K13 Induction currents (Faraday 1831)
Induction of an EMF by a moving
magnet
An electric current generates magnetism. The inverse process is also possible: Generation of a current by magnetism. The fundamental experiment involves a circuit, free of any electro-motoric force (EMF), with a coil A and a galvanometer G (Fig. 566). Push a steel magnet E quickly into the coil. Something unexpected happens: The needle of the galvanometer moves strongly, while the magnet moves, but immediately returns to its position of rest, that is, a current must have flown through the circuit. However, a current always demands presence of an EMF; apparently, the magnet generated an EMF as it penetrated the coil. If you now withdraw the magnet, something similar happens: During this motion - only as long as it lasts - current flows again through the galvanometer, but this time in the opposite direction. The experiment is unsuccessful when you use instead of a magnet an unmagnetic piece of iron. In other word, the magnetism has here an essential role.
The movement of the galvanometer becomes the stronger, but vanishes also the faster, the faster is the motion of the magnet. If you push the magnet slowly into the coil or withdraw it slowly, the deflections of the galvanometer become smaller, but last longer. If the galvanometer is sensitive enough, you see readily that current is always only present, but also only when the magnet is moved. You can then say: In general, moving magnetism generates in a neighbouring coil an EMF- in every nearby conductor. You call it induced EMF and the current, which it provokes, an induction current. The electric currents from power stations are induced currents.
Also in this situation, a solenoid with a current flowing through it can take the place of the magnetic bar. When you insert such a coil A (Fig. 567) and when you withdraw it, a current is induced in B (Volta-induction), first in the opposite direction of the primary current and then in the same direction. The already existing current is called primary, the induced current secondary. But current is generated in B not only when A is pushed into B. When you leave A inside B and intermittently close and open the current through A, every time current is induced in B; during closing, it is in the opposite direction of the primary current (closing current), during opening, in the direction of the primary current (opening current).
Moreover, if you go on keeping the current in A closed, but intermittently strengthen and weaken it, a current is generated every time in the one or the other direction (microphone). The coil with the current flowing through it does not even have to be inside the other coil, in order to induce a current; it is sufficient that it is located nearby. In fact, you do not even have to use a solenoid as every straight conductor A (Fig. 568) induces in another conductor B a current whenever a current grows or fades in it. Moreover: You need not push or withdraw a magnetic bar into or from the coil; you only must have magnetism develop or vanish, even only weaken and strengthen near it, in order to induce currents.
Telephone (Alexander Graham Bell 1847-1922 1877) and Microphone (Hughes David Edward Hughes 1831-1900 1878)
A current is also induced in the coil S (Fig. 569 upper) when the magnetic field in which it is located is changed as a thin iron plate P approaches or distances itself from the magnet N S. Even when it oscillates in front of it (Fig. 569 lower) like a drumhead, that is, only bends towards it and away from it, currents lasting a short time are induced in the coil. Their periodic rise and fall coincide with the periodic motion of the plate.
These facts form the foundation for the telephone. If you conduct the currents, generated in the coil N S around an iron bar E, the bar becomes magnetic and non-magnetic in the same rhythm as the oscillation of the plate. If there is opposite to it an about 0.5 mm thick iron plate P' , it is moved in the same rhythm, in which P moves. If P resonates with a given tone and oscillates, it does so with a certain period, corresponding to the tone, and P' does the same, that is, P' reproduces the tone. Hence the set-up of Fig. 569 can transmit sounds into the distance, that is, serve as a telephone.
The telephone is a device for the conversion and transmission of energy: The energy of the sound waves, which encounter P, is converted into the energy of the oscillation of P. These convert into the energy of the induced current, the current flows through the conductor to E, where its energy performs the work of magnetization of the iron core. It causes thereby the plate P' to oscillate and the energy of the oscillations of P' converts again into the oscillations which the telephonist hears as a sound. By conversion of the energy of sound waves into an electric current, which the wire takes away and keeps together, the telephone transmits the sound over larger distances..
Fig. 569 top presents only the principle of the telephone. In practice, a magnet is also used at E (it is intermittently strengthened and weakened by the input current), because otherwise only sounds would be reproduced at P', that his tele-hearing, but could not also convert sound energy into current energy, in order to be able to telephone. Moreover, a horseshoe magnet is used, in order to agitate the plate with both poles. The currents, generated by the layout in Fig. 569, are very weak due to the resistance of the telephone transmission line, whence the sounds are clear only for a distance of a few hundred meters. This is why the telephone is only used for tele-hearing. Another gadget - the microphone - is employed for telephoning (Fig. 570). It acts inductively by strengthening and weakening periodically an already present current. A board carries two carbon holders a and b between which rests losely a round carbon stick c. The current flows from the battery B through a, c and b to the telephone. If the carbon holders are protected against shocks, the current remains constant and the telephone is silent. However, if the sound plate A experiences a very small shock (ticking of a pocket watch), the transition resistances of the contacts ca and ab change. In this way, the current experiences synchronous oscillations which are heard in the telephone. In order to yet increase the (rather large) sensitivity of the microphone, you connect microphone, telephone and element as shown in Fig. 571. You employ an inductor PS, connect the microphone and the element E to the primary coil P and the telephone through the transmission line to the secondary coil S. Every station receives a microphone for speaking and a telephone for hearing.
During more intense use of the microphone, the contacts of the carbon stick suffer from heating. You increase therefore the contacts either by using more carbon sticks c (roller microphone) or - what was done in 1935 - you place in between the (oscillating) sound plate and a fixed plate, facing it, a large number of small carbon spheres (0.5 - 3 mm) which touch both plates (grain microphone). This increase in the number of contacts reduces heating and - which is important - improves the performance of the microphone.
