Fig. 1,284.—Series or cascade connection of condensers. Unlike terminals are joined together as shown. The total capacity of such connection is equal to the reciprocal of the sum of the reciprocals of the several capacities, that is, C = 1 ÷ (1 / c + 1 / c' + 1 / c")

Figs. 1,285 and 1,286.—Diagrams showing effect of condenser in direct and alternating current circuits. Each circuit contains an incandescent lamp and a condenser, one circuit connected to a dynamo and the other to an alternator. Since the condenser interposes a gap in the circuit, evidently in fig. 1,285 no current will flow. In the case of alternating current, fig. 1,286, the condenser gap does not hinder the flow of current in the metallic portion of the circuit. In fact the alternator produces a continual surging of electricity backwards and forwards from the plates of the condenser around the metallic portion of the circuit, similar to the surging of waves against a bulkhead which projects into the ocean. It should be understood that the electric current ceases at the condenser, there being no flow between the plates.

Fig. 1,287.—Diagram showing alternating circuit containing capacity. Formula for calculating the ohmic value of capacity or "capacity reactance" is X_c = 1 ÷ 2πfC, in which X_c = capacity reactance; π = 3.1416; f = frequency; C = capacity in farads (not microfarads). 22 microfarads = 22 ÷ 1,000,000 = .000022 farad. Substituting, X_c = 1 ÷ (2 × 3.1416 × 100 × .000022) = 72.4 ohms.

Fig. 1,288.—Pressure and current curves, illustrating lag. The effect of inductance in a circuit is to retard the current cycle, that is to say, if the current and pressure be in phase, the introduction of inductance will cause a phase difference, the current wave "lagging" behind the pressure wave as shown. In other words, inductance causes the current wave, indicated in the diagram by the solid curve, to lag behind the pressure wave, indicated by the dotted curve. Following the curves starting from the left end of the horizontal line, it will be noted that the current starts after the pressure starts and reverses after the pressure reverses; that is, the current lags in phase behind the pressure, although the frequency of both is the same.

Fig. 1,289.—Pressure and current curves illustrating lead. The effect of capacity in a circuit is to cause the current to rise to its maximum value sooner than it would otherwise do; capacity produces an effect exactly the opposite of inductance. The phase relation between current and pressure with current leading is shown graphically by the two armature positions in full and dotted lines, corresponding respectively to current and pressure at the beginning of the cycle.

Fig. 1,290.—Mechanical analogy of lag. If at one end force be applied to turn a very long shaft, having a loaded pulley at the other, the torsion thus produced in the shaft will cause it to twist an appreciable amount which will cause the movement of the pulley to lag behind that of the crank. This may be indicated by a rod attached to the pulley and terminating in a pointer at the crank end, the rod being so placed that the pointer registers with the crank when there is no torsion in the shaft. The angle made by the pointer and crank when the load is thrown on, indicates the amount of lag which is measured in degrees.

Fig. 1,291.—Diagram of circuit for example I.

Fig. 1,292.—Diagram of circuit for example II.

Fig. 1,293.—Steam engine analogy of current flow at zero pressure (see questions below). When the engine has reached the dead center point the full steam pressure is acting on the piston, the valve having opened an amount equal to its lead. The force applied at this instant, indicated by the arrow is perpendicular to the crank pin circle, that is, the tangential or turning component is equal to zero, hence there is no pressure tending to turn the crank. The latter continues in motion past the dead center because of the momentum previously acquired. Similarly, the electric current, which is here analogous to the moving crank, continues in motion, though the pressure at some instants be zero, because it acts as though it had weight, that is, it cannot be stopped or started instantly.

Fig. 1,294.—Diagram of the circuit for example I. Here the resistance is taken at zero, but this would not be possible in practice, as all circuits contain more or less resistance though it may be, in some cases, negligibly small.

Fig. 1,295.—Diagram of circuit for example II. As in example I, resistance is disregarded.

Fig. 1,296.—Diagram of circuit for example III.

Fig. 1,297.—Diagram showing alternating circuit containing resistance, inductance, and capacity. Formula for calculating the impedance of this circuit is Z = √(R² + (X_i - X_c)²) in which, Z = impedance; R = resistance; X_i = inductance reactance; X_c = capacity reactance. Example: What is the impedance when R = 4, X_i = 94.2, and X_c = 72.4? Substituting Z = √(4² + (94.2 - 72.4)²) = 22.2 ohms. Where the ohmic values of inductance and capacity are given as in this example, the calculation of impedance is very simple, but when inductance and capacity are given in milli-henrys and microfarads respectively, it is necessary to first calculate their ohmic values as in figs. 1,295 and 1,296.

