November 8, 1918. THE COLLIERY GUARDIAN. 967 THE POWER FACTOR. ni. (Specially Contributed.) The Effect of the Power Factor. In fig. 4 the lines OP, OP1, OP2, OP3 represent conventionally the output of an alternator and the input of an induction motor; the figure will be recog- nised as a variation of the conventional diagram used in the previous articles.* OP is the radius revolving round the centre O and making angles of increasing value with the base line OA. PN, P]? Nn P2, N2, and P3 N3 are perpendiculars dropped from the ends of the radii on the base line. OA represents the useful work done either by the alternator or by the motor. When the revolving radius coincides with it, the useful power is then the product of the pressure into the current. As the radius recedes from OA, and as the angle it makes with OA increases, so does the useful power decrease. ON, ON,, ON2 and ON3 represent the useful power when the angle of lag is represented by the angles PON, P1? ON,, P2, ON2 and P2ON3. These angles are the respective values of the angle <£. and the useful work in each case is found by multiplying the power (represented by the radius) by the cosine of the angle <£. It will be remembered that by trigonometry the cosine of any angle equals the base divided by the hypothenuse. In the case of the triangle PON, the cosine of the angle PON=—Multiplying both sides by OP we Ml OP get ON equals OP cosine PON, or OP cosine <£. It will be noted how, as the angle increases—as the radius moves away from the base line—the value of the length of the base line representing the useful work decreases, and from a table of cosines we find that cosine 0 = 1. This is the angle representing continuous current working, and those few cases in alternating current working where the power factor O 0 O Fig. 8. O NN,N? Na 18 25 37 45 60 66 16 25 37 45 60 66 18 25 37 45 60 66 18 25 37 45 60 66 Fig. 6. Fig. 4. Fig. 5, 90 0 Fig. 7 sc Fig. 4.—Conventional diagram representing output of an alternator or the input of an induction motor. Fig. 5.—Curve of cosine values. is unity and the current neither lags behind nor leads the pressure. Cosine 18 degs. equals 0-951, and represents the case in alternating current working where the bulk of the load consists of incandescent lamps and other apparatus very little subject to self- induction. Cosine of 25 degs. equals 0-9, is the case that is striven for in modern three-phase working by the addition, to plant that is already running, of some apparatus giving a leading power factor in order to counteract the lagging power factor due to the generators and motors. The cosine of 37 degs. is 0-798, practically 0-8, which is the standard figure taken by manufacturers for the output of their machines. In manufacturers’ catalogues it will be noted that alternators are listed to furnish, not so many kilowatts, but so many kilo-volt-amperes, with a power factor of 0-8. The cosine of 45 degs. is 0-707—a condition that is reported to be rather fre- quent in colliery working. The cosine of 60 degs. is 0-5, w’hich is also found in colliery working, and it is a very bad case. The cosine of 66 degs. is 0-40—a very bad case indeed, but one that is occasionally met with. These figures mean that the apparent outputs or inputs, kilo-volt-amperes, apparently furnished by the generator or a number of generators, and appa- rently delivered to a motor or to a group of motors, have to be multiplied by 0-95, 0-8, 0-7, etc., to deter- mine the actual useful output or input of the different machines. Fig. 5 is a curve of cosines prepared from the values of the cosine taken from the tables. The angles are represented by distances along the base line, and the value of the cosine, i.e., the power factor, is repre- sented by the ordinates, vertical heights of the different points in the curve above the base line. The use of the above curve, and the values given in table of cosines, will enable the useful output of any generator to be obtained from the apparent out- * Colliery Cuardian, August 23, 1918, p. 388 ; October 4, 1918, p. 706. put when either the angle of lag is known or the power factor itself. In modern generating stations there is now usually a power factor meter fixed on the main switchboard, which shows at any instant the resultant power factor of the service. In a later article the writer proposes to explain how the power factors of the different apparatus are combined to produce the resultant power factor. It would be useful to have a power factor meter attached to the switchboard of every important haulage or pumping plant underground. This would be another guide as to what the machines were actually doing, and it would show what steps should be taken to correct the power factor. By a few experiments that the elec- trical staff of the colliery could easily make also, when power factor meters are liberally applied, it would be seen when any particular motor was working with a low power factor, the conditions under which the low power factor arose, and in a great many cases it might be possible to raise the power factor by altering the working conditions of particular motors. From fig. 4 also the apparent output of any generator and the apparent input of any motor can be obtained; that is to say, the apparent output and input required respectively to give a certain useful output of electrical energy from the generator and of mechanical energy from the motor. By trigonometry again we know that the secant of an angle is found by dividing the hypothenuse by the base. Secant PON = . Multiplying ON we get OP =0N x secant PON ; that is to say, when the angle of lag and the useful output required are known, the apparent output (or in the case of a motor the apparent input) may be found by multiplying the useful output or input by the secant of the angle of lag. The same tables in engineering pocket books which give the values of the sines and cosines also give the values of the secant; for 0 deg. this is 1; for 18 degs. it is 1-051; for 25 degs. it is 1-103; for Fig. 6.—Curve of secant values. Fig. 7.—Curve showing value of wattless en-rgy with different values of <£. Fig. 8.—Curve of tangent values of 0" to 66°. 