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SHORTAGE OF ELECTRONS

-ELECTROSTATIC FIELD

SURPLUS OF ELECTRONS

i- CHARGING CURRENT

Figure M SIMPLE CAPACITOR

Illustrating the Imaginary lines of force representing the paths along which the repelling force of the electrons would act on a free electron located between the two capacitor plates.

a staudpipe having a cross section of 1 square inch and a 2 td. capacitor to a standpipe having a cross section of 2 square inches. The charge will represent a given volume of water, as the charge simply indicates a ceitain number of electrons. Suppose the water is equal to 5 gallons.

Now the potential energy, or capacity for doing work, of the 5 gallons of water will be twice as great when confined to the 1 sq. in. standpipe as when confined to the 2 sq. in. standpipe. Yet the volume of water, or charge is the same in either case.

Likewise a 1 fd. capacitor charged to 1000 volts possesses twice as much potential energy as does a 2 fd. capacitor charged to 500 volts, though the charge (expressed in coulombs: Q = CE) is the same in either case.

The Unit of Сорос- If the external circuit of itonce: The Forod the two capacitor plates is completed by joining the terminals together with a piece of wire, the electrons will rush immediately from one plate to the other through the external circuit and establish a state of equilibrium. This latter phenomenon explains the discharge of a capacitor. The amount of stored energy in a charged capacitor is dependent upon the charging potential, as well as a factor which takes into account the size of the plates, dielectric thickness, nature of the dielectric, and the number of plates. This factor, which is determined by the foregoing, is called the capaci-tanceoi a capacitor andis expressed in farads.

ТЪе farad is such a large unit of capacitance that it is rarely used in radio calculations, and the following more practical units have, theiefore, been chosen.

1 *сго-шсго/лга</= 1/1,000,000 of a micro farad, or .000001 microfarad, or crofarads.

1 micro-microfarad = one-millionth of one-millionth of a farad, or 10~ farads.

If the capacitance is to be expressed in microfarads in the equation given for energy storage, the factor С would then have to be divided by 1,000,000, thus:

С X E

Stored energy in joules =-

2 X 1,000,000

This storage of energy in a capacitor is one of its very important properties, particularly in those capacitors which are used in power supply filter circuits.

Dielectric Although any substance which has Materials the characteristics of a good insulator may be used as a dielectric material, commercially manufactured capacitors make use of dielectric materials which have been selected because their characteristics are particularly suited to the job at hand. Air is a very good dielectric material, but an air-spaced capacitor does not have a high capacitance since the dielectric constant of air is only slightly greater than one. A group of other commonly used dielectric mate-ials is listed in figure 15-

Certain materials, such as bakelite. Incite, and other plastics dissipate considerable energy when used as capacitor dielectrics.

\ microfarad-=-1/1,Ш,Ш of a farad, .000001 farad, or 10 farads.

 TABLE OF DIELECTRIC MATERIALS MATERIAL DIELECTRIC CONSTANT10 MC. POWER FACTOR 10 MC, SOFTENINGPOINT FAHRENHEIT ANILINE-FORMALDEHYDE RESIN 0,004 260 8ARIUM TITANATE IZOO CASTOR OIL 4.67 CELLULOSE ACETATE 0.04 180 GLASS.WINDOW POOR 2000- GLASS, PYREX 0.02 KEL-f FLUOROTHENE . - METHYL- METHACHYLATE-LUCITE 0.007 160 MICA 0.0003 MYCALEX, MYKROY 0.002 650 PHENOL-FORMALDEHYDE, LOW-LOSS YELLOW 0.015 270 PHENOL-FORMALDEHYDE BLACK BAKELITE 0.03 350° PORCELAIN 0,005 2600 POLYETHYLENE 2,25 0,0003 220° POLYSTYRENE 2.55 0.0002 175 QUA-RTZ, FUSED 0.0002 2600 RUBBER, HARD-EBONITE г.в 0,007 ISO* STEATITE e. r 0.003 2700- SULFUR 0.003 236° TEFLON 2. 1 0,02 TITANIUM DIOXIDE 100-175 0.0006 2700 TRANSFORMER OIL 0.003 UREA -FORMALDEHYDE 0.05 260 VINYL RESINS 0.02 200 WOOD, MAPLE POOR

FIGURE IS

This energy loss is expressed in terras of the power factor of the capacitor, which represents the portion of the input volt-amperes lost in the dielectric material. Other materials including air, polystyrene and quartz have a very low power factor.

