Communication Networks, Vol I - The Classical Theory Of Lumped Constant Networks [1, 10 ed.] 9333650652, 9789333650656

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COMMUNICATION NETWORKS

VOL.Il Tim CLASSICAL MORI 01 WES, C!!IU:SA.W I ILTIRS s3, JICel. lJJ !'cves. 6byl,cioth.

COMMUNICATION NETWORKS VOL. I THE CLASSICAL THEORY OF LUMPED CONSTANT NETWORKS

ERNST A. GUILLEMIN, PH.D. Profeuqr of E ~ Commii11-icaiiona, Ma,asachuutta l,W.itUU of Tuhnak,gy

JOHN WILEY & SONS, ! Ne. NEW YORK

LONDON

U!ij,1,/arorl

",.,,........ ,.,.. ..."'"' .

n;,r,,.,i.,°"f",-i,-,,.......

...-,.,..;.;..f/a.,.;/i,:lo-.

PREFACE

In writing the present volume, I have attempted to fill a need for &n introductory treatment of cla.si,ic&l network analysis which would be thorough enough to give the reader more than just an elementary knowledge of circuit theory. I believe that the moffl immediate use for a deeper understanding of network principles lies in the field of communications. Wherever illustrative ex~ amples are used, I have, therefore, placed the emphasis upon communic&tion networks. The existing texts on communication principles deal with the theory and prn,ctical significance of special types of networks commonly used in radio or telephone circuits. Progressive research, however, demands a more thorough grasp of the funda,. mental methods of attack on network problems in general. The sources where such information may be found are at present widely scattered throughout the literature on mathematics and mechanics, as well as in various adv&nood treatises on circuit theory. Besides being sc&ttered, these sources are rather difficult to read because they naturally presuppose a general understanding of the subject. The student who is beginning the study of network theory, therefore, finds it difficult to piece together a coherent and consistent picture out of such material as this. In the text I have not limited myself to the consideration of such information which bas a direct beari.Dg upon practical forms. The most abstract ideas have received 8.11 much space as the more applicable ones. This point of view is as essenti.&.1. in elementary courses as it is in advanood study; and i t applies to tech.meal as well as purely scientific curricula. Regarding the present&tion of the subject, I have deliberately chosen what seems offhand to be an illogical order. I have started with some general remarks regarding m&thematical methods and analogies; and instead of proceeding with a logica.1 presentation of fundamental principles, I have spent two chapten on the application of network principles to the single mesh circuit. This is done for the purpose of acqua~ting the reader with the general

iv

PRRFACE

subject. Without this acquaintance he would be unable to appreciate the res.sons for the proeedure in the general case. Generalization is meaningless unless one has some idea of what is being generalized. Perhaps I should he.ve spent more time on the discl.1$ion of special cases before starting with the methods of attack for the general case. I have a.ssumed, however, that the

reader would naturally supplement this "warming-up stage" with other material, either in the classroom or laboratory. The consideration of the general case I have attempted to put into as logicalaformaspossible. In the chapter on the evaluation of integration constant.a, I have given - besides the usual method- what I call the direct method. I do not believe that this method is generally known among engineers. The analytic form looka very formidable, but I hope that this will not disturb the reader too much.

The chapter on the vect-Or interpretation of the transient solution is given merely for the purpose of throwing additional light upon the natural behavior of the system. The vector method employed is well known in the study of linear transformations, but not generally known Ill network analysis. The subject of complex notation I have placed as promlllently in the foreground as possible. Its importance cannot be overemphasized in this kind of work because it obviates the necessity of opus.ting witb. the clumsy sines Md cosines and their explicit phsse angles. These matrers are particularly emphasized when Fourier se.ries enrer into the problem. Here the complex form is almost a necessity. The cha.pter on Heaviside's and the superposition formulae, I have D'.lB.de very brief because these more special Illil,tters are ably treated by other authors. I have included them here, however, to point out their relation to the classical theory and to give the student & working knowledge of them. The last chapWr on the Fourier representation of a periodic force function is included chiefly for the purpose of introducing the reader to the complex form and its uses. The presentation procoois from the usual sine and cosine form, instead of from an independent derivation, bees.use I assumed that this method of approach would be more easily understood by engineers. I have given a number of illustrations which show how much more con-

PREFACE

venient the oomplex form is in connection with network problems. The treatment of the microphone and condenser transmitter circuits is added in order to illustrate how the complex form is used for the solution of circuits in which the parameters are periodic functions of time. It may be well to say a few words here regarding the use of -called advanced. methods of attack in a book which is intended primarily as an introductory treatment. Methods are frequently designated as advanced merely because they are not in current use. To the student, the entire field is new; the advanred methods are no exception. If they afford a better understanding of the situation involved, then it is good pedagogy to introduce them into an elementary discussion. It is well for the teacher to bear in mind that methods which are very familiar to him are not necessa.ril.y the easiest for the student to grasp. In conclusion, I wish to state that I am chiefly indebted to Professor Arnold Sommerfeld, t hrough whose teaching I gained my first real insight into the philosophy of oscillatory systems, and particularly in the use of complex methods as compared to the trigonometric representation. I am also most grateful to the Electrical Engineering Department of the Massachusetts Insti• tute of Technology, through whose cooperative attitude this work has been made possible. In particular, I wish to express my thanks to Professor E. L. Bowles and Messrs. E. A. Johnson and F. M. Gager for suggestions and criticisms regarding the arrangement of material. Acknowledgment is also due to my wife, Sallie, who typed all of the oopy and otherwise helped prepare t he manuscript. E.A.G. C.ua1RIDG1t, MASI!.

Nomnber28,1930.

CONTENTS 1NTJl0D1JCT'.ION .•

CHAPTER I Tm: Ptm.owrHT OP Li""l!:AJI. EL!:CTRJc..u. NETWORKS . • .

