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Geometric-Analytic Theory o)Srans'ition in

ElectricalOLngDineer.IRn:E. FOLKLE BCOLIN}DERt, MEMBER,T IR1E

Summary-A geometric-analytic theory of transition is presentedand applied to circuit theory. A transition from one state to anotheris represented in a complex plane by two points which, by variation ofa parameter, approach each other, coalesce, and then separate alongtrajectories perpendicular to the original trajectories.

Three analogous cases are treated, namely

1) Movements of fixed points in the complex impedance planeand, the complex reflection coefficient plane (Smith chart),

2) Movements of poles in the complex frequency plane, and3) Movements of saddle points in the complex frequency plane.

In the analytic treatment, the linear fractional transformation(Moebius transformation) is used, which makes conformal graphicalmethods applicable in the geometric treatment. Such a method is, forexample, the isometric circle method.

By muapping stereographically the complex plane on the Riemannunit sphere, we see that a transition can be represented in threedimensions by the movements of two straight lines, each being thepolar of the other with respect to the sphere. The transition takesplace when both lines are perpendicular and tangent to the sphere ata point corresponding to the transition point.

I1. ISOMi-TRIC CIRCiESThe isometric circle is defined as the circle that is the

complete locus of poinlts in, the neighborhood of whichelengths are unaltered in magnitude by the Iinear fractionial tranisfor-mation

Iaw + b

w-)cW + d

ad - be = I

where w-=a+jv, w1=W +jv' aId a, b, e amid d aie com,-

plex constan-ts obeying the condition ad beIFrom (1) we obtain

dw' 1(2)

dw (cw + d)2

so that the isometric circle of the direct transformationis

1. INTRODtJCTION

COMPLEX quantity w=-+jv(j2 -1) can berepresented by a point with the coor(iinLates aand v in a plane. The use of such a "complex

plane" seems to have been-i suggested independently byWessel (1745-1818), Arganid (1768-1822), and Gauss(1777-1855). Ever since Steinmetz, in 1895, introducedthe coniplex plane in electrical enigineering, this sii-nplemiathematical tool has obtained extensive applications,especially in circuit theory.

In the present paper, the complex plane will be usedin a siniple geometric-a-alytic theory of tran-isitioni inelectrical engineeri-ng. The geometric part of the theoryutilizes two "isomlletric" circles which are usually usedin connection with a graphical method of transforniinga com-plex: quantitty by, means of the liiiear fractionaltiransformation (Moebius transformation), called the"isometric circle miethod. "I The analytic part conlsistsof the use of quadratic equations. Aifter a briif stuly ofthese tools, we are going to investigate some sillmple ex-amples of fixed point trajectories in the complex imped-ance and reflectioini coefficient planes, and pole andsaddlepoint trajectories in the complex frequency plane

* Original manuscript received by the IRE, L)ecember 10, 1958;revised manuscript received, March 2, 1959. Presented at the URSIIRE Fall Meeting, Pennsylvania State College, State College, Pa.,October 21-23, 1958.

t Electromagnetic Radiation Lab., Air Force Cambridge Res.Center, Bedford, Mass.

I E. F. Bolinder, "Impedance and polarizatiorn-ratio transforma-tions by a graphical method using the isometric circles," IRE FRANS.ON MICROWAVE THEORY AND TECHNIQUES, Vo1. MTT-4, pp. 176180; July, 1956.

Iw + d 1, ce# 0 (3)

Silmilarly, the in-iverse tran sforma tion

dw + b!

ew' aad .-b (4)

has the isometric circle

ICw't -(a I, c XO (1)Thus the isoCmXetric circle of the first direct transfornmia-tion, Cd, has its ceniter at Oa= dl and radius R,=I1/ e!c the isometric circle of the inverse transforma-tion, C,, has its center at 0=-a/c aind the samne radiusSee Fig. 1.

Mathematically, (1) is divided in-to two classes oftransformation.: the loxodromic transformation, char-acterized by a+dd-comp1ex, and the nonloxodromictransformation, characterized by a+d ireal. 'Ihe second class is further (divided in-to hyperbolic ( a 1 d| > 2),parabolic (a+d' +2), and elliptic (Qa+dI <2) tran.s-formationis.

