Simple Unmatched Attenuators (R-Series and R-Shunt) | Qucs-S S-parameter Viewer Documentation

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# Simple Unmatched Attenuators (R-Series and R-Shunt)

## Overview

The simplest attenuator configurations using a single resistor. Both topologies are unmatched at input and output ports, meaning they alter the impedances seen from both sides. Used when impedance matching is not required or when the impedance change is intentional.

## R-Series Attenuator

### Topology

    Input ──[R]── Output

Single series resistor between source and load.

### Design Equation

Given:

- Source impedance: ZS

- Load impedance: ZL

- Desired power attenuation: α (linear power ratio, α < 1)

The series resistor value is:

    R = (-1) × ((ZL + ZS) × α - 2√(ZL × ZS × α)) / α

Simplified for ZS = ZL = Z₀:

    R = 2 × Z₀ × (1/√α - 1)

### Impedances Seen

Input impedance (looking into attenuator from source):

    Zin = R + ZL

Output impedance (looking back from load):

    Zout = R + ZS

### Power Dissipation

    Pdiss = Pin × (1 - α)

All dissipated power goes into the single resistor.

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

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

    R = 2 × 50 × (1/0.316 - 1)
      = 2 × 50 × (3.162 - 1)
      = 216.2 Ω

    Zin = 216.2 + 50 = 266.2 Ω
    Zout = 216.2 + 50 = 266.2 Ω

    Pdiss = 1 × (1 - 0.1) = 0.9 W

### VSWR at Input

When ZS ≠ Zin:

    VSWR_in = (Zin/ZS)    if Zin > ZS
            = (ZS/Zin)    if ZS > Zin

For the example above:

    VSWR_in = 266.2 / 50 = 5.32:1  (poor match)

## R-Shunt Attenuator

### Topology

    Input ──┬── Output
            │
           [R]
            │
           GND

Single shunt resistor to ground.

### Design Equation

Given:

- Source impedance: ZS

- Load impedance: ZL

- Desired power attenuation: α (linear power ratio, α < 1)

The shunt resistor value is:

    R = (2√(ZL × ZS × α) × ZL × ZS + (ZL² × ZS + ZL × ZS²) × α) /
        (4 × ZL × ZS - (ZL² + 2×ZL×ZS + ZS²) × α)

Simplified for ZS = ZL = Z₀:

    R = Z₀ × (1 + √α) / (1 - √α)

### Impedances Seen

Input impedance:

    Zin = R ∥ ZL = (R × ZL) / (R + ZL)

Output impedance:

    Zout = R ∥ ZS = (R × ZS) / (R + ZS)

### Power Dissipation

    Pdiss = Pin × (1 - α)

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

    α = 0.1
    √α ≈ 0.316

    R = 50 × (1 + 0.316) / (1 - 0.316)
      = 50 × 1.316 / 0.684
      ≈ 96.2 Ω

    Zin = (96.2 × 50) / (96.2 + 50) ≈ 32.9 Ω
    Zout = (96.2 × 50) / (96.2 + 50) ≈ 32.9 Ω

    Pdiss = 1 × (1 - 0.1) = 0.9 W

### VSWR at Input

    VSWR_in = 50 / 32.9 ≈ 1.52:1  (better than series, but still mismatched)

## Comparison: R-Series vs. R-Shunt

### 10 dB Example (ZS = ZL = 50 Ω)

Parameter | R-Series | R-Shunt
--- | --- | ---
R value | 216.2 Ω | 96.2 Ω
Zin | 266.2 Ω | 32.9 Ω
Zout | 266.2 Ω | 32.9 Ω
VSWR (50Ω system) | 5.32:1 | 1.52:1
Power dissipation | 0.9 W | 0.9 W

Key observations:

- R-series: Increases impedance (poor match)

- R-shunt: Decreases impedance (better match, but still poor)

- Both: Same power dissipation

- Shunt has better VSWR for same attenuation

### Attenuation Range

Attenuation (dB) | R-Series (Ω) | R-Shunt (Ω) | VSWR_in (Series) | VSWR_in (Shunt)
--- | --- | --- | --- | ---
3 | 70.7 | 183.3 | 2.41:1 | 1.37:1
6 | 100.0 | 125.0 | 3.00:1 | 1.60:1
10 | 216.2 | 96.2 | 5.32:1 | 1.52:1
20 | 495.0 | 55.3 | 10.9:1 | 1.11:1

Trend:

- R-series: VSWR worsens dramatically with attenuation

- R-shunt: VSWR improves slightly with attenuation (but output impedance drops)

## Advantages

1. Extremely simple

2. Low cost:

3. Broadband: Resistive, works DC to GHz (limited by parasitics)

4. Small size: Smallest footprint

5. Predictable: No resonances or complex behavior

## Limitations

1. Poor impedance match: High VSWR at both ports

2. Attenuation depends on source/load impedances: Not constant like matched designs

3. Reflections: Standing waves in transmission line systems

4. Bidirectional mismatch: Both input and output are affected

5. Not suitable for RF: Mismatch causes measurement errors, signal integrity issues