COMPLEX QUANTITIES AND THEIR USE IN ELECTRICAL ENGINEERING.
BY CHAS. PROTEUS STEINllETZ.
1.-INTRODUCTION.
In the following, I shall outline a method of calcula.ting alternate current phenomena, which, I believe, differs from former methods essentially in so far, as it allows us to represent the alternate current, the sine-function of time, by a C<>n1Jtant numerical quantity, and thereby eliminates the independent variable "time" alt:ogether from the calculation of alternate current phenomena.
Herefrom results a considerable simplification of methods. Where before we had to deal with periodic functions of an independent variable, time, we have now to add, subtract, etc., constant quantities-a matter of elementary algebra-while problems like the discussion of circuits containing distributed capacity, which before involved the integration of differential equations containing two independent variables: " time " and " distance," are now reduced to a differential equation with one independent variable only, " distance," which can easily be integrated in its most general form.
Even the restriction to sine-waves, incident to this method, is no limitation, since we can recom1truct in the usual way the complex harmonic wave from its component ,:,iine-waves; though almost always the assumption of the alternate current as a true sine-wave is warranted by practical experience, and only und.er rather exceptional circumstances the higher harmonics become noticeable.
In the graphical treatment of alternate current phenomena
different representations have been used. It is a remarkable
fact, however, that the &implest graphical representation of
88

84

STBINJfBTZ ON OOJfPLBX QUANTITIES.

periodic functions, the common, well-known polar ooordim,a,t.e,; with Ums as angk or ampUt'ttAU, and the inatan:/o,1UHJtUJ 'VfM/UM of the function as .,.adi;i, vectnrea, which has proved its usefulness through centuries in other branches of science, and which is known to every mechanical engineer from the Zenner diagram of valve motions of the steam engine, and should consequently be known to every electrical engineer also, it is remarkable that this polar diagram has been utterly neglected, and even where it has been used, it has been misunderstood, and the sine-wave represented-instead of by one circle--by two circles, whereby the phase of the wave becomes indefinite, and hence the diagram

FIG. 1.
useless. In its place diagrams have been proposed, where revolving lines represent the instantaneous values by their projections upon a. fixed line, etc., which die.grams evidently are not able to give as plain and intelligible a conception of the variation of instantaneous values, as a curve with the instantaneous
values as radii, and the time as angle. It is easy to understand
then, that graphical calculations of alternate current phenomena have found almost no entrance yet into the engineering practice.
In graphical representations of alternate currents, we shall make use, therefore, of the Polar Coordinau SyBum, representing the time by the angle 'P a.s amplitude, counting from an

87'/fJINJIETZ ON OOMPLMX QUAN1T7'IE8.

35

initial radius o A chosen as zero time or s~rting point, in posi-

tive direction or counter-clockwise,* and representing the time of
= one complete period by one complete revolution or 360° 2 rr.

•

The instantaneous values of the periodic function are repre-
= sented by the length of the radii vectores o B r, correspond-

ing to the different angles 'P or times t, and every periodic

function is hereby represented by a closed curve (Fig. 1). At any

time t, represented by angle or amplitude tp, the inste.nt&neous

value of the periodic function is cut out on the movable radius

by its intersectionOB with the characteristic curve c of the func-

FIG. 9.
ti.on, and is positive, if in the direction of the radius, negative, if in opposition.
The 81/IUMD<W6 is represented by one circle (Fig. 2). The diameter o o of the circle, which represents the sine-wave, is called the intmunty of the sine-wave, and its amplitude,
= .A O B w, is called the ph<ue of the sine-wave.
The sine-wave is completely determined and characterized by inttmsity and phase.
It is obvious, that the ph<ue is of interest only as difference of pluue, where several waves of different phases are under consideration.
-Th.is direction of rotation has been chosen as positive, since it is the direct.ion of rotation of celestial bodies.

86

BTBINJIBTZ ON OOJfPLBX QUANTITIES.

Where only the vn,t,egra/, voJ,V61J of the sine-wave, and not it.s

imt,anuJlfUJO'U,8 'VO},U61J are required, the characteristic circle c of

the sine-wave can be dropped, and it.e diameter o o considered as

the repreeentatation of the sine-wave in the polar-diagram, and

in this case we can go a step further, and instead of using the

maJ1Jilm1t1,m 'lJlU/U,8 of the wave as it.e representation, 1188 the l!Jfect--

= ,vi • • •
ive

-1
VUAi't.ee,

wh"1ch

m .

t h e

sm. e

wave

.
18

maa:imum mlffll

Where, however, the characteristic circle is drawn with the
ffl!ecfAlv6 volJue 88 diameter, the ill.8tantaneo118 values, when taken
from the diagram, have to be enlarged by ,vi

0

Fie. 8.

We see herefrom, that :

.

"In polar coordinateJJ, t/uJ airuHJJave UJ represen:t«l in -in-

tensity and ph(J,IJe by a vector o o, and in coml;i,nVIUJ or du-

solving ai.ne-waveB, they a1·6 t<> be comlnned or diaaolved ~ the

paral.ulogram or polygon of aine-wa1.•es."

For the purpose of calculation, the sine-wave is represented

by two constant.e : 0, ai, intensity and phase.

In this case the combination of sine-waves by the Law of

Parallelogram, involves the use of trigonometric functions.

The sine-wave can be represented also by its rectan.gu/,a,r co-

ordin<uea, a and l> (Fig. 3), where :

8'1'BINJLllTZ ON <JOJIPLJIJX QU.ANTI'l'IBS.

8'7

a= Ocoetol

b=Osintof

Here a and bare the two rect,o;n,gul.,alr oomponen;ts of tM nne-
1oa/lJ6.
This representation of the sine-waves by their rectangular component.a a, and bis very useful in 80 far as it avoids the use of trigonometric functions. To combine sine-waves, we have simply t.o add or subtract their rectangular component.a. For instance, if a, and b are the rectangular component.a of one sine-
+ wave, a,1 and b1 those of another, the resultant or combined sine-
• wave has the rectangular component.a a+ a1 and b b1• To distinguish the horizontal and the vertical components of
sine-waves, 80 as not to mix them up in a calculation of any greater length, we may mark the ones, for instance, the vertical component.a, by a distinguishing index, as for instance, by the addition of the letter j, and may th118 represent the sine-wave by the expression:
a+jb

which means, that a is the horizontal, b the vertical component of the sine-wave, and both are combined t.o the resultant wave:

= 0 .Ya'+ ~2

which has the phase:

tan ai = a~-

Analogous, a - j b means a sine-wave with a as horizontal,

and - b as vertical component, etc.

.

For the :fi.rst,j is nothing but a distinguishing index without

numerical meaning.

A wave, differing in phase from the wave a,+ j b by 180°, or

one-half period, is represented in polar coordinates by a vect.or

of opposite direction, hence denoted by the algebraic expression :

- a - j b.

+ This means:
" K ultipyilfUJ t°M algebrOAC e:rpr688ion a j b of tM BVne-
= wa-ve by - 1, m.eanll reversi,'fU} t!UJ wave, or rotating it by 180°

one-half period.
= A wave of equal strength, but lagging 90° one-quarter
+ period behind a j b, has the horizontal component - b, and

88

STHINNETZ ON OOJIPLBX. QUANTITIES.

the vertical component a, hence .is represented algebraically by

the symbol:
ja-b.
+ :Multiplying, however: a j b by j, we get:

Ja+,:l2b

hence, if we define the-until now meaningless-symbol j so, as

to say, that:

hence:

l=-1
+ j (a j b) =j a - b,

we have:
+ " N.mtipling th.e algebraic ~pr68awn a j b of the R-ine-wat,e
• by j, 1TUJMl,8 rotatvng the wave by 90°, ur one-<J.'Ullll'fm' period, tluit

i8, •reto;rdi,ng tM wave by one-<J.'Ut0trter period."

In tbe same way :

"Multiplying by - j, 'IManB adv01Ming the wave by mie-

quarfm' period."

J..,, = -1 means:

j = t' - 1, that is:
+ "j u the im,agi,nary unit, and the 8'ine-wave ia repr~ 'by a
compk:J, imuJgi;n,ary quantity a .i b."

Herefrom we get the result :
u " In the polar diagratm of ti~, tM 8Vn6-'Wa'#Je repr68(m'1ea

i.n interutity a8 well O,JJ pluue by one compl,e.:r: quanti;f;g:

a+.i b,

= + where a is tM lwmont,al, b the ver'flwal component of the W(//Vt,
tM int.erunty u gvven by: 0 t'a2 '/J'I.

a;n,d, the ph.atJe by:

tan

ii>

=

! ,
a

amd it iR:

= a 0 cos ii>

b = 0 sin,;;

+ hence the wave: a j b COin, oho be eap768Bed by: + 0 (cos ii> j sin ii>)."

Since we have seen that sine-waves are combined by adding

their rectangular components, we have :

"Sine-wave1J are combined by adding their comp~ alge'lnaw

e;r_pr688'UJn.8.''

