Quarter-Wave Attenuators | Qucs-S S-parameter Viewer Documentation

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# 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

- RF Circuit Synthesis Tools