Theory and Calculation of Alternating Current Phenomena (1900) — part 3 of 19

The corresponding diagram is shown in Fig. 21. Obvi- ously, no exact numerical values can be taken from a par- allelogram as flat as OF1FF0^ and from the combination of vectors of the relative magnitudes 1:6: 100.

Hence the importance of the graphical method consists

34

ALTERNA TING-CURRENT PHENOMENA.

not so much in its usefulness for practical calculation, as to aid in the simple understanding of the phenomena involved.

24. Sometimes we can calculate the numerical values trigonometrically by means of the diagram. Usually, how- ever, this becomes too complicated, as will be seen by trying

Fig. 21.

to calculate, from the above transformer diagram, the ratio of transformation. The primary M.M.F. is given by the equation : —

ffo = Vfr2 + S^2 + 20^ sin Wi,

an expression not well suited as a starting-point for further calculation.

A method is therefore desirable which combines the exactness of analytical calculation with the clearness of the graphical representation.

Fig. 22.

25. We have seen that the alternating sine wave is represented in intensity, as well as phase, by a vector, Of, which is determined analytically by two numerical quanti- ties — the length, Of, or intensity ; and the amplitude, AOf, or phase <3, of the wave, /.

Instead of denoting the vector which represents the sine wave in the polar diagram by the polar coordinates,

S YMB OL1C ME T11OD.

35

/ and <3, we can represent it by its rectangular coordinates, a and b (Fig. 22), where —

a = fcos u> is the horizontal component,

b = I sin co is the vertical component of the sine wave.

This representation of the sine wave by its rectangular components is very convenient, in so far as it avoids the use of trigonometric functions in the combination or reso- lution of sine waves.

Since the rectangular components a and b are the hori- zontal and the vertical projections of the vector represent- ing the sine wave, and the projection of the diagonal of a parallelogram is equal to the sum of the projections of its sides, the combination of sine waves by the parallelogram

law is reduced to the addition, or subtraction, of their rectangular components. That is,

Sine waves are combined, or resolved, by adding, or subtracting, their rectangular components.

For instance, if a and b are the rectangular components of a sine wave, /, and a' and b' the components of another sine wave, /' (Fig. 23), their resultant sine wave, I0, has the rectangular components a0 — (a -f- a!}, and b0 = (b -f- b'}.

To get from the rectangular components, a and b, of a sine wave, its intensity, i, and phase, o>, we may combine a and b by the parallelogram, and derive, —

tan

36 AL TERN A TING-CURRENT PHENOMENA .

Hence we can analytically operate with sine waves, as with forces in mechanics, by resolving them into their rectangular components.

26. To distinguish, however, the horizontal and the ver- tical components of sine waves, so as not to be confused in lengthier calculation, we may mark, for instance, the vertical components, by a distinguishing index, or the addition of an otherwise meaningless symbol, as the letter /, and thus represent the sine wave by the expression, —

I=a

which now has the meaning, that a is the horizontal and b the vertical component of the sine wave /; and that both components are to be combined in the resultant wave of intensity, — _

/ = V^ + //2,

and of phase, tan <3 = b / a.

Similarly, a —jb, means a sine wave with a as horizon- tal, and — b as vertical, components, etc.

Obviously, the plus sign in the symbol, a -f- jb, does not imply simple addition, since it connects heterogeneous quan- tities — horizontal and vertical components — but implies combination by the parallelogram law.

For the present,/ is nothing but a distinguishing index, and otherwise free for definition except that it is not an .ordinary number.

27. A wave of equal intensity, and differing in phase from the wave a + jb by 180°, or one-half period, is repre- sented in polar coordinates by a vector of opposite direction, and denoted by the symbolic expression, — a — jb. Or —

Multiplying the symbolic expression, a + jb, of a sine wave by — 1 weans reversing' the wave, or rotating it through 180°, or one-half period.