The new ceramic dielectrics such as steatite (talc) and titanium dioxide products are especially suited for high frequency and high temperature operation. Ceramics based upon titanium dioxide have an unusually high dielectric constant combined with a low power factor. The temperature coefficient with respect to capacity of units made with this material depends upon the mixture of oxides, and coefficients ranging from zero to over -700 parts per million per degree Centigrade may be obtained in commercial production.

Mycalex is a composition of minute mica particles and lead borate glass, mixed and fired at a relatively low temperature. It is hard and brittle, but can be drilled or machined when water is used as the cutting lubricant.

Шса dielectric capacitors have a very low power factor and extremely high voltage breakdown per unit of thickness. A mica and copper-foil sandwich is formed under pressure to obtain the desired capacity value. The effect of temperature upon the pressures in the sandwich causes the capacity of the usual mica capacitor to have large, non-cyclic variations. If the copper electrodes are plated directly upon the mica sheets, the temperature coefficient can be stablized at about 20 parts per million per degree Centigrade. A process of this type is used in the manufacture of silver mica capacitors.

Paper dielectric capacitors consist of strips of aluminum foil insulated from each other by a thin layer of paper, the whole assembly being wrapped in a circular bundle. The cost of such a capacitor is low, the capacity is high in proportion to the size and weight, and the power factor is good. The life of such a ca-pacitoris dependent upon the moisture penetration of the paper dielectric and upon the applied d-c voltage.

Air dielectric capacitors are used in transmitting and receiving circuits, principally where a variable capacitor of high resetability is required. The dielectric strength is high, though somewhat less at radio frequencies than at 60 cycles. In addition, corona discharge at high frequencies will cause ionization of the air dielectric causing an increase in power loss. Dielectric strength may be increased by increasing the air pressure, as is done in hermetically sealed radar units. In some units, dry nitrogen gas may be used in place of air to provide a higher dielectric strength than that of air.

Likewise, the dielectrLc strength of an air

capacitor may be increased by placing the

unit in a vacuum chamber to prevent ionization of the dielectric.

The temperature coefficient of a variable air dielectric capacitor varies widely and is often non-cyclic. Such things as differential expansion of various parts of the capacitor, changes in internal stresses and different temperature coeffi-cients of various parts contribute to these variances.

Dielectric The capacitance of a capacitor is Constant determined by the thickness and nature of the dielectric material between plates. Certain materials offer a greater capacitance than others, depending upon their physical makeup and chemical constitution. This property is expressed by a constant K, called the dielectric constant. (K = 1 for air.)

Dielectric If the charge becomes too great Breakdown for a given thickness of a certain dielectric, the capacitor will break down, i.e., the dielectric will puncture. It is for this reason that capacitors are rated in the manner of the anaount of voltage they will safely withstand as well as the capacitance in microfarads. This rating is commonly expressed as the d-c working voltage.

Calculation of The capacitance of two parallel Copacitance plates is given with good accuracy by the following formula:

CIRCULAR PLATE CAPACITORS

CAPACITANCE FOR A GIVEN SPACING

T 2 3 Л 5 6 7 a 9 10 11 12 13 14

CAPACITANCE IN MICRO-MICROFARADS Figure 16

Through the uso of this chart it Is possible to determine the required plate diameter (with the the necessary spacing established by peak voltage considerations) for a circular-plate neutralizing capacitor. The copacitance given Is for a dielectric of air and the spacing given is between ad/aeent faces of the two plates.

HANDBOOK

Capacitive Circuits 33

PARALLEL CAPACITORS

SERIES CAPACITORS

CAPACITORS IN SERIES-PARALLEL Figure 17

CAPACITORS IN SERIES, PARALLEL, AND SERIES-PARALLEL

С =0.2248 xK X

where С = capacitance in micro-microfarads, К = dielectric constant of spacing material,

A = area of dielectric in square inches, t = thickness of dielectric in inches.

