Reflection Attenuator | Qucs-S S-parameter Viewer Documentation

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# Reflection Attenuator

## Overview

Wideband matched attenuator using a 3 dB 90° hybrid coupler with resistive terminations on the isolated ports. Provides excellent impedance match.

## Topology

            Port 1 ┌──────────────┐ Port 3
       Input ──────┤              ├────── Output
                   │   3dB 90°    │
                   │    Hybrid    │
            Port 2 │   Coupler    │ Port 4
                ┌──┤              ├──┐
                │  └──────────────┘  │
               [Ri]                 [Ri]
                │                    │
               GND                  GND

Hybrid coupler:

- Coupling factor: -3 dB (k = 0.7071 = 1/√2)

- Phase shift: 90° between coupled ports

- Ports 1-3 and 2-4 are isolated

Both termination resistors: Ri (equal value)

## Principle of Operation

1. Input signal enters Port 1

2. Hybrid splits power equally between Ports 2 and 4 (with 90° phase difference)

3. Resistors Ri reflect a portion of the power based on mismatch

4. Reflected signals recombine at Port 3 with controlled attenuation

5. Remaining power is absorbed in the resistors

## Design Equations

    L = 10^(-0.05 × Attenuation)

    Ri = Z₀ × (1 - L) / (1 + L)

Constraint: Requires Zin = Zout = Z₀ (bilateral matched design).

### Derivation

The S-parameter from Port 1 to Port 3 is:

    S₂₁ = (1/√2) × e^(jπ/2) × Γ₂ + (1/√2) × e^(jπ/2) × Γ₄

    where Γ = (Z₀ - Ri) / (Z₀ + Ri)

For equal resistors (Ri at Ports 2 and 4):

    S₂₁ = (Z₀ - Ri) / (Z₀ + Ri) × e^(jπ/2)

    |S₂₁| = |Z₀ - Ri| / (Z₀ + Ri)

Attenuation:

    α = -20 × log₁₀(|S₂₁|) = -20 × log₁₀(|Z₀ - Ri| / (Z₀ + Ri))

Solving for Ri when Ri < Z₀:

    10^(-0.05×α) = (Z₀ - Ri) / (Z₀ + Ri)

    Ri = Z₀ × (1 - 10^(-0.05×α)) / (1 + 10^(-0.05×α))

## Power Dissipation

Each resistor dissipates half of the total dissipated power:

    Pdiss_total = Pin × (1 - 10^(-0.1 × Attenuation))

    Pdiss_Ri = 0.5 × Pin × (1 - 10^(-0.1 × Attenuation))

Both resistors dissipate equal power.

## Example: 10 dB, Z₀ = 50 Ω, Pin = 1 W

    L = 10^(-0.05 × 10) = 10^(-0.5) ≈ 0.316

    Ri = 50 × (1 - 0.316) / (1 + 0.316)
       = 50 × 0.684 / 1.316
       ≈ 26.0 Ω

Power dissipation:

    Pdiss_total = 1 × (1 - 10^(-0.1 × 10))
                = 1 × (1 - 0.1)
                = 0.9 W

    Pdiss_R1 = Pdiss_R2 = 0.45 W each

## Resistor Values vs. Attenuation

Attenuation (dB) | Ri (Ω, Z₀=50Ω) | Pdiss per resistor (Pin=1W)
--- | --- | ---
3 | 39.4 | 0.25 W
6 | 31.2 | 0.37 W
10 | 26.0 | 0.45 W
20 | 20.5 | 0.50 W
30 | 18.9 | 0.50 W

Observation: As attenuation increases, Ri approaches Z₀/2 ≈ 25Ω, and power dissipation approaches 50% in each resistor.

## Advantages

1. Very wideband: DC to >20 GHz (limited only by hybrid coupler)

2. Excellent impedance match: Return loss >20 dB at all frequencies

3. Less sensitive to tolerances: Slight resistor value errors cause minimal performance degradation

4. High power handling: Limited by hybrid coupler, not resistors

## Limitations

1. Requires hybrid coupler:

   - Bulky at low frequencies (<100 MHz)

   - More expensive than resistor-only designs

   - Adds extra insertion loss

2. Higher cost: Hybrid coupler more expensive than three resistors

3. Matched at both: Cannot provide impedance transformation

## 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