CHAPTER V. 

SYMBOUC MBTHOD. 

23. The graphical method of representing alternating- 
current phenomena by polar coordinates of time affords the 
best means for deriving a clear insight into the mutual rela- 
tion of the different alternating sine waves entering into the 
problem. For numerical calculation, however, the graphical 
method is frequently not well suited, owing to the widely 
•different magnitudes of the alternating sine waves repre- 
sented in the same diagram, which make an exact diagram- 
matic determination impossible. For instance, in the trans- 
former diagrams (cf. Figs. 18-20), the different magnitudes 
•will have numerical values in practice, somewhat like E-^ = 
100 volts, and /j = 75 amperes, for a non-inductive secon- 
dary load, as of incandescent lamps. Thus the only reac- 
tance of the secondary circuit is that of the secondary coil, 
or, x\ = .08 ohms, giving a lag of eSj = 3.6^. We have 

also, 

fty = 30 turns. 

Uf, = 300 turns. 

(Fi = 2250 ampere-turns. 

$f = 100 ampere- turns. 

Er = 10 volts. 

£^ = 60 volts. 

Ei = 1000 volts. 

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

Hence the importance of the graphical method consists 


84 


ALTERNATING-CURRENT PHENOMENA. [§§24,25 


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 


f/g. 27. 


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

equation : — ' 

$Fo = V^H^i' + 2 IFSFi 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. 


Flq, 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, AO/, 
or phase <o, of the wave, /. 

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


§26] 


SYMBOLIC METHOD. 


35 


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

a = /cos u> is the horizontal component, 

^ = /sin 0) 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 


Fig, 23. 

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

Sifie waves arc combined^ or resolved^ by adding^ or 
subtractings their rectangular components. 

For instance, if a and b are the rectangular components 
of a sine wave, /, and d and b* the components of another 
sine wave, /' (Fig. 23), their resultant sine wave, /^, has the 
rectangular components a^ == (a •\- d), and b^^(p •\- b'). 

To get from the rectangular components, a and b, of a 
sine wave, its total intensity, /, and phase, <o, we may com- 
bine a and b by the parallelogram, and derive, — 


/= Vi^+Ta. 


tan (u = — . 
a 


86 AL TERN A TING-CURRENT PHENOMENA, [§§ 26» 27 

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 y, and thus 
represent the sine wave by the expression, — 

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

and of phase, tan ta = b j a. 

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

Obviously, the plus sign in the symbol, a +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 algebraic exprcssiotiy a '\-jby of a sine wave 
by —1 means 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 + jb, has (Fig. 24) the horizontal 


128] 


SYMBOLIC METHOD, 


8T 


component, — by and the vertical component, ^, and is rep- 
resented algebraically by the expression, ja — b. 
Multiplying, however, a + jb by 7, we get : — 

jii+rb\ 

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

we have — 

j{a+Jb) r=ja — b\ 

hence : — 

Multiplying the algebraic expression^ a -{-jb, of a sine wave 
by j means rotating the wave through 90^ y or one-quarter pe- 
riod ; that isy retarding the wave through one-quarter period. 


Fig. 24, 

Similarly, — 

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

since y^ = ~ 1, y = V— 1 ; 

that is, — 

j is the imaginary unity 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 ofy = 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 — , .^ 

a +jb, 


38 AL TERXA TING-CUKKENT PHENOMENA. [§ 29 

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


the phase by 


and 


/= V^^a^/,^, 


tan o) = - , 
a 


a -= t cos o), 
^ = /■ sin a> ; 


hence the wave a -^jh can also be expressed by — 

I (cos CO +y sin w), 

or, by substituting for cos w and sin ^ their exponential 
expressions, we obtain — 

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

Sine ivavcs may be combined or 7rsolved by adding or 
subtracting their complex algebraic expressions. 

For instance, the sine waves, — 

a +jb 
and 

combined give the sine wave — 

I^{a + a')+j{b + b'). 

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

29. If /= / +ji' 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 — 

r/= ri '\' jri\ 

If L is the inductance, and j: = 2 tt NL the reactance, 
the E.M.F. produced by the reactance, or the counter 


§20] SYMBOLIC METHOD. 39' 

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

jxl =jxi — xi\ 

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 — 

— ylr/= — Jxi + xi\ 

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

(r~»/; 

that is — 

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

Hence, li I = i +ji' is the current, the E.M.F. required 
to overcome the impedance, Z = r — jxy is — 

hence, since y^ = — 1 

£ = (ri -|- xi') + J (ri' — xi) ; 

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

Z r — jx 

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

7 ^ (^+J^(^+Jx) ^ er — /x , . /r + <f j: . 

or, if ^= ^? +yV' is the impressed E.M.F., and / = i +ji^ 
the current flowing in the circuit, its impedance is — 

Z^^ =^+J^ ^ (f +J^(' -Jn ^ ^i + ^i' I /i' - ^i' 


40 ALTERNATING-CURRENT PHENOMENA, [§§ 30, 31 

30. If C is the capacity of a condenser in series in 
a circuit of current 7=1 +/**', the E.M.F. impressed upon 

the terminals of the condenser is -£" = ~ , 90° behind 

the current ; and may be represented by —-^ — — - , or jx^ /, 

where x^^ = — • is the capacity reaciatice or condensance 

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, icNL = magnetic reactance. 

If C = capacity, Xx = , = capacity reactance, or conden- 
sance ; 

Z = r — j {x — jTi), is the impedance of the circuit. 

Ohm*s law is then reestablished as follows : 

^ = Z/, 7=^, Z = ^. 

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. 


§32] SYMBOLIC METHOD, 41 

Since, if a complex quantity equals zero, the real part as 
well as the imaginary part must be zero individually, if 

a +jb = 0, tf = 0, i^ = 0, 

and if both the RM.Fs. and currents are resolved, 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 towards 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 cannot be represented as 
a vector in the diagram. 

In what follows, complex quantities will always be de- 
noted by capitals, absolute quantities and real quantities by 
small 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, /, over an inductive line, we can now represent the 
impedance of the line by Z = r —jXy 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 — 

Eo = E + ZL 

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

Eo = E + ir —jix ; 

hence, with non-inductive load, or E ^ e, 

Eo={e + ir) —jix, 

ix 


Co = V(<r + it-y^ + (ix)\ tan w^ = J^^- 


y 


42 AL TERNA TING-CURRENT PHENOMENA. [§ 32 

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

Eo=^e—j/ + ir—jx)i 
= {e + ir)^j{/+ix\ 

.or ^0 = V(<?+/r)^+ (/ + ix)\ tan w^ = -^tif . 

e + tr 

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

Eo^{e+j/)+{r-jx)i 

e, = V(^ + *>)» + (/ - ixy, tan a<, = 7q-^ ; 

values which easily permit calculation. 


§33] TOPOGRAPinC METHOD, 43