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THE THOUGHT IS IN THE QUESTION THE INFORMATION IS IN THE ANSWER

HAWKINS
ELECTRICAL GUIDE
NUMBER
FIVE
QUESTIONS
ANSWERS
&
ILLUSTRATIONS

A PROGRESSIVE COURSE OF STUDY FOR ENGINEERS, ELECTRICIANS, STUDENTS AND THOSE DESIRING TO ACQUIRE A WORKING KNOWLEDGE OF

ELECTRICITY AND ITS APPLICATIONS

A PRACTICAL TREATISE

by
HAWKINS AND STAFF

THEO. AUDEL & CO. 72 FIFTH AVE. NEW YORK.

COPYRIGHTED, 1914,
BY THEO. AUDEL & CO.,
New York.

Printed in the United States.

TABLE OF CONTENTS
GUIDE NO. 5.

CHAPTER XLVI
ALTERNATING CURRENTS

The word "alternating" is used with a large number of electrical and magnetic quantities to denote that their magnitudes vary continuously, passing repeatedly through a definite cycle of values in a definite interval of time.

As applied to the flow of electricity, an alternating current may be defined as: A current which reverses its direction in a periodic manner, rising from zero to maximum strength, returning to zero, and then going through similar variations in strength in the opposite direction; these changes comprise the cycle which is repeated with great rapidity.

The properties of alternating currents are more complex than those of continuous currents, and their behavior more difficult to predict. This arises from the fact that the magnetic effects are of far more importance than those of steady currents. With the latter the magnetic effect is constant, and has no reactive influence on the current when the latter is once established. The lines of force, however, produced by alternating currents are changing as rapidly as the current itself, and they thus induce electric pressures in neighboring circuits, and even in adjacent parts of the same circuit. This inductive influence in alternating currents renders their action very different from that of continuous current.

Ques. What are the advantages of alternating current over direct current?

Ans. The reduced cost of transmission by use of high voltages and transformers, greater simplicity of generators and motors, facility of transforming from one voltage to another (either higher or lower) for different purposes.

Figs. 1,206 to 1,212.—Apparatus which operates successfully on a direct current circuit. The direct current will operate incandescent lamps, arc lamps, electric heating apparatus, electro-plating and typing bath, direct current motors; charge storage batteries, produce electro-chemical action. It will flow through a straight wire or just as freely through the same wire when wound over an iron bar.

Figs. 1,213 to 1,217.—Apparatus which operates successfully on an alternating circuit. The alternating current will operate incandescent lamps, arc lamps, electric heating apparatus, alternating current motors. It will flow through a straight wire with slightly increased retarding effect, but if the wire be wound on an iron bar its strength is greatly reduced.

The size of wire needed to transmit a given amount of electrical energy (watts) with a given percentage of drop, being inversely proportional to the square of the voltage employed, the great saving in copper by the use of alternating current at high pressure must be apparent. This advantage can be realized either by a saving in the weight of wire required, or by transmitting the current to a greater distance with the same weight of copper.

In alternating current electric lighting, the primary voltage is usually at least 1,000 and often 2,000 to 10,000 volts.

Ques. Why is alternating current used instead of direct current on constant pressure lighting circuits?

Ans. It is due to the greater ease with which the current can be transformed from higher to lower pressures.

Ques. How is this accomplished?

Ans. By means of simple transformers, consisting merely of two or more coils of wire wound upon an iron core.

Since there are no moving parts, the attention required and the likelihood of the apparatus getting out of order are small. The apparatus necessary for direct current consists of a motor dynamo set which is considerably more costly than a transformer and not so efficient.

Ques. What are some of the disadvantages of alternating current?

Ans. The high pressure at which it is used renders it dangerous, and requires more efficient insulation; alternating current cannot be used for such purposes as electro-plating, charging storage batteries, etc.

Fig. 1,218.—Application and construction of the sine curve. The sine curve is a wavelike curve used to represent the changes in strength and direction of an alternating current. At the left of the figure is shown an elementary alternator, consisting of a loop of wire ABCD, whose ends are attached to the ring F, and shaft G, being arranged to revolve in a uniform magnetic field, as indicated by the vertical arrows representing magnetic lines at equidistances. The alternating current induced in the loop is carried to the external circuit through the brushes M and S. The loop, as shown, is in its horizontal position at right angles to the magnetic field. The dotted circle indicates the circular path described by AB or CD during the revolution of the loop. Now, as the loop rotates, the induced electric pressure will vary in such a manner that its intensity at any point of the rotation is proportional to the sine of the angle corresponding to that point. Hence, on the horizontal line which passes through the center of the dotted circle, take any length as 08, and divide into any number of equal parts representing fractions of a revolution, as 0°, 90°, 180°, etc. Erect perpendiculars at these points, and from the corresponding points on the dotted circle project lines (parallel to 08) to the perpendiculars; these intersections give points, on the sine curve, for instance, through 2 at the 90° point of the revolution of the loop, and projecting over to the corresponding perpendicular gives 2'2, whose length is proportional to the electric pressure at that point. In like manner other points are obtained, and the curved line through them will represent the variation in the electric pressure for all points of the revolution. At 90° the pressure is at a maximum, hence by using a pressure scale such that the length of the perpendicular 2'2 for 90° will measure the maximum pressure, the length of the perpendicular at any other point will represent the actual pressure at that point. The curve lies above the horizontal axis during the first half of the revolution and below it during the second half, which indicates that the current flows in one direction for a half revolution, and in the opposite direction during the remainder of the revolution.

