L-Pad (1st Series) Attenuator

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).

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)

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