Feedback for noise reduction – a discovery in August 1927 by H. S. Black. Y(s) G(s) (Amplifier)  N(s) + S(s) But what about the stability with large open-loop gain amplifiers? (Electronic amplifiers in 1927 were of about 27th order!!)

Feedback System Stability and G(jw) Stability of the system is given by the zeros of 1+ kG(s). This was known (for the corresponding differential equation) since 1868 from the work of James Clarke Maxwell on Governors. If G(s) is known then it’s easy to check for the zeros of 1+kG(s) but we know only G(jw). (Electronic amplifiers of about 27 th order!!) We use the Argument Principle and for that we use the Nyquist Contour as the closed contour and map it with kG(s) to the Nyquist (or G(s))-Plane. Number of the encirclements, by the map in the G(s)-plane, of the (-1, j0) point is equal to z – p, where z is the number of RHP zeros of 1+kG(s) and p the number of RHP poles of 1+kG(s). The poles of 1+kG(s) are also the poles of G(s). G(s)k

Nyquist Map Re(s) Im(s) Re(G(s)) Im(G(s)) s-plane Nyquist Contour Nyquist Plane 1.Red part of the contour is mapped using G(jw) information. 2.Green part is the mirror image of the red part about the real axis (why?) 3.The blue part is the mapping of the Rexp(j  ) semicircle.