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