Alternating Current Principles.—In the operation of a direct current generator or dynamo, as explained in Chapter XIII, alternating currents are generated in the armature winding and are changed into direct current by the action of the commutator. It was therefore necessary in that chapter, in presenting the basic principles of the dynamo, to explain the generation of alternating currents at length, and the graphic method of representing the alternating current cycle by the sine curve. In order to avoid unnecessary repetition, the reader should carefully review the above mentioned chapter before continuing further. The diagram fig. 168, showing the construction and application of the sine curve to the alternating current, is however for convenience here shown enlarged (fig. 1,218). In the diagram the various alternating current terms are graphically defined.

Fig. 1,219—Diagram illustrating the sine of an angle. In order to understand the sine curve, it is necessary to know the meaning of the sine of an angle. This is defined as the ratio of the perpendicular let fall from any point in one side of the angle to the other side divided by the hypotenuse of the triangle thus formed. For instance, in the diagram, let AD and AE be the two sides of the angle φ, and DE a perpendicular let fall from any point D of the side AD to the other side AE. Then, the sine of the angle (written sin φ) = DE ÷ AD.
It is evident that if the perpendicular be let fall at a unit's distance from the
apex A, as at B,

This line BC is called the natural sine of the angle, and its values for different angles are given in the table on page 451.

Fig. 1,220.—Diagram illustrating the equation of the sine curve: y = sin φ. y is any ordinate, and φ, the angle which the coil makes with the horizontal line, corresponding to the particular value of y taken.

The alternating current, as has been explained, 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 comprises the cycle.

This series of changes can best be represented by a curve, whose abscissæ represent time, or degrees of armature rotation, and whose ordinates, either current or pressure. The curve usually chosen for this purpose is the sine curve, as shown in fig. 1,218, because it closely agrees with that given by most alternators.

The equation of the sine curve is

y = sin φ

in which y is any ordinate, and φ, the angle of the corresponding position of the coil in which the current is being generated as illustrated in fig. 1,220.

Ques. What is an alternation?

Ans. The changes which the current undergoes in rising from zero to maximum pressure and returning back to zero; that is, a single positive or negative "wave" or half period, as shown in fig. 1,221.

Fig. 1,221.—Diagram showing one alternation of the current in which the latter varies from zero to maximum and back to zero while the generating loop ABCD makes one half revolution.

Ques. What is the amplitude of the current?

Ans. The greatest value of the current strength attained during the cycle.

The foregoing definitions are also illustrated in fig. 1,218.

Fig. 1,222.—Diagram illustrating amplitude of the current. The current reaches its amplitude or maximum value in one quarter period from its point of zero value, as, for instance, while the generating loop moves from position ABCD to A'B'C'D'. At three-quarter revolution, the current reaches its maximum value in the opposite direction.

Ques. Define the term "period."

Ans. This is the time of one cycle of the alternating current.

Ques. What is periodicity?

Ans. A term sometimes used for frequency.

Frequency.—If a slowly varying alternating current be passed through an incandescent lamp, the filament will be seen to vary in brightness, following the change of current strength. If, however, the alternations take place more rapidly than about 50 to 60 per second, the eye cannot follow the variations and the lamp appears to burn steadily. Hence it is important to consider the rate at which the alternations take place, or as it is called, the frequency, which is defined as: the number of cycles per second.

Fig. 1,223.—Diagram of alternator and engine, illustrating frequency. The frequency or cycles per second is equal to the revolution of armature per second multiplied by one-half the number of poles per phase. In the figure the armature makes 6 revolutions to one of the engine; one-half the number of poles = 8 ÷ 2 = 4, hence frequency = (150 × 4 × 6) ÷ 60 = 60. The expression in the parenthesis gives the cycles per minute, and dividing by 60, the cycles per second.

In a two pole machine, the frequency is the same as the number of revolutions per second, but in multipolar machines, it is greater in proportion to the number of pairs of poles per phase.

Thus, in an 8 pole machine, there will be four cycles per revolution. If the speed be 900 revolutions per minute, the frequency is

The symbol ~ is read "cycles per second."

Ques. What frequencies are used in commercial machines?

Ans. The two standard frequencies are 25 and 60 cycles.

Fig. 1,224—Diagram answering the question: Why are alternators always built multipolar? They are made multipolar because it is desirable that the frequency be high. It is evident from the figure that to obtain high frequency would require too many revolutions of the armature of a bipolar machine for mechanical safety—especially in large alternators. Moreover a double reduction gear in most cases would be necessary, adding complication to the drive. Comparing the above illustration with fig. 1,223, shows plainly the reason for multipolar construction.

Ques. For what service are these frequencies adapted?

Ans. The 25 cycle frequency is used for conversion to direct current, for alternating current railways, and for machines of large size; the 60 cycle frequency is used for general distribution for lighting and power.

The frequency of 40 cycles, which once was introduced as a compromise between 25 and 60 has been found not desirable, as it is somewhat low for general distribution, and higher than desirable for conversion to direct current.

Fig. 1,225.—Diagram illustrating "phase." In wave, vibratory, and simple harmonic motion, phase may be defined as: the portion of one complete vibration, measured either in angle or in time, that any moving point has executed.

Ques. What are the advantages of low frequency?

Ans. The number of revolutions of the rotor is correspondingly low; arc lamps can be more readily operated; better pressure regulation; small motors such as fan motors can be operated more easily from the circuit.

