CHAPTER XVI 

POWER, AND DOUBLE-FREQUENCY QUANTITIES IN 

GENERAL 

135. Graphically, alternating currents and voltages are repre- 
sented by vectors, of which the length represents the intensity, 
the direction the phase of the alternating wave. The vectors 
generally issue from the center of coordinates. 

In the topographical method, however, which is more con- 
venient for complex networks, as interlinked polyphase circuits, 
the alternating wave is represented by the straight line between 
two points, these points representing the absolute values of 
potential (with regard to any reference point chosen as coordi- 
nate center), and their connection the difference of potential in 
phase and intensity. 

Algebraically these vectors are represented by complex quan- 
tities. The impedance, admittance, etc., of the circuit is a com- 
plex quantity also, in symbolic denotation. 

Thus current, voltage, impedance, and admittance are related 
by multiplication and division of complex quantities in the same 
way as current, voltage, resistance, and conductance are related 
by Ohm's law in direct-current circuits. 

In direct-current circuits, power is the product of current into 
voltage. In alternating-current circuits, if 

the product, 

is not the power; that is, multiplication and division, which are 
correct in the inter-relation of current, voltage, impedance, do 
not give a correct result in the inter-relation of voltage, current, 
power. The reason is, that E and / are vectors of the same fre- 
quency, and Z a constant numerical factor or "operator," which 
thus does not change the frequency. 

179 


180 ALTERNATING-CURRENT PHENOMENA 

The power, P, however, is of double frequency compared with 
E and /, that is, makes a complete wave for every half wave of 
E or 7, and thus cannot be represented by a vector in the same 
diagram with E and I. 

Po = EI is a quantity of the same frequency with E and /, and 

thus cannot represent the power. 

136. Since the power is a quantity of double frequency of E 
and /, and thus a phase angle, 6, in E and I corresponds to a 
phase angle, 2 6, in the power, it is of interest to investigate the 
product, EI, formed by doubling the phase angle. 

Algebraically it is, 

P = EI = {e' -i-je''){i' -\-ji'') 

Since j- = — 1, that is, 180° rotation for E and I, for the double- 
frequency vector, P, j" = + 1, or 360° rotation, and 

i X 1 = J, 
lXj=-j. 

That is, multiplication with j reverses the sign, since it denotes 
a rotation by 180° for the power, corresponding to a rotation of 
90° for E and 7. 
Hence, substituting these values, we have 

p = [EI] = {eH' + eii/ii) -]- Jie'H' - eH''). 
The symbol [£'7] here denotes the transfer from the frequency 
of E and 7 to the double frequency of P. 

The product, P = [£^7], consists of two components: the real 
component, 

pi = [EIY = (eH' + giizii); 
and the imaginary component, 

jpi ^ j[Eiy = j(e'H' - elf 11). 
The component, 

pi ^ [Ely ^ (giji + giiiii), 

is the true or "effective" power of the circuit, = EI cos (EI). 
The component, 

pi = [Ely = (giir - eifii), 
is what may be called Che "reactive power," or the wattless or 
quadrature volt-amperes of the circuit, = EI sin (£"7). 


DOUBLE-FREQUENCY QUANTITIES 181 

The real component will be distinguished by the index 1; the 
imaginary or reactive component by the index, j. 

By introducing this symbolism, the power of an alternatmg 
circuit can be represented in the same way as in the direct-cur- 
rent circuit, as the symbolic product of current and voltage. 

Just as the symbolic expression of current and voltage as com- 
plex quantity does not only give the mere intensity, but also the 
phase, 

^ =. gi +jeii 

^ = \e' + ell' 
tan 6 = —r, 

so the double-frequency vector product P = [EI] denotes more 
than the mere power, by giving with its two components, P^ = 
[Eiy and P' = [EIY, the true power volt-ampere, or "effective 
power," and the wattless volt-amperes, or "reactive power." 
If 

E = el + jell, 
/ = i^ + ji^\ 
then 


^ = Ve'' + el 


/ = ^/^ ' 


+ ^•ll 


and 


or 


pi := [Ely = (eH'i + eiizii), 
pi = [EI]i = (giH-i _ gi^-ii)^ 


pi' J^ pr-=. e^\i' + gii\-u= _^ gii\-i' _^ f,v^n' 

= (gi' + eii')(ii' + z'li') = (Ely = Pa^ 

where Pa = total volt-amperes of circuit. That is, 

The effective power, P\ and the reactive power, P\ are the two 

rectangular components of the total apparent power, Pa, of the 

circuit. 