Fundamental mechanical analogies. Mffhanical degreM or rreedom and theirnetwork ana.log. An&logouaenergyrel&tioil!I. :Kinds or motion. Mathematieal methods of attack. The physical side of t he mathematic.al method . lntegrntioncoru,t&nl.!! and their relation to initial conditioru,. The OCCIIITf!Doe of ooincident modes.

CHAPI"ER II THE Sum:u: MESH NJ:TWOII.K WITH CosSTA.,""l' ExCTTATIOS

30

The R, L cireu.it. The R, C cireu.it . The L, C circuit . Vector treatment for the L, C cir 0 is then: (195)

where the dt,-sign is, of course, dropped for the time being. The particular integral or steady-state response must be a harmonic function of angular frequency w. Hence we write for this integral: i,=1&"'

(196)

and substitute this assumption into (195), There results the condition equation:

,c.

(R+J-)1- E

(197)

84 SINGLE MESH CIRCUIT WITH HARMONIC EXCITATION

which is easily satisfied by making: (197a)

I~ ~ ·

R+je; Hence the steady-state current is given by;

i,"'(R..[R7i]· ,c.

(198)

If we a.gain define: (199)

and:

z-•-!,; - 1z1~

(200)

so that:

IZI - V R'+c~• )

(201)

and: 8-

-tan-{Rcw)

then our result (198) may be written: i, - J,,(tl .., 0.

l

(35-0)

(351)

161

EXAMPLES

The two conditions (350) and (351) are nothing but separate systems expressing the time-average voltage equilibrium conditions for the two frequencies :i and i 1-,t,,(U

so that (365) becomes: (J.1P>i{et.1+~,h)+ ep,;-;,.. PJ J. But the pair of modes :ve of the form: Pi"""

and:

-cc+ig

so that we have for t-he sum (365): IJ,(lllr"' {e.l~.1,

(k - I, 2, . .. n).

(377a)

Thus, all the amplitudes corresponding to the mode p 1 are expressed in tenns of a single arbitrary constant. Exaetly the same procedure can be followed for the remaining modes. For the rth mode we have:

Jiln = Cu(,l(J(rl

(k =- 1, 2, ... n).

(378)

The general expression for any amplitude in the set of equations (363) may be written: Ji(,l - Ca(,l(J(rl

l

I=

i arbitrary integer k-1,2, ... n . r - I, 2, . . 2n

(379)

The entire set is determined in terms of the 2n arbitrary constanta e, for example, could be considered as the integration constants. This is merely a matter of personal taste. From (379) we can obviously get: (380)

whe.re the amplitudes of normal functions for the sth mesh are siniled out. Substituting (380) back into (379), we have:

I

i - "bitnuy integ& ]

C (,) J,/~l•c:(,J•J,Vl

s""

"

k-1,2, .. . n r - 1,2, . . . 2n

"

i· (

38 l)

Thus all the J's are expressed in terms of those for the sth mesh, whlch may be considered the true integration oonstants. The manipula.tione from (379) to (381) are me.rely a. matter of formality. Aiiy 2n quantities could be singled out as integration

NEGATIVE POWER$ 11' DE.'TERMIXAXTAL EQUATION

195

constants. The important point is that there arc but 2n arbitrary quantities in the general transient solution for the n-mesb network no matter bow we express our solutions symbolically. To sum up, any net work with n degrees of freedom (independent meshes) will have 2n or less modes, the same number of integration constants (G's for example), and the same number of initial conditions in the form of mesh chn.rges and currents. 9. Coincident Modes. It is perfectly possible for the determinant.al equation to have equal roots, and hence for the network to possess coincident modes. The mathematical side of this question has been treated at the close of the first chapter. We shall show wh:i.t form this takes in connection with the above general network solution. Suppose that n1l the modes from s to 2 n inclusive are coincident, the rest being different from each other. Then the system of transient mesh-currcot solutions becomes instead of (363): ii

= J1(l)tf,' + .

+ J1(,-l)eJ,_,I + (Ji',•! + J ,r.,+I) • t + - · · + .ft(:.J · 1:..-•)etl + - + J ! a determinant&! equation which involves the same positive and negative powers of p, i.e., the determinanta.l equation will be a polynomial in p start,.. ing with P" and p&BSing continuously through p 0 top-. A simple illustration of this is given by the single R, L, C circuit which gives riseto:

D(p) "'Lp

+ R + time, thus obtaining:

LS+R~+b - CL

(386)

NEGATiVE POWERS IN DETERMINANTAL EQUATION'

197

It is now ready for integration by the usual method. Assuming; i ... Jell we get:

from which the determinants.I equation in the form (384) is evident. Another way of looking at this situation is to say that since (385) involves an integral, the solution for i must contain a constant of integration for this integral in addition to the constant which enters due to the integration of the differential tenn. Hence, the solution must contain two constants of integration, and therefore the system must possess two modes, which in turn requires the determinantal equation to be of the second degree The method just applied to the force-free equilibrium condition (385) for the simple circuit, can a.lso be carried out for the set of homogeneous equations representing the force-free equilibrium of the general network. The trouble is that it cannot be applied to the system (353) directly, except in the general case, for the following reason. Off-hand it would seem that one merely had to differentiate each equation in the set (3.53) until it became free from integrals, and then proceed in the usual fashion. This is not always correct. Suppose we consider as an illustration any t.wo-mesh network. For this the homogeneous system (353) becomes:

aui,+auii=O

a,1i, + ani, = O.

l

(388)

Before we can begin freeing the equilibrium condition from integrals, we must first obta.in a single equation in,·olving a. single unknown, i.e., we must eliminate between the pair of equations (388). Let us do this so as to retain. i,. Then, we multiply the first equation by ai::, the second by - au, and add, obtaining:

(a,_,a:, - au% - 0.

