Theory and Calculation of Alternating Current Phenomena (1900) — part 8 of 19

ence of phase between both as function of the distance from receiver circuit ; under the conditions,

E.M.F. at receiving end, 10,000 volts; hence, Ev =el = 10,000; current at receiving end, 65 amperes, with a power factor of .385.

that is, / = t\ + j // = 25 + 60 j ;

line constants per unit length,

r = 1, g = 2 X 10-5,

hence,

a = 4.95 x 10-3, ] 13 = 28.36 x 10 -3, j-

length of line corresponding to one complete period of the wave

x0 = L = — = 221.5 =

(^ of propagation. A = 1.012 - 1.206 y ) B = .812 + .794 / j

These values, substituted, give,

/= {£«x (47.3 cos /?x + 27.4 sin fix) — e-«*

(22.3 cos ftx + 32.6 sin fix)}

• y (e«x (27.4 cos ftx — 47.3 sin ftx) + €-«x

(32.6 cos y3x — 22.3 sin /3x)}; E = {eox (6450 cos /3x + 4410 sin j8x) + c-ax

(3530 cos fix + 4410 sin /?x)}

• y (eox (4410 cos /3x — 6450 sin £x) — e~ax (4410 cos ft x- 3530 sin /3x)};

tan 5, = ~ °-ljc + PS = _ .073, JJ« = - 4.2°.

120. As a further instance are shown the characteristic curves of a transmission line of the relative constants,

r\x\g>.b = % : 32 : 1.25 X 10 ~4 : 25 X 10 ~4, and e = 25,000, i = 200 at the receiving circuit, for the con- ditions,

a, non-inductive load in the receiving circuit, Fig. 87.

174

ALTERNATING-CURRENT PHENOMENA.

b, wattless receiving circuit of 90° lag, Fig. 88.

c, wattless receiving circuit of 90° lead, Fig. 89. These curves are determined graphically by constructing

the topographic circuit characteristics in polar coordinates as explained in Chapter VI., paragraphs 36 and 37, and de- riving corresponding values of current, potential difference and phase angle therefrom.

As seen from these diagrams, for wattless receiving cir- cuit, current and E.M.F. oscillate in intensity inversely to

ZJ

7

6sa

7

rig. 87.

DISTRIBUTED CAPACITY.

175

each other, with an amplitude of oscillation gradually de- creasing when passing from the receiving circuit towards the generator, while the phase angle between current and E.M.F. oscillates between lag and lead with decreasing am- plitude. Approximately maxima and minima of current co- incide with minima and maxima of E.M.F. and zero phase angles.

\

V

Fig. 88.

176

AL TERNA TING-CURRENT PHENOMENA.

For such graphical constructions, polar coordinate paper and two angles a and 8 are desirable, the angle a being the

angle between current and change of E.M.F., tan a = - = 4, and the angle 8 the angle between E.M.F. and change of

current, tan 8 = - = 20 in above instance. g

\

Fig. 89.

DISTRIBUTED CAPACITY.

177

With non-inductive load, Fig. 87, these oscillations of intensity have almost disappeared, and only traces of them are noticeable in the fluctuations of the phase angle and the relative values of current and E.M.F. along the line.

Towards the generator end of the line, that is towards rising power, the curves can be extended indefinitely, ap- proaching more and more the conditions of non-inductive circuit, towards decreasing power, however, all curves ulti- mately reach the conditions of a wattless receiving circuit, as Figs. 88 and 89, at the point where the total energy in-

t

a +120

ISSION LINE

V

Fig. 90.

put into the line has been consumed therein, and at this point the two curves for lead and for lag join each other as shown in Fig. 90, the one being a prolongation of the other, and the flow of power in the line reverses. Thus in Fig. 90 power flows from both sides of the line towards the point of zero power marked by 0, where current and E.M.F. are in quadrature with each other, the current being leading with regard to the flow of power from the left, and lagging with regard to the flow of power from the right side of the diagram.

