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)

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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  ✓

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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