IndieRF Match
L-network impedance matching calculator
All valid two-element L-networks for a complex source and load, with the series and shunt values, the nodal Q, and the return-loss band after snapping to real parts.
Two topologies
An L-network is one series reactance and one shunt susceptance. Topology A puts the series element at the load and the shunt at the source. It can cancel the load only when the load resistance is no larger than the source conductance will accept: RL ≤ 1/Gt. On a Smith chart normalized to a real source, that is the disk outside the r = Rs/Zref circle, including the circle itself. Topology B puts the shunt at the load and the series arm at the source, and it requires GL ≤ 1/Rs. Inside the forbidden disk that topology has no real solution.
Each allowed topology has two signs. One is the low-pass pair, series L with shunt C. The other is the high-pass pair, series C with shunt L. A series capacitor blocks DC. A network with no series capacitor and no shunt inductor passes DC.
Formulas
Write ZL = RL + jXL and YL = GL + jBL. The source Zs = Rs + jXs has target admittance Yt = 1/Zs* = Gt + jBt. Topology A:
Topology B:
A positive X is an inductor, L = X/ω. A negative X is a capacitor, C = −1/(ωX). A positive B is a capacitor, C = B/ω. A negative B is an inductor, L = −1/(ωB). If a discriminant is within 10⁻¹² of zero it is clamped, and the two signs become one. A reactance or susceptance smaller than about 10⁻⁹ relative to Zref is dropped, which is the single-element match.
Every accepted solution is re-evaluated with ABCD matrices. Series impedance is [[1, Z], [0, 1]] and shunt admittance is [[1, 0], [Y, 1]]. The cascade runs source to load. Zin = (A ZL + B) / (C ZL + D). The power-wave reflection is (Zin − Zs*) / (Zin + Zs). Ideal parts at the design frequency must sit under |Γ| of 10⁻⁹.
Computed check, Pozar Example 5.1
TODO(Jon): transcribe Pozar’s printed Example 5.1 answers, with the edition and page, before this table is read as a book comparison. The numbers below are computed by this engine and checked with the ABCD oracle. They are not a copy of the textbook line.
Load 200 − j100 Ω, real source and chart reference 100 Ω, 500 MHz. Only topology B is legal. The sweep used for snapping is 450–550 MHz so the E24 choice stays on the nearest values.
| Case | B and X | Parts | At 500 MHz |
|---|---|---|---|
| B+, ideal, low-pass | B = 0.002899 S, X = 122.47 Ω | 38.98nH, 922.8fF | matched, |Γ| under 10⁻⁹ |
| B−, ideal, high-pass | B = -0.006899 S, X = -122.47 Ω | 2.599pF, 46.14nH | matched, |Γ| under 10⁻⁹ |
| B+, E24, Q infinite | nearest E24 on a ±10% sweep | 39nH, 910fF | 46.06 dB RL, GT -0.0001 dB |
| B+, ideal parts, QL = 30, QC = 200 | same ideal X and B, constant Q | 38.98nH, 922.8fF | 33.79 dB RL, GT -0.191 dB |
| B+, E24, QL = 30, QC = 200 | snapped, then Q | 39nH, 910fF | 32.01 dB RL, GT -0.190 dB, 450MHz – 550MHz |
Finite Q does not change the topology solver. It is applied when the network is evaluated, so the ideal match and the lossy match stay distinct. The E24 row with infinite Q is still a conjugate match to well under a millidecibel of transducer loss. Adding QL = 30 and QC = 200 to the ideal parts drops return loss into the mid-30 dB range and costs about 0.19 dB of transducer gain. That 0.19 dB figure is computed, not taken from a book.
Bandwidth
Nodal Q is |X|/R on a series node and |B|/G on a shunt node. The rough bandwidth f0/Qn is labeled as an estimate. The number reported on each card is the contiguous span around the design frequency where return loss stays at or above the threshold you picked (6, 10, 14, or 20 dB). Edges are interpolated. If the span hits the end of the sweep, the card is limited by that end and the true band may be wider.
FAQ
Can an L-network match a complex source?
Yes. The source is matched to its conjugate. Topology A is allowed when the load resistance is at most 1/Gt, where Gt is the conductance of Zs*. Topology B is allowed when the load conductance is at most 1/Rs. With a complex source both topologies can be legal, so the list can hold four networks.
Which L-network has the highest bandwidth?
Rank by the return-loss band on the sweep, not by the f0/Q estimate. For a real-to-real step the two-element L has a nodal Q of sqrt(Rhigh/Rlow − 1). That Q is the minimum for two lossless elements, which is why a wider real-to-real match needs more than one L-section.
When would I use a Pi or T network?
A Pi or T is two L-sections through a virtual resistance, used when you want a higher nodal Q than the L-section minimum, often for harmonic filtering. This page does not calculate Pi or T networks. It stops at the L-section solutions.
What component Q should I enter?
Constant Q is the default: the inductor series resistance is ωL/Q and the capacitor ESR is 1/(ωCQ), both evaluated at each frequency with the Q you typed at the design frequency. The skin-effect option scales only the inductor, Q(f) = Q0 · sqrt(f/f0). Leave Q at 0 for an ideal part. A coil Q near 30 and a capacitor Q of a few hundred is a reasonable first pass for a lumped UHF or low-microwave match, then replace it with the datasheet.