For instance, the sine-waves :

a+Jb

aud

a1 +J "61

STEINJIETZ ON OOltlPLJCX QUANTITIES.

89

combined give the wave :
A + j B = (a + a1) + j (b + b1).

As seen, the combination of sine-waves is reduced hereby to

the elementary algebra of complex quantities.
= + If O o j c1 is a sine-wave of alternate current, and r is
the 1'88istance, the E. x. F. consumed by the resistance is in phase
with the current, and equal to current times resistance, hence

it is:
r O = r o + j r c1. = If L is the " coefficient of self-induction," or 8 2 1r N L
the "inductive resistance" or "ohmic inductance," which in the following shall be called the " inductance," the E. K. F. produced by the inductance (counter E. K. F. of self-induction) is equal to current times inductance, and lags 90° behind the cur-

rent, hence it is represented by the algebraic expreesion:

}80

and the x. K. F. required to overcome the inductance is consequently:
-j80

that is, 90_0 ahead of the current (or, in the usual expression, the

cm1,mt lags 90° behind the E. K. F.).

Hence, the E. K. F. required to overcome the .resistance r and

the inductance 8 is :

(r-j 8) 0

•

that is:

"I= r - j a i8 the e;,,pre11ai<m, of the imp«l,a;n,ce, in C<YI~
= = = lJ'UO!nfA-eia, where r rwt,a,n,ce, 8 2 1r 1( L im,ductomee." = + Hence, if O c j c1 is the current, the E. K. F. required to

overcome the impedance I= r - j a is:

= = + E I O (r - j a) (c j o'), hence, since J"2 = - 1 :

= + + (r o

8

0

1 )

j (r c1 - 8 c)

= + or, if E e j et is the impressed E. K. F., and I= r - j R is

the impedance, the current fl.owing through the circuit is :

O

_
-

E_e+
y- '1'

J e1
-J8

+ or, multi11lying numerator and denominator by (r j 11), to elim-

inate the imaginary from the denominator :

0

=

(e

+ + j e1) (r j r2+r

8)

=

e r - e1 8
r+al

•
+,

+ e1 r e a
r2+s2

40

8TEINJfBTZ ON OOJIPLRX QU.AN1.'ITIB8.

If K is the oapaci,ty of a. condenser, connected in series into
= + a circuit of current O c j 01, the :s. H.F. impressed upon
= t the .terminals of the condenser is E 2 rr K' a.nd lags 90°
behind the current, hence represented by :

E =J•21r0NK=J"kO.'

= 1
where k 2 1r N Kean be called the "oapaci,f;y inuluctantt"

or simply " inductam,ce" of the condenser. Capacity induc-

tance is of opposite sign to magnetic inductance.

That means:
= " ..(f r r68U'taln.,oe, = = = L coejficient qf aelf-vn.d:uotwn, hence s 2 1r NL i11-

ductance,

= K = capaeif;y, lum.oe k = 2 1r ~ K oapaei,ty imductanee,

I= r - j (s - k) ·iR the impedance qf the cirouit, and Ohm's

la·w is re-e-8uihlished :

E= I 0,

•

(J

=

E
1'

E I= '{J'

in a more gmi,eral form., lwwei•er, gim,ng not rm,/;y tM i ~ , but alHo the p/1.a./Jt qf the tr1°;ne-wooe.B, by tAe.ir e;r.presRion. in e01n-
pu1:_w_ q1.Ulln:•titu• s."
In the following we shall outline the applfoation of complex r1uantities to· various problems of alternate and •polyphase currents, and shall show that these complex quantities can be operated upon like ordinary algebraic numbers, so that for the solution of most of the problems of alternate and polyphase currents, elementary algebra is sufficient.
Al,gebraic operatwn8 with cornplex quantities:
J"2=-1
= a + j b + c (cos w j sin ,;;) = + o .y'a2 b2,
= ... b
tan a, a•

lY.l'BINllll'l'Z ON OOJLPLB:S. Qll.AN'J'ITIBS.

,1

+ = = = + If a j b a1 j b1, it must be: a a1, b b1•
Addition and aubtraction : •
(a + j b) ± (a' + j b1) = (a ± a1) + j (b ± b1).

Multiplication:

+ = + + (a+ j b) (a1 j b1

(a a1

b b1

j (a b1 b a1

)

-

)

).

Division:

= = a+Jb (a+Jb)(a1 - j b1) aa1 +bb1 .aFb-ab1

at +J b'

au+ bu

au+ 1)12 +J au+ b12·

Difference of phase between :
= + + a j b c (cos ,;; j sin w) and,
a1+ j b' = c1(ooe w1+ j sin w1) :

a tan '"'l - ta ... = = tan (,;;1 -cu)= 1 +:n,i.ita:;

1>1 l>

iii -

a b1 - b a1

b b• aa• +bb''

1 +a-a1

= :Multiplication by - 1 means reversion, or rotation by 180°

one-half period.

:Multiplication by j means rotation by 90°, or retardation by

one-quarter period.

Multiplication by - j means rotation by - 90°, or advance

by one-quarter period.
+ Multiplication by cos w j sin ciJ means rotation by angle,;;.

II. CIRCUITS CoN'fAI.NING RESISTANOE, lNDUOTANOE .A:ND
CAPACITY.
Having now established Ohm's law as the fundamental law of alternate currents, in its complex form :
E= IO, where it represents not only the ~ t y , but the phase of the electric quantities also, we can by simple application of Ohm's law -in the same way as in continuous current circuits, keeping in mind, however, that E, O, I are complex quantities--dissolve and calculate any alternate current circuit, or network of circuits, containing resistance, inductance, or capacity in any combination, without meeting with greater difficulties than are met with in continuous current circuits. Indeed, the continuous current distribution appeal'S as a particular case of the general problem, characterized by the disappearance of all imaginary terms.
As an instance, we shall apply this method to an indluetive

"9

8TBINJLBn ON 001iPLEX QU..tlf'J'ITIBB.

cilrO'tllit, alvwn,ted &g a ~ , am,d fed tl,,rovgl,, ~

'11UJlim,,, upon which a constant alternate E. K. F. is impreseed, 88

shown diagrammatically in Fig. 4.

= Let r resistance,

= L coefficient of self-induction, hence

= N L = a 2 1r

inductance, and :

= I = r - j a impedance of consumer circuit..

= Let r 1 = resistance of condenser lead.a,
K capacity, hence

= = k 2 1r ~ K

+j = = l 1 r 1

k

capacity inductance, e.nd: impedance of condenser circuit.

Let r0 = resistance,
= L 0 coefficient of self-induction, hence
= = BO 2 1r NL O inductance, and:

1.l'I&. 4:.

= = 10 r - j Bo impedance of the two main leads.
= Let Eu E. K. F. impressed upon the circuit.

E = We have then, if,

E. x. F. at ends of main leads, or at

tenninals of consumer and condenser circuit :

= Corrent m. consumer C.ircu.it, 0

E
7

Current in condenser circuit, 01 = ~

Hence, total current,

+ + J 1

= = 0 0 0 0 1 E{-}

)

J+ ~)1
= = Ji ( E. x. F. consumed in main leads E1 0 0 I 0

Hence, total E.M.F.

E., =.E+.E'=E{ 1 +J+7.}

STBINMBTZ ON <JOJl'PLBX QUAN'l'IT1148.

4S

+ = or, E. 11. F. at end of main leads, E Io I+E_i0~~ I Ii

= + E.11.F.consumedb1mam• ln~ .nAaE1

IE+o IoIo(I+ IiI) Ii

10

11

c = + Current m• consumer c1•rcu1•t,

= EI Io I+ EIo0 IIi1 1 Ii

= + Current

1•n condenser c1•rcm·t,

n
v1

7E;_ =fol+ E 0 I

I Ii

1011

Total current,

+ n _ C n _

Eo (I+ I;.)

+ Vo -

v1 - Io I+ Io Ii I Ii

Substituting herein the values,

Io= ro - j Bo

I= r-j s

Ii= r1 +i k

and,

10I+ 10Ii + 1Ii = a -j b,

where,

a =

+ r1 + 1'1 - lo lJ + llo } + 8 k

ro

'r

ro

'r

= + + r1 + b

80 'I'

80 r 1

lJ

'1' lJ

0 -

r0 k

-

r

k

t we get
= E iJ b'{[a(rr1+Bk)+b(r18-'l"k)]+j[b(rr1+ak)-a(r18-'l"k)]}

0 = a2 ~ b2 {(r1 a - k b) + j (r1 b + le a) }
t= 01 "2 b, {(r a + B b) + j (r b - a a) }

+ C.= at! lJ { [(r+ r1) a+ (s-k) b] + j[(r r1)b-(1J-k)a)] }

Ae an instance, we ma,- consider the case :

'

= E0 100 volts,

= = = r0 1 ohm}
,. = , = = 10 ohm

r 2 ohm} 10 ohm

r1 0

}

k 20 ohm

= 10 1-lOj

I-;- 2-lOj

ft= 20j

hence

a= 302

b =-30

Substituting these values, we get,
== + E0 100,
E 68.0 (.98 .17J),
= E 1 35.1 (.94 - .34J), = 0 6.6 (.10 + .99.7),

4.4

8TBiNJL.B'n ON <JOXPL.BX QUANTITI.B8.

