Cascaded L-Section Matching | Qucs-S S-parameter Viewer Documentation

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# Cascaded L-Section Matching

## Description

Cascaded L-section matching uses multiple L-sections in series to achieve impedance matching with broader bandwidth than a single L-section. Each section provides a partial impedance (real to real) transformation using lumped elements (inductors and capacitors).

## When to Use

- Lumped element implementation required

- Moderate bandwidth needed (broader than single L-section)

- Low to moderate frequencies (< 2 GHz typically)

- Compact size preferred over transmission lines

## Design Theory

The load and source impedances are connected through N intermediate resistance values, with each adjacent pair matched by an L-section.

### Intermediate Resistances

\[R_i = R_1^{(N-i)/N} \cdot R_2^{i/N}, \quad i = 1, 2, ..., N-1\]

where R₁ is the higher resistance and R₂ is the lower resistance.

### Quality Factor per Section

For section i (high-to-low transition):

\[Q_i = \sqrt{\frac{R_{i-1}}{R_i} - 1}\]

## Lowpass vs Highpass

Each L-section can be implemented in lowpass or highpass configuration, giving the network different filtering characteristics.

### Lowpass Configuration

Series inductors and shunt capacitors:

\[C_i = \frac{Q_i}{\omega R_{i-1}}, \quad L_i = \frac{Q_i R_i}{\omega}\]

Characteristics

- DC pass, high-frequency attenuation

### Highpass Configuration

Series capacitors and shunt inductors:

\[L_i = \frac{R_{i-1}}{\omega Q_i}, \quad C_i = \frac{1}{Q_i R_i \omega}\]

Characteristics

- DC block, low-frequency attenuation

## Parameters

Parameter | Description
--- | ---
Z0 | Source impedance (Ω)
ZL | Load impedance (real part) (Ω)
Frequency | Matching frequency (Hz)
Sections (N) | Number of L-sections (2-10)
Solution | Lowpass (1) or Highpass (2)

## Design Equations

### For Z0 > RL (Step-down)

Starting from source, each section:

1. Shunt element first (C for lowpass, L for highpass)

2. Series element second (L for lowpass, C for highpass)

### For Z0 < RL (Step-up)

Starting from source, each section:

1. Series element first

2. Shunt element second

Network is designed in reverse, working backwards from load.

### Component Values

Lowpass configuration (Z0 > RL):

\[C_{shunt,i} = \frac{Q_i}{\omega R_{i-1}}\]
\[L_{series,i} = \frac{Q_i R_i}{\omega}\]

Highpass configuration (Z0 > RL):

\[L_{shunt,i} = \frac{R_{i-1}}{\omega Q_i}\]
\[C_{series,i} = \frac{1}{Q_i R_i \omega}\]

## Bandwidth Considerations

### Q Factor per Section

With N sections, each has quality factor:

\[Q_{section} \approx Q_{total}^{1/N}\]

where Q_total is the Q of a single-section match.

## Example

Match 10Ω to 50Ω at 500 MHz (3-section Lowpass)

Given:

- Z0 = 50Ω

- RL = 10Ω

- f = 1000 MHz

- N = 3 sections

- Lowpass configuration

Results:

- Section 1: C₁ = 2.68 pF (shunt), L₁ = 3.92 nH (series)

- Section 2: C₂ = 4.59 pF (shunt), L₂ = 2.29 nH (series)

- Section 3: C₃ = 7.84 pF (shunt), L₃ = 1.34 nH (series)

Circuit topology:

    Port ──┬── L₁ ──┬── L₂ ──┬── L₃ ── Load(10Ω)
           │        │        │
           C₁       C₂       C₃
           │        │        │
          GND      GND      GND

## Reference

Bahl, I. J. “Fundamentals of RF and Microwave Transistor Amplifiers”, Wiley, 2009, pp. 169-170

## See Also

- RF Circuit Synthesis Tools