# Quarter Wave Transformer Matching

Quarter Wave Transformer Matching
For a TL in the sinusoidal steady state with an arbitrary resistive load

the input impedance of the right-hand TL is given as

Now imagine that we have a special length l =l1 / 4 of TL, as indicated in the figure above. At this frequency and physical length, the electrical length of the TL is

Consequently, for a l/ 4 - length TL, tanb1l ®¥. Using this result in (1) gives

This result is an interesting characteristic of TLs that are exactly l/4 long. We can harness this characteristic to design a matching network using a l/4-length section of TL. Note that we can adjust Z1 in (3) so that Zin = Z0 . In particular, from (3) with Zin = Z0 we find
ZL = ÖZ0 RL
In other words, a l/4 section of TL with this particular characteristic impedance will present a perfect match (T = 0) to the feedline (the left-hand TL) in the figure above. This type of matching network is called a quarter-wave transformer (QWT). Through the impedance transforming properties of TLs, the QWT presents a matched impedance at its input by appropriately transforming the load impedance. This is accomplished only because we have used a very special characteristic impedance Z1 , as specified in (4). Three disadvantages of QWTs are that:
1. A TL must be placed between the load and the feedline.
2. A special characteristic impedance for the QWT is required, which depends both on the load resistance and the characteristic impedance of the feedline.
3. QWTs work perfectly only for one load at one frequency. (Actually, it produces some bandwidth of “acceptable” VSWR on the TL, as do all real-life matching networks.)