Source: https://guthlabs.ai/docs/instruments/impedance-matching-network-designer-l-network/mechanism/

instrument mechanism / verified-clean

# Impedance Matching Network Designer (L-network) mechanism

The inspectable method record for Impedance Matching Network Designer (L-network), kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The stated standard method is valid only for positive, purely resistive source and load impedances and a fully specified L-network topology. The instrument should explicitly expose both dual solutions, define orientation and element placement, normalize frequency units, and special-case equal resistances. The existing honest label correctly identifies the result as an ideal lossless-component match, but it should also warn that the computed match is narrowband and assumes resistive terminations.

- The specification names standard Q-factor equations but does not define the actual topology for each network orientation, including whether the shunt element is placed on the source side or load side.

- The signs of the series and shunt reactances are unspecified. For a conventional resistive match, one valid solution uses a series inductor and shunt capacitor; the dual solution uses a series capacitor and shunt inductor.

- The equations assume purely resistive, positive source and load impedances. The input description does not state whether source/load reactances are forbidden or separately handled.

- For unequal positive resistances, the standard magnitude equations are Q=sqrt(R_high/R_low-1), |X_series|=Q*R_low, and |X_shunt|=R_high/Q. At R_source=R_load, Q=0 and R_high/Q is undefined even though no matching reactance is required.

## Units and conventions

- Source and load resistance values are in ohms.

- Frequency is positive and must have an explicitly defined unit; the test vectors below use Hz, MHz, or GHz as labeled.

- Inductance uses L=|X|/(2*pi*f), with f in hertz and X in ohms.

- Capacitance uses C=1/(2*pi*f*|X|), with f in hertz and X in ohms.

- The calculated Q is dimensionless and is the ideal reactive-network Q, not a component-loss or loaded-Q measurement.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "source_resistance_ohm": 50,
    "load_resistance_ohm": 200,
    "frequency_hz": 100000000,
    "network_orientation": "source_low_to_load_high",
    "solution": "series_L_shunt_C"
  },
  "expected_outputs": {
    "q": "sqrt(3) = 1.7320508075688772",
    "series_reactance_ohm": "50*sqrt(3) = 86.60254037844386",
    "shunt_reactance_ohm": "-200/sqrt(3) = -115.47005383792515",
    "series_inductance": "50*sqrt(3)/(2*pi*100000000) H = 137.8322 nH",
    "shunt_capacitance": "sqrt(3)/(400*pi*100000000) F = 13.7832 pF"
  },
  "reference_data": {
    "formula": "R_low=50, R_high=200; series L and shunt C",
    "frequency_unit": "Hz"
  },
  "working": "R_high/R_low=200/50=4. Q=sqrt(4-1)=sqrt(3)=1.7320508075688772. |X_series|=Q*R_low=sqrt(3)*50=86.60254037844386 ohm. |X_shunt|=R_high/Q=200/sqrt(3)=115.47005383792515 ohm. For the series-L/shunt-C solution, X_series=+86.60254037844386 and X_shunt=-115.47005383792515. L=X_series/(2*pi*f)=86.60254037844386/(2*pi*100000000) H=137.8322 nH. C=1/(2*pi*f*|X_shunt|)=1/(2*pi*100000000*115.47005383792515) F=13.7832 pF."
}
````

## Failure boundary

- Division by zero when source and load resistances are equal and the implementation evaluates R_high/Q.

- Invalid or nonphysical results if either resistance is zero, negative, non-finite, or if frequency is zero or negative.

- Ambiguous frequency units can produce errors of 10^3 or 10^6 in component values unless the UI and API enforce units.

- A single orientation label is insufficient unless it specifies the exact shunt location and reference direction for reactance signs.

- The ideal equations omit component Q, ESR, self-resonance, inductor winding resistance, capacitor ESR, PCB parasitics, bandwidth, and voltage/current limits.

- A match is not achievable with this simple two-reactive-element formulation for arbitrary complex source/load impedances unless additional complex-impedance equations and constraints are implemented.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/impedance-matching-network-designer-l-network/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/impedance-matching-network-designer-l-network/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/impedance-matching-network-designer-l-network/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche8/tranche8-results.json`