CHAPTER I. 

INTRODUCTION. 

1. In the practical applications of electrical energy, we 
meet with two different classes of phenomena, due respec- 
tively to the continuous current and to the alternating 
current. 

The continuous-current phenomena have been brought 
within the realm of exact analytical calculation by a few 
fundamental laws : — 

1.) Ohm's law : i = e j r, where r, the resistance, is a 
constant of the circuit. 

2.) Joule's law : P= i^r, where P is the rate at which 
energy is expended by the current, /, in the resistance, r. 

3.) The power equation : P^ = ei, where P^ is the 
power expended in the circuit of E.M.F., <?, and current, /. 

4.) Kirchhoff* s laws : 

a) The sum of all the E.M.Fs. in a closed circuit = 0, 
if the E.M.F. consumed by the resistance, ir^ is also con- 
sidered as a counter E.M.F., and all the E.M.Fs. are taken 
in their proper direction. 

b,) The sum of all the currents flowing towards a dis- 
tributing point = 0. 

In alternating-current circuits, that is, in circuits con- 
veying currents which rapidly and periodically change their 


2 AL TKRXA TING-CURRKXT PHEXOMEXA. [ § 2 

direction, these laws cease to hold. Energy is expended, 
not only in the conductor through its ohmic resistance, but 
also outside of it ; energy is stored up and returned, so 
that large currents may flow, impressed by high E.M.Fs., 
without representing any considerable amount of expended 
energy, but merely a surging to and fro of energy ; the 
ohmic resistance ceases to be the determining factor of 
current strength ; currents may divide into components, 
each of which is larger than the undivided current, etc. 

2. In place of the above-mentioned fundamental laws of 
continuous currents, we find in alternating-current circuits 
the following : 

Ohm's law assumes the form, i = c j Sy where r, the 
apparent resistance, or impcdaiue^ is no longer a constant 
of the circuit, but depends upon the frequency of the cur- 
rents ; and in circuits containing iron, etc., also upon the 
E.M.F. 

Impedance, ^, is, in the system of absolute units, of the 
same dimensions as resistance (that is, of the dimension 
L T~ * = velocity), and is expressed in ohms. 

It consists of two components, the resistance, r, and the 
reactance, x, or — , 

The resistance, r, in circuits where energy is expended 
only in heating the conductor, is the same as the ohmic 
resistance of continuous-current circuits. In circuits, how- 
ever, where energy is also expended outside of the con- 
ductor by magnetic hysteresis, mutual inductance, dielectric 
hysteresis, etc., r is larger than the true ohmic resistance 
of the conductor, since it refers to the total expenditure of 
energy. It may be called then the effective resistance. It 
is no longer a constant of the circuit. 

The reactance, ;r, does not represent the expenditure of 
power, as does the effective resistance, r, but merely the 
surging to and fro of energy. It is not a constant of the 


§3] I.XTRODUCriO.V, 3 

circuit, but depends upon the frequency, and frequently, 
as in circuits containing iron, or in electrolytic conductors, 
upon the E.M.F. also. Hence, while the effective resist- 
ance, /', refers to the energy component of E.M.F., or the 
E.M.F. in phase with the current, the reactance, ;r, refers 
to the wattless component of E.M.F., or the E.M.F. in 
quadrature with the current. 

3. The principal sources of reactance are electro-mag- 
netism and capacity. 

ELECTRO-MAGNETISM, 

An electric current, /, flowing through a circuit, produces 
a magnetic flux surrounding the conductor in lines of 
magnetic force (or more correctly, lines of magnetic induc- 
tion), of closed, circular, or other form, which alternate 
with the alternations of the current, and thereby induce 
an E.M.F. in the conductor. Since the magnetic flux is 
in phase with the current, and the induced E.M.F. 90°, or 
a quarter period, behind the flux, this E.M.F. of self-indHc- 
tancc lags 90°, or a quarter period, behind the current ; that 
is, is in quadrature therewith, and therefore wattless. 

If now <> = the magnetic flux produced by, and inter, 
linked with, the current / (where those lines of magnetic 
force, which are interlinked //-fold, or pass around n turns 
of the conductor, are counted ;/ times), the ratio, o//, is 
denoted by Z, and called self -inductance^ or the coejficient of 
sclfindiiction of the circuit. It is numerically equal, in 
absolute units, to the interlinkagcs of the circuit with the 
magnetic flux produced by unit current, and is, in the 
system of absolute units, of the dimension of length. In- 
stead of the self-inductance, Z, sometimes its ratio with 
the ohmic resistance, r, is used, and is called the Tivie- 
Constant of the circuit : 

r 


4 ALTERNATING-CURRENT PHENOMENA, [§3 

If a conductor surrounds with n turns a magnetic cir- 
cuit of reluctance, (R, the current, /, in the conductor repre- 
sents the M.M.F. of ni ampei'e-turns, and hence produces 
a magnetic flux of «//(ft lines of magnetic force, sur- 
rounding each n turns of the conductor, and thereby giving 
^ = n^i I (^ interlinkages between the magnetic and electric 
circuits. Hence the inductance is L = ^ / 1 = n^ / iR. 