A wave of equal intensity, but lagging 90°, or one- quarter period, behind a -f jb, has (Fig. 24) the horizontal

SYMBOLIC METHOD. 37

component, — b, and the vertical component, a, and is rep- resented symbolically by the expression, ja — b, Multiplying, however, a + jb by/, we get : —

therefore, if we define the heretofore meaningless symbol, j, by the condition, —

y2 = - i,

we have —

/(+/) =ja — 1>;

hence : —

Multiplying the symbolic expression, a -- jb, of a sine wave by j means rotating the wave through 90°, or one-quarter pe- riod ; tJiat is, retarding the wave through one-quarter period.

Fig. 24.

Similarly, —

Multiplying by — j means advancing the wave through one-quarter period.

since y'2 = — 1, j = V— 1 ;

that is, —

j is the imaginary unit, and the sine wave is represented by a complex imaginary quantity, a -+- jb.

As the imaginary unit j has no numerical meaning in the system of ordinary numbers, this definition of/ = V— 1 does not contradict its original introduction as a distinguish- ing index. For a more exact definition of this complex imaginary quantity, reference may be made to the text books of mathematics.

28. In the polar diagram of time, the sine wave is represented in intensity as well as phase by one complex quantity —

38 ALTERNATING-CURRENT PHENOMENA.

where a is the horizontal and b the vertical component of the wave ; the intensity is given by —

the phase by —

tan <o = - , a and

a = i cos to,

b = i sin w ;

hence the wave a +jb can also be expressed by — / (cos <i> --j sin <3),

or, by substituting for cos w and sin w their exponential

expressions, we obtain —

id™.

Since we have seen that sine waves may be combined or resolved by adding or subtracting their rectangular com- ponents, consequently : —

Sine waves may be combined or resolved by adding or subtracting their complex algebraic expressions.

For instance, the sine waves, —

a +jb and

combined give the sine wave —

7- (a +

It will thus be seen that the combination of sine waves is reduced to the elementary algebra of complex quantities.

29. If /= i +/z' is a sine wave of alternating current, and r is the resistance, the E.M.F. consumed by the re- sistance is in phase with the current, and equal to the prod- uct of the current and resistance. Or — rl ' — ri -- jri' .

If L is the inductance, and x = 2 TT NL the reactance, the E.M.F. produced by the reactance, or the counter

SYMBOLIC METHOD. 39

E.M.F. of self-induction, is the product of the current and reactance, and lags 90° behind the current ; it is, therefore, represented by the expression —

The E.M.F. required to overcome the reactance is con- , sequently 90° ahead of the current (or, as usually expressed,-** the current lags 90° behind the E.M.F.), and represented by the expression —

— jxl = — jxi -f- xi'.

Hence, the E.M.F. required to overcome the resistance, r, and the reactance, x, is —

that is —

Z = r — jx is the expression of the impedance of the cir- cuit, in complex quantities.

Hence, if / = i --ji' is the current, the E.M.F. required to overcome the impedance, Z = r — jx, is —

hence, sincey"2 = — 1

or, if E = e -- je' is the impressed E.M.F., and Z = r — jx the impedance, the current flowing through the circuit is : —

or, multiplying numerator and denominator by (r+jx) to eliminate the imaginary from the denominator, we have —

T _

or, if E = e --je' is the impressed E.M.F., and 7 = i ' -- ji' the current flowing in the circuit, its impedance is —

0 +./>') O'-./'') «'+^'' . ' ~ ei'

'

40 ALTERNATING-CURRENT PHENOMENA.

30. If C is the capacity of a condenser in series in a circuit of current I = i + //', the E.M.F. impressed upon

the terminals of the condenser is E = - - , 90° behind the current ; and may be represented by — - - , or jx^ /,

where x^ = - is the capacity reactance or condensatice 2 TT NC

of the condenser.

Capacity reactance is of opposite sign to magnetic re- actance ; both may be combined in the name reactance.

We therefore have the conclusion that

If r = resistance and L = inductance,

then x = 2 IT NL = magnetic reactance.

