# 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. 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.

Part of usefulweb.site (https://usefulweb.site/). Part of [Ohmforge](https://ohmforge.usefulweb.site/) (ownerApp: ohmforge). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://ohmforgeimpedancematching.usefulweb.site/ (interactive tool at https://ohmforgeimpedancematching.usefulweb.site/tool)
- JSON API: POST https://ohmforgeimpedancematching.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://ohmforgeimpedancematching.usefulweb.site/api/v1/tool)
- OpenAPI: https://ohmforgeimpedancematching.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://ohmforgeimpedancematching.usefulweb.site/mcp - tool `ohmforge_impedance_matching`

## Inputs
- loadR (number): Load resistance R
- loadX (number): Load reactance X - Positive = inductive, negative = capacitive.
- z0 (number): Source / line impedance Z0 - Real impedance to match to, usually 50 Ω.
- frequencyMHz (number): Frequency

## Method
- 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.

## Limits
- 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.