it may be assumed at 30,000 maxwells with cast-iron pole-pieces, and at 45,000 with wrought-iron pole-pieces. In multipolar machines, the figures are from 5000 to 7000 higher in each case. A formula for the length of active armature conductor is E' The value of k is determined by multiplying 10-8 by a factor ranging from 50 to 72, depending on the percentage of polar arc, i.e.. the percentage of the armature subtended by the pole-pieces. If the percentage of polar arc is 50 the factor is 50, if the percentage is 100 the factor is 72. V varies from 35 in a 1 kw. machine to 50 in a 200 or 300 kw. machine with a drum armature. With ring armatures, in high speed machines. V varies from 65 in a 1 kw. machine to 75 in a 25 kw.. 85 in a 300 kw, and 100 in a 5000 kw. machine. On low speed dynamos the figures are approximately one half the above. (E+ e). In series machines, under 1 kw.. e is from 40 to 20 per cent of E; in machines of from 1 to 25 kw., from 20 to 10 per cent; in 25 to 500 kw. machines. from 10 to 4 per cent; and in machines of over 500 kw. from 4 to 2.5 per cent of E. In shunt-wound machines e has approximately one half the value used in series machines; in compound-wound machines approximately three quarters the value used in series machines. The diameter of the armature core is found by means of the assumed velocity and the given revolutions per minute, D=(12×60V) ÷ (r.p.m. X). The area of the conductors on the armature depends on the amount of current to be carried. d2 300 I'÷p. In a series machine I' = I; in shunt and compound machines l'=I+i. The current consumed in the shunt field varies with the size of the machine approximately as follows kw. = 1 081 5 10 .061 .051 20 .041 50 .031 100 .02751 500 2000 .021 .0151 i In large machines it is better, in order to diminish the eddy currents, to make the armature conductors in the form of a cable, than to use single wires. If the conductor on the armature is a single wire the thickness of insulation varies from .012 to .020 inch, depending on the voltage. If the conductor is a cable, each strand is insulated with a thickness of from .005 to .01 inch and the entire cable is covered with insulation of thickness varying from .005 to .01 inch. = In a small machine with but a single layer of conductors on the armature Ll÷ N. (1.885,000D × h) ÷ d2. 2 (n2 X nз) ÷ n1; For drum armatures for ring armatures N N = (n2 X nз) n1. A general formula given by Wiener for the length of armature is The minimum number of bars on the commutator is Cmin Amperes: over 100 100-50 b 20. The number which may be used, provided it does not fall below C'min is For drum armatures the number of conductors attached to each commutator bar must be an even number. The quotient of C, obtained as above, by the largest even number which will give a result greater than Cmin is the proper number of commutator bars for drum armatures. For ring armatures it is the quotient of C by the largest number which will give a result greater than Cmin. In each case the divisor is the number of conductors which should be attached to each bar. The flux through the armature is: where m is a factor depending on the percentage of polar arc. Assuming 100 per cent and 50 per cent as the limits of polar are, the following are the respective values of m at those limits In bipolar, smooth armature. machines m 1.00 and .70; in bipolar, toothed armature machines m = 1.00 and .55; in smooth armature multipolar machines m = 100 and .625, with from 4 to 12 poles: m = 1.00 and .60 with from 14 and 20 poles. With toothed armatures the figures are slightly lower. The area of the field magnet cores depends on the flux to be generated. φ' = φ Χ λ. A value for A is assumed, which will vary with the size and type of machine. By means of this assumed value the principal dimensions of the magnetie circuit are calculated. The true value of A is next calculated by means of the formula The permeance of a path is its magnetic conductance. The stray paths are those across the pole-pieces, across the magnet cores and between the pole-pieces and the yoke joining the magnet cores. With the new value of A, p' is recalculated. If the true and assumed values of give a large difference in flux then the dimensions of the circuit must be changed and à recalculated. The areas of the various portions are found by dividing the total flux by the allowable flux density. The allowable flux densities in maxwells per square inch are as follows: Wrought iron, 90,000; cast steel, 85.000; east iron, 40,000. The various areas being known, the winding of the magnets is calculated as shown in the section on Strength of the Magnetic Field. EXAMPLE. Design a 200 K.W. bipolar, smooth drum armature, shunt dynamo, with wrought-iron pole-pieces, and cast iron magnet cores and yoke. Volts, 500; amperes, 400; R.P.M., 450. Assume Hi = D = = = .0251; percentage of = 420.7 feet. 40,000; V = 45; e = 03E; i polar arc = 85. Then E' 515; I' = 410 and k 68 X 10(515 X 1 X 100,000,000) ÷ (68 × 45 × 40,000) (12 X 60 X 45) ÷ (450 × 3.1416) = 22.91 inches. d2 300 X 410 ÷ 1 = 123,000. In this size of machine it is desirable to use cables. Each conductor may be composed of three cables in parallel, each composed of seven wires. A No. 12 B. & S. gauge wire has an area of 6530 cir. mils, and 7 X 3 X 6530 137,130, which is near enough to d2. To find d' Number of strands on a diameter = 3. Insulation on each strand .005; insulation of cable = .008: diameter No. 12 wire = .080808: d' 3 X (.0808 + 2 X .005) + (2 X .008) = .2884 inch. Assume h = .625; n1 = 3; na .625 .2884 = 2 + Then L inches. = 22.91 X 3.1416.2884 = 249; ns = (12 X 3 X 420.7) + (2 X 249) = 30.41 (249 X 23) ÷ 4 = 41 (too small); Cmin=515 X1: 10 = 51.5; C (249 × 2 ÷ 3) ÷ 2 = 83. .. C = 83. N (2 X 249) X 2 ÷ 3 = 332, 6 X 1 X 515 X 1,000,000,000 ÷ 332 X 450 = 20,683,000. Assume m = .94; B = 20,683,000 (3.1416 X 22.91 X 30.41 X .94) = 10777. To calculate & would require more space than can be spared bere. Assume à 1.34. Φ' 1.34 X 20,683,000 27,715,220. Area of magnet cores = 27,715,220 ÷ 40000 = 692 sq. inches. 