If C = capacity, x^ = - = capacity reactance, or conden- sance ;

Z = r — j (x — JCi), is the impedance of the circuit Ohm's law is then reestablished as follows :

, -, .

The more general form gives not only the intensity of the wave, but also its phase, as expressed in complex quantities.

31. Since the combination of sine waves takes place by the addition of their symbolic expressions, Kirchhoff's laws are now reestablished in their original form : —

a.} The sum of all the E.M.Fs. acting in a closed cir- cuit equals zero, if they are expressed by complex quanti- ties, and if the resistance and reactance E.M.Fs. are also considered as counter E.M.Fs.

b.) The sum of all the currents flowing towards a dis- tributing point is zero, if the currents are expressed as complex quantities.

SYMBOLIC METHOD. 41

If a complex quantity equals zero, the real part as well as the imaginary part must be zero individually, thus if a +jb = 0, a = 0, b = 0.

Resolving the E.M.Fs. and currents in the expression of Kirchhoff 's law, we find : —

a.} The sum of the components, in any direction, of all the E.M.Fs. in a closed circuit, equals zero, 'if the resis- tance and reactance are considered as counter E.M.Fs.

b.} The sum of the components, in any direction, of all the currents flowing to a distributing point, equals zero.

Joule's Law and the energy equation do not give a simple expression in complex quantities, since the effect or power is a quantity of double the frequency of the current or E.M.F. wave, and therefore requires for its representa- tion as a vector, a transition from single to double fre- quency, as will be shown in chapter XII.

In what follows, complex vector quantities will always be denoted by dotted capitals when not written out in full ; absolute quantities and real quantities by undotted letters.

32. Referring to the instance given in the fourth chapter, of a circuit supplied with an E.M.F., E, and a cur- rent, 7, over an inductive line, we can now represent the impedance of the line by Z = r — jx, where r = resistance, x = reactance of the line, and have thus as the E.M.F. at the beginning of the line, or at the generator, the

expression —

E0 = E + ZI.

Assuming now again the current as the zero line, that is, / = /, we have in general —

E0 = E -f ir —jix ;

hence, with non-inductive load, or E = e, E0=(e + ir) -jix,

• /r)2 + (/X)2, tan S>0 =

42 ALTERNATING-CURRENT PHENOMENA.

In a circuit with lagging current, that is, with leading E.M.F., E = e -je', and

-)2> tan <S0

e + />

In a circuit with leading current, that is, with lagging E.M.F., E = * +>', and

— /V) , tan w0 = values which easily permit calculation.

TOPOGRAPHIC METHOD. 43

CHAPTER VI.

TOPOGRAPHIC METHOD.

33. In the representation of alternating sine waves by vectors in a polar diagram, a certain ambiguity exists, in so far as one and the same quantity — an E.M.F., for in- stance — can be represented by two vectors of opposite direction, according as to whether the E.M.F. is considered as a part of the impressed E.M.F., or as a counter E.M.F. This is analogous to the distinction between action and reaction in mechanics.

Further, it is obvious that if in the circuit of a gener- ator, G (Fig. 25), the current flowing from terminal A over resistance R to terminal B, is represented by a vector OI (Fig. 26), or by /= i --ji', the same current can be con- sidered as flowing in the opposite direction, from terminal B to terminal A in opposite phase, and therefore represented by a vector OI-± (Fig. 26), or by 7l = — i —ji'>

Or, if the difference of potential from terminal B to terminal A is denoted by the E = e + je' , the difference of potential from A to B is El = — e — je' .

44

ALTERNA TING-CURRENT PHENOMENA.

Hence, in dealing with alternating-current sine waves, it is necessary to consider them in their proper direction with regard to the circuit. Especially in more complicated circuits, as interlinked polyphase systems, careful attention has to be paid to this point.

-*'

Fig. 28.

