For each Smith chart below, if you click on it you'll get it's pdf file for high quality printing.

Example matching problem using two tuning elements:

First step: don't worry about the stub lengths, first find the tuning susceptances:

To get started, recall that to get no reflection at plane B you must be on the "g = 1" circle in plane B'. We can move the "g=1" circle to plane A so that we can ultimately find Y1:

Now we have to get the load to plane A':

Now we must move on a circle of constant real part of Yload in plane A' to get onto the transformed g=1 circle in plane A; this will actually give us Y1:

Now for each possible choice of Y1 we need to move back to plane B'; this will gives us Y2: