# Finding Root Locus for 3 Real Poles and 1 Real Zero

Cuthbert Nyack
Applet below show how the root locus is determined from the magnitude and angle conditions.
The red blue boundary shows where the angle condition is satisfied. The closed loop poles for any K can be found by increasing K to that value and noting the starting point of the locus. This proceedure makes the closed loop poles at K equal to the an open loop system whose open loop poles have been moved to the closed loop poles with K.

A second approach to finding the closed loop poles for any K is to move the cyan x along the red blue boundary by adjusting sr and si until a point is found where the magnitude condition is satisfied as shown by the value of K.

To find the CL poles for K = 5, change the parameters to (5.0, 0.5, 1.0, 1.5, 2.0, sr, si) to see the starting points at ~ -0.55 ± ~j1.66 and at ~-1.8.

Alternatively set the parameters to (0.0, 0.5, 1.0, 1.5, 2.0, sr, si). With sr = -0.58 ± j1.67, K = ~5.0, f = ~-180°. With sr = - 1.844(by extrapolation), si = 0.0 , K = ~ 5 and f = -540°. The closed loop poles are therefore -1.844, -0.58 ± j1.66 in approximate agreement with the first method.

Gif image below show how applet should appear. 