L-section

Basic schematic for matching R₁ to R₂ with an L pad. R₁ > R₂, however, either R₁ or R₂ may be the source and the other the load. One of X₁ or X₂ must be an inductor and the other must be a capacitor.

L networks for narrowband matching a source or load impedance Z to a transmission line with characteristic impedance Z₀. X and B may each be either positive (inductor) or negative (capacitor). If Z/Z₀ is inside the 1+jx circle on the Smith chart (i.e. if Re(Z/Z₀)>1), network (a) can be used; otherwise network (b) can be used.^[2]

A simple electrical impedance-matching network requires one capacitor and one inductor. In the figure to the right, R₁ > R₂, however, either R₁ or R₂ may be the source and the other the load. One of X₁ or X₂ must be an inductor and the other must be a capacitor. One reactance is in parallel with the source (or load), and the other is in series with the load (or source). If a reactance is in parallel with the source, the effective network matches from high to low impedance.

The analysis is as follows.^[3] Consider a real source impedance of ${\displaystyle R_{1}}$ and real load impedance of ${\displaystyle R_{2}}$. If a reactance ${\displaystyle X_{1}}$ is in parallel with the source impedance, the combined impedance can be written as:

${\displaystyle {\frac {jR_{1}X_{1}}{R_{1}+jX_{1}}}}$

If the imaginary part of the above impedance is canceled by the series reactance, the real part is

${\displaystyle R_{2}={\frac {R_{1}X_{1}^{2}}{R_{1}^{2}+X_{1}^{2}}}}$

Solving for ${\displaystyle X_{1}}$

${\displaystyle \left\vert X_{1}\right\vert ={\frac {R_{1}}{Q}}}$.

${\displaystyle \left\vert X_{2}\right\vert =QR_{2}}$.

where ${\displaystyle Q={\sqrt {\frac {R_{1}-R_{2}}{R_{2}}}}}$.

Note, ${\displaystyle X_{1}}$, the reactance in parallel, has a negative reactance because it is typically a capacitor. This gives the L-network the additional feature of harmonic suppression since it is a low pass filter too.

The inverse connection (impedance step-up) is simply the reverse—for example, reactance in series with the source. The magnitude of the impedance ratio is limited by reactance losses such as the Q of the inductor. Multiple L-sections can be wired in cascade to achieve higher impedance ratios or greater bandwidth. Transmission line matching networks can be modeled as infinitely many L-sections wired in cascade. Optimal matching circuits can be designed for a particular system using Smith charts.

Load-end conditions

In a transmission line, a wave travels from the source along the line. Suppose the wave hits a boundary (an abrupt change in impedance). Some of the wave is reflected back, while some keeps moving onwards. (Assume there is only one boundary, at the load.)

Let

${\displaystyle V_{i}\,}$ and ${\displaystyle I_{i}\,}$ be the voltage and current that is incident on the boundary from the source side.

${\displaystyle V_{t}\,}$ and ${\displaystyle I_{t}\,}$ be the voltage and current that is transmitted to the load.

${\displaystyle V_{r}\,}$ and ${\displaystyle I_{r}\,}$ be the voltage and current that is reflected back toward the source.

On the line side of the boundary ${\displaystyle V_{i}=Z_{c}I_{i}\,}$ and ${\displaystyle V_{r}=-Z_{c}I_{r}\,}$ and on the load side ${\displaystyle V_{t}=Z_{L}I_{t}\,}$ where ${\displaystyle V_{i}\,}$, ${\displaystyle V_{r}\,}$, ${\displaystyle V_{t}\,}$, ${\displaystyle I_{i}\,}$, ${\displaystyle I_{r}\,}$, and ${\displaystyle I_{t}\,}$ are phasors.

At a boundary, voltage and current must be continuous, therefore

${\displaystyle V_{t}=V_{i}+V_{r}\,}$

${\displaystyle I_{t}=I_{i}+I_{r}\,}$

All these conditions are satisfied by

${\displaystyle V_{r}=\Gamma _{TL}V_{i}\,}$

${\displaystyle I_{r}=-\Gamma _{TL}I_{i}\,}$

${\displaystyle V_{t}=(1+\Gamma _{TL})V_{i}\,}$

${\displaystyle I_{t}=(1-\Gamma _{TL})I_{i}\,}$

where ${\displaystyle \Gamma _{TL}\,}$ is the reflection coefficient going from the transmission line to the load.

${\displaystyle \Gamma _{TL}={Z_{L}-Z_{c} \over Z_{L}+Z_{c}}=\Gamma _{L}\,}$^[4][5][6]

Source-end conditions

At the source end of the transmission line, there may be waves incident both from the source and from the line; a reflection coefficient for each direction may be computed with

${\displaystyle -\Gamma _{ST}=\Gamma _{TS}={Z_{s}-Z_{c} \over Z_{s}+Z_{c}}=\Gamma _{S}\,}$,

where Zs is the source impedance. The source of waves incident from the line are the reflections from the load end. If the source impedance matches the line, reflections from the load end will be absorbed at the source end. If the transmission line is not matched at both ends reflections from the load will be re-reflected at the source and re-re-reflected at the load end ad infinitum, losing energy on each transit of the transmission line. This can cause a resonance condition and strongly frequency-dependent behavior. In a narrow-band system this can be desirable for matching, but is generally undesirable in a wide-band system.