Pi and Tee Attenuators | Qucs-S S-parameter Viewer Documentation

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# Pi and Tee Attenuators

## Overview

Pi and Tee are the two most common bilateral matched attenuator topologies. Both provide perfect impedance match when Zin = Zout = Z₀.

## Pi Attenuator

### Topology

    Input ──┬──[R2]──┬── Output
            │        │
           [R1]     [R3]
            │        │
           GND      GND

Two shunt resistors (R1, R3) with one series resistor (R2).

### Design Equations

    L = 10^(Attenuation/10)

    R₂ = 0.5 × (L - 1) × √(Zin × Zout / L)

    R₁ = 1 / (((L + 1) / (Zin × (L - 1))) - (1 / R₂))

    R₃ = 1 / (((L + 1) / (Zout × (L - 1))) - (1 / R₂))

For matched case (Zin = Zout = Z₀):

    R₁ = R₃ = Z₀ × (L + 1) / (L - 1)
    R₂ = Z₀ × (L² - 1) / (2L)

### Power Dissipation

    Pdiss_R1 = Pin × Zin / R₁

    Pdiss_R2 = Pin × R₂ × (R₁ - Zin)² / (R₁² × Zin)

    K = R₁ × R₂ - Zin × (R₁ + R₂)
    Pdiss_R3 = Pin × K² / (R₁² × R₃ × Zin)

## Tee Attenuator

### Topology

    Input ──[R1]──┬──[R3]── Output
                  │
                 [R2]
                  │
                 GND

Two series resistors (R1, R3) with one shunt resistor (R2).

### Design Equations

    L = 10^(Attenuation/10)

    R₂ = (2 × √(Zin × Zout × L)) / (L - 1)

    R₁ = Zin × ((L + 1) / (L - 1)) - R₂

    R₃ = Zout × ((L + 1) / (L - 1)) - R₂

For matched case (Zin = Zout = Z₀):

    R₁ = R₃ = Z₀ × (L - 1) / (L + 1)
    R₂ = 2 × Z₀ × L / (L² - 1)

### Power Dissipation

    Pdiss_R1 = Pin × R₁ / Zin

    Pdiss_R2 = Pin × (R₁ - Zin)² / (R₂ × Zin)

    Pdiss_R3 = Pin × R₃ × (R₁ + R₂ - Zin)² / (Zin × R₂²)

## Example: 10 dB Matched Attenuator (Z₀ = 50 Ω)

Given:

- Attenuation = 10 dB

- Z₀ = 50 Ω

- Pin = 30 dBm (1 W)

Calculation:

    L = 10^(10/10) = 10

Pi Attenuator:

    R₁ = R₃ = 50 × (10 + 1) / (10 - 1) = 61.1 Ω
    R₂ = 50 × (10² - 1) / (2 × 10) = 247.5 Ω

    Pdiss_R1 ≈ 0.082 W
    Pdiss_R2 ≈ 0.018 W
    Pdiss_R3 ≈ 0.82 W

Tee Attenuator:

    R₁ = R₃ = 50 × (10 - 1) / (10 + 1) = 40.9 Ω
    R₂ = 2 × 50 × 10 / (10² - 1) = 10.1 Ω

    Pdiss_R1 ≈ 0.818 W
    Pdiss_R2 ≈ 0.082 W
    Pdiss_R3 ≈ 0.001 W

## Pi vs. Tee Comparison

Characteristic | Pi | Tee
--- | --- | ---
Series resistor value | Higher | Lower
Shunt resistor value | Lower | Higher
High attenuation | Series R₂ becomes very large | Shunt R₂ becomes very small
Low attenuation | Shunt R₁,R₃ become very large | Series R₁,R₃ become very small
Preferred for high Z₀ | Tee (smaller shunt R) | —
Preferred for low Z₀ | Pi (larger shunt R) | —

## Advantages

- Matched at both ports

- Broadband

- Easy to implement with chip resistors

## References

Vizmuller, P. (1995). RF Design Guide: Systems, Circuits, and Equations, Chapter 3. Artech House.

## See Also

- RF Circuit Synthesis Tools