178 DISTRIBUTED CAPACITY.

121. The  following  are  some  particular  cases : 
     

A.) Open circuit at end of lines : x = 0 : /! = 0.

hence,

E = i-r— ^{(eax + e-ax) cos/3x — y(cax — c-ax)sin/3x};

.£?.) Line grounded at end:

A — (a/\ -J- /?//) +/ (a// — ^zi) = -? -^-T-^{(eax — c-ax) cos/?x — >(eax + c~ax) sin)8x};

(T.) Infinitely long conductor :

Replacing x by — x, that is, counting the distance posi- tive in the direction of decreasing energy, we have,

x = oo : 7= 0, E = 0; hence

and

I = — - — ^£-°x(cos/Sx +ysin/3x),

'

revolving decay of the electric wave, that is the reflected wave does not exist.

The total impedance of the infinitely long conductor is

(q-yff) (g+M

• b? g* + b*

ALTERNATING-CURRENT PHENOMENA. 179

The infinitely long conductor acts like an impedance

7 _ °-K + P ?>c _ • fig — Q-bc

f*+v g* + K'

that is, like a resistance

combined with a reactance

We thus get the difference of phase between E.M.F. and current,

which is constant at all points of the line. If g = 0, x = 0, we have,

hence,

tan to = 1, or,

£ = 45° ;

that is, current and E.M.F. differ by £th period. D.) Generator feeding into closed circuit : Let x = 0 be the center of cable ; then,

hence : E — 0 at x = 0 ;

which equations are the same as in B, where the line is grounded at x = 0.

E.) Let the length of a line be one-quarter wave length;

and assume the resistance r and conductance g as negligible

180 AL TERN A TING-CURRENT PHENOMENA.

compared with x and bc.

r=0=g These values substituted in (11) give

a=0.

(3= V^

Let the E.M.F. at the receiving end of the line be assumed zero vector

£l = ei = E.M.F. and

fi — i'i + ji. — current at end of line x = 0 £0 = E.M.F. and

S0 = current at beginning of line

Substituting in (16) these values of El and 7: and also r = 0 = g, we have

From these equations it follows that

which values, together with the foregoing values of Ev Iv r, g, a, and /8, substituted in (14) reduce these equations to

— j (i\ +jiC) ~r s^

ALTERNATING-CURRENT TRANSFORMER, 181

Then at x

Hence also

•£"„ and 70 are both in quadrature ahead of <?x and 7j respectively.

Il = EQ y — = constant, if 7f0 = constant. That is, at

constant impressed E.M.F. E& the current 7X in the receiv- ing circuit of a line of one-quarter wave length is constant, and inversely (constant potential — constant current trans- formation by inductive line). In this case, the current 70 at the beginning of the line is proportional to the load el at the end of the line.

If XQ = lx = total reactance,

b0 = lbc = total susceptance of line, then

*<A> = 4-

Instance* = 4, bc = 20 X 10 ~5, E0 = 10,000 V. Hence / = 55.5, *0 = 222, b0 = .0111, 7j = 70.7, 70 = .00707 e.

122. An interesting application of this method is the determination of the natural period of a transmission line ; that is the frequency at which such a line discharges an accumulated charge of atmospheric electricity (lightning), or oscillates at a sudden change of load, as a break of cir- cuit.

182 ALTERNATING-CURRENT PHENOMENA.

The discharge of a condenser through a circuit contain- ing self-induction and resistance is oscillating (provided that the resistance does not exceed a certain critical value de- pending upon the capacity and the self-induction). That is, the discharge current alternates with constantly decreasing intensity. The frequency of this oscillating discharge de- pends upon the capacity, C, and the self-induction, L, of the circuit, and to a much lesser extent upon the resistance, so that if the resistance of the circuit is not excessive the fre- quency of oscillation can, by neglecting the resistance, be expressed with fair, or even close, approximation by the formula -

An electric transmission line represents a capacity as well as a self-induction ; and thus when charged to a certain potential, for instance, by atmospheric electricity, as by in- duction from a thunder-cloud passing over or near the line, the transmission line discharges by an oscillating current.

Such a transmission line differs, however, from an ordi- nary condenser, in that with the former the capacity and the self-induction are distributed along the circuit.