Phase.—As applied to an alternating current, phase denotes the angle turned through by the generating element reckoned from a given instant. Phase is usually measured in degrees from the initial position of zero generation.

If in the diagram fig. 1,225, the elementary armature or loop be the generating element, and the curve at the right be the sine curve representing the current, then the phase of any point p will be the angle φ or angle moved through from the horizontal line, the starting point.

Ques. What is phase difference?

Ans. The angle between the phases of two or more alternating current quantities as measured in degrees.

Ques. What is phase displacement?

Ans. A change of phase of an alternating pressure or current.

Figs. 1,226 and 1,227.—Diagram and sine curves illustrating synchronism. If two alternators, with coils in parallel planes, be made to rotate at the same speed by connecting them with chain drive or equivalent means, they will then be "in synchronism" that is, the alternating pressure or current in one will vary in step with that in the other. In other words, the cycles of one take place with the same frequency and at the same time as the cycles of the other as indicated by the curves, fig. 1,226. It should be noted that the maximum values are not necessarily the same but the maximum and zero values must occur at the same time in both machines, and the maximum value must be of the same sign. If the waves be distorted the maximum values may not occur simultaneously. See fig. 1,348.

Synchronism.—This term may be defined as: the simultaneous occurrence of any two events. Thus two alternating currents or pressures are said to be "in synchronism" when they have the same frequency and are in phase.

Ques. What does the expression "in phase" mean?

Ans. Two alternating quantities are said to be in phase, when there is no phase difference between; that is when the angle of phase difference equals zero.

Thus the current is said to be in phase with the pressure when it neither lags nor leads, as in fig. 1,228.

A rotating cylinder, or the movement of an index or trailing arm is brought into synchronism with another rotating cylinder or another index or trailing arm, not only when the two are moving with exactly the same speed, but when in addition they are simultaneously moving over similar portions of their respective paths.

Fig. 1,228—Pressure and current curves illustrating the term "in phase." The current is said to be in phase with the pressure when it neither lags nor leads.

When there is phase difference, as between current and pressure, they are said to be "out of phase" the phase difference being measured as in fig. 1,229 by the angle φ.

Fig. 1,229—Pressure and current curves illustrating the term "out of phase." The current is said to be out of phase with the pressure when it either lags or leads, that is when the current is not in synchronism with the pressure. In practice the current and pressure are nearly always out of phase.

When the phase difference is 90° as in fig. 1,231 or 1,232, the two alternating quantities are said to be in quadrature; when it is 180°, as in fig. 1,233, they are said to be in opposition.

When they are in quadrature, one is at a maximum when the other is at zero; when they are in opposition, one reaches a positive maximum when the other reaches a negative minimum, being at each instant opposite in sign.

Ques. What is a departure from synchronism called?

Ans. Loss of synchronism.

Figs. 1,230 to 1,233.—Curves showing some phase relations between current and pressure. Fig. 1,230, synchronism of current and pressure, expressed by the term "in phase," meaning simultaneous zero values, and simultaneous maximum values of the same sign; fig. 1,231, in quadrature, current leading 90°; fig. 1,232 in quadrature, current lagging 90°; fig. 1,233, in opposition, meaning that the phase different between current and pressure is 180°.

Maximum Volts and Amperes.—In the operation of an alternator, the pressure and strength of the current are continually rising, falling and reversing. During each cycle there are two points at which the pressure or current reaches its greatest value, being known as the maximum value. This maximum value is not used to any great extent, but it shows the maximum to which the pressure rises, and hence, the greatest strain to which the insulation of the alternator is subjected.

Fig. 1,234.—Elementary alternator developing one average volt. If the loop make one revolution per second, and the maximum number of lines of force embraced by the loop in the position shown (the zero position) be denoted by N, then each limb will cut 2N lines per second, because it cuts every line during the right sweep and again during the left sweep. Hence each limb develops an average pressure of 2N units (C.G.S. units), and as both limbs are connected in series, the total pressure is 4N units per revolution. Now, if the loop make f revolutions per second instead of only one, then f times as many lines will be cut per second, and the average pressure will be 4N f units. Since the C.G.S. unit of pressure is so extremely small, a much greater practical unit called the volt is used, which is equal to 100,000,000, or 10⁸ C.G.S. units is employed. Hence average voltage = 4Nf ÷ 10⁸. The value of N in actual machines is very high, being several million lines of force. The illustration shows one set of conditions necessary to generate one average volt. The maximum pressure developed is 1 ÷ .637 = 1.57 volts; virtual pressure = 1.57 × .707 = 1.11 volts.

Average Volts and Amperes.—Since the sine curve is used to represent the alternating current, the average value may be defined as: the average of all the ordinates of the curve for one-half of a cycle.

Ques. Of what use is the average value?

Ans. It is used in some calculations but, like the maximum value, not very often. The relation between the average and virtual value is of importance as it gives the form factor.

Virtual Volts and Amperes.—The virtual[1] value of an alternating pressure or current is equivalent to that of a direct pressure or current which would produce the same effect; those effects of the pressure and current are taken which are not affected by rapid changes in direction and strength,—in the case of pressure, the reading of an electrostatic voltmeter, and in the case of current, the heating effect.

[1] NOTE.—"I adhere to the term virtual, as it was in use before the term efficace which was recommended in 1889 by the Paris Congress to denote the square root of mean square value. The corresponding English adjective is efficacious; but some engineers mistranslate it with the word effective. I adhere to the term virtual mainly because the adjective effective is required in its usual meaning in kinematics to represent the resolved part of a force which acts obliquely to the line of motion, the effective force being the whole force multiplied by the cosine of the angle at which it acts with respect to the direction of motion. Some authors use the expression 'R.M.S. value' (meaning 'root mean square') to denote the virtual or quadratic mean value."—S. P. Thompson.

