Theory and Calculation of Alternating Current Phenomena (1900) — part 7 of 19

where : r = hysteretic resistance, xc — hysteretic condens- ance ; and the angle of dielectric hysteretic lag, tan a = b' / g = xc / r, are constants of the circuit, independent of E.M.F. and frequency. The E.M.F. is obviously inversely propor- tional to the frequency.

The true static dielectric hysteresis, observed by Arno as proportional to the 1.6th power of the density, will enter the admittance and the impedance as a term variable and dependent upon E.M.F. and frequency, in the same manner as discussed in the chapter on magnetic hysteresis.

To the magnetic hysteresis corresponds, in the electro- static field, the static component of dielectric hysteresis, following, probably, the same law of 1.6th power.

To the eddy currents in the iron corresponds, in the electrostatic field, the viscous component of dielectric hys- teresis, following the square law.

As a rule however, these hysteresis losses in the alter- nating electrostatic field of a condenser are very much smaller than the losses in an alternating magnetic field, so that while the latter exert a very marked effect on the de- sign of apparatus, representing frequently the largest of all the losses of energy, the dielectric losses are so small as to be very difficult to observe.

FOUCAULT OR EDDY CURRENTS. 147

To the phenomenon of mutual inductance corresponds, in the electrostatic field, the electrostatic induction, or in- fluence.

100. The alternating electrostatic field of force of an electric circuit induces, in conductors within the field of force, electrostatic charges by what is called electrostatic influence. These charges are proportional to the field strength ; that is, to the E.M.F. in the main circuit.

If a flow of current is produced by the induced charges, energy is consumed proportional to the square of the charge ; that is, to the square of the E.M.F.

These induced charges, reacting upon the main conduc- tor, influence therein charges of equal but opposite phase, and hence lagging behind the main E.M.F. by the angle of lag between induced charge and inducing field. They require the expenditure of a charging current in the main conductor in quadrature with the induced charge thereon ; that is, nearly in quadrature with the E.M.F., and hence consisting of an energy component in phase with the E.M.F. — representing the power consumed by electrostatic influence — and a wattless component, which increases the capacity of the conductor, or, in other words, reduces its capacity reactance, or condensance.

Thus, the electrostatic influence introduces an effective conductance, g, and an effective susceptance, b, — of the same sign with condenser susceptance, — into the equations of the electric circuit.

While theoretically g and b should be constants of the circuit, frequently they are very far from such, due to disruptive phenomena beginning to appear at high electro- static stresses.

Even the capacity condensance changes at very high potentials ; escape of electricity into the air and over the surfaces of the supporting insulators by brush discharge or electrostatic glow takes place. As far as this electrostatic

148 ALTERNATING-CURRENT PHENOMENA

corona reaches, the space is in electric connection with the conductor, and thus the capacity of the circuit is deter- mined, not by the surface of the metallic conductor, but by the exterior surface of the electrostatic glow surround- ing the conductor. This means that with increasing po- tential, the capacity increases as soon as the electrostatic corona appears ; hence, the condensance decreases, and at the same time an energy component appears, representing the loss of power in the corona.

This phenomenon thus shows some analogy with the de- crease of magnetic inductance due to saturation.

At moderate potentials, the condensance due to capacity can be considered as a constant, consisting of a wattless component, the condensance proper, and an energy com- ponent, the dielectric hysteresis.

The condensance of a polarization cell, however, begins to decrease at very low potentials, as soon as the counter E.M.F. of chemical dissociation is approached.

The condensance of a synchronizing alternator is of the nature of a variable quantity ; that is, the effective reactance changes gradually, according to the relation of impressed and of counter E.M.F., from inductance over zero to condensance.

Besides the phenomena discussed in the foregoing as terms of the energy components and the wattless compo- nents of current and of E.M.F., the electric leakage is to be considered as a further energy component ; that is, the direct escape of current from conductor to return con- ductor through the surrounding medium, due to imperfect insulating qualities. This leakage current represents an effective conductance, g, theoretically independent of the E.M.F., but in reality frequently increasing greatly with the E.M.F., owing to the decrease of the insulating strength of the medium upon approaching the limits of its disruptive strength.

