In a circuit containing several impedances in series, the ioint impedance is not the sum of the individual impedances, but is obtained by talcing the square root of the total added reactances squared plus the total added resistances, squared. That is, Impedance V al -I- R2 + 122)1 ( fil 27 as + 27 f102 The joint impedance of several impedances in parallel is found as follows. Construct a parallelogram from the reciprocals of .two of the impedances, each expressed in its proper phase relation. The direction of the diagonal will give the phase of the resultant im pedance and its reciprocal amount will give the reciprocal of its length. For more than two, the method of the polygon of forces is applied. The effect of self-induction varies with the fre quency of the current supplied, and as the square of the number of turns in a circuit. The self-induction in the armature of an alternator has two effects. The first is to produce a lag ging current and thus lower the terminal volt age, and the second is a demagnetizing effect. The current is thrown into such a phase that it produces lines of force directly opposed to the field and thus lowers the voltage by reducing the total flux. The effect of armature reaction depends upon whether the current is leading or lagging in phase. A lagging current lowers the voltage of an alternator and a leading current raises it.
All insulated conductors have the quality of being able to hold, stored on their surfaces, a certain quantity of static elec tricity, and are thus condensers. The charg ing and discharging of an alternating current circuit causes the current to flow from the generator into the line and then back into the generator again, with the frequency of the alternator, in order to keep up the static poten tial on the line. As this charging current is greatest when the rate of change of electromo tive force is greatest, a sinusoidal wave of capacity electromotive force with 90 degrees difference in phase from the machine electromo tive force is produced. This leads the active electromotive force by 90 degrees and is thus directly opposite to the electromotive force of self-induction. If we have a circuit in which the electromotive' force of self-induction is just equal to the capacity electromotive force, and these two parts of the circuit are in series, the effect of both is neutralized and we have, as in direct currents, W=EX C.
The The one piece of ap paratus that more than all else has made pos sible the electrical transmission of energy to long distances is the transformer. This is the apparatus that receives in one set of coils the dangerous potential of the line and transforms it into whatever potential is desired for lights or motors, which are supplied from an entirely separate winding. The transformer consists of a magnetic circuit of laminated iron or mild steel interlinked with two electric circuits, one, the primary, receiving electrical energy and the other, the secondary, delivering it to the con sumer. The effect of the iron is to make as many as possible of the lines of force set up by the primary current cut the secondary wind ing and there give rise to an electromotive force of the same frequency, but different vol tage.
Not only does the transformer make pos sible the transformation of voltages, hut it also permits of changing from one system to an other. Thus a single-phase primary may sup ply a three-wire Edison system, of course, with alternating current. A two-phase system can he changed to a three-phase or vice versa; a four-wire two-phase may make a three-wire two-phase, and many other useful combinations may he effected. The Scott connection for changing two-phase to three-phase, or the op posite, uses but two transformers. One has a ratio of, say 10 to 1, with a tap at the middle of its secondary coil. The other must then
have a ratio of 10 to .866=10 to V4. One ter minal of the secondary of this transformer is connected to the middle of the other secondary, and the remaining free ends of both seconda ries form the three terminals of a three-phase circuit. The value /4 is the altitude of an equilateral triangle of which the base is unity, and thus we may consider the current to be taken from the corners of an equilateral tri angle, which represent, in phase and potential difference, a true three-phase system. The cur rent in the transformer of secondary, .866 being the resultant of the other two-phases, is greater than under normal two-phase conditions; and, therefore, the windings must have about 15 Per cent more copper. If two similar transformers are used the secondary of each has taps giving 50 per cent and 86.7 per cent of full voltage. In many large installations, notably at Niagara Falls, we find two-phase generators feeding three-phase lines through Scott connected step up transformers. In small systems standard transformers may be used having ratios of 10 to 1 and 9 to 1 respectively, and the results will be quite satisfactory.
The Induction Acting upon the well-known fact that a copper disc could be made to revolve by rotating a horseshoe magnet so that the lines of force cut the disc, Ferraris, Tesla, Dobrosvolsky and others have developed the present type of induction motor. The credit for the first commercial application of the ro tating field caused by currents of displaced phase probably belongs to Tesla. At the pres ent day the value of these discoveries in the transmission and distribution of power can hardly be estimated. The induction motor is somewhat similar to the direct-current shunt motor. Both motors have field and armature windings. In both cases, also, the field is con nected directly across the mains. In the shunt motor the armature current is supplied through brushes and a commutator to the windings, while in the induction motor the armature cur rent is an indirect current, the field acting as the primary of a transformiT of which the armature is the secondary. In both motors the efficiency is inversely proportional 'to the arm ature resistance, as is also the speed regulation of the motors. The less the armature resist ance the higher the efficiency and the closer the regulation of speed between no load And full load. In practice, either element may be the one to revolve. The rotation is produced by the reaction of the armature, or indirect current, on the revolving magnetic field, which results in dragging the moving element around in order to keep up with the field flux, as it passes around the face of the primary windings. This field, being the resultant of two or more alternating fields of different phases, rotates with the polar frequency of the supplied voltage. The secondary winding is made up of copper bars set in slots in a laminated iron core and running across the armature parallel with the axis of rotation. This separating of the old copper disc into narrow bars constrains the current to flow into the hest direction for pro ducing torque and avoids the waste of the un constrained Foucault currents in the Arago disc, and thus makes the motor much more efficient. Sometimes the secondary windings are joined to heavy short-circuiting rings at both ends, resulting in the squirrel-cage type of motor; and in other cases the secondary wind ings are taken out through collector rings, if the secondary be the rotating element, and start ing resistances are inserted in series to lessen the reaction due to excessive starting current and thus improve the starting torque. When up to speed these resistances are cut out and the terminals short-circuited as in the squirrel-cage type.