Spark inductor (Induction coil )
The induction coil depends on Volta induction;
it converts a current of low tension but large intensity into a
current of high tension, but smaller intensity. Its design (Fig. 572) was largely due to Heinrich Daniel Rümkorff 1803-1877
and does not differ significantly from Fig, 567, which was employed to explain Volta induction.
The inducing current of the battery E flows through the
inner - primary - coil p, which contains for
amplification of the induction an iron bar S (more
correctly, a bundle of varnished iron wires). You shut and open
the circuit E all the time by means of an automatic
interrupter. Every time when the circuit is shut or opened, an
EMF arises in the secondary coil, which has one or another
direction depending on whether it arises from shutting or opening
of the primary current. Its magnitude depends on how many more
windings are on the secondary than on the primary coil, since every single
primary loop acts with induction on all secondary loops. The
EMFs, induced in the individual primary secondary coils, add up
and generate between the ends of the coils, connected to the
metal bars A and B, a very large
potential difference. It can become so large that it penetrates
in air distances of 1 m or more. Following
Fizeau, one introduces parallel to a condenser K,
which is to reduce as much as is possible the spark arising as
the current is opened and thereby to ensure the best possible drop-off of the
primary
current .
For example, an induction coil allows to force passage of electricity through very diluted gases, which demands a large EMF - cathode- and X-rays then arise.The inductor has been improved side by side with the development of X-ray techniques - especially the interrupter. One of its most common forms is the hammer interrupter (Fig. 572 right hand side). Opening the current pulls the head of the hammer towards the iron core S in the primary coil, separates the hammer handle from the contact P and thereby interrupts the current. Hence the core S loses its magnetism and the hammer handle - a spring - returns due to its elasticity to its old position, thus recloses the circuit, etc.
X-ray techniques demand much faster interruptions (several thousand per sec) and interruption of larger quantities of energy than the hammer can handle. Both are provided by the mercury-ray-interrupter (Boas) and the electrolyte interrupter of Artur Wehnelt 1871-1944 1899. In the first, a very fast, turbine like device (driven by an electric motor) sucks mercury vertically upwards and flings it horizontally as a (clock pointer like) very fast rotating jet against the inside of a ring of narrow conducting and isolating segments. The current flows from the mercury reservoir through the jet to the ring of segments.
The electrolyte interrupter is a decomposition cell with 30% sulphuric acid as conductor, a large plate of lead as cathode and a platinum wire, a few millimetres thick, as anode. Eletro-lytic formation of detonating gas and evaporation of the acid at the anode(due to the large current density at the small wire) interrupt the current for a brief instant. The extensions, which arise thereby instantaneouly, penetrate the vapour and gas layer as a spark, separate them violently, the acid closes again the current, etc.
Transformer for alternating current
The transformers of electrical
engineering are like inductors. You convert in them alternating
current of low tension into high tension current and vice
versa - the first, in order to transmit most economically
electric energy from the power
station to very far
away users, the second, in order to reconvert at
the user the dangerously high tension into safer low tension.
Transformers do not require for this purpose interrupters, the
alternating current induces already by its pulsation. The transformer is to convert the
energy with the smallest loss of energy. In essence, you achieve
this with closed iron ring cores, which hold the magnetic lines
of force together and increase thereby the induction. Fig. 573 shows the simplest type of transformer.
The primary coil A and the secondary coil B are
wound separated from each other on a ring of soft iron (high
permeability, small coercive force). As in the inductor, the
primary windings are a few thick wires, the secondary windings
many relatively much thinner wires. Current oscillations in A
cause a continuous change of the density of the lines of force in
the iron ring and thus induce EMFs in B. The greater is
the number of windings in the coil B compared with those
in the coil A, the larger is the induced EMF, but the
lower the current which can be drawn from B. The
fraction of the number of primary windings over those of the
secondary windings is called the step ratio
of the transformer. For example, if it is 100, the mean effective
tension is raised
ideally by a factor of 100, the current intensity, which can be
take from the secondary coil, is reduced to 1/100 of the current
flowing through the primary coil. Thus, if V1
and I1 are the effective tension and
intensity of the current in the primary coil, then for a step
ratio of 100 the secondary tension in the ideal case is V1
= 100/V2 and I2
= (1/100)I1. Hence, as had to be
expected, V1I 1=V2
I.2 , since in both cases the product of
tension and intensity are the measure for electric energy,
because you can draw from a 
transformer working without losses the primarily
expended electric energy secondarily. In reality, a small part of
the energy is always lost and converted into heat.
Like the currents of the alternating current machine (50 - 60 per sec), you can also transform the currents of a discharging Leyden jar (millions per sec) to higher tension (Tesla 1893). The transformer for this purpose differs essentially from the one for small numbers of oscillations: It does not contain iron, the primary coil has a few windings of thick copper wire, the secondary coil - it is the inner one - has very many thin windings of well insuolated material. Both are immersed in well insulating oil, since otherwise the insulation of the coils from each other is not sufficiently large to stop discharges between them. The tension, supplied by a Tesla transformer, is huge (by the way, physiologically harmless!) and leads to highly characteristic effects (glowing phenomena in evacuated glass vessels without electrodes). They only have theoretical value.
