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In order to save the trouble of calculating the inductance with various sizes of and distances between wires, the following table is given:
INDUCTANCE, IN MILLIHENRY'S PER MILE. FOR EACH OF TWO PARALLEL
Examples. — To show the use of the above formulas and table, let it be required to determine the inductance of an overhead line, 14 miles long, consisting of two No. 0000 copper wires, 48 inches apart. Since it is a metallic circuit with two conductors, the total inductance is due to 2 x U = 3 miles of wire. From (54) the inductance per mile is
/ 2 4\ —«
( Z = 80 5 + 740 lo;; J10 .
Substituting, in this expression, the value of the distance between the wires A = 48 inches, and the diameter of each wire, d— .46 inch, we have
L per mile = (80.5 + 740.3 log 10 ~ ' = .001799 henry.
This is equal to 1.799 millihenry, and is the same value as that given in the first column of the table, and shows how those figures were obtained. The total inductance of the circuit is 3 x 1.799 = 5.397 millihenrys.
Impedance of Circuits. — Having obtained the inductance of a given line or circuit, by calculation or from the table, the reactance from (38) is 2ir / L, and the impedance from (39) is V^2 + (2 -a f Lf. The resistance R may be found in the table on page 8. Tables are often given showing the impedance of lines; but in order to cover the various frequencies, sizes of wire, and distance apart, they become too bulky to include in the present volume.
Mutual Inductance of Circuits. — The inductive effect of one circuit upon another separate circuit is called mutual inductance. The most familiar example in electrical engineering is to be found in the action between the primary and secondary coils of a transformer, and will be considered later under that head. If two conductors run parallel to each other, as, for example, two overhead lines upon the same poles, an alternating current in one tends to induce an alternating E.M.F. in the other, the direction of which is opposite to that of the inducing current. Consequently two parallel alternating currents which are exactly in phase tend to oppose each other; but if they differ by 180° in phase, that is, flow in opposite directions at the same time, they tend to aid each other. The currents in two or more parallel wires leading from the same terminal of an alternating current source would have about the same phase, assuming their angles of lag to be nearly equal, hence they tend to oppose each other. This opposition has the effect of increasing the drops in voltage similar to that due to self-induction; in fact, the action of these wires upon one another is practically the same as that of one element of a wire upon the other elements, but in the latter case it is called self-induction.
In practice, two alternating currents from independent generators would not be likely to remain exactly in phase, except for a few seconds at a time, so that their mutual induction upon each other would produce opposing effects at one time, aiding effects at another, and so on as the phase changed. Supposing the frequency of one current to be 100 and of the other 100J periods per second, the difference would be one period in two seconds, so that the voltage on each circuit would be raised once and lowered once every two seconds, causing very objectionable flickering in incandescent lamps. This is avoided by increasing the difference in frequency between the two currents. For example, if one were raised 5 per cent and the other lowered 5 per cent, the difference would be 10 periods per second, and the fluctuations in voltage occurring at that rate would be hardly noticeable. It is better, however, to have a still higher rate of 15 or 20 per second in order not to be perceptible or injurious to the eye. It is also possible to eliminate this effect by arranging or transposing the wires as described later.
Means of Reducing Self-Inductance. — In equations (51) to (58) it is evident that self-inductance is decreased by diminishing A the interaxial distance between two wires forming a metallic circuit. This somewhat paradoxical fact is understood when we consider that self-induction is proportional to the number of magnetic lines linked with a circuit, as defined on page 116. Consequently, the greater the distance between the two wires which constitute it, the more lines will be enclosed. Hence the wires should be as close together as possible in order to reduce self-inductance, the limit being the distance necessary for proper insulation, and in the case of overhead wires they must be sufficiently far apart not to swing too near each other.
If two insulated wires are laid side by side, or twisted together, their self-inductance becomes insignificant ; and if concentric conductors are used, it disappears entirely, since the tendency to produce magnetic lines by one is neutralized by the other, the currents being equal and opposite. One wire, carrying an alternating current and running through an iron pipe, will have large self-inductance, on account of the great number of lines which are set up around it; but if both wires of a metallic circuit are put in the pipe, the self-inductance is very small.
