CHAPTER XXI. 

DIBTOBTIOX OF WAVS-SHAFE AND ITS CAUSES. 

212. In the preceding chapters we have considered 
the alternating currents and alternating E.M.Fs. as sine 
waves or as replaced by their equivalent sine waves. 

While this is sufficiently exact in most cases, under 
certain circumstances the deviation of the wave from sine 
shape becomes of importance, and no longer, and it may 
not be possible to replace the distorted wave by an equiv- 
alent sine wave, since the angle of phase displacement 
of the equivalent sine wave becomes indefinite. Thus it 
becomes desirable to investigate the distortion of the wave, 
its causes and its effects. 

Since, as stated before, any alternating wave can be 
represented by a series of sine functions of odd orders, the 
investigation of distortion of wave-shape resolves itself in 
the investigation of the higher harmonics of the alternating 
wave. 

In general we have to distinguish between higher har- 
monic^ of E.M.F. and higher harmonics of current. Both 
depend upon each other in so far as with a sine wave of 
impressed E.M.F. a distorting effect will cause distortion 
of the current wave, while with a sine wave of current 
passing through the circuit, a distorting effect will cause 
higher harmonics of E.M.F. 

213. In a conductor revolving with uniform velocity 
through a uniform and constant magnetic field, a sine wave 
of E.M.F. is induced. In a circuit with constant resistance 
and constant reactance, this sine wave of E.M.F. produces 


1214] DISTORTION OF WAVE-SHAPE. 321 

a sine wave of current. Thus distortion of the wave-shape 
or higher harmonics may be due to : lack of uniformity of 
the velocity of the revolving conductor ; lack of uniformity 
or pulsation of the magnetic field ; pulsation of the resis- 
tance ; or pulsation of the reactance. 

The first two cases, lack of uniformity of the rotation or 
of the magnetic field, cause higher harmonics of E.M.F. at 
open circuit. The last, pulsation of resistance and reac- 
tance, causes higher harmonics only with a current flowing 
in the circuit, that is, under load. 

Lack of uniformity of the rotation is of no practical in- 
terest as cause of distortion, since in alternators, due to 
mechanical momentum, the speed is always very nearly 
uniform. 

Thus as causes of higher harmonics remain : 

1st. Lack of uniformity and pulsation of the magnetic 
field, causing a distortion of the induced E.M.F. at open 
circuit as well as under load. 

2d. Pulsation of the reactance, causing higher harmonics 
under load. 

3d. Pulsation of the resistance, causing higher harmonics 
under load also. 

Taking up the different causes of higher harmonics we 
have : — 

Lack of Uniformity and Pulsation of the Magnetic Field, 

214. Since most of the alternating-current generators 
contain definite and sharply defined field poles covering in 
different types different proportions of the pitch, in general 
the magnetic flux interlinked with the armature coil will 
not vary as simply sine wave, of the form : 

* cos )3, 

but as a complex harmonic function. 

However, with an armature of uniform magnetic reluc- 
tance, in general the distortion caused by the shape of the 


822 AL TERN A TING-CURRENT PHENOMENA, [§ 214 

field poles is small and negligible, as for instance the curves 
Fig. 153 and Fig. 154 show, which represent the no-load 
and full-load wave of E.M.F. of a three-phase multitooth^ 
alternator. 

Even where noticeable, these harmonics can be consid- 
ered together with the harmonics due to the varj'ing reluc- 
tance of the magnetic circuit. 

In ironclad alternators with few slots and teeth per pole, 
the passage of slots across the field poles causes a pulsation 
of the magnetic reluctance, or its reciprocal, the magnetic 
inductance of the circuit. In consequence thereof the mag- 
netism per field pole, or at least that part of the magnetism 
passing through the armature, will pulsate with a frequency 
2 y if y = number of slots per pole. 

Thus, in a machine with one slot per pole, the instanta- 
neous magnetic flu.x interlinked with the armature con- 
ductors can be expressed by the equation : 

<^ = <l>cos)3{l + ccos[2)3- 0)]} 

where, <^ = average magnetic flux, 

c = amplitude of pulsation, 

and 0) = phase of pulsation. 

In a machine with y slots per pole, the instantaneous flux 
interlinked with the armature conductors will be : 

<^ = <I>cos)3{l + ccos[2y)3- 0)]}, 

if the assumption is made that the pulsation of the magnetic 
flux follows a simple sine law, as can be done approximately. 
In general the instantaneous magnetic flux interlinked 
with the armature conductors will be : 

<^ = * cos )3 {1 4- ci cos (2 p — ^,) + €o cos (4 )3 — wo) + . . . }, 

where the terms c^ is predominating if y = number of 
armature slots per pole. 

