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OCTOBER the modulus-frequency function, as indicated in Fig. 1. A further condition occurs when for 1<0.1578 the locus indents as shown in Fig. 2. Equation (1) may be written as where G g = -and S=s For the frequency response, with S=jx, where x = w / w is a nondimensional frequency, Wn3 w.with This modulus function has real unequal turning-point values, for the modulus varying as a function of x , providing (1 -2r)2 > 3 giving 1;<0.259 as the condition for which a resonance peak is exhibited.For an indented form of locus, designating y as the imaginary part of FGx), the change of y with x must be of the type shown in Fig. 3, Le., there must be three turning points. From (3);Differentiating with respect to x for the turning points gives 32' -(7 -*')e' + (5 -12r')~ -1 0, (6) with z =$. Here e and r are to be determined to give the condition where indentation just occurs. Fig. 4 shows the form of variation of y with x when this does not occur, e.g., for a frequency response locus as shown in Fig. 1. For the curve of Fig. 3 to degenerate to that of Fig. 4, the frequency values xl and 2 2 must coalesce, i.e., ( 6 ) must factorize to Equating coefficients of (6) and (7) gives and Eliminating C2 from (8) and (9) gives which factorizes to (al)(a -1.9594)(a -0.036) .(a + 1.182) = 0 (11) The root a = 1 gives r = O in (8) or (9) while a =0.036 gives a negative value. As z must be positive for physically-realizable frequency values, a = 1.9594 is the valid root. Substituting this value in (8) or (9) then gives 6=0.1578 as the transition condition. Quo d ra eac \ ' C -L Fig, 1. Noniadented Nyquist locus. Drawn for S -0.2. 8 -3. Lead I Fig. 2. Inder Y l t e d form. Drawn for f =O, A -1. -3 Fig. 3. Variation o f y with I for indented locus. Drawn for f =0.1.Hence, for rC0.1578 the Nyquist locus has the indented form shown in Fig. 2.The difficulties associated with these changes of shape of the Nyquist locus can be avoided by plotting the inverse locus, I Fig. 4. x-y variation: noniadented locus. D r a m for f ~0 . 2 .Fig. 5. Inverse Nyquist locus. derived from (3) as g P ( i X ) = jn(1x* +j2rn) (12) lt'riting the lhs as X+j Y gives X = -2r$ and Y=x(l-$), leading to Y' = -(1 + X ) tvalid for XSO. The inverse Nyquist diagram is therefore a parabola the shape of which does not vary with r; a typical curve is shown in Fig. 5.

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