By William K. Roots (Auth.)
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Table I lists the values of φ at, and an octave each side of, the break frequency. 6. Nyqutst Diagrams* The phasor in Fig. ) Thus this phasor can be represented by (1 -jv)l(l + v*) = 1/(1 +jv) = 1/(1 +jœT) 1/(1 + j«>T) is the relationship of output to input in an exponential lag when the input and output are both undamped and harmonic. It is called a transfer function. + See Nyquist. 6. NYQUIST DIAGRAMS 35 The transfer function is obtained by replacing all "ZJ's" in a transfer operator by "jœ's" and introducing the general symbols: G(D) for a transfer operator G(ja>) for a transfer function The length of the phasor in the frequency response diagram (Fig.
This low-frequency similarity can be explained as follows; if T represents the magnitude of both a transit delay and an exponential lag: 1. Geometrically the two loci 1/(1 + Ts) and exp(-Ts) are tangent at ω = 0. This is illustrated in Fig. 7. 2. 5 Locus of / w + 'D ΑΌ = πι■14 FIG. 7. Nyquist display of the frequency response of an exponential lag [1/(1 + Ts)] and a transit delay [exp( — Ts)], each of the same magnitude (T).
1 1 + TD -10° -20° i = i0 sin cot Θ = 0O sin(tüf + φ) A = eji0 = l/[(l + νψ*] φ = — arctan v v = ωΤ -30° -40° FIG. 13. Frequency-response locus of a first-order process (such as an expo nential lag). T h e amplitude ratio A and the phase angle φ ave polar coordinates of the transfer locus graduated in the nondimensional frequency v = ωΤ. T h e locus is a semicircle. 32 2 . THERMAL-PROCESS RESPONSE the tip of this phasor traces out a curve known as the frequency response locus. The frequency response locus of an exponential lag or any similar first-order process, which is subjected to unit sinusoidal input, is a semicircle.