L-Pad Attenuators | Qucs-S S-parameter Viewer Documentation

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# L-Pad Attenuators

## Overview

L-pad attenuators are essentially broadband L-section matching networks. They are used mainly for impedance transformation (e.g. in network analyzers 50 Ohm to 75 Ohm).

## L-Pad (First Series)

### Topology

    Input ──[R1]──┬── Output
                  │    Zout ≠ Zin
                 [R2]
                  │
                 GND

Series resistor R1 followed by shunt resistor R2.

### Design Equations

    α = 10^(-Attenuation/10)    (power ratio, α < 1)

    R₁ = -Zin × (α - 1) / (√α + 1)    (series)

    R₂ = -Zin × (α + √α) / (α - 1)    (shunt)

    Zout = R₂ × (R₁ + Zin) / (R₁ + R₂ + Zin)

### Power Dissipation

    Pdiss_R1 = Pin × (1 - √α)

    Pdiss_R2 = Pin × α × (1 - α) / (α + √α)

### Example: 10 dB, Zin = 50 Ω, Pin = 1 W

    α = 10^(-10/10) = 0.1
    √α ≈ 0.316

    R₁ = -50 × (0.1 - 1) / (0.316 + 1)
       = -50 × (-0.9) / 1.316
       ≈ 34.2 Ω

    R₂ = -50 × (0.1 + 0.316) / (0.1 - 1)
       = -50 × 0.416 / (-0.9)
       ≈ 23.1 Ω

    Zout = 23.1 × (34.2 + 50) / (34.2 + 23.1 + 50)
         ≈ 20.0 Ω

Power dissipation:

    Pdiss_R1 = 1 × (1 - 0.316) ≈ 0.684 W
    Pdiss_R2 = 1 × 0.1 × (1 - 0.1) / (0.1 + 0.316) ≈ 0.216 W

    Pout = 1 - 0.684 - 0.216 = 0.1 W  ✓ (10 dB attenuation)

## L-Pad (First Shunt)

### Topology

    Input ──┬──[R2]── Output
            │         Zout ≠ Zin
           [R1]
            │
           GND

Shunt resistor R1 followed by series resistor R2.

### Design Equations

    α = 10^(-Attenuation/10)    (power ratio, α < 1)

    R₁ = Zin / (1 - √α)         (shunt)

    R₂ = Zin × (1 - √α) / √α    (series)

    Zout = -Zin × (α - 2√α + 2) / (α - 2√α)

### Power Dissipation

    Pdiss_R1 = Pin × √α × (1 - 2√α + α) / (1 - √α)

    Pdiss_R2 = Pin × (1 - √α)

### Example: 10 dB, Zin = 50 Ω, Pin = 1 W

    α = 0.1
    √α ≈ 0.316

    R₁ = 50 / (1 - 0.316)
       ≈ 73.1 Ω

    R₂ = 50 × (1 - 0.316) / 0.316
       ≈ 108.2 Ω

    Zout = -50 × (0.1 - 2×0.316 + 2) / (0.1 - 2×0.316)
         = -50 × 1.468 / (-0.532)
         ≈ 138.0 Ω

Power dissipation:

    Pdiss_R1 = 1 × 0.316 × (1 - 0.632 + 0.1) / (1 - 0.316) ≈ 0.216 W
    Pdiss_R2 = 1 × (1 - 0.316) ≈ 0.684 W

    Pout = 1 - 0.216 - 0.684 = 0.1 W  ✓

## Comparison: First Series vs. First Shunt

### 10 dB Example (Zin = 50 Ω)

Parameter | First Series | First Shunt
--- | --- | ---
R₁ value | 34.2 Ω | 73.1 Ω
R₂ value | 23.1 Ω | 108.2 Ω
Zout | 20.0 Ω | 138.0 Ω
Power in R1 | 68.4% | 21.6%
Power in R2 | 21.6% | 68.4%

Key observations:

- First series: Zout < Zin (impedance step-down)

- First shunt: Zout > Zin (impedance step-up)

- Power distribution is reversed between topologies

### Output Impedance vs Topology

Attenuation (dB) | First Series Zout | First Shunt Zout
--- | --- | ---
3 | 35.4 Ω | 70.7 Ω
6 | 25.0 Ω | 100.0 Ω
10 | 20.0 Ω | 138.0 Ω
20 | 16.7 Ω | 500.0 Ω

- First series: Zout decreases with attenuation

- First shunt: Zout increases with attenuation