Another way to reduce the drop due to self-induction is to subdivide the conductor, using several smaller wires having the same total sectional area.
Example. — An overhead line 1 mile long consists of two No. 0000 wires forming a metallic circuit, the distance between the wires being 24 inches. One mile of No. 0000 has .258 ohms resistance at 20° C, so the resistance of the circuit is 2 x .258 = .51( i ohms. The self-inductance per mile from the table on page 130, is 1.576 millihenrys, or .003152 henrys for the circuit. At a frequency of 100 the impedance is + (628 x .003152)2 = 2.05 ohms.
With a current of 40 amperes the drop due to resistance is 40 x .516 = 20.64 volts, and the total drop is -10 x 2.05 — 82 volts. Using eight No. 6 wires in parallel the joint resistance would be .52 ohms, being almost exactly the same as before, or .52 x 8 = 4.16 ohms for each wire. Assuming the distance apart to be the same, or 24 inches for each pair, and neglecting the mutual inductance between the pairs, the self-inductance from the table would be 1.912 millihenrys per mile, or .00:3824 for the circuit, and the impedance yjAtiP + (.628~x .003824)* = 5 ohms. The current in each wire is 40 -r 8 = 5 amperes, so the resistance drop is 5 x 4.16 = 20.8 volts, and the total drop is 5x5= 25 ohms. In this case the resistance drop is practically the same as before, and the impedance drop is only 25 volts, or about 20 per cent greater than the simple resistance drop, while in the previous case it was 82 volts, or four times the resistance drop.
The above example proves the great reduction in inductance drop effected by subdividing the conductor. This is sometimes said to be due to the use of smaller wires, but this is not true; in fact, the inductance itself is increased by reducing the size of wire, as shown in the foregoing example, and in equation (53). In reality, the impedance drop in the second case would probably be greater than that calculated, on account of the mutual inductance between the corresponding wires of each pair; but this need not be very great if they are not put close together, and may be practically neutralized by arranging or transposing the wires, as explained under the next heading.
Another method of reducing the effect of inductance is to balance it by the effect of capacity. It was shown in connection with Fig. 90 and equations (46) and (49), that certain values of capacity in a circuit may completely or partially neutralize the reactance due to inductance.
In the Stanley electric power system condensers are used upon the circuit in connection with the motors to balance their inductance, so that the watless current is much reduced. In other words, the power-factor is raised, and the drop on the line is diminished. The amount of capacity, K, in farads, required to neutralize a certain inductance, L, in henrys, at a frequency, /, is obtained from (49) and has the following value:
K = TIM? (58>
Synchronous alternating current motors may also be used to balance inductance, since they have the effect of capacity in causing the current to lead when their field magnets are over excited. By regulating the field excitation the power-factor can be raised to practically 100 per cent. The same effect is produced by rotary converters, and will be considered more fully later in connection with the polyphase transmission and direct-current distribution system.
Means of Reducing Mutual Inductance. — The simplest plan is to increase the distance between the conductors, as already stated; but this is limited by practical considerations, such as requirements for carrying the wires on the same pole. In such cases the effect of mutual induction would be great if the wires were arranged as
B# #D H F
Figs. 104, 105, and 106. Arrangement of Conductors to Neutralize Induction.
represented in Fig. 104, in which A and B indicate the wires of one circuit, and C and D those of another parallel circuit. The wire A being very near the corresponding wire C of the other circuit, would tend to set up an opposing E.M.F. in it, and C would react upon A in a similar manner. The mutual induction of B and D would also have the same effect. If, however, the wires are placed equidistant, as shown in Fig. 105, any one conductor, E, will be acted upon equally by both wires, G and H, of the other circuit, since they are at the same distance from it. Consequently mutual induction between the two circuits is neutralized. This can also be accomplished by transposing the wires with respect to each other, at certain intervals, as shown in Fig. 106, in which the portion N of one wire, counteracts the effect of the part Q of the other wire of the same circuit. Consequently the inductive action upon the other circuit, JK and LAI, is nil.