Cubic-form of Characteristic Equation and Its Graphical Solution by Means of ζ-<i>h</i> Nomogram

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  • 3次特性方程式と根計算用ζ-<i>h</i>線図
  • Cubic-form of Characteristic Equation and Its Graphical Solution by Means of ^|^zeta;-h Nomogram

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Introduced in this paper are nomograms which can be used for solving third order characteristic equation which is Hurwitz and a study on critical conditions of third order feedback control system.<br>Characteristic equation of third order constant coefficient differential equation or characteristic equation of third order feedback control system is generally as follows:<br>as3+bs2+cs+d=0 …(a)<br>where s is parameter of Laplace transform. The equation (a), when each of its coefficients is not equal to zero, can be modified into<br>s3+2ζωns2+ωn2s+2ζhωn3=0 …(b)<br>By making s/ωn=p, the following is obtained:<br>p3+2ζp2+p+2ζh=0 …(c)<br>Three roots of the equatin (c)<br>p1=x+jy, p2=x-jy, p3=v …(d)<br>have the following relationships:<br>y2=3x2+4ζx+1 …(e)<br>y4+2x2y2+x4-(1+h)y2+(3h-1)x2+h=0 …(f)<br>v=-2(x+ζ) …(g)<br>h=-v/2ζ(v2+2ζv+1) …(h)<br>where x and v are real and y is real or imaginary.<br>Nomograms for solving the equation (c) can be obtained by means of the graphs of the equations (e), (f) and (h).

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