This formula indicates that the capacitance is directly proportional to the area of the plates and inversely proportional to the thickness of the dielectric (spacing between the plates). This simply means that when the area of the plate is doubled, the spacing between plates remaining constant, the capacitance will be doubled. Also, if the area of the plates remains constant, and the plate spacing is doubled, the capacitance will be reduced to half.

The above equation also shows that capacitance is directly proportional to the dielectric constant of the spacing material. An air-spaced capacitor that has a capacitance of 100 fifiid. in air would have a capacitance of 467 ifitd. when immersed in castor oil, because the dielectric constant of castor oil is 4.67 times as great as the dielectric constant of air.

Where the area of the plates is definitely set, when it is desired to know the spacing needed to secure a required capacitance,

A X 0.2248 X К

t =-

where all units are expressed just as in the preceding formula. This formula is not confined to capacitors having only square or

rectangular plates, but also applies when the plates are circular in shape. The only change will be the calculation of the area of such circular plates; this area can be computed by squaring the radius of the plate, then multiplying by 3-1416, or pi. Expressed as an equation:

A = 3.1416 xr

where r = radius in inches

The capacitance of a multi-plate capacitor can be calculated by taking the capacitance of one section and multiplying this by the number of dielectric spaces. In such cases, however, the formula gives no consideration to the effects of edge capacitance; so the capacitance as calculated will not be entirely accurate. These additional capacitances will be but a small part of the effective total capacitance, particularly when the plates are reasonably large and thin, and the final result will, therefore, be within practical limits of accuracy.

Capacitors in Equations for calculating ca-Parallel end pacitances of capacitors in par-in Series allel connections are the same

as those for resistors in series.

С = Ci + Q + . . . + c

Capacitors in series connection are calculated in the same manner as are resistors in parallel connection.

The formulas are repeated: (1) For two or more capacitors of unequal capacitance in series:

1 1 1

- + - + -

С, Q

1111

or -= - + - + -

С С. Cj с,

(2) Two capacitors of unequal capacitance in series:

С, x Cj

C + Cj

(3) Three capacitors of equal capacitance in series:

С =- where C, is the common capacitance.

(4) Three or more capacitors of equal capacitance in series.

Value of common capacitance

Number of capacitors in series (5) Six capacitors in series parallel:

+

Cj C2 C3 C5 Cj

EQUAL /

CAPaCITANCENT

. \ EtJUAL

RESISTANCE

Capacitors in A-C Wlien a capacitor is con-and D-C Circuits fleeted into a direct-current circuit, it will block the d.c, or stop the flow of current. Beyond the initial movement of electrons during the period when the capacitor is being charged, rhere will be no flow of current because the circuit is effectively broken by the dielectric of the capacitor.

Strictly speaking, a very small current may actually flow because the dielectric of the capacitor may not be a perfect insulator. This minute current flow is the leakage current previously referred to and is dependent upon the internal d-c resistance of the capacitor. This leakage current is usually quite noticeable in most types of electrolytic capacitors.

When an alternating current is applied to a capacitor, the capacitor will charge and discharge a certain number of times per second in accordance with the frequency of the alternating voltage. The electron flow in the charge and discharge of a capacitor when an a-c potential is applied constitutes an alternating current, in effect. It is for this reason that a capacitor will pass an alternating current yet offer practically infinite opposition to a direct current. These two properties are repeatedly in evidence in a radio circuit.

Voltage Rating Any good paper dielectric of Capacitors filter capacitor has such a in Series high internal resistance (in-

dicating a good dielectric) that the exact resistance will vary considerably from capacitor to capacitor even though they are made by the same manufacturer and are of the same rating. Thus, when 1000 volts d.c. is connected across two l-/zfd. 500-volt capacitors in series, the chances are that the voltage will divide unevenly and one capacitor will receive more than 500 volts and the other less than 500 volts.

Voltage Equalizing By connecting a half-Resistors megohm 1-watt carbon resistor across each capacitor, the voltage will be equalized because the resistors act as a voltage divider, and the internal resistances of the capacitors are so much higher (many megohms) that they have but little effect in disturbing the voltage divider balance.