• FOUCAULT OR EDDY CURRENTS. 149

101. In the foregoing, the phenomena causing loss of energy in an alternating-current circuit have been dis- cussed ; and it has been shown that the mutual relation between current and E.M.F. can be expressed by two of the four constants :

Energy component of E.M.F., in phase with current, and =

current X effective resistance, or r ; wattless component of E.M.F., in quadrature with current, and =

current 'X effective reactance, or x • energy component of current, in phase with E.M.F., and =

E.M.F. X effective conductance, or g ; wattless component of current, in quadrature with E.M.F., and =

E.M.F. X effective susceptance, or b.

In many cases the exact calculation of the quantities, r, x, g, b, is not possible in the present state of the art.

In general, r, x, g, b, are not constants of the circuit, but depend — besides upon the frequency — more or less upon E.M.F., current, etc. Thus, in each particular case it be- comes necessary to discuss the variation of r, x, g, b, or to determine whether, and through what range, they can be assumed as constant.

In what follows, the quantities r, x, g, b, will always be considered as the coefficients of the energy and wattless components of current and E.M.F., — that is, as the effec- tive quantities, — so that the results are directly applicable to the general electric circuit containing iron and dielectric losses.

Introducing now, in Chapters VII. to IX., instead of " ohmic resistance," the term " effective resistance," etc., as discussed in the preceding chapter, the results apply also — within the range discussed in the preceding chapter — to circuits containing iron and other materials producing energy losses outside of the electric conductor.

150 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XII.

POWER, AND DOUBLE FREQUENCY QUANTITIES IN GENERAL.

102. Graphically alternating currents and E.M.F's are represented by vectors, of which the length represents the intensity, the direction the phase of the alternating wave. The vectors generally issue from the center of co-ordinates.

In the topographical method, however, which is more convenient for complex networks, as interlinked polyphase circuits, the alternating wave is represented by the straight line between two points, these points representing the abso- lute values of potential (with regard to any reference point chosen as co-ordinate center) and their connection the dif- ference of potential in phase and intensity.

Algebraically these vectors are represented by complex quantities. The impedance, admittance, etc., of the circuit is a complex quantity also, in symbolic denotation.

Thus current, E.M.F., impedance, and admittance are related by multiplication and division of complex quantities similar as current, E.M.F., resistance, and conductance are related by Ohms law in direct current circuits.

In direct current circuits, power is the product of cur- rent into E.M.F. In alternating current circuits, if

The product,

P0 = EI= (Ml - *"/") +j (W

POWER, AND DOUBLE FREQUENCY QUANTITIES. 151

is not the power; that is, multiplication and division, which are correct in the inter-relation of current, E.M.F., impe- dance, do not give a correct result in the inter-relation of E.M.F., current, power. The reason is, that El are vec- tors of the same frequency, and Z a constant numerical factor which thus does not change the frequency.

The power P, however, is of double frequency compared with E and /, that is, makes a complete wave for every half wave of E or 7, and thus cannot be represented by a vector in the same diagram with E and /.

P0 = E I is a quantity of the same frequency with E and /, and thus cannot represent the power.

\

103. Since the power is a quantity of double frequency of E and /, and thus a phase angle w in E and / corre- sponds to a phase angle 2 w in the power, it is of interest to investigate the product E I formed by doubling the phase angle.

Algebraically it is,

P=EI= (* +>") (V1 +/z n) =

Since j* = - 1, that is 180° rotation for E and /, for the double frequency vector, P,j* = + 1, or 360° rotation, and

j x 1 =j 1 x>= -j

That is, multiplication with / reverses the sign, since it denotes a rotation by 180° for the power, corresponding to a rotation of 90° for E and /.

Hence, substituting these values, we have,

p = [El] = (W1 + ^V11) +/ (W1 - A'u)

The symbol [E /] here denotes the transfer from the frequency of E and / to the double frequency of P.

152 AL TERNA TING-CURRENT PHENOMEMA.

The product, P = \E /] consists of two components ; the real component,

JP1 = [EIJ = (W1 + e"in) and the imaginary component,

JPJ =j The component,

P1

is the power of the circuit, = E I cos (E /) The component, PJ =

is what may be called the " wattless power," or the power- less or quadrature volt-amperes of the circuit, = E /sin (El}.

The real component will be distinguished by the index 1, the imaginary or wattless component by the index/.

By introducing this symbolism, the power of an alternat- ing circuit can be represented in the same way as in the direct current circuit, as the symbolic product of current and E.M.F.