Fig. 1,235.—Maximum and average values of the sine curve. The average value of the sine curve is represented by an ordinate MS of such length that when multiplied by the base line FG, will give a rectangle MFSG whose area is equal to that included between the curve and base line FDGS.

Fig. 1,236.—Diagram illustrating "virtual" volts and amperes. The word virtual is defined as: Being in essence or effect, not in fact; not actual, but equivalent, so far as effect is concerned. As applied to the alternating current, it denotes an imaginary direct current of such value as will produce an effect equivalent to that of the alternating current. Thus, a virtual pressure of 1,000 volts is one that would produce the same deflection in an electrostatic voltmeter as a direct pressure of 1,000 volts: a virtual current of 10 amperes is that current which would produce the same heating effect as a direct current of 10 amperes. Both pressure and current vary continually above and below the virtual values in alternating current circuits. Distinction should be made between the virtual and "effective" values of an alternating current. See fig. 1,237. The word effective is commonly used erroneously for virtual. See note page [1,011].

The attraction (or repulsion) in electrostatic voltmeters is proportional to the square of the volts.

The readings which these instruments give, if first calibrated by using steady currents, are not true means, but are the square roots of the means of the squares.

Now the mean of the squares of the sine (taken over either one quadrant or a whole circle) is ½; hence the square root of mean square value of the sine functions is obtained by multiplying their maximum value by 1 ÷ √2, or by 0.707.

The arithmetical mean of the values of the sine, however, is 0.637. Hence an alternating current, if it obey the sine law, will produce a heating effect greater than that of a steady current of the same average strength, by the ratio of 0.707 to 0.637; that is, about 1.11 times greater.

If a Cardew voltmeter be placed on an alternating circuit in which the volts are oscillating between maxima of +100 and -100 volts, it will read 70.7 volts, though the arithmetical mean is really only 63.7; and 70.7 steady volts would be required to produce an equal reading.

Fig. 1,237.—Diagram illustrating virtual and effective pressure. If the coil be short circuited by the switch and a constant virtual pressure be impressed on the circuit, the whole of the impressed pressure will be effective in causing current to flow around the circuit. In this case the virtual and effective pressures will be equal. If the coil be switched into circuit, the reverse pressure due to self induction will oppose the virtual pressure; hence, the effective pressure (which is the difference between the virtual and reverse pressures) will be reduced, the virtual or impressed pressure remaining constant all the time. A virtual current is that indicated by an ammeter regardless of the phase relation between current and pressure. An effective current is that indicated by an ammeter when the current is in phase with the pressure. In practice, the current is hardly ever in phase with the pressure, usually lagging, though sometimes leading in phase. Now the greater this phase difference, either way, the less is the power of a given virtual current to do work. With respect to this feature, effective current may be defined as: that proportion of a given virtual current which can do useful work. If there be no phase difference, then effective current is equal to virtual current.

The matter may be looked at in a different way. If an alternating current is to produce in a given wire the same amount of effect as a continuous current of 100 amperes, since the alternating current goes down to zero twice in each period, it is clear that it must at some point in the period rise to a maximum greater than 100 amperes. How much greater must the maximum be? The answer is that, if it undulate up and down with a pure wave form, its maximum must be √2 times as great as the virtual mean; or conversely the virtual amperes will be equal to the maximum divided by √2. In fact, to produce equal effect, the equivalent direct current will be a kind of mean between the maximum and the zero value of the alternating current; but it must not be the arithmetical mean, nor the geometrical mean, nor the harmonic mean, but the quadratic mean; that is, it will be the square root of the mean of the squares of all the instantaneous values between zero and maximum.

Effective Volts and Amperes.—Virtual pressure, although already explained, may be further defined as the pressure impressed on a circuit. Now, in nearly all circuits the impressed or virtual pressure meets with an opposing pressure due to inductance and hence the effective pressure is something less than the virtual, being defined as that pressure which is available for driving electricity around the circuit, or for doing work. The difference between virtual and effective pressure is illustrated in fig. 1,237.

Ques. Does a given alternating voltage affect the insulation of the circuit differently than a direct pressure of the same value?

Ans. It puts more strain on the insulation in the same proportion as the maximum pressure exceeds the virtual pressure.

Fig. 1,238.—Current or pressure curve illustrating form factor. It is simply the virtual value divided by the average value. For a sine wave the virtual value is 1 / √2 times the maximum, and the average is 2 / π times the maximum, so that the form factor is π/2√2 or 1.11. The induction wave which generates an alternating pressure wave has a maximum value proportional to the area, that is, to the average value of the pressure wave. Hence the induction values corresponding to two pressure waves whose virtual values are equal, will be inversely proportional to their form factors. This is illustrated by the fact that a peaked wave causes less hysteresis loss in a transformer core than a flat topped wave, owing to the higher form factor of the peaked wave. See wave forms, figs. 1,245 to 1,248.

Form Factor.—This term was introduced by Fleming, and denotes the ratio of the virtual value of an alternating wave to the average value. That is

Ques. What does this indicate?

Ans. It gives the relative heating effects of alternating and direct currents, as illustrated in figs. 1,239 and 1,240.

That is, the alternating current will have about 11 per cent. more heating power than the direct current which is of the same average strength.