In F'ig. the isom-netr-ic circlc mIethod. is outlinied. Itconsists in the loxodromic case ot 1) an. inversion- in tIheisomctietric circle of direct tiranslormation C( (wW ) 2)a reflection. in the symm71-iaeti-y iniie L to the two circles,(Wi>W2) anid 3) a rotationi arounid the ceniter 0w of theisom-netric circle of the iniver se tra,nsfornimation throughai-i angle -2 aig (a+d), (w2>W'). In the non.loxodrorniccase a ]d real, so tlhat the third opeiratiorn is linm-imiat-eci (w2 w') xIn this work onily the isometric circlesand n.ot the isometric circie m-nethod itself will be usedl.

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Bolinder: Geometric-Analytic Theory of T'ransition in Electrical Engineering

+ j "states" is obtained. We are now going to study simpleexamples of how such a transition may occur in, con-nection with fixed point trajectories in the impedanceand reflection coefficient planes, and pole and saddle-point trajectories in the complex frequen,cy planie.

IV. FIXED POINT TRAJECTORIESImpedance and Reflection Coefficient TransformatlionsThrough Bilateral Two-Port NetworksThe input voltage V' and the input curren-it I' of a

bilateral two-port network can-i be expressed in the out-put voltage V and output current I by the followiniglinear equations:

'= aV + blI = cV + dIFig. 1-The isometric circle method. (10)

Wlh

w-plane, W U+jv

wWeW =lW2 A'B

ip 2

where, at a fixed frequenicy, the four complex constantssatisfy the reciprocity relations

ad - bc - 1. (Ii)If we put

W2h U

Vtif/

'

_=zW7e

Fig. 2-The roots of a quadratic equation plottedin a complex plane.

11I. QUADRATIC EQUATIONS

If we solve an arbitrary complex quadratic equation

Aw2- 2Bw + C = 0 (6)

where AL B, and C are real constants, we obtain

V1.IV2J

B VB2-ACI

A A (7)

(12)

we obtain

aZ + b

cZ ± dad - bc = 1. (13)

This is a linear fractional transformnation. By using theproperty that the imaginary axis of the Z-plane,Z R +jX, is invariant under a lossless transformation,we obtain for such a transformation

a'Z +jb"z7= d.jc"Z + d'

a'd' + bc"=t 1 (14)

In (7) B2-AC is called the discriminant. If the dis-criminant is positive, two real roots w,h and W2h are ob-tained which we can plot in the complex w plane. SeeFig. 2. If the discriminant is niegative, two conjugatecomplex roots wie and W2, are obtained:

Wie' B VAC- B2I> - -J

W2eY A A(8)

See Fig. 2. Finally, if the discriminant equals zero thetwo roots coalesce and we obtain

BWiLp = W2P = (9)

If we assume that the constants A, B, and C are func-tions of a certain paramieter, then, by varying the pa-rameter, wi and W2 may approach each other, coalesce,and separate along trajectories perpendicular to theoriginal trajectories. A "transition' between two

wherea = a' + ja")b = b' + jb"c = c + jc"d = d' + jdd"

The fixed points of (14) are:

ZfI a - d+v\(a'+d')2 --4Z2j 2jc"

(15)

(16)

By using (14) and the well known expression for thereflection coefficient

Z 1

-,A/2 -%/2r =

Z 1

V2 %/2

Z' 1

V\/2 /2

r /--\ fZ' 1_V +V

(17)

X_ i 0-

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PROCEEDINGS OF THE IRE

X LX

HYP PAR. ELL(a) (b) (c)Fig, 3--Fixed point trajectories in the complex

itmipedance platne,

Fig. 5-Fixed point trajectories of the Riemann unit sphere,

HYP PAR. ELL.