Just as the symbolic expression of current and E.M.F. as complex quantity does not only give the mere intensity, but also the phase,

£ =

jfc ==

P tan <f> = -j

so the double frequency vector product P = [E /] denotes more than the mere power, by giving with its two compo- nents P1 = [E I]1 and PJ = [E /]•>, the true energy volt- amperes, and the wattless volt-amperes.

If

E =

POWER, AND DOUBLE FREQUENCY QUANTITIES. 153

then

and

P1 =

or

2 2 22 22 22 22

+PJ =<* ,1 + *" /

where ^ = total volt amperes of circuit. That is,

The true power P1 and the wattless power P$ are the two rectangular components of the total apparent power Q of the circuit.

Consequently,

In symbolic representation as double freqi'ency vector pro- ducts, powers can be combined and resolved by the parallelo- gram of vectors just as currents and E.M.F's in graphical or symbolic representation.

The graphical methods of treatment of alternating cur- rent phenomena are here extended to include double fre- quency quantities as power, torque, etc.

P1

— =p = cos w = power factor.

PJ

— = q = sin w = inductance factor

of the circuit, and the general expression of power is,

= Q (cos co --j sin o>)

104. The  introduction  of  the  double  frequency  vector 
     

product P = \E I~\ brings us outside of the limits of alge-

154 ALTERNATING-CURRENT PHENOMENA.

bra, however, and the commutative principle of algebra, a X b = b X a, does not apply any more, but we have,

[El] unlike [IE] since

we have

[EIJ = [IEJ

[EI]J=-[IE]J

that is, the imaginary component reverses its sign by the interchange of factors.

The physical meaning is, that if the wattless power [E 7p is lagging with regard to E, it is leading with regard to/.

The wattless component of power is absent, or the total apparent power is true power, if

[EI]J = (W1 - A'11) = 0. that is,

or,

tan (E) = tan (/),

that is, E and / are in phase or in opposition.

The true power is absent, or the total apparent power wattless, if

[El]1 = (W1 + M* = 0

that is,

*" _ i1

7 ~ ~/» or,

tan E = — cot I

that is, E and / are in quadrature,

POWER, AND DOUBLE FREQUENCY QUANTITIES. 155

The wattless power is lagging (with regard to E or lead- ing with regard to /) if,

and leading if,

The true power is negative, that is, power returns, if,

We have,

[£, - 7] = [- E, 7] = -

that is, when representing the power of a circuit or a part of a circuit, current and E.M.F. must be considered in their proper relative phases, but their phase relation with the re- maining part of the circuit is immaterial. We have further

\EJT\ = -j [£, 7] = [E, iy -j \E, 7]1 \JE, 7] =j [E, 7] = - [E, Jy +j [E, 7]1 \jEjr\ = [£, 7] = [E7? +j [E, jy

105. If      7-  =  [^/J,        7>2  =  [E2/2]  .  .  .  Pn  =  [Enln} 
     

are the symbolic expressions of the power of the different parts of a circuit or network of circuits, the total power of the whole circuit or network of circuits is

7^' = TV + T'ijJ. . • • + TV

In other words, the total power in symbolic expression (true as well as wattless) of a circuit or system is the sum of the powers of its individual components in symbolic expression.

The first equation is obviously directly a result from the law of conservation of energy.

156 ALTERNATING-CURRENT PHENOMENA.

One result derived herefrom is for instance : If in a generator supplying power to a system the cur- rent is out of phase with the E.M.F. so as to give the watt- less power Pi, the current can be brought into phase with the generator E.M.F., or the load on the generator made non-inductive by inserting anywhere in the circuit an appa- ratus producing the wattless power — F$\ that is, compen- sation for wattless currents in a system takes place regardless of the location of the compensating device.

Obviously between the compensating device and the source of wattless currents to be compensated for, wattless currents will flow, and for this reason it may be advisable to bring the compensator as near as possible to the circuit to be compensated.

106. Like power, torque in alternating apparatus is a double frequency vector product also, of magnetism and M.M.F. or current, and thus can be treated in the same way.

In an induction motor, for instance, the torque is the product of the magnetic flux in one direction into the com- ponent of secondary induced current in phase with the magnetic flux in time, but in quadrature position therewith in space, times the number of turns of this current, or since the induced E.M.F. is in quadrature and proportional to the magnetic flux and the number of turns, the torque of the induction motor is the product of the induced E.M.F. into the component of secondary current in quadrature therewith in time and space, or the product of the induced current into the component of induced E.M.F. in quadra- ture therewith in time and space.