01 = 3.4 (.17 - .98 J),
= + 00 3.4 (.37 .98 ;),
where the complex quantities are represent.ed in the form c (oos
+ j w sin w), so that the numerical value in front of the paren-
thesis gives the effective intensity, the parenthesis gives the pluwJ of the alt.eme.te current or E. M. F.
This means: Of the 100 volts impressed, 35.1 volt.a are consumed by the leads, and 68.0 volts left at the end of the line.
The main current of 8.4 amperes divides into the consumer current of 6.6 amperes, and the condenser current of 3.4 amperes.
Increasing, however, the capacity K, that is reducing the capa<,._
k = ity inductance to 10, or I,.= 10 j, we get:

Hence:

a= 102,
b = o. E 0 = 100,
ll = + 100 (.98 .20 .1), = E 1 19.9 (.10 - .99 J), = 0 9.8j,
= 01 10 (.20 - .98 J), = 0 0 1.98.

Here, though the leads consume 19.9 volts, still the full

potential of 100 volts is left at their end.

1.98 amperes in the main line divide into two branch current.s,

of 9.8 and of 10 amperes. We have here one of the frequent

cases, where one alternat;e current divides into two branches, so

that either branch current is larger than the undivided or tot.al

current.

k = = Increasing the capacity still further to

5, or I,_ ~ j,

gives:

Hence:

a= 2,
b = 15. E 0 = 100,
= E 387 (.32 - .95 j), = + E 1 318 (- .03 j),
= + 0 33.0 (.99 18 J), = + 01 96.3 (1 .06 J). = + 00 63.6 (1 .03 .1),

That means, in the leads self-induction consumes an E. K. F. of

318 volts, and still 337 volts exist at the end of the line, giving

STEINMETZ ON OOJIPLEX QUANTITIES.

~

a rise of potential in the leads of 237 volts, due to the combined .

effect of self-induction and cape.city. The main current of 63.6 amperes divides into the two branch

currents of 33.0 and 96.3 amperes.

.

The current which passes over the line is far larger than the

current which in the absence of capacity would he permitted hy

the dead resistance of the line. While in this case 63.6 amperes

flow over the line, a continuous E. M. F. of 100 volts would send

1:;:. = only ro r 33.3 amperes over the line; and with an alter-
nating E. M. v., but ~thout capacity the current would be limited to 4.95 amperes only, since in this case :

o. + + = 0 = (r0

r )

-EJ0 •

(
80

)·=
8

3 •

-

loio

J .

4.95 (.15+.99 j).

Even by short-circuiting the line, we get only :

= . = + Oo

E0
ro-J8o

1

-

100 10 J•

=

10

(.1

.99 i),

or 10 amperes over the line.

Hence we have in this arrangement of a condenser shunted to

the inductive circuit and fed by inductive mains, the curious re-

sult that a short-eircuit at the terminals of the consumer circuit

reduces the line current to about one-sixth.

As a further instance, we may consider the problem :

"W/1.,,t iB the '11Ulndmwm power which can be transmitUJI}, over

an inductive Une into a n<m-i1iductvve reBiBtance, as lights, ,,n,d

lww f,ir can this o-utput be increa,sed by the UYe of 8liunted

capacity."

Let,

rO = resistance,

hence,

= = 80 = inductance,
Io rO - j 80 impedance of the line.

= Let r resistance of the consumer circuit, which is shunted

by the capacity inductance k.

r and k a.re to be determined as to make the power in the

receiving circuit: 0 2 r, a maximum.

In a contin11,0U8 current ci'rcuit the maximum output is

= 1\ reached, if r r 0 , or E • where .E~ is the E. M. F. at the

= begm• nm• g, .1L~' the E. M. 1''. at the end of tl1e l"me, an<l C' 2E-rou,
hence:

•

46

STBINMBTZ ON OOMPLBX QUANTITI.88.

P = ~~ Eor2o the maxi.mum output at efBc1•ency 50 per cent.
= = Hence, if Eo 100, r0 1, it is: P = 2,500 watts.
Very much less is the maximum output of an tilternate cur-
rent circuit. With an alt.ernate E. K. F. Ko, but without the use

of a condenser, the impedance of the whole circuit is:

+ 1 = r 0

- j r

10 ,

+ + Ko Eo (ro + r-j Bo)
hence the current: 0 = 1 = (ro r'f 1102

l _

E0

{

ro + r

•

Bo

+ - ¥(r0 +r)2 t102 ¥(r0 + r)2+ Bo2 + J ¥(ro + r)2+ 801 '

the E. K. F. at end of line:

E- O _

Eo r

+ ~ r0 r

•

_80

}

- r- t'(r0 +r)2+ao1 ( +'(r0 +r)2+a02 + J ¥(ro+r)2+ao' '

= + + hence the power: P

E O. = (ro

E

0

1
r
)2

'r

Bo 2•

The condition of maximum output is,

/JP= 0 Ir

that is, •

= 0 (r + r0'J + 801- 2 r (r + r 0), or,
= + r2 ro2 Bo2, + , r = .Y ro"' 02,

and the maximum output is,

J!,,~I

+ + P=~---:=;=:;;;;=;:;::;;;:; • 2 Jro t' r 02 B0 2f

+ .,.

+ - + , at the efficiency, 1'0

r

ti r02 802

r0

r t'

ll
0

,-

B0 2

= In the instance, E0 = 100, r0 = 1, B0 10 is:

= P 453 watts, against 2,500 watts with continuous currents.

If, however, we shunt the receiver circuit by capacity induc-

tance k, we have,

Leads,

= I 0 r0 - j s0,

Consumer circuit, J = r, a= 0,
= Condenser circuit, Ii j k, r 1 = 0,

hence, by substituting in the equations derived in the first part

of this chapter,

BTRINJIRTZ ON 001/PLEX QU.ANTITIES.

47

+ a= r0 r 80 k = + lJ 80 r - k (r0 r)

and,

= 0 a'E-t-kli' (-b +j a),

or, substituting,

and,

hence, power,

= = p OE= Eo2 k'r

E:2 k'r

+ + + a2 li' (ro r 80 k)2 (10 r -. k r0 - k r)2

The condition of the maximum output P is,

I-hPr= 0I,lPk-= 0

•

that is,
+ = + = + 'Jc' {ro2 1/) 'r (ro1 [Bo - k]') k 80 ro2 802

hence,

substituting this in P, we get :

P

--

- - , E.2 0 4 f"o

the same condition as for continuous current.

That means,

"' No mattw Aow large tM self-inductwn of an alternating

current circ-uit ill, l>g a proper use of ,hunted capacity the out-

put of tM circuit can always be railled to the same aB f<»' con-
ti1n1,0U1J et1,rren't8 ,· that is, the effect of self-induction upon the

o-utput can entirely.and completely be annihilated."

8TBINJ£ETZ ON <JOJlPLBX QU.ANTil'IEB.

III. THE ALTERNATE CURRENT TIU.N8FORM'.ER.

A. General, Rem,arks. ,

In the coils of an alternate current transformer, E. JI. F. is in-

duced by the alternations of the magnetism, which is produced

by the combined magnetizing effect of primary and secondary

current. If,

= X maximum magnetism, = N frequency (complete cycles per second), = 1i number of turns,

the tdfective int<mlli,flg of the induced E. :w:. F. is,

E = +'2r. n N .JI. lo-' = 4.44 n N .M 10-a

Hence, if E. 11. F., frequency and number of turns are given,

or chosen, this formula gives the 'ffl,(J{Dimum magnetism,

Eto• M= t"ir.nN

To produce the magnetism JI. of the transformer, a JI. 11. F.
Fis required, which is determined from the shape and the magnetic characteristic of the iron, in the usual manner.
At no load, or open secondary circmt, tbe 11. 11. F. Fis furnished by the "exciting current," improperly called the "leakage currentr'
The energy of this current is the energy commmed by hysteresis and eddy-currents in the iron ; its intensity represents t~e

M, M. F.
This current is not a aine-w01Ve, but is di8torted by hysteruu. It reaches it.s maximum t.ogether with the maximum of magnetism, but passes through zero long before the magnetism.
This exciting current can be dissolved in two components:
a sine-wO/Ve 000 of IKJtt,a,/, intensi,ty a11d Bq_'IJ,(U power witl,, tM Q:cit,ing current,, and a wattless complez higher Jw,rmonic.
Practically this separation is made by the electro-dyne.mometer. Connecting ammeter, voltmeter and wattmeter into the primary of an alternate current transformer, at open secondary circuit the instrument readings give the current 000 in intensity and phase, but suppre~s the higher harmonics.
In Fig. 5 such a. wave is shown in rectangular coordinates. The sine-wave of magnetism is represented by the dotted curve
X, the exciting current by the dist.orted curve c, which is sepa-
rated int.o the sine-wave 000 and the higher harmonic C.

