  In the case of alternating current, because of its peculiar behaviour, there are several effects which must be considered in making wiring calculations, which do not enter into the problem with direct current. Accordingly, in determining the size of wires, allowance must be made for 1. Self-induction; Most of these items have already been explained at such length, that only a brief summary of facts need be added, to point out their connection and importance with alternating current wiring. Induction.—The effect of induction, whether self-induction or mutual induction, is to set up a back pressure of spurious resistance, which must be considered, as it sometimes materially affects the calculation of circuits even in interior wiring. Self-induction is the effect produced by the action of the electric current upon itself during variations in strength. Ans. The shape of the circuit, and the character of the surrounding medium. If the circuit be straight, there will be little self-induction, but if coiled, the effect will become pronounced. If the surrounding medium be air, the self-induction is small, but if it be iron, the self-induction is considerable. Figs. 2,671 to 2,676.—The effect of self-induction. In a non-inductive circuit, as in fig. 2,672, the whole of the virtual pressure is available to cause current to flow through the lamp filament, hence it will glow with maximum brilliancy. If an inductive coil be inserted in the circuit as in fig. 2,674, the reverse pressure due to self-induction will oppose the virtual pressure, hence the effective pressure (which is the difference between the virtual and reverse pressures), will be reduced and the current flow through the lamp diminished, thus reducing the brilliancy of the illumination. The effect may be intensified to such degree by interposing an iron core in the coil as in fig. 2,676, as to extinguish the lamp. Ques. With respect to self-induction, what method should be followed in wiring? Ans. When iron conduits are used, the wires of each circuit should not be installed in separate conduits, because such arrangement will cause excessive self-induction. The importance of this may be seen from the experience of one contractor, who installed feeders and mains in separate iron pipes. Ques. What is mutual induction? Ans. Mutual induction is the effect of one alternating current circuit upon another. Fig. 2,677.—Measurement of self induction when the frequency is known. The apparatus required consists of a high resistance or electrostatic a.c. voltmeter, d.c. ammeter, and a non-inductive resistance. Connect the inductive resistance to be measured as shown, and close switch M, short circuiting the ammeter. Connect alternator in circuit and measure drop across R and across Xi. Disconnect alternator and connect battery in circuit, then open switch M and vary the continuous current until the drop across R is the same as with the alternating current, both measurements being made with the same voltmeter; read ammeter, and measure drop across Xi. Call the drop across Xi with alternating current E, and with direct current Ei, and the reading of the ammeter J. Then L = vE2 + Ei2 ÷ 2p f I. If the resistance Xi be known, and the ammeter be suitable for use with alternating current, the switch and R may be dispensed with. Then L = vE2 - Xi2 Ii2 ÷ 2p f I, where Ii is the value of the alternating current. The resistance of the voltmeter should be high enough to render its current negligible as compared with that through Xi. Ques. How is it caused? Ans. It is due to the magnetic field surrounding a conductor cutting adjacent conductors and inducing back pressures therein. This effect as a rule in ordinary installations is negligible. Transpositions.—The effect of mutual induction between two circuits is proportional to the inter-linkage of the magnetic fluxes of the two lines. This in turn depends upon the proximity of the lines and upon the general relative arrangement of the conductors. Fig. 2,678.—Transposition diagram for two parallel lines consisting of two wires each. Fig. 2,679.—Transposition diagram for three phase, three wire line, transposing at the vertices of an equilateral triangle. The line is originally balanced and becomes unbalanced on transposing, a procedure which should be resorted to only to prevent mutual induction. Fig. 2,680.—Transposition diagram of three phase, three wire line, with center arranged in a straight line. The inductive effect of one line upon another is equal to the algebraic sum of the fluxes due to the different conductors of the first line, considered separately, which link the secondary line. The effect of mutual induction is to induce surges in the line where a difference of frequency exists between the two currents, and to induce high electrostatic charges in lines carrying little or no current, such as telephone lines.
