i>Kiii.\\:roi\i iMi'iin.i.wf-: <)/• two-mesh cihcuits nis 



rt-stricted nK>is. l-dr >jx-(:iHl cast-^ of /en), pure inuiKin.iry, or inliniii' 

 r(mts, tlu" rorrespDnrlinp domains art- the limiting rapes of the Rcnernl 

 iii)main. clfM-rilx-d afx)\-e. Such limifinp: rases may rediire to a sinRli' 

 >-»'t;nu'nt or to a region hounded in pari !>>■ the line al inhnit>'. Tht' 

 homoneneous coordinates emploxj-d .ire \er>' useful in dealing; with 

 these sfR-rial cases. 



4. Ki(;iKi-.s ii.i.i siKAtiM, lilt: Domain i ii Poi.ics 



The preceding section presented a discussion of the domain of the 

 ixiles ass<K-i.ited with any four assigned roots, the domain being 

 plotted in terms of the coefficients of the denominator of the impe- 

 dance, that is, in terms of symmetric functions of the poles. In order 

 to show the mutual relations between the actual \alues of the roots 

 anil the poles, it is convenient to plot, in the up()er half of the complex 

 plane, the domain of one pole, the other pole being its conjugate. 

 This provides a complete representation for the case of complex poles. 

 In order tt) include the domain of real poles, an auxiliary graph can 

 l>e j)ro\ided to indicate pairs of points on the real axis. 



The mathematical analysis for this form of representation can be 

 obtained from that of the preceding section by substituting (ii+ii3 = 

 — b-> bt and »i</i3 = 6.T bi. For complex poles, li-> = ii + iv and ^3 = 11 — 11', 

 this transformation from the x. v plane to the 11. v plane is simply 

 2u= —X and u''+v- = y. Thus a conic in the .v. y plane becomes, in 

 general, a curve of the fourth degree in the 11. v plane. The analysis 

 of the curves obtained in the «, i' plane is not so simple as in the other 

 plane, but there is a decided advantage in the interpretation of the 

 results in this plane, since the coordinate «, the real part of the pole, 

 corresponds to the damping factor, and the coordinate i", the im- 

 aginarN' part of the pole, corresponds to the frequency factor. 



In the complex plane, the necessary conditions (29) -(;}!) rc(|uirc 

 the domain of comple.x poles to lie entireK' within the region bounded 

 by the vertical axis, a vertical line to the left of the axis, two circles 

 about the origin as center, and a circle through the origin with its 

 center on the real axis. Furthermore, the boundary cur\e of the 

 domain must Ik- tangent to each of these lines and circles, since the 

 corresponding conic (3) for rf = () is tangent to the corresp<inding 

 lines (4)-(6) for d = 0. 



For the special case of one root a positi\e pure imaginarN-. the 

 second root In-ing its conjugate, the domain in the upper half of the 

 complex plane reduces merely to the points on an arc of a circle with 

 its center on the real axis. If the third root is complex with a posi- 

 tive imaginary- part, the fourth root l)eing its conjugate, the domain 