Fig. 1,298.—Diagram of circuit for example II.

Fig. 1,299.—Diagram of circuit for example III.

Fig. 1,300.—Diagram of circuit for example IV.

Fig. 1,301.—Simple choking coil. There is an important difference in the obstruction offered to an alternating current by ordinary resistance and by reactance. Resistance obstructs the current by dissipating its energy, which is converted into heat. Reactance obstructs the current by setting up a reverse pressure, and so reduces the current in the circuit, without wasting much energy, except by hysteresis in any iron magnetized. This may be regarded as one of the advantages of alternating over direct current, for, by introducing reactance into a circuit, the current may be cut down with comparatively little loss of energy. This is generally done by increasing the inductance in a circuit, by means of a device called variously a reactance coil, impedance coil, choking coil, or "choker." In the figure is a coil of thick wire provided with a laminated iron core, which may be either fixed or movable. In the first case, the inductance, and therefore also the reactance of the coil, is invariable, with a given frequency. In the second case, the inductance and consequent reactance may be respectively increased or diminished by inserting the core farther within the coil or by withdrawing it, as was done in fig. 1,266, the results of which are shown in fig. 1,302.

Fig. 1,302.—Impedance curve for coil with variable iron core. The impedance of an inductive coil may be increased by moving an iron wire core into the coil. In making a test of this kind, the current should be kept constant with an adjustable resistance, and voltmeter readings taken, first without the iron core, and again with 1, 2, 3, 4, etc., inches of core inserted in the coil. By plotting the voltmeter readings and the positions of the iron core on section paper as above, the effect of inductance is clearly shown.

Fig. 1,303.—Diagram of a resonant circuit. A circuit is said to be resonant when the inductance and capacity are in such proportion that the one neutralizes the other, the circuit then acting as though it contained only resistance. In the above circuit X_i = 2πfL = 2 × 3.1416 × 100 × .01 = 6.28 ohms; X_c = 1 ÷ (2 × 3.1416 × 100 × .000253) = 6.28 ohms whence the resultant reactance = X_i - X_c = 6.28 - 6.28 = 0 ohms. Z = √(R² + (X_i - X_c)²) = √(7² + 0²) = 7 ohms.

Fig. 1,304.—Application of a choking coil to a lighting circuit. The coil is divided into sections with leads running to contacts similar to a rheostat. Each lamp is provided with an automatic short-circuiting cutout, and should one, two, or more of them fail, a corresponding number of sections of the choking apparatus is put in circuit to take the place of the broken lamp or lamps, and thus keep the current constant. It must not be supposed that this arrangement of lamps, etc. is a general one; it being adopted to suit certain special conditions.

Fig. 1,305.—Curve showing variation of current by increasing the frequency in a circuit having inductance and capacity. The curve serves to illustrate the "critical frequency" or frequency producing the maximum current. The curve is obtained by plotting current values corresponding to different frequencies, the pressure being kept constant.

Fig. 1,306.—Section of conductor illustrating "skin effect" or tendency of the alternating current to distribute itself unequally through the cross section of the conductor as shown by the varied shading flowing most strongly in the outer portions of the conductor. For this reason it has been proposed to use hollow or flat conductors instead of solid round wires. However with frequency not exceeding 100 the skin effect is negligibly small in copper conductors of the sizes usually employed. Where the conductor is large or the frequency high the effect may be judged by the following examples calculated by Professor J. J. Thomson: In the case of a copper conductor exposed to an electromotive force making 100 periods per second at 1 centimetre from the surface, the maximum current would be only .208 times that at the surface; at a depth of 2 centimetres it would be only .043; and at a depth of 4 centimetres less than .002 part of the value at the surface. If the frequency be a million per second the current at a depth of 1 millimetre is less than one six-millionth part of its surface value. The case of an iron conductor is even more remarkable. Taking the permeability at 100 and the frequency at 100 per second the current at a depth of 1 millimetre is only .13 times the surface value; while at a depth of 5 millimetres it is less than one twenty-thousandth part of its surface value. The disturbance of current density may be looked upon as a self-induced eddy current in the conductor. It necessarily results in an increase of ohmic loss; as compared with a steady current: proportional to the square of the total current flowing and consequently gives rise to an apparent increase of ohmic resistance. The coefficient of increase of resistance depends upon the dimensions and the shape of the cross section, the frequency and the specific resistance. A similar but distinct effect is experienced in conductors due to the neighborhood of similar parallel currents. For example in a heavy multicore cable the non-uniformity of current density in any core may be considered as partly due to eddy currents induced by the currents in the neighboring cores and partly to the self-induced eddy current. It is only the latter effect which should rightly be considered as comprised under the term skin effect.