Thus if

E1 = £ +jea- — induced E.M.F. in one direction in space.

72 = z1 +j z11 = secondary current in the quadrature di- rection in space,

POWER, AND DOUBLE FREQUENCY QUANTITIES. 157

the torque is

By this equation the torque is given in watts, the mean- ing being that T = \E /]•>' is the power which would be exerted by the torque at synchronous speed, or the torque in synchronous watts.

The torque proper is then

where

/ = number of pairs of poles of the motor.

In the polyphase induction motor, if 72 = il +/zu is the secondary current in quadrature position, in space, to E.M.F. Ej.

The current in the same direction in space as El is /! =y72 = — z11 +//1; thus the torque can also be ex- pressed as

158 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XIII.

DISTRIBUTED CAPACITY, INDUCTANCE, RESISTANCE, AND LEAKAGE.

107. As far as capacity has been considered in the foregoing chapters, the assumption has been made that the condenser or other source of negative reactance is shunted across the circuit at a definite point. In many cases, how- ever, the capacity is distributed over the whole length of the conductor, so that the circuit can be considered as shunted by an infinite number of infinitely small condensers infi nitely near together, as diagrammatically shown in Fig. 83.

iiiimiiiiumiiiT

TTTTTTTTTT.TTTTTTTTTT

i

Fig. 83. Distributed Capacity.

In this case the intensity as well as phase of the current, and consequently of the counter E.M.F. of inductance and resistance, vary from point to point ; and it is no longer possible to treat the circuit in the usual manner by the vector diagram.

This phenomenon is especially noticeable in long-distance lines, in underground cables, and to a certain degree in the high-potential coils of alternating-current transformers for very high voltage. It has the effect that not only the E.M.Fs., but also the currents, at the beginning, end, and different points of the conductor, are different in intensity and in phase.

Where the capacity effect of the line is small, it may with sufficient approximation be represented by one con-

DISTRIBUTED CAPACITY. 159

denser of the same capacity as the line, shunted across the line. Frequently it makes no difference either, whether this condenser is considered as connected across the line at the generator end, or at the receiver end, or at the middle.

The best approximation is to consider the line as shunted at the generator and at the motor end, by two condensers of \ the line capacity each, and in the middle by a con- denser of | the line capacity. This approximation, based on Simpson's rule, assumes the variation of the electric quantities in the line as parabolic. If, however, the capacity of the line is considerable, and the condenser current is of the same magnitude as the main current, such an approxi- mation is not permissible, but each line element has to be considered as an infinitely small condenser, and the differ- ential equations based thereon integrated. Or the pheno- mena occurring in the circuit can be investigated graphically by the method given in Chapter VI. § 37, by dividing the circuit into a sufficiently large number of sections or line elements, and then passing from line element to line element, to construct the topographic circuit characteristics.

108. It is thus desirable to first investigate the limits of applicability of the approximate representation of the line by one or by three condensers.

Assuming, for instance, that the line conductors are of 1 cm. diameter, and at a distance from each other of 50 cm., and that the length of transmission is 50 km., we get the capacity of the transmission line from the formula —

C = 1.11 X 10 -«K/ -=- 4 loge 2 d/ 8 microfarads, where

K = dielectric constant of the surrounding medium = 1 in air ;

/ = length of conductor = 5 x 106 cm. ;

d = distance of conductors from each other = 50 cm. ;

8 = diameter of conductor = 1 cm.

Since C = .3 microfarads,

the capacity reactance is x — 106 / 2 TT NC ohms,

160 ALTERNATING-CURRENT PHENOMENA.

where N '= frequency; hence, at N = 60 cycles,

x = 8,900 ohms ;

and the charging current of the line, at E = 20,000 volts, becomes, ^ = E / x = 2.25 amperes.

The resistance of 100 km of line of 1 cm diameter is 22 ohms ; therefore, at 10 per cent = 2,000 volts loss in the line, the main current transmitted over the line is

2,000 / = -^- = 91 amperes,

representing about 1,800 kw.

In this case, the condenser current thus amounts to less than 2^ per cent., and hence can still be represented by the approximation of one condenser shunted across the line.

If the length of transmission is 150 km., and the voltage, 30,000,

capacity reactance at 60 cycles, x = 2,970 ohms ;

charging current, i0 = 10.1 amperes ;

line resistance, r = 66 ohms ;

main current at 10 per cent loss, 7= 45.5 amperes.

