On a couple of tests, we actually had to analytically draw our smith charts on graph paper, then use it for then next problem. Say you want 15 dB return loss a

On a couple of tests, we actually had to analytically draw our smith charts on graph paper, then use it for then next problem. Say you want 15 dB return loss at some frequency, well there are two impedances points for that. One with the impedance locus outside Z0, and another with the locus encompassing Z0. The latter has broader bandwidth, but that intuition is lost without a Smith or polar plot.