Carbon resistors of the inexpensive type are not particularly accurate (not being designed for precision service); therefore it is

Figure 18

SHOWING THE USE OF VOLTAGE EQUALIZING RESISTORS ACROSS CAPACITORS CONNECTED IN SERIES

advisable to check several on an accurate ohmmeter to find two that are as close as possible in resistance. The exact resistance is unimportant, just so it is the same for the two resistors used.

Capacitors In When two capacitors are con-Series on A.C. nected in series, alternating voltage pays no heed to the relatively high internal resistance of each capacitor, but divides across the capacitors in inverse proportion to the capacitance. Because, in addition to the d.c, across a capacitor in a filter or audio amplifier circuit there is usually an a-c or a-f voltage component, it is inadvisable to series-connect capacitors of unequal capacitance even if dividers are provided to keep the d.c. within the ratings of the individual capacitors.

For instance, if a 500-volt l-/ifd. capacitor is used in series with a 4-;xfd. 500-volt capacitor across a 250-volt a-c supply, the 1-jufd. capacitor will have 200 volts a.c. across it and the 4-/xfd. capacitor only 50 volts. An equalizing divider to do any good in this case would have to be of very low resistance because of the comparatively low impedance of the capacitors to a.c. Such a divider would draw excessive current and be impracticable.

The safest rule to follow is to use only capacitors of the same capacitance and voltage rating and to install matched high resistance proportioning resistors across the various capacitors to equalize the d-c voltage drop across each capacitor. This holds regardless of how many capacitors are series-connected.

Electrolytic Electrolytic capacitors use a very Capacitors thin film of oxide as the dielectric, and are polarized; that is, they have a positive and a negative terminal which must be properly connected in a circuit; otherwise, the oxide will break down and the capacitor will overheat. The unit then will no longer be of service. When electrolytic capacitors are connected in series, the positive terminal is always connected to the positive lead of the power supply; the negative terminal of

HANDBOOK

М agnetism 35

the capacitor connects to the positive terminal of the next capacitor in the series combination-The method of connection for electrolytic capacitors in series is shown in figure 18. Electrolytic capacitors have very low cost per microfarad of capacity, but also have a large power factor and high leakage; both dependent upon applied voltage, temperature and the age of the capacitor. The modern electrolytic capacitor uses a dry paste electrolyte embedded in a gauze or paper dielectric. Aluminium foil and the dielectric are wrapped in a circular bundle and are mounted in a cardboard or metal box. Etched electrodes may be employed to increase the effective anode area, and the total capacity of the unit.

The capacity of an electrolytic capacitor is affected by the applied voltage, the usage of the capacitor, and the temperature and humidity of the environment. The capacity usually drops with the aging of the unit. The leakage current and power factor increase with age. At high frequencies the power factor becomes so poor that the electrolytic capacitor acts as a series resistance rather than as a capacity.

2-4 Magnetism

and Electromagnetism

The common bar or horseshoe magnet is familiar to most people. The magnetic field which surrounds it causes the magnet to attract other magnetic materials, such as iron nails or tacks. Exactly the same kind of magnetic field is set up around any conductor carrying a current, but the field exists only while the current is flowing.

Magnetic Fields Before a potential, or voltage, is applied to a conductor there is no external field, because there is no general movement of the electrons in one direction. However, the electrons do progressively move along the conductor when an e.m.f. is applied, the direction of motion depending upon the polarity of the e.m.f. Since each electron has an electric field about it, the flow of electrons causes these fields to build up into a resultant external field which acts in a plane at right angles to the direction in which the current is flowing. This field is known as the magnetic field.

The magnetic field around a current-carrying conductor is illustrated in figure 19- The direction of this magnetic field depends entirely upon the direction of electron drift or current flow in the conductor. When the flow is toward the observer, the field about the conductor is clockwise; when the flow is away from the observer, the field is counter-clockwise. This is easily remembered if the left hand is clenched, with the thumb outstretched

й й L\

ELECTRON DRIFT -SWITCH

Figure 19 UEFT-HAND RULE

Showing the dinctlon of the magnetic linos of force produced around a conductor carrying an electric current.

and pointing in the direction of electron flow. The fingers then indicate the direction of the magnetic field around the conductor.

