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Ohmforge L-Network Impedance Matching (Smith Chart)

Match a load to 50 Ω with two parts: every L-network solution, part values, Q and a Smith chart.

Ω
Ω

Positive = inductive, negative = capacitive.

Ω

Real impedance to match to, usually 50 Ω.

MHz
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Try an example: Pozar example 5.1: 200 − j100 Ω to 100 Ω at 500 MHz · Low-impedance load 10 + j5 Ω to 50 Ω at 100 MHz · Short 433 MHz antenna 12.5 − j60 Ω to 50 Ω

More about this tool

How it works

Enter a load impedance R + jX, the source or line impedance and the frequency. The tool returns every lossless two-element L-network that matches them — shunt-first and series-first, with inductor and capacitor values — the nodal Q, the VSWR ≤ 2 bandwidth and a Smith chart that shows each match path from the load to the centre. Each solution is verified by evaluating the network's input impedance.

  • Analytic L-section design after Pozar §5.1. Shunt-first (shunt part across the load, series part towards the source): B = (X_L ± √(R_L/Z0)·√(R_L² + X_L² − Z0·R_L)) / (R_L² + X_L²), then the series reactance cancels what is left; possible when |Z_L|² ≥ Z0·R_L (always for R_L > Z0). Series-first (series part next to the load, shunt part towards the source): X = ±√(R_L(Z0 − R_L)) − X_L, B = ±√((Z0 − R_L)/R_L)/Z0; possible when R_L < Z0.
  • Parts at f: X > 0 → L = X/ω, X < 0 → C = −1/(ωX); B > 0 → C = B/ω, B < 0 → L = −1/(ωB).
  • Each solution is checked by computing the input impedance of the finished network with complex arithmetic and once more by parsing its network expression with Ohmforge's impedance calculator.
  • Nodal Q = |X|/R at the node between the two parts. The VSWR ≤ 2 band is found by bisection on |Γin(f)| = 1/3, with the load modelled as R_L plus the inductor or capacitor that has X_L at the design frequency.
  • The Smith chart is normalised to Z0; series parts move along circles of constant resistance, shunt parts along circles of constant conductance.

Ideal, lossless parts: real inductors have resistance and self-resonance, capacitors have ESR and ESL, and PCB pads add parasitics that matter above a few hundred MHz. A real load's impedance changes with frequency differently from the simple R + L or R + C model used for the bandwidth. Tune on the bench with a VNA.

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Related tools

Design an L-network to match an antenna or amplifier to 50 Ω

Get every two-part L-network that matches a load impedance to your source, with inductor and capacitor values, Q, bandwidth and a Smith chart.

What it is for

A small antenna, a chip amplifier input or a filter rarely presents exactly 50 Ω. With an inductor and a capacitor arranged as an L you can transform almost any load to your source impedance at one frequency. The question is which arrangement and which values.

Enter the Load resistance R, the Load reactance X, the Source / line impedance Z0 and the Frequency. The tool returns every lossless L-network that matches them. Shunt at the load networks put the shunt part across the load and the series part towards the source. Series at the load networks put the series part next to the load. You get the part values, the nodal Q, the frequency range where VSWR stays at or below 2, and a Smith chart of each match path.

It does not pick real parts for you. Real inductors and capacitors have losses and self-resonance, and pads add parasitics. A real load also changes with frequency differently from the simple model used for the bandwidth.

Worked example

Pozar's textbook example 5.1 matches 200 − j100 Ω to a 100 Ω line at 500 MHz. Because R is above Z0, only shunt-at-the-load networks work, and the tool finds both. Solution 1 is a series L of 38.98 nH from the source, then a shunt C of 922.8 fF across the load. Solution 2 is a series C of 2.599 pF and a shunt L of 46.14 nH. Both have a nodal Q of 1.22. Pozar's printed values (38.8 nH, 0.92 pF, 46.1 nH, 2.61 pF) differ slightly only because the book rounds the reactance to 122 Ω.

A short 433 MHz antenna of 12.5 − j60 Ω on a 50 Ω radio has four solutions. Solution 3, a shunt C of 12.71 pF on the source side and a series L of 29.95 nH to the antenna, has the lowest Q (1.73). Its VSWR ≤ 2 band runs from 408.4 MHz to 458.4 MHz.

How to read the result

The L-network solutions table lists the topology, the series part, the shunt part, the nodal Q and the VSWR ≤ 2 band. How to build them says the same in words, source side first.

On the Smith chart the load is the black dot and the match is the centre. Series parts move the impedance along a circle of constant resistance; shunt parts move it along a circle of constant conductance. The small circle marks the point between the two parts.

The code block gives each network as an expression for Ohmforge's impedance calculator, so you can sweep it over frequency there.

Tips and common mistakes

  • Prefer the solution with the lowest Q. It gives the widest band and is the least sensitive to part tolerances.
  • A solution with a series capacitor also blocks DC; one with a shunt inductor shorts DC to ground. Pick the one that suits your bias.
  • Very small values (under about 1 pF or 1 nH) are hard to build. Choose another solution or a different topology.
  • If no network has a low enough Q for your bandwidth, use two L-sections in cascade.
  • Measure the load with a VNA at the real frequency. A datasheet value or a guess is only a starting point.

FAQ

Why are there sometimes two and sometimes four solutions?

Shunt-at-the-load networks need |ZL|² ≥ Z0·RL, and series-at-the-load networks need R below Z0. Loads that meet both conditions get four solutions.

What does the nodal Q tell me?

It is the ratio |X|/R at the point between the two parts. Roughly, the matched bandwidth shrinks as Q grows.

Can I match to an impedance that is not real?

No. Z0 is a real source or line impedance.

Guide last updated:

For scripts and AI agents

  • JSON API: POST https://ohmforgeimpedancematching.usefulweb.app/api/v1/calculate (same calculation as this page)
  • Interface description: OpenAPI 3.1 · AI guide: llms.txt
  • MCP server (Streamable HTTP): https://ohmforgeimpedancematching.usefulweb.app/mcp · tool name ohmforge_impedance_matching
Example request
curl -s -X POST https://ohmforgeimpedancematching.usefulweb.app/api/v1/calculate \
  -H 'content-type: application/json' \
  -d '{"loadR":200,"loadX":-100,"z0":100,"frequencyMHz":500}'
Feedback · ideas & bug reports welcome