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of the circuit on itself, or of one portion of the circuit on another portion of the same circuit, in which cases it is called self-induction; or it may be due to the action of one circuit upon another independent circuit, in which case it is called mutual induction. The former is the one generally considered in transmission, and will be treated first.

The unit of inductance, or the "coefficient of self- or mutual induction," is called the henry, which is the inductance of a circuit when the E.M.F. induced in it is one volt, while the inducing current varies at the rate of one ampere per second. For example, if a counter E.M.F. of one volt is set up in coil when the current is increased at the rate of one ampere per second, then the self-inductance of that coil is one henry.

The physical cause of the phenomenon of self-induction is the fact that a current flowing in a conductor tends to produce magnetic lines of force around itself. If the conductor is a helix of wire, the lines produced by each turn pass through that turn and through most of the others, so that the total flux through the helix is large. When the current varies, the lines of force also vary in number, and necessarily cut the turns of wire, thereby setting up an E.M.F. in the latter. With increasing current this E.M.F. is counter, and opposes the flow; with decreasing current it aids it; but when the current is steady no E.M.F. is induced, since the lines of force do not vary or cut the conductor. In the case of mutual induction, it is evident that a second coil B in the neighborhood will be cut by the lines of force produced by the first, tending to set up an E.M.F. in B, which will cause a current to flow in it, or will oppose or aid a current already flowing, according to the relative directions of the lines of force and the current.

Inductance was defined by the Chicago Electrical Congress of 1893 in terms of the E.M.F. generated, but it is also proportional to the number of turns of wire and to the average flux through each when unit current is flowing. This is similar to the first definition, since the production of a certain number of lines of force by one ampere in one second tends to generate a certain E.M.F.

A third definition of inductance may be based upon the electromagnetic energy stored in a coil when a unit current is flowing, which energy is proportional to the square of the flux density, other things being equal.

These three definitions may be summed up as follows:

Three Definitions of Inductance. - Calling L the inductance in henrys, e and i the instantaneous values of the E.M.F. in volts and the current in amperes respectively, n the instantaneous value of the average flux through each turn of wire, Z the number of turns,

di

W the energy in joules and the time rate of variation of the current, we have:

dt

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It has been shown that the

Reactance due to Self-Induction. effect of inductance in an alternating current circuit is to oppose the flow of current on account of the counter E.M.F. which it sets up. This opposition may be considered as an apparent resistance, and is called reactance to distinguish it from true ohmic resistance. The value of the reactance due to inductance is given by the following expression, in which fis the frequency in periods per second, and L is the inductance measured in henrys

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The result obtained gives the equivalent or apparent resistance in ohms.

Example. A coil of wire having a self-inductance of 25 millihenrys = .025 henry is supplied with an alternating current at a frequency of 100 periods per second. Its reactance, assuming its ohmic resistance to be negligible, would be 2 TfL 2 X 3.1416 X 100 X .025 15.7 ohms.

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Such a coil would have the same apparent resistance as a non-inductive circuit of 15.7 true ohms, and if connected to an alternating current source giving 1000 volts at 100 frequency, the effective current flowing through the coil 1000 would be 15.7

63.7 amperes.

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Impedance due to Resistance and Inductance. Actual circuits always have resistance as well as inductance, and in most cases the former cannot be neglected. The combined effect of resistance

and inductance is called impedance to distinguish it from the other two, and has the following value in ohms (apparent resistance).

Impedance

=

√ R2 + (2 πfL)2.

(40)

Example. A coil of wire has a resistance of 20 ohms and an inductance of .025 henry. For an alternating current having a frequency of 100 the impedance of the coil is

√ R2 + (2 π ƒL)2 = √ 202 + 15.72 = 25,4 ohms.

1000

Supplied with 1000 volts the coil would receive a current = 39.3

amperes.

25.4

The relations expressed analytically in (40) are evidently repre

sented graphically by the right

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angle triangle in Fig. 82. The resistance R in ohms is laid off on a convenient scale to form the base, the reactance 2 #ƒL is laid off also in ohms to form the perpendicular, and the impedance in ohms is found by measuring the hypothenuse of the triangle, since it is equal to the square root of the sum of the squares of the other two.