This general equation includes also the effect of lack 
of uniformity of the magnetic flux. 


5 2141 DISTORTION Oh' WAVE-SflArE. 323 


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110 


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Fig. 753. Mo-maH Waut of E.M.F. of WicJt/tooU Tknt^flniar. 


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F/9^ IH. full-iiMa Want of £.M.F. of MultltootH Tkna-fHattr. 


824 AL TERN A TING-CURRENT PHENOMENA, [§216 

In case of a pulsation of the magnetic flux with the 
frequency 2y, due to an existence of y slots per pole in the 
armature, the instantaneous value of magnetism interlinked 
with the armature coil is : 

<^ = <l> cos )3 {1 + c cos [2 y )3 - w]}. 

Hence the E.M.F. induced thereby : 

dt 

= - V2 7riV^«<I> — {cos)3(l + €COS[2y)3~w])}. 

dp 
And, expanded : 

e = V2 TT A'//* {sin P + c ^y~-^ sin [(2 y - 1) ^ _ <i] 

Hence, the pulsation of the magnetic flux with the 
frequency 2 y, as due to the existence of y slots per pole, 
introduces two harmonics, of the orders (2 y — 1) and 

(2 7+1). 

215. If y = 1 it is : 

e = V2»-7V^«* {sin/3 + isin (/3 - w) + *l!sin (3)3 - <S)}; 

that is : In a unitooth single-phaser a pronounced triple 
harmonic may be expected, but no pronounced higher 
harmonics. 

Fig. 155 shows the wave of E.M.F. of the main coil of 
a standard monocyclic alternator at no load, represented by : 

e = ^{sin p - .242 sin (3 )3 - G.3) - .046 sin (5 p - 2.6) 
+ .068 sin (7 p - 3.;5) - .027 sin (^ P - 10.0) - .018 sin 
{np- Q,^) + .020 sin (13)3 - 8.2)}; 

hence giving a pronounced triple harmonic only, as expected. 
If y = 2, it is : 

^ = V^> TT Nn 4> \ sin P + ^^y sin (ti p - Q,) + '^ sin (5 p - (d)\ 


5216] DISTORTION OF WAVE-SHAPE. 325 

the no-load wave of a unitooth quarter-phase machine, hav- 
ing pronounced triple and quintuple harmonics. 
If y = 3, it is : 

e= -•/2,rNn^ j sin^ +^ sin (5^— 5) + Lf sin (7 ;8 - i) j . 

That is; In a unitooth three-phaser, a pronounced quin- 
tuple and septuple harmonic may be expected, but no pro- 
nounced triple harmonic 


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Fig. 156 shows the wave of E.M.F. of a standard unitooth 
three-phaser at no load, represented by : 

e= E {sin ^ - .12 sin (a j8 - 2.3) - .23 sin (5^- 1.5) + .134 sin 

{7 P- fi.2) - .(H)2 sin (9 ^ -f 27.7) - .046 sin (11 ^ — 
5.5) +.(131 sin (13/3 - 61.5)}. 

Thus giving a pronounced quintuple and septuple and 
a lesser triple harmonic, probably due to the deviation of 
the field from uniformity, and deviation of the pulsation 
of reluctance from sine shape. 


ALTEKNATINC-CURRENT PHENOMENA. [S 215 


^5."." ttl 1 \ c. .,.,-,» 


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rf mm* of f.ir./: a/ I/nHaoU nnt-nlnm »IUtKtttor. 


In general, if the pulsation of the magnetic inductance 
is denoted by the general expression : 

the instantaneous magnetic fiux is: 

* = *cos/3i 1 +5x^^-^05(2 7^-5^) I ■ 
t 1 ' 

= * j cos )3 + -^ cos (0 - i,) + 51 [f cos {{2 y + 1) 
hence, the E.M.F. 

[,,sin((2y + l)^-a,) + .,^,siii((2y+l)^-V)]! 


§§216,217] DISTORTIOiV OF WAVE-SHAPE. 327 

Pulsation of Reactance. 

216. The main causes of a pulsation of reactance are : 
magnetic saturation and hysteresis, and synchronous motion. 
Since in an ironclad magnetic circuit the magnetism is not 
proportional to the M.M.F., the wave of magnetism and 
thus the wave of E.M.F. will differ from the wave of cur- 
rent. As far as this distortion is due to the variation of 
permeability, the distortion is symmetrical and the wave 
of induced E.M.F. represents no power. The distortion 
caused by hysteresis, or the lag of the magnetism behind 
the M.M.F., causes an unsymmetrical distortion of the wave 
which makes the wave of induced E.M.F. differ by more 
than 90° from the current wave and thereby represents 
power, — the power consumed by hysteresis. 