If an alternating current voltmeter be placed upon a circuit in which the volts range from +100 to -100, it will read 70.7 volts, although the arithmetical average, irrespective of + or-sign, is only 63.7 volts. If the voltmeter be connected to a direct current circuit, the pressure necessary to give the same reading would be 70.7 volts.

Figs. 1,239 and 1,240.—Relative heating effects of alternating and direct currents. If it takes say five minutes to produce a certain heating effect with alternating current at say 63.7 average volts, it will take 33 seconds longer with direct current at the same pressure, that is, the alternating current has about 11 per cent. more heating power than the direct current of the same average pressure. The reader should be careful not to get a wrong conception of the above; it does not mean that there is a saving by using alternating current. When both voltmeters read the same, that is, when the virtual pressure of the alternating current is the same as the direct current pressure, the heating effect is of course the same.

Ques. What is the relation between the shape of the wave curve and the form factor?

Ans. The more peaked the wave, the greater the value of its form factor.

A form factor of units would correspond to a rectangular wave; this is the least possible value of the form factor, and one which is not realized in commercial machines.

Figs. 1,241 to 1,244.—Various forms of pressure or current waves. Figs. 1,241 to 1,243 show the general shape of the waves produced by some alternators used largely for lighting work and having toothed armatures. The effect of the slots and shape of pole pieces is here very marked. Fig. 1,244 shows a wave characteristic of large alternators designed for power transmission and having multi-slot or distributed windings.

Wave Form.—There is always more or less irregularity in the shape of the current waves as met in practice, depending upon the construction of the alternator.

The ideal wave curve is the so called true sine wave, and is obtained with a rate of cutting of lines of force, by the armature coils, equivalent to the swing of a pendulum, which increases in speed from the end to the middle of the swing, decreasing at the same rate after passing the center. This swing is expressed in physics, as "simple harmonic motion".

Figs. 1,245 and 1,246.—Resolution of complex curves into sine curves. The heavy curve can be resolved into the simpler curves A and B shown in No. 1, the component curves A and B have in the ratio of three to one; that is, curve B has three times as many periods per second as curve A. All the curves, however, cross the zero line at the same time, and the resultant curve, though curiously unlike either of them, has a certain symmetry. In No. 2 the component curves, besides having periods in the ratio of three to one, cross the zero line at different points. The resultant curve produced is still less similar to its components, and is curiously and unsymmetrically humped. At first sight it is difficult to believe that such a curious curve could be resolved into two such simple and symmetrical ones. In both figures the component curves are sine curves, and as the curves for sine and cosine functions are exactly similar in form, the simplest supposition that can be made for the variation of pressure or of current is that both follow a sine law.

Fig. 1,247.—Reproduction of oscillograph record of wave form of alternator with one coil per phase per pole. Here the so-called "super-imposed harmonic" is clearly indicated.

Fig. 1,248.—Reproduction of oscillograph record of Wagner alternator having three coils per phase per pole.

The losses in all secondary apparatus are slightly lower with the so called peaked form of wave. For the same virtual voltage, however, the top of the peak will be much higher, thereby submitting the insulation to that much greater strain. By reason of the fact that the losses are less under such wave forms, many manufacturers in submitting performance data on transformers recite that the figures are for sine wave conditions, stating further that if the transformers are to be operated in a circuit more peaked than the sine wave, the losses will be less than shown.

The slight saving in the losses of secondary apparatus, obtained with a peaked wave, by no means compensates for the increased insulation strains and an alternator having a true sine wave is preferred.

Ques. What determines the form of the wave?

Ans. 1. The number of coils per phase per pole, 2, shape of pole faces, 3, eddy currents in the pole pieces, and 4, the air gap.

Ques. What are the requirements for proper rate of cutting of the lines of force?

Ans. It is necessary to have, as a minimum, two coils per phase per pole in three phase work.

Ques. What is the effect of only one coil per phase per pole?

Ans. The wave form will be distorted as shown in fig. 1,247.

Ques. What is the least number of coils per phase per pole that should be used for two and three phase alternators?

Ans. For three phase, two coils, and for two phase, three coils, per phase per pole.

Single or Monophase Current.—This kind of alternating current is generated by an alternator having a single winding on its armature. Two wires, a lead and return, are used as in direct current.

An elementary diagram showing the working principles is illustrated in fig. 1,249, a similar hydraulic cycle being shown in figs. 1,250 to 1,252.

Fig. 1,249.—Elementary one loop alternator and sine curve illustrating single phase alternating current. There are three points during the revolution at which there is no current: at 0° the position shown, 180°, and 360°; in other words, at the beginning, middle point and end of the cycle. The current reaches a maximum at 90°, reverses at 180°, and reaches a maximum in the reverse direction at 270°.

Two Phase Current.—In most cases two phase current actually consists of two distinct single phase currents flowing in separate circuits. There is often no electrical connection between them; they are of equal period and equal amplitude, but differ in phase by one quarter of a period. With this phase relation one of them will be at a maximum when the other is at zero. Two phase current is illustrated by sine curves in fig. 1,253, and by hydraulic analogy in figs. 1,254 and 1,255.

