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

Skip to content

# L-Section Matching

## Description

The L-section is the simplest lumped-element matching network, consisting of one series and one shunt reactive element (inductor or capacitor). It provides narrowband matching at a single frequency.

## When to Use

- Narrowband applications

- Simple, low-cost solution needed

- Load has moderate Q factor

## Design Equations

### Case 1: Z0 > RL (Shunt-Series Configuration)

\[X = \pm\sqrt{R_L(Z_0 - R_L)} - X_L\]
\[B = \pm\frac{\sqrt{(Z_0 - R_L)/R_L}}{Z_0}\]

### Case 2: Z0 < RL (Series-Shunt Configuration)

\[B = \frac{X_L \pm \sqrt{R_L/Z_0}\sqrt{R_L^2 + X_L^2 - Z_0 R_L}}{R_L^2 + X_L^2}\]
\[X = \frac{1}{B} + \frac{X_L Z_0}{R_L} - \frac{Z_0}{B R_L}\]

where:

Parameter | Description
--- | ---
X | Series reactance
B | Shunt susceptance
RL | Load resistance
XL | Load reactance
Z0 | System impedance

### Component Values

For series element:

- If X < 0: Capacitor, \(C = -1/(\omega X)\)

- If X > 0: Inductor, \(L = X/\omega$\)

For shunt element:

- If B > 0: Capacitor, \(C = B/\omega\)

- If B < 0: Inductor, \(L = -1/(\omega B)\)

## Parameters

Parameter | Description
--- | ---
Z0 | Source impedance (Ω)
ZL | Load impedance (R + jX) (Ω)
Frequency | Matching frequency (Hz)
Solution | Solution 1 or 2 (determines high/low-pass characteristic)

## Solutions

Each L-section match has two solutions:

- Solution 1: Typically provides low-pass response

- Solution 2: Typically provides high-pass response

Select based on desired filtering characteristics or to avoid DC short/open conditions.

## Limitations

- Narrowband (typically < 10% fractional bandwidth)

- No control over Q factor

- Cannot match purely reactive loads (RL = 0)

- Limited rejection outside passband

## Example

Match 25Ω to 50Ω at 1 GHz

Input data

Parameter | Value
--- | ---
Z0 | 50Ω
ZL | 25Ω
frequency | 1 GHz

Results

Solution 1 (Z0 > RL, shunt-series):

Parameter | Value
--- | ---
Shunt capacitor | 3.18 pF
Series inductor | 3.98 nH

Circuit topology:

    Port ──┬── L (3.98 nH) ── Load (25Ω)
           │
           C (3.18 pF)
           │
          GND

## Reference

Vizmuller, P. “RF Design Guide: Systems, Circuits, and Equations”, Artech House, 1995

## See Also

- RF Circuit Synthesis Tools