The cover image was created by the transcriber and is placed in the public domain.
THE THOUGHT IS IN THE QUESTION THE INFORMATION IS IN THE ANSWER
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
COPYRIGHTED, 1914,
BY
THEO. AUDEL & CO.,
New York.
Printed in the United States.
ALTERNATING CURRENTS | 997 to 1,066 |
---|---|
The word "alternating"—advantages of alternating current—direct current apparatus; alternating current apparatus—disadvantages of alternating current—alternating current principles—the sine—application and construction of the sine curve—illustrated definitions: cycle, alternation, amplitude, period, periodicity, frequency—commercial frequencies—advantages of low frequency—phase—phase difference—phase displacement—synchronism—"in phase"—curves illustrating "in phase" and "out of phase"—illustrated definitions: in phase; in quadrature, current leading; in quadrature, current lagging; in opposition—maximum volts and amperes—average volts and amperes—elementary alternator developing one average volt—virtual volts and amperes—effective volts and amperes—relation between shape of wave and form factor—wave form—oscillograph wave form records—what determines wave form—effect of one coil per phase per pole—single phase current; hydraulic analogy—two phase current; hydraulic analogy—two phase current distribution—three phase current; hydraulic analogy; distribution—inductance—the henry—inductive and non-inductive coils—hydraulic analogy of inductance—inductance coil calculations—ohmic value of inductance—capacity: hydraulic analogy—the farad—specific inductive capacity—condenser connections—ohmic value of capacity—lag and lead—mechanical analogy of lag—lag measurement—steam engine analogy of current flow at zero pressure—reactance—examples—choking coil—impedance curve—resonance—critical frequency—skin effect.[Pg iv] | |
ALTERNATING CURRENT DIAGRAMS | 1,067 to 1,100 |
Definitions: impressed pressure, active pressure, self-induction pressure, reverse pressure of self-induction—rate of change in current strength—properties of right angle triangles—equations of the right triangle—representation of forces by wires—parallelogram of forces; the resultant—circuits containing resistance and inductance—graphical method of obtaining the impressed pressure—equations for ohmic drop and reactance drop—examples—diagram for impedance, angle of lag, etc.—circuits containing resistance and capacity—capacity in series, and in parallel—amount of lead—action of condenser—the condenser pressure—capacity pressure—equation for impedance—- examples and diagrams—circuits containing resistance, inductance, and capacity—impedance equation—examples and diagrams—equation for impressed pressure—examples and diagrams. | |
THE POWER FACTOR | 1,101 to 1,124 |
Definition of power factor—true watts—- apparent watts—ferry boat analogy of power factor—limits of power factor—effect of lag or lead—how to obtain the power curve—nature of the power curve—synchronism of current and pressure; power factor unity—case of synchronism of current and pressure with power factor less than unity—steam engine analogy of power factor—"wattless current;" power factor zero—examples of phase difference nearly 90 degrees—mechanical analogy of wattless current—why the power factor is equal to cos φ—graphical method of obtaining the active component—examples and diagrams—effect of capacity—diagrams illustrating why the power factor is unity when there is no resultant reactance in the circuit—usual value of power factor—power factor test—how alternators are rated; kva.—curves illustrating power factor—how to keep the power factor high—why power factor is important in station operation—wattmeter method of three phase power measurement. | |
ALTERNATORS | 1,125 to 1,186 |
Uses of alternators—classes of alternator—single phase alternators; essential features; width of armature coils—[Pg v]elementary single phase alternator—polyphase alternators—uses for two and three phase current—elementary three phase alternator—starting difficulty with single phase motors—six and twelve phase windings—belt or chain driven alternators—sub-base and ratchet device for tightening the belt—horse power transmitted by belts—best speeds for belts—advantages of chain drive; objections—direct connected alternator—"direct connected" and "direct coupled" units—revolving armature alternators; their uses—revolving field alternators—marine view showing that motion is purely a relative matter—essential parts of revolving field alternator—the terms "stator" and "rotor"—inductor alternators: classes, use, defects—hunting or surging in alternators—amortisseur windings—monocyclic alternators—diagram of connections—teaser coil—armature reaction—distortion of field—strengthening and weakening effects—superpositions of fields—three phase reactions—magnetic leakage—field excitation of alternators—self-excited alternator—direct connected exciter—gear driven exciters—slow speed alternators—fly wheel alternators—high speed alternators—water wheel alternators—construction of rotor—turbine driven alternators—construction—step bearing—alternators of exceptional character—asynchronous alternators—image current alternators—extra high frequency alternators—self-exciting image current alternators. | |
CONSTRUCTION OF ALTERNATORS | 1,187 to 1,266 |
Essential parts of an alternator—field magnets—methods of excitation: self-excited, separately excited, compositely excited—magneto—construction of stationary magnets—revolving field—slip rings—spider for large alternator—provision for shifting armature to give access to field—armatures—core construction—advantages of slotted core armatures—armature windings—classification: revolving and stationary windings—half coil and whole coil windings—concentrated or uni-coil winding; features; waveform—distributed or multi-coil windings: breadth of coil, partial and fully distributed coils—the Kapp coefficient—general equation for voltage—wire, strap, and bar windings—condition, governing type of inductor—coil covering—single and double layer multi-wire inductors and methods of placing them[Pg vi] on the core—insulation—core stamping—single and multi-slot windings—- arrangement in slot of two layer bar winding—table of relative effectiveness of windings—single phase windings—advantage of half coil winding—two phase windings—shape of coil ends—three phase windings—shape of coil ends—kind of coil used with three phase windings—grouping of phases—two phase star connection—two phase mesh connection—three phase star connection—winding diagrams with star and Δ connections—three phase Δ connection—three phase winding with "short" coils—three phase lap winding star connection—three phase wave winding star connection—output of star and delta connected alternators—gramme ring armatures showing three phase star and mesh connections with direction of currents in the coils—features of star connection—characteristics of delta connection—proper ends to connect to star point—determination of path and value of currents in delta connection—points to be noted with Y connection—diagram of Y connection with return wire—chain or basket winding—skew winding—fed-in winding—imbricated winding—spiral winding—mummified winding—shuttle winding—creeping winding—turbine alternator winding: how the high voltage is obtained with so few poles; table of frequency and revolutions—turbine alternator construction—form of armature generally used—two pole radial slot field—parallel slot field—difficulty experienced with revolving armatures—how the field design is modified to reduce centrifugal force—examples of revolving fields. |
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,[Pg 998] facility of transforming from one voltage to another (either higher or lower) for different purposes.
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.
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.
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.
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.
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.
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
8 | 900 | |||
× | = | 60 ~ | ||
2 | 60 |
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.
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.
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.
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.
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.
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.
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.
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.
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.
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[Pg 1014] 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.
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
virtual value | .707 | |||||
form factor | = | = | = | 1.11 | ||
average value | .637 |
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.
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.
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".
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.
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[Pg 1021] by sine curves in fig. 1,253, and by hydraulic analogy in figs. 1,254 and 1,255.
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.
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.
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.
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.
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[Pg 1027] 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.
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[Pg 1028] 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.
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[Pg 1029] 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.
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.
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.
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:
magnetic flux × turns | ||
henrys | = | |
current × 100,000,000 |
or
N × T | |||
L | = | (1) | |
108 |
where
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.
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.
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.
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
L = 4π2r2n2μ ÷ (l × 109) (2)
for a coil having an iron core. In the above formulæ:
EXAMPLE.—An air core coil has an average radius of 10 centimeters and is 20 centimeters long, there being 500 turns, what is the inductance?
Substituting these values in formula (1)
L = 4 × (3.1416)2 × 102 × 5002 ÷ (20 × 109) = .00494 henry
Ques. Is the answer in the above example in the customary form?
Ans. No; the henry being a very large unit, it is usual to express inductance in thousandths of a henry, that is, in milli-henrys. The answer then would be .04935 × 1,000 = 49.35 milli-henrys.
EXAMPLE.—An air core coil has an inductance of 50 milli-henrys; if an iron core, having a permeability of 600 be inserted, what is the inductance?
The inductance of the air core coil will be multiplied by the permeability of the iron; the inductance then is increased to
50 × 600 = 30,000 milli-henrys, or 30 henrys.
Ohmic Value of Inductance.—The rate of change of an alternating current at any point expressed in degrees is equal to the product of 2π multiplied by the frequency, the maximum current, and the cosine of the angle of position θ; that is (using symbols)
rate of change = 2πfImaxcos θ.
The numerical value of the rate of change is independent of its positive or negative sign, so that the sign of the cos φ is disregarded.
The period of greatest rate of change is that at which cos φ has the greatest value, and the maximum value of a cosine is when the arc has a value of zero degrees or of 180 degrees, its value corresponding, being 1. (See fig. 1,037, page 1,068.)
The pressure due to inductance is equal to the product of the rate of change by the inductance; that is, calling the inductance L,[Pg 1038] the pressure due to it at the point of maximum value or
Emax = 2πfImax × L (1)
Now by Ohm's law
Emax = RImax (2)
for a current Imax, hence substituting equation (2) in equation (1)
RImax = 2πfImax × L
from which, dividing both sides by Imax, and using Xi for R
Xi = 2πfL (3)
which is the ohmic equivalent of inductance.
The frequency of a current being the number of periods or waves per second, then, if T = the time of a period, the frequency[Pg 1039] of a current may be obtained by dividing 1 second by the time of a period; that is
one second | 1 | ||||
frequency | = | = | (4) | ||
time of one period | T |
substituting 1 / T for f in equation (3)
L | |||
Xi | = | 2π | |
T |
Capacity.—When an electric pressure is applied to a condenser, the current plays in and out, charging the condenser in alternate directions. As the current runs in at one side and out at the other, the dielectric becomes charged, and tries to discharge itself by setting up an opposing electric pressure. This opposing pressure rises just as the charge increases.
A mechanical analogue is afforded by the bending of a spring, as in fig. 1,279, which, as it is being bent, exerts an opposing force[Pg 1040] equal to that applied, provided the latter do not exceed the capacity of the spring.
Ques. What is the effect of capacity in an alternating circuit?
Ans. It is exactly opposite to that of inductance, that is, it assists the current to rise to its maximum value sooner than it would otherwise.
Ques. Is it necessary to have a continuous metallic circuit for an alternating current?
Ans. No, it is possible for an alternating current to flow through a circuit which is divided at some point by insulating material.
Ques. How can the current flow under such condition?
Ans. Its flow depends on the capacity of the circuit and accordingly a condenser may be inserted in the circuit as in fig. 1,286, thus interposing an insulated gap, yet permitting an alternating flow in the metallic portion of the circuit.
Ques. Name the unit of capacity and define it.
Ans. The unit of capacity is called the farad and its symbol is C. A condenser is said to have a capacity of one farad if one coulomb (that is, one ampere flowing one second), when stored[Pg 1042] on the plates of the condenser will cause a pressure of one volt across its terminals.
The farad being a very large unit, the capacities ordinarily encountered in practice are expressed in millionths of a farad, that is, in microfarads--a capacity equal to about three miles of an Atlantic cable.
It should be noted that the microfarad is used only for convenience, and that in working out problems, capacity should always be expressed in farads before substituting in formulæ, because the farad is chosen with respect to the volt and ampere, as above defined, and hence must be used in formulæ along with these units.
For instance, a capacity of 8 microfarads as given in a problem would be substituted in a formula as .000008 of a farad.
The charge Q forced into a condenser by a steady electric pressure E is
Q = EC
in which
Ques. What is the material between the plates of a condenser called?
Ans. The dielectric.
Ques. Upon what does the capacity of a condenser depend?
Ans. It is proportional to the area of the plates, and inversely proportional to the thickness of the dielectric between the plates, a correction being required unless the thickness of dielectric be very small as compared with the dimensions of the plates.
The capacity of a condenser is also proportional to the specific inductive capacity of the dielectric between the plates of the condenser.
Specific Inductive Capacity.—Faraday discovered that different substances have different powers of carrying lines of electric force. Thus the charge of two conductors having a given difference of pressure between them depends on the medium between them as well as on their size and shape. The number indicating the magnitude of this property of the medium is called its specific inductive capacity, or dielectric constant.
The specific inductive capacity of air, which is nearly the same as that of a vacuum, is taken as unity. In terms of this unit the following are some typical values of the dielectric constant: water 80, glass 6 to 10, mica 6.7, gutta-percha 3, India rubber 2.5, paraffin wax 2, ebonite 2.5, castor oil 4.8.
In underground cables for very high pressures, the insulation, if homogeneous throughout, would have to be of very great thickness in order to have sufficient dielectric strength. By employing material of high specific inductive capacity close to the conductor, and material of lower specific inductive capacity toward the outside, that is, by grading the insulation, a considerably less total thickness affords equally high dielectric strength.
Ques. How are capacity tests usually made?
Ans. By the aid of standard condensers.
Ques. How are condensers connected?
Ans. They may be connected in parallel as in fig. 1,283, or in series (cascade) as in fig. 1,284.
Condensers are now constructed so that the two methods of arranging the plates may conveniently be combined in one condenser, thereby obtaining a wider range of capacity.
Ques. How may the capacity of a condenser, wire, or cable be tested?
Ans. This may be done by the aid of a standard condenser, trigger key, and an astatic or ballistic galvanometer.
In making the test, first obtain a "constant" by noting the deflection d, due to the discharge of the standard condenser after a charge of, say, 10 seconds from a given voltage. Then discharge the other condenser, wire, or cable through the galvanometer after 10 seconds charge, and note the deflection d'. The capacity C' of the latter is then
d' | ||||
C' | = | C | × | |
d |
in which C is the capacity of the standard condenser.
Ohmic Value of Capacity.—The capacity of an alternating current circuit is the measure of the amount of electricity held by it when its terminals are at unit difference of pressure. Every such circuit acts as a condenser.
If an alternating circuit, having no capacity, be opened, no current can be produced in it, but if there be capacity at the break, current may be produced as in fig. 1,286.
The action of capacity referred to the current wave is as follows: As the wave starts from zero value and rises to its maximum value, the current is due to the discharge of the[Pg 1046] capacity, which would be represented by a condenser. In the case of a sine current, the period required for the current to pass from zero value to maximum is one-quarter of a cycle.
At the beginning of the cycle, the condenser is charged to the maximum amount it receives in the operation of the circuit.
At the end of the quarter cycle when the current is of maximum value, the condenser is completely discharged.
The condenser now begins to receive a charge, and continues to receive it during the next quarter of a cycle, the charge attaining its maximum value when the current is of zero intensity. Hence, the maximum charge of a condenser in an alternating circuit is equal to the average value of the current multiplied by the time of charge, which is one-quarter of a period, that is
maximum charge = average current × ¼ period (1)
Since the time of a period = 1 ÷ frequency, the time of one-quarter of a period is ¼ × (1 ÷ frequency), or
¼ period = ¼ f (2)
f, being the symbol for frequency. Substituting (2) in (1)
maximum charge = Iav × ¼f (3)
The pressure of a condenser is equal to the quotient of the charge divided by the capacity, that is
charge | |||
condenser pressure | = | (4) | |
capacity |
Substituting (3) in (4)
⎛ | 1 | ⎞ | Iav | ||||||||
condenser pressure | = | ⎜ | Iav | × | ⎟ | ÷ | C | = | (5) | ||
⎝ | 4f | ⎠ | 4fC |
But, Iav = Imax × 2 / π, and substituting this value of Iav in equation (5) gives
Imax × 2 / π | Imax | ||||
condenser pressure | = | = | (6) | ||
4fC | 2πfC |
This last equation (6) represents the condenser pressure due to capacity at the point of maximum value, which pressure is opposed to the impressed pressure, that is, it is the maximum reverse pressure due to capacity.
Now, since by Ohm's law
E | ||||
I | = | = | I × R | |
R |
and as
Imax | 1 | |||
= | Imax | × | ||
2πfC | 2πfC |
it follows that 1 / (2πfC) is the ohmic value of capacity, that is it expresses the resistance equivalent of capacity; using the symbol Xc for capacity reactance
1 | |||
---|---|---|---|
Xc | = | (7) | |
2πfC |
EXAMPLE.—What is the resistance equivalent of a 50 microfarad condenser to an alternating current having a frequency of 100?
Substituting the given values in the expression for ohmic value
1 | 1 | 1 | ||||||
Xc | = | = | = | = | 31.8 ohms. | |||
2πfC | 2 × 3.1416 × 100 × .000050 | .031416 |
If the pressure of the supply be, say 100 volts, the current would be 100 ÷ 31.8 = 3.14 amperes.
Lag and Lead.—Alternating currents do not always keep in step with the alternating volts impressed upon the circuit. If there be inductance in the circuit, the current will lag; if there be capacity, the current will lead in phase. For example, fig. 1,288, illustrates the lag due to inductance and fig. 1,289, the lead due to capacity.
Ques. What is lag?
Ans. Lag denotes the condition where the phase of one alternating current quantity lags behind that of another. The term is generally used in connection with the effect of inductance in causing the current to lag behind the impressed pressure.
Ques. How does inductance cause the current to lag behind the pressure?