In determining the frequency of the oscillating discharge of such a transmission line, a sufficiently close approximation is obtained by neglecting the resistance of the line, which, at the relatively high frequency of oscillating discharges, is small compared with the reactance. This assumption means that the dying out of the discharge current through the influence of the resistance of the circuit is neglected, and the current assumed as an alternating current of ap- proximately the same frequency and the same intensity as the initial waves of the oscillating discharge current. By this means the problem is essentially simplified.

Let / = total length of a transmission line, r = resistance per unit length, x = reactance per unit length = 2 ?r NL.

DISTRIBUTED CAPACITY. 183

where L = coefficient of self-induction or inductance per unit

length ;

g = conductance from line to return (leakage and dis- charge into the air) per unit length ; b = capacity susceptance per unit length = 2 TT NC where C = capacity per unit length.

x = the distance from the beginning of the line,

We have then the equations : The E.M.F.,

(^eax _ ^e-ax) CQS £x _j (4€

g — jb I + ^e~ax) sin /3x the current,

1 ^ (Aea* + ^e~ax) COS /3x — y (^4e

where,

,(14.)

(r1 + ^c2) + (^r -

' (11.)

c = base of the natural logarithms, and A and B integration constants.

Neglecting the line resistance, r = 0, and the conduc- tance (leakage, etc.), g=0, gives,

These values substituted in (14) give, J-

= J-(A - B} cos ^fbx^ -j (A + H) sin

/ = -4= J (^ + -ff) cos V^x — y (<4 - B) sin

; J

184 ALTERNATING-CURRENT PHENOMENA.

If the discharge takes place at the point : x = 0, that is, if the distance is counted from the discharge point to the end of the line ; x = /, hence :

At x = 0, E = 0, Atx=/, 7=0.

Substituting these values in (25) gives,

For x = 0,

^-7^ = 0 A = B

which reduces these equations to,

E = — — sin Nbx x

b \

7= -^4^= cos V&t: x

VA* I

and at x = 0,

At x = /, / = 0, thus, substituted in (26),

cos V^/ = 0 (28.)

hence :

V^/^2**1)", 1 = 0,1, 2,... (29.)

that is, *Jbx I is an odd multiple of ^ • And at x = /,

2t

O A

Substituting in (29) the values,

we have,

hence,

^=M + l (31.)

4/VCZ

DISTRIBUTED CAPACITY. 185

the frequency of the oscillating discharge, where k = 0, 1, 2. . . .

That is, the oscillating discharge of a transmission line of distributed capacity does not occur at one definite fre- quency (as that of a condenser), but the line can discharge at any one of an infinite number of frequencies, which are the odd multiples of the fundamental discharge frequency,

*-I7^z (32'>

Since

C0 = 1C = total capacity of transmission line, )

L0 = IL = total self-inductance of transmission line, J ^ ''

we have,

2,£ + 1

-= the frequency of oscillation, (34.)

or natural period of the line, and

NI — - - the fundamental,

or lowest natural period of the line. From (30), (33), and (34),

b = 2irNC= 2/ \T0 (36-)

and from (29),

V^ = (2^2f/)7r- <37')

These substituted in (26) give,

f- (38.) 4/7 (2£ + l)7rx

/=(2TTi)-^cosL^H

The oscillating discharge of a line can thus follow any of the forms given by making k — 0, 1, 2, 3 . . .in equation (38).

Reduced from symbolic representation to absolute values

186 ALTERNATING-CURRENT PHENOMENA.

by multiplying E with cos 2 * Nt and / with sin 2 TT A7/ and omitting j, and substituting A7" from equation (34), we have,

(2£+l)7rx

— sin — JT— - — -cos 2/

where ^4 is an integration constant, depending upon the initial distribution of voltage, before the discharge, and / = time after discharge.

123. The fundamental discharge wave is thus, for k = 0,

124.   Lo        .       .        7TX  7T/ 
     

-^ A sin 7^ C0 2/

. o . . 7TX

= — \ -^ A sin 7^— cos

TT V

4 / - _, 7T X 7T/

fi = — A cos 7n- sin -

With this wave the current is a maximum at the begin- ning of the line : x = 0, and gradually decreases to zero at the end of the line : x = /.

The voltage is zero at the beginning of the line, and rises to a maximum at the end of the line.

Thus the relative intensities of current and potential along the line are as represented by Fig. 91, where the cur- is shown as /, the potential as E.