87'EIN.MJJJTZ ON 00.V:PLEX QUA.N'l'ITIES

49

As seen, the higher harmonic is small, even in a closed circuit

transformer, compared with the exciting current 000, and since 000 itself is only a few per cent. of the whole primary current, the higher harmonic can for all practical purposes be suppressed.

All tests made on transformers by electro-dynamometer

methods suppress the higher harmonic anyway.

Representing the exciting current by a sine-wave 000 of equal effective intensity and equal power with the distorted wave, the

exciting current is advanced in phase age.inst the magnetism by
an angle a, which may be called the "angk ef hysuretic advance ef phase." This angle a is very small in all open circuit

transformers, but may be as large as 40° to 50° in closed circuit

transformers.

-t 1 _[_:

-~_h J- I - r-- • 1

r-1

'

I -i I : -·_, :-j

-C

II I

r , - ,·

I

M

.

l -L I I

L_ L I I L ~ -.

Li--~~! --1~P ! __._~_.--'~-

I

C
. '
II ~
~ I~ . I L -~

Fm. 5.

We can now in the usual manner dissolve the sine-wave of ex-

citing current 000 into its two rectangular components: h, the "hysuretic energy current" at right angles with the

magnetism, hence in phase with the induced E. ~1. F., and, there-

fore repruenting consumption qf energy ; and,

g, the "magnetizing current" in phase with the magnetism,

hence at right angles with the induced E. :M. F., and, therefore,

wattkas.
= h 000 sin a, and can be calculated from the loss of energy

Ly hysteresis (and eddies), for it is:

= h energy co.nsumed by hysteresi11

primary E. x. F,

1

so

STEINMETZ ON COMPLEX QU.ANTITIES.

And since 000 can be calculated from shape and characteristic of the iron, the angle of hysteretic advance of phase a is given by:

sin a= ohoo·
= The magnetizing current g 000 cos a does not consume
energy (except by resistance), and can be supplied by a condenser of suitable capacity shunted to the transformer.
Since in the closed circuit transformer h, which cannot be supplied by a condenser, is not much smaller than 000, there is no advantage in using a condenser on a closed circuit transformer. In an open circuit transformer, however, or transformer· motor, 000 is very much larger than h, and a condenser may be of advantage in reducing the exciting current from 000 to h.
B.-The Ofosed Circuit Lighting Traniform,er.

The alternate current transformer with closed magnetic cir-
cuit, when feeding into a non-inductive resistance, as lights, can
be characterized by four constants :
= p resistance loss as fraction of the total trans£ormed power: = resistance loes
p total power at full load.
= s hysteretic loss as fraction of the total tre.m~formed power:

_
e -

bysteretic Joss total power

t" f }l
a u

IOad •

= a E. M. F. of self-induction as fraction of total E. ». F.:

= 8(.'}f·induction
t1 total E. 11. F. at full load.
= r magnetizing current as fraction of total current :

= r

magnetizin1t curreut -total current

st

full

load.

Denoting In primary:
no
ro
Oo .E~
If
1
00
we have then:

In secondary :

and

n1

" " "
" "

= number of turns.
= resistances. = currents. = induced E. M. F.'s.
= E. ::.v. F.'s at terminals.
= currents at full load.

BTBINMBTZ ON COMPLEX QU.4NTITIES.

:n

8 a= .M' where

8 = magnetism leaking between primary and secondary. = .M magnetism surrounding both primary and secondary.

= h

!

001'

g
t" =7J;r

Hence at the fraction IJ of full lead.

{J = 01
0/

choosing the induced E. x. F. as the real axis of coordinates,

the magnetism as the imaginary axis of CO(!rdinates), we have,

= Primary exciting current, 000 h + j g

0 = Primary current correspond- }
ing to secondary current Oi, f

n1 0 n0 1

hence, t.otal primary current,

and, ratio of current.s,

Since, however,

= + • + • ) + 0.oo

I,,

J g = 0.o1 ( £

J ?'

=

n-1

+ r, 1
VJ

(

!

J•'t') = -ni

OY c 1

'J.j r ,

no

no

t.

we have, substituted,

Ratio of 0urr8'nfAJ.

= + + ~o
(; 1

f 1l-i } 1
no

_!
,4;

j

!._
/j

I~ .

+ ) +(; ) = ~ 1/(1 ~ 2

2
or,formediumandlargeload,

=

nn

.
0

{

1

+

!
{}

_

+2~IP}

The E. :M. F. at the secondary terminals is,

= Ee

Ea -

01 1'( = E. { 1 -

pi } 2 .

at the primary terminals,

= = ti E

Eo+ rl.lo,('ro - J• Bo)

E,o

1) 1+

P

{}
2

- J•

iiif

52

STEINME'l'Z ON COMPLEX QUANTITIES.

hence, since

~o = - L"

{Jo E

1,

n1

• i Ratio of E. M. F.'s at termiwJJs. = :; {1 + /Hl - j a{}}

= no ~ 1 + /' {} + 62, {P_ l

11,1 '

2 {

Difference of phase w between E. M. F. at primary terminals

and primary currents.

Since we have seen, that multiplying a complex quantity by
+ (cos <u j sin oj), means rotating its vector by angle ,ti, the

difference of phase between primary current and E. K. F., ii i@,

given by, or,

+ C0 = a (cos w j sin cu) E
a (cos w. + J•sm• w•) = -(·EJo-·

where wis the difference of phase, and a a constant.

= !~ Since in the present case the secondary current is in phssc
with the secondary E. M. F., it is, b

combining this with the foregoing, we have,
1~ a b (cos w+ j sin cu) = ~;

hence,

(!!:!) + -=-) = a b cos w

2
(-~!_) (1 - p 8

= 2

no

,.,

no

r( a Rill = ( ::

+; )

l,

UJ

(1

lJ

and,

J)ifference qf phnRf' betuwm, pr/mary cu.rren'. and E. M. F. at
tNm in ala.

= + tan w a lJ !:_ ' ,"}

S'l'l!lINMHTZ ON COMPLEX QU.AN'J.'ITIEB.

58

hence:

" With vMyi,ng load IJ, the dijference of phase wur th~ lag,

1/ = = first decreaBea, reache, a minilmum at a IJ !. or lJ

r ,

/'}

a

and af'terwarda increa868 again."

At light loads it is mainly the magnetizing current r, at large

load the self-induction a, which determine the lag.

= + ,; The formula, tan w a {J

is only an approximation, and

ceases to hold for any light load, where we have to use the complete expression.

alJ+~
+; tan ,,, = - - - -I} 1 -pi IJ.

+ ; ), The 8jfteiency is, 1 - (P ij

and the •

1/ = hence a minimum at, lJ

E , the point of maximum efficiency.

f>

Let, as an instance, be :

no= 10 ni

p = .02
a= .06

= e .03
= 1" .08

hence,
= at full load, {J. 1,

u; + + = 0. .1 (1 .oa = .0032) .1033

+ + :: = 10 (1 .02 = .0018) 10.2~

= + = = ~ tan ,,, .06 .08 .14, or, ,u

0 ,

= at 100% overload,{} 2,

== energy factor, cos "' .tt9

-i: + + = .1 (1 .015 .0008) = .l0lti

= + + ~
.E,

10 (1

.04

.0072) = 10.-t-7

BTJIINJIBTZ ON OOJIPLRX QUAN1T1'IB8.

= tan a, .12 + .04 = .16, or, "' 9°,
= energy factor, cos "' .99
at one-half load :

= iJ =

.5:

Oo
7Y =

.1 (1 +

.06 +

.
.0128)

.1073.

'

li1

= = EEro, 10 (1 + .01 + .0005) 10.11.

tantii=.03+.16=.19, or ii=ll0 ,energy factor: cos ii=.98. at one-tenth load:
= (Ofot 1 (1 + .8 + .32) = .162.
or more exactly,
= + + = .1 t'(l .3)2 .82 .153.
EKoi = 10 (1 + .002 + .0000) = 10.02.
= tan •cii .006 + .8 = .806.
or more exactly,
=1-•0.0o6o+2+·8.s=·62, or"-'=s2°, energy factor: cos a..,=.Sa",
at open secondary :
tan w= :~: = 2.67, or w= 70°, energy factor: cos w= .34,

the minimum leg takes place at :

1/· = -/J

= 08
.06

1.155,

or 15i per cent. overload, and is:

ta.nw=.0693+.o693=.1386,orw=7.9°,energy factor: cos,.i=.99,

the efficiency is a maximum at:

= = -/J•

V A

1.03 ·-

1.225 .