34. Let, for instance, in Fig. 27, an interlinked three- phase system be represented diagrammatically, as consist- ing of three E.M.Fs., of equal intensity, differing in phase by one-third of a period. Let the E.M.Fs. in the direction

Fig. 27

from the common connection O of the three branch circuits to the terminals A19 A2,AB, be represented by Elt E2, £3. Then the difference of potential from A2 to A± is £z — £lf

since the two E.M.Fs., El and

are connected in cir-

cuit between the terminals A, and A*, in the direction,

TOPOGRAPHIC METHOD. 45

Al — O — A2; that is, the one, Ez, in the direction OA2, from the common connection to terminal, the other, JS1, in the opposite direction, A^O, from the terminal to common connection, and represented by — El. Conversely, the dif- ference of potential from A1 to Az is El — Ez.

It is then convenient to go still a step farther, and drop, in the diagrammatic representation, the vector line altogether ; that is, denote the sine wave by a point only,, the end of the corresponding vector.

" Looking at this from a different point of view, it means that we choose one point of the system — for instance, the common connection O — as a zero point, or point of zero potential, and represent the potentials of all the other points of the circuit by points in the diagram, such that their dis- tances from the zero point gives the intensity ; their ampli- tude the phase of the difference of potential of the respective point with regard to the zero point ; and their distance and amplitude with regard to other points of the diagram, their difference of potential from these points in intensity and phase.

Fig. 28.

Thus, for example, in an interlinked three-phase system with three E.M.Fs. of equal intensity, and differing in phase by one-third of a period, we may choose the common con- nection of the star-connected generator as the zero point, and represent, in Fig. 28, one of the E.M.Fs., or the poten-

46

AL TERN A TING-CURRENT PHENOMEMA.

tial at one of the three-phase terminals, by point Er The potentials at the two other terminals will then be given by the points Ez and E& which have the same distance from O as Ev and are equidistant from E± and from each other. The difference of potential between any pair of termi- nals — for instance E^ and E2 — is then the distance EZEV or E±EV according to the direction considered.

35. If now the three branches OEV ~OEZ and "OEW of the three-phase system are loaded equally by three currents equal in intensity and in difference of phase against their

THUEE-PHA8E 8V8TEM 48° LAO

BALANCED THREE-PHASE SYSTEM

NON-INDUCTIVE LOAD E°

Fig. 29.

E.M.Fs., these currents are represented in Fig. 29 by the vectors 07^ = 072 = Ofs = I, lagging behind the E.M.Fs. by angles E.O^ = EZOIZ = EZOI& = Q.

Let the three-phase circuit be supplied over a line of impedance Z± = r^ —jx\ from a generator of internal im- pedance Z0 = x0 -jx0.

In phase OEV the E.M.F. consumed by resistance r^ is represented by the distance E^EJ = Irv in phase, that is parallel with current OIV The E.M.F. consumed by re- actance #! is represented by E^Ej' = Ixv 90° ahead of cur-

TOPOGRAPHIC METHOD.

47

rent OIr The same applies to the other two phases, and it thus follows that to produce the E.M.F. triangle E^E^E^ at the terminals of the consumer's circuit, the E.M.F. tri- angle E^E^E? is required at the generator terminals.

Repeating the same operation for the internal impedance of the generator we get E"E'" = Iroi and parallel to OIV E'"E° = Ixoy and 90° ahead of ~OTV and thus as triangle of (nominal) induced E.M.Fs. of the generator E°E£E°.

In Fig. 29, the diagram is shown for 45° lag, in Fig. 30 for noninductive load, and in Fig. 31 for 45° lead of the currents with regard to their E.M.Fs.

BALANCED THREE -PHASE SYSTEM

45° LEAD

THREE-PHASE CIRCUIT

80°LA»

TRANSMISSION LINE'

WITH DISTRIBUTED

CAPACITY, INDUCTANCB

RESISTANCE AUD LEAKAQB

•I,

Fig. 31.

Fig. 32.

