CHAPTER XLVI
ALTERNATING CURRENTS
The word "alternating" is used with a large number of electrical and magnetic quantities to denote that their magnitudes vary continuously, passing repeatedly through a definite cycle of values in a definite interval of time.
As applied to the flow of electricity, an alternating current may be defined as: _A current which reverses its direction in a periodic manner, rising from zero to maximum strength, returning to zero, and then going through similar variations in strength in the opposite direction_; these changes comprise the cycle which is repeated with great rapidity.
The properties of alternating currents are more complex than those of continuous currents, and their behavior more difficult to predict. This arises from the fact that the magnetic effects are of far more importance than those of steady currents. With the latter the magnetic effect is constant, and has no reactive influence on the current when the latter is once established. The lines of force, however, produced by alternating currents are changing as rapidly as the current itself, and they thus induce electric pressures in neighboring circuits, and even in adjacent parts of the same circuit. This inductive influence in alternating currents renders their action very different from that of continuous current.
~Ques. What are the advantages of alternating current over direct current?~
Ans. The reduced cost of transmission by use of high voltages and transformers, greater simplicity of generators and motors, facility of transforming from one voltage to another (either higher or lower) for different purposes.
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 ÷ √2̅, 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 √2̅ times as great as the virtual mean; or conversely the virtual amperes will be equal to the maximum divided by √2̅. In fact, to produce equal effect, the equivalent direct current will be a kind of mean between the maximum and the zero value of the alternating current; but it must not be the arithmetical mean, nor the geometrical mean, nor the harmonic mean, but the _quadratic_ mean; that is, it will be the _square root of the mean of the squares_ of all the instantaneous values between zero and maximum.
~Effective Volts and Amperes.~--Virtual pressure, although already explained, may be further defined as the pressure _impressed_ on a circuit. Now, in nearly all circuits the impressed or virtual pressure meets with an opposing pressure due to inductance and hence the _effective_ pressure is something less than the virtual, being defined as _that pressure which is available for driving electricity around the circuit, or for doing work_. The difference between virtual and effective pressure is illustrated in fig. 1,237.
~Ques. Does a given alternating voltage affect the insulation of the circuit differently than a direct pressure of the same value?~
Ans. It puts more strain on the insulation in the same proportion as the maximum pressure exceeds the virtual pressure.
~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 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 √2̅ times the voltage in either phase, and the current in the line used as common return will be √2̅ 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 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 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 √3̅ × 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 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
L = coefficient of self induction in henrys; N = total number of lines of force threading a coil when the current is one ampere; T = number of turns of coil.
If a coil had a coefficient of self-induction of one henry, it would mean that if the coil had one turn, one ampere would set up 100,000,000, or 10⁸, lines through it.
The henry[2] is too large a unit for use in practical computations, which involves that the millihenry, or ¹/₁₀₀₀th 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 = 4π²_r²n²_ ÷ (_l_ × 10⁹) (1)
for a thin coil with air core, and
L = 4π²_r²n²_μ ÷ (_l_ × 10⁹) (2)
for a coil having an iron core. In the above formulæ:
L = inductance in henrys; π = 3.1416; _r_ = average radius of coil in centimeters; _n_ = number of turns of wire in coil; μ = permeability of iron core; _l_ = length of coil in centimeters.
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 = 2π_f_Iₘₐₓ_cos 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, the pressure due to it at the point of maximum value or
Eₘₐₓ = 2π_f_Iₘₐₓ × L (1)
Now by Ohm's law
Eₘₐₓ = RIₘₐₓ (2)
for a current Iₘₐₓ, hence substituting equation (2) in equation (1)
RIₘₐₓ = 2π_f_Iₘₐₓ × L
from which, dividing both sides by Iₘₐₓ, and using Xᵢ for R
~Xᵢ = 2π_f_L~ (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 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 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 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
Q = charge in coulombs; E = electric pressure in volts; C = capacity of condenser in farads.
~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 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 × ¼_f_ (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) 4_f_ 4_f_C
But, Iₐᵥ = Iₘₐₓ × 2 / π, and substituting this value of Iₐᵥ in equation (5) gives
Iₘₐₓ × 2 / π Iₘₐₓ condenser pressure = ----------- = ------ (6) 4_f_C 2π_f_C
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 = ---, or E = I × R R
and as
Iₘₐₓ 1 ------ = Iₘₐₓ × ------ 2π_f_C 2π_f_C
it follows that 1 / (2π_f_C) 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) 2π_f_C~
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. 2π_f_C 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° 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 2π_f_L 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 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. 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 = 2π_f_L = 2 × 3.1416 × 60 × .5 = 188.5 ohms
The quantity 2π_f_L 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 = 2π_f_LI (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 = 2π_f_LI = 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 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 2π_f_C 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:
1. Ohmic resistance; 2. Inductance reactance; 3. Capacity reactance.
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 2π_f_L for the reactance as follows: (using symbols for impedance and resistance)
Z = √(R² + (2π_f_L)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 = 2π_f_L = 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 2π_f_L, and the reactance of capacity in inverse proportion to the quantity 2π_f_C. 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 = 2π_f_L = 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 2π_f_C 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 = 2π_f_L, and for capacity reactance, Xc = 1 ÷ (2π_f_C); accordingly if capacity and inductance in a circuit be equal, that is, if the circuit be resonant
1 2π_f_L = ------ (1) 2π_f_C
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.