Fig. 2,678 shows this method. The length L' should be an even factor of L so that to every section of the line transposed there corresponds an opposing section. Fig. 2,681.—Capacity effect in single phase transmission line. The effect is the same as would be produced by shunting across the line at each point an infinitesimal condenser having a capacity equal to that of an infinitesimal length of circuit. For the purpose of calculating the charging current, a very simple and sufficiently accurate method is to determine the current taken by a condenser having a capacity equal to that of the entire line when charged to the pressure on the line at the generating end. The effect of capacity of the line is to reduce the pressure drop, that is, improve the regulation, and to decrease or increase the power loss depending on the load and power factor of the receiver. Fig. 2,682.—Capacity effect in a three phase transmission line. It is the same as would be produced by shunting the line at each point by three infinitesimal condensers connected in star with the neutral point grounded, the capacity of each condenser being twice that of a condenser of infinitesimal length formed by any two of the wires. The effect of capacity on the regulation and efficiency of the line can be determined with sufficient accuracy in most cases by considering the line shunted at each end by three condensers connected in star, the capacity of each condenser being equal to that formed by any two wires of the line. An approximate value for the charging current per wire is the current required to charge a condenser, equal in capacity to that of any two of the wires, to the pressure at the generating end of the line between any one wire and the neutral point. The self inductance of lines is readily calculated from the following formula: L = .000558 {2.303 log (2A ÷ d) + .25} per mile of circuit where L = inductance of a loop of a three phase circuit in henrys. Note.—The inductance of a complete single phase circuit = L × 2 ÷ v3. A = distance between wires;
Capacity.—In any given system of electrical conductors, a pressure difference between two of them corresponds to the presence All circuits have a certain capacity, because each conductor acts like the plate of a condenser, and the insulating medium, acts as the dielectric. The capacity depends upon the insulation. For a given grade of insulation, the capacity is proportional to the surface of the conductors, and universally to the distance between them. A three phase three wire transmission line spaced at the corners of an equilateral triangle as regards capacity acts precisely as though the neutral line were situated at the center of the triangle. The capacity of circuits is readily calculated by applying the following formulae:
in which C = Capacity in micro-farads; sc = Specific inductive capacity of insulating material; D = Inside diameter of lead sheath; d = Diameter of conductor; h = Distance of conductors above ground; A = Distance between wires. Frequency.—The number of cycles per second, or the frequency, has a direct effect upon the inductance reactance in an alternating current circuit, as is plainly seen from the formula. Xi = 2p f L In the case of a transmission line alone; the lower frequencies are the more desirable, in that they tend to reduce the inductance drop and charging current. The inductance drop is proportional to the frequency. The natural period of a line, with distributed inductance and capacity, is approximately given by P = 7,900 / vLC where L is the total inductance in millihenrys, and C, the total capacity in micro-farads. Accordingly some lower odd harmonic of the impressed frequency may be present which corresponds with the natural period of the line. When this obtains, oscillations of dangerous magnitude may occur. Such coincidences are less likely with the lower harmonics than with the higher. Ques. What effect has "skin effect" on the current? Ans. It is equivalent to an increase of ohmic resistance and therefore opposes the current. Figs. 2,683 to 2,687.—Skin effect and shield effect. Fig. 2,683, section of conductor illustrating skin effect or tendency of the alternating current to distribute itself unequally through the cross section of a conductor as shown by the varied shading, which represents the current flowing most strongly in the outer portions of the conductor. For this reason it has been proposed to use hollow or flat instead of solid round conductors; however, with frequency not exceeding 100, the skin effect is negligibly small in copper conductors of the sizes usually employed. In figs. 2,684 and 2,685, or 2,686 and 2,687, if two adjacent conductors be carrying current in the same direction, concentration will occur on those parts of the two conductors remote from one another, and the nearer parts will have less current, that is to say, they will be shielded. In this case, the induction due to one conductor will exert its opposing effect to the greatest extent on those parts of the other conductor nearest to it; this effect decreasing the deeper the latter is penetrated. After crossing the current axis, the induction will still decrease in magnitude, but will now aid the current in the conductor. Hence, the effect of these two conductors on one another will make the current density more uniform than is the case where the two conductors adjacent to one another are carrying current in opposite directions, as in figs. 2,685 and 2,686, therefore, the resistance and the heating for a given current will be smaller. If the two return conductors be situated on the line passing through the center of the conductors just considered, the effect will be to still further concentrate the current; the distribution symmetry will be further disturbed, and the resistance of the conductor system increased. It is therefore difficult to say which of the two cases considered holds the advantage so far as increasing the resistance is concerned. The case, however, in which the phases are mixed has much the smaller reactive drop. If the conductor be large, or the frequency high, the central portion of the conductor carries little if any current, hence the resistance is therefore greater for alternating current than for direct current. Ques. For what condition may "skin effect" be neglected? Ans. For frequencies of 60 or less, with conductors having a diameter not greater than 0000 B. & S. gauge. Ques. How is the "skin effect" calculated for a given wire? Ans. Its area in circular mils multiplied by the frequency, gives the ratio of the wire's ohmic resistance to its combined resistance. That is to say, the factor thus obtained multiplied by the resistance of the wire to direct current will give its combined resistance or resistance to alternating current. The following table gives these ratio factors for large conductors.