Figs. 1,250 to 1,252.—Hydraulic analogy illustrating the difference between direct (continuous) and alternating current. In fig. 1,250 a centrifugal pump C forces water to the upper pipe, from which it falls by gravity to the lower pipe B and re-enters the pump. The current is continuous, always flowing in one direction, that is, it does not reverse its direction. Similarly a direct electric current is constant in direction (does not reverse); though not necessarily constant in value. A direct current, constant in both value and direction as a result of constant pressure, is called "continuous" current. Similarly in the figure the flow is constant, and a gauge D placed at any point will register a constant pressure, hence the current may be called, in the electrical sense, "continuous." The conditions in fig. 1,251 are quite different. The illustration represents a double acting cylinder with the ends connected by a pipe A, and the piston driven by crank and Scotch yoke as shown. In operation, if the cylinder and pipe be full of water, a current of water will begin to flow through the pipe in the direction indicated as the piston begins its stroke, increasing to maximum velocity at one-quarter revolution of the crank, decreasing and coming to rest at one-half revolution, then reversing and reaching maximum velocity in the reverse direction at three-quarter revolution, and coming to rest again at the end of the return stroke. A pressure gauge at G will register a pressure which varies with the current. Since the alternating electric current undergoes similar changes, the sine curve will apply equally as well to the pump cycle as to the alternating current cycle.

Fig. 1,253.—Elementary two loop alternator and sine curves, illustrating two phase alternating current. If the loops be placed on the alternator armature at 90 magnetic degrees, a single phase current will be generated in each of the windings, the current in one winding being at its maximum value when the other is at zero. In this case four transmission conductors are generally used, two for each separate circuit, and the motors to which the current is led have a double winding corresponding to that on the alternator armature.

If two identical simple alternators have their armature shafts coupled in such a manner, that when a given armature coil on one is directly under a field pole, the corresponding coil on the other is midway between two poles of its field, the two currents generated will differ in phase by a half alternation, and will be two phase current.

Ques. How must an alternator be constructed to generate two phase current?

Ans. It must have two independent windings, and these must be so spaced out that when the volts generated in one of the two phases are at a maximum, those generated in the other are at zero.

In other words, the windings, which must be alike, of an equal number of turns, must be displaced along the armature by an angle corresponding to one-quarter of a period, that is, to half the pole pitch.

Figs. 1,254 and 1,255.—Hydraulic analogy illustrating two phase alternating current. In the figure two cylinders, similar to the one in fig. 1,251, are shown, operated from one shaft by crank and Scotch yoke drive. The cranks are at 90° as shown, and the cylinders and connecting pipes full of water. In operation, the same cycle of water flow takes place as in fig. 1,251. Since the cranks are at 90°, the second piston is one-half stroke behind the first; the flow of water in No. 1 (phase A) is at a maximum when the flow in No. 2 (phase B) comes to rest, the current conditions in both pipes for the entire cycle being represented by the two sine curves whose phase difference is 90°. Comparing these curves with fig. 1,253, it will be seen that the water and electric current act in a similar manner.

Figs. 1,254 and 1,255.—Hydraulic analogy illustrating two phase alternating current. In the figure two cylinders, similar to the one in fig. 1,251, are shown, operated from one shaft by crank and Scotch yoke drive. The cranks are at 90° as shown, and the cylinders and connecting pipes full of water. In operation, the same cycle of water flow takes place as in fig. 1,251. Since the cranks are at 90°, the second piston is one-half stroke behind the first; the flow of water in No. 1 (phase A) is at a maximum when the flow in No. 2 (phase B) comes to rest, the current conditions in both pipes for the entire cycle being represented by the two sine curves whose phase difference is 90°. Comparing these curves with fig. 1,253, it will be seen that the water and electric current act in a similar manner.

The windings of the two phases must, of course, be kept separate, hence the armature will have four terminals, or if it be a revolving armature it will have four collector rings.

As must be evident the phase difference may be of any value between 0° and 360°, but in practice it is almost always made 90°.

Ques. In what other way may two phase current be generated?

Ans. By two single phase alternators coupled to one shaft.

Ques. How many wires are required for two phase distribution?

Ans. A two phase system requires four lines for its distribution; two lines for each phase as in fig. 1,253. It is possible, but not advisable, to reduce the number to 3, by employing one rather thicker line as a common return for each of the phases as in fig. 1,256.

Fig. 1,256.—Diagram of three wire two phase current distribution. In order to save one wire it is possible to use a common return conductor for both circuits, as shown, the dotted portion of one wire 4 being eliminated by connecting across to 1 at M and S. For long lines this is economical, but the interconnection of the circuits increases the chance of trouble from grounds or short circuits. The current in the conductor will be the resultant of the two currents, differing by 90° in phase.

If this be done, the voltage between the A line and the B line will be equal to √2 times the voltage in either phase, and the current in the line used as common return will be √2 times as great as the current in either line, assuming the two currents in the two phases to be equal.

Ques. In what other way may two phase current be distributed?

Ans. The mid point of the windings of the two phases may be united in the alternator at a common junction.

Figs. 1,257 to 1,259.—Various two phase armature connections. Fig. 1,257, two separate circuit four collector ring arrangement; fig. 1,258, common middle connection, four collector rings; fig. 1,259, circuit connected in armature for three collector rings. In the figures the black winding represents phase A, and the light winding, phase B.

This is equivalent to making the machine into a four phase alternator with half the voltage in each of the four phases, which will then be in successive quadrature with each other.

Ques. How are two phase alternator armatures wound?

Ans. The two circuits may be separate, each having two collector rings, as shown in fig. 1,257, or the two circuits may be coupled at a common middle as in fig. 1,258, or the two circuits may be coupled in the armature so that only three collector rings are required as shown in fig. 1,259.