Each electron adds its field to the total external magnetic field, so that the greater the number of electrons moving along the conductor, the stronger will be the resulting field.

One of the fundamental laws of magnetism is that like poles repel one another and unlike poles attract one another. This is true of current-carrying conductors as well as of permanent magnets. Thus, if two conductors areplaced side by side and the current in each is flowing in the same direction, the magnetic fields will also be in the same direction and will combine to form a larger and stronger field. If the current flow in adjacent conductors is in opposite directions, the magnetic fields oppose each other and tend to cancel.

The magnetic field around a conductor may be considerably increased in strength by winding the wire into a coil. The field around each wire then combines with those of the adjacent turns to form a total field through the coil which is concentrated along the axis of the coil and behaves externally in a way similar to the field of a bar magnet.

If the left hand is held so that the thumb is outstretched and parallel to the axis of a coil, with the fingers curled to indicate the direction of electron flow around the turns of the coil, the thumb then points in the direction of the north pole of the magnetic field.

The Magnetic Circuit

In the magnetic circuit, the units which correspond to current, voltage, and resistance in the electrical circuit are flux, magnetomotive force, and reluctance.

Flux, Flux As a current is made up of a drift Density of electrons, so is a magnetic field made up of lines of force, and the total number of lines of force in a given magnetic circuit is termed the flux. The flux

depends upon the material, cross section, and

length of the magnetic circuit, and it varies directly as die current flowing in the circuit.

The unit of flux is the maxwell, and the sym-h>ol is the Greek letter ф (phi).

Flux density is the number of lines of force per unit area. It is expressed in gauss if the unit of area is the square centimeter (1 gauss = 1 line of force per square centimeter), or in lines per square inch. The symbol for flux density is В if it is expressed in gausses, or В if expressed in lines per square inch.

Magnetomotive The force which produces a Force flux in a magnetic circuit

is called magnetomotive force. It is abbreviated m.m.f. and is designated by the letter P. The unit of magnetomotive force is the gilbert, which is equivalent to 1.26 x N1, where N is the number of turns and I is the current flowing in the circuit in amperes.

The m.m.f. necessary to produce a given flux density is stated in gilberts per centimeter (oersteds) (H), or in ampere-turns per inch (H).

Reluctance Magnetic reluctance corresponds to electrical resistance, and is the property of a material that opposes the creation of a magnetic flux in the material. It is expressed in rels, and the symbol is the letter R. A material has a reluctance of 1 rel when an m.m.f. of 1 ampere-turn (N1) generates a flux of 1 line of force in it. Combinations of reluctances are treated the same as resistances in finding the total effective reluctance. The specific reluctance of any substance is its reluctance per unit volume.

Except for iron and its alloys, most coimnon materials have a specific reluctance very nearly the same as that of a vacuum, which, for all practical purposes, may be considered the same as the specific reluctance of air.

Ohms Law for The relations between flux. Magnetic Circuits magnetomotive force, and reluctance are exactly the same as the relations between current, voltage, and resistance in the electrical circuit. These can be stated as follows:

R Ф

where ф - flux, F = m.m.f., and R = reluctance.

Permeability Permeability expresses the ease with which a magnetic field may be set up in a material as compared with the effort required in the case of air. Iron, for example, has a permeability of around 2000 times that of air, which means that a given amount of magnetizing effect produced in an iron core by a current flowing through a coil of wire will produce 2000 times the flux density that the same magnetizing effect would pro-

duce in air. It may be expressed by the ratio B/H or B/H. In other words,

В

where ц is the premeability, В is the flux density in gausses, В is the flux density in lines per square inch, H is the m.m.f. in gilberts per centimeter (oersteds), and H is the m.m.f. in ampere-turns per inch. These relations may also be stated as follows:

В В

Н =- or Н=-, and В=Нд or В = Ни

It can be seen from the foregoing that permeability is inversely proportional to the specific reluctance of a material.