The next higher discharge frequency, for : k — 1, gives :

47. [Ln . . 3v_

(41.)

4/ " " - '

/, = o- A cos

n 7 27

DISTRIBUTED CAPACITY.

187

Here the current is again a maximum at the beginning of the line : x = 0, and gradually decreases, but reaches

zero at one-third of the line : x = _, then increases again, in

o

Fig.

H----0

Fig.

\

\

\

\1

Figs. 91-93.

188 ALTERNATING CURRENT-PHENOMENA.

the opposite direction, reaches a second but opposite maxi-

2/

mum at two-thirds of the line : x = ^— , and decreases to

o

zero at the end of the line. There is thus a nodal point of current at one-third of the line.

The E.M.F. is zero at the beginning of the line : x = 0,

rises to a maximum at one-third of the line : x = - , de-

2/ 3 creases to zero at two-thirds of the line : x = IT > and rises

again to a second but opposite maximum at the end of the line: x = /. The E.M.F. thus has a nodal point at two- thirds of the line.

The discharge waves : k = 1, are shown in Fig. 92, those with k = 2, with two nodal points, in Fig. 93.

Thus k is the number of nodal points or zero points of current and of E.M.F. existing in the line (not counting zero points at the ends of the line, which of course are not nodes).

In case of a lightning discharge the capacity C0 is the capacity of the line against ground, and thus has no direct relation to the capacity of the line conductor against its return. The same applies to the inductance L0.

If d = diameter of line conductor,

D = distance of conductor above ground, and / = length of conductor,

the capacity is,

1.11 x 10-6/ ,.

~

the self-inductance,

The fundamental frequency of oscillation is thus, by substituting (42) in (35),

DISTRIBUTED CAPACITY. 189

That is, the frequency of oscillation of a line discharging to ground is independent of the size of line wire and its distance from the ground, and merely depends upon the length / of the line, being inversely proportional thereto.

We thus get the numerical values,

Length of line

10 20 30 40 50 60 80 100 miles. = 1.6 3.2 4.8 6.4 8 9.6 12.8 16 x 106 cm..

hence frequency,

N-i = 4680 2340 1560 1170 937.5 780 585 475 cycles-..

As seen, these frequencies are comparatively low, and especially with very long lines almost approach alternator frequencies.

The higher harmonics of the oscillation are the odd! multiples of these frequencies.

Obviously all these waves of different frequencies repre- sented in equation (39) can occur simultaneously in the oscillating discharge of a transmission line, and in general the oscillating discharge of a transmission line is thus of the form,

(by substituting: ak = * j

where a^ as ay . . . are constants depending upon the initial distribution of potential in the transmission line, at the moment of discharge, or at / = 0, and calculated there- from.

190 AL TERN A TING-CURRENT PHENOMENA .

124. As an instance the following discharge equation of a line charged to a uniform potential e over, its entire length, and then discharging at x = 0, has been calculated.

The harmonics are determined up to the 11 — that is, av

•a& #5> av a9> an-

These six unknown quantities require six equations, which

/ 2/ 3/ 4/ 5/ 6/ are given by assuming E = e for x = g, ,,,,_.

At / = 0, E = e, equation (44) assumes the form

4 / HT ( . TTX , . 3 TTX e = — V £? j «i sm 27 + *3 sm~27 + ' ' ' ' + *u

(45.)

/ 2/ 6/

Substituting herein for x the values : - , — , . . . —

gives six equations for the determination of av <73 . . . an. These equations solved give,

E = e (1.26 sin w cos $ + .40 sin 3 w cos 3 <f» + .22 sin ^ 5 w cos 5 <^ + .12 sin 7 o> cos 7 <£ + .07 sin 9 co cos 9 ^ + .02 sin 11 o> cos 11 ^>

5

L0

cos 5 <o sin 5 <^ + .12 cos 7 o> sin 7 <£ + .07 cos 9 to sin 9 <£ + .02 cos 11 o> sin 11 </>

7 = e i/5 (1.26 cos o> sin <£ + .40 cos 3 w sin 3 <£ + .22

V 7rt

,(46.)

where,

"-57 1

„ r<47')

Instance, .