(a) (b) (c)

Fig. 4-Fixed point trajectories in the comnplex reflectioncoefficient-plane (Stm-ith chart).

we obtaini

AF + C*CF+A*

where an asterisk indicates a conjugate cotmplexquantity, and

A [(a' + d')--j(b +Gt)[(a' d') j(b'

Fixed Point Trajectories in the Z Plane and the SmithChart, and on the Riemann Sphere

Since the distance betweeni the centers of the two iso-m-ietric circles in the nonloxodromic case is (a'+-d')Wj"?while the sum of the two radii is 2/c", it follows that thehyperbolic case is obtained, if the two circles are ex-

ternial; the parabolic case, if they are tangent; aid theelliptic case, if they intersect. In the Z platne theisometric circles have their ceniters on the imagrinary axis

in the lossless case. A simple examuple indicating the iso-metric circle positionls of a lossless exponentiallyNtapered transiiiissioni line is shown in F ig. 3. A geo-

ml-etric interpretation of (16) slows that in the hyper-bolic or below cutoff case, the fixed points are obtainedIV here a circle, orthogonal to the two isometric circles,cuts the imnaginary axis. Iin the parabolic or cuAtoff case,

the fixed points coalesce at the point of tangetncy of thetw o isometric circles. Finally, in the elliptic or abovecutoff case, the fixed points constitute the crossoverpoinlts of the two isometric circles. We find, that if wevary a parameter, for example, the length of the line orthe frequency, the fixed points move oni the imnaginaryaxis, coalesce, and then separate perpendicularly to theorigii1al trajectories following the ullit circle.The corresponiding trajectories in the Smith chart are

showni in Fig. 4. Here the fixed poin-ts move oni the ui-sitcircle in the hyperbolic case, coalesce, and then separateperpendicularly followiing the im:aginary axis of theF-plane (F PA=F, 0) in the elliptic case

If we map the Z plane stereographically oii the uiiitRiemaun sphere with the top of the sphere, (x, y, z)

(0, 0, 1), as projection center, the Smrlith chart is oh-taimed in the yz plane by an additional stereographicprojectioni from the poitnt (- 1, 0, 0). 4 See Fig. 5 Byplotting the fixed point trajectories of the lossless exponieiitially tapered transmissioin line studied above oithe sphere, we find that by usiiig three dimilenlsions thediscon-tinuous mnovements of the fixed points ii twodinieiisions are replaced by a continluous 1niovei-n-eiit. Iithe elliptic case, the fixed points miove oni the unit circlein the xy plane. We connect the poinits by a stiaight li ieLi, parallel to the x axis. This line has a polar lihue LJ2parallel to the z axis. W\7heni the line Lie m-ioves towardsthe point (0, 1, 0), L2e moves towards the samve poi;nt.In the parabolic case both lines, niow called Li, and L2,are tangent to the sphere. If the hiiies continue thenrmovemen:ts the hyperbolic case is obtained. Nosw the

2 F, Steinier, "Die Anwen-duing der Riemannscheii Zahleiikugelund ihrer Projektioneni in der Wechselstromtechnivk," Radiowelt, vol.1, pp. 23-26; October, 1946.

3 H. A. Wheeler, "Geometric relations in circle diagrams of trans-mission-line imnpedance," Wheeler Monographs, Wheeler Labs., GreatNeck, N. Y., vol. 1, no. 4,1948.

4G. A. Deschamps, "Geometric siewpoints in the representationiof waveguides and wavegtuide jun-ictionis,7' Proc. Symp. on MkodernNetwork Synthesis, Polytechnic Inst. of Brooklyi, Bklyfn, N. y. vol.1, pp. 277-295; April, 1952.

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Bolinder: Geometric-A nalytic Theory of Transition in Electrical Engineering

first line, called Llh, falls outside the sphere, while theseconid line L2k cuts the sphere in the fixed points situ-ated on the unit circle in the yz plane. Thus a smoothtransition between the elliptic and hyperbolic states isobtained.

V. POLE TRAJECTORIES

The Resonant Circuit

The input impedanice of a simple parallel resonancecircuit consistinig of an inductance L with a series re-sistance r, and a capacitance C with a coniductanice G(see Fig. 6) is

1 s+r/LZ(s) C 2 (20)

C S2 + (rlL + G/C)s + (I + rG)ILCwhere s is the complex frequen-cy, s=-+jw. Eq.can be written

Cd)

(a)

(A)

Ip

(20)(c)

L C

(b))tA

/

g(d)Fig. 7-Pole trajectories in the complex frequency plane.

1 1

Z(S) sG/C+(1+rG)/LC

s + r/L

(21)

G--f

Fig. 6 Resonant circuiit.