As seen, the induced generator E.M.F. and thus the generator excitation with lagging current must be higher, with leading current lower, than at non-inductive load, or conversely with the same generator excitation, that is the same induced generator E.M.F. triangle E°E£E°, the E.M.Fs. at the receiver's circuit, Ev Ez, E9 fall off more with lagging, less with leading current, than with non- inductive load.

36. As further instance may be considered the case of a single phase alternating current circuit supplied over a cable containing resistance and distributed capacity.

48 ALTERNATING-CURRENT PHENOMENA.

Let in Fig. 33 the potential midway between the two terminals be assumed as zero point 0. The two terminal voltages at the receiver circuit are then represented by the points E and El equidistant from 0 and opposite each other, and the two currents issuing from the terminals are rep- resented by the points / and I1, equidistant from 0 and opposite each other, and under angle & with E and El respectively.

Considering first an element of the line or cable next to the receiver circuit. In this an E.M.F. EEl is consumed by the resistance of the line element, in phase with the current OI, and proportional thereto, and a current //x con- sumed by the capacity, as charging current of the line element, 90° ahead in phase of the E.M.F. OE and propor- tional thereto, so that at the generator end of this cable element current and E.M.F. are OI^ and OEl respectively.

Passing now to the next cable element we have again an E.M.F. E1EZ proportional to and in phase with the current OI^ and a current IJZ proportional to and 90° ahead of the E.M.F. OEV and thus passing from element to element along the cable to the generator, we get curves of E.M.Fs. e and e1, and curves of currents i and il, which can be called the topographical circuit characteristics, and which corre- spond to each other, point for point, until the generator terminal voltages OE0 and OE0l and the generator currents OI0 and OIJ are reached.

Again, adding 'E~Er' = I0r0 and parallel OI0 and E"E° = I0x0 and 90° ahead of ~OIM gives the (nominal) induced E.M.F. of the generator OE°, where Z0 = r0 — jx0 = inter- nal impedance of the generator.

In Fig. 33 is shown the circuit characteristics for 60° lag, of a cable containing only resistance and capacity.

Obviously by graphical construction the circuit character- istics appear more or less as broken lines, due to the neces- sity of using finite line elements, while in reality when calculated by the differential method they are smooth curves.

TOPOGRAPHIC METHOD.

49

37. As further instance may be considered a three-phase circuit supplied over a long distance transmission line of distributed capacity, self-induction, resistance, and leakage.

Let, in Fig. 38, O£v ~OEy ~OEZ = three-phase E.M.Fs. at receiver circuit, equidistant from each other and = E.

Let OIV Oly Of3 = three-phase currents in the receiver circuit equidistant from each other and = /, and making with E the phase angle <3.

Considering again as in § 35 the transmission line ele- ment by element, we have in every element an E.M.F. consumed by the resistance in phase with the current n^ proportional thereto, and an E.M.F. E^, Ef con-

sumed by the reactance of the line element, 90° ahead of the current OIV and proportional thereto.

In the same line element we have a current IJ^ in phase with the E.M.F. OEV and proportional thereto, representing the loss of energy current by leakage, dielectric hysteresis, etc., and a current ^V/', 90° ahead of the E.M.F. OEV and proportional thereto, the charging current of the line ele- ment as condenser, and in this manner passing along the line, element by element, we ultimately reach the generator terminal voltages E°, E°, Es°, and generator currents //, /2°, 78°, over the topographical characteristics of E.M.F. ev ev es, and of current iv z'2, z'3, as shown in Fig. 33.

The circuit characteristics of current i and of E.M.F. e

50

ALTERNATING-CURRENT PHENOMENA.

correspond to each other, point for point, the one giving the current and the other the E.M.F. in the line element.

TRANSMISSION

WITH DISTRIBUTED

CAPACITY, INDUCTANCE

RESISTANCE AND LEAKAGE 90° LAO

Fig. 34.

