L-Network Impedance Matching & Smith Chart

Method, formulas and references

Problem

A lossless two-element network transfers all the available power from a source Zs = Rs + jXs to a load ZL = RL + jXL when its input impedance, with the load connected, is the complex conjugate of the source impedance:

For a real source (Xs = 0) this is Pozar's L-section problem of matching ZL to a line of impedance Z0 = Rs. The load must have some resistance; a purely reactive load cannot absorb power, so it cannot be matched.

Two topologies

Shunt part next to the load (Pozar Fig. 5.2a). The shunt susceptance B moves the load admittance YL = GL + jBL until the real part of its inverse equals Rs, then the series reactance X cancels what is left:

It exists when GL·Rs ≤ 1. Series part next to the load (Pozar Fig. 5.2b). With the target admittance Yt = 1/Zs* = Gt + jBt, the series reactance moves the load until the real part of its admittance equals Gt, then the shunt part adds the missing susceptance:

It exists when RL·Gt ≤ 1. With a real source these reduce to Pozar's equations: the first case is the load inside the r = 1 (1 + jx) circle, or outside it with enough reactance that its conductance is below 1/Z0; the second is RL < Z0. When both apply there are four networks, otherwise two. When a square root is zero, or a part comes out as zero, the network has one part (for example RL = Rs only needs the load reactance cancelled) and duplicates are dropped. Each result is checked by computing Zin from the parts.

Component values

At ω = 2πf₀, a positive series reactance is an inductor L = X/ω and a negative one a capacitor C = −1/(ωX). A positive shunt susceptance is a capacitor C = B/ω and a negative one an inductor L = −1/(ωB). A series inductor with a shunt capacitor is a low-pass network; a series capacitor with a shunt inductor is a high-pass network. The low-pass form helps with harmonic suppression; the high-pass form also blocks DC.

Q and bandwidth

The node quality factor is Qn = |X|/R of the impedance at a node of the network. The page reports the largest Qn of the load, the node between the two parts and the input. For resistive terminations it is set by the impedance ratio alone:

Ludwig and Bogdanov give the loaded Q of an L-section as Qn/2, so the half-power bandwidth is roughly 2f₀/Qn. The reported bandwidth does not rely on that estimate: it is the band around f0 where the swept return loss stays at or above the chosen limit, found by stepping outwards from f0 and refining each edge by bisection. Higher Q means a narrower band; the Bode–Fano criterion (Pozar §5.9) limits how much any network can widen it for a reactive load.

Frequency sweep

The sweep uses ideal, lossless parts. The source and load impedances entered at f0 are modelled as their resistance in series with the inductor or capacitor that has the entered reactance at f0, so their reactance changes with frequency. The plotted quantity is the power-wave reflection coefficient at the source:

1 − |Γ|² is the fraction of the available power that reaches the load. For a real source this is the ordinary reflection coefficient.

Standard values

Each part is rounded to the series value just below and just above it (IEC 60063 E12, E24 or E96), and of the up to four combinations the one with the highest return loss at f0 is shown. Inductors are commonly sold in E12 or E24 steps; small RF capacitors in E24 steps or finer.

Smith chart

The chart is normalised to Rs: z = Z/Rs and Γ = (z − 1)/(z + 1). With a real source the matched point is the centre. With a complex source it is z = 1 − jXs/Rs, still on the r = 1 circle, and is marked with a ring. A series reactance moves the point along a circle of constant resistance; a shunt susceptance moves it along a circle of constant conductance (the admittance grid is the impedance grid rotated by 180°). The arcs are computed point by point from the actual part values.

Limits

Real parts have loss, self-resonance, pad and via parasitics and tolerance; above a few hundred megahertz a chip inductor or capacitor can differ noticeably from its nominal value. Check the selected parts with their S-parameter models, and keep each part's self-resonant frequency well above f0. A high-Q network is more sensitive to all of this.

Data sources

1. D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012
   §2.4 (Smith chart, Γ = (z − 1)/(z + 1)), §5.1 (L-section matching, eqs. for Fig. 5.2a and 5.2b, Example 5.1 used in the tests), §5.9 (Bode–Fano limit).
2. K. Kurokawa, “Power Waves and the Scattering Matrix”, IEEE Trans. Microwave Theory Tech., vol. 13, no. 2, pp. 194–202, 1965
   Power-wave reflection coefficient (Z − Zs*)/(Z + Zs) for a complex source impedance.
3. R. Ludwig, G. Bogdanov, RF Circuit Design: Theory and Applications, 2nd ed., Pearson Prentice Hall, 2009
   §8.1 (two-component matching networks, node quality factor Qn = |X|/R and the relation QL = Qn/2 for L-sections).