The fundamental law of electro-magnetic induction is, 
that the E.M.F. induced in a conductor by a varying mag- 
netic field is the rate of cutting of the conductor through 
the magnetic field. 

Hence, if / is the current, and L is the inductance of 
a circuit, the magnetic flux interlinked with a circuit of 
current, /, is Lt, and 4 NLt is consequently the average 
rate of cutting ; that is, the number of lines of force cut 
by the conductor per second, where JV= frequency, or 
number of complete periods (double reversals) of the cur- 
rent per second. 

Since the maximum rate of cutting bears to the average 
rate the same ratio as the quadrant to the radius of a 
circle (a sinusoidal variation supposed), that is the ratio 
IT / 2 -5- 1, the maximum rate of cutting is 2^ N, and, conse- 
quently, the maximum value of E.M.F. induced in a cir- 
cuit of maximum current strength, i, and inductance, Z, is, 

e=:2wJVLt\ 

Since the maximum values of sine waves are proportional 
(by factor V2) to the effective values (square root of mean 
squares), if i = effective value of alternating current, c = 
2ir NLi is the effective value of E.M.F. of self-inductance, 
and the ratio, e I i = 2 ir A^Ly is the magnetic reactance : 

Thus, if r = resistance, x,,^ — reactance, c = impedance, — 

the E.M.F. consumed by resistance is: fi = ir ; 
the E.M.F. consumed by reactance is : t\ = Lv ; 


1 4] INTRODUCTION, 5 

and, since both E.M.Fs. are in quadrature to each other, 
the total E.M.F. is — 


that is, the impedance, ^, takes in alternating-current cir- 
cuits the place of the resistance, r, in continuous-current 
•circuits. 

CAPACITY. 

4. If upon a condenser of capacity, C, an E.M.F., e, is 
impressed, the condenser receives the electrostatic charge, Ce. 

If the E.M.F., e, alternates with the frequency, iV, the 
average rate of charge and discharge is 4 A^, and 2w JV the 
maximum rate of charge and discharge, sinusoidal waves sup- 
posed, hence, i = 2ir NCe the current passing into the con- 
denser, which is in quadrature to the E.M.F., and leading. 

It is then : — ^e = t = 


the capacity reactance, or condensance. 

Polarization in electrolytic conductors acts to a certain 
extent like capacity. 

The capacity reactance is inversely proportional to the 
frequency, and represents the leading out-of-phase wave ; 
the magnetic reactance is directly proportional to the 
frequency, and represents the lagging out-of-phase wave. 
Hence both are of opposite sign with regard to each other, 
and the total reactance of the circuit is their difference. 


X — x^ x^. 


The total resistance of a circuit is equal to the sum of 
all the resistances connected in series ; the total reactance 
of a circuit is equal to the algebraic sum of all the reac- 
tances connected in series ; the total impedance of a circuit, 
however, is not equal to the sum of all the individual 
impedances, but in general less, and is the resultant of the 
total resistance and the total reactance. Hence it is not 
permissible directly to add impedances, as it is with resist- 
ances or reactances. 


6 AL TEKNA TI\G-CURKENT PHENOMENA, [§§ 5, 6 

A further discussion of these quantities will be found in 
the later chapters. 

5. In Joule's law, P = /^r, r is not the true ohmic 
resistance any more, but the " effective resistance ; " that 
is, the ratio of the energy component of E.M.F. to the cur- 
rent. Since in alternating-current circuits, besides by the 
ohmic resistance of the conductor, energy is expended, 
partly outside, partly even inside, of the conductor, by 
magnetic hysteresis, mutual inductance, dielectric hystere- 
sis, etc., the effective resistance, r, is in general larger than 
the true resistance of the conductor, sometimes many times 
larger, as in transformers at open secondary circuit, and is 
not a constant of the circuit any more. It is more fully 
discussed in Chapter VII. 

In alternating-current circuits, the power equation con- 
tains a third term, which, in sine waves, is the cosine of 
the difference of phase between E.M.F. and current : — 

p^ = (i cos <f>. 

Consequently, even if v and / are both large, P^ may be 
very small, if cos <^ is small, that is, <^ near 90°. 

Kirchhoff's laws become meaningless in their original 
form, since these laws consider the E.M.Fs. and currents 
as directional quantities, counted positive in the one, nega- 
tive in the opposite direction, while the alternating current 
has no definite direction of its own. 

6. The alternating waves may have widely different 
shapes ; some of the more frequent ones are shown in 
a later chapter. 