Saturation Permeability is similar to electric conductivity. There is, however, one important difference: the permeability of magnetic materials is not independent of the magnetic current (flux) flowing through it, although electrical conductivity is substantially independent of the electric current in a wire. When the flux density of a magnetic conductor has been increased to the saturation point, a further increase in the magnetizing force will not produce a corresponding increase in flux density.

Calculations To simplify magnetic circuit calculations, a magnetization curve may be drawn for a given unit of material. Such a curve is termed a B-H curve, and may be determined by experiment. When the current in an iron core coil is first applied, the relation between the winding current and the core flux is shown at A-B in figure 20. If the current is then reduced to zero, reversed, brought back again to zero and reversed to the

MAGNETIZING FORCE

H --

Figure 20 TYPICAL HYSTERESIS LOOP (B-H CURVE = A-B)

Showing relationship between the current in the winding of an iron core inductor and the core flu*. A direct current Hawing through the inductance brings the mognetic state of the core to some point on the hysteresis loop, such as C.

HANDBOOK

Inductance

original direction, the flux passes through a typical hysteresis loop as shown.

Residual Magnetism; The magnetism remaining Retentivity in a material after the

magnetizing force is removed is called residual magnetism. Reten-tivity is the property which causes a magnetic material to have residual magnetism after having been magnetized.

Hysteresis; Hysteresis is the character-

Coercive Force istic of a magnetic system which causes a loss of power due to the fact that a negative magnetizing force must be applied to reduce the residual magnetism to zero. This negative force is termed coercive force. By negative magnetizing force is meant one which is of the opposite polarity with respect to the original magnetizing force. Hysteresis loss is apparent in transformers and chokes by the heating of the core.

Inductance If the switch shown in figure 19 is opened and closed, a pulsating direct current will be produced. When it is first closed, the current does not instantaneously rise to its maximum value, but builds up to it. While it is building up, the magnetic field is expanding around the conductor. Of course, this happens in a small fraction of a second. If the switch is then opened, the current stops and the magnetic field contracts quickly. This expanding and contracting field will induce a current in any other conductor that is part of a continuous circuit which it cuts. Such a field can be obtained in the way just mentioned by means of a vibrator interrupter, or by applying a.c. to the circuit in place of the battery. Varying the resistance of the circuit will also produce the same effect. This inducing of a current in a conductor due to a varying current in another conductor not in acutal contact is called electromagnetic induction.

Self-inductance If an alternating current flows through a coil the varying magnetic field around each turn cuts itself and the adjacent turn and induces a voltage in the coil of opposite polarity to the applied e.m.f. The amount of induced voltage depends upon the number of turns in the coil, the current flowing in the coil, and the number of lines of force threading the coil. The voltage so induced is known as a counter-e.m.f. or back-e.m.f., and the effect is termed self-induction. When the applied voltage is building up, the counter-e.m.f. opposes the rise; when the applied voltage is decreasing, the counter-e.m.f. is of the same polarity and tends to maintain

the current. Thus, it can be seen that self-induction tends to prevent any change in the current in the circuit.

The storage of energy in a magnetic field is expressed in joules and is equal to (Ll)/2. (A joule is equal to 1 watt-second. L is defined immediately following.)

The Unit of Inductance is usually denoted by Inductance; the letter L, and is expressed in The Henry henrys. A coil has an inductance of 1 henry when a voltage of 1 volt is induced by a current change of 1 ampere per second. The henry, while commonly used in audio frequency circuits, is too large for reference to inductance coils, such as those used in radio frequency circuits; millihenry or microhenry is more commonly used, in the following manner;

1 henry = 1,000 millihenrys, or 10 milli-henrys.

1 millihenry - 1/1,000 of a henry, .001 henry, or 10 henry.

I microhenry = 1/1,of a henry, or .000001 henry, or 10 * henry.

I microhenry = 1/1,000 of a millihenry, .001 or 10 millihenrys.

1,000 microhenrys = 1 millihenry.

Mutual Inductance When one coil is near another, a varying current in one will produce a varying magnetic field which cuts the turns of the other coil, inducing a current in it. This induced current is also varying, and will therefore induce another current in the first coil. This reaction between two coupled circuits is called mutual induction, and can be calculated and expressed in henrys. The symbol for mutual inductance is M. Two circuits thus joined are said to be inductively coupled.