(389)

In the same way we could obtain: (auai:: - aul)l1 ,,. 0.

(390)

198

TRANSIENT SOLUTION FOR THE GEX~RAL Xh'TWORK

These are the separate homogeneous equations in i1 and i2 alone. Note incidentally that these are simply the determinant of the system (388) multiplied by the respective mesh~urrent for which the equation is desired. That is if we let;

(391) where the subscript d denotes that t.bis is a deOOrmi.aant whose elements are differential-integral operators, then {389) and (390) are simply: (389a) (390a) o.nd respectively. This is generally correct for the n-mesh system (353) as the reader can easily demonstro.te for himself by means of t he above method or by the theory of determinant solutions. Let us proceed to our point, however, with the two-mesh case as an example. Consider first the network as gh·en by Fig. 66. Here all three kinds of para.meters are present in each branch. Therefore:

a11-L11~+Ru+s11f di

a,, -

ai,

=

L,,i+

R11

(392)

+ S,!J dt

a,:=L.rai+R::+8=J dt 90

that (389), for example, becomes:

(Lul.,z - £,,1) ~ +

+ (R11Rn -

(Lullsi + ~11 R111 + SuLn

2 LuR11) ~

+ SuLn -

2 SuL.1) i

J

+(Ru8=+Yll- 2R11Su) f i1dt+(S118= - S12 1) f i1dtdtc 0. (393) The same equation is obtained for i,. It is clear that two differentiations are necessa.ry in order to free this equation from integrals, thus resulting in a differential equation of the fourth order, so that the determinant.al equation becomes one of the fourth degree. But now let us contrast this result with what we obtain for the

NEGATIVE POWERS IX DETERMIXAXTAL EQU.~TION

199

two-mesh network illustrated in Fig. 67. Here only one of the nwshcs contains capacitance independently, i.e., ~ and S11 a.re both zero. Introducing this fact into (393), we see that all the terms are retained except the double integral at the end. This would also vanish if 811 - Sa "' 0, or if Sn = Si: = 8=, i.e., if the only condenser wl're in the last or the middle branch respectively instead. of being in the first. Hence in such a case only one differentiation is necessary in order to clear the resulting equation of integrals although two differentiations would be n~essary in the

rr+rn F:m.66

F'IG.6i

given system (388) if the only condenser were in t he oommon branch. Strictly speaking, t-his method of elimination is the rigorous procedure for integrating the given set. We chose the direct method in this chapter because it is much simpler. The only difficulty with the direct method arises when the determinants.I equation comes out with negative powers of p, or appears to start with too low a positive power. This, however, can easily be rectified by simply multiplying it through afterward by the proper power of p. The question which we have to settle now is what thia proper pow-eris. From what we have just been saying, this is quite simple to answer. Namely, the determinantal equation must be multiplied through by that power of p which equals the number of meshes which contain capacitance independently, i.e., in which charge may be independently specified. This is not always equal to that power which will just eliminate all the negative powers of p, otherwise we should not have spent all this time and spa.oe on such a simple matter. To illustrate this point consider the two-mesh network of Fig. 68. For this the coefficients ea become:

200

TltAK81ENT 80Ll!TION FOR THE GEN"ERAL NETWORK tu -

Lip+ Sp-I)

(394)

Sp- 1

ti:= -

en-L,.p+,5p-• oothat: D(p)

= t11'21

ti, 1 -

-

L1L,p 1

±~,

+ S(L. + Li).

(395)

TW!I polynomial is already free from negative powers of p, but the correct determinants! equation for this network requires the introduction of another power of p beeause we have one mesh in which

~ - - - . --

(Z

L,

1

-----,

S

2

1•

L,

chargema.ybemdependently,,..,died Theeo,rect equation 1s L,L,p'

'

+ S(Li. + 4)p ~ 0. (396)

Ftu. 68

It must be a cubic because we ha.ve three initial conditions to specify altogether, namely initial current and charge in mesh f l, and initial current in mesh f 2. The system has three modes, which are the roots of (396). These are:

''p, -- : J..;sci. + L.> -,;;r;-

P•- _,✓S('l.t

I

(397)

L,).

Thus we see tha.t the system of Fig. 68 possesses one pai:r of conjugate imaginary modes, and one real mode whose ·value jg rero. This zero mode IllJI.Y seem trivial, but fundamentally it is aa important as the other two. The transient solutions take the form:

i, "" J 1 : J2(1l ..,

Cu :

cull)

(405)

and: J1(f) :

Ji(fi ,,. Cu(l) : C,ill

=-

en(II : c,u(II_

(406)

Substituting the values from (400) into (405) and (406), we get: (407)

and: Ltp,+Rr Jp,= - ~ J 11l/

(408)

For the special case (403) these become: J,(•) . llf. . !Li . . . . tli Jlil- VRi- Vr;=ni

(407a)

and:

where n1 and flt are the turns in the two coils respeetively. Hence, we see that for the special case (403), the amplitudes for the mode p1 are roughly inversely proportional to the ratio of turns on the two coils. For the mode Pl they are roughly inversely proportional to the negative ratio of turns. Summarizing, we see that the natural behavior of the network of Fig. 69 is composed of a superposition of two exponentially decaying functions in both meshes. When the coupling is tight, the

204

TRANSIENT SOLtrrION FOR THE GENERAL NETWORK

modes are widely different. evaluates t-0:

For the limit M---. ~ . (402)

and :

Pi"" -co.

