CHAPTER XII. POWER, AND DOUBLE FREQUENCY QUANTITIES IN GENERAL. 102. Graphically alternating currents and E.M.F's are represented 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 co-ordinates. In the topographical method, however, which is more convenient for complex networks, as interlinked polyphase circuits, the alternating wave is represented by the straight line between two points, these points representing the abso- lute values of potential (with regard to any reference point chosen as co-ordinate center) and their connection the dif- ference of potential in phase and intensity. Algebraically these vectors are represented by complex quantities. The impedance, admittance, etc., of the circuit is a complex quantity also, in symbolic denotation. Thus current, E.M.F., impedance, and admittance are related by multiplication and division of complex quantities similar as current, E.M.F., resistance, and conductance are related by Ohms law in direct current circuits. In direct current circuits, power is the product of cur- rent into E.M.F. In alternating current circuits, if The product, P0 = EI= (Ml - *"/") +j (W POWER, AND DOUBLE FREQUENCY QUANTITIES. 151 is not the power; that is, multiplication and division, which are correct in the inter-relation of current, E.M.F., impe- dance, do not give a correct result in the inter-relation of E.M.F., current, power. The reason is, that El are vec- tors of the same frequency, and Z a constant numerical factor which thus does not change the frequency. 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 /. P0 = E I is a quantity of the same frequency with E and /, and thus cannot represent the power. \ 103. Since the power is a quantity of double frequency of E and /, and thus a phase angle w in E and / corre- sponds to a phase angle 2 w in the power, it is of interest to investigate the product E I formed by doubling the phase angle. Algebraically it is, P=EI= (* +>") (V1 +/z n) = Since j* = - 1, that is 180° rotation for E and /, for the double frequency vector, P,j* = + 1, or 360° rotation, and j x 1 =j 1 x>= -j That is, multiplication with / reverses the sign, since it denotes a rotation by 180° for the power, corresponding to a rotation of 90° for E and /. Hence, substituting these values, we have, p = [El] = (W1 + ^V11) +/ (W1 - A'u) The symbol [E /] here denotes the transfer from the frequency of E and / to the double frequency of P. 152 AL TERNA TING-CURRENT PHENOMEMA. The product, P = \E /] consists of two components ; the real component, JP1 = [EIJ = (W1 + e"in) and the imaginary component, JPJ =j The component, P1 is the power of the circuit, = E I cos (E /) The component, PJ = is what may be called the " wattless power," or the power- less or quadrature volt-amperes of the circuit, = E /sin (El}. The real component will be distinguished by the index 1, the imaginary or wattless component by the index/. By introducing this symbolism, the power of an alternat- ing circuit can be represented in the same way as in the direct current circuit, as the symbolic product of current and E.M.F. Just as the symbolic expression of current and E.M.F. as complex quantity does not only give the mere intensity, but also the phase, £ = jfc == P tan = -j so the double frequency vector product P = [E /] denotes more than the mere power, by giving with its two compo- nents P1 = [E I]1 and PJ = [E /]•>, the true energy volt- amperes, and the wattless volt-amperes. If E = POWER, AND DOUBLE FREQUENCY QUANTITIES. 153 then and P1 = or 2 2 22 22 22 22 +PJ =<* ,1 + *" / where ^ = total volt amperes of circuit. That is, The true power P1 and the wattless power P$ are the two rectangular components of the total apparent power Q of the circuit. Consequently, In symbolic representation as double freqi'ency vector pro- ducts, powers can be combined and resolved by the parallelo- gram of vectors just as currents and E.M.F's in graphical or symbolic representation. The graphical methods of treatment of alternating cur- rent phenomena are here extended to include double fre- quency quantities as power, torque, etc. P1 — =p = cos w = power factor. PJ — = q = sin w = inductance factor of the circuit, and the general expression of power is, = Q (cos co -\-j sin o>) 104. The introduction of the double frequency vector product P = \E I~\ brings us outside of the limits of alge- 154 ALTERNATING-CURRENT PHENOMENA. bra, however, and the commutative principle of algebra, a X b = b X a, does not apply any more, but we have, [El] unlike [IE] since we have [EIJ = [IEJ [EI]J=-[IE]J that is, the imaginary component reverses its sign by the interchange of factors. The physical meaning is, that if the wattless power [E 7p is lagging with regard to E, it is leading with regard to/. The wattless component of power is absent, or the total apparent power is true power, if [EI]J = (W1 - A'11) = 0. that is, or, tan (E) = tan (/), that is, E and / are in phase or in opposition. The true power is absent, or the total apparent power wattless, if [El]1 = (W1 + M* = 0 that is, *" _ i1 7 ~ ~/» or, tan E = — cot I that is, E and / are in quadrature, POWER, AND DOUBLE FREQUENCY QUANTITIES. 155 The wattless power is lagging (with regard to E or lead- ing with regard to /) if, and leading if, The true power is negative, that is, power returns, if, We have, [£, - 7] = [- E, 7] = - that is, when representing the power of a circuit or a part of a circuit, current and E.M.F. must be considered in their proper relative phases, but their phase relation with the re- maining part of the circuit is immaterial. We have further \EJT\ = -j [£, 7] = [E, iy -j \E, 7]1 \JE, 7] =j [E, 7] = - [E, Jy +j [E, 7]1 \jEjr\ = [£, 7] = [E7? +j [E, jy 105. If 7- = [^/J, 7>2 = [E2/2] . . . Pn = [Enln} are the symbolic 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 7^' = TV + T'ijJ. . • • + TV In other words, the total power in symbolic expression (true as well as wattless) of a circuit or system is the sum of the powers of its individual components in symbolic expression. The first equation is obviously directly a result from the law of conservation of energy. 156 ALTERNATING-CURRENT PHENOMENA. One result derived herefrom is for instance : If in a generator supplying power to a system the cur- rent is out of phase with the E.M.F. so as to give the watt- less power Pi, the current can be brought into phase with the generator E.M.F., or the load on the generator made non-inductive by inserting anywhere in the circuit an appa- ratus producing the wattless power — F$\ that is, compen- sation for wattless currents in a system takes place regardless of the location of the compensating device. Obviously between the compensating device and the source of wattless currents to be compensated for, wattless currents will flow, and for this reason it may be advisable to bring the compensator as near as possible to the circuit to be compensated. 106. 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 com- ponent of secondary induced 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 induced E.M.F. is in quadrature and proportional to the magnetic flux and the number of turns, the torque of the induction motor is the product of the induced E.M.F. into the component of secondary current in quadrature therewith in time and space, or the product of the induced current into the component of induced E.M.F. in quadra- ture therewith in time and space. Thus if E1 = £ +jea- — induced E.M.F. in one direction in space. 72 = z1 +j z11 = secondary current in the quadrature di- rection in space, POWER, AND DOUBLE FREQUENCY QUANTITIES. 157 the torque is By this equation the torque is given in watts, the mean- ing being that T = \E /]•>' is the power which would be exerted by the torque at synchronous speed, or the torque in synchronous watts. The torque proper is then where / = number of pairs of poles of the motor. In the polyphase induction motor, if 72 = il +/zu is the secondary current in quadrature position, in space, to E.M.F. Ej. The current in the same direction in space as El is /! =y72 = — z11 +//1; thus the torque can also be ex- pressed as 158 ALTERNATING-CURRENT PHENOMENA.