Fig. 1,260.—Elementary three loop alternator and sine curves, illustrating three phase alternating current. If the loops be placed on the alternator armature at 120 magnetic degrees from one another, the current in each will attain its maximum at a point one-third of a cycle distant from the other two. The arrangement here shown gives three independent single phase currents and requires six wires for their transmission. A better arrangement and the one generally used is shown in fig. 1,261.

Fig. 1,261.—Elementary three wire three phase alternator. For the transmission of three phase current, it is not customary to use six wires, as in fig. 1,260, instead, three ends (one end of each of the loops) are brought together to a common connection as shown, and the other ends, connected to the collector rings, giving only three wires for the transmission of the current.

Three Phase Current.—A three phase current consists of three alternating currents of equal frequency and amplitude, but differing in phase from each other by one-third of a period. Three phase current as represented by sine curves is shown in fig. 1,260, and by hydraulic analogy in fig. 1,262. Inspection of the figures will show that when any one of the currents is at its maximum, the other two are of half their maximum value, and are flowing in the opposite direction.

Figs. 1,262 and 1,263.—Hydraulic analogy illustrating three phase alternating current. Three cylinders are here shown with pistons connected through Scotch yokes to cranks placed 120° apart. The same action takes place in each cylinder as in the preceding cases, the only difference being the additional cylinder, and difference in phase relation.

Ques. How is three phase current generated?

Ans. It requires three equal windings on the alternator armature, and they must be spaced out over its surface so as to be successively ⅓ and ⅔ of the period (that is, of the double pole pitch) apart from one another.

Ques. How many wires are used for three phase distribution?

Ans. Either six wires or three wires.

Six wires, as in fig. 1,260, might be used where it is desired to supply entirely independent circuits, or as is more usual only three wires are used as shown in fig. 1,261. In this case it should be observed that if the voltage generated in each one of the three phases separately E (virtual) volts, the voltage generated between any two of the terminals will be equal to √3 × E. Thus, if each of the three phases generate 100 volts, the voltage from the terminal of the A phase to that of the B phase will be 173 volts.

Fig. 1,264.—Experiment illustrating self-induction in an alternating current circuit. If an incandescent lamp be connected in series with a coil made of one pound of No. 20 magnet wire, and connected to the circuit, the current through the lamp will be decreased due to the self-induction of the coil. If now an iron core be gradually pushed into the coil, the self-induction will be greatly increased and the lamp will go out, thus showing the great importance which self-induction plays in alternating current work.

Inductance.—Each time a direct current is started, stopped or varied in strength, the magnetism changes, and induces or tends to induce a pressure in the wire which always has a direction opposing the pressure which originally produced the current. This self-induced pressure tends to weaken the main current at the start and prolong it when the circuit is opened.

The expression inductance is frequently used in the same sense as coefficient of self-induction, which is a quantity pertaining to an electric circuit depending on its geometrical form and the nature of the surrounding medium.

If the direct current maintains the same strength and flow steadily, there will be no variations in the magnetic field surrounding the wire and no self-induction, consequently the only retarding effect of the current will be the "ohmic resistance" of the wire.

If an alternating current be sent through a circuit, there will be two retarding effects:

1. The ohmic resistance;

2. The spurious resistance.

Fig. 1,265.—Non-inductive and inductive resistances. Two currents are shown joined in parallel, one containing a lamp and non-inductive resistance, and the other a lamp and inductive resistance. The two resistances being the same, a sufficient direct pressure applied at T, T' will cause the lamps to light up equally. If, however, an alternating pressure be applied, M will burn brightly, while S will give very little or no light because of the effect of the inductance of the inductive resistance.

Ques. Upon what does the ohmic resistance depend?

Ans. Upon the length, cross sectional area and material of the wire.

Ques. Upon what does the spurious resistance depend?

Ans. Upon the frequency of the alternating current, the shape of the conductor, and nature of the surrounding medium.

Fig. 1,266.—Inductance test, illustrating the self-induction of a coil which is gradually increased by moving an iron wire core inch by inch into the coil. The current is kept constant with the adjustable resistance throughout the test and readings taken, first without the iron core, and again when the core is put in the coil and moved to the 1, 2, 3, 4, etc., inch marks. By plotting the voltmeter readings and the position of the iron core on section paper, a curve is obtained showing graphically the effect of the self-induction. A curve of this kind is shown in fig. 1,302.

Ques. Define inductance.

Ans. It is the total magnetic flux threading the circuit per unit current which flows in the circuit, and which produces the flux.

In this it must be understood that if any portion of the flux thread the circuit more than once, this portion must be added in as many times as it makes linkage.

Inductance, or the coefficient of self-induction is the capacity which an electric circuit has of producing induction within itself.

Inductance is considered as the ratio between the total induction through a circuit to the current producing it.

Ques. What is the unit of inductance?

Ans. The henry.

Ques. Define the henry.

Ans. A coil has an inductance of one henry when the product of the number of lines enclosed by the coil multiplied by the number of turns in the coil, when a current of one ampere is flowing in the coil, is equal to 100,000,000 or 10⁸.

An inductance of one henry exists in a circuit when a current changing at the rate of one ampere per second induces a pressure of one volt in the circuit.

Ques. What is the henry called?

Ans. The coefficient of self-induction.

Fig. 1,267.—Diagram illustrating the henry. By definition: A circuit has an inductance of one henry when a rate of change of current of one ampere per second induces a pressure of one volt. In the diagram it is assumed that the internal resistance of the cell and resistance of the connecting wires are zero.