Wb.en the coupling is loose, the modes approach each other. For the limit M-,. 0, (402) gives rise to: p,= - ~

and:

... -t

The same general tendencies are true for the special condition (403), of course. For this case we can also roughly determine the relative magnitudes of the normal function amplitudes in the two meshes, and find that these are approximately inversely proportional to the ratio of turns, this ratio being positive for pi and negative for the mode p,;. The na.tural function corresponding to -pi will in general not be as important as that for Pi, because the former will decay so much faster, especially when the leakage flux iB small as i t is in most transformers. All this, of course, does not yet give us a complete picture of the net behavior of this network, because the stea.dy state is still missing, and the magnitudes of the transient current amplitudes h&ve not yet been evaluat.ed. T his la.tier operation will be taken up in the next ch&pter. We wish t-o point out, however, how much information can be obtained relative to the natural behavior of a network without carrying the analysis through to a finished result. The student should further notice how much more useful a.n a.nalytic investigation, such as the above, is than a specific numerical. one. It would have been much ea.sier for us to demonstrate a numerical case with 1).

(495)

(495o)

23.5

ILLUSTR.A Tiox.-:

Thr~ cqu:itions may be use't+2-Hl7 t'-3.5-10llt'+6·1016r-- 1.0'25•1()10t'+ ...,\

2!

3!

41

5!

··.) (501)

23H

THE E\".\LL\TH 1;,; OF J:-/TEGRATION CONSTAN'I\S

It i~ quitr cvi,Jent that they converge very poorly. The above fi\"f' tl'rms sufficl' to calculate the behavior only up t.o about t 1.5. 10-• &•(•onds. This is still considerably less than the largest time constant in,·oh·ed. Hence, the Taylor expansion method of solution is very unsatisfactory in this case. Let us turn our attention to anot her network for which the Taylor expansion may ,•ery well be used to represent the form of thr solution during the initial stages. This is illustrated in Fig. 76. It is the same as the network given in Fig. 74 ucept for the introduction of the elnstancc in the first mesh. Ut us keep the

r

·□.□·· R,

FiG.76

same values for the corresponding inductances and resistances as in the preceding problem, and introduce the value S1 = 1()3 darafs for the elast nnoo. \Ve will proceed to obtain the normal function solution first. The first mesh has a weight of two, and the second has a weight of one. Hence, the network possesses three modes, and the determlnantal equation ~ill be a. cubic. We have: C11 Cn

= .001 p + 5 + l(lllp-1) = .001 p "'" Ct!

(502)

r,::=.002p+10. Hence: D(p)

= (.001 p + 5 + lO'p- )(.000p + IO) 1

- (.001 p)!. (503)

The first mesh contains independent capacitance, ao that D(p) must be multiplied through by p in forming the determinantal equation. We have: '(r

+ 2 • l()'p2 + 2 • lQilp + JOU

>=

0.

(504)

We expect from the relatively mall resistances that the network is fully oscillatory, i.e., that it posae;ses one real mode and one

237

ILLUSTR,\TIONS

pair of conjugate complex modes.

real root of (504} terms, thus:

The first approximation to the may be found by neglecting all but thi:? fa.st two 2• l0 11 p+ 101.l "' Q.

This gives;

(-505}

p - -5000.

If we divide this root into (504), we find that the remainder is negligible. Hence (505) is already dose enough to the correct value. The remaining quadratic is:

-pl+ 1.5 · 10-p + 2 · 10 = 0. 11

(506)

Its roots are: p =- -7500±j447000. Hence the modes of our given network may be summarized as follows: Pi=- -5000

'/>! - -7500

+ j447000

I

(507)

Pi "" - i500 - j 447000.

Since there are no steady-state currents in either of the meshes, the complete solutions for current after closing the switch (t • 0) become;

: : j:::::=:: ! ~j:::~::: t ~;:::=;::~~:::}

(508)

or in terms of the true integration constants we have:

z: g;:::;g;:;~:: !~g:;:g::;:=:tg:::~~==~:=;::} I~."' ,."' I

(509}

where Cu 1•) and C1/•) are the minors of:

for the mode p ..,

(510)

~

We have three integration oonsta.nta to evaluate, namely (l(I), Gfll, and (JJ>. For this purpose we h.we three initial conditions to specify, namely i10, iio, and q10- We will proceed with the dirt>ct method of eva.luation first . The initia.J. current conditions may

238

THE EVALUATION OF IXTEGR.ATION CONSTANTS

be substituted into the equations (509) directly. The initial charge in mesh #1 cannot be substituted until we have fonued the charge equation for mesh #l. This is done by integrating the corresponding current equation once with respect to time. This

(SU)

Note the introduction of the term 10-•E in the aOOve equation. This is the steady-state charge in mesh #1. The steady-state current is .zero, but the charge is equal to the capacity times the SU'ady '\--Oltage E. If the impressed force had been harmonic instead of constant, then the current equation would have contained a steady-state term, and the corresJX)nding steady-state charge would have been the integral of the current term as given by the first equation (441). It is only in the case of a constantapplied voltage that the steady-state charge must be inserted by inspection, as was done aOOve. If there is any doubt about what the form of this term is, the steady-state charge should be determined for a. harmonic-impresood force having the same magnitude, and the limit: Frequency-> 0 carried out. This is hardly ever necessary however, since the steady-state cha.rge is always equal to the contour capacity times the voltage impressed on the oontourof the mesh in question. If there is no steady-state impressed voltage in that mesh, then the at.early-state mesh charge is zero. We will assume initial rest conditions for the evaluation of the integration constants. Then setting t - 0 in equations (509) and (511), we have:

C11< 1>G - N, sin 35° lt( 1l

because Ji(Tl and are the projections of the normal amplitude upon the axes 1 and 2 respectively. But in the last chapW we found that: J,< 11 == O.lE. Hence it follows that:

N1

=~- 0.122B

(562)

which iB the amplitude of the mode pi in its normal coordinate.. Its position is in the second quadrant. Its projection upon axis 2iB: Ji(U = 0.122 E sin 35° = 0.0707 E