Length of line, / = 25 miles = 4 x 106 cm. Size of wire : No. 000 B. & S. G., thus : d = 1 cm. Height above ground : D — 18 feet = 550 cm. Let e = 25,000 volts = potential of line in the moment of -discharge.

DISTRIBUTED CAPACITY. 191

We then have,

E = 31,500 sin w cos <fr + 10,000 sin 3 <o cos 3 <£ + 5500 sin

5 o> cos 5 <J> + 3000 sin 7 o> cos 7 <£ -j- 1750 sin 9 o> cos

9 <£ + 500 sin 11 w cos 11 <£. /= 61.7 cos w sin <£ + 19.6 cos 3 o> sin 3 <£ + 10.8 cos 5 « sin

5 <£ + 5.9 cos 7 CD sin 7 </> + 3.4 cos 9 to sin 9 <£ + 1.0

cos 11 <o sin 11 <J>.

<o= .39 .r 10 -6 </> = 1.18/ 10+4

A simple harmonic oscillation as a line discharge would require a sinoidal distribution of potential on the trans- mission line at the instant of discharge, which is not proba- ble, so that probably all lightning discharges of transmission lines or oscillations produced by sudden changes of circuit conditions are complex waves of many harmonics, which in their relative magnitude depend upon the initial charge and its distribution — that is, in the case of the lightning dis- charge, upon the atmospheric electrostatic field of force.

The fundamental frequency of the oscillating discharge of a transmission line is relatively low, and of not much higher magnitude than frequencies in commercial use in alternating current circuits. Obviously, the more nearly sinusoidal the distribution of potential before the discharge, the more the low harmonics predominate, while a very un- equal distribution of potential, that is a very rapid change along the line, as caused for instance by a sudden short circuit rupturing itself instantly, causes the higher harmo- nics to predominate, which as a rule are more liable to cause excessive rises of voltage by resonance.

125. As has been shown, the electric distribution in a transmission line containing distributed capacity, self-induc- tion, etc., can be represented either by a polar diagram with the phase as amplitude, and the intensity as radius vector, as in Fig. 34, or by a rectangular diagram with the

192 ALTERNATING-CURRENT PHENOMENA.

distance as abscissae, and the intensity as ordinate, as in Fig. 35 and in the preceding paragraphs.

In the former case, the consecutive points of the circuit characteristic refer to consecutive points along the trans- mission line, and thus to give a complete representation of the phenomenon, should not be plotted in one plane but in front of each other by their distance along the transmission line. That is, if 0, 1, 2, etc., are the polar vectors in Fig. 34, corresponding to equi-distant points of the transmission line, 1 should be in a plane vertically in front of the plane of 0, 2 by the same distance in front of 1, etc.

In Fig. 35 the consecutive points of the circuit charac- teristic represent vectors of different phase, and thus should be rotated out of the plane around the zero axis by the angles of phase difference, and then give a length view of the same space diagram, of which Fig. 34 gives a view along the axis.

Thus, the electric distribution in a transmission line can be represented completely only by a space diagram, and as complete circuit characteristic we get for each of the lines a screw shaped space curve, of which the distance along the axis of the screw represents the distance along the transmis- sion line, and the distance of each point from the axis rep- resents by its direction the phase, and by its length the intensity.

Hence the electric distribution in a transmission line leads to a space problem of which Figs. 34 and 35 are par- tial views. The single-phase line is represented by a double screw, the three-phase line by a triple screw, and the quarter- phase four-wire line by a quadruple screw. In the symbolic expression of the electric distribution in the transmission line, the real part of the symbolic equation represents a pro- jection on a plane passing through the axis of the screw, and the imaginary part a projection on a plane perpendicular to the first, and also passing through the axis of the screw.

ALTERNATING-CURRENT TRANSFORMER. 193

CHAPTER XIV.

THE ALTERNATING-CURRENT TRANSFORMER.

126. The simplest alternating-current apparatus is the transformer. It consists of a magnetic circuit interlinked with two electric circuits, a primary and a secondary. The primary circuit is excited by an impressed E.M.F., while in the secondary circuit an E.M.F. is induced. Thus, in the primary circuit power is consumed, and in the secondary a corresponding amount of power is produced.