The henry is the coefficient by which the time rate of change of the current in the circuit must be multiplied, in order to give the pressure of self-induction in the circuit.

The formula for the henry is as follows:

or

where

• L = coefficient of self induction in henrys;
• N = total number of lines of force threading a coil when the current is one ampere;
• T = number of turns of coil.

If a coil had a coefficient of self-induction of one henry, it would mean that if the coil had one turn, one ampere would set up 100,000,000, or 10⁸, lines through it.

Figs. 1,268 to 1,270.—Various coils. The inductance effect, though perceptible in an air core coil, fig. 1,268, may be greatly intensified by inserting a core made of numerous pieces of iron wire, as in fig. 1,269. Fig. 1,270 shows a non-inductive coil. When wound in this manner, a coil will have little or no inductance because each half of the coil neutralizes the magnetic effect of the other. This coil, though non-inductive, will have "capacity." It would be useless for solenoids or electromagnets, as it would have no magnetic field.

The henry[2] is too large a unit for use in practical computations, which involves that the millihenry, or ¹/_1,000th henry, is the accepted unit. In pole suspended lines the inductance varies as the metallic resistance, the distance between the wires on the cross arm and the number of cycles per second, as indicated by accepted tables. Thus, for one mile of No. 8 B. & S. copper wire, with a resistance of 3,406 ohms, the coefficient of self-induction with 6 inches between centers is .00153, and, with 12 inches, .00175.

[2] NOTE.—The American physicist, Joseph Henry, was born in 1798 and died 1878. He was noted for his researches in electromagnetism. He developed the electromagnet, which had been invented by Sturgeon in England, so that it became an instrument of far greater power than before. In 1831, he employed a mile of fine copper wire with an electromagnet, causing the current to attract the armature and strike a bell, thereby establishing the principle employed in modern telegraph practice. He was made a professor at Princeton in 1832, and while experimenting at that time, he devised an arrangement of batteries and electromagnets embodying the principle of the telegraph relay which made possible long distance transmission. He was the first to observe magnetic self-induction, and performed important investigations in oscillating electric discharges (1842), and other electrical phenomena. In 1846 he was chosen secretary of the Smithsonian Institution at Washington, an office which he held until his death. As chairman of the U. S. Lighthouse Board, he made important tests in marine signals and lights. In meteorology, terrestrial magnetism, and acoustics, he carried on important researches. Henry enjoyed an international reputation, and is acknowledged to be one of America's greatest scientists.

Fig. 1,271.—Hydraulic-mechanical analogy illustrating inductance in an alternating current circuit. The two cylinders are connected at their ends by the vertical pipes, each being provided with a piston and the system filled with water. Reciprocating motion is imparted to the lower pulley by Scotch yoke connection with the drive pulley. The upper piston is connected by rack and pinion gear with a fly wheel. In operation, the to and fro movement of the lower piston produces an alternating flow of water in the upper cylinder which causes the upper piston to move back and forth. The rack and pinion connection with the fly wheel causes the latter to revolve first in one direction, then in the other, in step with the upper piston. The inertia of the fly wheel causes it to resist any change in its state, whether it be at rest or in motion, which is transmitted to the upper piston, causing it to offer resistance to any change in its rate or direction of motion. Inductance in the alternating current circuit has precisely the same effect, that is, it opposes any change in the strength or direction of the current.

Ques. How does the inductance of a coil vary with respect to the core?

Ans. It is least with an air core; with an iron core, it is greater in proportion to the permeability[3] of the iron.

[3] NOTE.—The permeability of iron varies from 500 to 1,000 or more. The permeability of a given sample of iron is not constant, but decreases in value as the magnetizing force increases. Therefore the inductance of a coil having an iron core is not a constant quantity as is the inductance of an air core coil.

The coefficient L for a given coil is a constant quantity so long as the magnetic permeability of the material surrounding the coil does not change. This is the case where the coil is surrounded by air. When iron is present, the coefficient L is practically constant, provided the magnetism is not forced too high.

In most cases arising in practice, the coefficient L may be considered to be a constant quantity, just as the resistance R is usually considered constant. The coefficient L of a coil or circuit is often spoken of as its inductance.

Fig. 1,272.—Experiment showing effect of inductive and non-inductive coils in alternating current circuit. The apparatus is connected up as shown; by means of the switch, the lamp may be placed in parallel with either the inductive or non-inductive coil. These coils should have the same resistance. Pass an alternating current through the lamp and non-inductive coil, of such strength that the lamp will be dimly lighted. Now turn the switch so as to put the lamp and inductive coil in parallel and the lamp will burn with increased brilliancy. The reason for this is because of the opposition offered by the inductive coil to the current, less current is shunted from the lamp when the inductive coil is in the circuit than when the non-inductive coil is in the circuit. That is, each coil has the same ohmic resistance, but the inductive coil has in addition the spurious resistance due to inductance, hence it shunts less current from the lamp than does the non-inductive coil.

Ques. Why is the iron core of an inductive coil made with a number of small wires instead of one large rod?

Ans. It is laminated in order to reduce eddy currents and consequent loss of energy, and to prevent excessive heating of the core.

Ques. How does the number of turns of a coil affect the inductance?

Ans. The inductance varies as the square of the turns.

That is, if the turns be doubled, the inductance becomes four times as great.

The inductance of a coil is easily calculated from the following formulæ:

L = 4π²r²n² ÷ (l × 10⁹) (1)

for a thin coil with air core, and