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