Consequently, 

In symbolic representation as double-frequency vector products, 

powers can be combiyied and resolved by the parallelogram of 

vectors just as currents and voltages in graphical or sijmbolic 

representation. 


182 ALTERNATING-CURRENT PHENOMENA 

The graphical methods of treatment of alternating-current 
phenomena are here extended to include double-frequency 
quantities, as power, torque, etc. 

2? = p- = cos 6 — power-factor. 

P' 

q = p- = sin 6 ^ induction factor 

■* a 

of the circuit, and the general expression of power is 
P = Pa(p-\- jq) = Pa (cos 9 -\-j sin 6). 

137. The introduction of the double-frequency vector product, 
P = [EI], brings us outside of the limits of algebra, however, 
and the commutative principle of algebra, a X b = h X a, 
does not apply any more, but we have 

[EI] unlike [IE] 

since 

[^7] = [EIV-hj[Eiy 

[IE] = [IE]'-{-j[IEV = [Ely - j[EIV, 

we have 

[EI]' = [IE]' 

[Ely = - [lEY 

that is, the imaginary component reverses its sign by the inter- 
change of factors. 

The physical meaning is, that if the reactive power, [£'/]', 
is lagging with regard to E, it is leading with regard to 7. 

The reactive component of power is absent, or the total 
apparent power is effective power, if 

[Ely = (e'H' - eH'') = 0; 

that is, 

e^ _ i^ 
e' ~ T' 
or, 

tan {E) = tan (7); 

that is, E and 7 are in phase or in opposition. 

The effective power is absent, or the total apparent power 
reactive, if 

[Ely = {eH' + e'H'') = 0; 


DOUBLE-FREQUENCY QUANTITIES 183 


that is, e^ 


or, 

tan E = — cot / ; 

that is, E and / are in quadrature. 

The reactive component of power is lagging (with regard to 
E or leading with regard to /) if 

[EI]'> 0, 
and leading if 

[EIV< 0. 

The effective power is negative, that is, power returns, if 

[EIY< 0. 
We have, 

[E, -!] = [- E, I] = - [EI] 
[-E,-I] = -^[EI]" 

that is, when representing the power of a circuit or a part of a 
circuit, current and voltage must be considered in their proper 
relative phases, but their phase relation with the remaining 
part of the circuit is immaterial. 
We have further, 

[E,3l] = -j[E,I] = [E,IV -j[E,IY 
[jEJ] =j[E,I] = -[E,'iy+j[E,'lV 
[jEj'l] = [^,/i = [EIV'Vj[E,IV' 

138. Expressing voltage and current in polar coordinates; 

E = e^-\- je^^ — e (cos a -\- j sin a) 
I = i^ -{- ji^^ = i (cos |8 + J sin /3) 

gives the vector power: 

P = et { (cos a cos /3 + ^'^ sin a sin jS) + (j sin a cos /3 + cos a j sin /3)} 

and since, by the change to double frequency: 

+ i' = + 1 . 

+ «i = - ja 
it is: 
P = ei { (cos a cos /3 + sin a sin /3) + j(sin a sin jS — cos a cos /?) } 

P = ei {cos (cc — j8) + j sin (a — /3)} 


184 ALTERNATING-CURRENT PHENOMENA 

and: 

the effective power: 

P^ = ei cos (a — 0) 
the reactive power: 

pi = ei sin (a — /3) 

We thus must note the distinction: 

E = ZI = {r -\- jx) (i^ + ji^^) = zi (cos y -\- j sin 7) (cos j8 + j sin /3) 
= {rp — a;^■ll) + j (n^^ + a-z^ = sz {cos (7 + /3) + j sin (7 + /?) } 
and: 