Ans. It tends to prevent changes in the strength of the current. When two parts of a circuit are near each other, so that one is in the magnetic field of the other, any change in the strength of the current causes a corresponding change in the magnetic field and sets up a reverse pressure in the other wire.
This induced pressure causes the current to reach its maximum value a little later than the pressure, and also tends to prevent the current diminishing in step with the pressure.
Ques. What governs the amount of lag in an alternating current?
Ans. It depends on the relative values of the various pressures in the circuit, that is, upon the amount of resistance and inductance which tends to cause lag, and the amount of capacity in the circuit which tends to reduce lag and cause lead.
Ques. How is lag measured?
Ans. In degrees.
Thus, in fig. 1,288, the lag is indicated by the distance between the beginning of the pressure curve and the beginning of the current curve, and is in this case 45°.
Ques. What is the physical meaning of this?
Ans. In an actual alternator, of which fig. 1,288 is an elementary diagram showing one coil, if the current lag, say 45°[Pg 1052] behind the pressure, it means that the coil rotates 45° from its position of zero induction before the current starts, as in fig. 1,288.
EXAMPLE I.—A circuit through which an alternating current is passing has an inductance of 6 ohms and a resistance of 2.5 ohms. What is the angle of lag?
Substituting these values in equation (1), page 1,053,
6 | ||||
tan φ | = | = | 2.4 | |
2.5 |
Referring to the table of natural sines and tangents on page 451 the corresponding angle is approximately 67°.
EXAMPLE II.—A circuit has a resistance of 2.3 ohms and an inductance of .0034 henry. If an alternating current having a frequency of 125 pass through it, what is the angle of lag?
Here the inductance is given as a fraction of a henry; this must be reduced to ohms by substituting in equation (3), page 1,038, which gives the ohmic value of the inductance; accordingly, substituting the above given value in this equation
inductance in ohms or Xi = 2π × 125 × .0034 = 2.67
Substituting this result and the given resistance in equation (1), page 1,053,
2.67 | ||||
tan φ | = | = | 1.16 | |
2.3 |
the nearest angle from table (page 451) is 49°.
Ques. How great may the angle of lag be?
Ans. Anything up to 90°.
The angle of lag, indicated by the Greek letter φ(phi), is the angle whose tangent is equal to the quotient of the inductance expressed in ohms or "spurious resistance" divided by the ohmic resistance, that is
reactance | 2πfL | ||||
tan φ | = | = | (1) | ||
resistance | R |
Ques. When an alternating current lags behind the pressure, is there not a considerable current at times when the pressure is zero?
Ans. Yes; such effect is illustrated by analogy in fig. 1,293.
Ques. What is the significance of this?
Ans. It does not mean that current could be obtained from a circuit that showed no pressure when tested with a suitable voltmeter, for no current would flow under such conditions. However, in the flow of an alternating current, the pressure[Pg 1054] varies from zero to maximum values many times each second, and the instants of no pressure may be compared to the "dead centers" of an engine at which points there is no pressure to cause rotation of the crank, the crank being carried past these points by the momentum of the fly wheel. Similarly the electric current does not stop at the instant of no pressure because of the "momentum" acquired at other parts of the cycle.
Ques. On long lines having considerable inductance, how may the lag be reduced?
Ans. By introducing capacity into the circuit. In fact, the current may be advanced so it will be in phase with the pressure or even lead the latter, depending on the amount of capacity introduced.
There has been some objection to the term lead as used in describing the effect of capacity in an alternating circuit, principally on the ground that such expressions as "lead of current," "lead in phase," etc., tend to convey the idea that the effect precedes the cause, that is, the current is in advance of the pressure producing it. There can, of course, be no current until pressure has been applied, but if the circuit has capacity, it will lead the pressure, and this peculiar behavior is best illustrated by a mechanical analogy as has already been given.
Ques. What effect has lag or lead on the value of the effective current?
Ans. As the angle of lag or lead increases, the value of the effective as compared with the virtual current diminishes.
Reactance.—The term "reactance" means simply reaction. It is used to express certain effects of the alternating current other than that due to the ohmic resistance of the circuit. Thus, inductance reactance means the reaction due to the spurious resistance of inductance expressed in ohms; similarly, capacity reactance, means the reaction due to capacity, expressed in ohms.[Pg 1055] It should be noted that the term reactance, alone, that is, unqualified, is generally understood to mean inductance reactance, though ill advisedly so.
The resistance offered by a wire to the flow of a direct current is expressed in ohms; this resistance remains constant whether the wire be straight or coiled. If an alternating current flow through the wire, there is in addition to the ordinary or "ohmic" resistance of the wire, a "spurious" resistance arising from the development of a reverse pressure due to induction, which is more or less in value according as the wire be coiled or straight. This spurious resistance as distinguished from the ohmic resistance is called the reactance, and is expressed in ohms.
Reactance, may then be defined with respect to its usual significance, that is, inductance reactance, as the component of the impedance which when multiplied into the current, gives the wattless component of the pressure.
Reactance is simply inductance measured in ohms.
EXAMPLE I.—An alternating current having a frequency of 60 is passed through a coil whose inductance is .5 henry. What is the reactance?
Here f = 60 and L = .5; substituting these in formula for inductive reactance,
Xi = 2πfL = 2 × 3.1416 × 60 × .5 = 188.5 ohms
[Pg 1056]The quantity 2πfL or reactance being of the same nature as a resistance is used in the same way as a resistance. Accordingly, since, by Ohm's law
E = RI (1)
an expression may be obtained for the volts necessary to overcome reactance by substituting in equation (1) the value of reactance given above, thus
E = 2πfLI (2)
EXAMPLE II.—How many volts are necessary to force a current of 3 amperes with frequency 60 through a coil whose inductance is .5 henry? Substituting in equation (2) the values here given
E = 2πfLI = 2π × 60 × .5 × 3 = 565 volts.
The foregoing example may serve to illustrate the difference in behaviour of direct and alternating currents. As calculated, it requires 565 volts to pass only 3 amperes of alternating current through the coil on account of the considerable spurious resistance. The ohmic resistance of a coil is very small, as compared with the spurious resistance, say 2 ohms. Then by Ohm's law I = E ÷ R = 565 ÷ 2 = 282.5 amperes.
Instances of this effect are commonly met with in connection with transformers. Since the primary coil of a transformer has a high reactance, very little current will flow when an alternating pressure is applied. If the same transformer were placed in a direct current circuit[Pg 1057] and the current turned on it would at once burn out, as very little resistance would be offered and a large current would pass through the winding.
EXAMPLE III.—In a circuit containing only capacity, what is the reactance when current is supplied at a frequency of 100, and the capacity is 50 microfarads?
1 | ||||||
50 microfarads | = | 50 | × | = | .00005 farad | |
1,000,000 |
capacity reactance, or
1 | 1 | |||||
Xc | = | = | = | 31.84 ohms | ||
2πfC | 2 × 3.1416 × 100 × .00005 |
Impedance.—This term, strictly speaking, means the ratio of any impressed pressure to the current which it produces in a conductor. It may be further defined as the total opposition in an electric circuit to the flow of an alternating current.
All power circuits for alternating current are calculated with reference to impedance. The impedance may be called the combination of:
The impedance of an inductive circuit which does not contain capacity is equal to the square root of the sum of the squares of the resistance and reactance, that is
impedance = √(resistance2 + reactance2) (1)
EXAMPLE I.—If an alternating pressure of 100 volts be impressed on a coil of wire having a resistance of 6 ohms and inductance of 8 ohms, what is the impedance of the circuit and how many amperes will flow through the coil? In the example here given, 6 ohms is the resistance and 8 ohms the reactance. Substituting these in equation (1)
Impedance = √(62 + 82) = √(100) = 10 ohms.
The current in amperes which will flow through the coil is, by Ohm's law using impedance in the same way as resistance.
volts | 100 volts | |||||
current | = | = | = | 10 amperes. | ||
impedance | 10 ohms |
The reactance is not always given but instead in some problems the frequency of the current and inductance of the circuit. An expression to fit such cases is obtained by substituting 2πfL for the reactance as follows: (using symbols for impedance and resistance)
Z = √(R2 + (2πfL)2) (2)
EXAMPLE II.—If an alternating current, having a frequency of 60, be impressed on a coil whose inductance is .05 henry and whose resistance is 6 ohms, what is the impedance?
Here R = 6; f = 60, and L = .05; substituting these values in (2)
Z = √(62 + (2π × 60 × .05)2) = √(393) = 19.8 ohms.
EXAMPLE III.—If an alternating current, having a frequency of 60, be impressed on a circuit whose inductance is .05 henry, and whose capacity reactance is 10 ohms, what is the impedance?
Xi = 2πfL = 2 × 3.1416 × 60 × .05 = 18.85 ohms
Z = Xi - Xc = 18.85 - 10 = 8.85 ohms
When a circuit contains besides resistance, both inductance and capacity, the formula for impedance as given in equation (1), page 1,058, must be modified to include the reactance due to capacity, because, as explained, inductive and capacity reactances work in opposition to each other, in the sense that the reactance of inductance acts in direct proportion to the quantity 2πfL, and the reactance of capacity in inverse proportion to the quantity 2πfC. The net reactance due to both, when both are in the circuit, is obtained by subtracting one from the other.
To properly estimate impedance then, in such circuits, the following equation is used:
impedance = √(resistance2 + (inductance reactance - capacity reactance)2)
or using symbols,
Z = √(R2 + (Xi - Xc)2) (3)
EXAMPLE IV.—A current has a frequency of 100. It passes through a circuit of 4 ohms resistance, of 150 milli-henrys inductance, and of 22 microfarads capacity. What is the impedance?
a. The ohmic resistance R, is 4 ohms.
b. The inductance reactance, or
Xi = 2πfL = 2 × 3.1416 × 100 × .15 = 94.3 ohms.
(note that 150 milli-henrys are reduced to .15 henry before substituting in the above equation).
c. The capacity reactance, or
1 | 1 | |||||
Xc | = | = | = | 72.4 ohms | ||
2πfC | 2 × 3.1416 × 100 × .000022 |
(note that 22 microfarads are reduced to .000022 farad before substituting in the formula. Why? See page 1,042).
Substituting values as calculated in equation (3), page 1,060.
Z = √(42 + (94.2 - 72.4)2) = √(491) = 22.2 ohms.
Ques. Why is capacity reactance given a negative sign?
Ans. Because it reacts in opposition to inductance, that is it tends to reduce the spurious resistance due to inductance.
In circuits having both inductance and capacity, the tangent of the angle of lag or lead as the case may be is the algebraic sum of the two reactances divided by resistance. If the sign be positive, it is an angle of lag; if negative, of lead.
Resonance.—The effects of inductance and capacity, as already explained, oppose each other. If inductance and capacity be present in a circuit in such proportion that the effect of one neutralizes that of the other, the circuit acts as though it were purely non-inductive and is said to be in a state of resonance.
For instance, in a circuit containing resistance, inductance, and capacity, if the resistance be, say, 8 ohms, the inductance 30, and the capacity 30, then the impedance is
√(82 + (302 - 302)) = √(82) = 8 ohms.
The formula for inductance reactance is Xi = 2πfL, and for capacity reactance, Xc = 1 ÷ (2πfC); accordingly if capacity and inductance in a circuit be equal, that is, if the circuit be resonant
1 | |||
2πfL | = | (1) | |
2πfC |
from which
1 | |||
f | = | (2) | |
2π√(CL) |
Ques. What does equation (1) show?
Ans. It indicates that by varying the frequency in the proper way as by increasing or decreasing the speed of the alternator, the circuit may be made resonant, this condition being obtained when the frequency has the value indicated by equation (2).
Ques. What is the mutual effect of inductance and capacity?
Ans. One tends to neutralize the other.
Ques. What effect has resonance on the current?
Ans. It brings the current in phase with the impressed pressure.
It is very seldom that a circuit is thus balanced unless intentionally brought about; when this condition exists, the effect is very marked, the pressure rising excessively and bringing great strain upon the insulation of the circuit.
Ques. Define "critical frequency."
Ans. In bringing a circuit to a state of resonance by increasing the frequency, the current will increase with increasing frequency until the critical frequency is reached, and then the current will decrease in value for further increase of frequency. The critical frequency occurs when the circuit reaches the condition of resonance.
Ques. How is the value of the current at the critical frequency determined?
Ans. By the resistance of the circuit.
Skin Effect.—This is the tendency of alternating currents to avoid the central portions of solid conductors and to flow or pass mostly through the outer portions. The so-called skin effect becomes more pronounced as the frequency is increased.
Ques. What is the explanation of skin effect?
Ans. It is due to eddy currents induced in the conductor.
Consider the wire as being composed of several small insulated wires placed closely together. Now when a current is started along these separate wires, mutual induction will take place between them, giving rise to momentary reverse pressures. Those wires which are nearer the center, since they are completely surrounded by neighboring wires, will clearly have stronger reverse pressures set up in them than those on or near the outer surface, so that the current will meet less opposition near the surface than at the center, and consequently the flow will be greater in the outer portions.
Ques. What is the result of skin effect?
Ans. It results in an apparent increase of resistance.
The coefficient of increase of resistance depends upon the dimensions and the shape of the cross section, the frequency, and the specific resistance.
Hughes, about 1883, called attention to the fact that the resistance of an iron telegraph wire was greater for rapid periodic currents than for steady currents.
In 1888 Kelvin showed that when alternating currents at moderately high frequency flow through massive conductors, the current is practically confined to the skin, the interior portions being largely useless for the purpose of conduction. The mathematical theory of the subject has been developed by Kelvin, Heaviside, Rayleigh, and others.
Whenever an alternating pressure is impressed on a circuit, part of it is spent in overcoming the resistance, and the rest goes to balance the reverse pressure due to self-induction.
The total pressure applied to the circuit is known as the impressed pressure, as distinguished from that portion of it called the active pressure which is used to overcome the resistance, and that portion called the self-induction pressure used to balance the reverse pressure of self-induction.
The intensity of the reverse pressure induced in a circuit due to self-induction is proportional to the rate of change in the current strength.
Thus a current, changing at the rate of one ampere per second, in flowing through a coil having a coefficient of self-induction of one henry, will induce a reverse pressure of one volt.
Ques. Describe how the rate of change in current strength varies, and how this affects the reverse pressure.
Ans. The alternating current varies from zero to maximum strength in one-quarter period, that is, in one-quarter revolution of the generating loop or 90° as represented by the sine curve in fig. 1,307. Now, during, say, the first 10 degrees of rotation (from 0 to A), the current jumps from zero value to A', or 4[Pg 1068] amperes, according to the scale; during some intermediate 10 degrees of the quarter revolution, as from B to C, the current increases from B' to C' or 2½ amperes, and during another 10 degrees as from D to E, at the end of one-quarter revolution where the sine curve reaches its amplitude, it rises and falls ½ ampere. It is thus seen that the rate of change varies from a maximum when the current is least, to zero when the current is at its maximum. Accordingly, the reverse pressure of self-induction being proportional to the rate of change in the current strength, is greatest when the current is at zero value, and zero when the current is at its maximum.
This relation is shown by curves in fig. 1,308, and it should be noted that the reverse pressure and current are 90° apart in phase. For this reason many alternating current problems may be solved graphically by the use of right angle triangles, the sides, drawn to some arbitrary scale, to represent the quantities involved, such as resistance, reactance, impedance, etc.
Properties of Right Angle Triangles.—In order to understand the graphical method of solving alternating current problems, it is necessary to know why certain relations exist between the sides of a right angle triangle. For instance, in every right angle triangle:
The square of the hypothenuse is equal to the sum of the squares on the other two sides.
That is, condensing this statement into the form of an equation:
hypothenuse2 = base2 + altitude2 (1)
the horizontal side being called the base and the vertical side, the altitude.
This may be called the equation of the right angle triangle.
Ques. Why is the square of the hypothenuse of a right angle triangle equal to the sum of the squares of the other two sides?
Ans. This may be explained with the aid of fig. 1,309. Draw a line AB, 4 inches in length and erect a perpendicular BC, 3 inches in height; connect A and C, giving the right angle triangle ABC. It will be found that AC the hypothenuse of this triangle is 5 inches long. If squares be constructed on all[Pg 1071] three sides of the triangle, the square on the hypothenuse will have an area of 25 sq. ins.; the square on the base, 16 sq. ins., and the square on the altitude, 9 sq. ins. Then from the figure 52 = 42 + 32, that is 25 = 16 + 9.
Repeating equation (1), it is evident from the figure that
hypothenuse2 | ⎫ | ⎧ | base2 + altitude2 | ⎫ | |
⎬ | = | ⎨ | ⎬ | ||
52 | ⎭ | ⎩ | 42 + 32 | ⎭ |
that is,
25 = 16 + 9.
