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    ELECTRICAL AND

    ELECTRONIC PRINCIPLES

    Introduction to Electronic

    fundamentals

    EE001-3-0

    Lecture 5

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    •  Electronic devices such as diodes, transistors, and

    integrated circuits are made of semiconductor

    materials.

    •  To understand how these devices work, we need a basic knowledge of the atoms structure and the interaction

    of atomic particles.•  An important concept introduced in this chapter is that of

    the p-n junction that is formed when two different types ofsemiconductor material are joined.•  p-n  junction is fundamental to the operation of devices

    such as diode and certain types of transistor. 

    Basic Semiconductor Physics

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    Semiconductors

    Semiconductors 

    • All materials are made up of atoms.

    • These atoms contribute to the electrical propertiesof a material, including its ability to conductelectrical current.

    • or purposes of discussing electrical properties, anatom can be represented by the valence shell and a

    core that consists of all the inner shells and thenucleus.

    • This concept is illustrated in figure !." for acarbon atom.

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    Semiconductors

     

    • #arbon is used in many types of resistors.

    •  $otice that the carbon atom has four electrons in the valence

    shell and two electrons in the inner shell %&'.• The nucleus consists of si( protons and si( neutrons so the )! 

    indicates the positive charge of the ! protons.

    • The simplified representation shows the four valence electronsand a core with a net charge of )* %! for the nucleus and + for

    the two inner shell electrons'.

    Fig.6.3 Carbon atom

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    #onductors

    #onductors•  A conductor is a material that easily conducts electrical

    current.

    • The best conductors are single element materials, such

    as copper, silver, gold, and aluminum, which arecharacteri-ed by atoms with only one valence electron

    very loosely bound to the atom.

    • These loosely bound valence electrons can easily breakaway from their atoms and become free electrons.

    • Therefore, a conductive material has many free

    electrons that, when moving in a net direction, make up

    the current.

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    nsulators

    nsulators

    • An insulator is a material that does not conduct

    electrical current under normal conditions.

    • /ost good insulators are compounds rather than

    single element materials.

    • 0alence electrons are tightly bound to the atoms1

    therefore, there are very few free electrons in an

    insulator.

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    2asic Semiconductor physics

    Semiconductors

    • A semiconductor is a material that is between conductorsand insulators in its ability to conduct electrical current.

    • A semiconductor in its pure %intrinsic' state is neither a

    good conductor nor a good insulator.• The most common single element semiconductors are

    silicon, germanium, and, carbon.

    • #ompound semiconductors such as gallium arsenide arealso commonly used.

    • The single element semiconductors are characteri-ed byatoms with four valence electrons.

    • or a semiconductor material if the temperature isincreased the resistance will decrease since it has negativetemperature coefficient.

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    ntrinsic semiconductor 

    • An intrinsic crystal is one that has no impurities.

    Conduction in semiconductor

    • An intrinsic %pure' silicon crystal at roomtemperature derives heat %thermal' energy fromthe surrounding air, causing some valenceelectrons to gain sufficient energy to jump the gapfrom the valence band into the conduction band,

     becoming free electrons not bound to any oneatom but free drift.

    • ree electrons are also called conductionelectrons.

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    Electron 6ole+pair generation

    Fig.6.6 Electron Hole-pair generation

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    Electron 76ole current

    Fig 6. Electron ! Hole current

    Electron and hole current

    • 3hen a voltage is applied across a piece of

    intrinsic silicon, as shown in figure !.8.

    • The thermally generated free electrons in the

    conduction band, are now easily attracted

    toward the positive end.

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    Electron 76ole current

    • This movement of free electrons is one type ofcurrent in a semiconductor material and is calledelectron current. 

    • Another type of current occurs at the valence

    level, where the holes created by the free electronse(ist.

    • Electrons remaining in the valence band are stillattached to their atoms and are not free to moverandomly in the crystal structure as are the freeelectrons.

    • 6owever, a valence electron can move into nearbyhole, with little change in its energy level, thusleaving another hole where it came from.

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    Electron 76ole current

    • Effectively the hole has moved from one place toanother in the crystal structure, as illustrated in figure!.9. This is called a hole current.

    Fig6." Electron and hole mo#ement

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     $+type and :+type semiconductor 

    n+type and p+type semiconductor 

    • The conductivity of silicon and germanium can be

    drastically increased by the controlled addition of

    impurities %pure' semiconductor material.• This process, called doping, increases the number

    of current carriers %electrons or holes', thus

    increasing the conductivity and decreasing the

    resistivity.

     The two categories of impurities are n-t$pe and p-

    t$pe. 

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     $+type and :+type semiconductor 

    n+type semiconductor  

    • To increase the number of conduction band electrons

    in intrinsic silicon, penta#alent impurity atoms are

    added.• These are atoms with five valence electrons such as

    arsenic %As', phosphorus %:', bismuth %2i', and

    antimony %Sb'. 

