CHAPTER XXXII. QUARTER-PHASE SYSTEM. 294. In a three-wire quarter-phase system, or quarter- phase system with common return wire of both phases, let the two outside terminals and wires be denoted by 1 and 2> the middle wire or common return by 0. It is then : EI = E = E.M.F. between 0 and 1 in the generator. Ez=jE = E.M.F. between 0 and 2 in the generator. Let: ./i and 72 = currents in 1 and in 2, 70 = current in 0, Z-L and Zz = impedances of lines 1 and 2, Z0 = impedance of line 0. Yl and Y2 = admittances of circuits 0 to 1, and 0 to 2, // and //= currents in circuits 0 to 1, and 0 to 2, Eia.-ndE2'= potential differences at circuit 0 to 1, and 0 to 2. it is then, 7, -f 78 + 70 = 0 ) «v or, I0 =-(/; + 72) j that is, 70 is common return of 7: and 72. Further, we have, El =JE - 72 Z0 + 0Z0 =jE - 72 (Z2 + Z0) - A and A = K, E{ (3) 484 AL TERNA TING-CURRENT PHENOMENA. Substituting (3) in (2) ; and expanding : *•/ - *• _ l + F2Z2 + F2Z0(l-y) _ '. (4) • 2 • /1_l_VX_l_V7'W'l_l_V7_l_V5^ V V '7 2 \*- i * 1^0 "T" *\**\)\~ i * i **• T * i^ij — *i *J ^o Hence, the two E.M.Fs. at the end of the line are un- equal in magnitude, and not in quadrature any more. 295. SPECIAL CASES : A. Balanced System. Z0 = Z / V2 ; F, = F2 = F Substituting these values in (4), gives : i + 1 + V2-yrz ' 1 + V2 (1 + V2) FZ + (1 + V2) F2Z; _ E 1 + (1.707 - .707/) FZ • 1 + 3.414 FZ + 2.414 F2Z2 (5) V2 ~J • 1 + V2 (1 + V2) FZ + (1 + V2) F2Z2 _ . ^ 1 + (1.707 + .707.;) FZ ' 1 + 3.414 FZ + 2.414 F2Z2 Hence, the balanced quarter-phase system with common return is unbalanced with regard to voltage and phase rela- tion, or in other words, even if in a quarter-phase system with common return both branches or phases are loaded equally, with a load of the same phase displacement, nevertheless the system becomes unbalanced, and the two E.M.Fs. at the end of the line are neither equal in magnitude, nor in quadrature with each other. QUARTER-PHASE SYSTEM. B. One branch loaded, one unloaded. 485 a.) b.) Substituting these values in (4), gives : i + V2 — y b.} l + FZ a.) £1 = E V2 1 + V2 V2 j 2.414 + 1.414 YZ *+'*f = /^l4-1.707FZ 1+^1^ • 1 + 1.707 FZ -t I ^/O 1 + F2 V2 , FZ 1 + + V2 2.414 + 1.414 FZ (6) 486 AL TERNA TING-CURRENT PHENOMENA. These two E.M.Fs. are unequal, and not in quadrature with each other. But the values in case a.) are different from the values in case b.}. That means : The two phases of a three-wire quarter-phase system are unsymmetrical, and the leading phase 1 reacts upon the lagging phase 2 in a different manner than 2 reacts upon 1. It is thus undesirable to use a three-wire quarter-phase system, except in cases where the line impedances Z are negligible. In all other cases, the four-wire quarter-phase system is preferable, which essentially consists of two independent single-phase circuits, and is treated as such. Obviously, even in such an independent quarter-phase system, at unequal distribution of load, unbalancing effects may take place. If one of the branches or phases is loaded differently from the other, the drop of voltage and the shift of the phase will be different from that in the other branch ; and thus the E.M.Fs. at the end of the lines will be neither equal in magnitude, nor in quadrature with each other. With both branches however loaded equally, the system remains balanced in voltage and phase, just like the three- phase system under the same conditions. Thus the four-wire quarter-phase system and the three- phase system are balanced with regard to voltage and phase at equal distribution of load, but are liable to become un- balanced at unequal distribution of load ; the three-wire quarter-phase system is unbalanced in voltage and phase, even at equal distribution of load. APPENDICES. APPENDIX I. ALGEBRA OF COMPLEX IMAGINARY QUANTITIES. INTRODUCTION. 296. The system of numbers, of which the science of algebra treats, finds its ultimate origin in experience. Directly derived from experience, however, are only the absolute integral numbers ; fractions, for instance, are not directly derived from experience, but are abstractions ex- pressing relations between different classes of quantities. Thus, for instance, if a quantity is divided in two parts, from one quantity two quantities are derived, and denoting these latter as halves expresses a relation, namely, that two of the new kinds of quantities are derived from, or can be combined to one of the old quantities. 