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L-Network Impedance Matching

RF & Wireless

Which two stock parts match this antenna to 50 Ω, and what VSWR do they give across the band?

Inputs

The frequency ZL was measured at.
Hz
2.44 GHz
Hz
2.4 GHz
Hz
2.483 GHz
Real only: the feed line, usually 50 Ω.
Ω
Real part of ZL at f, from the antenna datasheet or a VNA with the pads fitted.
Ω
Imaginary part of ZL at f, from the same place. Negative is capacitive.
Ω
Best is the lowest worst-case VSWR in the band with stock values.
Part series

Results

Matching network
6.8pF series at the load + 3.3nH shunt at the source
Pass

Checked: VSWR ≤ your 2:1 limit from 2.4 GHz to 2.4835 GHz, failing past 1.1 × it (design margin): worst 1.037:1 at 2.4835 GHz, with E24 parts
Show working
  1. Node QQ = √(R_s / R_L − 1) = √(50 / 25 − 1) = 1.000
  2. Series partX = −√(R_L (R_s − R_L)) − X_L = −√(25 × 25) + 15 = −10 Ω; C = 1 / (2π f |X|) = 1 / (2π × 2.44 GHz × 10 Ω) = 6.523 pF, nearest E24 6.8 pF
  3. Shunt partB = −√((R_s − R_L) / R_L) / R_s = −√(25 / 25) / 50 = −0.02 S; L = 1 / (2π f |B|) = 1 / (2π × 2.44 GHz × 0.02 S) = 3.261 nH, nearest E24 3.3 nH
  4. Worst VSWR in bandVSWR = (1 + |Γ|) / (1 − |Γ|), largest of 101 points from 2.4 GHz to 2.4835 GHz = 1.037:1 at 2.4835 GHz
Worst VSWR in band 

at 2.4835 GHz
1.037:1
Return loss at f RL
40.03dB
Mismatch loss at f ML
0.000432dB
All results (6)
VSWR at f
1.020:1
|Γ| at f
0.0100
Zin at f
49.185−j0.560Ω
Node Q Q
1.000
Exact parts 

before rounding to the series
6.523pF series at the load + 3.261nH shunt at the source
Fractional bandwidth 

(fhi − flo) / f
3.42%
Every solution
NetworkExact partsStock partsWorst VSWR
Series L at the load, shunt C, low-pass1.091:1
Series C at the load, shunt L, high-pass1.037:1
Series C at the load, shunt L, high-passSeries L at the load, shunt C, low-pass

Accuracy

Source
  • D. M. Pozar
  • C. Bowick

D. M. Pozar, Microwave Engineering the equation

C. Bowick, RF Circuit Design the equation

Precision
Exact arithmetic on ideal, lossless parts, with Z_L held at its value at f across the band.

Real parts add loss, self-resonance and pad parasitics, and a real antenna moves across the band. The measured Z_L usually dominates, because it changes with the board, the ground clearance and the enclosure.

Verified against

2 independent cases. See the working.

This is a design aid. The engineer remains responsible for the design and for checking the standard itself.

Parameter sweep

Vary one input over a range and see the answer and verdict at each step, as a table and a curve.Pro

Worst-case corners

Put a tolerance on each input and get the worst-case band around the answer.Pro

Parts that track move together.Inputs in the same group move together, like the parts of a matched network. Leave independent parts ungrouped: that is the safe choice.

Circuit

The principle

An L network is two reactive parts, one in series and one in shunt, that turn a load impedance into the source resistance at one frequency. This page finds every L network that matches ZL to Rs, rounds each part to a stock value, and checks the VSWR the stock parts give across your band. ZL is the load at f, as measured, held constant across the band.