Since the same magnetic circuit is interlinked with both electric circuits, the E.M.F. induced per turn must be the same in the secondary as in the primary circuit ; hence, the primary induced E.M.F. being approximately equal to the impressed E.M.F., the E.M.Fs. at primary and at sec- ondary terminals have approximately the ratio of their respective turns. Since the power produced in the second- ary is approximately the same as that consumed in the primary, the primary and secondary currents are approxi- mately in inverse ratio to the turns.

127. Besides the magnetic flux interlinked with both electric circuits — which flux, in a closed magnetic circuit transformer, has a circuit of low reluctance — a magnetic cross-flux passes between the primary and secondary coils, surrounding one coil only, without being interlinked with the other. This magnetic cross-flux is proportional to the current flowing in the electric circuit, or rather, the ampere- turns or M.M.F. increase with the increasing load on the transformer, and constitute what is called the self-induc- tance of the transformer ; while the flux surrounding both

194 ALTERNATING-CURRENT PHENOMENA.

coils may be considered as mutual inductance. This cross- flux of self-induction does not induce E.M.F. in the second- ary circuit, and is thus, in general, objectionable, by causing a drop of voltage and a decrease of output. It is this cross-flux, however, or flux of self-inductance, which is uti- lized in special transformers, to secure automatic regulation, for constant power, or for constant current, and in this case is exaggerated by separating primary and secondary coils. In the constant potential transformer however, the primary and secondary coils are brought as near together as possible, or even interspersed, to reduce the cross-flux.

As will be seen by the self-inductance of a circuit, not the total flux produced by, and interlinked with, the circuit is understood, but only that (usually small) part of the flux which surrounds one circuit without interlinking with the other circuit.

128. The alternating magnetic flux of the magnetic circuit surrounding both electric circuits is produced by the combined magnetizing action of the primary and of the secondary current.

This magnetic flux is determined by the E.M.F. of the transformer, by the number of turns, and by the frequency. If

<£ = maximum magnetic flux, N= frequency, n = number of turns of the coil ;

the E.M.F. induced in this coil is

E= V2 • JVfc * 10 -8 = 4.44 .Afo 10 -'volts;

hence, if the E.M.F., frequency, and number of turns are determined, the maximum magnetic flux is

To produce the magnetism, $, of the transformer, a M.M.F. of 5 ampere-turns is required, which is determined

ALTERNATING-CURRENT TRANSFORMER. 195

by the shape and the magnetic characteristic of the iron, in the manner discussed in Chapter X.

For instance, in the closed magnet circuit transformer, the maximum magnetic induction is ($> = & /S, where S = the cross-section of magnetic circuit.

129. To induce a magnetic density, ($>, a M.M.F. of 3CTO ampere-turns maximum is required, or, 3COT / V2 ampere- turns effective, per unit length of the magnetic circuit ; hence, for the total magnetic circuit, of length, /,

/3C & = — :r- ampere-turns ;

« *V2 where n = number of turns.

At no load, or open secondary circuit, this M.M.F., CF, is furnished by the exciting current, T00, improperly called the leakage current, of the transformer ; that is, that small amount of primary current which passes through the trans- former at open secondary circuit.

In a transformer with open magnetic circuit, such as the "hedgehog" transformer, the M.M.F., &, is the sum of the M.M.F. consumed in the iron and in the air part of the magnetic circuit (see Chapter X.).

The energy of the exciting current is the energy con- sumed by hysteresis and eddy currents and the small ohmic loss.

The exciting current is not a sine wave, but is, at least in the closed magnetic circuit transformer, greatly distorted by hysteresis, though less so in the open magnetic circuit transformer. It can, however, be represented by an equiv- alent sine wave, f00, of equal intensity and equal power with the distorted wave, and a wattless higher harmonic, mainly of triple frequency.

Since the higher harmonic is small compared with the

196 ALTERNATING-CURRENT PHENOMENA.

total exciting current, and the exciting current is only a small part of the total primary current, the higher harmonic .can, for most practical cases, be neglected, and the exciting current represented by the equivalent sine wave.