Condition imposed on the lines of force for induction of EMF
The various conditions under which induction currents arise have a single source. In order to determine it, consider the magnetic field and its lines of force. In Fig. 574, N and S are the poles of a magnet, K a closed frame or ring made out of wire. Let the field be uniform, its strength be determined by the number of lines of force, which pass through 1 cm² of an area perpendicular to the lines. If the frame K is parallel to the lines, no (0) lines of force pass through it, if it is perpendicular to them, the largest possible (maximum) number of lines pass through it; in every other position, a number of lines between 0 and the maximum pass through it. As long as the angle between the plane of the frame and the direction of the lines of force is the same, also this number of lines of force does not change. While you bring the frame from one position into another one, in general, the number of lines of force passing through it changes; only as you displace it in its plane, the number does not change, because then the angle between the plane and the direction of the lines of force remains the same.
We will now reformulate the law of induction as follows: If you move the frame so that the number of lines of force surrounded by it changes, an EMF arises during the motion; if the number of lines of force remains unchanged, no EMF is generated. Instead of moving the frame in the field of the magnet, we can move the field of the magnet and hold on to the frame; we can also hold fixed both parts, the frame and the magnet, and change the number of lines of force by weakening and strengthening the magnetic field. Both bring about that at one time less, at another time more lines of force pass through the frame. Hence we can formulate the law of induction as follows: As long as the number of lines of force passing through the frame changes, an EMF is generated.
We wonder now whether it is essential during the induction experiment that the wire forms a frame. Imagine the frame to be very large, so that only a small part of it, say, its short side enters the magnetic field, while the other sides lie by far outside the active field. If we now displace the frame in an arbitrary direction - but not parallel to the lines of force! - all our assumptions regarding the generation of an EMF are met. But the EMF is now not like earlier induced in the entire frame, but only in that part of it which enters the magnetic field. Hence the frame itself has not an essential role; for generation of EMF, only a section of a conductor entering the field is required. Hence we can further generalize the law of induction and say: An EMF is generated in every conductor as it intersects lines of force or while it is intersected by lines of force.
Direction of induced current (Lenz's Rule)
We return to Fig. 574. If we turn the frame about its axis,
we change all along the number of lines of 
force passing through it and an EMF is
generated. In closed frames, it induces a current. Now cut
one side of the frame, solder to the free ends wires, lead them
along the axis out of the field and connect them to a volt meter
to measure the induced EMF, while the frame turns - quickly or
slowly, clockwise or anti-clockwise. Our first question concerns the direction of the EMF,
that is, the direction in which the current flows through the
volt meter, if we turn the frame in the direction of the curved
arrow in Fig. 574. This question is answered by the Rule
of Lenz 1834, and indeed in a very general form,
applicable to all induction processes: The induced current has always the direction in which it impedes
by its magnetic action on the present field the motion, to which
it owes its induction.
For example, turn the frame in the direction of the curved arrow in Fig. 574 by 90º. Then a current will flow through the frame which we can imagine to have been replaced by a small magnet, directed perpendicularly to the plane of the frame. According to Lenz's Rule, the current has a direction such that the motion of the frame is impeded. This means that the small magnet must have the direction that its North pole faces the South pole S in Fig. 574, its South pole the North pole of the inducing magnet, because only in this way will it try to stop the enforced motion. Hence the current must flow in the direction of the arrow drawn on the lower side of the frame in Fig. 574.
Another example is offered by Fig. 566. If we push the magnet E, with its North pole facing forward, a little bit into the coil, a current is induced in it, which must be directed so that the upper end of the coil becomes the North pole, because only then will the motion of the magnet be impeded. When we pull out the magnet, then there must arise at the upper end of the coil a South pole, which wants to hold on to the magnet and thereby impede the motion.
The mechanical work, which you must perform in order to overcome this resistance, is not lost; it reappears as the energy of the induced current. When the coil circuit (Fig. 566) or the frame (Fig. 574) are not closed, the induced EMF cannot perform work, since no current develops. Then you do not obseve an impediment of the moved conductor or the magnet is observed.
The dynamo has developed in steps from the wire frame, rotating in a magnetic field (Fig. 574) - more accurately - the armature of the dynamo. In view of their historical importance, we refer to the double-T-armature of von Siemens 1857, the ring-armature, named after Zénobe Théophile Gramme and already invented 1860 by Antonio Pacinotti 1841-1912, and the drum-armature, developed out of the double-T-armature of Friedrich Hefner-Alteneck 1845-1904 1872. In the first of these (Fig. 575), the wire frame is a coil of insulated wire, wound around along an iron cylinder with I-formed cross-section, parallel to the axis of the cylinder, imbedded in groves. In the Gramme-ring (Fig. 576), the frame or better the system of frames comprises a large number of interconnected coils, wound around an iron ring; they rotate with it in the field. The drum armature (Fig. 577) avoids like the Gramme-ring the strong pulsations of the EMF of the T-armature and exploits every coil uniformly for induction: The coils are wound on to a cylindrical iron drum, parallel to the axis of the cylinder and, like in the case of the double-T armature, are embedded in groves (in the Gramme-ring, the halves of the coils, lying on the inside of the ring, act less than those on the outside rotating closer to the poles.) The field magnets in the dynamos have been formed correspondingly to the different forms of the armatures; the iron cores on which the armature coils are wound, strengthen the effect of the field on the coils. The double-T armature was used 1935 only for the simplest technical applications (signals), the Gramme-ring has only historical value, since it was the starting stage of today's dynamo. More or less, the drum armature was in 1935 the anchor of dynamos.
We ask now what is the magnitude of the induced EMF and examine first whether there arises a difference through use of conductors of different thickness and different materials (copper, iron, etc.). For this purpose, we make equally large frames out of different types of wire and place them in the magnet field of Fig. 574. Every time, we rotate the frame at the same rate by the same angle and measure simultaneously with the volt meter the induced EMF. In this way, we discover: The thickness and material of the wire has no effect. This independence of induction from the nature of the current had already been demonstrated by Faraday; naturally, it applies only to the EMF; the strength of the current would differ in a closed frame, depending on the line resistance.