297. Directly derived from experience is the operation of counting or of numeration. a, a + 1, a + 2, a + 3 . . . . Counting by a given number of integers : b integers introduces the operation of addition, as multiple counting : a + b = c. It is, a + b = b + a, 490 APPENDIX 7. that is, the terms of addition, or addenda, are interchange- able. Multiple addition of the same terms : a -+- a -\- a -+- . . . + a = c b equal numbers introduces the operation of multiplication : a x b = c. It is, a X b = b X a, that is, the terms of multiplication, or factors, are inter- changeable. Multiple multiplication of the same factors : aX aX aX . . • X a = c b equal numbers introduces the operation of involution : Since ab is not equal to #", the terms of involution are not interchangeable. 298. The reverse operation of addition introduces the operation of subtraction : If a + 6 = f, it is c — b = a. This operation cannot be carried out in the system of absolute numbers, if : b> c. Thus, to make it possible to carry out the operation of subtraction under any circumstances, the system of abso- lute numbers has to be expanded by the introduction of the negative number: _ « = (_ 1) X «, .where (- 1) is the negative unit. Thereby the system of numbers is subdivided in the COMPLEX IMAGINARY QUANTITIES. 491 positive and negative numbers, and the operation of sub- traction possible for all values of subtrahend and minuend. From the definition of addition as multiple numeration, and subtraction as its inverse operation, it follows : c - (- b) = c + b, thus: (-l)X (-!) = !; that is, the negative unit is defined by, (—I)2 = 1. 299. The reverse operation of multiplication introduces the operation of division : If a X b = c, then - = a. b In the system of integral numbers this operation can only be carried out, if b is a factor of c. To make it possible to carry out the operation of division under any circumstances, the system of integral numbers has to be expanded by the introduction of infraction: :©. where - is the integer fraction, and is defined by : T- x b = 1. 300. The reverse operation of involution introduces two new operations, since in the involution : the quantities a and b are not reversible. Thus V^ = sin [— /?]) it is Associate numbers: a + jb = r (cos ft +/ sin /3) = and b + ja = r ( cos 1 ^ — (3\ -f j sin \7- it is (a+jb)(b+ja)=j(a*+P) If a+jb = a' +jb', it is a = af If a +J/= 0 ; it is a = 0, 304. Addition and Subtraction : Multiplication : (a +jb) (a' +jb') = (aa1 - b b') +j(ab' + b a') or r (cos ^3 + / sin ft) X r' (cos /? + / sin ftf) = r r' (cos [£ -p ^]+ysin[/3 + ^]); or re J* X r'^'07 = rr'ef& + M. Division : Expansion of complex imaginary fraction, for rationaliza- tion of denominator or numerator, by multiplication with the conjugate quantity : COMPLEX IMAGINARY QUANTITIES. 495*" a+jb = (a+jb}(a' -jb'} = (aar+ bb'} +j (b a' - ab'} -jb'} , *" + *" (a! -f j b'} (a — jb} (a a' + b b'} +j(ab' — b a') ' or, _ r ^_p ^ _ ^ . r' or> r involution : (a +jbY = {r (cos evolution : -v/^- (cos /8 + y sin 305. Roots of the Unit : =+l, -1; \ 180 / 3W \ MO ^ ^-1 raT X — — TWO — J j»W8Q \ / \ . ___ ^. •^-i \ / _^ — ->T=- Vy /.\ -' -~ t 0 en atin . g E 135 cc M.F X " [ E =5 ^ stf> 1 4afs 2° a periodic function, and a function decreasing in geometric proportion with the time. The latter is the exponential function Af~gt. 309. Thus, the general expression of the oscillating current is /= ^/-0'COS (2-rrNt — S), since A'-** = A' A-'* = U~bt. Where e = basis of natural logarithms, the current may be expressed 7= i(.~bt cos (2-n-JVf— «) = ze-a* cos (<#> - £), where <#> = %-nNt; that is, the period is represented by a complete revolution. OSCILLATING CURRENTS. 499 In the same way an oscillating electromotive force will be represented by E = etra* cos O — 5). Such an oscillating electromotive force for the values e = 5, a = .1435 or «- 2™ = .4, £ = 0, is represented in rectangular coordinates in Fig. 207, and in polar coordinates in Fig. 208. As seen from Fig. 207, the oscillating wave in rectangular coordinates is tangent to the two exponential curves, Fig. 208. 