R

Fig. 82. Graphical Representat on.

Lag of Current due to Inductance. Besides opposition or reactance to an alternating current, inductance also causes the latter to lag behind the E.M.F. which produces it. The curve EF in Fig. 83 represents the waves of an alternating E.M.F. impressed upon a circuit containing ohmic resistance without inductance or capacity. In such a case the resulting current will reach its maximum as well as zero values at the same instants as the E.M.F., and may be represented by the curve CD. If now a self-induction coil be introduced into the circuit in series with the resistance, the current waves will lag with respect to those of E.M.F.; that is, the maximum current will flow a little later than the instant of maximum E.M.F., as indicated by the dotted curve GH. The amount of this lag is measured as an angle called the angle of lag, assuming one complete period to correspond to 360°. In Fig. 83 the current wave is shown as having its zero value one-eighth of a period, or 45° behind the zero E.M.F., and the same for the maximum and other corresponding points, hence the angle of lag is 45°.

The tangent of the angle of lag with a given resistance R and inductance L in the circuit is

tan =

reactance
resistance

=

2πfL
R

(40a)

Referring to Fig. 82, it is evident that the tangent of the angle is equal to 2πfL÷R; therefore represents the angle of lag, which may be easily determined graphically in this way. It is apparent, from Fig. 82, that the angle of lag is small if the resistance is

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large compared with the inductance L, unless the frequency is high. It is a fact also, that however large the inductance or frequency, and however small the resistance, the angle of lag can never be greater than a right angle, or 90°. This is evident in (40a), since = 90° when its tangent is infinity.

Example. A circuit has a resistance of 2 ohms and an inductance of .0016 henry. What is the angle of lag for an alternating current having a

frequency of 100?

2πfL = 2 × 3.1416 X 100 X .0016=2 ohms.

This

The resistance R is also 2 ohms, therefore tan == 1 and is 45°. is the condition shown in Fig. 83, the current wave GH (dotted) being 45° behind the E.M.F. wave EF. The curve CD represents the current that would flow if a wire of 2 ohms resistance without inductance were supplied with 100 volts alternating E.M.F., the current at any instant having one half the numerical value of the E.M.F. its effective value being 100÷250 amperes with no lag. The addition of .0016 henry inductance produces a reactance of 2 ohms, which combined with the resistance of 1 ohms, makes an impedance of ✔22+22= 2.82 ohms, which is much less than their arithmetical sum.

The current is 1002.82 = 35.5 amperes, so that the effect of inductance is to diminish the current, and cause it to lag as shown by comparing curves CD and GH in Fig. 83.

Determination of the Power of an Alternating Current. — In a circuit containing ohmic resistance only, the current wave C does not lag with respect to the E.M.F. wave E, and the power is represented by the curve PQ

P

E

360°

D

in Fig. 84. At any instant the power in watts is the product of the E.M.F. and current at that instant, but for convenience these values (curve PQ) are plotted on a smaller scale than E and C. The power is positive at all times, since the product of the positive values of E and C as well as the negative values of D and E are always positive, and its effective value is the Vmean2 of these products, which is simply the product of the effective E.M.F., and current, as read on a volt and an ampere-meter, that is

Fig. 84. Power of Alternating Current with no Lag.

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With inductance in the circuit, the current lags behind the E.M.F. and the power may be represented by the curve PRQS in Fig. 85. The negative values R and S of the power are due to the fact that the current C is positive when the E.M.F. is negative, or vice versa; hence the actual power is reduced, being the

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algebraic sum of these quantities. When the reactance is very great compared with the resistance, the current lags 90°; and the negative power at T and U, in Fig. 86, is equal S to the positive power at P and Q, so that the actual power is zero. All that occurs is a charging and in the coil, the amount re

discharging of electro-magnetic energy turned being nearly equal to that stored. It should be noted that the frequency of the power curves in Figs. 85 and 86 is twice that

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