This equivalent sine wave, 7^, leads the wave of mag- netism, 3>, by an angle, a, the angle of hysteretic advance of phase, and consists of two components, — the hysteretic energy current, in quadrature with the magnetic flux, and therefore in phase with the induced E.M.F. = I00 sin a; and the magnetizing current, in phase with the magnetic fluXj and therefore in quadrature with the induced E.M.F., and so wattless, = I00 cos a.

The exciting current, 700, is determined from the shape and magnetic characteristic of the iron, and number of turns ; the hysteretic energy current is —

Power consumed in the iron

I00 sin a

Induced E.M.F.

130. Graphically, the polar diagram of M.M.Fs. ot a transformer is constructed thus :

Fig. 94.

Let, in Fig. 94, O® = the magnetic flux in intensity and phase (for convenience, as intensities, the effective values are used throughout), assuming its phase as the vertical;

ALTERNATING-CURRENT TRANSFORMER. 197

that is, counting the time from the moment where the rising magnetism passes its zero value.

Then the resultant M.M.F. is represented by the vector QS, leading O<b by the angle &O® = a.

The induced E.M.Fs. have the phase 180°, that is, are plotted towards the left, and represented by the vectors OZT; and OE±.

If, now, ft' = angle of lag in the secondary circuit, due to the total (internal and external) secondary reactance, the secondary current II , and hence the secondary M.M.F., JF1= «j /L, will lag behind £•[ by an angle ft1, and have the phase, 180° + ft', represented by the vector O^1. Con- structing a parallelogram of M.M.Fs., with Off as a diag- onal and Oif1 as one side, the other side or O'S0 is the primary M.M.F., in intensity and phase, and hence, dividing by the number of primary turns, n0, the primary current is /.-./..

To complete the diagram of E M.Fs. , we have now, —

In the primary circuit :

E.M.F. consumed by resistance is 70r0, in phase with fot and represented by the vector OEr0 •

E.M.F. consumed by reactance is IoX0, 90° ahead of I0, and represented by the vector OEx0 ;

E.M.F. consumed by induced E.M.F. is E', equal and oppo- site to E'o, and represented by the vector Off.

Hence, the total primary impressed E.M.F. by combina- tion of OEr0, OEx0, and OE' by means of the parallelo- gram of E.M.Fs. is,

E0 = ~OE0,

and the difference of phase between the primary impressed E.M.F. and the primary current is

ft0 = E0O50. In the secondary circuit :

Counter E.M.F. of resistance is 1^ in opposition with Iv and represented by the vector OJS'r^ ;

198

AL TERNA TING-CURRENT PHENOMENA,

90° behind 7X, and

Counter E.M.F. of reactance is represented by the vector OE^x^

Induced E.M.Fs., E( represented by the vector OE-[.

Hence, the secondary terminal voltage, by combination of OEr^ OEx{ and OE^ by means of the parallelogram of

E.M.Fs. is -==•

A = M»II

and the difference of phase between the secondary terminal voltage and the secondary current is

As will be seen in the primary circuit the " components of impressed E.M.F. required to overcome the counter E.M.Fs." were used for convenience, and in the secondary circuit the "counter E.M.Fs."

Er,

Fig. 95. Transformer Diagram with 80° Lag in Secondary Circuit.

131. In the construction of the transformer diagram, it is usually preferable not to plot the secondary quantities, current and E.M.F., direct, but to reduce them to corre- spondence with the primary circuit by multiplying by the ratio of turns, a = n0/ nv for the reason that frequently primary and secondary E.M.Fs., etc., are of such different

AL TERA?A TING-CURRENT TRANSFORMER.

19!)

magnitude as not to be easily represented on the same scale; or the primary circuit may be reduced to the sec- ondary in the same way. In either case, the vectors repre- senting the two induced E.M.Fs. coincide, or OE-^ = OE^.

Fig. 96. Transformer Diagram with 50° Lag in Secondary Circuit.

Figs. 96 to 107 give the polar diagram of a transformer having the constants —

r0 = .2 ohms, x0 = .33 ohms, f! = .00167 ohms, *! = .0025 ohms, g0 = .0100 mhos,

for the conditions of secondary circuit,

= .0173 mhos, = 100 volts, = 60 amperes, =10 degrees. ?