310. In polar coordinates, the oscillating wave is repre- sented in Fig. 208 by a spiral curve passing the zero point twice per period, and tangent to the exponential spiral, The latter is called the envelope of a system O.L oscillat- ing waves of which one is shown separately, with the same constants as Figs. 207 and 208, in Fig. 209. Its character- 500 APPENDIX II. istic feature is : The angle which any concentric circle makes with the curve y — ee~a, is tan a = which is, therefore, constant ; or, in other words : " The envelope of the oscillating current is the exponential spiral, which is characterized by a constant angle of intersection Fig. 209. Fig. 210. with all concentric circles or all radii vectores." The oscil- lating current wave is the product of the sine wave and the exponential or loxodromic spiral. 311. In Fig. 210 let j/ = e€~a represent the expo-' nential spiral ; let z = e cos (<£ — a) represent the sine wave ; and let E = ef.-** cos (<£ — w) represent the oscillating wave. We have then tan y3 = Ed* _ — sin (<£ — w) — a cos COS (<£ — oi) = — {tan (<^> — £) + a} ; — to) OSCILLATING CURRENTS. 501 that is, while the slope of the sine wave, z = e cos (<£ — w), is represented by tan y = — tan (<£ — w), the slope of the exponential spiral y = ei'0* is tan a = — a = constant. That of the oscillating wave E = *?e~a* cos (<£ — to) is tan /3 = — {tan (<£ — w) + a} . Hence, it is increased over that of the alternating sine wave by the constant a. The ratio of the amplitudes of two consequent periods is A is called the numerical decrement of the oscillating wave, a the exponential decrement of the oscillating wave, a the angular decrement of the oscillating wave. The oscillating wave can be represented by the equation £ = ec-**™" cos ($ — 5). In the instance represented by Figs. 181 and 182> we have A = .4, a = .1435, a = 8.2°. Impedance and Admittance. 312. In complex imaginary quantities, the alternating wave * = e cos (* - ffl) is represented by the symbol E = e (cos w -\-j sin w) = cos ( — w) can be expressed by the symbol E = e (cos w -\-j sin w) dec a = (e± -\-j'e^) dec a, where a = tan a is the exponential decrement, a the angular decrement, e~27ra the numerical decrement. 502 APPENDIX II. Inductance. 313. Let r = resistance, L = inductance, and x = 2 IT N L = reactance. In a circuit excited by the oscillating current, /= /£-«* cos (<£ — w) = /(cos to +y sin w) dec a = (*i -\-J*z) dec a, where /i = / cos w, /2 = / sin £>, a = tan a. We have then, The electromotive force consumed by the resistance r of the circuit ^ The electromotive force consumed by the inductance L of the circuit, Ef**L—~*iNI&t = *—. dt d<$> d<$> Hence Ex = — xif.~a^> (sin ( — fy -\- a cos (<£ — w)} xi(.~a^ . ,. „ , N = sin (^> — w -f- a). COS a Thus, in symbolic expression, £x = - °^—{— sin (w — a) +/ cos (w — a)} dec a COS a = — x i (a -f y ) (cos w + 7 sin a>) dec a ; that is, Ex = — x I (a +/') dec a . Hence the apparent reactance of the oscillating current circuit is, in symbolic expression, X = x (a +y') dec a. Hence it contains an energy component ax, and the impedance is Z = (r — X) dec a = {r — x (a +/')} dec a = (r — ax —jx) dec a. Capacity. 314. Let r = resistance, C = capacity, and xc = 1 /2-n-JVC = capacity reactance. In a circuit excited by the oscillating OSCILLATING CURRENTS. 503 current /, the electromotive force consumed by the capacity Cis or, by substitution, Ex = x I * e~a* cos (<£ {sin (<£ — w) — a COS (<£ — oi 2 (1 + 02) COS a hence, in symbolic expression, sin ( — u> — a) ; = 2 (« + /) (cos w +y sin w) dec a ; hence, that is, the apparent capacity reactance of the oscillating circuit is, in symbolic expression, dec 315. We have then: In an oscillating current circuit of resistance r, induc- tive reactance x, and capacity reactance xc , with an expo- nential decrement a, the apparent impedance, in symbolic expression, is : *' 1 +a2/ V 1 +** = ra — jxa; 504 APPENDIX 77. and, absolute, Admittance. 316. Let /=/e-a*cos^_£)==current< Then from the preceding discussion, the electromotive force consumed by resistance r, inductive reactance x, and capa- city reactance xc , is cos $ — r — ax — a*e — sin (<£ — = iza(.~a^ cos (<£ — w + 8), where tan 8 = i_^ , a r — ax — —. -Xf substituting & + 8 for G, and ^ = /^a we have cos <> — I = — e~a* cos (<#> — w — 8) ,1 \ cos 8 / i ~\ i sin 8 . / , = e e. a