In the right angle triangle, the following relations also hold:
base2 | = | hypothenuse2 | - | altitude2 | (2) |
(42 | = | 52 | - | 32) | |
altitude2 | = | hypothenuse2 | - | base2 | (3) |
(32 | = | 52 | - | 42) |
In working impedance problems, it is not the square of any of the quantities which the sides of the triangle are used to represent that is required, but the quantities themselves, that is, the sides. Hence extracting the square root in equations (1), (2) and (3), the following are obtained:
hypothenuse | = | √(base2 + altitude2) | (4) |
(5 | = | √(42 + 32)) | |
base | = | √(hypothenuse2 - altitude2) | (5) |
(4 | = | √(52 - 32)) | |
altitude | = | √(hypothenuse2 - base2) | (6) |
(3 | = | √(52 - 42)) |
Representation of Forces by Lines.—A single force may be represented in a drawing by a straight line, 1, the point of application of the force being indicated by an extremity of the line, 2, the intensity of the force by the length of the line, and 3, the direction of the force by the direction of the line, an arrow head being placed at an extremity defining the direction.
Thus in fig. 1,310, the force necessary to balance the thrust on the steam piston may be represented by the straight line f whose length measured on any convenient scale represents the intensity of the force, and whose direction represents the direction of the force.
Composition of Forces.—This is the operation of finding a single force whose effect is the same as the combined effect of two or more given forces. The required force is called the resultant of the given forces.
The composition of forces may be illustrated by the effect of the wind and tide on a sailboat as in fig. 1,311. Supposing the boat be acted upon by the wind so that in a given time, say half an hour, it would be moved in the direction and a distance represented by the line AB, and that in the same time the tide would carry it from A to C. Now, lay down AB, to any convenient scale, representing the effect of the wind, and AC that of the tide, and draw BD equal and parallel to AC, and CD equal and parallel to AB, then the diagonal AD will represent the direction and distance the boat will move under the combined effect of wind and tide.
Ques. In fig. 1,311 what is the line AD called?
Ans. The resultant, that is, it represents the actual movement of the boat resulting from the combined forces of wind and tide.
Ques. What are the forces, AB and AD in fig. 1,311, represented by the sides of the parallelogram, and which act upon a body to produce the resultant, called?
Ans. The components.
EXAMPLE.—Two forces, one of 3 lbs. and one of 4 lbs. act at a point a in a body and at right angles, what is the resultant?
Take any convenient scale, say 1 in. = 1 lb., and lay off (fig. 1312.) AB = 4 ins. = 4 lbs.; also, AC (at right angles to AB) = 3 ins. = 3 lbs. Draw CD and BD parallel to AB and AC respectively, and join AD. The line AD is the resultant of the components AB and AC, and when measured on the same scale from which AB and AC were drawn will be found to be 5 inches long, which represents 5 lbs. acting in the direction AD.
Circuits containing Resistance and Inductance.—In circuits of this kind where the impressed pressure encounters[Pg 1075] both resistance and inductance, it may be looked upon as split up into two components, as already explained, one of which is necessary to overcome the resistance, and the other, the inductance. That is, the impressed pressure is split up into
The active pressure is in phase with the current.
The self induction pressure is at right angles to the current and 90 degrees ahead of the current in phase.
Ques. Why is the active pressure in phase with the current?
Ans. The pressure used in overcoming resistance is from Ohm's law, E = RI. Hence, when the current is zero, E is zero, and when the current is a maximum E is a maximum. Hence, that component of the impressed pressure necessary to overcome the resistance must be in phase with the current.
Ques. Why is this?
Ans. Since the reverse pressure of self induction is 90° behind the current, the component of the impressed pressure necessary to overcome the reverse pressure of self induction, being opposite to this, will be represented as being 90° ahead of the current.
The distinction between the reverse pressure of self-induction, that is, the induced pressure, and the pressure necessary to overcome self-induction should be carefully noted. They are two equal and opposite forces, that is, two balancing forces just as is shown in fig. 1,310. Here, in analogy, the thrust of the piston may represent the induced pressure and the equal and opposite force indicated by the arrow f, the component of the impressed pressure necessary to balance the induced pressure.
The Active Pressure or "Ohmic Drop."—The component of the impressed pressure necessary to overcome resistance, is from Ohm's law:
active pressure = ohmic resistance × virtual current
that is
Ea = RoIv (1)
this is the "ohmic drop" and may be represented by a line AB, fig. 1,314 drawn to any convenient scale, as for instance, 1 in. = 10 volts.
The Self-induction Pressure or "Reactance Drop."—The component of the impressed pressure necessary to overcome the induced pressure, is from Ohm's law:
inductance pressure = inductance reactance × virtual current;
that is,
Ei = XiIv (2)
Now the reactance Xi, that is the spurious resistance, is obtained from the formula
Xi = 2πfL (3)
as explained on page 1,038, and in order to obtain the volts necessary to overcome this spurious resistance, that is, the "reactance drop" as it is called, the value of Xi in equation (3) must be substituted in equation (2), giving
Ei = 2πfLI (4)
writing simply I for the virtual pressure.
Since the pressure impressed on a circuit is considered as made up of two components, one in phase with the current and one at right angles to the current, the component Ei or "reactance drop" as given in equation (4) maybe represented by the line BC in fig. 1,314, at right angles to AB, and of a length BC, measured with the same scale as was measured AB, to correspond to the value indicated by equation (4).
[Pg 1078]EXAMPLE.—In an alternating circuit, having an ohmic drop of 5 volts, and a reactance drop of 15 volts, what is the impressed pressure?
With a scale of say, ¼ inch = one volt, lay off, in fig. 1,315, AB = 5 volts = 1¼ in., and, at right angles to it, BC = 15 volts = 15/4 or 3¾ ins. Join AC; this measures 3.95 inches, which is equivalent to 3.95 × 4 = 15.8 volts, the impressed pressure. By using good paper, such as bristol board, a 6H pencil, engineers' scale and triangles or square, such problems are solved with precision. By calculation impressed pressure = √(52 + 152) = 15.8 volts. Note that the diagram is drawn with the side BC horizontal instead of AB—simply to save space.
EXAMPLE.—In an alternating circuit, having an ohmic drop of 5 volts and an impressed pressure of 15.8 volts, what is the reactance drop?
In fig. 1,317, draw a horizontal line of indefinite length and at any point B erect a perpendicular AB = 5 volts. With A as center and radius of length equivalent to 15.8 volts, describe an arc cutting the horizontal line at C. This gives BC, the reactance drop required, which by measurement is 15 volts.
EXAMPLE.—An alternating current of 10 amperes having a frequency of 60, is impressed on a circuit containing a resistance of 5 ohms and an inductance of 15 milli-henrys. What is the impressed pressure?
The active pressure or ohmic drop is 5 × 10 = 50 volts.
The inductance reactance or Xi = is 2 × 3.1416 × 60 × .015 = 5.66 ohms. Substituting this and the current value 10 amperes in the formula for inductance pressure or reactance drop (equation 2 on page 1,077) gives Ei = 5.65 × 10 = 56.5 volts.
In fig. 1,321, lay off AB = 50 volts, and BC = 56.6 volts. Using a scale of 20 volts to the inch gives AB = 2.5 ins., and BC = 2.83 ins. Joining AC gives the impressed voltage, which by measurement is 75.4 volts.
In some problems it is required to find the impedance of a circuit in which the ohmic and spurious resistances are given. This is done in a manner similar to finding the impressed pressure.
Ohmic resistance and spurious resistance or inductance reactance both tend to reduce an alternating current. Their combined action or impedance is equal to the square root of the sum of their squares, that is,
impedance = √(resistance2 + reactance2)
This relation is represented graphically by the side of a right angle triangle as in fig. 1322, in which the hypothenuse corresponds to the impedance, and the sides to the resistance and reactance.
EXAMPLE.—In a certain circuit the resistance is 4 ohms, and the reactance 3 ohms. What is the impedance?
[Pg 1082]In fig. 1,323, lay off, on any scale AB = 4 ohms and erect the perpendicular BC = 3 ohms. Join AC, which gives the impedance, and which is, measured with the same scale, 5 ohms.
EXAMPLE.—A coil of wire has a resistance of 20 ohms and an inductance of 15 milli-henrys. What is its impedance for a current having a frequency of 100?
The ohmic value of the inductance, that is, the reactance is
2πfL = 2 × 3.1416 × 100 × .015 = 9.42 ohms.
In fig. 1,324, lay off, on any scale, AB = 20 ohms, and the perpendicular BC to length = 9.42 ohms. Join AC, which gives the impedance, which is, measured on the same scale, 22.1 ohms.
EXAMPLE.—What is the angle of lag in a circuit having a resistance of 4 ohms and a reactance of 3 ohms?
Construct the impedance diagram in the usual way as in fig. 1,325, then the angle included between the impedance and resistance lines [Pg 1083](denoted by φ) is the angle of lag, that is, the angle BAC. By measurement with a protractor it is 37 degrees. By calculation the tangent of the angle of lag or
BC | 3 | ||||
tan φ | = | = | or .75 | ||
AB | 4 |
From the table on page 451, the angle is approximately 37°.
Circuits containing Resistance and Capacity.—The effect of capacity in an alternating current circuit is to cause the current to lead the pressure, since the reaction of a condenser, instead of tending to prolong the current, tends to drive it back.
Careful distinction should be made between capacity in series with a circuit and capacity in parallel with a branch of a circuit. The discussion here refers to capacity in series, which means that the circuit is not continuous but the ends are joined to a condenser, as shown at the right in fig. 1,326, so that no current can flow except into and out of the condenser.
Ques. In circuits containing resistance and capacity upon what does the amount of lead depend?
Ans. Upon the relative values of the resistance and the capacity reactance.
Ques. Describe the action of a condenser when current is applied.
Ans. When the current begins to flow into a condenser, that is, when the flow is maximum, the back pressure set up by the condenser (called the condenser pressure) is zero, and when the flow finally becomes zero, the condenser pressure is a maximum.
Ques. What does this indicate?
Ans. It shows that the phase difference between the wave representing the condenser pressure and the current is 90°, as illustrated in fig. 1,327.
Ques. Is the condenser pressure ahead or behind the current and why?
Ans. It is ahead of the current. The condenser pressure, when the condenser is discharged being zero, the current enters[Pg 1086] at maximum velocity as at A in fig. 1,327, and gradually decreases to zero as the condenser pressure rises to maximum at B, this change taking place in one-quarter period. Thus the condenser pressure, which opposes the current, being at a maximum when the current begins its cycle is 90° ahead of the current, as is more clearly seen in the last quarter of the cycle (fig. 1,327).
Ques. What is the phase relation between the condenser pressure and the pressure applied to the condenser to overcome the condenser pressure?
Ans. The pressure applied to the condenser to overcome the condenser pressure, or as it is called, the capacity pressure, must be opposite to the condenser pressure, or 90° behind the current.
In circuits containing resistance and capacity, the total pressure impressed on the circuit, or impressed pressure, as it is called, is made up of two components:
1. The active pressure, or pressure necessary to overcome the resistance;
The active pressure is in phase with the current.
2. The capacity pressure, or pressure necessary to overcome the condenser pressure,
The capacity pressure is 90 degrees behind the current.
Problems involving resistance and capacity are solved similarly to those including resistance and inductance.
The Active Pressure or "Ohmic Drop."—This, as before explained is represented, in fig. 1,329, by a line AB, which in magnitude equals, by Ohm's law, the product of the resistance multiplied by the current, that is,
Ea = RoIv (1)
The Capacity Pressure or "Reactance Drop."—This component of the impressed pressure, is, applying Ohm's law,
capacity pressure = capacity reactance × virtual current.
Ec = XcIv (2)
That is, the expression for capacity reactance Xc, that is, for the value of capacity in ohms is, as explained on page 1,048,
1 | |||
Xc | = | (3) | |
2πfC |
Substituting this value of Xc in equation (2) and writing I for virtual current.
I | |||
---|---|---|---|
Ec | = | (4) | |
2πfC |
CAUTION—The reader should distinguish between the 1 (one) in (3) and the letter I in (4); both look alike.
Since the capacity pressure is 90° behind the current, it is represented in fig. 1,329, by a line BC, drawn downward, at right angles to AB, and of a length corresponding to the capacity pressure, that is, to the reactance drop.
The Impressed Pressure.—Having determined the ohmic and reactance drops and represented them in the diagram, fig. 1,329, by lines AB and BC respectively, a line AC joining A and C, will then be the resultant of the two component pressures, that is, it will represent the impressed pressure or total pressure applied to the circuit.
In the diagram it should be noted that the active pressure is called the ohmic drop, and the capacity pressure, the reactance drop.
[Pg 1089]EXAMPLE.—A circuit as shown in fig. 1,330 contains a resistance of 30 ohms, and a capacity of 125 microfarads. If an alternating current of 8 amperes with frequency 60 be flowing in the circuit, what is the ohmic drop, the reactance drop, and the impressed pressure?
The ohmic drop or active pressure is, substituting in formula (1) on page 1,087,
Ea = 30 × 8 = 240 volts
which is the reading of voltmeter A in fig. 1,330.
The reactance drop or
I | 8 | |||||
Ec | = | = | = | 170 volts | ||
2πfC | 2 × 3.1416 × 60 × .000125 |
in substituting, note that the capacity C of 125 microfarads is reduced to .000125 farad.
Using a scale of say 1 inch = 80 volts, lay off in fig. 1,331, AB equal to the ohmic drop of 240 volts; on this scale AB = 3 inches. Lay off at right angles, BC = reactance drop = 170 volts = 2.125 inches. Join AC, which gives the impressed voltage, (that is the reading of voltmeter I in fig. 1,330,) which measures 294 volts.
By calculation, impressed pressure = √(2402 + 1702) = 294 volts.
[Pg 1090]EXAMPLE.—In the circuit shown in fig. 1,330, what is the angle of lead?
The tangent of the angle of lead is given by the quotient of the reactance divided by the resistance of the circuit. That is,
reactance | reactance drop | ||||||
tan φ | = | = | |||||
resistance | resistance drop | ||||||
Ec | I | ||||||
tan φ | = | = | ÷ | Ea | (1) | ||
Ea | 2πfC |
The tangent is given a negative sign because lead is opposed to lag and because the positive value is assigned to lag. Substituting in (1)
170 | 2.125" | |||||
tan φ | = | or | = | -.71 | ||
240 | 3" |
the angle corresponding is approximately 35¼° (see table page 451).
Circuits Containing Inductance and Capacity.—The effect of capacity in a circuit is exactly the opposite of inductance, that is, one tends to neutralize the other. The method of representing each graphically has been shown in the preceding figures. Since they act oppositely, that is 180° apart, the reactance due to each may be calculated and the values thus found, represented by oppositely directed vertical lines: the inductance resistance upward from a reference line, and the capacity resistance downward from the same reference line. The difference then is the resultant impedance. This method is shown in fig. 1,332, but it is more conveniently done as in fig. 1,333.
EXAMPLE.—In a circuit, as in fig. 1,334, containing an inductance of 30 milli-henrys and a capacity of 125 microfarads, how many volts must be impressed on the circuit to produce a current of 20 amperes having a frequency of 100.
The inductance reactance is
Xi = 2πfL = 2 × 3.1416 × 100 × .03 = 18.85 ohms.
Substituting this and the current value of 20 amperes in the formula for inductance pressure
Ei = RiI = 18.85 × 20 = 377 volts.
[Pg 1092]Reducing 125 microfarads to .000125 farad, and substituting in the formula for capacity pressure
I | 20 | |||||
Ec | = | = | = | 255 volts. | ||
2πfC | 2 × 3.1416 × 100 × .000125 |
A diagram is unnecessary in obtaining the impressed pressure since it is simply the difference between inductance pressure and capacity pressure (the circuit being assumed to have no resistance), that is
impressed pressure = Ei - Ec = 377 - 255 = 122 volts.
EXAMPLE.—A circuit in which a current of 20 amperes is flowing at a frequency of 100, has an inductance reactance of 18.25 ohms, and a capacity of 125 microfarads. What is the impedance?
The reactance due to capacity is
1 | 1 | |||||
Xc | = | = | = | 12.76 ohms. | ||
2πfC | 2 × 3.1416 × 100 × .000125 |
The impedance of the circuit then is the difference between the two reactances, that is impedance = inductance reactance - capacity reactance, or
Z = Xi - Xc = 18.25 - 12.76 = 5.49 ohms.
Circuits Containing Resistance, Inductance, and Capacity.—When the three quantities resistance, inductance, and capacity, are present in a circuit, the combined effect is easily[Pg 1093] understood by remembering that inductance and capacity always act oppositely, that is, they tend to neutralize each other. Hence, in problems involving the three quantities, the resultant of inductance and capacity is first obtained, which, together with the resistance, is used in determining the final effect.
Capacity introduced into a circuit containing inductance reduces the latter and if enough be introduced, inductance will be neutralized, giving a resonant circuit which will act as though only resistance were present.
Ques. What is the expression for impedance of a circuit containing resistance, inductance and capacity?
Ans. It is equal to the square root of the sum of the resistance squared plus the square of inductance reactance minus capacity reactance.
This is expressed plainer in the form of an equation as follows:
impedance = √(resistance2 + (inductance reactance - capacity reactance)2)
or, using symbols,
Z = √(R2 + (Xi - Xc)2) (1)
Ques. If the capacity reactance be larger than the inductance reactance, how does this affect the sign of (Xi-Xc)2?
Ans. The sign of the resultant reactance of inductance and capacity will be negative if capacity be the greater, but since in the formula the reactance is squared, the sign will be positive.