The condenser current is thus about 22 per cent of the main current, and the approximate calculation of the effect of line capacity still fairly accurate.

At 300 km length of transmission it will, at 10 per cent, loss and with the same size of conductor, rise to nearly 90 per cent, of the main current, thus making a more explicit investigation of the phenomena in the line necessary.

In most cases of practical engineering, however, the ca- pacity effect is small enough to be represented by the approx- imation of one ; viz., three condensers shunted across the line.

109. A.} Line capacity represented by one condenser shunted across middle of line.

Let —

Y = g + j b = admittance of receiving circuit ; z = r — j x = impedance of line ; be = condenser susceptance of line.

DISTRIBUTED CAPACITY.

161

Denoting, in Fig. 84,

the E.M.F., viz., current in receiving circuit by £, It

the E.M.F. at middle of line by £',

the E.M.F., viz., current at generator by E0)I0\

If

We have,

Fig. 84. Capacity Shunted across Middle of Line.

. = I-jbcE'

E\ \ (r

Jbe(r-Jx) ., (r-jxy(

~~

or, expanding,

[(* - bc} - (rg+

-jx)

I (r-jx)(g+jt)-} 2 Jf

110. ^.)  Z«W  capacity  represented  by  three  condensers^ 
     

in the middle and at the ends of the line. Denoting, in Fig. 85,

the E.M.F. and current in receiving circuit by £, 7,

the E.M.F. at middle of line by £' ',

162

ALTERNATING-CURRENT PHENOMENA.

the current on receiving side of line by /', the current on generator side of line by 7", the E.M.F., viz., current at generator by £0, f0,

Iff

_L I

85. Distributed  Capacity. 
    

otherwise retaining the same denotations as in A.), We have, 7 =

2" = 1' -

As will be seen, the first terms in the expression of E0 and of I0 are the same in A.) and in B.).

DISTRIBUTED CAPACITY. 163

111. C.) Complete investigation of distributed capacity, inductance, leakage, and resistance.

In some cases, especially in very long circuits, as in lines conveying alternating power currents at high potential over extremely long distances by overhead conductors or un- derground cables, or with very feeble currents at extremely high frequency, such as telephone currents, the consideration of the line resistance — which consumes E.M.Fs. in phase with the current — and of the line reactance — which con- sumes E.M.Fs. in quadrature with the current — is not sufficient for the explanation of the phenomena taking place in the line, but several other factors have to be taken into account.

In long lines, especially at high potentials, the electro- static capacity of the line is sufficient to consume noticeable currents. The charging current of the line condenser is proportional to the difference of potential, and is one-fourth period ahead of the E.M.F. Hence, it will either increase or decrease the main current, according to the relative phase of the main current and the E.M.F.

As a consequence, the current will change in intensity as well as in phase, in the line from point to point ; and the E.M.Fs. consumed by the resistance and inductance will therefore also change in phase and intensity from point to point, being dependent upon the current.

Since no insulator has an infinite resistance, and as at high potentials not only leakage, but even direct escape of electricity into the air, takes place by " silent discharge," we have to recognize the existence of a current approximately proportional and in phase with the E.M.F. of the line. This current represents consumption of energy, and is therefore analogous to the E.M.F. consumed by resistance, while the condenser current and the E.M.F. of inductance are wattless.

Furthermore, the alternate current passing over the line induces in all neighboring conductors secondary currents,

164 ALTERNATING-CURRENT PHENOMENA.

which react upon the primary current, and thereby intro- duce E.M.Fs. of mutual inductance into the primary circuit. Mutual inductance is neither in phase nor in quadrature with the current, and can therefore be resolved into an energy component of mutual inductance in phase with the current, which acts as an increase of resistance, and into a wattless component in quadrature with the current, which decreases the self-inductance.

This mutual inductance is not always negligible, as, for instance, its disturbing influence in telephone circuits shows.

The alternating potential of the line induces, by electro- static influence, electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges on the line wires. This electrostatic influence re- quires the expenditure of a current proportional to the E.M.F., and consisting of an energy component, in phase with the E.M.F., and a wattless component, in quadrature thereto.