. 02

or 22i per cent. overload, and is :

1-.0245- .0245 = .951,or 95.1 percent.

0.-Genera~ Equati-011,8 of .Alternate Current Tramfqr,ner.
The foregoing considerations will apply strictly only to the
r closed circuit transformer, where p, a2, e, are so small that their

STEINXE1."Z ON OOMPLEX QUANTITIES.

M

product.s a.nd higher powers may be neglected when feeding into

a non-inductive resist.ance.

The open circuit transformer, a.nd in general the transformer

feedi~ into a.n inductive circuit-in which case a and r become

of greatly increased importance-requires a. fuller consideration.

Let:

= n0 = r 0

and
"

n1 r1

number of turns, resI.S..w... .nce,

= = = 80 21r N L 0 and 81 2 1r N L1

self-inductances, hence:

= L, = r 0 - j 80 and 11 = r 1 - j _81 impedances of the two

transformer coils.

The secondary terminals may be connected to a. circuit of re-

sistance 1-l and inductance 8, hence of impedanr.e I= R - j 8.

Then we have :

Magnetism:

j .M.

Secondary induced E.M.F.: .E,_ = ¥2 ;r n1 .lV .M·10-·.

Primary induced E. M. F.: E0 = t -2 Tr n0 1V M 10.-• -nno, .E,_.

Secondary current:

C1 -

B.E,_L- -- (J- l+r1)-Ej(. S+s1)

or: where:

+ E~ (S s,l + b = (It + r1)2 + (8 a2)2

Primary current correspondin{;? hereto:

+ 0

=

-n 1
no

01

=

n- 1 a
no

•
'I

n -

1

b.

• ·no

Primary exciting current:
= Ooo h +} g,
hence, total primary current:

or: where:

l~ = 0 + l~0 = (.n,1 a + h) + J• (n;;_1 1, + fl)

0

0

+ 00 = o j d,

---

56

8TBINJ£BTZ ON COMPLEX QUANTITIES.

E. M. F. consumed by primary impedance :
+ 0 0 I 0 = (c j d) (r0 - j B0 ) + + = (c r0 d B0) j (d r0 - C a0)•

. E.M.F. consumed by secondary impedance:

0, I. = (a+ j b) (r1 - j a1)
+ + = (a r 1 b a1) j (b r1 - a a1).

hence, E. :M. F. at secondary terminals:

+ + _ r;o
E C - fit -

()" .l _
1 l-

E
1

j l

1

-

(a 1·1

b 81) j (b r 1 -
E;

a a1) }
•

E. :M. F. at primary terminals:

.

+ + + +j E = Eo 00 I;, = E { l (c r0 d B0)

(d 1"0 - C 80 ) } •

Substituting now in Oi, 00 , E~, Ethe values of a, b, c, d, we

get:

8econdarg cmTen t:

Ki (R + r1)

+ . E.., (8 a1)

O,=(R + r 1)2 + (8 + s1)2 + J (R + r,)2 + (8 + a.)"·

Primary current :

j ~Ei(R+r,)

l

j ~1Ei(8+B1)

l

f O.,= ( (R~r1f+(8+a1) + h + j ( (R~1·i)''+8 +a1)1+g) •

E. M. F, at secondary te-rmin,(l.ls,

E. _ ' -

E 1

j l

1 -

ri( R+r1)+s1(S +s,) } .E { Sr1-RB1 } (R+r,)2+(8-f-s1l -J 1 (R-tr1)2+(8+a1)2

E. M. F. at primary term,inal&,

E

-[nu

Ei

{ i+(

n1

2
) ru(R

+

·r1! +

Bt,(S +

&1)} +(r/i +

8<(1)7

n1

1io (ll+r,)2+(8 +si)2

...l

+J•

[ n1
nu

E 1

{

1·0 (S (R

+ +

s,) - s" (R + r1) l + r1)2 + (S + s1)2 f

(r. g _ O

•- h)]
vu

the general equati<>n8 of the alternate current transform.er,

repre8enti11g the currents and E. M. F.'B in intemnty and plia88.

In general, the percent8.e,cre of resistance in inductance will be

the same, or can without noticeab]e error be assumed the 88.Jlle

in primary as in secondary circuit.

That means,

STHINJIIITZ ON OOJfPLBX QUANTIT/88.

substituting this, we get, E. M:. J'. at a6CO'llitary 'tlmnitiallJ,
= Ee E1 [1 - A - j B]
E. M:. F. at primary U'rminah,
B} E = : Ei {1 + A + j + (r0 h ·+ Bog) + j (r0 g - Bo h)

where,

.A = r1 (R + r1) + R1 (8 + 81)

(R +

+ 2
r1)

(8 +

2 8 1)

+ + B _

r1 8- 81 R

- (R r1t (8+ s1)2

Therefore we get for the closed circuit transformer, feeding
= into a non-inductive resistance, 8 0 .

= + ..El
.E;

no
n,

{ 1

+

1,

o22 fl

n,{ OO.o= n

1+e+~ +°'O'} 2 2

at full load.

IV. DISTRIBUTED CA.PA.CITY, INDUOTANOE, LEAKAGE A.ND
RE818TANOE.
In many cases, especially in long circuits, as lines conveying altern&t.e power currents at high potentials over long distances by overhead conductors or underground cables, or very feeble currents at extremely high frequency, as telephone currents, the consideration of the rwtance-which consumes E. M. F. in phase with the current-and of the vnductance -which consumes E. M:. F. in quadrature with the current-is not sufficient for the explanation of the phenomena. taking place in the line, but seven} other factors have to be taken into account.
In long lines, especially at high potentials, the electro~tatic
capa,cwg of the line is sufficient to consume noticeable currents.
The charging current of the line-condenser is proportional to the difference of potential, and one-quarter period ahead of the E. x. F. Hence it will either increase or decrease the main current, according to the relative phase of the me.in current and the E. JI. F.
In consequence hereof, the current will change in the line •from point to point, in intensity as well as phase, and the E. M.

GB

BTEINHETZ ON OOJIPLEX QUANTITIES.

F.'s consumed by resistance and inductance wi11, therefore, change also from point to point, being dependent upon the current.
In considering the effect of capacity, it is not permissible, however, to neglect the inductance, since in overhead lines the inductance is usually at least of the same magnitude as the condenser effect, and is not negligible in concentric cables even. In the latter, however, and to a lesser extent everywhere else, still other factors have to be considered.
The line consumes not only currents in quadrature with the E. :M. F., but also curren-ta in phase with the E. M. F.
Since no insulator has an infinite resistance, and at higher potentials not only leakage, but even direct eacape of ilectrit:ity 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. x. 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 R. x._F. of inductance are wattless.
Furthermore, the alternate current passing over the line induces in all neighboring conductors secondary currents, which react upon the primary current and thereby introduce E. M. F.'s of mutual, inductarwe into the primary circuit.
Mutual inductance is neither in phase nor in quadrature with the current, and can, therefore, be dissolved into an 1:nergy component of mutual inductance-which acts like an increase of resistance-in phase with the current, and a wattleas oomp<ment, in quadrature with the current-which decreases the self-inductance.
The mutual inductAnce is by no means negligible, as for instance, its disturbing influence in telephone circuits shows.
The alternate potential of the line induces by electro,tatic iln,.. jl.1.umce electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges in the line wires. This electrostatic influence requires 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 alternate electro-magnetic :field of force, set up by the line current, causes in some materials a Joss of energy by elecflro-magnetic hystereRia, requiring the expenditure of an E. M. F. in phase with the current, which acts like an increase of resis-.

STHINJfETZ ON OOMPLHX QU.ANTITIES.

59

t&nce. The wattless component of this •E. x. F. disappears under "inductance," or rather we must say, that the hya'ttlremc E. x. F. is the eM'rfl'!I component of ind·uctance. This magnetic hysteresis loss may take place in the conductor proper, if iron wires are used, and will then be very serious at high frequencies, as with telephone current.a, or it may take place in the iron armor of the cable, etc.
The effect of the "eddy owrren'ts" is referred to ~!ready under "mutual inductance," whose energy component it is.
The alternating electrostatic field of force, expends energy in dielectric.a by what I called" diekctric hysteresis." In concentric cables, where the electrostatic gradient in the dielectric is comparatively large, the dielectric hysteresis may at high potentials even consume more energy than the ohmic resistance.
The dielectric hysteresis appears in the circuit as consumption of a current, whose component in phase with the :E. K. F. is the " diekct,ric 8'Tl,6rgy cu,rrent "-the component in quadrature with the E. K. F. disappears in the "condenaer C'Urrent," whose energy component the dielectric energy current is.
Besides this, there is the increase of ohmic resistance due to un8f_fU,(1,l diatriJ>,u,tion of C1trrent, which, however, is practically never large enough to be noticeable.
Hence we have the phenomena.: Rwtance--consumes E. K. F. in phase with current. &l,f-vruluetance, and its energy component dectro-rnagn,etic hyaterw. Af'U,f/u,'ll ir,,d,ucta,nce, and its energy component eddg currenta. L6aka,ge--consumes current in phase with E. K. F. Capacity, and it.a energy component dulecflric hysterem.
I"tftuenoe. This gives, as the most general case, per unit length of line;
= E. x. F.'s CO'fl,IJUm,ed in phase with the C'ltrrent 0, (llnd r 0,
representing consumption of energy and due to : Resilltance, and its increase by unequal current distribution. Energy component of ailf-ind!uction, or electro-magnetic lvgsterma. Energy component of tnlutual, inductance, or indvced
C'urrents.
= E. x. F.'s ConlJ'UIITJ,6d in quadrature with the current 0, and
B 0, being wattless, and due to: SelfJVTUJ,uct,ance.