The magnitude of the mutual inductance depends upon the shape and size of the two circuits, their positions and distances apart, and the premeability of the medium. The extent to

 ч

Figure 21

MUTUAL INDUCTANCE

The quanr/f}r M represents the mutual Inductance between the two colls Lj and L.

INDUCTANCE OF SINGLE-LAYER SOLENOID COILS

R N 9R+10L

MICROHENRIES

TURNS

WHERE : R = RADIUS OF COIL TO CENTER OF WIRE L = LENGTH OF COIL N = NUMBER OF TURNS

Figure 22 FORMULA FOR CALCULATING INDUCTANCE

Through fha use of the aquation and the sketch shown obove fhe Inductance of single-layer solenoid coils can be calculated with an ac> curacy of about one per cent for the types of colls normally used In the h-f and v-h-f range.

which two inductors are coupled is expressed by a relation known as coefficient of coupling. This is the ratio of the mutual inductance actually present to the maximum possible value.

The formula for mutual inductance is L = Li +Lj -I- 2M when the coils are poled so that their fields add. When they are poled so that their fields buck, then L = L, -i- Lj - 2M (figure 21).

Inductors in Inductors in parallel are com-Porollel bined exactly as are resistors in parallel, provided that they are far enough apart so that the mutual inductance is entirely negligible.

Inductors in Inductors in series are additive. Series just as are resistors in series,

again provided that no mutual inductance exists. In this case, the total inductance L is:

L = Li + Lj -I-........etc.

Where mutual inductance does exist:

L =L, +Lj + 2M,

where M is the mutual inductance.

This latter expression assumes that the coils are connected in such a way that all flux linkages are in the same direction, i.e., additive. If this is not the case and the mutual linkages subtract from the self-linkages, the following formula holds:

L = Li -b Lj - 2M,

where M is the mutual inductance.

Core Material Ordinary magnetic cores cannot be used for radio frequencies because the eddy current and hysteresis losses in the core material becomes enormous

as the frequency is increased. The principal use for conventional magnetic cores is in the audio-frequency range below approximately 15,000 cycles, whereas at very low frequencies (50 to 60 cycles) their use is mandatory if an appreciable value of inductance is desired.

An air core inductor of only 1 henry inductance would be quite large in size, yet values as high as 500 henrys are commonly available in small iron core chokes. The inductance of a coil with a magnetic core will vary with the amounr of current (both a-c and d-c) which passes through the coil. For this reason, iron core chokes that are used in power supplies have a certain inductance rating at a predetermined value of d-c.

The premeability of air does not change with flux density; so the inductance of iron core coils often is made less dependent upon flux density by making part of the magnetic path air, instead of utilizing a closed loop of iron. This incorporation of an air gap is necessary in many applications of iron core coils, particularly where the coil carries a considerable d-c component. Because the permeability of air is so much lower than that of iron, the air gap need comprise only a small fraction of the magnetic circuit in order to provide a substantial proportion of the total reluctance.

Iron Cared Inductors Iron-core inductors may at Radio Frequencies be used at radio frequencies if the iron is in a very finely divided form, as in the case of the powdered iron cores used in some types of r-f coils and i-f transformers. These cores are made of extremely small particles of iron. The particles are treated with an insulating material so that each particle will be insulated from the others, and the treated powder is molded with a binder into cores. Eddy current losses are greatly reduced, with the result that these special iron cores are entirely practical in circuits which operate up to 100 Mc. in frequency.

2-5 RC and RL Transients

A voltage divider may be constructed as shown in figure 23- Kirchhoffs and Ohms Laws hold for such a divider. This circuit is known as an RC circuit.