We ask next: How does the induced EMF change with the field strength H and what is the effect, if the conductor of length l intersects the lines of force at the angle y with the velocity u? The important aspect is shown by the layout of the following experiment (Fig. 578). You place into a uniform magnetic field (with straight, parallel and equidistant lines of force ) a plane, rectangular frame klmn, made out of metal rails. Rail m can be pushed along l and n parallel to itself, so that the length of l can be changed. At a definite position, you fix m, so that l is constant. Rail l itself can be pushed along k and m; it is the conductor, which we will move during the experiment. A voltmeter is introduced at s; it measures through the conducting rails k and m the tension at the ends of l. - We place this frame into the field so that the lines of force, which may run horizontally, are perperpendicular to it (Fig. 579 a, b). As long as l is at rest, the electrometer does not move. However, if we move l in the direction of the arrow starting at P (Fig. 579 b), so that it intersects the lines of force perpendicularly, the electrometer moves, that is, an EMF is induced in l.
How large is the EMF under these special conditions? Denote the field intensity by H, the length of the conductor by l and its velocity by u. The experiment tells us that the induced EMF is proportional to H·l·u. Hence, if you double l and do not change H and u, the EMF is also doubled. If you also double u and only leave H as it is, the EMF is four times as large; finally, if we also double H, it becomes eight times as large. - However, the EMF does not only depend on H, l and u. In the experiment, the motion and position of the conductor are at right angle to the lines of force. However, if you place the frame into the field so that while l is perpendicular to the lines of force (Fig. 579 c), the plane of the frame is at an acute angle y, that is, l moves at an inclination (say, from the right hand side above to the left hand side below), not all of the velocity u acts, but only its projection on the dashed line, along which l moved previously. The projection is ucosy, and the expression H·l·u becomes H·l·u·cosy.
If you return the frame to its original position and rotate it about its vertical axis, so that, seen from above, it has the direction to the lines of force as in Fig. 579 d, not the entire length l of the conductor acts, but only the projection of l on the dashed line, at which it was initially (Fig. 579 a). Denote the angle between the conductor and the lines of force by j, so that this projection is lsinj and H·l·u becomes H·lsinj·u. If you now move the frame so that it diverts from the first experimental set-up (Fig. 579 a) by the direction, characterized by the angle j as well as the angle y, you arrive at the most general expression for the induced EMF: H·lsinj··ucosy. You see that it vanishes when j =0º or when y=90º, that is, when l lies parallel or moves parallel to the lines of force. Its maximum occurs when j is a right angle and y is zero, that is, in the case of Fig. 579 b, which we will consider in more detail.
The expression H·lsinj··ucosy, to which the induced EMF is proportional, is equal to the number of lines of force, intersected by l. For example, if j·= 90º, y = 0º, l·u (cm²) is the rectangle, covered by l while it displaces by u cm. Since the field strength is H, you have H lines of force per 1 cm², l·u·H lines of force per l·u cm²; this number of lines of force is indeed equal to the number, to which in this case the induced EMF is proportional.
If l moves perpendicularly to the lines of force along itself with the velocity u and H is the field strength, then an EMF arises in l proportional to H·l·u. You can express H, l and u in cm, g and sec, hence also the EMF, that is, you can measure it in absolute units (like the current strength). Now let the field have the intensity H = 1 Gauß, the conductor the length l = 1 cm and the velocity be 1 cm/sec in the direction j = 90º and y = 0º. The magnitude of the EMF, which then arises in it, will be called the absolute unit of EMF. - It is so small that, for example, the EMF of an Daniell-element is 107 million times this unit. In order to practically employ this unit would be like expressing kilometre long distances in hundredths of millimetres. Hence the practical unit Volt has been introduced, which is 100 million times the absolute unit. You define: One Volt equals 108 absolute units of EMF. By this definition, under the above conditions, the induced EMF E = H·l·u absolute units or H·l·u·10-8 Volt.
The Volt is also the legal unit for the EMF. But you would encounter almost insurmountable problems if you wanted to practically derive the exact value of the Volt unit by induction measurements. Hence the legal definition of Volt has been based on quite different considerations. You employ the relationship which, according to Ohm's law, holds between the three units for current, tension and resistance. The unit of current - 1 Amp - and the unit of resistance - 1 Ohm - are relatively simply realized quantities, whence one Volt has been defined: One Volt is the EMF which evokes in a conductor with the resistance of 1 Ohm a current of 1 Amp. - Next, we consider the unit of resistance, which also is required for the realization of the Volt unit.
Electric resistances and measurements of resistance
A conductor has the absolute unit of resistance, if the absolute unit of EMF generates in it the absolute unit of current.. The unit, defined in this manner, is very small compared to resistances which occur in practice. Hence one has introduced as practical unit the resistance, which is 109 as large - the Ohm. However, in order to avoid absolute tension- and current- measurements during the control of the in practice employed Ohm-normal, which demand considerable experimental effort, the Ohm has been defined by law as follows: The Resistance of a mercury column of length 106.3 cm and 1 mm² cross-section. (This length of 106.3 cm takes this legal unit as close as possible to 109 times the absolute unit.)
