Quarter-wave impedance transformer

[3] The device can be used as a component in a filter, and in this application it is sometimes known as an inverter because it produces the mathematical inverse of an impedance.

[1] Filters incorporating λ/4 inverters are only suitable for narrow band applications.

This is because the impedance transformer line only has the correct electrical length of λ/4 at one specific frequency.

The further the signal is from this frequency the less accurately the impedance transformer will be reproducing the impedance inverter function and the less accurately it will be representing the element values of the original lumped-element filter design.

At the input to the line the reflected voltage adds to the incident voltage and the reflected current subtracts (because the wave is travelling in the opposite direction) from the incident current.

which is the same as the condition for dual impedances; Similar properties can be realized using either a "T" or "PI" network consisting of lumped elements each of which has a reactance equal to the Zo of the simulated one-quarter wavelength (λ), transmission line.

[8] This realization of the transformer is useful at lower frequencies where a quarter-wave transmission line would be impractically long.

The quarter wave transformer is a subset of series line (section) matching methods.

Using a transmission line as an impedance transformer.
The lumped-element low-pass filter (top) can be converted to a design that eliminates the inductors and contains capacitors only by the use of J -inverters, resulting in a mixed lumped-element and distributed-element design.
Quarter-wave transformers are illustrated in an impedance Smith chart . Looking towards a load through a length l of lossless transmission line, the normalized impedance changes as l increases, following the blue circle. At l =λ/4 , the normalized impedance is reflected about the centre of the chart.
Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Black dots represent electrons, and arrows show the electric field . A quarter-wavelength away from the open-circuit, the current and voltage oscillations are exactly the same as at a short-circuit, and vice versa. This reflects the fact that open circuit ( Z =∞ ) is dual to short circuit ( Z =0 ).
The lumped equivalent of the above transmission line. This is one of four possible realizations of the network.