EXAMPLE.—What is the impedance in a circuit having 25 ohms resistance, 30 ohms inductance reactance, and 40 ohms capacity reactance?
To solve this problem graphically, draw the line AB, in fig. 1,337, equal to 25 ohms resistance, using any convenient scale.
At B draw upward at right angles BC = 30 ohms; draw from C downward CC' = 40 ohms. This gives -BC' (= BC - CC') showing the capacity reactance to be 10 ohms in excess of the inductance reactance. Such a [Pg 1095]circuit is equivalent to one having no inductance but the same resistance and 10 ohms capacity reactance.
The diagram is completed in the usual way by joining AC giving the required impedance, which by measurement is 26.9 ohms.
By calculation, Z = √(252 + (30 - 40)2) = √(252 + (-10)2) = 26.9.
Form of Impedance Equation without Ohmic Values.—
Using the expressions 2πfL for inductance reactance and 1 / (2πfC) for capacity reactance, and substituting in equation (1) on page 1,093 gives the following:
Z = √(R2 + (2πfL - 1 / (2πfC))2) (2)
which is the proper form of equation (1) to use in solving problems in which the ohmic values of inductance and capacity must be calculated.
EXAMPLE.—A current has a frequency of 150. It passes through a circuit, as in fig. 1,339, of 23 ohms resistance, of 41 milli-henrys inductance, and of 51 microfarads capacity. What is the impedance?
The inductance reactance or
Xi = 2πfL = 2 × 3.1416 × 150 × .041 = 38.64 ohms
(note that 41 milli-henrys are reduced to .041 henry before substituting in the above equation).
The capacity reactance, or
1 | 1 | |||||
Xc | = | = | = | 20.8 ohms | ||
2πfC | 2 × 3.1416 × 150 × .000051 |
(note that 51 microfarads are reduced to .000051 farad before substituting in the above equation).
Substituting the values as calculated for 2πfL and 1 / (2πfC) in equation (2)
Z = √(232 + (38.64 - 20.8)2) = 29.1 ohms.
To solve the problem graphically, lay off in fig. 1,340, the line AB equal to 23 ohms resistance, using any convenient scale. Draw upward and at right angles to AB the line BC = 38.64 ohms inductance reactance, and from C lay off downward CC' = 20.8 ohms capacity reactance. The resultant reactance[Pg 1097] is BC' and being above the horizontal line AB shows that inductance reactance is in excess of capacity reactance by the amount BC'. Join AC' which gives the impedance sought, and which by measurement is 29.1 ohms.
In order to obtain the impressed pressure in circuits containing resistance, inductance and reactance, an equation similar to (2) on page 1,095 is used which is made up from the following:
Eo | = | RI | (3) |
Ei | = | 2πfLI | (4) |
I | |||
Ec | = | (5) | |
2πfC |
When all three quantities, resistance, inductance, and capacity are present, the equation is as follows:
impressed pressure | = | √(ohmic drop2 + (inductive drop - capacity drop)2) | |
Eim | = | √(Eo2 + (Ei - Ec)2) | (6) |
Substituting in this last equation (6), the values given in (3), (4) and (5)
Eim | = | √(R2I2 + (2πfLI - (I / (2πfC)))2) | |
= | I√(R2 + (2πfL - (1 / (2πfC)))2) | (7) |
Ques. What does the quantity under the square root sign in equation (7) represent?
Ans. It is the impedance of a circuit possessing resistance, inductance, and capacity.
Ques. Why?
Ans. Because it is that quantity which multiplied by the current gives the pressure, which is in accordance with Ohm's law.
EXAMPLE.—An alternator is connected to a circuit having, as in fig. 1,341, 25 ohms resistance, an inductance of .15 henry, and a capacity of 125 microfarads. What pressure must be impressed on the circuit to allow 8 amperes to flow at a frequency of 60?
The ohmic drop is
Eo = RI = 25 × 8 = 200 volts.
The inductance drop is
Ei = 2πfLI = 2 × 3.1416 × 60 × .15 × 8 = 452 volts
The capacity drop is
I | 8 | |||||
Ec | = | = | = | 170 volts. | ||
2πfC | 2 × 3.1416 × 60 × .000125 |
Substituting the values thus found,
impressed pressure | = | √(Eo2 + (Ei - Ec)2) |
= | √(2002 + (452 - 170)2) | |
= | √(2002 + 2822) | |
= | √(119524) | |
= | 345.7 volts. |
The determination of the power in a direct current circuit is a simple matter since it is only necessary to multiply together the volts and amperes to obtain the output in watts. In the case of alternating current circuits, this holds true only when the current is in phase with the pressure—a condition rarely found in practice.
When the current is not in phase with the pressure, the product of volts and amperes as indicated by the voltmeter and ammeter must be multiplied by a coefficient called the power factor in order to obtain the true watts, or actual power available.
There are several ways of defining the power factor, any of which requires some explanation. The power factor may be defined as: The number of watts indicated by a wattmeter, divided by the apparent watts, the latter being the watts as measured by a voltmeter and ammeter.
The power factor may be expressed as being equal to
true power | true watts | true watts | ||
= | = | |||
apparent power | apparent watts | volts × amperes |
Ques. What are the true watts?
Ans. The watts as measured by a wattmeter.
Ques. What are the apparent watts?
Ans. The watts obtained by multiplying together the simultaneous voltmeter and ammeter readings.
Ques. What is usually meant by power factor?
Ans. The multiplier used with the apparent watts to determine how much of the power supplied is available.
Ques. Upon what does the power factor depend?
Ans. Upon the relative amounts of resistance inductance and capacity contained in the circuit.
Ques. How does the power factor vary in value?
Ans. It varies from one to zero.
The power factor, as will be shown later, is equal to the cosine of the angle of phase difference; its range then is from one to zero because these are the limiting values of the cosine of an angle (neglecting the + or-sign).
Ques. What is the effect of lag or lead of the current on the power factor?
Ans. It causes it to become less than one.
How to Obtain the Power Curve.—Since under any phase condition, the power at any instant is equal to the product of the pressure multiplied by the current at that instant, a curve may be easily plotted from the pressure and current curves, giving the instantaneous values of the power through a complete cycle.
In fig. 1,344, from the zero line of the current and pressure curves, draw any ordinate as at F cutting the current curve at G and the pressure[Pg 1104] curve at G'. The values for current and pressure at this point are from the scale, 2 amperes and 3.7 volts. Since watts = amperes × volts, the ordinate FG is to be multiplied by ordinate FG' that is,
2 × 3.7 = 7.4.
Project up through F the ordinate FG" = 7.4, and this will give one point on the power curve.
Similarly at another point, say M, where the current and pressure are maximum
MS | × | MS' | = | MS", | that is |
3 | × | 5 | = | 15 |
giving S" another point on the curve. Obtaining several points in this way the power curve is then drawn through them as shown.
Ques. Why is the power curve positive in the second half of the period when there are negative values of current and pressure?
Ans. Because the product of two negative quantities is positive.
Ques. Does fig. 1,344 represent the usual way of drawing a power curve?
Ans. Since ordinates of the power curve are products of the current and pressure ordinates, they will be of inconvenient[Pg 1105] length if drawn to the same scale; it is therefore customary to use a different scale for the power ordinates, as in fig. 1,345.
The illustration is lettered identical with fig. 1,344, with which it should be compared.
Synchronism of Current and Pressure; Power Factor Unity.—The current and pressure would be in phase as represented in fig. 1,346 were it possible to have a circuit containing resistance only. In actual practice all circuits contain at least a small amount of reactance.
A circuit supplying nothing but incandescent lamps comes very nearly being all resistance, and may be so considered in the discussion here. Fig. 1,347 illustrates a circuit containing only resistance. In such a circuit the pressure and current (as shown in fig. 1,346) pass through zero and through their maximum values together.
Multiplying instantaneous values of volts and amperes will give the power curve, as before explained, whose average value is half-way between the zero line and the maximum of the curve; that part of the power curve above the line of average power WW, exactly filling the open space below the line WW. That is,
average power | = | maximum power ÷ √2 |
= | maximum voltage × maximum current | |
√2 | ||
= | virtual voltage × virtual current. |
This latter is simply the product of the voltmeter and ammeter readings which gives the watts just the same as in direct current.
Ques. What should be noticed about the power curve?
Ans. Its position with respect to the zero line; it lies wholly above the zero line which denotes that all the power delivered to the circuit except that dissipated by friction is useful, that is, the[Pg 1107] power factor is unity. Hence, to keep the power factor as near unity as possible is one of the chief problems in alternating current distribution.
Ques. Can the power factor be less than unity if the current and pressure be in phase?
Ans. Yes, if the waves of current and voltage be distorted as in fig. 1,348.
Effect of Lag and Lead.—In an alternating circuit the amount of power supplied depends on the phase relationship of the current and pressure. As just explained, when there is synchronism of current and pressure, that is, when they are in phase (as in fig. 1,346) the power factor is unity, assuming no distortion of current and pressure waves. In all other cases the power factor is less than unity that is, the effect of lag or lead is to make the power factor less than unity.
The effect of lag on the power factor may be illustrated by fig. 1,349, in which the angle between the pressure and current, or the angle of lag is taken as 40°, corresponding to a power factor of .766. Plotting the power curve from the products of instantaneous volts and amperes taken at various points, the power curve is obtained, a portion of which lies below the horizontal line. The significance of this is that at certain times, the current is flowing in the opposite direction to that in which the impressed pressure would send it. During this part of the period conditions are reversed, and the power (indicated by the shaded area), instead of being supplied by the source to the circuit, is being supplied by the circuit to the source.
This condition is exactly analogous to the case of a steam engine, expanding the steam below the back or exhaust pressure, a condition sometimes caused by the action of the governor in considerably reducing[Pg 1110] the cut off for very light load. An indicator diagram of such steam distribution is shown in fig. 1,351. This gives a negative loop in the diagram indicated by the shaded section.
It must be evident that the average pressure of the shaded loop portion of the diagram must be subtracted from that of the other portion, because during the expansion below the exhaust pressure line, the back pressure is in excess of the forward pressure exerted on the piston by the expanding steam, and the engine would accordingly reverse its motion, were it not for the energy previously stored up in the fly wheel in the form of momentum, which keeps the engine moving during this period of back thrust. Evidently the shaded area must be subtracted from the positive area to obtain the net work done during the stroke. Hence following the analogy as far as possible if M work (watts) be done during each revolution (cycle) when steam does not expand below back pressure (when current and pressure are in phase), and S negative work (negative watts) be done when steam expands below back pressure (when there is lag), the efficiency (power factor) is (M - S) ÷ M.
"Wattless Current;" Power Factor Zero.—When the power factor is zero, it means that the phase difference between the current and the pressure is 90°.
The term wattless current, as understood, does not indicate an absence of electrical energy in the circuit; its elements are there,[Pg 1111] but not in an available form for external work. The false power due to the so called wattless current pulsates in and out of the circuit without accomplishing any useful work.
An example of wattless current, showing that the power factor is zero is illustrated in fig. 1,353. Here the angle of lag is 90°, that is, the current is 90° behind the pressure.
The power curve is constructed from the current and pressure curves, and, as shown in the diagram, it lies as much below the zero line as above, that is, the two plus power areas which occur during each period are equal to the two negative (shaded) power areas, showing that the circuit returns as much energy as is sent out. Hence, the total work done during each period is zero, indicating that although a current be flowing, this current is not capable of doing external work.
Ques. Is the condition as just described met with in practice?
Ans. No.
Ques. Why not?
Ans. The condition just described involves that the circuit have no resistance, all the load being reactance, but it is impossible to have a circuit without some resistance, though the[Pg 1112] resistance may be made very small in comparison to the reactance so that a close approach to wattless current is possible.
Ques. Give some examples where the phase difference is very nearly 90°.
Ans. If an alternator supplies current to a circuit having a very small resistance and very large inductance, the current would lag nearly 90° behind the pressure. The primary current of a transformer working with its secondary on an open circuit is a practical example of a current which represents very little energy.
Ques. When the phase difference between the current and pressure is 90°, why is the current called "wattless"?
Ans. Because the product of such a current multiplied by the pressure does not represent any watts expended.
A man lifting a weight, and then allowing it to descend the same distance to its initial position, as shown in figs. 1,355 to 1,357, presents a mechanical analogy of wattless current.
[Pg 1113]Let the movement of the weight represent the current and the weight the pressure. Then calling the weight 10 pounds (volts), and the distance two feet (amperes). The work done by the man (alternator) on the weight in lifting it is
10 pounds × 2 feet = 20 foot pounds (1)
(10 volts × 2 amperes = 20 watts.)
The work done on the man by the weight in forcing his hand down as his muscles relax is
10 pounds × 2 feet = 20 foot pounds (2)
(10 volts × 2 amperes = 20 watts.)
From (1) and (2) it is seen that the work done by the man on the weight is equal to the work done by the weight on the man, hence no useful work has been accomplished; that is, the potential energy of the weight which it originally possessed has not been increased.
Why the Power Factor is equal to Cos φ.—In the preceding figures showing power curves for various phase relations between current and pressure, the curves show the instantaneous values of the fluctuating power, but what is of more importance, is to determine the average power developed.
When the current is in phase with the pressure, it is a simple matter, because the power or
watts = amperes × volts
that is, the product of the ammeter and voltmeter readings will give the power. However, the condition of synchronism of current and pressure hardly ever exists in practice, there being more or less phase difference.
When the current is not in phase with the pressure, it is considered as made up of two components at right angles to each other.
1. The active component, in phase with the pressure;
2. The wattless component, at right angles to the pressure.
With phase difference between current and pressure the product of ammeter and voltmeter readings do not give the true[Pg 1115] power, and in order to obtain the latter, the active component of the current in phase with the pressure must be considered, that is,
true power = volts × active amperes (1)
The active component of the current is easily obtained graphically as in fig. 1,358.
With any convenient scale draw AB equal to the current as given or read on the ammeter, and AC, equal to the pressure, making the angle φ between AB and AC equal to the phase difference between the current and pressure.
From B, draw the line BD perpendicular to AC, then BD will be the wattless component, and AD (measured with the same scale as was used for AB) the active component of the current, or that component in phase with the pressure.
Hence from equation (1)
true power = AC × AD (2)
Now in the right triangle ABD
AD | ||
= | cos φ | |
AB |
from which
AD = AB cos φ (3)
Substituting this value of AD in equation (2) gives
true power = AC × AB cos φ (4)
Now the power factor may be defined as: that quantity by which the apparent watts must be multiplied in order to give the true power. That is
true power = apparent watts × power factor (5)
Comparing equations (4) and (5), AC × AB in (4) is equal to the apparent watts, hence, the power factor in (5) is equal to cos φ. That is, the power factor is numerically equal to the cosine of the angle of phase difference between current and pressure.
EXAMPLE I.—An alternator supplies a current of 200 amperes at a pressure of 1,000 volts. If the phase difference between the current and pressure be 30°, what is the true power developed?
In fig. 1,359, draw AB to scale, equal to 200 amperes, and draw AC of indefinite length making an angle of 30° with AB. From B, draw BD perpendicular to AC which gives AD, the active component, and which measured with the same scale as was used in laying off AB, measures 173.2 amperes. The true power developed then is
true watts = 173.2 × 1,000 = 173.2 kw.
The true power may be calculated thus:
From the table cos 30° = .866, hence
true watts = 200 × 1,000 × .866 = 173.2 kw.
EXAMPLE II.—If in an alternating current circuit, the voltmeter and ammeter readings be 110 and 20 and the angle of lag 45°, what is the apparent power and true power?
The apparent power is simply the product of the current and pressure readings or
apparent power = 20 × 110 = 2,200 watts
The true power is the product of the apparent power multiplied by the cosine of the angle of lag. Cos 45° = .707, hence
true power = 2,200 × .707 = 1,555.4 watts.
Ques. Does the power factor apply to capacity reactance in the same way as to inductance reactance?
Ans. Yes. The angles of lag and of lead, are from the practical standpoint, treated as if they lay in the first quadrant of the circle. Even the negative sign of the tangent φ when it occurs is simply used to determine whether the angle be one of lag or of lead, but in finding the value of the angle from a table it is treated as a positive quantity.
Ques. In introducing capacity into a circuit to increase the power factor what should be considered?
Ans. The cost and upkeep of the added apparatus as well as the power lost in same.
Ques. How is power lost in a condenser?
Ans. The loss is principally due to a phenomenon known as dielectric hysteresis, which is somewhat analogous to magnetic hysteresis. The rapidly alternating charges in a condenser placed in an alternating circuit may be said to cause alternating polarization of the dielectric, and consequent heating and loss of energy.
Ques. When is inductance introduced into a circuit to increase the power factor?
Ans. When the phase difference is due to an excess of capacity.
EXAMPLE.—A circuit having a resistance of 3 ohms, and a resultant reactance of 4 ohms, is connected to a 100 volt line. What is: 1, the impedance, 2, the current, 3, the apparent power, 4, the angle of lag, 5, the power factor, and 6, the true power?
1. The impedance of the circuit.
Z = √(32 + 42) = 5 ohms.