Only the circuit characteristics of the first phase are shown as ^ and z'r As seen, passing from the receiving end towards the generator end of the line, potential and

TRANSMISSION LINE

WITH DISTRIBUTED CAPACITY, INDUCTANCE RESISTANCE AND LEAKAGE

Fig. 35.

current alternately rise and fall, while their phase angle changes periodically between lag and lead.

TOPOGRAPHIC METHOD. 51

37. a. More markedly this is shown in Fig. 34, the topo- graphic circuit characteristic of one of the lines with 90° lag in the receiver circuit. Corresponding points of the two characteristics e and i are marked by corresponding figures 0 to 16, representing equidistant points of the line. The values of E.M.F., current and their difference of phase are plotted in Fig. 35 in rectangular co-ordinates with the distance as abscissae, counting from the receiving circuit towards the generator. As seen from Fig. 35, E.M.F. and current periodically but alternately rise and fall, a maximum of one approximately coinciding with a minimum of the other and with a point of zero phase displacement.

The phase angle between current and E.M.F. changes from 90° lag to 72° lead, 44° lag, 34° lead, etc., gradually decreasing in the amplitude of its variation.

52 ALTERNATING-CURRENT PHENOMENA.

CHAPTER VII.

ADMITTANCE, CONDUCTANCE, SUSCEPTANCE.

38. If in a continuous-current circuit, a number of resistances, ?, r%, r3, . . . are connected in series, their joint resistance, R, is the sum of the individual resistances

If, however, a number of resistances are connected in multiple or in parallel, their joint resistance, R, cannot be expressed in a simple form, but is represented by the expression : —

= J_ l JL + J_ +

/*! /*2 ^3

Hence, in the latter case it is preferable to introduce, in- stead of the term resistance, its reciprocal, or inverse value, the term conductance, g = 1 / r. If, then, a number of con- ductances, g^, g^, gz, . . . are connected in parallel, their joint conductance is the sum of the individual conductances, or G = gl + gz + gz + . . . When using the term con- ductance, the joint conductance of a number of series- connected conductances becomes similarly a complicated expression —

Hence the term resistance is preferable in case of series connection, and the use of the reciprocal term conductance in parallel connections ; therefore,

The joint resistance of a number of series-connected resis- tances is equal to the sum of the individual resistances ; the

ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 53

joint conductance of a number of parallel-connected conduc~ tances is equal to the sum of the individual conductances.

39. In alternating-current circuits, instead of the term resistance we have the term impedance, Z = r —Jx, with its two components, the resistance, r, and the reactance, x, in the formula of Ohm's law, E = IZ. The resistance, r, gives the component of E.M.F. in phase with the current, or the energy component of the E.M.F., Ir; the reactance, x, gives the component of the E.M.F. in quadrature with the current, or the wattless component of E.M.F., Ix ; both combined give the total E.M.F., —

Since E.M.Fs. are combined by adding their complex ex- pressions, we have :

The joint impedance of a number of series-connected impe- dances is the sum of the individual impedances, when expressed in complex quantities.

In graphical representation impedances have not to be added, but are combined in their proper phase by the law of parallelogram in the same manner as the E.M.Fs. corre- sponding to them.

The term impedance becomes inconvenient, however, when dealing with parallel-connected circuits ; or, in other words, when several currents are produced by the same E.M.F., such as in cases where Ohm's law is expressed in the form,

-I-

It is preferable, then, to introduce the reciprocal of impedance, which may be called the admittance of the circuit, or

-*•

As the reciprocal of the complex quantity, Z = r —jx, the admittance is a complex quantity also, or Y = g+jb;

54 ALTERNATING-CURRENT PHENOMENA.

it consists of the component g, which represents the co- efficient of current in phase with the E.M.F., or energy current, gEt in the equation of Ohm's law, —

and the component b, which represents the coefficient of current in quadrature with the E.M.F., or wattless com- ponent of current, bE.

g is called the conductance, and b the susceptance, of the circuit. Hence the conductance, g, is the energy com- ponent, and the susceptance, b, the wattless component, of the admittance, Y = g -f jb, while the numerical value of

admittance is —

y = Vr1 + P ;

the resistance, r, is the energy component, and the reactance, x, the wattless component, of the impedance, Z — r — jx, the numerical value of impedance being —

z = VV' + x\

40. As shown, the term admittance implies resolving the current into two components, in phase and in quadra- ture with the E.M.F., or the energy current and the watt- less current ; while the term impedance implies resolving the E.M.F. into two components, in phase and in quad- rature with the current, or the energy E.M.F. and the wattless E.M.F.