The positions of the poles, s,j anid sp2, are obtainied bymaking the denominator equal to zero

-s,G/C - (1 + rG)/LCSp (22)

sp + r/L

Eq. (22) is analogous to the equation used in the pre-

ceding section in obtaining the fixed points Zfl and Zf2of-the linear fractionial transformation (13). To obtainan exact analogy, the coefficien-ts in (22) have to obeythe condition ad bc 1. Eq. (22) then transforms into

-s,GVL/CI- (1 + rG)/V\LCsPVLC + rvC/L

two circles intersect [Fig. 7(a) ], two complex conijugatepoles are obtained, corresponding to the damiiped oscil-latory case. If the two circles are tangent [Fig. 7(b)]7two coalescing real poles are obtained, correspondinig to

the critically damped case. If, finally, the two circlesare external [Fig. 7(c) J, two real poles are obtained as

the crossover points of the real axis anid a circle that isorthogonal to the two given circles, corresponding tothe aperiodic case. In all cases on-ie zero of (20) is situatedat infinity, 501 , and another at S02=- r/L.

Thus, we find that by varying the conductance G, forexample, the poles follow the fixed circle with its cen-terat -r/L, coalesce, an-d then separate perpendicuilarlyfollowing the real axis. In various textbooks, we findthat the pole trajectories showni in Fig. 7(d) are ob-tained for an rLC circuit by varying the resistance r.

This figure is immediately explained by the graphicalmethod that has been described. With G= 0, one circle isfixed with its center at the origin of the s plalne, and theother moves along the negative real axis as r is varied.The trajectories are therefore the real axis anid the fixedcircle.

If we map the s plane stereographically on the unitRiemann sphere, the example chosen can be treated bythe polar theory of the precedinig sectioni. See Fig. 8.

The pole trajectories on the sphere are in-idicated byheavy lines.

Pole Trajectories in the Complex Frequency Plane and on

the Riemann SphereIn the Z plane, the positions of the fixed points were

obtained from the positions of the isometric circles.Analogous conditions yield two circles in the s planewith centers at -d/c=-rIL and a/ac=-GIC, bothhaving the radius 1/j c 1/VLLC-wo, the resonance

(radian) frequency. The positions of these circles im-Mediately specify the pole positions. See Fig. 7. If the

VI. SADDLEPOINT TRAJECTORIES

The Inverse Laplace Transform Treated by the Saddle-point Method

Saddlepoints of the second order play an importantrole in methods of approximate integration of the in-verse Laplace transform

1f(t) =-- F(s)es Ids; s a' + jco27rj

(24)

0c

\I 0

0 1

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PROCEEDINGS OF I'HE IRE

LK,

Fig. 8-Pole tiajectories on- the Rien-ianni unlit sphere.

w

x

s-plane

)== Br,Fig. 9-Example of poles, zeros, branch-cuts, and the Bromnwich-1

contour of integration in the complex frequency plane.

0f

Fig. 10-P=O aind Q-O lines throuLgh a saddlepointof seconod order.

wjo6t)I

0 2 64-6

(a) (b)Fig. 11-(a) The Bessel function of the first kind anid of order zero,

(b) Saddlepoint trajectories in the complex frequency plane

steepest descent. These paths are founld by- expantdigW(s, t) in a Taylor series in the viciniity of the saddle-poinlt s'.5 We put

IC(s, t) - I(Vs, 1) P + jQ (27)

In (24), t indicates time, anid y, the conitour of in-tegra-tioni in the s plane. This contour of integrationi may bea straight line parallel to the co axis ("Bromwich-1"conitour) (see Fig. 9) or the topologically equivalentconitour folded aroutnd the siingularities in the left halfof the s plane ("Bromwich -2" contour).