The determination of the electrical resistance of a conductor in Ohm is among the many routine measurements: You compare it with a resistance the magnitude of which in Ohm is known. You employ for this purpose a set of resistances, which is analogous to the set of weights, which you employ for a comparison of masses, that is, for weighing. Wire coils B, wound bi-filarly (cf. Fig. 582), are so dimensioned, that they have a given resistance, say, 10, 5, etc. Ohm, are interconnected (Fig. 580). With the aid of such a set of resistances, you can introduce an arbitrary resistance into a circuit b; for example, in Fig. 581, 1275 Ohm, by pulling out the plugs E at 1000, 200, 50, 20 and 5, leaving all others where they are (that is, short circuiting all others). The current flows everywhere through the heavy rails except at the locations, where the plugs E were removed and it must therefore flow through the thin wires.
Self-induction. Excess-current
A conductor, through which flows current, lies always in a magnetic field - the field, aroused by the current. Every change of its intensity - including especially switching on and off - changes this field and the change of the field of force acts on the conductor, through which the current passes: It induces in it an EMF (Faraday). Because this induction acts back on the conductor, it is referred to as self-induction, and the arising current is called excess-current.
The excess-current is always directed so that it impedes the change of the current, to which it owes its generation (Lenz's Rule). For example, if you close a current or strengthen it, it does not reach immediately its full strength, or, if you strengthen it, it does not reach immediately its full intensity, but only gradually; the excess-current delays its growth; if you cut a circuit, the current does not vanish instantly; at sufficient strength, it creates at the point of interruption a strong spark - thus, the excess-current also delays the vanishing of the current. Self-induction is strongest when the conductor has many, densely spaced windings, all of which act in the same direction on the external field, like in the case of a solenoid (Fig. 542) and especially when the solenoid surrounds an iron core like in the spark inductor (Fig. 572), because it changes especially strongly the field of lines of force. You suppress self-induction of a coil by forming the windings as in Fig. 582, that is, bi-filarly. The windings then conduct the current in neighbouring windings in opposite directions so that their magnetic fields cancel each other. In coils with large resistances, their charge capacity nevertheless disturbs. In order to make it as small as possible, you wind coils from 500 Ohm upwards in the manner shown in Fig. 583: You wind narrow layers of of a few windings and change after every layer their direction.
Also, during self-induction, the magnitude of the EMF, induced in unit time, depends on the number of lines of force, intersected by the induced conductor during unit time. It depends on how fast changes the current strength, since the number of lines of force changes with the strength of the current. That is why it is much larger during opening of a circuit (opening-excess-current) than during closing it (closing-excess-current). In fact, during closing of a circuit, the arising current is impeded by the EMF of the self-induction; it rises only slowly from zero to its full strength, it has at the instant of closing the strength zero, but an instant later by no means the full strength. It is different during opening of a circuit. Immediately before, its strength is full, an instant later zero. For this reason, the EMF of the opening-current is many times larger than that of the closing current - it may be so large that it bridges the opening location by a spark in which the separated ends of the conductor melt. - The bridging extends the duration of the primary current, makes the drop (to zero) less steep and decreases thereby the induction tension of the current opening. Hence one lets in the spark inductor the electricity, which would discharge in the spark at opening, flow into the condenser, from which it escapes at the next closure into the circuit.
Moreover, the form of the conductor has great influence. If you use the same wire once stretched out linearly, another time as solenoid, and change both times the current equally quickly, the solenoid may have hundred times, even one thousand times larger induction than the straight wire. If you wind the wire bi-filarly into a spool, no induction occurs. Every coil, in fact, every conductor is characterized in this respect by the ratio between the EMF of the excess-current and the rate of change of the current. This ratio is the self-induction-coefficient; it is defined as that EMF, which is induced in the conductor itself, when the the current flowing in it changes in unit time by the current unit. If it changes in 1 second by 1 Amp and the coil is such (in form, length, cross-section and number of windings) that the EMF of the excess-current is 1 Volt, one says: The self-induction-coefficient of this coil is one (1) Henry. This measure is enormously large; in practice, you reckon with 1/1000 Henrys.
The self-induction normal at the German Government Installation, the nominal value of which is 1 Henry, is uni-filarly wound on a marble cylinder with a diameter of 89 mm and height of 33 mm. The thickness of the wire is 0.5 mm, the number of windings 2894, their internal diameter 89 mm, their external diameter 155 mm, their height 33 mm, their resistance 94 Ohm.
The material of the wire, like in the case of induction, does not influence the self-induction, but it must not be ferro-magnetic, that is, iron, nickel, cobalt.
So far, we have only spoken of wires. But what applies to wires, is also
valid for metal sheets and other components made out of metal.
The direction of the induced currents always corresponds to Lenz's Rule. Since sheets contain an unlimited number of
current tracks, induced EMF provokes in
them always currents in different
directions and of different strengths (eddy currents). Their energy converts into heat which
raises the temperature of the conductor. For example, the primary
coil of a spark inductor is rapidly opened and closed in
sequence. In order to limit as much as possible the eddy currents
( currents of Foucault) which arise in the magnetic core, you
do not use as core a solid bar, but a bundle of thin wires,
covered with varnish to insulate
them from each other.
At times, you compose machine parts, which could be manufactured
as a single unit, out of layered sheets, in order to avoid eddy
currents by the partitioning of the metal. You subdivide the body
so that the interfaces, for example, those of sheets, are
perpendicular to the direction in which the currents would run.