264

VECTOR IXTERPllETATlOX OJ,' TRA.:.'\~IEXT SOLUTION

which checks with what was found pre\i.ously. From this it is clear that the amplitude eormponding to the normal mode Pi in its own cOOrdinate i.s i\'i, and that the projections of this amplitude upon the axC'S I and 2 give the transient mcsh-eurrcnt amplitudes in the meshffi I a.nd 2 rE'Speetivcly corresponding to this mode. Considering the mode ~ for this problem, we have t.he vector group whose components are specified by (ii60). This group is composed of ci:I) with eomponents: cu are not the normal amplitudes directly, but are proportional to them, as mentioned above. Also the minors c..r..l are not the cosines of the angles between the directions of the normal amplitudes and the coordinate k directly, but are proportional to them. However, since the system of reference axes is orthogonal, we have:

l)

Hence:

266

VEC.."TOR I:\7"ERPR.l!.'TATIOX OF TRANSIENT OOLUTION

Similarly:

In general the normal amplitude will be llc,tennincd by: N.

= G'•l

\.1 (Cnc,Jf

+ ((.'i,.(•Jf + · · + (C,~ 1•1f.

(567)

If we denote the angle bet1'-ecn the direction of the normal amplitude N, and the axis k by Pion then it follows that: (568) From these considerations we then have: (569)

From these equations N, always has the same algebraic sign as and cos pi,, has the sign of C;1,(•>. If this is observed, then the algebraic signs of the J's will come out correct. These formulae {567), (568), and (569) are given merely to Q(•J,

roWld out the theory regarding normal coordinates and normal amplitudes. In the solution of a problem it is not necessary to det.ennine either the normal amplitudes (507) or the directioO cosines (568), except for the sake of interest. It might be well to emphasize again, that when a pair of conjugate modes are considered, that pair together defines one nonnal coordinate. Since this pair of modes gives rise to a pair of conjugate e,cponent ial tenns, the actual amplitude of this normal oscille.tion will be 2 IN,I, where II i.e the number of one of the modes. For such complex modes, N, by (567) will be complex, and ao will cos p1o, by (568); i.e., both of these quantities will have a magnitude and an angle. The sum of these angles is the angle of the corresponding J,c,>, and determines the time phase of the component nonna1 oscillation in mesh lk. This time phase is different in the different meshes a.s pointed out previously. If it were the aa.m.e in all the meshes, then cos pi,, would be real and not complex. A complex oosine is, of course, hard to visualize. It means that when the nonnal amplitude N, is projected u pon va.rious reference OOOrclinat.es, not only the magnitude of the oscillation, hut also its ti.me phase is affected. 7. Realization of Normal Meshes. From the above it appears as t hough the,e nonnal colirdin.'ltNI (':tn only be realized physically

REALIZATION OF NORMAL MESHES

267

in mechanical syatems. In an electrical network they are in general mere fictitious meshes which are introduced into the picture because of the mathematical analogy between the mechanical and the electrica.l systems. The situation need not always be entirely fictitious, however. It is possible to so construct a network as to make one or more of the actual meshes become normal coordinates or normal meshes. We then say that these meshes have been normalized. Since a norma.l coordinate isolates its own mode, it follows that these normalized meshes will do the same. We shall now discuss how this may be done. From the mechawcal side, the possibility of coincidence between a normal OOOrdinate and a reference coordinate is readily appreciated. We said above, that these normal coordinates or diredions actually exist in mechanical systems. Suppose we have located one of these. Then we can orient our Cartesian reference axes so that one of them, say axis I, coincides with the normal direction. Since the reference axes are all normal to ea.ch other, it then follows that this particular normal COOrdinat-e will be at right angles to all the axes except axis 1 w:ith which it coincides. Hence, this normal oscillation ean have no projections upon the other reference coordinates, and is therefore completely confined to axisl. If we should be able t.o make an actual mesh in an ele\·c cquiyaJent circuit.. From tbis discussion it iB apparent that this particular equi\'alent cireuit is generally dissymmetrical, i.e., that In general:

Z,1 ~Zs.

If we dcm:mG(,le'-'

it - .;' Cur.->G(•iet.J

(873)

'

i1"",!i C11+,).

. (p,-p.) . F,i_p,)

(926) which is another form in which Heaviside's formula is found. Note tb.&t the function Fis the one which bears the indices relating to the location of the force and the mesh in which the current is calculated. The formula (926) ma.y be altered still more by iiltroducing the determinantal notation. In fact there are an endless number of additional modificatiOilB. The form (926) is, however, about as oonvenient a one 11.8 may be obtained. It no longer contains any derivatives, but expresses the solution directly in terms of the modes and the function F. As an illustration of the form (926), let us consider the pair of tuned coupled circuits of Fig. 114, but for zero resistance, and calculate the current in the second mesh. Then we have:

HEAVISIDE'S EXPANSION FORMULA

zt _

367

p(cuen - c,21"-11) -ci:

= p(cu + C11) (cu

- C,i)

- -~p [ (L + M)p+~] [ (L - M)p +zk]

- _V,,;,."'(p'+C(L~M))(p'+C(L~>f)) - - IJ ;;rl'AP (p

+ i"'1) (p -

j,..,,_) (p

+ ;wt) (p -

jw,,). (927)

Here it is obvious that:

P(p) - -V,,;,,,"'-

(928)

In substituting into {926) we note that the terms come in conjugate pairs. Hence, we shall consider only: and: and then take twice the rea1 portion of the result. i - 2 ot. [ 2j,..,,_(:f.:1::,) fi'::_M1)

+ 2jw,,{:,~~:)

= (IJ-~(w,,1-1JJ11) ("'1 sin 1JJJ.t- -~(1JJJ.siil1JJJ.l-1JJtsill/JJtf)

We

have:

fv~MJ

w,, sin IJJtf)

(929)

which checks with the result (854). With the work thll5 highly organized, the formula does seem to be shorter than the ordinary method. However, with the same degree of orga.nizs.tion, the usual method for obtai.lling the solution is just a.s brief. There may be some difference of opinion on this point, and the reader may, therefore, draw his own conclusions after he bas beoome sufficiently familiar with the various methods. The form (926), incidentally, makes it evident that the formula cannot be applied when coincident roots appear, because this would cause two or more faclors in the denominator to be a.like. Hence, the denominator would vanish for one of these root.s, ao

368

50:\IE J;\IPQR'l'Al\T x1.-rWOHK J•OHMULAE

th:l.t the entire expression would Lccome infinite. Note also that a zero root of: Z(p) - 0 lead.s to an infinite result because this means two zero roots for: Z*(p) - 0.

