This circuit is for a QRP (low power) antenna tuner, a.k.a. a transmatch, for use in the short wave amateur radio bands from 3-30 Mhz. It allows a wide variety of antennas to be connected to a low power transmitter. When the circuit is properly tuned, the maximum transmitter power will be delivered to the antenna. It is used in conjunction with a standing wave ratio (SWR) meter. This is a fairly generic antenna tuner circuit.
With coax-fed antennas, the best location for an antenna tuner is where the coax feeds the antenna. Logistically, this may cause the tuner to be in a location that is difficult or impossible for the operator to tune so most people locate the tuner on the transmitter side of the coax.
Set the capacitance switch to 0 (center), adjust the variable capacitor to the mid point. Adjust the inductor to near the minimum inductance. Briefly transmit a CW carrier and observe the SWR reading. If your transmitter has an adjustable output power level, start the adjustment with low power then increase the power and fine-tune when a good match has been found.
Increase the inductance until the minimum reflected power and maximum forward power is observed. Adjust the capacitance for the best SWR reading. It is necessary to go back and forth on the adjustments to find the best match. If the best match is found with the capacitor at the max value, switch in either the 270pf or 510 pf parallel capacitors and re-adjust the variable capacitor and inductor.
Be careful not to leave the transmitter on for too long in the unmatched condition, doing so can damage the output transistors. If your transmitter has a variable output power, tune up at low power then re-adjust the circuit at full power. If you are using a vacuum tube transmitter, don't worry about it.
Caution: higher power transmitters can generate high voltages within this circuit, don't touch any of the wires when the transmitter is operating. If the roller inductor's adjustment shaft is connected to the inductor's wiring, the shaft should be mounted so that it does not come in contact with the metal box. The set screw on the inductor's knob may be electrically hot during use, you may want to cover it with a drop of plastic glue or silicone after tightening.
See Wikipedia's Antenna tuner article for more background information and Ulrich L. Rohde's (N1UL) antenna tuner circuit which can tune a wider range of antenna impedances. Ulrich Rohde has also published a more detailed article (zip) in German.