29.8 inches. ALTERNATING CURRENTS.* The advantages of alternating over direct currents are: 1. Greater simplicity of dynamos and motors, no commutators being required; 2. The feasibility of obtaining high voltages, by means of static transformers, for cheapening the cost of transmission; 3. The facility of transforming from one voltage to another, either higher or lower, for different purposes. A direct current is uniform in strength and direction, while an alternating current rapidly rises from zero to a maximum, falls to zero, reverses its direction, attains a maximum in the new direction, and again returns to zero. This series of changes can best be represented by a curve the abscissas of which represent time and the ordinates either current or electromotive force (e.m.f.). The curve usually chosen for this purpose is the sine curve, Fig. 172; the best forms of alternators give a curve that is a very close approximation to the sine curve, and all calculations and deductions of formulæ are based on it. The equation of the sine curve is y = sin x, in which y is any ordinate, and x is the angle passed over by a moving radius vector. In After the flow of a direct current has been once established, the only opposition to the flow is the resistance offered by the conductor to the passage of current through it. This resistance of the conductor, in treating of alternating currents, is sometimes spoken of as the ohmic resistance. The word resistance, used alone, always means the ohmic resistance. alternating currents, in addition to the resistance, several other quantities, which affect the flow of current, must be taken into consideration. These quantities are inductance, capacity, and skin effect. They are discussed under separate headings. The current and the e.m.f. may be in phase with each other, that is, attain their maximum strength at the same instant, or they may not, depending on the character of the circuit. In a circuit containing only resistance, the current and e.m.f. are in phase; in a circuit containing inductance the e.m.f. attains its maximum value before the current, or leads the current. In a circuit containing capacity the current leads the e.m.f. If both capacity and inductance are present in a circuit, they will tend to neutralize each other. Maximum, Average, and Effective Values. The strength and the e.m.f. of an alternating current being constantly varied, the maximum value of either is attained only for an instant in each period. The maximum values are little used in calculations, except in deducing formulæ and for proportioning insulation, which must stand the maximum pressure. The average value is obtained by averaging the ordinates of the sine curve representing the current, and is 2 or 0.637 of the maximum value. The value of greatest importance is the effective, or square root of the mean square," value. It is obtained by taking the square root of the mean of the squares of the ordinates of the sine curve. The effective value is the value shown on alternating-current measuring instruments. The prod 66 uct of the square of the effective value of the current and the resistance of circuit is the heat lost in the circuit. The comparison of the maximum, average, and effective values is as follows: =1.41XEEffec. = EEffec. EMax. X0.707; EAver.-EMax. X0.637; EMax. Frequency.-The time required for an alternating current to pass through one complete cycle, as from one maximum point to the next (a and b, Fig. 172) is termed the period. The number of periods in a second is termed the frequency of the current. Since the current changes its direction twice in each period, the number of reversals or alternations is double the frequency. A current of 120 alternations per second has a period of 1/60 *Only a very brief treatment of the subject of alternating currents can be given in this book. The following works are recommended as valuable for reference: Alternating Currents and Alternating Current Machinery, by D. C. and J. P. Jackson; Standard Polyphase Apparatus and Systems, by M. A. Oudin; Polyphase Electric Currents, by S. P. Thompson; Electric Lighting, by F. B. Crocker, 2 vols.; Electric Power Transmission, by Louis Bell; Alternating Currents, by Bedell and Crehore; Alternating-current Phenomena, by Chas. P. Steinmetz. The two last named are highly mathemat and a frequency of 60 The frequency of a current is equal to one half the number of poles on the generator, multiplied by the number of revolutions per second Frequency is denoted by the letter f The frequencies most generally used in the United States are 25, 40, 60, 125, and 133 cycles per second The Standardization Report of the A LEE recommends the adoption of three frequencies, viz. 25 60. and 120. With the higher frequencies both transformers and conductors will be less costly in a circuit of a given resistance, but the capacity and inductance effects in each will be increased, and these tend to increase the cost. With high frequencies it also becomes difficult to