2. The current.
current = volts ÷ impedance = 100 ÷ 5 = 20 amperes.
3. The apparent power.
apparent power = volts × amperes = 100 × 20 = 2,000 watts.
4. The tangent of the angle of lag.
tan φ = reactance ÷ resistance = 4 ÷ 3 = 1.33. From table of natural tangents (page 451) φ = 53°.
5. The power factor.
The power factor is equal to the cosine of the angle of lag, that is, power factor = cos 53° = .602 (from table).
6. The true power.
The true power is equal to the apparent watts multiplied by the power factor, or
true power | = | volts | × | amperes | × | cos φ | ||
= | 100 | × | 20 | × | .602 | = | 1,204 watts. |
Ques. Prove that the power factor is unity when there is no resultant reactance in a circuit.
Ans. When there is no reactance, tan φ which is equal to reactance ÷ resistance becomes 0 ÷ R = 0. The angle φ[Pg 1119] (the phase difference angle) whose tangent is 0 is the angle of 0 degrees. Hence, the power factor which is equal to cos φ = cos 0° = 1.
Ques. What is the usual value of the power factor in practice?
Ans. Slightly less than one.
Ques. Why is it desirable to keep the power factor near unity?
Ans. Because with a low power factor, while the alternator may be carrying its full load and operating at a moderate temperature, the consumer is paying only for the actual watts which are sent over the line to him.
For instance, if a large alternator supplying 1,000 kilowatts at 6,600 volts in a town where a number of induction motors are used on the line be operating with a power factor of say .625 during a great portion of the time, the switchboard instruments connected to the alternator will give the following readings:
Voltmeter 6,600 volts; ammeter 242.4 amperes; power factor meter .625.
The apparent watts would equal 1,600,000 watts or 1,600 kilowatts, which, if multiplied by the power factor .625 would give 100,000,000 watts or 1,000 kilowatts which is the actual watts supplied. The alternator and line must carry 242.4 amperes instead of 151 amperes and the difference 242.4 - 151 = 91.4 amperes represents a wattless current flowing in the circuit which causes useless heating of the alternator.
[Pg 1121]The mechanical power which is required to drive the alternator is equivalent to the actual watts produced, since that portion of the current which lags, is out of phase with the pressure and therefore requires no energy.
Ques. How are alternators rated by manufacturers in order to avoid disputes?
Ans. They usually rate their alternators as producing so many kilovolt amperes instead of kilowatts.
Ques. What is a kilovolt ampere (kva)?
Ans. A unit of apparent power in an alternating current circuit which is equal to one kilowatt when the power factor is equal to one.
The machine mentioned on page 1,120 would be designed to carry 151 amperes without overheating and also carry slight overloads for short periods. It would be rated as 6.6 kilo volts and 151 amperes which would equal approximately 1,000 kilowatts when the power factor is 1 or unity, and it should operate without undue heating. Now the lower the power factor becomes, the greater the heating trouble will be in trying to produce the 1,000 actual kilowatts.
Ques. How can the power factor be kept high?
Ans. By carefully designing the motors and other apparatus and even making changes in the field current of motors which are already installed.
Ques. How is the power factor determined in station operation?
Ans. Not by calculation, but by reading a meter which forms one of the switchboard instruments.
Ques. When is the power factor meter of importance in station operation, and why?
Ans. When rotary converters are used on alternating current lines for supplying direct currents and the sub-station operators are kept busy adjusting the field rheostat of the rotary to maintain a high power factor and prevent overheating of the alternators during the time of day when there is the maximum demand for current or the peak of the load.
EXAMPLE.—An alternator delivers current at 800 volts pressure at a frequency of 60, to a circuit of which the resistance is 75 ohms and .25 henry.
Determine: a, the value of the current, b, angle of lag, c, apparent watts, d, power factor, e, true power.
a. Value of current
pressure | E | |||
current | = | = | ||
impedance | √(R2 + (2πfL)2) | |||
800 | ||||
= | = | 6.7 amperes | ||
√(752 + (2 × 3.1416 × 60 × .25)2) |
b. The angle of lag
reactance | 2πfL | 2 × 3.1416 × 60 × .25 | ||||||
tan φ | = | = | = | = | 1.25 | |||
resistance | R | 75 | ||||||
φ | = | angle of lag | = | 51° 15´ | (from table, page 451). |
c. The apparent power
apparent power | = | volts × amperes | = | 800 × 6.7 | = | 5,360 watts |
= | 5.36 kva. |
d. The power factor
power factor | = | cosine of the angle of lag | ||
= | cos 51° 15´ | = | .626. |
e. The true power
true power | = | apparent power | × | power factor | ||
= | 5,360 | × | .626 | = | 3,355 watts. |
Use of Alternators.—The great increase in the application of electricity for supplying power and for lighting purposes in industry, commerce, and in the home, is due chiefly to the economy of distribution of alternating current.
Direct current may be used to advantage in densely populated districts, but where the load is scattered, it requires, on account of its low voltage, too great an investment in distributing lines. In such cases the alternator is used to advantage, for while commutators can be built for collecting direct current up to 1,000 volts, alternators can be built up to 12,000 volts or more, and this voltage increased, by step up transformers of high economy, up to 75,000 or 100,000 volts. Since the copper cost is inversely as the square of the voltage, the great advantage of alternating current systems is clearly apparent.
The use of alternating current thus permits a large amount of energy to be economically distributed over a wide area from a single station, not only reducing the cost of the wiring, but securing greater economy by the use of one large station, instead of several small stations.
The higher voltages generated by alternators enables the transmission of electrical energy to vastly greater distances than possible by a direct current system, so that the energy from many waterfalls that otherwise would go to waste may be utilized.
Classes of Alternator.—There are various ways of classifying alternators. They may be divided into groups, according to: 1, the nature of the current produced; 2, type of drive; 3, method of construction; 4, field excitation; 5, service requirements, etc.
From these several points of view, alternators then may be classified:
1. With respect to the current, as:
2. With respect to the type of drive, as:
3. With respect to construction, as:
4. With respect to mode of field excitation, as:
5. With respect to service requirements, as:
Single Phase Alternators.—As a general rule, when alternators are employed for lighting circuits, the single phase machines are preferable, as they are simpler in construction and do not generate the unbalancing voltages often occurring in polyphase work.
Ques. What are the essential features of a single phase alternator?
Ans. Fig. 1,370 shows an elementary single phase alternator. It consists of an armature, with single phase winding, field[Pg 1128] magnets, and two collector rings and brushes through which the current generated in the armature passes to the external circuit.
Ques. In what respect do commercial machines differ mostly from the elementary alternator shown in fig. 1,370, and why?
Ans. They have a large number of poles and inductors in order to obtain the desired frequency, without excessive speed, and electromagnets instead of permanent magnets.
Ques. In actual machines, why must the magnet cores be spaced out around the armature with considerable distance between them?
Ans. In order to get the necessary field winding on the cores, and also to prevent undue magnetic leakage taking place, laterally from one limb to the next of opposite sign.
Ques. Is there any gain in making the width of the armature coils any greater than the pole pitch, and why?
Ans. No, because any additional width will not produce more voltage, but on the contrary will increase the resistance and inductance of the armature.
Polyphase Alternators.—A multiphase or polyphase alternator is one which delivers two or more alternating currents differing in phase by a definite amount.
For example, if two armatures of the same number of turns each be connected to a shaft at 90 degrees from each other and revolved in a bipolar field, and each terminal be connected to a collector ring, two separate alternating currents, differing in phase by 90 degrees, will be delivered to the external circuit. Thus a two phase alternator will deliver two currents differing in phase by one-quarter of a cycle, and similarly a three phase alternator (the three armatures of which are set 120 degrees from each other) will deliver three currents differing in phase by one-third of a cycle.
In practice, instead of separate armatures for each phase, the several windings are all placed on one armature and in such sequence that the currents are generated with the desired phase difference between them as shown in the elementary diagrams 1,372 and 1,373 for two phase current, and figs. 1,374 and 1,375 for three phase current.
Ques. What use is made of two and three phase current?
Ans. They are employed rather for power purposes than for lighting, but such systems are often installed for both services.
Ques. How are they employed in each case?
Ans. For lighting purposes the phases are isolated in separate circuits, that is, each is used as a single phase current. For driving motors the circuits are combined.
Ques. Why are they combined for power purposes?
Ans. On account of the difficulty encountered in starting a motor with single phase current.
Ferarris, of Italy, in 1888 discovered the important principle of the production of a rotating magnetic field by means of two or more
alternating currents displaced in phase from one another, and he thus made possible by means of the induction motor, the use of polyphase currents for power purposes.
Ques. What is the difficulty encountered in starting a motor with single phase current?
Ans. A single phase current requires either a synchronous motor to develop mechanical power from it, or a specially constructed motor of dual type, the idea of which is to provide a method of getting rotation by foreign means and then to throw in the single phase current for power.
Six Phase and Twelve Phase Windings.—These are required for the operation of rotary converters. The phase difference in a six phase winding is 60 degrees and in a twelve[Pg 1134] phase winding 30 degrees. A six phase winding can be made out of a three phase winding by disconnecting the three phases from each other, uniting their middle points at a common junction, as shown by diagram fig. 1,376. This will give a star grouping with six terminals.
In the case of a mesh grouping, each of the three phases must be cut into two parts and then reconnected as shown in fig. 1,377.
As the phase difference of a twelve phase winding is one-half that of a six phase winding, the twelve phases may be regarded as a star grouping of six pairs crossed at the middle point of each pair as shown in fig. 1,378, or in mesh grouping for converters they may be arranged as a twelve pointed polygon. They may also be grouped as a combination of mesh and star as shown in fig. 1,379, which, however, is not of general interest.
Belt or Chain Driven Alternators.—The mode in which power is transmitted to an alternator for the generation of current is governed chiefly by conditions met with where the machine is to be installed.
In many small power stations and isolated plants the use of a belt drive is unavoidable. In some cases the prime mover is already installed and cannot be conveniently arranged for direct connection, in others the advantage to be gained by an increase in speed more than compensates for the loss involved in belt transmission.
There are many places where belted machines may be used advantageously and economically. They are easily connected to an existing source of power, as, for instance, a line shaft used for driving other machinery, and for comparatively small installations they are lower in first cost than direct connected[Pg 1136] machines. Moreover, when connected to line shaft they are run by the main engine which as a rule is more efficient than a small engine direct connected.
Where there is sufficient room between pulley centers, a belt is a satisfactory medium for power transmission, and one that is largely used. It is important that there be liberal distance between centers, especially in the case of generators or motors belted to a medium or slow speed engine, because, owing to the high speed of rotation of the electric machines, there is considerable difference in their pulley diameters and the drive pulley diameter; hence, if they were close together, the arc of contact of the belt with the smaller pulley would be appreciably reduced, thus diminishing the tractive power of the belt.
Ques. What provision should be made in the design of an alternator to adapt it to belt drive?
Ans. Provision should be made for tightening the belt.
Ques. How is this done?
Ans. Sometimes by an idler pulley, but usually by mounting the machine on a sub-base provided with slide rails, as in fig. 1,381, the belt being tightened by use of a ratchet screw which moves the machine along the base.
Ques. Give a rule for obtaining the proper size of belt to deliver a given horse power.
Ans. A single belt travelling at a speed of one thousand feet per minute will transmit one horse power; a double belt will transmit twice that amount.
This corresponds to a working strain of 33 lbs. per inch of width for single belt, or 66 lbs. for double belt.
Many writers give as safe practice for single belts in good condition a working tension of 45 lbs. per inch of width.
Ques. What is the best speed for maximum belt economy?
Ans. From 4,000 to 4,500 feet per minute.
EXAMPLE.—What is the proper size of double belt for an alternator having a 16 inch pulley, and which requires 50 horse power to drive it at 1,000 revolutions per minute full load?
The velocity of the belt is
circumference in feet | × | revolutions | = | feet per minute | ||
16 | ||||||
× | .1416 | × | 1,000 | = | 4,188. | |
12 |
Horse power transmitted per inch width of double belt at 4,188 feet speed
4,188 | ||||
2 | × | = | 8.38. | |
1,000 |
Width of double belt for 50 horse power
50 ÷ 8.38 = 5.97, say 6 inch.
Ques. What are the advantages of chain drive?
Ans. The space required is much less than with belt drive, as the distance between centers may be reduced to a minimum. It is a positive drive, that is, there can be no slip. Less liability of becoming detached, and, because it is not dependent on frictional contact, the diameters of the sprockets may be much less than pulley diameter for belt drive.
Ques. What are some objections?
Ans. A lubricant is required for satisfactory operation, which causes more or less dirt to collect on the chain, requiring[Pg 1141] frequent cleaning; climbing of teeth when links and teeth become worn; noise and friction.
Direct Connected Alternators.—There are a large number of cases where economy of space is of prime importance, and to meet this condition the alternator and engine are direct connected, meaning, that there is no intermediate gearing such as belt, chain, etc., between engine and alternator.
One difficulty encountered in the direct connection of engine and alternator is the fact that the most desirable rotative speed of the engine is less than that of the alternator. Accordingly a compromise is made by raising the engine speed and lowering the alternator speed.
[Pg 1142]The insistent demand for direct connected units in the small and medium sizes, especially for direct current units, was the chief cause resulting in the rapid and high development of what is known as the "high speed automatic engine."
Increasing the engine speed means that more horse power is developed for any given cylinder dimensions, while reducing the speed of the generator involves that the machine must be larger for a given output, and in the case of an alternator more poles are required to obtain a given frequency, resulting in increased cost.
The compactness of the unit as a whole, simplicity, and general advantages are usually so great as to more than offset any additional cost of the generator.
Ques. What is the difference between a direct connected and a direct coupled unit?
Ans. A direct connected unit comprises an engine and generator permanently connected; direct coupling signifies that[Pg 1143] engine and generator are each complete in itself, that is, having two bearings, and are connected by some device such as friction clutch, jaw clutch, or shaft coupling.
Revolving Armature Alternators.—This type of alternator is one which has its parts arranged in a manner similar to a dynamo, that is, the armature is mounted on a shaft so it can revolve while the field magnets are attached to a circular frame and arranged radially around the armature, as shown in fig. 1,389. It may be single or polyphase, belt driven, or direct connected.
Ques. When is the revolving type of armature used and why?
Ans. It is used on machines of small size because the pressure generated is comparatively low and the current transmitted by the brushes small, no difficulty being experienced in collecting such a current.
Ques. Could a dynamo be converted into an alternator?
Ans. Yes.
Ques. How can this be done?
Ans. By placing two collector rings on one end of the armature and connecting these two rings to points in the armature winding 180° apart, as shown in fig. 1,390.
Ques. Would such arrangement as shown in fig. 1,390 make a desirable alternator?
Ans. No.
Alternating current windings are usually different from those used for direct currents. One distinction is the fact that a simple open coil winding may be, and often is, employed, but the chief difference is the intermittent action of the inductors.
In a direct current Gramme ring winding a certain number of coils are always active, while those in the space between the pole pieces are not generating. In this way a practically steady pressure is produced by a large fraction of the coils.
In the case of an alternator all of the coils are either active or inactive at one time. Hence, the winding need cover only as much of the armature as is covered by the pole pieces.
Revolving Field Alternators.—In generating an electric current by causing an inductor to cut magnetic lines, it makes no difference whether the cutting of the magnetic lines is effected by moving an inductor across a magnetic field or moving the magnetic field across the inductor.
Motion is purely a relative matter, that is, an object is said to move when it changes its position with some other object regarded as stationary; it may be moving with respect to a[Pg 1147] second object, and at the same time be at rest with respect to a third object. Thus, a dory has a speed of four miles per hour in still water; if it be run up stream against a current flowing four miles per hour it would move at that speed with respect to the water, yet remain at rest with respect to the earth.
It must be evident then that motion, as stated, being a purely relative matter, it makes no difference whether the armature of a generator move with respect to the field magnets, or the field magnets move with respect to the armature, so far as inducing an electric current is concerned.
For alternators of medium and large size there are several reasons why the armature should be stationary and the field magnets revolve, as follows:
1. By making the armature stationary, superior insulation methods may be employed, enabling the generation of current at very much higher voltage than in the revolving armature type.
2. Because the difficulty of taking current at very high pressures from collector rings is avoided.
The field current only passes through the collector rings. Since the field current is of low voltage and small in comparison with the main current, small brushes are sufficient and sparking troubles are avoided.
3. Only two collector rings are required.
4. The armature terminals, being stationary, may be enclosed permanently so that no one can come in contact with them.
Ques. What names are usually applied to the armature and field magnets with respect to which moves?
Ans. The "stator" and the "rotor."
The terms armature and field magnets are to be preferred to such expressions. An armature is an armature, no matter whether it move or be fixed, and the same applies to the field magnets. There is no good reason to apply other terms which do not define the parts.
Ques. Explain the essential features of a revolving field alternator.
Ans. The construction of such alternators is indicated in the diagram, fig. 1,394. Attached to the shaft is a field core, which carries the latter, consisting of field coils fitted on pole pieces which are dovetailed to the field core. The armature is built into the frame and surrounds the magnets as shown. The field current, which is transmitted to the magnets by slip rings and brushes, consists of direct current of comparatively low pressure, obtained from some external source.