The alternating electromagnetic field of force set up by the line current produces in some materials a loss of energy by magnetic hysteresis, or an expenditure of E.M'.F. in phase with the current, which acts as an increase of re- sistance. This electromagnetic hysteretic loss may take place in the conductor proper if iron wires are used, and will then be very serious at high frequencies, such as those of telephone currents.

The effect of eddy currents has already been referred to under "mutual inductance," of which it is an energy component.

The alternating electrostatic field of force expends energy in dielectrics by what is called dielectric hysteresis. In concentric cables, where the electrostatic gradient in the dielectric is comparatively large, the dielectric hysteresis may at high potentials consume considerable amounts of energy. The dielectric hysteresis appears in the circuit

DISTRIBUTED CAPACITY. 165

as consumption of a current, whose component in phase with the E.M.F. is the dielectric energy current, which may be considered as the power component of the capacity current.

Besides this, there is the increase of ohmic resistance due to unequal distribution of current, which, however, is usually not large enough to be noticeable.

112. This gives, as the most general case, and per unit length of line :

E.M.Fs. consumed in phase with the current I, and = rl, representing consumption of energy, and due to : Resistance, and its increase by unequal current distri- tribution ; to the energy component of mutual inductance; to induced currents ; to the energy component of self-inductance ; or to electromag- netic hysteresis. E.M.Fs. consumed in quadrature with the current I, and

= x I, wattless, and due to : Self-inductance, and Mutual inductance. Currents consumed in phase with the E.M.F., E, and = gE, representing consumption of energy, and due to :

Leakage through the insulating material, including silent discharge; energy component of electro- static influence ; energy component of capacity, or of dielectric hysteresis. Currents consumed in quadrature to the E.M.F., E, and

= bE, being wattless, and due to : Capacity and Electrostatic influence.

Hence we get fo'ur constants : —

Effective resistance, r, Effective reactance, x, Effective conductance, g, Effective susceptance, b — — bc,

1GG ALTERNATING-CURRENT PHENOMENA.

per unit length of line, which represent the coefficients, per unit length of line, of

E.M.F. consumed in phase with current ; E.M.F. consumed in quadrature with current ; Current consumed in phase with E.M.F. ; Current consumed in quadrature with E.M.F.

113. This line we may assume now as feeding into a receiver circuit of any description, and determine the current and E.M.F. at any point of the circuit.

That is, an E.M.F, and current (differing in phase by any desired angle) may be given at the terminals of receiving cir- cuit. To be determined are the E.M.F. and current at any point of the line ; for instance, at the generator terminals. Or, Zl=rl— JXl ;

the impedance of receiver circuit, or admittance,

and E.M.F., E0, at generator terminals are given. Current and E.M.F. at any point of circuit to be determined, etc.

114. Counting now the distance, x, from a point, 0, of the line which has the E.M.F.,

•Ei = e\ + Je\i and the current : /i = i\ +///,

and counting x positive in the direction of rising energy, and negative in the direction of decreasing energy, we have at any point, X, in the line differential, dx :

Leakage current : JEgdx', Capacity current : — j E bc d x ;

hence, the total current consumed by the line element, dx, is dl= E(g-jbc}d*, or,

d-t=E(g-jbc\ (1)

E.M.F. consumed by resistance, Ird*\ E.M.F. consumed by reactance, — j

DISTRIBUTED CAPACITY. 107

hence, the total E.M.F. consumed in the line element, ^/x, is dE = I (r — j'x) </x, or, ffi. -I(f-jx). (2)

These fundamental differential equations :

*L-E(g-jt,),\ (1)

(2)

are symmetrical with respect to / and E. Differentiating these equations : d*I dE ,

and substituting (1) and (2) in (3), we get :

(4)

(5) the differential equations of E and L

115. These differential equations are identical, and con- sequently I and E are functions differing by their limiting conditions only.

These equations, (4) and (5), are of the form :

(6)

and are integrated by

W = tf 6rx,

where e is the basis of natural logarithms ; for, differen- tiating this, we get,

168 ALTERNATING-CURRENT PHENOMENA.

hence, z>2 = (g — j bc) (r — jx) ;

(7)

or, v = ± V (g - Jbe) (r — joe) \

hence, the general integral is :

tr*.«e+«-M«r«« (8)

where a and b are the two constants of integration ; Substituting

r-«-/0 (9)

into (7), we have,

(a -JP)* = (g - jbc) (r - jx) ; or,

therefore, _ f

);-' (10)

Vl/2 6 - e

/3= Vl/2 substituting (9) into (8) :

= a-cax (cos/3x — /sin^Sx) + ^cax (cos/3x +y sin/3x) ; «/ = (a£«x + /5>eax) cos)8x — y(aeax — ^«-ax) sin /3x (12)

which is the general solution of differential equations (4) and (5)

Differentiating (8) gives :

hence, substituting, (9) : (a —JP) {(a

x}. (13)

Substituting now / for w, and substituting (13) in (1), and writing,

DISTRIBUTED CAPACITY.