If you surround a suspended magnetic bar a, which can rotate in a horizontal plane, by a fixed. thick, undivided metal housing, for example, out of copper, and cause it to oscillate, it induces in the wall of the housing currents which, according to the Rule of Lenz, impede the motion of the magnet. You can suppress in this manner the vibrations of the magnetic needle in galvanometers, so that the needle moves to its new position of rest without oscillations (aperiodically). - If you move a metal sheet between the poles of a very strong magnet to and fro, you feel the interaction of the magnet with the eddy currents in the sheet as braking action. You generate such an action intentionally in the eddy current brake of a dynamometer, which serves the same purpose as Prony's brake dynamometer: The place of mechanical friction is then taken by the resistance generated by eddy currents. You let a massive metal disk (like a flywheel) rotate at its circumference past a strong electromagnet, in order to generate this resistance. You measure the resulting load in the same way as in Prony's brake dynamometer.
The currents of the electric industry (illumination, energy transfer, metallurgy) are always induction currents. They are generated by a dynamo. In principle, the basis of every dynamo (in many different realizations) is a system of conductors, which rotate in a magnetic field at rest, as illustrated by Fig. 574, or also a resting system of conductors, about which a magnet field rotates. Hence, in principle, a dynamo is an arrangement by which with the aid of the rotating relative motion of two inductively linked systems mechanical energy is converted into electric energy.
Basically, it is important that a periodic alternating tension (cf.
below) arises by these means, illustrated by Fig.
574, whence the prototype of the dynamo is the alternating
current generator. However, with the aid of a switching device
within the machine - the commutator
k - on which slip two contact plates (brushes), you can
force the current to have always the same direction (Fig. 575), that is, convert the machine
into a generator of direct current.
You generate the field always with electro-magnets, which demand direct current. The direct current dynamo can generate its field on its own. According to the electro-dynamic principle, discovered by von Siemens 1867, every iron has naturally enough magnetism to induce - although only weak - current. If you employ initially this current to amplify the natural magnetism of the iron and to induce by the amplified magnetism stronger current and continue with this alternating action, you eventually bring the iron to magnetic saturation and then receive a large induced current. The field of an alternating current dynamo must be excited by a specific direct current dynamo.
As explained earlier, the magnitude of the induced EMF depends on the strength of the field, on the length of the conductor in the field, on its velocity and on the velocity, at which the number of intersected lines of force changes. A perfection of these details are purely technical tasks.
Initially, the direct current dynamo dominated in electrical
engineering, because it seemed as if this was necessary for arc
lamps and energy transmission. Since 1900, the alternating
current dynamo has dominated, because its construction for very 
large performance and especially for high tension is
simpler; also since then illumination by arc lamps offered no
problems and the ability of the alternating current to be
transformed made it suitable for long power lines. The problems
of energy transmission were overcome above all by the rotary
current generator, which bypasses all other motors by its
efficiency and economy. The underlying physical principle is the
rotary field of Farraris, a rotating
magnetic field, which you can generate by combination
of several alternating currents which are displaced
with respect to each other in
a certain manner.
If you turn the armature of a dynamo without current (without a circuit attached), the continuation of the rotation demands much less energy than is required when current passes through the machine, in correspondence with the law of conversation of energy. In a closed circuit arise induction currents in the windings of the armature, which as a result of their electro-dynamics action tends to impede the motion of the armature in the magnetic field (Lenz's Rule). You can overcome this resistance to the motion by amplification of the rotatory moment at the axis of the armature. Hence an armature (at rest), if a current is sent through its coils (form another source of electricity), must start to rotate in the magnetic field, and indeed in a direction which, in the presence of equal current direction, is opposite to the rotation of the armature of the dynamo. This is the basic idea of the electric motor. (Dynamos convert mechanical energy into electric energy, motors convert electric energy into mechanical energy.)
While the armature of the electric motor rotates, its magnet field acts on it like in a dynamo, that is, it induces in it a current, which is oppositely directed to the current supplied to the motor. Also this impeding action is demanded by the law of the conservation of energy. If you feed from a battery with the EMF E a current of strength J while you hold the armature fixed and then release it, the current drops when the armature has reached its full velocity, say to J0. The loss of energy E(J - J0) is the energy of the induced counter current. Since before the start of rotation the current strength is much higher than afterwards E(J - J0), you may not, when you start the motor, feed immediately the full current strength, since then the coils will be overloaded. You can only supply the current in steps, depending on the rate of rotation, to the maximum value (employing a topping resistance (starter).
A current, the strength and direction of which changes constantly after equally large time intervals in the same manner that it has graphically a more or less pronounced wave form (Fig. 585), is called an alternating current. The time interval T, after the passage of which the current strength has the same magnitude and its direction is the same, is called its period. The change in time of the current corresponds in the simplest case to the sine function. In Fig. 585, the instantaneous value i of a one-wave sine current has been entered as ordinate, the time t as abscissa; the equation of this curve is: i = Jm sin w t, where Jm is the peak value - the amplitude of the current - and w a constant. It has the following significance: At the end of the period T, i must have again the same magnitude and the same direction as at the start of T. According to the above equation for i, this is so if wT = 2p, that is, w = 2p/T. The number 1/T tells us how many periods there are during one second; it is called the frequency and denoted by n, whence 1/T=n and w=2pn. w is called the circular frequency; it is the number of periods during 2p seconds. The current strength (effective current strength) is the square root of the mean of the values of i². For the current, represented by the sine curve, the effective current strength is Je = Jm/(2)1/2 = 0.707·Jm. The tension varies periodically like the strength, the effective tension is Ee = Em/(2)1/2 = 0.707·Em. Fig. 585 presents the graph of the alternating current for i: Let the straight line of length Jm rotate in the plane of the drawing as radius about the centre of the circle and project, as shown, the radius at each angle on to the vertical. You obtain thus the curve with the equation i = Jm sin w t. The angle w t = 2p nt is called the phase of the current.