-\-j sin oi) dec a, + sin OSCILLATING CURRENTS. 505 or, substituting, r — ax — I =E I- dec a. 317. Thus in complex quantities, for oscillating cur- rents, we have : conductance, susceptance, admittance, in absolute values, / o i To 1 in symbolic expression, Y=g+J» 1 + a2/ \ 1 + a2 ' Since the impedance is Z = ir — ax — we have 506 APPENDIX II. that is, the same relations as in the complex quantities in alternating-current circuits, except that in the present case all the constants ra , xa , za , g, z, y, depend upon the dec- rement a. Circuits of Zero Impedance, 318. In an oscillating-current circuit of decrement a, of resistance r, inductive reactance x, and capacity reactance xc, the impedance was represented in symbolic expression by -jxa = ! + «» or numerically by Thus the inductive reactance x, as well as the capacity reactance xc, do not represent wattless electromotive forces as in an alternating-current circuit, but introduce energy components of negative sign a — ax — - - x : 1 + a2 that means, " In an oscillating-current circuit, the counter electro- motive force of self-induction is not in quadrature behind the current, but lags less than 90°, or a quarter period; and the charging current of a condenser is less than 90°, or a quarter period, ahead of the impressed electromotive force." 319. In consequence of the existence of negative en- ergy components of reactance in an oscillating-current cir- cuit, a phenomenon can exist which has no analogy in an alternating-current circuit ; that is, under certain conditions the total impedance of the oscillating-current circuit can equal zero : In this case we have r - ax 0 ; x -- ^— = 0, - — c 1 + a2 1 + fla OSCILLATING CURRENTS. 507 substituting in this equation x = 2 TT NL • xc = and expanding, we have a That is, " If in an oscillating-current circuit, the decrement 1 and the frequency N = r/4iraL, the total impedance of the circuit is zero ; that is, the oscillating current, when started once, will continue without external energy being impressed upon the circuit." 320. The physical meaning of this is : " If upon an electric circuit a certain amount of energy is impressed and then the circuit left to itself, the current in the circuit will become oscillating, and the oscillations assume the fre- quency N = r/4:7raL, and the decrement 1 That is, the oscillating currents are the phenomena by which an electric circuit of disturbed equilibrium returns to equilibrium. This feature shows the origin of the oscillating currents, and the means to produce such currents by disturbing the equilibrium of the electric circuit ; for instance, by the discharge of a condenser, by make and break of the circuit, by sudden electrostatic charge, as lightning, etc. Obviously, the most important oscillating currents are 508 APPENDIX II. those flowing in a circuit of zero impedance, representing oscillating discharges of the circuit. Lightning strokes usually belong to this class. Oscillating Discharges. 321. The condition of an oscillating discharge is Z = 0, that is, ~ ~ / .1 r 2aL 2Z~ ~1' If r = 0, that is, in a circuit without resistance, we have a = 0, Af = 1 / 2 TT VZT ; that is, the currents are alter- nating with no decrement, and the frequency is that of resonance. If 4 H r2 C - 1 < 0, that is, r > 2 V2T/T, a and N become imaginary ; that is, the discharge ceases to be os- cillatory. An electrical discharge assumes an oscillating nature only, if r < 2 V/, / C. In the case r = 2 VZ, / C we have « = oo , ./V = 0 ; that is, the current dies out without oscillation. From the foregoing we have seen that oscillating dis- charges, — as for instance the phenomena taking place if a condenser charged to a given potential is discharged through a given circuit, or if lightning strikes the line circuit, — are denned by the equation : Z = 0 dec a. Since / = (/V+y/a) dec a, Er = Ir dec a, Ex = -x I (a +/) dec a, Exc= _^L_/(- a +/) dec a, we have r-aX--^—Xc = ^ I + a? hence, by substitution, Exc= x /(— a +/) dec a. OSCILLATING CURRENTS. 