In practice both effects are always superimposed ; that 
is, in a ferric inductance, a distortion of wave-shape takes 
place due to the lack of proportionality between magnetism 
and M.M.F. as expressed by the variation of the permea- 
bility in the hysteretic cycle. 

This pulsation of reactance gives rise to a distortion 
consisting mainly of a triple harmonic. Such current waves 
distorted by hysteresis, with a sine wave of impressed 
E.M.F., are shown in Figs. 66 to 69, Chapter X., on Hy- 
steresis. Inversely, if the current is a sine wave, the mag- 
netism and the E.M.F. will differ from sine shape. 

For further discussion of this distortion of wave-shape 
by hysteresis, Chapter X. may be consulted. 

217. Distortion of wave-shape takes place also by the 
pulsation of reactance due to synchronous rotation, as dis- 
cussed in chapter on Reaction Machines. 

In Figs. 148 and 149, at a sine wave of impressed 
E.M.F., the distorted current waves have been constructed. 
Inversely, if a sine wave of current, 

/ = /cos P, 


328 ALTERNATING-CURRENT PHENOMENA. [§217 

passes through a circuit of synchronously varying reac- 
tance; as for instance, the armature of a unitooth alterna- 
tor or synchronous motor — or, more general, an alternator 
whose armature reluctance is different in different positions 
with regard to the field poles — and the reactance is ex- 
pressed by 

Ar=jr{l + ccos(2)3-a>)}; 

or, more general, 

^=^|l+^€,cos(2y)3-cu,)|; 

the wave of magnetism is 

^= cos)3 = -! cos)3-|-VirtyCosi3cos(2y^~«,) > 


= ^r^T-S ^^^'^ + ^^^^ ^^ - «i) + 5E [^ cos((2y + 1) 

^ - "r) + ^^^cos((2y + i))8 - a, + i)J|; 

hence the wave of induced E.M.F. 

dt dfi 

= ^{sinj8 + ^sm(j3-a.)+^ ^^ + ^ [«, sin ((2 y + 1) 

)3 - cu, ) + c, + 1 s i n ( (2 y + 1 ) )3 - ^, + 1 ) ] I ; 

that is, the pulsation of reactance of frequency, 2y, intro- 
duces two higher harmonics of the order (2y— 1), and 

(2y + l. 

If A'=.r{l + ecos (2)3-0))}, 
"is ^ _ ^^ |cos/8 + icos(y3- a) + f;Cos(3^- a)| ; 

27rA«C w J > 

. =^|sin)3+isin()3-5i) + |5sin(3)3-<o)J- 

Since the pulsation of reactance due to magnetic satu- 
ration and hysteresis is essentially of the frequency, 2N, 


§§218,210] DISTORTION OF WAVE-SHAPE. 329 

— that is, describes a complete cycle for each half-wave of 
current, — this shows why the distortion of wave-shape by 
hysteresis consists essentially of a triple harmonic. 

The phase displacement between e and /, and thus the 
power consumed or produced in the electric circuit, depend 
upon the angle, w, as discussed before. 

218. In case of a distortion of the wave-shape by 
reactance, the distorted waves can be replaced by their 
equivalent sine waves, and the investigation with suffi- 
cient exactness for most cases be carried out under the 
assumption of sine waves, as done in the preceding chapters. 

Similar phenomena take place in circuits containing 
polarization cells, leaky condensers, or other apparatus 
representing a synchronously varying negative reactance. 
Possibly dielectric hysteresis in condensers causes a dis- 
tortion similar to that due to magnetic hysteresis. 

Pulsation of Resistance, 

219. To'a certain extent the investigation of the effect 
of synchronous pulsation of the resistance coincides with 
that of reactance ; since a pulsation of reactance, when 
unsymmetrical with regard to the current wave, introduces 
an energy component which can be represented by an 
"effective resistance." 

Inversely, an unsymmetrical pulsation of the ohmic 
resistance introduces a wattless component, to be denoted 
by "effective reactance." 

A typical case of a synchronously pulsating resistance is 
represented in the alternating arc. 

The apparent resistance of an arc depends upon the 
current passing through the arc ; that is, the apparent 

resistance of the arc = potential difference between electrode, j^ j^- ^ 

current ^ 

for small currents, low for large currents. Thus in an 
alternating arc the apparent resistance will vary during 


330 AL TERXA TIXG-CURRENT PHEXOMEXA. [3219 

ever)' half-wave of current between a maximum value at 
zero current and a minimum value at maximum current, 
thereby describing a complete cycle per half-wave of cur- 
rent. 

Let the effective value of current passing through the 
arc be represented by /. 