Corona Effect.—When two wires, having a great difference of pressure are placed near each other, a certain phenomenon occurs, which is called corona effect. When the spacing or distance between the wires is small and the difference of pressure in the wires very great, a continuous passage of energy takes place through the dielectric or atmosphere, the amount of this energy may be an appreciable percentage of the power transmitted. Therefore in laying out high pressure transmission lines, this effect must be considered in the spacing of the wires. Ques. How does the corona effect manifest itself? Ans. It is visible at night as a bluish luminous envelope and audible as a hissing sound. Ques. What is the critical voltage? Ans. The voltage at which the corona effect loss takes place. Ques. Upon what does the critical voltage depend? Ans. Upon the radius of the wires, the spacing, and the atmospheric conditions. Fig. 2,688.—Electromagnetic and electrostatic fields surrounding the conductors of a transmission line. The electromagnetic field is represented by lines of magnetic force that surround the conductors in circles, (the solid lines), and the electrostatic field by (dotted) circles passing from conductor to conductor across at right angles to the magnetic circles. For any given size of wire and distance apart of wires there is a certain voltage at which the critical density or critical gradient is reached, where the air breaks down and luminosity begins—the critical voltage where corona manifests itself. At still higher voltages corona spreads to further distances from the conductor and a greater volume of air becomes luminous. Incidentally, it produces noise. Now to produce light requires power and to produce noise requires power. Air is broken down and is heated in breaking down, and to heat also requires power; therefore, as soon as corona forms, power is consumed or dissipated in its formation. When this phenomenon occurs on the conductors of an alternating current circuit a change takes place in relation to current and voltage. On the wires of an alternating current transmission line, at a voltage below that where corona forms—at a voltage where wires are not luminous—considerable current, more or less depending on voltage and length of wire, flows into the circuit as capacity current or charging current. The critical voltage increases with both the diameter of the wires, and the spacing. The losses due to corona effect increase very rapidly with increasing pressure beyond the critical voltage. The magnitude of the losses as well as the critical voltage is affected, by atmospheric conditions, hence they probably vary with the particular locality, and the season of the year. Therefore, for a given locality, a voltage which is normally below the critical point, may at times be above it, depending upon changes in the weather. Such elements as smoke, fog, moisture, or other particles (dust, snow, etc.) floating in the air, increase the losses; rain, however, apparently has no appreciable effect upon the losses. It follows then that in the design of a transmission line, the atmospheric conditions of the particular locality through which the line passes should be considered. Ques. How should live wires be spaced? Ans. They should be so spaced as to lessen the tendency to leakage and to prevent the wires swinging together or against towers. The spacing should be only sufficient for safety, since increased spacing increases the self-induction of the line, and while it lessens the capacity, it does so only in a slight degree. The following spacing is in accordance with average practice.