50 £' The two constants, t\ and z'2, of the discharge, are deter- mined by the initial conditions, that is, the electromotive force and the current at the time t = 0. 322. Let a condenser of capacity C be discharged through a circuit of resistance r and inductance L. Let e = electromotive force at the condenser in the moment of closing the circuit, that is, at the time t — 0 or <£ = 0. A.t this moment the current is zero ; that is, 7=//2, /1==0. Since Exe= •*/(— a +/) dec a = e at = 0, we have x /2 Vl + a2 = e or /2 = = . x V 1 + a2 Substituting this, we have, I —j — e dec a, Er =je r dec a, x Vl + a2 x Vl + az Ex = e (1 -ja) dec a, ^c= e (1 +/ «) dec a, Vl + «8 Vl + a2 the equations of the oscillating discharge of a condense of initial voltage e. Since x = 2 *• N L, 1 we have x = hence, by substitution, l — dec a, .510 APPENDIX II. E - ef\fC -f^r-, — — rr~ \/ ~r~ 47TZ the final equations of the oscillating discharge, in symbolic expression. Oscillating Current Transformer. 323. As an instance of the application of the symbolic method of analyzing the phenomena caused by oscillating currents, the transformation of such currents may be inves- tigated. If an oscillating current is produced in a circuit including the primary of a transformer, oscillating currents will also flow in the secondary of this transformer. In a transformer let the ratio of secondary to primary turns be/. Let the secondary be closed by a circuit of total resistance, i\= r{ -\- TJ", where 1\ = external, 1\' = internal, resistance. The total inductance Ll = Z/ -f /,/', where Z/ = external, Zj" = internal, inductance ; total capacity, Cv Then the total admittance of the secondary circuit is ) dec a = where xl= 2irJVLl= inductive reactance: xcl = \l1-jrNC ' = capacity reactance. Let rQ = effecive hysteretic resistance, Z = inductance ; hence, x^ = Z-n-N LQ = reactance ; hence, admittance of the primary exciting circuit of the transformer ; that is, the admittance of the primary circuit at open secondary circuit. As discussed elsewhere, a transformer can be considered as consisting of the secondary circuit supplied by the im- pressed electromotive force over leads, whose impedance is OSCILLATING CURRENTS. 511 equal to the sum of primary and secondary transformer im- pedance, and which are shunted by the exciting circuit, out- side of the secondary, but inside of the primary impedance. Let r = resistance ; L = inductance ; C = capacity ; hence' x = 2 TT NL = inductive reactance, xc = 1 / 2 TT N C = capacity reactance of the total primary circuit, including the primary coil of the transformer. If EI = EI dec a denotes the electromotive force induced in the secondary of the transformer by the mutual magnetic flux ; that is, by the oscillating magnetism interlinked with the primary and secondary coil, we have Iv = E^ Yl dec a = secondary current. Hence, // = / 7X dec a = pEJ Yl dec a = primary load current, or component of primary current corresponding to secondary current. Also, 70 = - 2j/ F0 dec a = primary / ' exciting current ; hence, the total primary current is /= // + 70 = £-'{Fo +/2 Y,} dec a. E' E' = -^-i- dec a = induced primary electromotive force. / Hence the total primary electromotive force is E = (£' + /Z) dec a = £L (1 + Z F0 +/2Z Y,} dec a. P In an oscillating discharge the total primary electro- motive force E = 0 ; that is, or, the substitution a 1 + (r0 - ax0) -. . 0. 512 APPENDIX II. Substituting in this equation, ^r=2 it N C, xc = ~L/'2 etc., we get a complex imaginary equation with the two constants a and N. Separating this equation in the real and the imaginary parts, we derive two equations, from which the two constants a and N of the discharge are calculated. 324. If the exciting current of the transformer is neg- ligible, — that is, if YQ = 0, the equation becomes essentially simplified, — I a \ . I x \ (r — a x xc 1 — j ( x — I 1+/2v 1 + *8 i v Ljt^l=0; that is, or, combined, — (r, -2aXl) +/2 (r-2 ax) = 0, Substituting for xlt x, xel, xei we have +/aZ) i+/V / 4(A+/ +/2Z) V (n +/V)2 (Ci !} dec a, 7 =pEi YI dec a, /! = ^/ F! dec a, the equations of the oscillating-current transformer, with E{ as parameter.