It must be understood, however, that the conductance is not the reciprocal of the resistance, but depends upon the resistance as well as upon the reactance. Only when the reactance x = 0, or in continuous-current circuits, is the conductance the reciprocal of resistance.

Again, only in circuits with zero resistance (r = 0) is the susceptance the reciprocal of reactance ; otherwise, the susceptance depends upon reactance and upon resistance.

The conductance is zero for two values of the resistance : —

1.) If r = QO , or x = oo , since in this case no current passes, and either component of the current = 0.

ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 55

2.) If r = 0, since in this case the current which passes through the circuit is in quadrature with the E.M.F., and thus has no energy component.

Similarly, the susceptance, b, is zero for two values of the reactance : —

1.) If x = oo , or r = oo .

2.) If * = 0.

From the definition of admittance, Y ' = g + jbt as the reciprocal of the impedance, Z = r — jxy

we have Y — — , or, g -f- jb =

Z r —jx

or, multiplying numerator and denominator on the right side

by(r

hence, since

(r-jx) (r +» = r2 + x* = z\

x r . . x

, and conversely

By these equations, the conductance and susceptance can be calculated from resistance and reactance, and conversely. • Multiplying the equations for^- and r, we get : —

gr =

hence,

an j _ 1 1 ) the absolute value of

y V^"2 + b* ' ) impedance ; 1 1 ) the absolute value of

admittance.

56

AL TERNA TING-CURRENT PHENOMENA.

41. If, in a circuit, the reactance, *-, is constant, and the resistance, r, is varied from r = 0 to r = oo , the susceptance, b, decreases from b = 1 / x at r = 0, to # = 0 at r = cc ; while the conductance, g — 0 at r = 0, increases, reaches a maximum for r = x, where g — 1 / 2 r is equal to the susceptance, or g = b, and then decreases again, reaching g = 0 at r = oo .

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In Fig. 36, for constant reactance ^- = .5 ohm, the vari- ation of the conductance, g, and of the susceptance, b, are shown as functions of the varying resistance, r. As shown, the absolute value of admittance, susceptance, and conduc- tance are plotted in full lines, and in dotted line the abso- lute value of impedance,

ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 57

Obviously, if the resistance, r, is constant, and the reac- tance, x, is varied, the values of conductance and susceptance are merely exchanged, the conductance decreasing steadily from g = 1 / r to 0, and the susceptance passing from 0 at x = 0 to the maximum, b = 1 / 2 r = g =1 / '2 x at x = r, and to b = 0 at x = GO .

The resistance, r, and the reactance, x, vary as functions of the conductance, g, and the susceptance, b, in the same manner as g and b vary as functions of r and x.

The sign in the complex expression of admittance is always opposite to that of impedance ; this is obvious, since if the current lags behind the E.M.F., the E.M.F. leads the current, and conversely.

We can thus express Ohm's law in the two forms —

E = IZ, I =£Y,

and therefore —

The joint impedance of a number of series-connected im- pedances is equal to the sum. of the individual impedances ; the joint admittance of a number of parallel-connected admit- tances, if expressed in complex quantities, is equal to the sum of the individual admittances. In diagrammatic represen- tation, combination by the parallelogram law takes the place of addition of the complex quantities.

58 ALTERNATING-CURRENT PHENOMENA.

CHAPTER VIII.

CIRCUITS CONTAINING RESISTANCE, INDUCTANCE, AND CAPACITY.