Resistance of Wires.—For quick calculation the following method of obtaining the resistance (approximately) of wires will be found convenient: 1,000 feet No. 10 B. & S. wire, which is about .1 inch in diameter (.1019), has a resistance of one ohm, at a temperature of 68° F. and weighs 31.4 pounds. A wire three sizes larger, that is No. 7, has twice the cross section and therefore one-half the resistance. A wire three sizes smaller than No. 10, that is No. 13, has one-half the cross section and therefore twice the resistance. Thus, starting with No. 10, any number three sizes larger will double the cross sectional area and any wire three sizes smaller will halve For alternating current, the combined resistance, that is, the total resistance, including skin effect, is obtained by multiplying the resistance, as found above by the "ratio factor" (see table page 1,894). Figs. 2,689 to 2,692.—Triangles for obtaining graphically, impedance, impressed pressure, etc., in alternating current circuits. For a full explanation of this method the reader is referred to Guide 5, Chapter XLVII on Alternating Current Diagrams. A thorough study of this chapter is recommended. Impedance.—The total opposition to the flow of electricity in an alternating current circuit, or the impedance may be resolved into two components representing the ohmic resistance and the spurious resistance; these components have a phase difference of 90°, and they may be represented graphically by the two legs of a right angle triangle, of which the hypothenuse represents the impedance. Similarly, the volts lost or "drop" in an alternating circuit may be resolved into two components representing respectively These components have a phase difference of 90° and are represented graphically similar to the impedance components. This has been explained at considerable length in Chapter XLVII (Guide V). Fig. 2,693.—Mechanical analogy of power factor, as exemplified by a locomotive "poling" a car off a siding. The car and locomotive are shown moving in parallel directions, and the pole AB, inclined at an angle ?. Now, if the length of AB be taken to represent the pressure exerted on the pole by the locomotive, then the imaginary lines AC and BC, drawn respectively parallel and at right angles to the direction of motion will represent respectively the useful and no energy (wattless) components; that is to say, if the pressure AB be applied to the car at an angle ?, only part of it, AC, is useful in propelling the car, the other component, BC, being wasted in tending to push the car off the track at right angles to the rails, being resisted by the flanges of the outer wheels. Power Factor.—When the current falls out of step with the pressure, as on inductive loads, the power factor becomes less than unity, and the effect is to increase the current required for a given load. Accordingly, this must be considered in calculating the size of the wires. As has been explained, the current flowing in an alternating current circuit, as measured by an ammeter, can be This apparent current, as is evident from the triangle, exceeds the active current and lags behind the pressure by an amount represented by the angle ? between the hypothenuse and leg representing the energy current as shown in fig. 2,694. Fig. 2,694.—Diagram showing that the apparent current is more than the active current, the excess depending upon the angle of phase difference. Fig. 2,695.—Diagram showing components of impedance volts. Compare this diagram with figs. 2,689 and 2,671, and note that the term "reactance" is the difference between the inductance drop and the capacity drop if the circuit contain capacity, for instance, if inductance drop be 10 volts and capacity drop be 7 volts then reactance 10-7 = 3 volts. Ques. What determines the heating of the wires on alternating current circuits with inductive loads? Ans. The apparent current, as represented by the hypothenuse of the triangle in fig. 2,694. Ques. How is the apparent current obtained? Ans. Divide the true watts by the product of the power factor multiplied by the voltage. Example.—A certain circuit supplies 20 kw. to motors at 220 volts and .8 power factor. What is the apparent current?
Ques. What else, besides power factor, should be considered in making wire calculations for motor circuits? Ans. The efficiency of the motor, and the heavy starting current. The product of the efficiency of the motor multiplied by the power factor gives the apparent efficiency, which governs the size of the wires, apparatus, etc., necessary to feed the motors. Allowance should be made for the heavy starting current required for some motors to avoid undue drop.