    • As illustrated in igure !.;, each pentavalent atom

    %antimony in this case' forms covalent bonds with

    four adjacent silicon atoms.

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     $+type

    Fig 6.% n-t$pe semiconductor

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     $+type

    • our of the antimony atom

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    :+type

    :+type semiconductor  

    • To increase the number of holes in intrinsic silicon,tri#alent impurity atoms are added.

    • These are atoms with three electrons such asaluminum %Al', boron %2', indium %n', and gallium%=a'. 

    • As illustrated in igure !.>?, each trivalent atom%boron, in this case' forms covalent bonds with four

    adjacent silicon atoms.• All three of the boron atom

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    :+type

    Fig 6.10 p-t$pe semiconductor

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    :+type

    • 2ecause the trivalent atom can take an electron, it

    is often referred to as an atom.

    • The number of holes can be carefully controlled by the number of trivalent impurity atoms added

    to the silicon.

    • A hole created by this doping process is not

    accompanied by a conduction %free' electron.

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    :$+ junction diode

    The p+n junction

    • A p+type material consists of silicon atoms andtrivalent impurity atoms such as boron.

    • The boron atom adds a hole when it bonds withthe silicon atoms.

    • 6owever, since the number of protons and thenumber of electrons are e5ual throughout thematerial, there is no net charge in the material and

    so it is neutral.• An n+type silicon material consists of silicon

    atoms and pentavalent impurity atoms such asantimony.

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    :$+ junction diode

    • As it is seen earlier, an impurity atom releases anelectron when it bonds with four atoms.

    • Since there is still an e5ual number of protons andelectrons throughout the material, there is no net

    charge in the material and so it is neutral.• f a piece of intrinsic silicon is doped so that half is

    n+type and the other half is p+type, a pn-&unction forms between the two regions as indicated in

    igure !.>?.• The p region has many holes.

    • The n region has many free electrons.

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    :$+ junction diode

    Fig 6.10 Formation o' pn &unction

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    @epletion regionormation of the depletion region

    • The free electrons in the n region are randomly drifting

    in all directions.

    • At the instant of the pn junction formation, the free

    electrons near the junction in the n region begin todiffuse across the junction into the p region where they

    combine with holes near the junction, as shown in

    igure !.>>%a'.

    • 2efore the pn junction is formed, recall that there are as

    many electrons as protons in the n+type material making

    neutral in terms of net charge.

    • The same is true for the p+type material.

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    @epletion region

    Fig 6.11 Formation o' repletion region

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    @epletion region

    • 3hen the pn junction is formed, the n region loses

    free electrons as they diffuse across the junction.

    • As the electrons move across the junction, the p

    region loses holes as the electrons and holescombine.

    • This creates a layer of negative charges near the

     junction.

    • These two layers of positive and negative charges

    form the depletion region, as shown in figure

    !.>>%b'.

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    @epletion region

    • &eep in mind that the depletion region is formed

    very 5uickly and is very thin compared to the n

    region and p region.

    • The width of the depletion region in figure !.>> is

    e(aggerated for purposes of illustration.

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    2arrier potential

    2arrier potential

    • Any time there is a positive charge and a negative

    charge near each other, there is a force acting on the

    charges as described by #oulomb

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    2arrier potential

    • That is, e(ternal energy must be applied to get the electrons tomove across the barrier of the electric field in the depletion

    region.

    • The potential difference of the electric field across the depletion

    region is the amount of energy re5uired to move electronsthrough the electric field.

    • This potential difference is called as barrier potential and is

    e(pressed in #olts.

    • The barrier potential of a pn junction depends on several factors,including the type of semiconductor material, the amount of

    doping, and the temperature.

    • Typical barrier potential is appro(imately 0.( 'or silicon and

    0.3( 'or germanium at C°#.

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    2arrier potential

    Biasing the PN junction

    Forward biasReverse bias

    Figure 6.1) *iasing pn &unction

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    orward biasing

    orward bias

    • To bias a pn junction, apply an e(ternal dc voltage across it.

    • orward bias is the condition that allows current through a

     pn junction.• igure !.>" shows a dc voltage source connected by

    conductive material across a pn junction in the direction to

     produce forward bias.

    • This e(ternal bias voltage is designated as 02AS.

     $otice that the negative side of 02AS is connected to the n

    region and positive side is connected to the p region.

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    orward biasing

    Figure 6.13 For+ard biasing a pn &unction diode

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    orward biasing

    • This is one of the re5uirement for forward bias.

    • A second re5uirement is that the 02AS must be greater than

    the barrier potential.