37 degs. it is 1-252; for 45 degs. it is 1-414; for 60 degs. it is 2-0; and for 66 degs. it is 2-458. Fig. 6 is a curve of secants set out in the same manner as the curve of cosines. The useful output has to be multiplied by these figures to find the apparent kilo-volt-amperes that must be furnished by the generator or delivered to the motor in order that any given useful number of kilowatts may be delivered to the distribution system in which it is to work, and that the desired useful horse-power may be furnished by a motor. The Wattless Current. Unfortunately, this is not the whole of the trouble brought into the problem of the transmission of power by means of alternating currents due to the operation of the power factor. There is also what is known as the wattless current, and as it flows at the same pres- sure as the useful current it may be called wattless or useless energy. The wattless current flows through the coils of the generators and motors, and through the cables connecting them, but it is displaced in time by a quarter of a period from the useful current; it is in quadrature with the useful current, and that is why it is useless. The trouble is that the wattless current, as it flows through the conductors in the armatures of the generators and through the stator coils of the motors, liberates heat during its passage through these con- ductors, exactly in the same manner as if it were doing useful work. It will be remembered that the heat liberated by an electrical current when flowing through a conductor is measured by the formula H = C2Rf ; where H is the heat liberated in time t, C is the current flowing in amperes, and R is the resistance of the conductor in ohms. The actual number of B.Th.U. delivered to a conductor whose resistance is known, by a current of known strength flowing through it for a given time, can be calculated. The point the writer wishes to be noted here is that the heating effect varies directly as the square of the current strength and as the resistance of the conductor. When copper conductors are heated their resistance increases, and so the number of B.Th.U. liberated by currents flowing through the armature coils of alternators, for in- stance, tends to increase with the time during which the current is flowing. The wattless currents therefore, by reason of the heat liberated in the conductors through which they flow, tend to lower the efficiency of both generators and motors, owing to the increase of the resistance of the conductors on the armature and stator coils, and, in addition, the heat liberated by them tends to lessen the strength of the useful current the con- ductors can carry—that is to say, to lessen the output of these generators and motors. The output of a generator and the input of a motor are measured by the pressure at the terminals of either, multiplied by the useful currents flowing. Currents flowing through any conductor forming part of a generator or motor are limited by the insulation, practically by the ability of the conductors to carry the current without damaging the insulation. The insulation of the coils upon the generators or motors is one of the very weakest points of the whole elec- trical system. All insulators employed for this pur- pose are mechanically weak; none of them is able to stand high temperatures without deterioration, and consequently the currents flowing in the coils of those machines have to be limited to such values that the heating effect will not deteriorate the insulation. With wattless currents claiming a certain share of the channel, so to speak, through which the currents flow, the useful currents have to be decreased, and so the useful outputs of generators and motors are diminished by the direct operation of the power factor, and by the secondary action of the heat liberated by the wattless currents. The triangles shown in fig. 4 enable the wattless current, or, rather, the wattless energy, to be calcu- lated when either the total apparent output and the angle , or the useful output and the angle <£, are known. The sine of any angle being equal to the perpendicular divided by the hypothenuse in the triangle PON, the sine of the angle PON = — Multiplying both sides by OP, we have PN = OP sine PON. In the conventional diagram shown in fig. 4, PN, PxNj, P2N2 and P3N3 represent the watt- less current or wattless energy, OP, OPr, etc., rep- resenting the total apparent output. It follows, therefore, that the wattless current, or the wattless energy, can be found for any value of the angle by multiplying the total apparent output by sine Again, the tables of sines given in the engineering pocket book shows that the sine of 0 deg. equals 0; this is again the case where the power factor is unity, as with continuous currents. The sine of 18 degs. is 0-309; of 25 degs., 0-422; of 37 degs., 0-601; of 45 degs., 0-707; of 60 degs., 0-866; and of 66 degs., 0-913. It will be seen, again, how the value of the wattless energy increases with the angle . With such a small value for the angle as 18 degs. representing a power factor of 0-95, the wattless energy is already 30 per cent., as against the useful energy 95 per cent.; this is explained later. With the standard power factor, as given in manufacturers’ catalogues 0-8, represent- ing a value of 37 degs. for the angle , the wattless energy is 60 per cent., against the useful energy of 80 per cent. With the value of the angle = 45 degs., the wattless energy is equal to the useful energy; with the angle = 60 degs., the wattless energy is 87 per cent., as against the useful energy 50 per cent. ; with the angle = 66 degs., the wattless energy is 91 per cent., as against the useful energy 40 per cent. And the figures go on, the wattless energy steadily increasing and the useful energy steadily decreasing, the one in proportion to the value of the sine of the angle , and the other in proportion to the value of the cosine, till when the angle is 90 degs. the wattless energy is 100 per cent., and there is no useful energy at all. It is a paradoxical case, of course, that never occurs, but in the early days of the distribution by alternating currents for town lighting something very closely approaching that condition was reached. Fig. 7 is a curve showing the value of the wattless energy, with different values for the angle . The point that will probably strike any student of the subject here is: we have the com- bined values of useful and wattless currents exceeding the apparent total output or inpjit. This arises from the fact that the- currents, as explained above, are in quadrature, their relation to each other being represented by the relation to the lines PN and ON, PjNl and ONX, etc. This brings us to another method of finding the value of any one of the three quantities, the total apparent output or input, the useful output or input, and the wattless energy. When any two of them are known, the third can be found by the aid of geometry. In any right-angled triangle the square of the hypothenuse is equal to the sum of the squares of fhe other two sides. In the triangle PON, OP2 = ON2 plus PN2, and similarly with the other triangles. If, then, we have the total apparent output and the useful output, we can find the wattless energy, this being equal to the difference between the square of the total apparent output and the square of the useful output. Thus if we have a total apparent output of, say, 100 kilo-volt-amperes, and a useful output of 80 kilowatts, the difference between the square of 100 and the square of 80 is 3,600, and the square root of that is 60. This is the value of the wattless com- ponent, and a reference to a table of sines and cosines will show that this is the value of sine when the cosine of angle = 0-8. This rule may be applied to find the value of the wattless component with any other apparent output and any other useful output. It is sufficient if one of the two quantities be known— the apparent output or the useful output, providing that the value of the angle is known. Reference