Ques. What are the usual power factors encountered on commercial circuits? Ans. A mixed load of incandescent lamps and induction motors will have a power factor of from .8 to .85; induction motors above .8 to .85; incandescent and Nernst lamps .98; arc lamps, .85. Wire Calculations.—In the calculation of alternating current circuits, the two chief factors which make the computation different from that for direct current circuits, is induction and power factor. The first depends on the frequency, and physical condition of the circuit, and the second upon the character of the load. Ques. Under what conditions may inductance be neglected? Figs. 2,696 to 2,698.—Example of wiring showing where inductance is negligible, and where it must be considered in wire calculations. Ans. In cases where the wires of a circuit are not spaced over an inch apart, or in conduit work, where both wires are in the same conduit. Under these conditions the calculation is the same as for direct current after making proper allowance for power factor. Ques. Under what conditions must induction be considered? Ans. On exposed circuits with wires separated several inches, particularly in the case of large wires. Sizes of Wire.—The size of wire for any alternating circuit may be determined by slightly modifying the formula used in direct current work, and which, as derived in Guide No. 4, page 748, is
The quantity 21.6, is twice the resistance (10.8) of a foot of copper wire one mil in diameter (mil foot). This resistance (10.8) is multiplied by 2, giving the quantity 21.6, because the length of a circuit, or feet in the formula, is given as the "run" or distance one way, that is, one-half the total length of wire in the circuit, must be multiplied by 2 to get the total drop, viz.:
It is sometimes however convenient to make the calculation in terms of watts. Formula (1) may be modified for such calculation. In modifying the formula, the "drop" should be expressed in percentage instead of actual volts lost, that is, instead of the difference in pressure between the beginning and the end of the circuit. In any circuit the loss in percentage, or
from which
Substituting equation (2) in equation (1)
Equation (3) is modified for calculation in terms of watts as follows: The power in watts is equal to the applied voltage multiplied by the current, that is to say, the power is equal to the volts at the consumer's end of the circuit multiplied by the current, or simply watts = volts × amperes from which
Figs. 2,699 to 2,703.—Stranded copper cables. For conductors of large areas and in the smaller sizes where extra flexibility is required it becomes necessary to employ stranded cables made by grouping a number of wires together in either concentric or rope form. The concentric cable as here illustrated is formed by grouping six wires around a central wire thereby forming a seven wire cable. The next step is the application in a reverse direction of another layer of 12 wires and a nineteen wire cable is produced. This is again increased by a third layer of eighteen wires for a 37 wire cable and a fourth layer of 24 wires for a 61 wire cable. Successive layers, each containing 6 more wires than that preceding, may be applied until the desired capacity is obtained. The cuts show sectional views of concentric cables each formed from No. 10 B. & S. gauge wires. Substituting this value for the current in equation (3) and remembering that the pressure taken is the volts at the consumer's end of the line
in which M is a coefficient which has various values according to the kind of circuit and value of the power factor. These values are given in the following table:
NOTE.—The above table is calculated as follows: For single phase M = 2,160 ÷ power factor2 × 100; for two phase four wire, or three phase three wire, M = ½ (2,160 ÷ power factor2)× 100. Thus the value of M for a single phase line with power factor .95 = 2,160 ÷ .952 × 100 = 2,400. It must be evident that when 2,160 is taken as the value of M, formula (6) applies to a two wire direct current circuit and also to a single phase alternating current circuit when the power factor is unity. In the table the value of M for any particular power factor is found by dividing 2,160 by the square of that power factor for single phase and twice the square of the power factor for two phase and three phase. Ques. For a given load and voltage how do the wires of a single and two phase system compare in size and weight, the power factor being the same in each case? Ans. Since the two phase system is virtually two single phase systems, the four wires of the two phase systems are half the size of the two wires of the single phase system, and accordingly, the weight is the same for either system.
NOTE.—This table is for finding the value of the current in line, using the formula I=W÷(E×T), in which I = current in line; E = voltage between main conductors at receiving or consumers' end; W = watts. For instance, what is the current in a two phase line transmitting 1,000 watts at 550 volts, power factor .80? I = 1,000 ÷ (550 × 1.60) = 1.13. Ques. Since there is no saving in copper in using two phases, what advantage has the two phase system over the one phase system? Ans. It is more desirable on power circuits, because two phase motors are self-starting. That is to say, the rotating magnetic field that can be produced by a two phase current, permits an induction motor to start without being equipped with any special phase splitting devices which are necessary on single phase motors, because the oscillating field produced by a single phase current does not produce any torque on a squirrel cage armature at rest. Ques. For equal working conditions, what is the comparison between the single, two and three phase system as to size and weight of wires? Ans. Each wire of the three phase system is half the size of one of the wires of the single phase system, hence the weight of copper required for the three phase system is 75% of that required for the single phase system. Since in the two phase system half of the load is carried by each phase, each wire of the three phase system is the same size as one of the wires of the two phase system, hence, the copper required by the three phase system is 75% of that required by the two phase system.