Inductor Alternators.—In this class of alternator both armature and field magnets are stationary, a current being induced in the armature winding by the action of a so called inductor in moving through the magnetic field so as to periodically vary its intensity.
Ques. What influence have the inductors on the field flux?
Ans. They cause it to undulate; that is, the flux rises to a maximum and falls to a minimum value, but does not reverse.
Ques. How does this affect the design of the machine as compared with other types of alternator?
Ans. With a given maximum magnetic flux through each polar mass, the total number of armature turns necessary to produce a given pressure is twice that which is required in an alternator having an alternating flux through its armature windings.
Ques. Is the disadvantage due to the necessity of doubling the number of armature turns compensated in any way?
Ans. Yes, the magnetic flux is not reversed or entirely changed in each cycle through the whole mass of iron in the[Pg 1152] armature, the abrupt changes being largely confined to the projections on the armature surface between the coils.
Ques. What benefit results from this peculiarity?
Ans. It enables the use of a very high magnetic flux density in the armature without excessive core loss, and also the use of a large flux without an excessive increase in the amount of magnetic iron.
The use of a large flux permits a reduction in the number of armature turns, thus compensating, more or less, for the disadvantage due to the operation of only one-half of the armature coils at a time.
Classes of Inductor Alternator.—There are two classes into which inductor alternators may be divided, based on the mode of setting of their polar projections:
1. Homopolar machines;
2. Heteropolar machines.
Homopolar Inductor Alternators.—In this type the positive polar projections of the inductors are set opposite the negative polar projections as shown in fig. 1,402. When the polar projections are set in this manner, the armature coils must be "staggered" or set displaced along the circumference with respect to one another at a distance equal to half the distance from the positive pole to the next positive pole.
Heteropolar Inductor Alternators.—Machines of this class are those in which the polar projections are themselves staggered, as shown in fig. 1,403, and therefore, do not require the staggering of the armature coils. In this case, a single armature of double width may be used, and the rotating inductor then acts as a heteropolar magnet, or a magnet which presents alternatively positive and negative poles to the armature, instead of presenting a series of poles of the same polarity as in the case of a homopolar magnet.
Use of Inductor Alternators.—Morday originally designed and introduced inductor alternators in 1866. They are not the prevailing type, as their field of application is comparatively narrow. They have to be very carefully designed with regard to magnetic leakage in order[Pg 1154] to prevent them being relatively too heavy and costly for their output, and too defective with respect to their pressure regulation, other defects being heavy eddy current losses and inferior heat conductance.
Hunting or Singing in Alternators.—Hunting is a term applied to the state of two parallel connected alternators running out of step, or not synchronously, that is, "see sawing." When the current wave of an alternator is peaked and two machines are operated in parallel it is very difficult to keep them in step, that is in synchronism. Any difference in the phase relation which is set up by the alternation will cause a local or synchronizing current to flow between the two machines and at times it becomes so great that they must be disconnected.
Alternators which produce a smooth current wave and are maintained at uniform speed by properly designed governors,[Pg 1155] operate fairly well in parallel, but are not entirely free from hunting, and other means are provided to overcome the difficulty.
When heavy copper flanges, called dampers, are put over the polar projections or copper bars laid in grooves on the pole face and short circuited by connecting rings (called amortisseur winding), the powerful induced currents which are produced when the alternators get out of step tend to quickly re-establish the phase relation.
Two examples of a field provided with amortisseur[4] winding are shown in figs. 1,404 and 1,405.
[4] NOTE.—Amortisseur windings are often erroneously called "squirrel cage" windings on account of similarity of construction. The latter term should be reserved for its proper significance as being the name of the type of armature winding generally used for induction motors, the name being suggested by the resemblance of the finished armature to the wheel of a squirrel cage. A comparison of figs. 1,405 and 1,746 will show the distinction. In a squirrel cage winding there is a large number of bars uniformly spaced; an amortisseur winding consists of a comparatively small number of bars, usually unevenly spaced, that is they are divided into groups with considerable space between the groups, as in fig. 1,405, and less pronounced in fig. 1,404. The bars are short circuited by rings the same as in squirrel cage winding.
Monocyclic Alternators.—This type of alternator was designed prior to the introduction of the polyphase systems, to overcome the difficulties encountered in the operation of single phase alternators as motors. A single phase alternator will not start from rest as a motor, but must first be started and brought up to the proper speed before being connected with single phase mains. This condition constituted a serious difficulty in all cases where the motor had to be stopped and started at comparatively frequent intervals.
The monocyclic alternator is a single phase machine provided with an additional coil, called a teaser coil, wound in two phase relationship with, and connected to the center of the main single[Pg 1158] phase coil. It is provided with three collector rings; two for the single phase coil, and one for the free end of the teaser coil.
By this arrangement ordinary single phase incandescent lighting can be accomplished by means of a single pair of wires taken from the single phase coil. Where three phase motors have to be operated, however, a third wire, called the power wire, which is usually smaller than the main single phase wires is carried to the point at which the motor is located, and by the use of two suitably connected transformers three phase currents are obtained from the combined single phase and power wires for operating the motors.
Fig. 1,406 shows the connections of the monocyclic system and it is only necessary to carry the teaser wire into buildings where motors are to be used.
Armature Reaction.—Every conductor carrying a current creates a magnetic field around itself, whether it be embedded in iron or lie in air. Armature inductors, therefore, create magnetic fluxes around themselves, and these fluxes will, in part, interfere with the main flux from the poles of the field magnet. The effect of these fluxes is:
1. To distort the field, or
2. To weaken the field.
These disturbing fluxes form, in part, stray fluxes linked around the armature inductors tending to choke the armature current.
Ques. Explain how the field becomes distorted by armature reaction.
Ans. Considering a slotted armature and analyzing the electrical conditions as the inductors move past a pole piece, it will be observed: 1, when the coil is in the position shown in[Pg 1161] fig. 1,411, the current will be zero, assuming no armature self-induction, consequently for this position the armature coil has no disturbing effect upon the field set up by the field magnet; 2, when the inductors have moved under the pole face, as in fig. 1,412, currents will be induced in them, and they will tend to set up a magnetic field as indicated by the dotted lines, and in direction, by the arrow heads. The effect of this field will be to distort the main field, strengthening one side of the pole and weakening the other side.
Ques. Explain how the field becomes weakened by armature reaction.
Ans. In all armatures there is more or less inductance which causes the current to lag behind the pressure a corresponding amount. Accordingly, the current does not stop flowing at the same instant that the pressure becomes zero, therefore, when the coil is in the position of zero pressure, as in fig. 1,413,[Pg 1162] the current is still flowing and sets up a magnetic field which opposes the main field as indicated by the dotted arrows, thus weakening the main field.
Ques. In what kind of armature is this effect especially pronounced?
Ans. In slotted armatures provided with coils of a large number of turns.
Ques. What would be the effect if the current lead the pressure?
Ans. It would tend to strengthen the field as shown in fig. 1,414.
The value of the armature ampere turns which tend to distort and to diminish or augment the effect of the ampere turns on the field magnet is sometimes calculated as follows:
.707 × I × T × P | ||
A | = | |
s |
in which
This value of ampere turns, combined at the proper phase angle with the field ampere turns gives the value of the ampere turns available for producing useful flux.
Single Phase Reactions.—Unlike three phase currents, a single phase current in an alternator armature produces a periodic disturbance of the flux through the machine. In the magnet system this disturbance is of twice the normal frequency, while in the armature core it is the
same as the normal frequency. In both cases the eddy currents which are set up, produce a marked increase in the load losses, and thus tend to give the machine a higher temperature rise on single phase loading.
Designers continue to be singularly heedless of these single phase reactions, resulting in many cases of unsatisfactory single phase alternators. Single phase reactions distort the wave form of the machine.
Three Phase Reactions.—The action of the three phase currents in an alternator is to produce a resultant field which is practically uniform, and which revolves in synchronism with the field system. The resultant three phase reaction, because of its uniformity, produces no great increase in the load losses of the machine, the small additional losses which are present being due to windings not being placed actually in space at 120°, and to the local leakage in the teeth.
Magnetic Leakage.—In the design of alternators the drop of voltage on an inductive load is mainly dependent upon the magnetic leakages, primary and secondary. They increase with the load, and, what is of more importance, they increase with the fall of the power factor of the circuit on which they may be working. This is one reason why certain types of alternator,[Pg 1166] though satisfactory on a lighting circuit, have proved themselves unsatisfactory when applied to a load consisting chiefly of motors.
The designer must know the various causes which contribute to leakage and make proper allowance.
In general, to keep the leakage small, the pole cores should be short, and of minimum surface, the pole shoes should not have too wide a span nor be too thick, nor present needless corners, and the axial length of the pole face and of the armature core should not be too great in proportion to the diameter of the working face.
To keep the increase of leakage between no load and full load from undue magnitude, it is required that armature reactions[Pg 1167] shall be relatively small, that the peripheral density of the armature current (ampere-conductors per inch) be not too great, and that the pole cores be not too highly saturated when excited for no load.
The general character of the stray field between adjacent poles is shown in figs. 1,427 and 1,428 for straight poles and those having shoes.
Field Excitation of Alternators.—The fields of alternators require a separate source of direct current for their excitation, and this current should be preferably automatically controlled. In the case of alternators that are not self exciting, the dynamo which generates the field current is called the exciter.
The excitation of an alternator at its rated overload and .8 power factor would not, in some cases, if controlled by hand, exceed 125 volts, although, in order to make its armature voltage respond quickly to changes in the load and speed, the excitation of its fields may at times be momentarily varied by an automatic regulator between the limits of 70 and 140 volts.
The exciter should, in turn, respond at once to this demand upon its armature, and experience has shown that to do this its shunt fields must have sufficient margin at full load to deliver momentarily a range from 25 to 160 volts at its armature terminals.
It is obvious from the above that an exciter suitable for use with an automatic regulator must commutate successfully over a wide range in voltage, and, if properly designed, have liberal margins in its shunt fields and magnetic circuits.
Alternator fields designed for and operated at unity power factor have often proved unsatisfactory when the machines were called upon to deliver their rated kva. at .8 power factor or lower. This is due to the increased field current required at the latter condition and results, first, in the overheating of the fields and, second, in the necessity of raising the direct current exciting voltage above 125 volts, which often requires the purchase of new exciters.
Ques. What is a self-excited alternator?
Ans. One whose armature has, in addition to the main winding, another winding connected to a commutator for furnishing direct field exciting current, as shown in fig. 1,430.
Ques. How is a direct connected exciter arranged?
Ans. The exciter armature is mounted on the shaft of the alternator close to the spider hub, or in some cases at a distance sufficient to permit a pedestal and bearing to be mounted between the exciter and hub. In other designs the exciter is placed between the bearing and hub.
Figs. 1,432 and 1,433 are examples of direct connected exciter alternators, in fig. 1,432 the exciter being placed between the field hub and bearing, and in fig. 1,433, beyond the bearing.
Ques. What is the advantage of a direct connected exciter?
Ans. Economy of space.
This is apparent by comparing figs. 1,432 and 1,433 with fig. 1,434, which shows a belted exciter.
Ques. What is the disadvantage of a direct connected exciter?
Ans. It must run at the same speed as the alternator, which is slower than desirable, hence the exciter must be larger for a[Pg 1171] given output than the gear driven type, because the latter can be run at high speed and accordingly be made proportionally smaller.
Ques. What form of gear is generally used on gear driven exciters?
Ans. Belt gear.
Ques. What are the advantages of gear driven exciters?
Ans. Being geared to run at high speed, they are smaller and therefore less costly than direct connected exciters. In large[Pg 1172] plants containing a number of alternators one exciter may be used having sufficient capacity to excite all the alternators, and which can be located at any convenient place.
Ques. What is the disadvantage of gear driven exciters?
Ans. The space occupied by the gear.
In the case of a chain drive very little space is required, but for belts, the drive generally used, there must be considerable distance between centers for satisfactory transmission.
Slow Speed Alternators.—By slow speed is here understood relatively slow speed, such as the usual speeds of reciprocating engines. A slow speed alternator is one designed to run at a speed slow enough that it may be direct connected to an engine.[Pg 1173] Such alternators are of the revolving field type and a little consideration will show that they must have a multiplicity of field magnets to attain the required frequency.
In order that there be room for the magnets, the machine evidently must be of large size, especially for high frequency.
EXAMPLE.—How many field magnets are required on a two phase alternator direct connected to an engine running 240 revolutions per minute, for a frequency of 60?
An engine running 240 revolutions per minute will turn
240 ÷ 60 = 4 revolutions per second.
A frequency of 60 requires
60 ÷ 4 = 15 cycles per phase per revolution, or
15 × 2 = 30 poles per phase.[Pg 1174] Hence for a two phase alternator the total number of poles required is
30 × 2 = 60.
It is thus seen that a considerable length of spider rim is required to attach the numerous poles, the exact size depending upon their dimensions and clearance.
Fly Wheel Alternators.—The diameter of the revolving fields on direct connected alternators of very large sizes becomes so great that considerable fly wheel effect is obtained, although the revolutions be low. By giving liberal thickness to the rim[Pg 1175] of the spider, the rotor then answers the purpose of a fly wheel, hence no separate fly wheel is required. In fact, the revolving element resembles very closely an ordinary fly wheel with magnets mounted on its rim, as illustrated in fig, 1,437.
High Speed Alternators.—Since alternators may be run at speeds far in excess of desirable engine speeds, it must be evident that both size and cost may be reduced by designing them for high speed operation.
Since the desired velocity ratio or multiplication of speed is so easily obtained by belt drive, that form of transmission is generally used for high speed alternators, the chief objection being the space required. Accordingly where economy of space is not of prime importance, a high speed alternator is usually installed, except in the large sizes where the conditions naturally suggest a direct connected unit.
An example of high speed alternator is shown in fig. 1,438. Machines of this class run at speeds of 1,200 to 1,800 or more, according to size.
No one would think of connecting an alternator running at any such speed direct to an engine, the necessary speed reduction proper for engine operation being easily obtained by means of a belt drive.
Water Wheel Alternators.—In order to meet most successfully the requirements of the modern hydro-electric plant, the alternators must combine those characteristics which result in high electrical efficiency with a mechanical strength of the moving elements which will insure uninterrupted service, and an ample factor of safety when operating at the relatively high speeds often used with this class of machine.
When selecting an alternator for water wheel operation a careful analysis of the details of construction should be made in[Pg 1178] order to determine the relative values which have been assigned by the designers to the properties of the various materials used. Such analysis will permit the selection of a type of machine best adapted to the intended service and which possesses the required characteristics of safety, durability and efficiency.
The large use of electric power transmitted by means of high pressure alternating current has led to the development of a large number of water powers and created a corresponding demand for alternators suitable for direct connection to water wheels.
Ques. Name two forms of water wheel alternator.
Ans. Horizontal and vertical.
Examples of horizontal and vertical forms of water wheel alternator are shown in figs. 1,439 and 1,440.
Ques. How should the rotor be designed?
Ans. It should be of very substantial construction.
Ques. Why?
Ans. Because water wheel alternators are frequently required to operate safely at speeds considerably in excess of normal.
Ques. What special provision is made for cooling the bearings?
Ans. They are in some cases water cooled.
Turbine Driven Alternators.—Although the principle of operation of the steam turbine and that of the reciprocating engine are decidedly unlike, the principle of operation of the high speed turbine driven alternator does not differ from that of generators designed for being driven by other types of engine or by water wheels. There are, therefore, with the turbine driven alternator no new ideas for[Pg 1181] the operator who is familiar with the older forms to acquire.
It must be obvious that the proportions of such extra high speed machines must be very different from those permissible in generators of much slower speeds.
Ques. How does a turbine rotor differ from the ordinary construction?
Ans. It is made very small in diameter and unusually long.
Ques. Why?
Ans. To reduce vibration and centrifugal stresses.
Ques. What are the two classes of turbine driven alternators?
Ans. They are classed as vertical or horizontal.
Ques. How do they compare?
Ans. The vertical type requires less floor space than the horizontal design, and while a step bearing is necessary to carry the weight of the moving element, there is very little friction in the main bearings.
The horizontal machine, while it occupies more space, does not require a step bearing.
Ques. Describe a step bearing.
Ans. It consists of two cylindrical cast iron plates bearing upon each other and having a central recess between them into which lubricating oil is forced under considerable pressure by a steam or electrically driven pump, the oil passing up from beneath.
Ques. What auxiliary is generally used in connection with a step bearing?
Ans. A weighted accumulator is sometimes installed in
[Pg 1183] connection with the oil pipe as a convenient device for governing the step bearing pumps, and also as a safety device in case the pumps fail.
Alternators of Exceptional Character.—There are a few types of alternator less frequently encountered than those already described. The essentials of such machines are here briefly given.
