Quarter-Wave Attenuators

Overview

Narrowband matched attenuators using a quarter-wavelength (λ/4) transmission line section combined with resistive elements. Provides attenuation at a specific frequency with excellent match.

Quarter-Wave Series Attenuator

Topology (Distributed)

Input ──┬──[λ/4 TL]──┬── Output
        │    Z₀      │
       [R]          [R]
        │            │
       GND          GND
         └──[Z₀]──┘
             │
            GND

Series resistor R in parallel with λ/4 shunt line.

Topology (Lumped Equivalent)

Input ──┬──[L]──┬── Output
        │       │
       [C]     [R]
        │   │   │
       GND [R] GND
            │
        [Z₀][C]
         │   │
        GND GND

Design Equations

R = Zin / (10^(0.05 × Attenuation) - 1)

l_λ/4 = c / (4 × f)

Zout = R + Zin × (R + Zin) / (2R + Zin)

For lumped implementation (low frequency):

ω₀ = 2πf
L = Zin / ω₀
C = 1 / (Zin × ω₀)

Power Dissipation

K = (R + Zin)²

Pdiss_R1 = Pin × Zin × R / K    (shunt at input)
Pdiss_R2 = Pin × R² / K         (series path)
Pdiss_R3 = Pin × Zin × R / K    (shunt via λ/4)

Quarter-Wave Shunt Attenuator

Topology (Distributed)

Input ──[R]──┬── Output
             │
         [λ/4 TL]
             Z₀
          │  │  │
        [Z₀][R][Z₀]
          │  │  │
         GND GND GND

Shunt resistors with λ/4 line providing transformation.

Design Equations

R = Zin × (10^(0.05 × Attenuation) - 1)

l_λ/4 = c / (4 × f)

Zout = R + Zin × (R + Zin) / (2R + Zin)

For lumped implementation:

L = Zin / ω₀
C = 1 / (Zin × ω₀)

Power Dissipation

K = (R + Zin)²

Pdiss_R1 = Pin × Zin × R / K
Pdiss_R2 = Pin × Zin² / K
Pdiss_R3 = Pin × Zin × R / K

Example: 10 dB at 2 GHz, Z₀ = 50 Ω

Quarter-Wave Series

R = 50 / (10^(0.05 × 10) - 1)
  = 50 / (10^0.5 - 1)
  = 50 / 2.162
  ≈ 23.1 Ω

λ/4 = (3×10⁸) / (4 × 2×10⁹)
    = 37.5 mm

Zout ≈ 35.8 Ω

Resistor values:

  • R1 (input shunt): 23.1 Ω

  • R2 (series): 23.1 Ω

  • R3 (λ/4 shunt): 23.1 Ω

  • λ/4 line: Z₀=50Ω, l=37.5mm

Quarter-Wave Shunt

R = 50 × (10^0.5 - 1)
  ≈ 108.1 Ω

λ/4 = 37.5 mm

Zout ≈ 141.0 Ω

Resistor values:

  • R1 (first shunt): 108.1 Ω

  • R2 (λ/4 bottom, Z₀): 50 Ω

  • R3 (second shunt): 108.1 Ω

  • λ/4 line: Z₀=50Ω, l=37.5mm

Limitations

  • Narrowband (typically 20-30% fractional BW)

  • Physical size large at lower frequencies

References

[1] Doherty, W. E., & Joos, R. D. (1998). The PIN Diode Circuit Designer’s Handbook, Chapter 5: Reflection Attenuators. Microsemi Corp.

See Also