If a circuit only contains an Ohm resistance, that is, a resistance which straight line cables have, then the current and tension in the phase agree with each other; they attain their extreme value, pass through zero, reverse their direction simultaneously, as is shown by Fig. 586.It differs when the circuit is a coil with self-induction. The periodic change of the alternating current induces then in the coil a secondary current, which acts in the opposite direction, whence self-induction prevents the current from attaining with the tension its maximum value and passing through the zero value simultaneously - it delays the current strength; the tension precedes it by a certain angle - the phase angle j, as is shown in Fig. 587. This angle lies between the current and tension vectors, if we employ the same diagram, in order to present simultaneously the curves for the intensity and tension; j indicates the distance by which the two curves are shifted with respect to each other.
How large is the
effective tension which generates in a circuit with R
Ohm resistance and L Henry inductivity an effective
current of Je Amp? This tension is: Ee = Je(R² + w²L²)1/2 Volt. The
phase angle j is given by tanj = wL/R.
If R = 0, tanj = or j = p/2, that is,
he phase of the current lags behind the tension by 90º. This is almost so for high frequencies in coils with
very high inductivity and very small resistance.
Thus, Ohm's law is valid for alternating current only when L = 0, for only then J = E/R (and only then that is, the current is in phase with the tension). Self-induction acts like a resistance. It j = 0, depresses the current intensity below the value according to Ohm's law. Even if the Ohm resistance would be zero (R = 0), the current is not, as would be the case for direct current, infinitely large (shorted circuit), but is equal to E/wL. Lw is called the inductive resistance of the conductor.
For example, you can see with an incandescent lamp, which is inserted after a coil and gives a bright light, that self-induction acts like a resistance. It loses brilliance as the self-induction of the coil is increased by pushing an iron core into it. Because of this resistance, electrical engineers employ this effect (in order to reduce the current strength in alternating current circuits) instead of induction free resistances throttle coils with large inductivity and small Ohm resistance. They weaken the current by means of the electro-motoric counterforce, which arises in them, but consume almost no energy, because their Ohm resistance is so small that the Joule heat generated in them is of no consequence. They only consume as much energy as the demagnetization of the iron core in them demands.
The phase shift of the current with respect to the tension complicates the computation of the performance of alternating current. It is determined by multiplication of the effective values of the current strength and tension with each other and adding the factor cosj, whence U = Ee·Je ·cosj, where tanj = wL/R. - Hence, depending on the magnitude of cosj - the efficiency factor - the current performs over a given time interval different amounts of work. If j = 90º and cosj = 0 (in reality, j is never exactly 90º), that is, the curves of the current and tension are displaced with respect to each other by 90º, they have in the first quarter period the same sign, in the second quarter opposite signs, in the third quarter again the same sign, etc.; the performance of work of every ensuing quarter period therefore annuls the preceding one - the mean value Ee·Je is the same for every quarter period, but alternatingly positive and negative, every following quarter cancels the preceding one. In this case, the current does not perform work (it yields no Watt): It returns the work, which it has taken during a quarter period from the current generator, during the next quarter period to the circuit. For example, such a current is an alternating current, which flows through an iron-free self-induction coil with minimal Ohm resistance. (This is the reason why throttle coils with thick wire consume also for large EMF of the alternating current source no energy, although they weaken the current.)
A phase shift of
current and tension also occurs in an alternating current
circuit, if we insert instead of an self-induction coil a
condenser. It charges itself, then discharges, charges again in
the opposite direction, etc.: Hence a current flows in the
circuit (in a direct current circuit, a condenser cannot maintain a current), whence in
an alternating current circuit a condenser forms a secondary
current source. We obtain again two curves, shifted with respect
to each other, but this time the tension lags behind the current: It reaches its maximum later, passes
later through zero than the current, is during one part of the
period oppositely directed to the current, while it has the same
direction during another part. If the condenser has the capacity C,
the effective tension Ee = Em(R² + 1/w²C²)1/2. The phase
angle is given by tanj =
-1/wCR. For R = 0, tanj = -or j =
-p/2, that is, the phase of the current is
by 90º ahead of the tension in the condenser. (This condition is
met exactly in the case of good air condensers.)
If an alternating current circuit contains simultaneously the inductivity L and the capacity C, then the phase angle j is given by tanj = 1/R(wL - 1/wC). If wL=1/wC, that is, w = (1/LC)1/2, that is, the period of the source of alternating current is related to the inductivity and capacity of the circuit in this special manner, something remarkable happens. For the sake of emphasis, we concentrate on the work performed by the current in a conductor, in which the current and tension are out of phase. In this special case tanj = 0, whence the current and tensions are out of phase by a quarter period: Hence the current is Wattless, that is, it returns in the next period the work, which it has consumed during any quarter period. The coil and condenser are, so to say, during one period energy consumer, during the next period energy supplier, etc. The most important aspect is: The coil (the condenser) draws, while the condenser (the coil) yields, because the tension and current are in the coil and the condenser out of phase (+p/2 and -p/2) in opposite directions. Effectively, the condenser and coil alternatingly toss the energy to each other: During one quarter period, the coil consumes the energy, returned by the condenser, during the next quarter period consumes the condenser the energy returned by the coil, etc.: The electric energy flows to and fro between them - one says that it oscillates; it is in this sense, that one speaks of electric oscillations. An electric oscillation comprises a periodic conversion and reconversion of electro-magnetic energy from the one form into the other: The electro-magnetic energy, present in the circuit, finds itself at one definite instant totally in the coil as energy of its magnetic field, a quarter period later totally in the condenser as energy of its electric field, again one quarter period later totally in the coil, etc.