42. Having, in the foregoing, reestablished Ohm's law and Kirchhoff's laws as being also the fundamental laws of alternating-current circuits, when expressed in their com- plex form,

E = ZS, or, / = YE,

and *%E = 0 in a closed circuit,

S/ = 0 at a distributing point,

where E, I, Z, Y, are the expressions of E.M.F., current, impedance, and admittance in complex quantities, — these values representing not only the intensity, but also the phase, of the alternating wave, — we can now — by application of these laws, and in the same manner as with continuous- current circuits, keeping in mind, however, that E, I, Z, Y, are complex quantities — calculate alternating-current cir- cuits and networks of circuits containing resistance, induc- tance, and capacity in any combination, without meeting with greater difficulties than when dealing with continuous- current circuits.

It is obviously not possible to discuss with any com- pleteness all the infinite varieties of combinations of resis- tance, inductance, and capacity which can be imagined, and which may exist, in a system or network of circuits ; there- fore only some of the more common or more . interesting combinations will here be considered.

1.) Resistance in series with a circuit.

43. In a constant-potential system with impressed E.M.F.,

o = e. +/V, E. =

RESISTANCE, INDUCTANCE, CAPACITY. 59

let the receiving circuit of impedance

Z = r —jx, z = Vr2 + x'2,

be connected in series with a resistance, r0 . The total impedance of the circuit is then

Z + r0 = r + r0—jx\ hence the current is

---

•" Z + r0 r+r0 -jx (r + r0)2 -f *2 '

and the E.M.F. of the receiving circuit, becomes E = IZ = ^° (r ~J^ = ^°

or, in absolute values we have the following : — Impressed E.M.F.,

current,

zr zr

V(r + ;-0)2 + x2 -Vz2 + E.M.F. at terminals of receiver circuit,

E = EnJ >* + *2 . Eo

Vs2 + 2rr0 + r02 difference of phase in receiver circuit, tan w = - ;

difference of phase in supply circuit, tan o>0 =

since in general,

tan (phase) = ^aginary component ^ real component

a.} If x is negligible with respect to r, as in a non-induc- tive receiving circuit,

1= -=3_

r+ r.

and the current and E.M.F. at receiver terminals decrease steadily with increasing r0 .

60 ALTERNATING-CURRENT PHENOMENA.

b.} If r is negligible compared with x, as in a wattless receiver circuit,

7= E° , £ = £. X -

or, for small values of r0 ,

/=— °, ^ = ^0;

that is, the current and E.M.F. at receiver terminals remain approximately constant for small values of r0, and then de- crease with increasing rapidity.

44. In the general equations, x appears in the expres- sions for / and E only as xz, so that / and E assume the same value when x is negative, as when x is positive ; or, in other words, series resistance acts upon a circuit with leading current, or in a condenser circuit, in the same way as upon a circuit with lagging current, or an inductive circuit.

For a given impedance, z, of the receiver circuit, the cur- rent /, and E.M.F:, E, are smaller, as r is larger; that is, the less the difference of phase in the receiver circuit.

As an instance, in Fig. 37 is shown the E.M.F., E, at the receiver circuit, for E0 = const. = 100 volts, s = 1 ohm ; hence / = E, and —

a.) r0 = .2 ohm (Curve I.) b.) r0 = .8 ohm (Curve II.)

with values of reactance, x = V^2 — r2, for abscissae, from x = + 1.0 to x = — 1.0 ohm.

As shown, / and E are smallest for x = 0, r = 1.0, or for the non-inductive receiver circuit, and largest for x = ± 1.0, r = 0, or for the wattless circuit, in which latter a series resistance causes but a very small drop of potential.

Hence the control of a circuit by series resistance de- pends upon the difference of phase in the circuit.

For r0 = .8, and x = 0, x = + .8, x = — .8, the polar diagrams are shown in Figs. 38 to 40.

RESISTANCE, INDUCTANCE, CAPACITY.

61

2.) Reactance in series witJi a circuit. 45. In a constant potential system of impressed E.M.F.,