Thus, whcne\·er the dctenninantal equation for the .network contains a zero root, the formula does not apply. This would occur if we considered an inductance alone. Then: Z(p) - Lp Z'(p) - Lp' so that: Z*(p)

= Lp' "" 0

contains two coincident roots at p = 0. However, practically, we never have inductance without also having some resistance, so that this exception is not serious. To summarize the limitations of Heaviside's formula, we may state that it is not applicable for the cases where the initial conditions arc arbitrary, or where is a. constant, or where =O bas coincident roots. Otherwise it may be applied in any of the forms given above. Of these (914) is the briefest analytically, and (926) is the most convenient for numerical application. 2. The Superposition Formula- 1 This formula is based upon the linearity of the network. It proceeds from the fact that the net behavior of the system at any instant is due to the linear superposition of all responses which have occurred up to that time, counting from some arbitrary starting point. Thus, if we should subject a network to an initial shock at t = 0, and then follow by subsequent applications of forces at specified times, the net behavior at any time would be a sum of the various results which have occurred up to that time, with due allowance for the time at which each individual response started. This idea. is used in order to determine the behavior of a network when a force function of arbitrary form is impressed. So far we have considered only constant and harmonic forces. The object of the superposition formula. is to give us a means for handling any kind of an impressed force.

z•

lForathoroughdiscussioneeeV.Bush,Lc.,pp.56e.1.

z•

THE SUPERPOSITION FORMULA

369

Before giving the derivation of the formula proper, we must introduce a new kind of function which forms the starting point for this work. In the preceding paragraphs we have shown how the response of a network may be determined when a constant force is suddenly applied at some point. Suppose we consider the magnitude of this force to be unity, that it is applied in mesh '8, and that the response is desired in mesh f k. The function of time which represents this response. is denoted by; Au(t) (930) and is called by Carson1 the indicial admittance of mesh Is with respect to mesh f k. If the network is initially at rest, i.e., if the currents and charges are initially zero, then, due to t.he reciprocity theorem, the meshes 11 and k may be interchanged. The indicisJ

'"' ---,-f(o)

Fto.128

admittance is thus seen to be the response of the network resulting from the sudden application of a unit force. It is the response per unit of applied force, with proper regard to the points of application and obaervation, as indicated by the indices. When this function is known, the response for any form of the applied force may be determined as follows. Suppose we consider an applied force of the form illustrated in Fig. 128. Here I - Ois the switching instant. At this instant the force bas the valuef(O), i.e., it jumps from zero to f(O) instantly. • A.l.E,E. Trana. Vol. as, p. 345, 1919.

370

SOME IMPORTANT N~'"I'WORK FOR.~ULAE

After this initial jump, it follows the continuous curve as indicated. It is a common concept in calculus that this continuous variation may be considered as made up of a succession of differential increments. In exaggerated form, this gives rise to the step-.functioo illustrated in Fig. 128a. This function, made up of differential steps df, at differential time intervals dt, exactly replaces the original function in Fig. 128. /(f)

FIG. 128,i

Now consider, for example, the differential force. increment which take$ place at a time I = >., and is denoted by: (dfl,-~-

(931)

When this is suddenly impressed upon the network, the corresponding response is given by:

.A(t - >.) - (d/)1-~-

(932)

The indices on .A are left off for convenience. The reason that the argument of A is t - >. and not just t is because this response doee not come into existence until the time t - >.. That is, the expreeeion {932) is valid only fort ii: >., and must reduce to A(O) for I - >.. Sinoe >. is an arbitrary point on the time scale, &nywhere between .zero and eome specified. time t1 the expression (932) represents a typical differential respon.ae which may be initiated at any time between zero and t by the oontinuoU8ly varying force /(!). Therefore, if we wieh to know what the net response at the

THE SUPERPOSITION FORMULA

371

time t is, then we have merely to &dd up all the differential responses from zero tot, and add to this the initial response at the instsnt t - 0. This latter response is obviously given by: f(O) · A (t). (933)

The sum of a.II the differential responses from zero to t becomes an integral of the exprefflion (932) withrespeet to the variable A, which is the time at which ea.ch differential response is initiated. In order to form this integral, we note that;

m. . ·,, l

(df)•• • =/'(A) · dA

(934)

where the prime indicates the differentiation with respect to time.

Thua, we have: i,i.(t) - f (O) · Aa(t)

+

J:

1

A,.(t - >.)/'(A)d>..