Then the instantaneous value of current, assuming the 
current wave as* sine wave, is represented by 

/ = V2sin)3; 

and the apparent resistance of the arc, in first approxima- 
tion, by 

R = r{y -f ccos2<^); 

thus the potential difference at the arc is 

e^iR=^ / V2r sin <^ (1 + €C0s2 <^) 

= r/V2J/'l-^\sin<^ + ^sin3<^j. 

Hence the effective value of potential difference, 


and the apparent resistance of the arc, 


E f ? 

The instantaneous power consumed in the arc is, 
/ = /> = 2 r/* ^ Z' 1 - ^A sin* <^ + i sin <^ sin 3 <^ 

Hence the effective power. 


§220] DISTORTION OF WAVESHAPE. 331 

The apparent power, or volt amperes consumed by the 
arc, is, 

thus the power factor of the arc, 

1-i 

■' IE 


that is, less than unity. 

220. We find here a case of a circuit in which the 
power factor — that is, the ratio of watts to volt amperes 
— differs from unity without any displacement of phase ; 
that is, while current and E.M.F. are in phase with each 
other, but are distorted, the alternating wave cannot be 
replaced by an equivalent sine wave ; since the assumption 
of equivalent sine wave would introduce a phase displace- 
ment, 

cos w =y 

of an an^le, tu, whose sign is indefinite. 

As an instance are shown, in Fig. 157 for the constants, 

/= 12 
r= 3 
€ =.9 

the resistance, 

CR = 3 {1 + .9 cos 2 )5) ; 

the current, 

/ =17sin)3; 

the potential difference, 

<? = 28 (sin ^ + .82 sin 3 )3). 

In this case the effective E.M.F. is 

E = 25.5 ; 


332 ALTERNATING-CURRENT FHENOilENA. [{221 

the apparent resistance, 

^0 = 2.13; 
the power, 

y = 244 ; 

the apparent power. 


the power factor, 


/ = .796. 


/^ ^^ z 


LA t 


^^^^- t '^^ ^nL 


/ 3S I- yl S Z j\ 


1 fX-:.Jl , '^ I^ 


I, S3 ItA \ 1 t 


U K^^ ^^ V V 7 


/" V_/ \ ^_^ 7 


^ 5 / 


^ LK '^t- ^ 


i ~^3 t^ 3 


3 i I-T I 


/ ■ ARUBLi R StSTANC : \ / ' / 


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v' .-!.!.. . J.-lH) V^ ^7^ 


T 


ng. 157. PtrloHlcallii Varying Sflilana. 

As seen, with a sine wave of current the E.M.F. wave 
in an alternating arc will become double-peaked, and rise 
very abruptly near the zero values of current. Inversely, 
with a sine wave of E.M.F. the current wave in an alter- 
nating arc will become peaked, and very flat near the zero 
values of E.M.F. 

221. In reality the distortion is of more complex nature; 
since the pulsation of resistance in the arc does not follow 


§232] 


DISTORTION OF WAVESHAPE. 


838 


a simple sine law of double frequency, but varies much 
more abruptly near the zero value of current, making 
thereby the variation of E.M.F. near the zero value of 
current much more abruptly, or, inversely, the variation 
of current more fiat. 

A typical wave of potential difference, with a sine wavie 
of current passing through the arc, is given in Fig. 158.* 


- 


, 


, 


ij J, 1 '. : 


z_ 


l\ 


II 


a^^^l^ 


4 


n 


^ 


_ 


_ 


_ 


-. 


_ 


- 


- 


r 


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^~ 


— 


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1 » 


/' 


h i 1. li II 


Mr 


F 


' 


si! 


»- 


l' 


° 


- 


- 


1. A. C. dynamo e. m 


'• 


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-*-iAl 


II. w.Hi. 


^ 1 


\l 


' 


/ 


1 


^ 


\ 


f 


I 


1 1 


1 M ' 1 


flg. IS8. Electric Arc 

222. The value of t, the amplitude of the resistance 
pulsation, largely depends upon the nature of the electrodes 
and the steadiness of the arc, and with soft carbons and a 
steady arc is small, and the power factor / of the arc near 
unity. With hard carbons and an unsteady arc, t rises 
greatly, higher harmonics appear in the pulsation of resis- 
tance, and the power factor y falls, being in extreme cases 
even as low as .6. 

The conclusion to be drawn herefrom is, that photo- 
metric tests of alternating arcs are of little value, if, besides 
current and voltage, the power is not determined also by 
means of electro-dynamometers. 

* Frnm Amt-ricnn Instilule of Electrical Engineers, I'tansaclions, 18t>0, p. 
370. Toliey and Walbiidge, on Ihe Stanley Alternate Arc Dynamo. 


ALTERNA TING-CUKRENT PHENOMENA.