80

STllINMETZ ON OOJIPLBX QU.ANTITIEB.

Mut'U,(J,i, vn,d/11,C"ta,nce.
= Ourr(NU(J oon8'lJ/fll,6Q, in pluue with t~ E. K. F. E anul, IJ E,

representing consumption of energy, and due to:

Leoi,age through the insulation, including silent discharge.

Energy component of capacwJ/, or diekctric hyat6rellu.

Energy component of ilectr011tatic injftu,ence.
= OtllN'enta consumed in quadrature with the E. M. F. E and

x E, being we.ttless, and due to :

Oapaoi,ty.

ElectroflUl,tic influence.

Hence we get four constants:

r, a, {J, x.

representing the coefficient, per unit length of line, of:

E. M. F.'s consumed in phase with current,

r.

E. M. F.'s consumed in quadrature with current, 8.

Currents consumed in phase with E. K. F., IJ.

Currents consumed in quadrature with E. M. F., x.

This line we may assume now as feeding into a receilver C'ircuit

of any deacription, and determine current and E. K. F. at any

point of the circuit :

That is:

E. K. F. and current (differing in phase by any desired angle)

may be given at the terminals of the receiver circuit. To be

determined is the E. M. F. and the current e.t any point of the

line, for instance e.t the generator terminals.

Or:
= Impedance 1 R - j S of receiver circuit, and ,E, M. F. E 0 at
generator terminals are given. Current and E.M.F. at any point

of circuit are to be determined, etc.

The cases, which are usually and solely treated:
= 1. Current 0 at end of line, tbat is open circuit.
= 2. E. K. F. 0 at end of line, that is line grounded, and

3. Line of infinite length

are evidently of little practical interest, but of importance is

only the case of a line feeding into an inductive or non-induc-

tive receiver circuit.

Of the four line constants, r, 8, {}, 21:, usually:

r is me.inly the resistance, per unit length of line.
= = a is mainly 2 r. N L, where L coefficient of self-induction,

per unit length of line.

7

STEINMETZ ON UOM]'LEX QUANTITib:S.

61

= , ,'J iB mainly ~ where i the insulation resistance, per unit 1,

length of line.
= = x is mainly 2 r. N K, where K the capacity, per unit length

of line.

·o Counting now the distance 21 from a point of the line, which

has the E.M.F., the current,

E.
01

= =

e1 c1

+ j + j

e/
c/

and counting 21 positive in the direction of rising energy,

counting 21 negative in the direction of decree.sing energy,

we have at any point 21, in the line differential d 21:

Leakage current,

E {} d :c

Capacity current, - j Ex d 21

hence, total current consumed by d 21:

= d O E (fJ - j x) d 21, or:

.

dO=E(lJ-jx)

(1.)

d 21

E. K. F. consumed hy resistance,

C' r d .v

E. K. F. consumed by inductance, - j Os d ;_•

hence, total E. K. F. consumed by d ir.:
d E = 0 (r - j a) d .v, or:

= d- daE,

O( • )
TJB

(2.)

These Fundamental Differential EquationH (1.) and (2.) are
symmetrical in Oand E.
Differentiating these equations :

d-d2w- 0 _-dd- .vE (lJ _ J•X:)

(3.)

= dd2reE

d O

.

d :c (r -J a)

and substituting (3.) in (1.) and (2.), gives:

2 ~:

= E (fJ - j x:) (r - j s)

(4.)

dd2aiO2 -_ 0 (tJ - j x) (r - j a)

(5.)

The Differential Equation of O and of E. Tlieae Dii/ferential, Equations are identical, and consequently 0 and E are functions differing by their limiting conditUJ'l'l4
orJ,y.

62

STEINJIBTZ ON COMPLEX QU.ANTITIES.

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

= 2
~ ; v, (IJ - j x) (r - j a)

(6.)

and are integrated by :

= 1D a e"x

where e is the base of natural logarithms. For, differentiating this, we get:

hence:

= = d2 to ,,p a e"x # w
dar
= # (IJ - j x) (r - j a)

or:

= v ± t' (IJ - j x) (r - j 1)

hence, the comp'let8 int,egrol is ;
= + w a ,+r,x b r"x

where a and b are the two constant.a of integration. Substituting :
V 7 a-j p

in (7.), we have:

= (a - j {1)' (IJ - j x) (r - j a), or:

= <r-{/'

IJr-xs

}

2a{1

= IJs+xr

herefrom : rr + fl' = t' (lP + ~) (r + tr)

and:

v' = a

½J ii'(,~ + r) (r2 + tr) + (IJ r -

v 11 = ½t ii' (,~2 + r) <r2 + s2) - (/} r -

(7.) (8.)
(9.)
(10.)
l x ;) (11.) f "s)

substituting (9.) in (8.):

(a-j{tµ:

-(a-jb);r

w=at

+be

=

. a:e
a e (cos

{1 aJ-j

sin

p aJ)

~
+be (cos

p x

+ j

sin

/J

a,)

= + a;r

-<lZ

az

-ar

w (a e b e ) cos {1 w - j (a e - b e ) sin {1 a,

(12.)

the general solution of differential equations (4.) and (5.) Differentiating (8.) gives:

= d w

u

-N:

d a: v (a r - b e )

.

STEINMETZ ON OOMPLEX QUANTITIES.

68

hence, substituting (9.):

= dw
d

(a-j

a.z -az

az ~

P)t(ae -be )cospa:-j(ac +be )sin

pa,,

1(13.)

0

substituting now O for w, and substituting (13.) in (1.), and writing:

(a-j P) a= A
= (a-j P) b B

we get the

0

1 0MUn"a/, In.f;egral, E ~ of the Prob"lem.
1 ·pH.A eaz+Be-az)cospa:-j(A eaz-Be-az)sinpa:J

a-J 1

az

-az

aa,

-a:r

(14.)

E {}-J.X HAe -.Be )cosp.v-j(.A e +Be )sinpwf .J

where A and Bare the Constants of Integration.

If :

+ 0. = c1 j c/ is the current,

= + .E,. e1 j e/ is the E. M. F.,

{15.)

= at the point : a: 0,

We get, substituting (15.) in {14.)

= 2A I(ac1+Pci1)+(De1+ze.1) l+}{(ac11-Pc1)+(1?e/-xc,1H l (l6.)
2B= {(ac1+Pc/HDe1+xe11)}+j{(ac,1-Pc1)-(IJe11-xe1)! f
= If: I R - j 8 is the impedance of the receiver circuit, and

Eo = eo+Jeo1

(17.)

is the E. K. F. at the dynamo terminals, and
= = l length of line, we get at: a, 0:

hence:

O= A +B.
a-JP
E=.A-B
P-Jx

I _ -

IC_ 0 -

A A

-+

B B

a -j p p- j x'

.
or·

A-B=IP-}x

A+ B a -j p

(lS.)

= and at : a, l :

1 , al

-al

al

-al

l

Eo= {l-jx { (.Ae -Be )cos/U-j(.Ae +Be )sin 19l, {19.)

64:

ST/JJINMETZ ON COMPLEX QU.Alt"Tl1'IES.

Equations (18.) ·and (19.) determine the constants A and B,, which, substituted in (14.), give the final integral equations.
= The length : Xo 2pn: is a complete wave le~gth, that means,

= in the distance ~ 2 ; the phase of current and E. x. F. repeat, 1 in half this distance they arc just opposite. Hence the remarkable condition exists in a very long line, that at different points at the same time the currents flow in opposite directions, and the E. M. F.'s are opposite.
The JJifference of plw,,.e between current and E. M. F. at any point of the line is determined by :
+ = I(cos w j sin w) ft,

where I is a constant.
Hence, "' varies from point to point, oscillating around a medium position '"oo, which it approaches at infinity.
This difference of phase, towards which current and E. M. F. tend at infinity, is determined by :

+ = ~: J (cos '"oo j sin w00 )

= -a.r

or, substituting for 000 and Eoc their values, sine£

O, and

{tJ:
A e (cos pa, - sin pa,) cancels:

J (cos w00

(20.)