Time Constant- When switch S in figure 23 is RC and RL placed in position 1, a volt-Circuits meter across capacitor С will indicate the manner in which the capacitor will become charged through the resistor R from battery B. If relarively large values are used for R and C, and if a v-t voltmeter which draws negligible current is used

®

ш <

у I-.-

 =1=:: [-7 / -

®

O Q. -

55 О-0

0 J. I 2 3

TIUEL, INTERMSOF TIME CONSTANT RC

UjitJ -ant/)

©

0 T 2 3

TIME Lj IN TCRMS OF TIME CONSTANT RC

Figure 23

TIME CONSTANT OF AN R-C CIRCUIT

Shown at (A) Is the circuit upon which Is based the curves of (B) and (C). (B) shows the rate at which capacitor С will charge from the Instant at which switch S Is placed In position 7. (C) shows the discharge curve of capacitor С from the Instant at which switch 5 Is placed in position 3.

to measure the voltage e, the rate of charge of the capacitor may actually be plotted with the aid of a stop watch.

Voltage Gradient It will be found that the voltage e will begin to rise rapidly from zero the instant the switch is closed. Then, as the capacitor begins to charge, the rate of change of voltage across the capacitor will be found to decrease, the charging taking place more and more slowly as the capacitor voltage e approaches the battery voltage E. Actually, it will be found that in any given interval a constant percentage of the remaining difference between e and E will be delivered to the capacitor as an increase in voltage. A voltage which changes in this manner is said to increase logarithmically, or is said to follow an exponential curve.

Time Constont A mathematical analysis of the charging of a capacitor in this manner would show that the relationship between the battery voltage E and the voltage across the capacitor e could be expressed in the following manner:

e = E (I - e~ t/RC)

where e,E,R, and С have the values discussed above, f ~ 2.716 (the base of Naperian or natural logarithms), and t represents the time which has elapsed since the closing of the switch. With t expressed in seconds, R and С

Figure 24 TYPICAL INDUCTANCES

The Jorge inductance Is a 1000-watt transmitting coil. To the right and left of this coil are small r-f chokes. Several varieties of low power capability colls are shown below, along with various types of r-f

chokes intended for high-frequency operation.

R (including D,C. resistance °f inductor l)

5 80 о 40

TIME t, IN TERMS OF TIME CONSTANT

К

Figure 25

TIME CONSTAMT OF AM R-L CIRCUIT

Note that the time constant for the Increase In current through on R-L circuit is identical to the rate of increase in voltage ocross fhe capacitor in an R-C circuit.

may be expressed in farads and ohms, or R and С may be expressed in microfarads and megohms. The product RC is called the time constant of the circuit, and is expressed in seconds. As an example, if R is one megohm and С is one microfarad, the time constant RC will be equal to the product of the two, or one second.

When the elapsed time t is equal to the time constant of the RC network under consideration, the exponent of e becomes -1. Now is equal to I/e, or 1/2.716, which is 0.368. The quantity (1 - О.368) then is equal to 0.632. Expressed as percentage, the above

means that the voltage across the capacitor will have increased to 63-2 per cent of the battery voltage in an interval equal to the time constant or RC product of the circuit. Then, during the next period equal to the time constant of the RC combination, the voltage across the capacitor will have risen to 63-2 per cent of the remaining difference in voltage, or 86.5 per cent of the applied voltage E.

RL Circuit In the case of a series combination of a resistor and an inductor, as shown in figure 25, the current through the combination follows a very similar law to that given above for the voltage appearing across the capacitor in an RC series circuit. The equation for the current through the combination is:

i =- (1 e-tR/L) R

where i represents the current at any instant through the series circuit, E represents the applied voltage, and R represents the total resistance of the resistor and the d-c resistance of the inductor in series. Thus the time constant of the RL circuit is L/R, with R expressed in ohms and L expressed in henrys.

Voltage Decay \КЪеп the switch in figure 23 is moved to position 3 after the capacitor has been charged, the capacitor voltage will drop in the manner shown in figure 2З-С. In this case the voltage across the capacitor will decrease to 36.8 per cent of the initial voltage (will make 63-2 per cent of the total drop) in a period of time equal to the time constant of the RC circuit.

TYPICAL IRON-CORE INDUCTANCES

At the right is an upright mounting filter choke intended for use in low powered transmitters and audio equipment. At the center is a hermetically sealed inductance for use under poor env/ronmenta/ conditions. To the left is an inexpensive receiving-type choke, with a small iron-core r-f choke directly in front of it.

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