Application of the describing function technique in a single-loop feedback system with two nonlinearities

A graphic procedure is presented which allows the describing function technique to be extended to a single-loop feedback system with two nonlinearities. The graphic technique is very simple and immediately allows qualitative answers, or quantitative answers subject to the usual errors and restrictions of the describing function technique, to be obtained regarding the presence of limit cycles, regions of stability, instability, etc. The method essentially is as follows. A plot of
G_{1}(j\omega) G_{2}(j_\omega)
in Fig. 1 vs. ω is made, and the point of intersection of
G_{1}(j\omega) G_{2}(j\omega)
with the negative real axis is noted, for example, at
G_{1}(j\omega^{\ast}) G_{2}(j\omega^{\ast}) =-1/\Gamma, \Gamma > 0
. By plotting |G_{d_{1}}(A1)| vs. A
1
in the second quadrant, and
|G_{d_{2}}(A_{2})|
vs. A
2
in the fourth quadrant, it is possible to plot a curve (relating |G_{d_{1}}| vs. |G_{d_{1}}|) in the first quadrant. If this curve intersects
|G_{d_{1}}| |G_{d_{2}}| = \Gamma
, a limit cycle exists in the system. If no intersection takes place, then no limit cycle exists in the system.