Transmission Lines: Quarter-Wave Transformer

Quarter-Wave Transformer

Recall our formula for the input impedance of a transmission line of length L with characteristic impedance Z0 and connected to a load with impedance ZA:

An interesting thing happens when the length of the line is a quarter of a wavelength:

The above equation is important: it states that by using a quarter-wavelength of transmission line, the impedance of the load (ZA) can be transformed via the above equation. The utility of this operation can be seen via an example.

Example. Match a load with impedance ZA=100 Ohms to be 50 Ohms using a quarter-wave transformer, as shown below.

Solution: The problem is to determine Z0 (the characteristic impedance of our quarter-wavelength transmission line) such that the 100 Ohm load is matched to 50 Ohms. By applying the above equation, the problem is simple:

Hence, by using a transmission line with a characteristic impedance of 70.71 Ohms, the 100 Ohm load is matched to 50 Ohms. Hence, if a transmitter has an impedance of 50 Ohms and is trying to deliver power to the load (antenna), no power will be reflected back to the transmitter. In general, impedance matching is very important in RF/microwave circuit design. It is relatively simple at a single frequency, but becomes very difficult if wideband impedance matching is desired.

This technique is commonly employed with patch antennas. Circuits are printed as shown in the following figure. A 50 Ohm microstrip transmission line is matched to a patch antenna (impedance typically 200 Ohms or more) via a quarter-wavelength microstrip transmission line with the characteristic impedance chosen to match the load.

Because the quarter-wavelength transmission line is only a quarter-wavelength at a single frequency, this is a narrow-band matching technique. In the next section, we'll look at more uses of transmission lines.