169

we get,

/• ( Jfax. i >

?e-«)cosj8x-y(y ?«-«)cos/8x-y(y

-•

• — •/_> < \ ' a — 7/5

sin /2x} ;

'** 1 K^" i

S — J^c sin ySxf ;

where ^4 and ^ are the constants of integration. Transformed, we get,

/= J Aea* (cos )8x — j sin 0x) + Bf.~™

a — JP ( '

(cos /?x +/ sin /8x) > 1

^4eax (cos /8x — y sin

^-.

(cos /3x +y sin y8x)

Thus the waves consist of two components, one, with factor ^eax, increasing in amplitude toward the generator, the other, with factor ^e-ax, decreasing toward the genera- tor. The latter may be considered as a reflected wave.

At the point x = 0.

a-j/3 A-B

n

Thus m (cos to — j sin G) = -—

and,

m = amplitude.

w = angle of reflection.

These are the general integral equations of the problem.

116. If —

/! = /! + /// is the current { is the E.M.F.

at point, x

(15)

170 ALTERNATING-CURRENT PHENOMENA.

by substituting (15) in (14), we get : 2 A = {(a t\ + ft //) + (gev + bc ^')

(16)

2 B = {(a /! + /? //) - (ge, + /;c ,/)}

• /{(«//- 0/0 -(^I'-^ a and ft being determined by equations (11).

117. H Z — R — j X is the impedance of the receiver circuit, E0 = e0 + j >0' is the E.M.F. at dynamo terminals (17), and / = length of line, we get at

hence

g — jbc or

a-; ft At X = /,

E0

sin/?/}. (19)

Equations (18) and (19) determine the constants A and B, which, substituted in (14), give the final integral equations.

The length, X0 = 2 TT / ft is a complete wave length (20), ,vhich means, that in the distance 2 IT / ft the phases of the components of current and E.M.F. repeat, and that in half this distance, they are just opposite.

Hence the remarkable condition exists that, in a very long line, at different points the currents at the same time flow in opposite directions, and the E.M.Fs. are opposite.

118. The difference of phase, w, between current, /, and E.M.F., Ey at any point, x, of the line, is determined by

DISTRIBUTED CAPACITY. 171

the equation,

Z?(cos«+/sin£) =y, : \j JsTI71

where Z> is a constant.

Hence, w varies from point to point, oscillating around a medium position, wx, which it approaches at infinity.

This difference of phase, C>x, towards which current and E.M.F. tend at infinity, is determined by the expression,

^(cos . .. , (/

or, substituting for E and /their values, and since e~a* = 0, and A eax (cos ft x — j sin ft x), cancels, and

D (cos tow +/sin oioc) = — 2-p-

hence, tan ^ = ~a° c + ^ • (21)

This angle, Stx, = 0 ; that is, current and E.M.F. come more and more in phase with each other, when

abc — fig — 0 ; that is,

a -T- ft — g -r- bc , or,

2a/3 !2^/ 5 substituting (10), gives,

hence, expanding, r -4- ^ = ^ -f- ^c ; (22)

that is, tJie ratio of resistance to inductance equals the ratio of leakage to capacity.

This angle, wx, = 45° ; that is, current and E.M.F. differ by £th period, if — a bc + fig = a.g + pbc ; or,

which gives : rg + x bc = 0. (23)

172

ALTERNA TING-CURRENT PHENOMENA.

That is, two of the four line constants must be zero; cither g and x, or g and bc.

The case where g = 0 = x, that is a line having only resistance and distributed capacity, but no self-induction is approximately realized in concentric or multiple conductor cables, and in these the phase angle tends towards 45° lead for infinite length.

119. As an instance, in Fig. 86 a line diagram is shown, with the distances from the receiver end as abscissae. The diagram represents one and one-half complete waves, and gives total effective current, total E.M.F., and differ-

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DISTRIBUTED CAPACITY. 173