Asynchronous Alternators.—In these machines, the rotating magnet, which, with definite poles, is replaced by a rotor having closed circuits. In general construction, they are similar to asynchronous induction motors having short circuited rotors; for these alternators, when operating as motors, run at a speed slightly below synchronism and act as generators when the speed is increased above that of synchronism. Machines of this class are not self-exciting, but require an alternating or polyphase current previously supplied to the mains to which the stationary armature is connected.
Asynchronous alternators may be advantageously used in central stations that may be required to sustain a very sudden increase of load. In such cases, one or more asynchronous machines might be kept in operation as a non-loaded motor at a speed just below synchronism until its output as a generator is required; when by merely increasing the speed of the engine it will be made to act as a generator, thus avoiding the delays usually occurring before switching in a new alternator.
Image Current Alternators.—When the generated frequency of alternators excited by low frequency currents is either the sum or the difference of the excitation and rotation frequencies, any load current flowing through the armature of the machine is exactly reproduced in its field circuit. These reproduced currents are characteristic of all types of asynchronous machines, and are called "image currents," as they are actually the reflection from the load currents delivered by the armature circuit.
As the exciter of a machine of this type carries "image currents" proportional to the generated currents, its size must be proportional to the capacity of the machine multiplied by the ratio of the excitation and generated frequencies; therefore, in the commercial machines, the excitation frequency is reduced to the minimum value possible; from two to five cycles per second being suitable for convenient employment.
These machines as heretofore constructed are not self-exciting, but as the principle of image current enables the construction of self-exciting alternators, it will be of advantage to have a general understanding of the separately excited machine under different conditions of excitation.
When the generated frequency of the machine is equal to the difference of the excitation and rotation frequencies, the magnetization of the machine is higher under a non-inductive load than under no load. This is principally due to the ohmic resistance of the field circuit, which prevents the image current from entirely neutralizing the magnetomotive force of the armature current. In other words, the result of the magnetomotive force of the armature and image currents not only tends to increase the no load magnetization of the machine at non-inductive load, but depresses the original magnetization at inductive load, so that the terminal voltage of the machine increases with non-inductive load, and decreases with inductive load.
Again, the generated frequency is equal to the sum of the excitation and rotation frequencies, the resistance of the field circuit reacts positively; that is, it tends to decrease the magnetization, and consequently the terminal voltage of the machine at both inductive and non-inductive loads.
In the constant pressure machine, the two effects are combined and opposed to one another.
The connections of two alternators with diphase excitation are shown by fig. 1,446.
Extra High Frequency Alternators.—Alternators generating currents having a frequency up to 10,000 or 15,000 cycles per second have been proposed several times for special purposes, such as high frequency experiments, etc. In 1902 Nikola Tesla proposed some forms of alternators having a large number of small poles, which would generate currents up to a frequency of 15,000 cycles per second.
Later, the Westinghouse Company constructed an experimental machine of the inductor alternator type for generating currents having a frequency of 10,000 cycles per second. This machine was designed by Samms. It had 200 polar projections with a pole pitch of only 0.25 inch, and a peripheral speed of 25,000 feet per minute. The armature core was built up of steel ribbon 2 inches wide and 3 mils thick. The armature had 400 slots with one wire per slot, and a bore of about 25 inches. The air gap was only 0.03125 inch. On constant excitation the voltage dropped from 150 volts at no-load to 123 volts with an output of 8 amperes.
Self-Exciting Image Current Alternators.—The type of machine described in the preceding paragraph can be made self-exciting by connecting each pair of brushes, which collect the current from the armature, with a field coil so located that the flux it produces will be displaced by a predetermined angle depending on the number of phases required, as shown by fig. 1,447. The direction of the residual magnetism of the machine is shown by the arrows A, A.
When the armature is rotated, a pressure will be generated between the brushes 2 and 4, and a current will flow from C through the coils XX to B, producing a flux through the armature at right angles to the residual magnetism and establishing a resultant magnetic field between D, B, and D, C. This field will generate a pressure between the brushes 1 and 3, and a current will flow D through XX to E in such a direction that it will at first be opposed to the residual magnetism, and afterward reverse the direction of the latter. At the moment the residual magnetism becomes zero, the only magnetism left in the machine will be due to the currents from the brushes 2 and 4, and their field combining with the vertical reversed field will produce a resultant polar line between B and E. As these operations are cyclic, they will recur at periodic intervals, and the phenomena will become continuous. The negative field thus set up in the air gap of the machine will cut the conductors of the stator and will be cut by the conductors of the rotor in such a manner that the electromotive forces generated between the brushes of the armature will be equal and opposite to those between the terminals of the stator.
The construction of alternators follows much the same lines as dynamos, especially in the case of machines of the revolving armature type. Usually, however, more poles are provided than on direct current machines, in order to obtain the required frequency without being driven at excessive speed.
The essential parts of an alternator are:
and in actual construction, in order that these necessary parts may be retained in proper co-relation, and the machine operate properly there must also be included:
Field Magnets.—The early forms of alternator were built with permanently magnetized steel magnets, but these were later discarded for electromagnets.
Alternators are built with three kinds of electromagnets, classed according to the manner in which they are excited, the machines being known as,
Ques. What is a self-excited alternator?
Ans. One in which the field magnets are excited by current from one or more of the armature coils, or from a separate winding (small in comparison with the main winding), the current being transformed into direct current by passing it through a commutator.
Fig. 1,453 shows an armature of a self-excited machine, the exciting current being generated in a separate winding and passed through a commutator.
Ques. For what class of service are self-exciting alternators used?
Ans. They are employed in small power plants and isolated lighting plants where inductive loads are encountered.
Ques. What is a separately excited alternator?
Ans. One in which the field magnets are excited from a small dynamo independently driven or driven by the alternator shaft, either direct connected or by belt as shown in fig. 1,455.
Ques. What is a compositely excited alternator?
Ans. A composite alternator is similar to a compound wound dynamo in that it has two field windings. In addition to the regular field coils which carry the main magnetizing current from the exciter, there is a second winding upon two or upon all of the[Pg 1192] pole pieces, carrying a rectified current from the alternator which strengthens the field to balance the losses in the machine, and also if so desired, the losses on the line as shown in fig. 1,456.
Ques. What is a magneto?
Ans. A special form of alternator having permanent magnets for its field, and used chiefly to furnish current for gas engine ignition and for telephone call bells.
Details of construction and operation are shown in figs. 1,458 to 1,461.
Ques. What are the two principal types of field magnet?
Ans. Stationary and revolving.
Ques. What is the usual construction of stationary field magnets?
Ans. Laminated pole pieces are used, each pole being made up of a number of steel stampings riveted together and bolted or preferably cast into the frame of the machine. The field coils are machine wound and carefully insulated. After winding they are taped to protect them from mechanical injury. Each coil is then dipped in an insulating compound and afterwards baked to render it impervious to moisture.
Ques. Describe the construction of a revolving field.
Ans. The entire structure or rotor consists of a shaft, hub or spider, field magnets and slip rings. The magnet poles consist of laminated iron stampings clamped in place by means of through bolts which, acting through the agency of steel end plates, force the laminated stampings into a uniform, rigid mass. This mass is magnetically subdivided into so many small parts that the heating effect of eddy currents is reduced to a minimum. The cores are mounted upon a hub or spider either by dovetail construction or by means of through bolts, according to the centrifugal force which they must withstand in operation, either method permitting the easy removal of any particular[Pg 1198] field pole if necessary. The field coils are secured upon the pole pieces either by horns in one piece with the laminations, or separate and bolted. All the coils are connected in series, cable leads connecting them to slip rings placed on the shaft.
Ques. What are slip rings?
Ans. Insulated rings mounted upon the alternator shaft to receive direct current for the revolving field, as distinguished from collector rings which collect the alternating currents generated in an alternator of the revolving armature type.
In construction provision is made for attaching the field winding leads. The rings are usually made of cast iron and are supported[Pg 1199] mechanically upon the shaft, but are insulated from it and from one another.
The current is introduced by means of brushes as with a commutator. Carbon brushes are generally used.
A good design of slip ring should provide for air circulation underneath and between the rings.
Ques. What form of spider is used on large alternators?
Ans. It is practically the same form as a fly wheel, consisting of hub, spokes, and rim to which the magnets are bolted.
[Pg 1201] On alternators of the fly wheel type the spider rim is made of sufficient weight to obtain full fly wheel effect, thus making a separate fly wheel unnecessary.
Armatures.—In construction, armatures for alternators are similar to those employed on dynamos; they are in most cases simpler than direct current armatures due to the smaller number of coils, absence of commutator with its multi-connections, etc. Alternator armatures may be classified in several ways:
1. With respect to operation, as
2. With respect to the core, as
Ring and disc armatures are practically obsolete and need not be further considered. A ring armature has the inherent defect that the copper inside the ring is inactive.
Disc armatures were employed by Pacinotti in 1878, and afterwards adopted by Brush in his arc lighting dynamos.
The design failed for mechanical reasons, but electrically it is, in a sense, an improvement upon the Gramme ring, in that inductors on both sides of the ring are active, these being connected together by circumferential connectors from pole to pole, thus, corresponding to the end connectors on modern drum armatures.
3. With respect to the core surface, as
In early dynamos the armature windings were placed upon an iron core with a smooth surface. A chief disadvantage of this arrangement is that the magnetic drag comes upon the inductors and tends to displace them around the armature. To prevent
this, projecting metal pieces called driving horns were fixed into the core so as to take the pressure, but they proved unsatisfactory. This defect together with the long air gap necessary in smooth core construction resulted in the type being displaced by slotted core armatures.
A slotted core is one whose surface is provided with slots or teeth which carry the inductors, as shown in the accompanying illustrations, and is the type almost universally used. The inductors are laid in the slots, the sides and bottoms of which are first carefully insulated by troughs of mica-canvas, micanite or other suitable insulating material.
Ques. What are the advantages of slotted core armatures?
Ans. The teeth protect the inductors, retain them in place against the electrical drag and centrifugal force, and the construction permits a reduction of air gap to a minimum, thus reducing the amount of copper required for the field.
Armature Windings.—In general, the schemes for armature windings for alternators are simpler than those for direct current machines, as in the majority of cases the inductors are an even multiple of the number of poles, and the groupings are usually symmetrical with respect to each pole or each pair of poles. Furthermore, as a general rule, all the inductors of any one phase are in series with one another; therefore, there is only one circuit per phase, and this is as it should be, since alternators are usually required to generate high voltages. These general principles establish the rule, that in the circuit in a single phase armature, and in the individual circuits in a polyphase armature, the winding is never re-entrant, but the circuits have definite endings and beginnings. In exceptional cases, as those of polyphase converters, re-entrant circuits are employed, and the armature windings are so constructed that a commutator can be[Pg 1208] connected to them exactly as in direct current machines. These armatures are usually of the lap wound drum type.
Alternator windings are usually described in terms of the number of slots per phase per pole. For instance, if the armature of a 20 pole three phase machine have 300 slots, it has 15 slots per pole or 5 slots per each phase per pole, and will be described as a five slot winding. Therefore, in order to trace the connections of a winding, it is necessary to consider the number of slots per pole for any one phase on one of the following assumptions:[Pg 1209] 1, that each slot holds one inductor; 2, that there is one side of a coil in each slot; and 3, that one side of a coil is subdivided so as to permit of its distribution in two or more adjacent slots.
The voltage depends upon the number of inductors in a slot, but the breadth coefficient and wave form are influenced by the number of slots per pole, and not by the number of inductors within the slots.
Classification of Windings.—The fact that alternators are built in so many different types, gives rise to numerous kinds of armature winding to meet the varied conditions of operation. In dividing these forms of winding into distinctive groups, they may be classified, according to several points of view, as follows:
1. With respect to the form of the armature, as:
2. With respect to the mode of progression, as:
3. With respect to the relation between number of poles and number of coils, as:
4. With respect to the number of slots, as:
5. With respect to the form of the inductors, as:
6. With respect to the number of coils per phase per pole, as:
7. With respect to the kind of current delivered, as:
8. With respect to the shape of the coil ends, as:
In addition to these several classes of winding, there are a number of miscellaneous windings of which the following might be mentioned:
Ques. Define a revolving and a stationary winding.
Ans. The words are self-defining; a winding is said to be revolving or stationary according as the armature forms the rotor or stator of the machine.
Ques. What is the significance of the terms lap and wave as applied to alternator windings?
Ans. They have the same meaning as they do when applied to dynamo windings.
These are described in detail in Chapter XVIII. Briefly a lap winding is one composed of lap coils; a wave winding is one which roughly resembles in its diagram, a section of waves.
Half Coil and Whole Coil Windings.—The distinction as to whether the adjacent sides of consecutive coils are placed together under one pole or whether they are separated a distance equal to the pole pitch, gives rise to what is known as half coil and whole coil windings.
A half coil or hemitropic winding is one in which the coils in any phase are situated opposite every other pole, that is, a winding in which there is only one coil per phase per pair of poles, as in fig. 1,488.
A whole coil winding is one in which there is one coil per phase per pole, as in fig. 1,489, the whole (every one) of the poles being subtended by coils.
Concentrated or Uni-Coil Winding.—Fig. 1,492 shows the simplest type of single phase winding. It is a one slot winding and is sometimes called "monotooth" or "uni-coil" winding. The surface of the armature is considered as divided into a series of large teeth, one tooth to each pole, and each tooth is wound with one coil, of one or more turns per pole. Since[Pg 1215] all the turns of the coil are placed in single slots, the winding is called "concentrated."
Ques. What are the features of concentrated windings?
Ans. Cheap construction, maximum voltage for a given number of inductors. Concentrated windings have greater armature reaction and inductance than other types hence the terminal voltage of an alternator with concentrated winding falls off more than with distributed winding when the current output is increased. An alternator, therefore, does not have as good regulation with concentrated winding as with distributed winding.
Ques. What should be noted with respect to concentrated windings?
Ans. A concentrated winding, though giving higher voltage[Pg 1217] than the distributed type with no load, may give a lower voltage than the latter at full load.
Ques. What is the wave form with a concentrated winding?
Ans. The pressure curve rises suddenly in value as the armature slots pass under the pole pieces, and falls suddenly as the armature slots recede from under the pole pieces.
Distributed or Multi-Coil Windings.—Instead of winding an armature so it will occupy only one slot per phase per pole, it may be spread out so as to fill several slots per phase per pole. This arrangement is called a distributed winding.
To illustrate, fig. 1,496 represents a coil of say fifteen turns. This could be placed on an armature just as it is, in which case only one slot would be required for each side, that is, two in all. In place of this thick coil, the wire could be divided into several coils of a lesser number of turns each, arranged as in fig. 1,497; it is then said to be partially distributed, or it could be arranged as in fig. 1,498, when it is said to be fully distributed.
A partially distributed winding, then, is one, as in fig. 1,499, in which the coil slots do not occupy all the circumference of the armature; that is, the core teeth are not continuous.
A fully distributed winding is one in which the entire surface of the core is taken up with slots, as in fig. 1,500.
Ques. In a distributed coil what is understood by the breadth of the coil?
Ans. The distance between the two outer sides, as B in figs. 1,497 and 1,498.
Ques. How far is it advisable to spread distributed coils of a single phase alternator?
Ans. There is not much advantage in reducing the interior[Pg 1220] breadth much below that of the breadth of the pole faces, nor is there much advantage in making the exterior breadth greater than the pole pitch.
Undue spreading of distributed coils lowers the value of the Kapp coefficient (later explained) by reducing the breadth coefficient and makes necessary a larger number of inductors to obtain the same voltage.
The increase in the number of inductors causes more armature self-induction. From this point of view, it would be preferable to concentrate the winding in fewer slots that were closer together. This, however, would accentuate the distorting and demagnetizing reactions of the armature. Accordingly, between these two disadvantages a compromise is made, as to the extent of distributing the coils and spacing of the teeth, the proportions assigned being those which experience shows best suited to the conditions of operation for which the machine is designed.
The Kapp Coefficient.—A volt or unit of electric pressure is defined as the pressure induced by the cutting of 100,000,000 or 108 lines of force per second. In the operation of an alternator the maximum pressure generated may be expressed by the following equation:
πfZN | |||
Emax | = | (1) | |
108 |
in which
The maximum value of the pressure, as expressed in equation (1), occurs when θ = 90°.
The virtual value of the volts is equal to the maximum value divided by √2, or multiplied by ½ √2, hence,
½ √2 × πfZN | 2.22fZN | ||||
Evirt | = | = | (2) | ||
108 | 108 |
This is usually taken as the fundamental equation in designing alternators. It is, however, deduced on the assumptions that[Pg 1222] the distribution of the magnetic flux follows a sine law, and that the whole of the loops of active inductors in the armature circuit acts simultaneously, that is to say, the winding is concentrated.
In practice, the coils are often more or less distributed, that is, they do not always subtend an exact pole pitch; moreover, the flux distribution, which depends on the shaping and breadth of the poles, is often quite different from a sine distribution. Hence, the coefficient 2.22 in equation (2) is often departed from, and in the general case equation (2) may be written
kfZN | |||
---|---|---|---|
Evirt | = | (3) | |
108 |
where k is a number which may have different values, according to the construction of the alternator. This number k is called the Kapp coefficient because its significance was first pointed out by Prof. Gisbert Kapp.