= This angle wf'f) O, that is, current and 1<:. M. F. come more = and more in phase with each other, if: {1 lJ - a :r O, that is :
a + f1 = /J + :r, or:
x ,1."-{P _ /P-x2
- 2 ,1. /i - ~ /J

+ - r subst1. tut.mg (10.), gi.ves:

/j r -
_(I

x 8

V8

X'I'

lP 2

-ff
IZ

,

S1'HINMIC1'Z ON OUXPLEX QU.J.NTITJE.-!.

hence, expanded :

= r + 8 D + x.

(21.)

that iR:

"The ratio of remtance t,o 111ul,U,()i,anoe equab., t!UJ ratio of

lealeage to capacity."

I :,--~-

- ·-~

DIF'l'ERENC.

\

., OF PttA81!,
+ao '

-+-.ao I

I

't

~.
0

-t;.-.

I

-10 I ;

I •

I I

-20 .

__., J

\

roi.n
Ill 000

\. , ~-°' 1

, - ...

I

I

- \b

/4

\ If - \

•

I r~ ,.._,,

--

L_

I

looo

'\, i
I',_ "",....

'e/

r

-

h...i I

··~

-I

--

~.!!!!!

J

0 000

,/ ,~ ..--'""

8 0,0,0

-40l

)

AIIP ~290- : t.'ooo

~-~/'
" ~

I . ,
I
I

I 1':: ~
/ ~o : 2 000

/

IOO !0,000

I

/,,. "\

I

I
E.
I

' --- I,
K /

c-

: /l

~\ y..,, 1/

I - - ,,,..... "--

J

1,/
, _I

__, v'

,. c--

v"'

, /

/

_ . .,J

, , I

... ,...,.1

r=I •=4

••=5b,ooo

k'=I! ,000

180 1,000 110 8 000 140 4,000 1IO 2,000 100 0 000 IO 8 000
s-o r!,,oo-o
I 40 4,000

H_- o I

4

T

3L 4

RO 2,000

!?L

, 3L

4

l 2

ll'IG. 6.

= Tliis angle w00 45°, that is, current and E. M. F. differ by

= + one-eighth period, if: D -

D

that is :

p

a

x

a

p

x,

a D+x
7-0~

or:

(22.)

__ ____..

..,

6e

STJCINMETZ ON COMPLIGX QUAN1.'I'l'lHH.

that is:
"Two of the four line constants must be = O, either ll and z
or I} and s." As an instance, in Fig. 6 a line diagram is sbown, with the
distances from the receiver end as abecisse. Figure 6 represents one and a half complete waves, and gives t.otal effective current, t.otal E. M:. F., and difference of phase between both, u • functions of the distance from receiver circuit, under the condi-
tions:
= = E. v. F. at receiving end: 10,000 volts, hence: Ei. e1 10,000.
Current at receiving end: 65 amperes at .385 energy coefficient, that is:
= + = + 01 01 j 01l ~5 60 .f.

Line constants per unit length :

r =1

= t'i ~ X 10-5

= = s 4 . it 20 X to➔

hence:

= a 4.95 X 10--s

= p 28.36 X 10-a

+ = a·i {P .829 X 10-1

= p = = = :vu L ~ ,r j21.5 length of lm. e, corresponding to

one complete period of wave propagation. A= 1.012 -1.206j
= + B .812 .71l4}
These values substituted give:

0 { /\47.3cosP.r+~7.4sinp.l,)-c~(22.8cos,1.r+32.6~in 11;r)}

~

~

l

+J {e (27.4cosp3'-47Jlsinp.r)+e (:-J2.6cos;1;;r-22.3sin19-z) 1

E: { e~(6450cosp.c+4410sin/~x)+e•-a.r(3580co8j9il'-4410sin11.:r) f

+j { ,cu,(4410cos,9a-:-6450sin19a-)-r-az( 4410cos,9..r-8580sin,1a-) :-

P tan ,000

_
-

{J - a it _
---- -
alJ+/1x

• -

•0.'."..1',

};THIN.tllaTZ ON l'tAMPLh.'X QUAA'TJTI.BH.

67

S0111,e Particu.l,ar Oaae11.

...1. Open Oircuit at End of Line.

= ::r 0.

o. = 0.

= + + A (lj. e1 x e.1) j (/J e11 - x e.) = - B

hence:

= . + • 1

• cu -a.r

a.r

-aJi

f

fa..'

A { (£

p £ ) cos 11 :;r - j (£ - £ ) sin a, (

t'J-J x .

.

+ , ) {' =

1 . ~ A 1j (,a.r -

(l. - J /J '

, -uJ) cos 11 a, - j (1az

-a.r sin 11 z }

B. Line G-round,ed at llnd.

= -.r 0.

l!,~ = 0.

A = (a c, + /1 c.1) + j (a c/ - Pc1) = B,

hence:

= -~ . + 1

. IU'

-a.e

a.r

-a,:

•

I(

t

-J K A

·', (,
,

-

e

) cos /1 a, - j (t

t ) sin fJ z lf

1

. n..r -a.r

a.r -a.r •

t + C

=

a--- ;-.

.,
fJ

. .:1

J (a

£ ) cos /J .r -j"(• - t ) sin /J z :-

0. Irifim,uely Long (}onducu»-11.

.Replacing a, by - 'J'J, that is, counting dist;ance positive in the

direction of decreasing energy, we have:

= a,

®:

(! = 0,

E = 0:

hence: R=O

and:

= + ,· J

-cu

E

'J
t

- J•

X

A

t

(cos fj (I)

sin p w)

p 1

-az

0-=

a--

- .'.j J ,1

A

,

(cos j1 a,+ j sin w)

revolving decay of the wave.

The rota/, impedance of the infinite circuit is:

l=E C

- a-}/~
- ;~ -.i X

r + _ (a li /J x) -.i (11 lJ - ax)

+ - --

tP

"The infinitely long conductor acts like an impedance

l I I

68

81'EINJIE1'Z ON OOMPLEX QU.Alt'TI1'IE&

+ I = a 0I} 1+.tri x: -

.7· ~ lJ - a x: that . is like a resist.ance

r7-2 + r '

'

+ R

=

a ~I} +fr1 ", combined with· an

inductance S =

~ -~ - ":/:' v-+~-

Herefrom we get the difference of phase between E. M. F. and

current :

= - = tan co

8
R

)IJ-ax:
a___D_,,--+-----=11,..x...:._

which is constant at e.11 points of the line:

=\I'":., = = If: /J 0, s 0, we have: a= 19

_, hence:

= tan co 1, or:

= ,o 45°.

that hi, current and v.. M. F. differ by one-eighth period.

Flo. 7.
I). Generat,or Fl!.eding into Closed Ci·rc·uit:
= Let ((J 0 be the center of the cahle. It is then:
Ox.= O_x,
Ex. = - E;'_x ,
= = hence: .E' 0 at ~ 0,
that means, the equatious are the 88.me as in B., where the line
= is grounded at .r 0.
V. PoLYPHASJt; 8YSTJ1;1t1s.
In polyphase systems, we have two ways of connecting then circuits of an n-phase generator with each other and with the line.
1. The atar connection, represented diagrammatically in Fig. ·7, where the n-circuits, containing E. M. F.'s differing from each

STEIN.METZ ON COJIPLEX QU.ANTITIBS.

69

other b,v ~ of a period, are connected together at one end into a
n neutral pf>flfl:t 0-which may either be grounded or not-while the other ends of the circuit a.re connected to th~ line-wires, and:
2. The ring connect-ion, represented by Fig. 8, where then generator circuits are connected in closed circuit, and the n line wires connected to the points of contact of adjacent circuits.
Outside of the generato.r the two systems are identical. The consumer circuits may now either be connected between any pairs or sets of line-'\\ires, or between the wires and a neutral point. O', which may be grounded, or connected to the neutral point of the generator 0. 1. Let now, in the 11tar connectinn.. of generator, Ebe the E.

lil. 1-·. of one branch of the generator, and let 1, 2, . . . n be the generator circuits.

Since the E. M.. F.'i; of adjacent circuits differ by_!_ of a period, n

= •2 ,r

n 1 -

,

an d

rol•,~Qt•ion

by

2:

is represented algebraice.Uy by mul-

tiplication with:

= l=C082n-,r+J·S· ll2ln,-r

¥"1

(1.)

The E. M. F. in an~· circuit i is:

E;=e;E

(2.)

Hence, if O; is the current in circuit i, and I is the impedance per generator circuit, we have:

70

S'l'H1NK/J.'7'Z ON fJOMPLEX QU~NTITIES. •

E. v . .,._ at terminal i of generat;or:

E',. = E,.- 0,-I=e,. E- O,.I

(3.)

And the E. M. F. at the end of a. line of impedance I,., con-
nected to terminal i:

E",- = E,-- 0,-(I +I;)= e,. ll- O,(I + I,-) (4.)

Let now E.x denote the difference of potential between any

pair of terminals i and x,.

where:

ll;x = - Ex;

(5._1

we have~

E. v. F. of generat;or, acting between terminals i, and x:

= E;x (e; - i.1-) /f

(6.)