(93S)

Note that the integration is with respect to the variable time >. at which each differential response starts. The result is a function of t, as it should be. This is the superposition formula. It expresses the response for any form of the applied force f(t). It is not an explicit solution until the integral is evaluated. When the force function/(t) is given in analytic form, the integral maybe evaluated provided it is not too complicated. Otherwise the integration must either be performed graphically, i.e., in steps, or by means of a suitable form of integrating ma.chine. When /(t) is not given analytica.lly, then the graphical or ma.chine integra,. tion pl'Ol'le89 is the only one possible. The rea.l importance of the result (935) is th.at it expresses the response for any arbitrary applied force in terms of the response per unit of a constant applied force. Hence, the solution for any type of voltage is expressible in terms of the d.c. solution. This fact .is very useful in network analysis. The superposition formula (935), like Heaviside's formula, may be transformed into a number of equivalent forms, which may be better adapted to meet the requirement.s of particuls.r problems. We shall now give the derivation of the more important of these. Suppose we introduce in the integral of (935) the change of variable indicated by:

372

SO~IE IMPORTANT NETWORK FORMULAE

x- o.

(936)

d>. • -JO

(937)

tThen:

where tis treated a.s a constant, which it is so far as the integration in (935) is concerned. The function under the integral sign then

becomes: - A(!)j'(t - !)di

so that: J:1A (t - >.)f(>.)d). ,... -

r

(938)

A(6)f(t - 6)d0

... J:1A(O)f'(t - O)dO and if we now write ). for Oagain, we have instead of (935):

f,., :: f(O)Au(t ) + J:1j'(t -

>.)Au,(X)d X

(939)

which is one of the alternative forms. Two additional forms are obtained from (935) and (939) respectively by carrying out an integration by parts. The ml[. known formula for this is:

J u.av= uv - J vdu..

(940)

For (935), for example, we let:

a::

(941)

pg)~>.~> }

Then since the differentiation and integration is with respect to ). we have: (042) Hence, the integral in (935) becomes:

f. A(t 1

>.)f(>.) d).

= [ A (t

- >.)/(X)J:

= A (O)f(t)

+ J:i(>.)A'(t-X)d't..

- A (l)/(0)

+ f.!(>)A'(t - X)d>. (043)

THE SUPERPOSITION FORi\!ULA

373

Substituting this into (935), we have:

iu

= A,.(O)f(t)

+

j )C>.)Au'(t - A) dA

(944)

whlch is the desired alternative. In exactly the same manner we obtain from (939): i,t

= A,i(O)f(t) +

.fo A '(A)f(t 1

A) dA.

(945)

Still other forms are obtained when we make use of the mathematical theory regarding the differentiation of functions under an integral sign. Briefly, the situation regarding this process is as folloW5. Suppose we consider the following function defined byan integral:

(946) where/ is a function of the two variables t and A, and the limit.I! of integration are functions of t. It will be seen that any one of the integrals involved in the above formulae have thls general form. For this situation it is shown that:

~ - 1:• ¥id>- +f(t,h,.) -~+J(t,h,) -~ -

(947)

This result is very plausible because the differentiation with respect to t must affect the values at the limits, since these are functions of time. That is, ,J, must be looked upon as being a function of the variable.s t, h,, and h,., thus:

,J,

whence:

= ~(t,h1,h,.)

~-::+;t~'+:~-

The form (947) lll thus obtained. With this in mind, consider:

~1

1

A(t-A)/(A) d>.

Here we put:

=

J:

1

A '(t -A)/(A)dA +A(O)f(t).

(948)

374

SO!.)A{>.) d>. • J )'(t->.)A(>.) d>. + f(O)A(t).

(949)

Comparing (948) and (949) with (944) and {939) respectively, nseethat:

and: ip

=

iJ)(t -

>.)A,1(>.)d>.

(951)

which a.re two new forms for the superposition formula. The chief difference between the last two and the preceding fonns is the fact that the functions under the integral sign are not differentiated, the differentiation being done after the integration has been carried out. This may be advantageous since in some cases the process of differentiating makes a function more complicated, 80 that the subsequent integration becomes more difficult. It may, of course, be that the differentiation simplifies the function, in which case one of the other four forms would be preferable. In general no statement can be made as to the preference of any one of the above fonns over a.ny other. This must be decided for each specific case. Such decisions require experience and fa,. miliarity with the various forms. Sometimes the best form is found only after a number of trials. We shall now consider several examples in order to illustrate aome of the features discumied above. It is interesting t.o show how, by means of the superposition formula, the solution for a harmonic force may be obtained when that for a constant force is given. Take, for example, a simple R, L circuit. For this we know that:

,.

A(>.)

,. 1-/T·

(952)

The force function is given by: J(I)

• "-IE"'I.

(953)

Here we note that differentiating A(>.) will actually simplify the

THE SUPERPOSITION FORMULA

multing function under the integral. form (945). We have: A{O) - 0

A'(>.) d>. 80

375

Hence, we shall use the

.

q

d>.

,.

th.at: i

= J:' ot.[E~..(Mljg.d).

= dt.{!&"'[ e-(~)>-ax}

-ot•/R-:;: ,-(~), t) - ...

(958)

This integral is easily evaluat.ed. We get after taking the real portion:

i=l

R~r~LC(J1[e~-~CC6gt+'";Pmngt)l

(959)

Usually: a 80

-P« I

'

that: i;::;: 1 -

RGJC! LC{P(~ -

e-"' 008g().

(959a)

This result ia particula.rly interesting when the decrement of the circuit equals the decrement of the foroe function, i.e., when: a - ~-

(960)

i - 4 £~R~ ~ (l-ooegt).

(961)

Then (9S9a) becomes:

THE SUPERPOSITION FORMULA

377

Fig. 130 illustrates this case. The current oscillates and decays, but always remains positive. If a < {J, then the oscillation will go below the a.xis for a. short time during each period.

Another interesting type of force function is that illu.strated in It is given analytically by the expression:

Fig. 131.

J(t) -

E.-,..

(962)

F:IG.131

We sha.11 use the form (935) a.gain. Here we have:

f~I)~.°-

E...- (1 - PX) d>. }

("3)

Consider this foroe impressed upon the R, L, C circuit for which ·the indicial admittance is given by (957). Then we get:

so:.m

378

i -

-

t:.JPORHNT NETWORK FORMULAE

m..

