats the load.
  • Shorted or open stubs act as pure L or C.
• Mismatch gives reflection Γ = (ZL − Z0)/(ZL + Z0), standing waves and VSWR, so matching is required.
• Skin effect (δ = 1/√(πfμσ)) and dielectric loss raise the attenuation. Open lines radiate, so coax, waveguide, microstrip or stripline are used.
• Higher-order modes appear when the cross-section approaches λ/2.

Summary table

Example

A 7.5 cm shorted line at 1 GHz (λ = 30 cm) is λ/4 long, so it looks like an open circuit. At 50 Hz the same piece of wire is a dead short.