Difference of potential between generat;or tenninals i and x:

= E',-x (,J- r) If-- l(l~ - Ox)

(7.)

Difference of potential between lines i and x:
= E";z (e' - r) E- I(O, - Ox) - (/,. 0, - Ix Ox) (8.)

If now Oix represents the current, which passes from line i to x (and which is determined by the impedance Lx of the apparatus connected between i and x:

CY,. .

_
x

.E;x -1-,x.

and if Ou denote~ the current passing from line i to neutral

point

01 ,

we

have:

II

= 0,- ~ .1· (]ix

(9.)

0

Furthermore, if the ueu\ral points O and O' are insulated.

II
~' 01 = 0

II
~; 0,-"= O
1
If, however, the neutral point O a11d O' are grounded, or connected oogether:

II

II

~; 0,-= ~; O,."

(11.)

2. In the case of the ring connected generator, the generator

j

STEINJIE1.'Z ON OOJIPLBX QU.ANTITJES.

71

Take the place of the E. v. F.'s
E;;+i
of the star connection, hence the E. M. F. between any pair of terminals i and x is:

,·

All tl).e other considerations remain essentially the same, so that:
"Any polyphase system of the E. M. F.'s:
= = = ,._
E; ei E, i 1, 2, . . . n, a t' 1 (13.)

can be dissolved by Ohm's law:

E= OI

(14.)

Fm. 9.

and Kirchhoff's laws:

l = E 0 in any closed circuit,

(15.)

l = O 0 at any point of distribution." (16.)

It would carry me too far for the scope of this pa.per, to enter further into the general theory of the polyphase systems, and it may be sufficient therefore, to show in a particular instance, taken from the threephase system, what remarkable phenomena. can be expected in polypha.se systems.

Urwalanceil Thruphaae System.
let, in a threephase system, Fig. 9, with star connected generator,
E, e E~ t2 E be the E. 11.. ll'..'11 of the three generator branches, where:

•
8TEIN.ME1'Z· ON COMPLEX Ql'.ANTT7'1E8.

a =8y- l = - 1 +J. t'3
2

_ -1-.i +f8

-

2

= Le(I impedance per generator branch, = L impedance per line,

= and)et one pair of lines be connected by an impedance 12• We have then, if C the current flowing in this loaded

branch-the two other branches being unloaded, or open-that

is, the system "unbalanced.''

i,;. M. F. in generator circuits:

E

'

,E

,,E

} (17.)

Potentials)t generator terminali,: l!.,'-01 eE+ OJ ,'E

l
J (18.)

Potentials at end of line~:

+ E-0(1 /1)}
, E+ O(I+ 11) (19.)

EaE

Hence, differences of potential at generator terminals :

E (1 - e) - 2 0 I -loaded branch. }

+ , E (1 - ,) 0 I }

+ 01 ,,J'..'(l - ,)

-unloaded branches.

(20.)

Difference of potential at ends of line :

+ / E (1 - c) - 2 0 (I 1) -loaded branch. }

+ eE(l-s)+C(I+/1) t
,"E(l _ ,) 0 (I+ 11) i -unloaded branches.

(21.)

Hence, current in loaded branch:
= + O .E.. (1 - E) - 2 () (1 I,~
Ii
or, expanded:

= + rt
l.

12

E

(l 2

(-I

+1)

L

)

,

as

was

to

be expected, sm• ce

(22.)

+ + / It 2 (/ 1) is the total impedance, E (1 - e) the E. M. F. of

this circuit.

Rubstituting (22.) in (20.) and (21.), we get:

l"1'HINJl81'Z ON <JOJIPLJ,,'X QUAN7'1Tll!l8.

78

(23.)

Difference of potential at ends of line:

+ + Jf(t -- a) (1- 2 (I+ Ii) --) -loaded brand1.

..

12 2 (/ /1)

\I

+ +1
E(l - t) (, 12 2t;t 11))..

l (24.)

-unloaded hran<"he..14.

fi) + ~ 1
E(l - t) ( £2+ 12 211

These are three diffe.rent values. That means: '· Loading in a three phase system one branch only, the potentials of the two unloaded branches become unequal also." It is self evident, that this phenomenon of unbalancing does not take place in the three phase system only, but just a.swell in any other polyphase system, and that the amount of unbalancing depends upon the constants of the circuit, hence, can by a proper arrangement be reduced to almost nil, or can be exaggerated
greatly by an improper choice of circuit constant.
As an instance, we may consider the numerical example: Generator Jt~. 'M. F. 100 volts between terminals, henee:

_E(l - a)= too

Resistance per generator branch,

.r)l ohms.

Inductance per generator branch,

.05 ohms.

Hence, impedanc~ per generator branch, .01 - .05j.

Case 1. Non-inductive line of .1 ohms. I '• • Non-inductive load of .1 ohms. \

Case 2. Non-inductive line of .1 ohms. , Inductive load of - j ohms. ,

Case 3. Inductive line of Inductive load of -
( 1a.se 4. Inductive line of Non-inductive load of

.lj ohms.,
.i ohms. !
.lj ohms. f. .1 ohms. ,

74

S1'1GINltlHTZ ON OOMPL.KX QUAN7'I1'IKS.

Substituting these values in equations (22.), {23.) and (24.) we
get (writing all the quantities in the form, c (cos"'+ j sin w):

1. Non-inductive line and non-induetive load:; Ii= .1, ft·= 1:

0=~1.6(.99+.0Sj)

Ei= 98.0( J.m+.osJ)

E;=81.H( .99+.osJ)

E,,= 95.9(-.51+.86})

~= 92.6(-.44+.90/)

Ea= 102.9(-.47-.88})

1;=98;7(-.4-1-.911)

= = 2. Non-inductive line, inductive load: L .1, f., - j :

0=89.0(.20+.98})

E,~90.9

h;= 89.1( .98-.2-0j)

~=9'7.8(-.47 +.88j)

I~= 104.9(-.42+.92})

Ea= 97. 8(-.47-.88})

A;= 89.~(-.49-.86})

a. Inductive line and inductive load: Ii= -.tj, ]',= - j :

C= 7:l.O(.Ot+jJ

E~=92.!i(l- .OV)

h.;=78.4(1- .Olj)

fl..;= 98.8( -.4:7+.s~j)

.1~=95.6( -.41 +.91J)

~=98.7( -.4-8-.8~j)

1~=94-.ti( -.41-.91j)

= 4. Inductive line, non-inductive load: 1. = - .1 j, ~ 1:

0=94.0(.96+.2&1)
E. = 95.~ .»»+.o~j)
~= 95.2(-.50+.86j)

= E; 94.1( .96+.2s;)
I½= 86.1(-.52+.85j)

fi~= 102.7(-.4:7-.88j)

fa_;= 109.6(-.41-.9lj)

Remarkable is in 1. and in 4-. the rise. qf poter,tial in the line in the branch .1;..

Apparently these values look rather irregular, wmetimes the one. sometimes the other unloaded branch being higher. Look-

ing closer into it, however, we can not fail to see the regularity

displayed in the variation of potential, which makes it poBSible to control this phenomenon.

Lynn, Mt8ll., .July, 1898.

D1scuss10N.

PRoJ.o'. MACJ.o'ARLAN 1<; :-1 wish to make a remark in regard t:o

the fundamental principle of the use of complex quantities.

The letter j was first introduced as a distinguishing index with-

= - out mathematical meaning, and afterwards defined b~ the equa-

tion l

1. Such definition is ambiguous, for 1t refers to

orthogonal projection of a straight line upon another straight line,

and the right angle may be at the former straight line or at the

]atter. Tfte latter case is the ordinary meaning.

STHIN.V/1:1.'Z ON COMPLEX QU.ABTITIES.

75

It is not true that algebra is limited or bounded by the ordi-

nary complex quantity. There is a more general complex quan-

tity which applies to space, and of which the complex quantity

in a plane is only a special case. MR. STEINKKTZ :-In introducing j, first as distinguishing index

and then defining it as +" - 1 my object was to intreduce the

complex quantity in an elementary and graphical manner, without

reference to higher mathematics. To make the reasoning more
= complete, I might have added that the definition j .V 1
<loes not contradict the original definition of j as index without

numerical meaning, since in the range of ordinary numbers

t1 - 1 is meaningless.

.

From the mathematical standpoint the complex <\uantity can

directly be introduced without further explanation, smce in pure

1nathematics, for instance, the theory of functions, the plane, is

known as the standard representation of the complex quantity.

Referring to Prof. Maefarlane's last remark, my meaning is

that the complex quantity is the last and most general <dgebraic

number. No further generalization of numbers exists which

fulfills the fundamental condition of ~braic numbers, that if a
product is zero one of the factors must'be zero. This is the rea-

son why the complex quantity of the higher order does not prove

ae useful in space as the algebraic complex quantity in the plane.

The following paper was then read :