The root locus

The crosses are the poles before feedback; the gold dots are the poles with feedback. Leave T at 0 and drag K slowly up from the far left: the gold dots leave the crosses, travel along the gold lines and always sit where gold meets ice blue. The motor's two poles move towards each other, meet at −1 when K = 1 and then part, one up and one down; the response at top right starts to overshoot and swing. Switch to one more pole and two of the dots bend towards the line (the coral dashes); past K = 12 they cross it and the swings grow. The bottom-right corner counts the poles right of the line twice, once by solving for them and once from Routh's table, and the two counts always agree. Now drag T up: the push also takes in how fast the error is changing, and a zero slides in from the left and pulls the paths back; from T = 1/4 on (the zero past −4), no K, however large, takes them across the line. On the stick, set T to 1: the pole on the right has to be pulled across by K, and past K = 1/2 the stick stands. Drag T back to 0, pushing only on how far it leans, and it swings forever.