current tends to drive it back. The reactance is therefore written as 1/pK, and the angle will be such that tan 4-1/pKR. The impedance will be VR2+1/p2K2. If both inductance and capacity are present, tan

=(pL-1/pK)/R; the reactance will be pL-1/pK; and the impedance VR2+ (pL-1/pK)2.

Since capacity and inductance produce opposite effects they can be used to neutralize one another. They exactly balance if L=1/p2K. In that case the circuit is noninductive and the currents simply obey Ohm's law.

474. Choking Coils. It will be seen that if in a circuit there is little resistance, and much reactance, the current will depend on the reactance. For example if p(=2n) were, say, 1000 and L=10 henries while R was only 1 ohm, the resistance part of the impedance would be negligible and the law would become

E
C=
PL

Self-induction coils with large inductance and small resistance are sometimes used to impede alternate currents, and are called choking coils, or impedance coils.

If the current were led into a condenser of small capacity (say K = microfarad, then 1/pK=10,000), the current running in and out of the condenser would be governed only by the capacity and frequency, and not by the resistance, and would have the value

C=EpK.

475. Alternate-current Power. If to measure the

power supplied to a motor, or other part of an alternate current circuit, we measure separately with amperemeter and voltmeter the amperes and volts, and then multiply together the readings we obtain as the apparent watts a value often greatly in excess of the true watts, owing to the difference in phase, of which the instruments take no account. The true power (watts) is in reality W=CV cos, where C and V are the virtual values, and

the angle of lag. But the latter is usually an unknown quantity. Hence recourse must be had to a suitable watt-meter; the usual form being an electrodynamometer (Art. 438) specially constructed so that the high-resistance circuit in it shall be non-inductive.

Whenever the phase-difference (whether lag or lead) is very large the current, being out of step with the volts, is almost wattless. This is the case with currents flowing through a choking coil or into a condenser, if the resistances are small.

476. High Frequency Currents. - The reactive effects of inductance and capacity increase if the frequency is increased. The frequency used in electric lighting is from 50 to 120 cycles per second. If high frequencies of 1000 or more cycles per second are used the reactions are excessive. In such cases the currents do not flow equally through the cross-section of the conducting wire, but are confined mainly to its outer surface, even thick rods of copper offering great impedance. Even at a frequency of 100 the current at a depth of 12 millimetres from the surface is (in copper) only about of its value in the surface layers. In iron wires the depth of the skin for value is about 1 millimetre. For such rapid oscillations as the discharge of a Leyden jar, where the frequency is several millions, the conducting skin is probably less than of a millimetre thick. Hollow tubes in such cases conduct just as well as solid rods of same outer diameter. The conductance is proportional not to section but to perimeter.

Whenever a current is not distributed equally in the cross-section of any conductor there is a real increase in the resistance it offers; the heating effect being a minimum when equally distributed. The fact that the oscillatory currents are greatest at the skin gives the strongest support to the modern view that the energy in an electric circuit is transmitted by the surrounding medium and not through the wire (see Art. 519 on energy-paths).

477. Alternate-current Electromagnets. — When an alternate current is sent through a coil it produces an alternating magnetic field. An iron core placed in the alternating field will be subjected to a periodic alternating magnetization. Electromagnets for alternate currents must have their iron cores laminated to avoid eddy currents; and owing to their choking action are made with fewer turns of wire than if designed for continuous currents of equal voltage. They repel sheets of copper owing to the eddy currents which they set up in them; the phase of these eddy currents being retarded by their self-induction. Elihu Thomson, who studied these repulsions, constructed some motors based on this principle. A solenoid, with a laminated iron plunger, if supplied with alternate currents at constant voltage, has the remarkable property of attracting the core with much greater force when the core is protruding out than when it is in the tube. This also is owing to the choking action.

LESSON XLIV. — Alternate-current Generators

478. Alternators. The simple alternator (Fig. 243), with its two slip-rings for taking off the current, is merely typical. In practice machines are wanted which will deliver their currents at pressures of from 1000 to 5000 volts, with frequencies of from 50 to 120 cycles per second.

Slower frequencies are unsuitable for lighting, though applicable for power transmission. High voltages are common with alternate currents because (when using transformers) of the economy (Art. 447) thereby effected in the copper mains. Under these conditions almost all alternators are designed as multipolar machines; and as the perfect insulation required in the armatures is more readily attained if these parts are stationary it is common to fix them, and instead to rotate the field-magnet. The latter is separately excited with a small continuous current led in through slip-rings. One advantage of alternate current machines over continuous current dynamos is that there is no commutator.

Amongst the various types of alternators may be mentioned the following:-(1) Magnet rotating internally and consisting of a number of poles, alternately N and S, pointing radially outwards; armature external, fixed, and consisting of a number of coils wound either upon an iron ring (Gramme), or upon inwardly projecting iron poles (Ganz), or set against the inner face of an iron core (Elwell-Parker), or embedded in holes just within the face of an iron core (Brown). In all cases where iron cores are used in armatures it is carefully laminated. (2) Magnet fixed externally and consisting of a number of alternate poles pointing radially inwards; armature internal, revolving, consisting of a number of coils wound either upon the surface of a cylindrical iron core (Westinghouse, Thomson-Houston) or fixed upon radially projecting poles (Hopkinson). (3) Magnet fixed externally and consisting of two crowns of alternate poles, alternately N and S, projecting toward one another and nearly meeting, so making a number of magnetic fields between them; armature revolving, and without iron, consisting of a number of flat coils mounted together as a sort of star disk, revolve in the narrow gaps between the poles (Siemens, Ferranti).

Another form, known as Mordey's alternator, largely

used in England, is depicted in Fig. 259. The thin armature coils are fixed, in an external stationary ring, between two crowns of poles revolving on each side of them. These poles are, however, all N poles on one side, and all S poles on the other, being projections of two massive iron pole-pieces fixed on the shaft against a huge

internal bobbin, thus constituting a solid simple form of field-magnet. On the end of the shaft is a small continuous-current dynamo as exciter.

In Fig. 260 is given a view of the central generating station for the electric lighting of the City of London. Two kinds of alternators (Thomson-Houston and Mordey) are used. The cut shows one of the latter driven by an 800 horse-power steam-engine. Each of these machines has 40 poles in each crown, and can deliver 250 amperes at 2200 volts.