J:'e(~;;,.) d>.

.f (1-,6t\!~~:--:.

~ -Ee~~+nll

e(,,,-#-J,)).{l - ff"-)

d>..

(964)

After evaluating the integral, we have:

i_

(R

--1/4 _[{a-jg)

jg~;a+jg)e,sin (,.,,+;-•.).

(966a)

(966b)

Although these forms require less space, they are in every respect as cumbersome to handle as (966). They require the formulae (967) plus (968) for the determination of the amplitudes and phases. There is only one way out of this dilemma, and that is through the use of the exponential function. We know, for example, that: (970)

and: (970a)

If we substitute these expressions into (966) and arrange terms, we have: J(t) =-

aa+J~r•~iP,el""'+a. ~jp.e--i-'}.

a.

~ iP, -

(971)

,.

Now we note by (967) that:

;;

J:~co~ at

c912J

,.

and:

a,1jp, • t i--;;f(t)ei-'dt

(972a)

where we made use of the relation:

It is important to note that (972) and (972a) are conjugat.es, and that the only difference between the two integra.la is the difference in the algebraic sign of the index , . T his suggests thAt we introduce the new notation:

a,,..a,~jp. ("73)

COMPLEX FOURIER SERIES

Then obviously:

383

,. a, - ,;;J: •i(t)e--i-,ldt.

(9i4)

Note that the single formula (974) now replaces the two formulae (972) and (972a), since the latter is obtained from (974) by merely changing the sign of "· Also, note that for v "" 0, (974) betomes :

,.

ao=i;J: "1wa1 which is identical with the first formula (967). Hence, (974)

aloo gives the constant component of the series, i.e. : (975)

In terms of these results we ma.y now write the expres.5ion (971):

f(t)~,'i:.,,..,

(976)

in which the two terms int-he sum of (971), as well as the constant term ao, are included by virtue of the fact that the index v now takes on all negative integer values, as well as all positive integer values, inclusive of the value zero. Thus, the result (976) identically replaces (966), when the coefficients are determined by (974). This is called the complex Fourier series. Note how compact it is a.s compared to the clumsy form (966). Note also that only the one formula. (974) is necessa.ry for the determination of the coefficients as compared to the three formulae (967). The complex form haB so many advantages over the s.ine and cosine form that it is hard to appreciate them all at once. Processes of differentiation and int-egration with the form (976) are as simple as they are for a single exponential function. Another important point from the standpoint of network theory is the fact that no harmonic phase angles appear explicitly in the complex form; yet they are contained in it, namely in the complex character of the coefficients. Due to the fact that they are thus hidden from view, they CllllllOt bother us when they are not Wll.Ilted, as, for instance, during an integration process. After the integration is complete, the phases of the various harmonic components are

384

THE TREAT:-.IEXT OF PERIODIC FORCE FUNCTIONS

immediately appa~nt by inspection of the resulting coefficients. This \1ill be apprcdatcd more fullr later on. Kote also that :i.lthough the separate terms in (976) are complex, the sum is not complex, but :l real function. This is due to the fact that the terms come in conjugate pairs, each pair being given by the positive :i.nd negative mlues of the index ~ for a certain integer. Such a pair of conjugate terms togr.iier represent one harmonic i:omponenl. Thus, consider, for example, the pair of conjugate terms:

(977) Ea.ch term is a vector which rotates \1iU1 an angular velocity equal tow times the order of the harmonic. This pair of rotating vectors is illustrated in Fig. 134. Du, to the fact that they are conju-

ri:\:t:a~::~: :~:!

the projection of the vector "/. upon the real axis in Fig. 133. The magnitude of the vectors in Fig. 134 is hn..lf of the magnitude of 'Y.. The angle which the pair of vectors make with Fm. 134 the horizontal is ,fl,, and is the same in msgnitude as in the Fig. 133 or as given by (968) . Thus, we can see by inspection of these two figures that t he equations (973) are correct. The essence of the complex form (976) is, that ea.ch harmonic component is represented by a pair of oppositely rotating vectors. This is an old method for representing a harmonic function. The same device is ma.de use of in the theory of the single phase induction motor where the harmonically pulsating field is replaced by two oppositely rotating fields. The phase angle of a harmonic component is given by the angle of the corresponding complex coefficient as illustrated in Fig. 134. In particular when the phase angle of a, is minus ninety degrees, then the conjugate pair of vectors represents a sine fllllction. When the angle of a,. jg U!ro, then the pair defines a cosine function. The actual amplitude of

COMPLEX FOURIER SERIES

385

any harmonic component is :i.lw.'.lys t11ice the magnitude of the corresponding a,. These matters will become clearer by means of some illustrative examples. Consider the square function gi\-en in Fig. 13.5. It is assumed to continue in the same fashion in both directions. The origin for the time is arbitrarily chosen as indicated. The positive /(/)

Fto. 135

and negative amplitudes are equ.'.11 and denoted by E. By (974) we then have:

.

,.

a.-~[[,-,~ dt - . {;-,-;-, " ]

(978)

which iB equivalent to: 0.,

-f;.(1- ~1'r)J:;;j~M dt.

(978a)

Integrating and substituting limit!!, we have:

a.= 2

!"

( l - e-1'")'.

(979)

The factor (I - e-i••")! is rather int.eresting. For " equal to zero or any even integer, the factor vanishes, which indicates that the function of Fig. 135 contains no constant component and no

386

THE TREAT:\!Er'>T OF PERIODIC FORCE FUNCTIONS

e\·en hannonics. For any odd integer value of v, the factor is always equal to four. Henc{', for the odd harmonic components we get: 2E a,.--,--• {979a)

,.,

The oomplete Fourier series is, therefore:

!(3 Inleitnd, lO Integrntion coru