The simplest form, however, is the sine wave, shown in 
Fig. 1, or, at least, a wave very near sine shape, which 
may be represented analytically by : — 


27r 


/ = / sin =^ (/ - A) = /sin 2 tt A" (/ - A) ; 


§6] 


INTRODUCTION. 


where / is the maximum value of the wave, or its ampH- 
tiiiif ; 7" is the time of one complete cyclic repetition, or 
the period of the wave, or X = \ j T is the frequency or 
number of complete periods per second; and Z, is the time, 
where the wave is zero, or the epoch of the wave, generally 
called the phase* 

Obviously, " phase " or " epoch " attains a practical 
meaning only when several waves of different phases are 
considered, as "difference of phase," When dealing with 
one wave only, we may count the time from the moment 


fig. 1. Sliu Want. 

where the wave is zero, or from the moment of its maxi- 
mum, and then represent it by : — 

or, ( = / cos 2 n- JV/. 

Since it is a univalent function of time, that is, can at a 
given instant have one value only, by Fourier's theorem, 
any alternating wave, no matter what its shape may be, 
can be represented by a series of sine functions of different 
frequencies and different phases, in the form : — 

/ = /, sin 2 x.V(/- /,) + /, sin 4 «-.r(/ - i,) 
+ Asin6w/^(f—/t)+ . . . 

• " Eporh " is ihc lime where a pcrioiljc fiinclion teaches n certain value. 


"phas, 


s Ihc 


datum position, of a periodic f 

curccnl phenomena only difierent ways ol e 


iiiyuhi 


tvich r 


Both are in allernate- 
1 pressing the same thing. 


AL TERNA TING-CURRENT PHENOMENA. 


[S6 


where /|, /j, /g, . . . are the maximum values of the differ-. 
ent components of the wave, ty. /j, t^ . . . the times, where 
the respective components pass the zero value. 

The first term, /j sin 2 » ,V {t — /,), is called the fun- 
datneniai wave, or the^rf/ harmonic; the further terms are 
called the higher liarmonics, or "overtones," in analogy to 
the overtones of sound waves. /, sin Imr N {t — /J is the 
a"' harmonic. 

By resolving the sine functions of the time differences, 
i ~ f^, t ~ f^ . . . , we reduce the general expression of 
the wave to the form : 

i = Ai sin 2 vNt + A, sin 4 ttNI + A, sin C irNl + . . . 


1^ 5^ 


i \ i \ 


£ \. t V 


t ^ 4t ^ 


^'^ -.'^ ^ 


^ T ^ 


^ itz 


-.'^ i 


/^ 2 Htei* vfUcwt fnn 

The two half-waves of each period, the positive uaax-c 
and the negative wave (counting in a definite direction in 
the circuit), are almost always identical. Hence the even 
higher harmonics, which cause a difference in the shape of 
the two half-waves, disappear, and only the odd harmonics 
exist, except in very special cases. 

Hence the general alternating current is expressed by ; 


+ As 


rr A' (/—/,) -f-/,sin f.,rA'(/- 


/."» 


n2a-AV-{- .-/,sin6«-A>-|-^,sip. VA-wNt-\ 
a 2 uNt -f j9, cos 6 wNi + B^ cos 10 uNt -\ 


17] 


INTRODUCTION. 


Such a wave is shown in P'ig. 2, while Fig. 3 shows a 
wave whose half-waves are different. Figs. 2 and 3 repre- 
sent the secondary currents of a RuhmkorfF coil, whose 
secondary coil is closed by a hi^h external resistance : Fig. 
3 is the coil operated in the usual way, by make and break 
of the primary battery current ; Fig. 2 is the coil fed with 
reversed currents by a commutator from a battery. 

7. Self-inductance, or electro-magnetic momentum, which 
is always present in alternating-current circuits, — to a 
large extent in generators, transformers, etc., — tends to 


, 


\ 


■s 


^ 


, 


--, 


■-=, 


^' 


A 


1 


/ 


t 


^^ 


L 


L 


l__L 


Li- 


J— 


LL 


_U__I 


Flj. 3. Want ultti EatB 


suppress the higher harmonics of a complex harmonic wave 
more than the fundamental harmonic, and thereby causes 
a general tendency towards simple sine shape, which has 
the effect, that, in general, the alternating currents in our 
light and power circuits are sufficiently near sine waves 
to make the assumption of sine shape permissible. 

Hence, in the calculation of alternating-current phe- 
nomena, we can safely assume the alternating wave as a 
sine wave, without making any serious error ; and it will be 
sufficient to keep the distortion from sine shape in mind 
as a possible, though improbable, disturbing factor, which 


10 AL TERN A TING-CURRENT PHENOMENA. [ § 7 

generally, however, is in practice negligible — perhaps with 
the only exception of low-resistance circuits containing large 
magnetic reactance, and large condensances in series with 
each other, so as to produce resonance effects of these 
higher harmonics. 


INSTANTANEOUS AND INTEGRAL VALUES.