The construction of phase angle and gain loci by j[omega]-axis transformation

Some formulas were derived to construct the phase angle loci and the gain loci by the shifting technique. These formulas give the angular frequencies and corresponding gains for the phase angle loci and the angular frequencies for the gain loci for each shift of the j[omega] -axis. The oscillatory conditions for the phase angle loci were determined by applying the Continued Fraction Expansion to the ratio of the polynomials obtained from the real and imaginary part of the phase angle loci equation. From the Continued Fraction Expansion, two sets of formulas were obtained; one set for [phi] = [upsilon][pi] , ([upsilon] = 0,1,2,3, ...) and one set for [phi][not equal][upsilon][pi] . Each set consists of two formulas to determine the angular frequency and gain. By these formulas the angular frequencies and corresponding gains for a particular value of phase angle [phi] are determined. Two formulas were derived for construction of the gain loci by the shifting technique; one for a polynomial form and one for a factored form of the phase angle loci equation. By this formula the angular frequencies, for a particular value of gain K , are determined for each shift of the j[omega]-axis.