Series part next to the load
X=±RL(Rs−RL)−XLB=±1RsRs−RLRLX = \pm\sqrt{R_L (R_s - R_L)} - X_L \qquad B = \pm\frac{1}{R_s}\sqrt{\frac{R_s - R_L}{R_L}}
Pozar, for RL ≤ Rs, with the same sign in both.
Shunt part next to the load
Rp=RL2+XL2RLQ=RpRs−1B=±QRp+XLRL2+XL2X=±QRsR_p = \frac{R_L^{2} + X_L^{2}}{R_L} \qquad Q = \sqrt{\frac{R_p}{R_s} - 1} \qquad B = \pm\frac{Q}{R_p} + \frac{X_L}{R_L^{2} + X_L^{2}} \qquad X = \pm Q R_s
For Rp ≥ Rs, with the same sign in both. The load is taken in parallel form and the shunt part first cancels its susceptance. This gives the same B and X as Pozar's closed form.
Parts from the reactances
L=Xω,  C=−1ωXC=Bω,  L=−1ωBω=2πfL = \frac{X}{\omega},\; C = \frac{-1}{\omega X} \qquad C = \frac{B}{\omega},\; L = \frac{-1}{\omega B} \qquad \omega = 2\pi f
A positive X is a series inductor and a negative one a series capacitor. A positive B is a shunt capacitor and a negative one a shunt inductor. Each part is rounded to the nearest value in the series.
Input impedance with the stock parts
Zin=11ZL+jX+jBZin=11ZL+jB+jXZ_{in} = \frac{1}{\frac{1}{Z_L + jX} + jB} \qquad Z_{in} = \frac{1}{\frac{1}{Z_L} + jB} + jX
The series part next to the load, then the shunt part next to the load. X and B are worked out again at each frequency from the stock parts, with ZL held at its value at f.
Reflection and VSWR
Γ=Zin−RsZin+RsVSWR=1+∣Γ∣1−∣Γ∣Γ = \frac{Z_{in} - R_s}{Z_{in} + R_s} \qquad \t{VSWR} = \frac{1 + |Γ|}{1 - |Γ|}
Pozar. The worst VSWR is the largest over 101 points spread evenly across the band.
Return loss and mismatch loss
RL=−20log⁡10∣Γ∣ML=−10log⁡10(1−∣Γ∣2)\t{RL} = -20\log_{10}|Γ| \qquad \t{ML} = -10\log_{10}\left(1 - |Γ|^{2}\right)
Pozar: the load takes 1 − |Γ|² of the power the source offers. Mismatch loss is that fraction as a loss.
Node Q
Q=RsRL−1    (series next to the load)Q=RpRs−1    (shunt next to the load)Q = \sqrt{\frac{R_s}{R_L} - 1} \;\; \t{(series next to the load)} \qquad Q = \sqrt{\frac{R_p}{R_s} - 1} \;\; \t{(shunt next to the load)}
Bowick. The higher the Q, the narrower the band the network matches.
  • RsR_ssource resistance: the feed line, usually 50 Ω
  • ZL,RL,XLZ_L, R_L, X_Lload impedance at f, its resistance and its reactance (negative is capacitive)
  • XXseries reactance of the network
  • BBshunt susceptance of the network
  • RpR_pthe load's resistance in parallel form
  • QQnode Q of the network
  • ω\omegaangular frequency, 2πf
  • ZinZ_{in}input impedance the source sees
  • ΓΓreflection coefficient at the source
  • VSWR\t{VSWR}voltage standing wave ratio
  • RL,ML\t{RL}, \t{ML}return loss and mismatch loss, in dB
More detail

Which networks exist

With the series part next to the load, a match exists when RL ≤ Rs. With the shunt part next to the load, it exists when the load's parallel resistance Rp is at least Rs. Both can hold, and each form has two signs, so a load has up to four L networks (Pozar). The page lists each one once.

Low-pass or high-pass

A series inductor with a shunt capacitor is low-pass, and also attenuates a transmitter's harmonics. A series capacitor with a shunt inductor is high-pass, and keeps DC off the load. Every load has one of each. The others use two capacitors or two inductors. Best is the network whose stock parts give the lowest worst VSWR in the band.

Bandwidth

An L network's Q is set by the two resistances it joins, so its bandwidth is not a free choice. Bowick gives three-part π and T networks for a Q of your own choosing.

Stock parts and the board

Rounding each part to its series moves the match off f, and the worst VSWR across the band shows what that costs. Below about 0.5 nH or 0.5 pF a part is comparable to its pads and a short trace, so the board sets the match as much as the part does. The 0.5 nH and 0.5 pF of the page's note are a design choice.

ZL is one measured point, held constant across the band. Measure it on the board with the pads fitted, fit the parts, then measure S11 again and tune.
References: D. M. Pozar, Microwave Engineering: L networks, the reflection coefficient, SWR and return loss. C. Bowick, RF Circuit Design: the L network, its Q, and the three-part π and T networks. Consult them for stub and broadband matching. Related: Controlled Impedance & Delay, for the 50 Ω feed line.

Engine version ⁨1.26.4⁩