operate alternators in parallel. A low frequency current cannot be used on lighting circuits, as the lights will flicker when the frequency drops below a certain figure. For arc lights the frequency should not be less than 40. For incandescent lamps it should not be less than 25. If the circuit is to supply both power and light a frequency of 60 is usually desirable. For power transmission to long distances a low frequency, say 25, is considered desirable, in order to lessen the capacity effects If the alternating current is to be converted into direct current for lighting purpose, a low frequency may be used as the frequency will then have no effect on the lights. Inductance. -A current flowing through a conductor produces a magnetic flux around the conductor. If the current be changed in strength or direction, the flux is also changed, producing in the conductor an e.m.f. α E. CURRENT FIG. 172. b whose direction is opposed to that of the current in the conductor. This counter e.m f. is the counter e.m.f. of inductance It is proportional to the rate of change of current, provided that the permeability of the medium around the conductor remains constant. The unit of induct. ance is the henry symbol L. A circuit has an inductance of one henry if a uniform variation of current at the rate of one ampere per second produces a counter e.m.f. of one volt. The effect of inductance on the circuit is to cause the current to lag behind the e.m.f. as shown in Fig. 172, in which abscissas represent time, and ordinates represent e.m.f. and current strengths respectively. Capacity. Any insulated conductor has the power of holding a quantity of static electricity. This power is termed the capacity of the body. The capacity of a circuit is measured by the quantity of electricity in it when at unit potential. It may be increased by means of a condenser. A condenser consists of two parallel conductors, insulated from each other by a non-conductor. The conductors are usually in sheet form. The unit of capacity is a farad, symbol C. A condenser has a capacity of one farad when one coulomb of electricity contained in it produces a difference of potential of one volt. The farad is too large a unit to be conveniently used in practice, and the micro-farad is used instead. The effect of capacity on a circuit is to cause the e.m.f. to lag behind the current. Both inductance and capacity may be measured with a Wheatstone bridge by substituting for a standard resistance a standard of inductance or a standard of capacity. Power Factor.-In direct-current work the power, measured in watts, is the product of the volts and amperes in the circuit. In alternating-current work this is only true when the current and e.m.f. are in phase. If the current either lags or leads, the values shown on the volt and ammeters will not be true simultaneous values. Referring to Fig. 172, it will be seen that the product of the ordinates of current and e.m.f. at any particular instant will not be equal to the product of the effective values which are shown on the instruments. The power in the circuit at any instant is the product of the simultaneous values of current and e.m.f., and the volts and amperes shown on the recording instruments must be multiplied together and their product multiplied by a power factor before the true watts are obtained. This power factor, which is the ratio of the volt-amperes to the watts, is also the cosine of the angle of lag or lead of the current. Thus PIXEX power factor = IXEX cos where is the angle of lag or lead of the current. The A watt-meter, however, gives the true power in a circuit directly. method of obtaining the angle of lag is shown below, in the section on Impedance Polygons. Reactance, Impedance, Admittance. In addition to the ohmic resistance of a circuit there are also resistances due to inductance, capacity, and skin effect. The virtual resistance due to inductance and capacity is termed the reactance of the circuit. If inductance only be present in the circuit, the reactance will vary directly as the inductance. If capacity only be present, the reactance will vary inversely as the capacity. Inductive reactance =2xjL. The total apparent resistance of the circuit, due to both the ohmic resistance and the reactance, is termed the impedance, and is equal to the square root of the sum of the squares of the resistance and the reactance. Impedance = Z=√R2+(2πfL)2 when inductance is present in the circuit. Impedance = Z = √ R2+ (2=1c)2 when capacity is present in the circuit. Admittance is the reciprocal of impedance, = 1÷Z. If both inductance and capacity are present in the circuit, the reactance of one tends to balance that of the other; the total reactance is the algebraic sum of the two reactances; thus, 1 1 2 Total reactance = X = 2πfL 2πfC = In all cases the tangent of the angle of lag or lead is the reactance divided by the resistance. In the last case Skin Effect.-Alternating currents tend to have a greater density at the surface than at the axis of a conductor. The effect of this is to make the virtual resistance of a wire greater than its true ohmic resistance. With low frequencies and small wires the skin effect is small, but it becomes quite important with high frequencies and large wires. The following table, condensed from one in Foster's "Electrical Engineers' Pocket-book," shows the increase in resistance due to skin effect. Skin-effect Factors for Conductors carrying Alternating Currents. For virtual resistance, multiply ohmic resistance by factor from this table. |