= [(e^ + je^O, (^^ + i*^0] = ^*" [(cos a + j sin a), (cos /3 +i sin /3)] 
= (eifi + e^/ii) + j {e^H^ - eH^^) = ei {cos (a - /3) + 

J sin (a - /3) } 

139. If Pi = [EJl], -P2 = [^2/2] . . . Pn = [Enin] 

are the syraboHc expressions of the power of the different parts 
of a circuit or network of circuits, the total power of the whole 
circuit or network of circuits is 

P = Pi + P2 

pi ^ pi, + pi, 

P> = P2'' + P2'' 

In other words, the total power in symbolic expression (effect- 
ive as well as reactive) of a circuit or system is the sum of the 
powers of its individual components in symboHc expression. 

The first equation is obviously directly a result from the law 
of conservation of energy. 

One result derived herefrom is, for instance: 

If in a generator supplying power to a system the current is 
out of phase with the e.m.f. so as to give the reactive power 
P', the current can be brought into phase with the generator 
e.m.f. or the load on the generator made non-inductive by in- 
serting anywhere in the circuit an apparatus producing the react- 
ive power — P'; that is, compensation for wattless currents in a 
system takes place regardless of the location of the compensating 
device. • 

Obviously, wattless currents exist between the compensating 
device and the source of wattless currents to be compensated 
for, and for this reason it may be advisable to bring the com- 
pensator as near as possible to the circuit to be compensated. 


+ . 

. . . +Pn, 

2+ . 

. . . +Pn\ 

. +PJ. 

DOUBLE-FREQUENCY QUANTITIES 185 

140. Like power, torque in alternating apparatus is a double- 
frequency vector product also, of magnetism and m.m.f. or 
current, and thus can be treated in the same way. 

In an induction motor, for instance, the torque is the product 
of the magnetic flux in one direction into the component of 
secondary current in phase with the magnetic flux in time, but 
in quadrature position therewith in space, times the number of 
turns of this current, or since the generated e.m.f. is in quad- 
rature and proportional to the magnetic flux and the number 
of turns, the torque of the induction motor is the product of 
the generated e.m.f. into the component of secondary current 
in quadrature therewith in time and space, or the product of 
the secondary current into the component of generated e.m.f. 
in quadrature therewith in time and space. 

Thus, if 

E^ = e^ -\r je'^^ = generated e.m.f. in one direction in space, 

1 2 = i^ + ji^^ = secondary current in the quadrature direction 

in space, 

the torque is „ ri-rrT n-i ,-ii 

^ D =- [EI]' = e^V - eVK 

By this equation the torque is given in watts, the meaning 
being that D = [EI]' is the power which would be exerted by 
the torque at synchronous speed, or the torque in synchronous 
watts. 

The torque proper is then 

D --^ 

^°- 27r/p' 

where 

p = number of pairs of poles of the motor. 

/ = frequency. 
In the polyphase induction motor, if I2 = i^ + ji^^ is the 
secondary current in quadrature position, in space, to e.m.f. Ei, 
the current in the same direction in space as £"1 is I\ = jli = 
— i^^ + i^^; thus the torque can also be expressed as 
B = [EJiY = e^H' - eH'K 
It is interesting to note that the expression of torque, 

D = [EI]', 
and the expression of power, 

P = [EI]\ 


186 ALTERNATING-CURRENT PHENOMENA 

are the same in character, but the former is the imaginary, the 
latter the real component. Mathematically, torque, in syn- 
chronous watts, can so be considered as imaginary power, and 
inversely. Physically, "imaginary" means quadrature compo- 
nent; torque is defined as force times leverage, that is, force 
times length in quadrature position with force; while energy is 
defined as force times length in the direction of the force. Ex- 
pressing quadrature position by ''imaginary," thus gives torque 
of the dimension of imaginary energy; and ''synchronous watts," 
which is torque times frequency, or torque divided by time, thus 
becomes of the dimension of imaginary power. Thus, in its 
complex imaginary form, the vector product of force and length 
contains two quadrature components, of which the one is energy, 
the other is torque: 

F -^ [f,l] = [f,lV^j[f,l]' 
and 

[/, lY = energy 
[/, l\' = torque. 


SECTION IV 
INDUCTION APPARATUS