If 1/wC = wL, the energy balance of the coil and the condenser are equally large; each gives as much as it takes, the mean coil energy and the mean condenser energy are the same. Only as much of the electric energy is lost to heat as corresponds to the Ohm resistance of the circuit. Hence, if we make the Ohm resistance effectively equal to zero, there is no need for the source of current for the maintenance of the current. Once it has been started, it can be removed without the interchange of energy between the coil and the condenser stopping - like a pendulum, which swings without friction, never stops its motion and does not require external excitation. (This cannot be realized perfectly, since there does not exist a current source totally without resistance nor a completely frictionless pendulum.) If you do not remove the source of current, the amount of energy in the circuit increases continuously, since that already contained in it remains without a loss and the current supply sends continuously more energy. This results in overloading, which is dangerous for the conductor and demands some protective device.
The special aspect of the case wL = 1/wC lies in the relationship between L and C which, on the one hand, characterize the circuit and the magnitude of w, on the other hand, the source of current. On the one hand, w² = 1/LC, on the other hand, L w = 2p/T, where T is the period of the source of alternating current, whence T = 2p(LC)1/2 seconds, provided L is given in Henrys and C in Farads. hence we conclude: If the current has the period T = 2p(LC)1/2 sec (the period number 1/2p(LC)1/2), the energy contained in the circuit flows between the condenser of C Farad capacity and the coil of L Henry inductively during T = 2p(LC)1/2 sec once to and fro, that is, it converts itself once entirely into magnetic energy of the coil and again into the electric energy of the condenser. You call 2p(LC)1/2 sec the period of oscillation of the electro-magnetic energy, and indeed of the characteristic oscillations of this circuit, specified by L Henry and C Farad. With this period, which in this special case is also that of the source of the alternating current, the energy oscillates the energy between the condenser and the coil. However, the source of alternating current is not involved in the maintenance of the oscillation (whence it is the characteristic oscillation and characteristic period of the circle); also when it is switched off, the energy continues to swing in the circuit, consisting of the coil and condenser. This is why one says: The circuit is in resonance with the generator. It behaves like a tuning fork which begins to sound by resonance and continues to sound, also when the first source of the tone has ceased to act. The expression resonance is the more justified, as the circuit involving the inductivity L Henry and the capacity C Farad need, in order to oscillate, only lie in the field of the alternating current source with the period T = 2p(LC)1/2 sec without being connected with it.
However, the energy oscillates only when the Ohm resistance R of the circuit lies below a certain limiting value. Otherwise, the oscillation is aperiodic (like that of a pendulum in a very viscous fluid). A theoretical calculation shows that
For R = 0, T = 2p(LC)1/2 . Hence, if
the Ohm resistance R is of no
consequence, the simple formula applies. Otherwise, it depends on
whether R²/4L² 
1/LC, that is, R 2(LC)1/2
(when T does not have a finite value, there occurs no
oscillation whatsoever) or whether R < 2(LC)1/2
(when an oscillation occurs with a period which depends on the
resistance). If the resistance R contributing with
respect to 2(LC)1/2 is effective, it extends
the period of the oscillation.
Damping cannot be separated from electric oscillations; in order to maintain the oscillations, the energy lost by damping must be supplied all the time. Also maintenance of resonant oscillations demands the source of alternating current in the circuit which also contains the coil and condenser; however, it need only supply the energy lost by damping.
On the other hand,
the energy also swings between the coil and condenser when Lw
1/wL.
However, these forced oscillations of the circuit, for the
maintenance of which the source of current is indispensable - the
circuit behaves like a pendulum, which does not only oscillate
due to Earth's attraction (free), but due to another force which
forces it into oscillations other than its natural ones.
Most sources of alternating current yield currents, which consist of a fundamental oscillation and harmonic overtones. You can keep away from the circuit the overtones - all of them or selected ones - by introducing induction coils and condensers of definite proportions in a certain manner (Karl Wilhelm Wagner 1901-1977). Figs. 589-591 show a throttle chain, a condenser chain and a filter chain: The first suppresses all oscillation for which R > 1/(LC)1/2, the second all for R<1/(LC)1/2; the first removes from an alternating current overtones, the second a higher frequency from a lower one, the third only lets pass a definite range of frequencies, bounded above and below. The filter chains of Wagner serve as sound filter in an electro-acoustic system comprising a micophone and loudspeaker. Fig. 592 (below) shows the striking action of a filter chain for the suppression of overtones (above) of a source of current with a distorted tension curve (or a composite sound curve).
Telephone communications over large distances are affected by the electric properties of their lines. The alternating current, which the microphone feeds into a line, experiences increasing damping as the line becomes longer. This can be reduced by reduction of the resistance of the line, that is, by thickening the line; yet, sufficient strengthening of long lines becomes impossible due to cost. However, the damping can be reduced in quite a different way. The theory yields for the damping an expression, in which the term (R/2)(C/L)1/2 is decisive and a reduction of which is required. R is the resistance of the line, C its capacity, L its self-induction (in Henry). The building of lines is such that they have comparatively large capacity and small self-induction, whence, in order to decrease the value of (C/L)1/2, one must increase the self-induction L. For overland lines, this is done by the method of Michael Pupin 1858-1935 1900: You introduce coils with large self-induction into the lines at certain accurately computed distances; underground cables employ apart from Pupin-coils the method of Krarup (1902): You raise their self-induction by uniformly wrapping around the conductor a layer of fine iron wire. Only the Pupin-coil and Krarup-cable have made world-wide telephone communication possible.