Breaking weight of wire ÷ area = breaking weight per square inch. Breaking weight per square inch × area = breaking weight of wire. The weight of copper wire is 1-1/7 times the weight of iron wire of same diameter. EXAMPLE.—What size wires must be used on a single phase circuit 2,000 feet in length to supply 30 kw. at 220 volts with loss of 4%, the power factor being .9? The formula for circular mils is
Substituting the given values and the proper value of M from the table, in (1)
Referring to the accompanying table of the properties of copper wire, the nearest larger size wire is No. 1 B. & S. gauge having an area of 83,690 circular mils. TABLE OF THE PROPERTIES OF COPPER WIRE Giving weights, length and resistances of wires of Matthiessen's Standard Conductivity for both B. & S. G. (Brown & Sharpe Gauge) and B. W. G. (Birmingham Wire Gauge) from Transactions October 1903, of the American Institute of Electrical Engineers.
Drop.—In order to determine the drop or volts lost in the line, the following formula may be used
in which the % loss is a percentage of the applied power, that is, the power delivered to the consumer and not a percentage of the power at the alternator. "Volts" is the pressure at the consumer's end of the circuit.
The coefficient S has various values as given in the accompanying tables. As will be seen from the table, the value of S to be used depends upon the size of wire, spacing, power factor and frequency. These values are accurate enough for all practical purposes, and may be used for distances of 20 miles or less and for voltages up to 25,000. The capacity effect on very long high voltage lines, makes this method of determining the drop somewhat inaccurate beyond the limits above mentioned.
EXAMPLE.—A circuit supplying current at 440 volts, 60 frequency, with 5% loss and .8 power factor is composed of No. 2 B. & S. gauge wires spaced one foot apart. What is the drop in the line? According to the formula
Substituting the given values, and value of S as obtained from the table for frequency 60
Current.—As has been stated, the effect of power factor less than unity, is to increase the current; hence, in inductive circuit
and
Substituting (2) in (1)
Fig. 2,704.—Rope type of stranded copper cable which is used when a high degree of flexibility is required. The construction of this cable is the stranding together of seven groups, each containing seven wires and producing a total of 49 wires. In cases when a greater carrying capacity is desired than can be obtained through the use of the 7 × 7 or 49 wire cable, the number of groups is increased to nineteen thereby making a total of 133 wires (19 × 7). EXAMPLE.—A 50 horse power 440 volt motor has a full load efficiency of .9 and power factor of .8. How much current is required? Since the brake horse power of the motor is given, it is necessary to obtain the electrical horse power, thus
which in watts is 55.5 × 746 = 41,403 which is the actual load, and from which
EXAMPLE.—A 50 horse power single phase 440 volt motor, having a full load efficiency of .92 and power factor of .8, is to be operated at a distance of 1,000 feet from the alternator. The wires are to be spaced 6 inches apart and the frequency is 60, and % loss 5. Determine: A, electrical horse power; B, watts; C, apparent load; D, current; E, size of wires; F, drop; G, voltage at the alternator. A. Electrical horse power
or, 54.3 × 746 = 40,508 watts
B. Watts watts = E.H.P. × 746 = 54.3 × 746 = 40,508 C. Apparent load
D. Current
E. Size of wires
From table page 1,907, nearest size larger wire is No. 00 B. & S. gauge. F. Drop
NOTE.—Values of S are given on page 1910. G. Voltage at alternator alternator pressure = (volts at motor + drop) = 440 + 25.74 = 465.7 volts. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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