WAe can write (24) in the following form:

and determine the linie Q - 0 for xxwhich the real part ofW(s, t) -W(s8, I) is miaxinmumi niiegative. In F g. 10 thicregioni of P <0 arounid a saddlepoitnt of the second oidei-is shadowed. The line Q 0 in that region cotnstitutcsthe line of steepest descent

Let us study a simlTple example.5 rhe Bessel funt-ctionof the first kinid anid of order zero, J,,(t), plotted inFig. 11(a) has the Laplace trauisfoirm- 1/V/s2H+1, so that

f(t) = -- eIn F(s)+stds =

If we put

dw(s, t)=as

1J1\ew sttis. (25)ftl) JO()

I

2i.. ~,_/ _

Here

(26)

the roots of (26) correspond to points, called saddlepoints, in the s plane. These points, which are used itna theory of approximate integration-"the saddlepointmethod" (first inivenited by B. Rieinanni in 1863)-follow certaini trajectories in the s plane wheni the timiiet varies. The contour of integration is chaniged unitil itpasses through the saddlepoints along the paths of

aW(s7 I)as 2 + 1

5 M. V. Cerrillo, "On the evaluation of initegrals of the type1

f (TI 28T2,4- - T4) F(s@ ew(s 71,T2,2,r

I

I (28)

+ t 0 (29,)

"7nds

and the mechanism of formatioii of transient phenomena,' in "AnElemen-tary Introduction to the Theorv of the Saddlepoint Methodof Ititegration," Res. Lab. of Electronics, M.I.T. Cambridge, Mass.Tech. Rep. No. 55:2a; May 3, 19530

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Bolinder: Geometric-Analytic Theory of T'ransition in Electrical Engineering

yields

(ss/t) - 1ss =

Ss(30)

or

SS1}4 1 ± 2

=8-2t \2t/(31)

Eq. (30) is analogous to the equation used in SectionIV in obtaining the fixed points Zfl and Zf2 of the liiiearfractional transformation (13).

Saddlepoint Trajectories in the Complex Frequency Planeand on the Riemann Sphere

In the Z plane, the positionis of the fixed points wereobtained from the positions of the isometric circles.Analogous conditions yield, for our chosen saddlepointexample, two circles in the s plane with ceniters at-d/c=0 and a/c-l /t, both having the radius I/| cI 1.The positions of these circles imimediately specify thesaddlepoint positions. See Fig. 11 (b). For t 0; oniesaddlepoint is situated at the origin of the s plane andanother at infinity. For small t, when the two circlesare external, the saddlepoint positions are obtained asthe crossover points between the circle orthogonal tothe two circles and the real s axis. At t = 2, the two orig-inal circles are tangent and the two saddlepointscoalesce at s = (1, 0), thereby creating a saddlepoint ofthe third order.5 When t > 2 the two circles initersect andthe saddlepoints separate perpendicularly to the ao axisfollowing the unit circle in the s plane. Finally, whent-> co, the saddlepoints approach the points s = (0, ±j).

It is in-teresting to note that the velocities with whichthe saddlepoints travel along the a axis increase as thepoinlts approach the transition point s =(1, 0) and be-come infinite at that point. The velocities with whichthey separate are again infilnite. If, however, we map thecomplex frequenacy planie stereographically on theRienmannIi unit sphere and introduce the polar theory ofSections IV and V, we find that the polars pass with finitevelocities through the transition- point (x, y, z) = (1, 0, 0).

Fig. 12 -Saddlepoint trajectories on the Rieirnanu- nniiit sphere.

The fact that the saddlepoints miove approximately inthe tanigent plane at that poinit, accoun-its for the greatvelocities of the saddlepoints near to s-=(1, 0) in the splane. In Fig. 12, the saddlepoint trajectories oIni theRiemann sphere are indicated by heavy lines.

VII. CONCLUSION

In the geometric-analytic theory of transition pre-sented, we have used the complex plane and the Rie-mann sphere in the geometric treatment, and the theoryof quadratic equations in the analytic treatment. Byselecting simple examples, similar trajectories were ob-tained for two fixed poinits, poles, or saddlepoints in thecomplex planie anid on the sphere. All poinits were ob-tained by equating a quadratic to zero. Transitions be-tween two states were studied by using the coomplexplane and the Riemannl- sphere, the latter conisidered tobe imbedded in a three-dimenisionial space.The geomietric-an-alytic theory conistitutes an integral

part of a mnore genieral tranisition theory in physics. T[hemaini purpose of the theory presenited is to provide astimulus for continued research onX the ase of m-oderngeom-netry in electrical en-iginieering.

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