20° lead in Fig. 99. 50° lead " 100. 80° lead " 101.

ft' = 80° lag in Fig. 95.

50° lag " 96.

20° lag " 97.

O, or in phase, " 98.

As shown with a change of /?/ the other quantities E0, Iv I0, etc., change in intensity and direction. The loci de- scribed by them are circles, and are shown in Fig. 102, with the point corresponding to non-inductive load marked. The part of the locus corresponding to a lagging secondary

200 ALTERNATING-CURRENT PHENOMENA.

Fig. 97. Transformer Diagram with 20° Lag in Secondary Circuit

Fig. 98. Transformer Diagram with Secondary Current in Phase with E.M.F.

Fig. 99. Transformer Diagram with 20° Lead in Secondary Current.

ALTERNATING-CURRENT TRANSFORMER. 201

(To EO

Fig. 100. Transformer Diagram with 50° Lead in Secondary Circuit.

Fig. 101. Transformer Diagram with 80° Lead in Secondary Circuit.

Fig. 102.

202

AL TERNA TING-CURRENT PHENOMENA.

current is shown in thick full lines, and the part correspond- ing to leading current in thin full lines.

132. This diagram represents the condition of constant secondary induced E.M.F., £"/, that is, corresponding to a constant maximum magnetic flux.

By changing all the quantities proportionally from the diagram of Fig. 102, the diagrams for the constant primary impressed E.M.F. (Fig. 103), and for constant secondary terminal voltage (Fig. 104), are derived. In these cases, the locus gives curves of higher order.

Fig. 103.

Fig. 105 gives the locus of the various quantities when the load is changed from full load, /j = 60 amperes in a non-inductive secondary external circuit to no load or open circuit.

a.) By increase of secondary resistance ; b.} by increase of secondary inductive reactance ; c.) by increase of sec- ondary capacity reactance.

As shown in a.), the locus of the secondary terminal vol- tage, J5lt and thus of E0, etc., are straight lines; and in b.) and c.}, parts of one and the same circle a.} is shown

AL TERNA TING-CURRENT TRANSFORMER.

203

in full lines, b.} in heavy full lines, and c.} in light full lines. This diagram corresponds to constant maximum magnetic flux ; that is, to constant secondary induced E.M.F. The diagrams representing constant primary impressed E.M.F. and constant secondary terminal voltage can be derived from the above by proportionality.

Fig. 104.

133. It must be understood, however, that for the pur- pose of making the diagrams plainer, by bringing the dif- ferent values to somewhat nearer the same magnitude, the constants chosen for these diagrams represent, not the mag- nitudes found in actual transformers, but refer to greatly exaggerated internal losses.

In practice, about the following magnitudes would be found :

r0 = .01 ohms ; x0 = .033 ohms ; ri = .00008 ohms j

#! = .00025 ohms ; g0 = .001 ohms ; b0 = .00173 ohms ;

that is, about one-tenth as large as assumed. Thus the changes of the values of E0, Elt etc., under the different conditions will be very much smaller.

204

ALTERNATING-CURRENT PHENOMENA.

Symbolic Method.

134. In symbolic representation by complex quantities the transformer problem appears as follows :

The exciting current, 700, of the transformer depends upon the primary E.M.F., which dependance can be rep- resented by an admittance, the " primary admittance," °f tne transformer.

Fig. 105.

The resistance and reactance of the primary and the secondary circuit are represented in the impedance by

Z0=r0- jx0, and Zl=rl- j xl .

Within the limited range of variation of the magnetic density in a constant potential transformer, admittance and impedance can usually, and with sufficient .exactness, be considered as constant.

Let

n0 = number of primary turns in series ; #1 = number of secondary turns in series ; a = — = ratio of turns ;

Y0 = g0 4- jb0 = primary admittance

Exciting current . ~i I

Primary counter E.M.F. '

.VVWvVl

rw^ww

ALTERNATING-CURRENT TRANSFORMER. 205

Z0 = r0 — j x0 = primary impedance 7. — —

E.M.F. consumed in primary coil by resistance and reactance. ^ -n-f '" ' j**/.

Primary current ~ /

Z± = r± —jx1= secondary impedance

__ E.M.F. consumed in secondary coil by resistance and reactance . Secondary current