The value of k is further influenced by a "breadth coefficient" or "winding factor."
The effect of breadth in distributed windings is illustrated in figs. 1,506 to 1,508.
Wire, Strap, and Bar Windings.—In the construction of alternators, the windings may be of either wire, strap, or bar, according to which is best suited for the conditions to be met.
Ques. What conditions principally govern the type of inductor?
Ans. It depends chiefly upon the current to be carried and the space in which the inductor is to be placed.
Ques. What kind of inductors are used on machines intended for high voltage and moderate current?
Ans. The winding is composed of what is called magnet wire, with double or triple cotton insulation.
Ques. Where considerable cross section is required how is a wire inductor arranged?
Ans. In order that the coil may be flexible several small wires in multiple are used instead of a single large wire.
Ques. How is the insulation arranged on inductors of this kind?
Ans. Bare wire is used for the wires in parallel, insulation being wrapped around them as in fig. 1,510.
This construction reduces the space occupied by the wires, and the insulation serves to hold them in place.
Ques. What precaution is taken in insulating a wire wound coil containing a large number of turns?
Ans. On account of the considerable difference of pressure between layers, it is necessary to insulate each layer of turns as well as the outside of the coil, as shown in fig. 1,513.
Ques. Do distributed coils require insulation between the separate layers?
Ans. Since they are subdivided into several coils insulation between layers is usually not necessary.
Ques. How is a coil covered?
Ans. It is wound with a more or less heavy wrapping of tape depending upon the voltage.
Linen tape of good quality, treated with linseed oil, forms a desirable covering. Where extra high insulation is required the tape may be interleaved with sheet mica.
Ques. Is the insulation placed around the coils all that is necessary?
Ans. The slots into which the coils are placed, are also insulated.
Ques. How are bar windings sometimes arranged?
Ans. In two layers, as in fig. 1,523.
Single and Multi-Slot Windings.—These classifications correspond to concentrated and distributed windings, previously described. In usual modern practice, only two-thirds of the total number of slots (assuming the spacing to be uniform)
[Pg 1231] of a single phase armature are wound with coils. The reason for this may be explained by aid of fig. 1,524, which shows an armature with six slots per pole, four of which are wound. Owing to the different positions of, say, coils A and B, there will be a difference in phase between the pressure generated in them and consequently the resultant pressure of the two coils joined in series will be less than the sum of the pressure in each coil.
Fig. 1,525 shows the pressure plotted out as vector quantities, and the table which follows gives the relative effectiveness of windings with various numbers of slots wound in series.
The figures in the last column of the table show that a large increase in the weight of active material is required if the[Pg 1232] inductors in a single phase machine are to be distributed over more than two-thirds the pole pitch. Again, if much less than two-thirds of the surface be wound, it is more difficult to provide a sine wave of pressure.
Slots wound in series | Pressure across coils | Winding coefficient | Quantity of copper to produce same pressure |
---|---|---|---|
1 | 1 | 1 | 1 |
2 | 1.93 | .97 | 1.03 |
3 | 2.73 | .91 | 1.10 |
4 | 3.34 | .84 | 1.19 |
5 | 3.72 | .74 | 1.35 |
6 | 3.86 | .64 | 1.56 |
Ques. What other advantage besides obtaining a sine wave is secured by distributing a coil?
Ans. There is less heating because of the better ventilation.
Single Phase Windings.—There are various kinds of single phase winding, such as, concentrated, distributed, hemitropic, etc. Fig. 1,527 shows the simple type of single phase winding. It is a "one slot" winding, that is, concentrated coils are used.
The armature has the same number of teeth as there are poles, the concentrated coils being arranged as shown. In designing[Pg 1234] such a winding, the machine, for example, may be required to generate, say, 3,000 volts, frequency 45, revolutions 900 per minute.
These conditions require 720 inductors in series in the armature circuit, and as the armature is divided into six slots corresponding to the six poles, there will be 120 inductors per slot, and the coil surrounding each of the six teeth on the surface of the armature will consist of 60 turns. The connections must be such as to give alternate clockwise and counter-clockwise winding proceeding around the armature.
Ques. In what other way could the inductors be arranged in concentrated coils?
Ans. They could be grouped in three coils of 120 turns each, as shown in fig. 1,528.
When thus grouped the arrangement is called a hemitropic winding, as previously explained.
Ques. What is the advantage, if any, of a half coil winding?
Ans. In single phase machines a half coil winding is equivalent, electrically, to a monotooth winding, and, therefore, is not of any particular advantage; but in three phase machines, it has a decided advantage, as in such, a concentrated winding yields a higher pressure than a distributed winding.
Two Phase Armature Windings.—This type of winding can be made from any single phase winding by providing another set of slots displaced along the surface of the armature to the extent of one-half the pole pitch, placing therein a duplicate winding.
For instance: If the six pole monotooth, single phase winding, shown in fig. 1,527, be thus duplicated, the result will be the one slot two-phase winding shown in fig. 1,529, which will have twelve slots, and will require four slip rings, or two rings for each phase.
By connecting up the two windings in series, the machines could be used as a single phase, with an increase of voltage in the ratio of 1.41 to 1.
Ques. How must the coils be constructed for two phase windings?
Ans. They must be made of two different shapes, one bent up out of the way of the other, as in fig. 1,534.
There are numerous kinds of two phase windings; the coils may be concentrated or distributed, half coil or whole coil, etc. Fig. 1,530 shows a two phase winding with four slots per pole, and fig. 1,533 one with six slots per pole.
Three Phase Armature Windings.—On the same general principle applicable to two phase windings, a three phase winding can be made from any single phase winding, by placing three identical single phase windings spaced out successively along the[Pg 1240] surface of the armature at intervals equal to one-third and two-thirds, respectively, of the double pole pitch, the unit in terms of which the spacing is expressed, being that pitch, which corresponds to one whole period.
Each of the three individual windings must be concentrated into narrow belts so as to leave sufficient space for the other windings between them. This limits the breadth or space occupied by the winding of any one phase to one-third of the pole pitch.
Ques. How are three phase coil ends treated?
Ans. They may be arranged in two ranges, as in fig. 1,538, or in three ranges, as in fig. 1,539.
Ques. What kind of coil must be used for three phase windings in order that the ends may be arranged in only two ranges?
Ans. Hemitropic or half coils; that is, the number of coil per phase must be equal to one-half the number of pole.
Grouping of Phases.—In the preceding diagrams, the general arrangement of the coils on the armature surface are shown for the numerous classes of winding. In polyphase alternators the separate windings of the various phases may be grouped in two ways:
1. Star connection;
2. Mesh connection.
Ques. Describe the two phase star connection.
Ans. In this method of grouping, the middle points of each of the two phases are united to a common junction M, and the[Pg 1244] four ends are brought out to four terminals a, a', b, b', as shown in fig. 1,544, or in the case of revolving armatures, to four slip rings.
Ques. What does this arrangement give?
Ans. It is practically equivalent to a four phase system.
Ques. How is the two phase mesh connection arranged?
Ans. In this style of grouping, the two phases are divided[Pg 1245] into two parts, and the four parts are connected up in cyclic order, the end of one to the beginning of the next, so as to form a square, the four corners of which are connected to the four terminals a, b, a', b', as shown in fig. 1,545, or in the case of revolving armatures, to four slip rings.
Ques. Describe a three phase star connection?
Ans. In three phase star grouping, one end of each of the three circuits is brought to a common junction M, usually insulated, and the three other ends are connected to three terminals a, b, c, as shown in fig. 1,546, or in the case of revolving armatures to three slip rings.
Ques. What other name is given to this connection, and why?
Ans. It is commonly called a Y connection or grouping owing to the resemblance of its diagrammatic representation to the letter Y.
Ques. How is a three phase mesh connection arranged?
Ans. The three circuits are connected up together in the form of a triangle, the three corners are connected to the three terminals, a, b, c, as shown in fig. 1,549, or in the case of revolving armatures to three slip rings.
Ques. What other name is given to this style of connection, and why?
Ans. It is commonly called a delta grouping on account of the resemblance of its diagrammatic representation to the Greek letter Δ.
In polyphase working, it is evident that by the use of four equal independent windings on the armature, connected to eight terminals or slip rings, a two phase alternator can be built to supply currents of equal voltage to four independent circuits. Likewise, by the use of three equal independent windings, connected to six terminals or slip rings, a three phase alternator can be made to supply three independent circuits.
This is not the usual method employed in either case, however, as the star grouping or mesh grouping methods of connection not only gives the same results, but also, in star grouping, a greater plurality of voltages for the same machine, and a higher voltage between its main terminals.
Radial diagrams of the arrangement and connections of Y grouping of lap windings and wave windings for three phase alternators are shown by figs. 1,551 and 1,552.
Ques. In three phase star grouping, what is the point where the phases join, called?
Ans. The star point.
Ques. In a three phase star connected alternator what is the voltage between any two collector rings?
Ans. It is equal to the voltage generated per phase multiplied by √3 or 1.732.
Ques. In a three phase star connected alternator what is the value of the current in each line?
Ans. The same as the current in each phase winding.
Ques. What is the value of the total output in watts of a star connected alternator?
Ans. It is equal to the sum of the outputs of each of the three phases. When working on a non-inductive load, the total output of a star connected alternator is equal to √3 multiplied by the product of the line current and line voltage.
Ques. What is the value of the line voltage in a three phase delta connected alternator?
Ans. It is equal to the voltage generated in each phase.
Ques. What is the value of the line current in a three phase delta connected alternator?
Ans. It is equal to the current in each phase multiplied by √3.
Ques. What is the total output of a three phase delta connected alternator working on a non-inductive load?
Ans. The total watts is equal to √3 multiplied by the product of the line current and the line voltage.
Ques. What are the features of the star connection?
Ans. It gives a higher line voltage than the delta connection for the same pressure generated per phase, hence it is suited for machines of high voltage and moderate current.
The delta connection gives a lower line voltage than the star[5] connection for the pressure generated per phase, and cuts down the current in the inductors; since the inductors, on this account, may be reduced in size, the delta connection is adapted to machines of large current output.
[5] NOTE.—In the star connected armature the proper ends to connect to the common terminal or star point are determined as follows: Assume that the inductor opposite the middle of a pole is carrying the maximum current, and mark its direction by an arrow. Then the current in the inductors on either side of and adjacent to it will be in the same direction. As the maximum current must be coming from the common terminal, the end toward which the arrow points must be connected to one of the rings, while the other end is connected to the common terminal. The current in the two adjacent inductors evidently must be flowing into the common terminal, hence the ends toward which the arrows point must be connected to the common terminal, while their other ends are connected to the remaining two rings.
Ques. How is the path and value of currents in a delta connected armature determined?
Ans. Starting with the inductors of one phase opposite the middle of the poles, assume the maximum current to be induced[Pg 1254] at this moment; then but one-half of the same value of current will be induced at the same moment in the other two phases, and its path and value will best be shown by aid of fig. 1,560, in which X may be taken as the middle collector ring, and the maximum current to be flowing from X toward Z. It will be seen that no current is coming in through the line Y, but part of the current at Z will have been induced in the branches b and c.
Ques. Since most three phase windings can be connected either Y or delta, what should be noted as to the effects produced?
Ans. With the same winding, the delta connection will stand 1.732 as much current as the Y connection, but will give only 1 ÷ 1.732 or .577 as much voltage.
Chain or Basket Winding.—One disadvantage in ordinary two-range windings is that two or three separate shapes of coil are required. The cost of making, winding, and supplying spares would be less if one shape of coil could be made to do for all phases. One way of accomplishing this is by the method of chain winding, in which the two sides of each coil are made of different lengths, as shown in fig. 1,563, and bent so that they can lie behind one another.
In the case of open slots the coils may be former wound and afterwards wedged into their places.
In chain winding the adjacent coils link one another as in a chain (hence, the name); the winding is similar to a skew coil[Pg 1257] winding. This plan of winding is supposed to have some advantage in keeping coils of different phases further separated than the two range plan.
Skew Coil Winding.—In this type of winding the object is to shape the coils so that all may be of one pattern. This is accomplished by making the ends skew shape as shown in figs. 1,566 to 1,568.
Fed-in Winding.—This name is given to a type of winding possible with open or only partially closed slots, in which coils previously formed are introduced, only a few inductors at a time if necessary. They are inserted into the slots from the top, the slot being provided with a lining of horn fibre or other suitable material, which is finally closed over and secured in place by means of a wedge, or by some other suitable means. An example of a fed-in winding is shown in figs. 1,566 and 1,568.
Imbricated Winding.—This is a species of spiral coil winding in which the end connections are built up one above the other, either in a radial, or in a horizontal direction.
The winding is used especially on the armatures of turbine alternators and dynamos.
Spiral Winding.—This is a winding in which "spiral" coils, as shown in fig. 1,569, are used. The spiral form of coil is very extensively used for armature windings of alternators.
Mummified Winding.—The word mummified as applied to a winding is used to express the treatment the coils of the winding receive in the making; that is, when a winding, after being covered with tape or other absorbent material, is saturated in an insulating compound and baked until the whole is solidified, it is said to be mummified.
Shuttle Winding.—This type of winding consists of a single coil having a large number of turns, wound in two slots spaced 180° apart. It was originally used on Siemens' armature and is now used on magnetos, as shown in figs. 1,459 to 1,461.
Creeping Winding.—Another species of winding, known as a creeping winding is applicable to particular cases.
If three adjacent coils, each having a pitch of 120 electrical degrees, be set side by side, they will occupy the same breadth as 4 poles, and, by repetition, will serve for any machine having a multiple of 4 poles, but cannot be used for machines with 6, 10 or 14 poles. Fig. 1,571 shows this example.
In the same way 9 coils, each of 160 electrical degrees, will occupy the same angular breadth as 8 poles.
Further, 9 coils of 200 electrical degrees will occupy the same angular breadth as 10 poles.
Now of these 9 coils, any three contiguous ones are nearly in phase, if wound alternately clockwise and counter-clockwise.
For the 8 pole machine, the phase difference between adjacent coils is 20 degrees.
For the 10 pole machine, the phase difference is also 20 degrees.
The cosine of 20 degrees is .9397, consequently, if 3 adjacent coils be united in series, their joint pressure will be 2.897 multiplied by that of the middle one of the three.
The 9 coils may therefore be joined up in three groups of 3 adjacent coils, for the three phases.
By repetition, the same grouping will suit for any machine having[Pg 1262] a multiple of 8 or of 10 poles. These two cases are illustrated in figs. 1,572 and 1,573. In the figures, the coils are represented as occupying two slots each, but they might be further distributed.
Turbine Alternator Winding.—For the reason that steam turbines run at so much higher speed than steam engines, the construction of armatures and windings for alternators intended to be direct connected to turbines must be quite different from those driven by steam engines. Accordingly, in order that the frequency be not too high, turbine driven alternators must have very few poles—usually two or four, but rarely six.
The following table will show the relation between the revolutions and frequencies for the numbers of poles just designated.
Frequency | Revolutions | ||
---|---|---|---|
2 pole | 4 pole | 6 pole | |
25 | 1,500 | 750 | 500 |
60 | 3,600 | 1,800 | 1,200 |
100 | 6,000 | 3,000 | 2,000 |
From the table, it is evident that a large number of poles is not permissible, considering the high speed at which the turbine must be run.
Ques. How is the high voltage obtained with so few poles?
Ans. There must be either numerous inductors per slot or numerous slots per pole.
Ques. What form of armature is generally used?
Ans. A stationary armature.
Ques. What difficulty is experienced with revolving armatures?
Ans. The centrifugal force being considerable on account[Pg 1266] of the high speed, requires specially strong construction to resist it, consequently closed or nearly closed slots must be used.
Ques. How is the design of the rotor modified so as to reduce the centrifugal force?
Ans. It is made long and of small diameter.
Some examples of revolving fields are shown in figs. 1,579 to 1,584. Figs. 1,577 and 1,578 show some construction details of a stationary armature of turbine alternator.
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ELECTRICAL GUIDE, NO. 1
Containing the principles of Elementary Electricity, Magnetism, Induction, Experiments, Dynamos, Electric Machinery.
ELECTRICAL GUIDE, NO. 2
The construction of Dynamos, Motors, Armatures, Armature Windings, Installing of Dynamos.
ELECTRICAL GUIDE, NO. 3
Electrical Instruments, Testing, Practical Management of Dynamos and Motors.
ELECTRICAL GUIDE, NO. 4
Distribution Systems, Wiring, Wiring Diagrams, Sign Flashers, Storage Batteries.
ELECTRICAL GUIDE, NO. 5
Principles of Alternating Currents and Alternators.
ELECTRICAL GUIDE, NO. 6
Alternating Current Motors, Transformers, Converters, Rectifiers.
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Alternating Current Systems, Circuit Breakers, Measuring Instruments.
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Alternating Current Switch Boards, Wiring, Power Stations, Installation and Operation.
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Telephone, Telegraph, Wireless, Bells, Lighting, Railways.
ELECTRICAL GUIDE, NO. 10
Modern Practical Applications of Electricity and Ready Reference Index of the 10 Numbers.
Silently corrected simple spelling, grammar, and typographical errors.
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