    • A fundamental picture of what happens when a pn

     junction is forward biased is shown in igure !.>*.

    • 2ecause like charges repel, the negative side of the bias

    voltage source pushesB the free electrons, which are themajority carriers in the n region, towards the pn junction.

    • This flow of free electrons is called electron current.

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    orward biasing

    Figure 6.1 For+ard biasing pushes the electrons

    to the pn &unction

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    orward biasing

    The effect of forward bias on the depletion region

    • As more electrons flow into the depletion region, the number of

     positive ions is reduced.

    • As more holes effectively flow into the depletion region on the

    other side of the pn junction, the number of negative ions isreduced.

    • This reduction in positive and negative ions during forward bias

    causes the depletion region to narrow, as indicated in igure !.>C.

    Figure 6.15 E''ect o' 'or+ard bias

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    orward biasing

    Effect of the barrier potential during forward bias

    • 3hen forward bias applied, the free electrons are provided with

    enough energy from the bias voltage source to overcome the

     barrier potential and effectively climbB the energy hill and cross

    the depletion region.

    • The energy that the electrons re5uire in order to pass through the

    depletion region is e5ual to the barrier potential.

    • This energy loss results in a voltage drop across the pn junction

    e5ual to the barrier potential as shown in figure !.>C%b'.

    • An additional small voltage drop occurs across the p and n

    regions due to the internal resistance of the material.

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    4everse bias

    • As illustration of what happens when pn junction is reverse+ biased is shown in figure !.>!.

    •  $otice that the positive side of 02AS is connected to the n

    region of the pn junction and the negative side is connected

    to the p region.

    • Also note that the depletion region is shown wider than inforward bias or e5uilibrium.

    • 2ecause unlike charges attract, the positive side of the bias

    voltage source pullB the free electrons, which are themajority carriers in the n region, away from the pn junction.

    • n the n region, as the electrons flow toward the positive sideof the voltage source, additional positive ions are created.This results in a widening of the depletion region.

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    4everse bias

    Figure 6.16 E''ect o' re#erse bias

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    4everse bias

    • n the p region, electrons from the negative side

    of the voltage source enter as valence electrons

    and move from hole to hole toward the depletion

    region where they create additional negativeions.

    • This results in a widening of the depletion region

    and a depletion of majority carriers.• The flow of valence electrons can be viewed as

    holes being pulledB toward the positive side.

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    4everse current4everse current

    • The e(tremely small current that e(ists in reverse bias after

    the transition current dies out is caused by the minority

    carriers in the n and p regions that are produced by thermally

    generated electron+hole pairs.

    • The small number of free minority electrons in the p region

    are pushedB toward the pn junction by the negative bias

    voltage.

    • 3hen these electrons reach the wide depletion region, they

    fall down the energy hillB and combine with the minorityholes in the n region as valence electrons and flow toward

    the positive bias voltage, creating a small hole current.

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    4everse current• The conduction band in the p region is at a higher energy level

    than the conduction band in the n region.

    • Therefore, the minority electrons easily pass through the

    depletion region because they re5uire no additional energy.

    • 4everse current is illustrated in figure !.>8

    Figure 6.1

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    Current-voltage characteristic

    #urrent+voltage characteristic of a :$ junction• 3hen a forward bias voltage is applied across a silicon pn

     junction, there is current through the junction.

    • This current is called the 'or+ard current and is

    designated .

    • igure !.>9 illustrates what happens as the forward bias

    voltage is increased positively from ?0.

    • The resistor is used to limit the forward current to a valuethat will not overheat the pn junction and cause damage.

    • 3ith ?0 across the pn junction, there is no forward current,

    as indicated in figure !.>9%a'.

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    #urrent+voltage characteristic

    • As the bias voltage is gradually increased, the forward current andthe voltage across the pn junction gradually increase, as shown in

     part %b'.

    • A portion of the applied bias voltage is dropped across the limiting

    resistor.• 3hen the applied bias voltage is increased to a value the voltage

    across the pn junction reaches appro(imately ?.80, the forward

    current begins to increase rapidly.

    • As you continue to increase the bias voltage, the current continuesto increase very rapidly, but the voltage across the pn junction

    increases very gradually above ?.80, as illustrated in figure !.>9%c'.

    • This small increase in the pn junction voltage above the barrier

     potential is due to the dynamic resistance.

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    #urrent+voltage characteristic

    Figure 6.1" current-#oltage measurement o' a diode

    (a

    )

    (b)

    (c)

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    Current-voltage characteristic /

    forward bias plot

    Figure 6.1% For+ard bias cur#e

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    @ynamic resistance

    @ynamic resistance

    • nlike a linear resistance, the resistance of the forward+

     biased pn material is not constant over the entire curve.

    • 2ecause the resistance as you move along the +0 curve,

    it is called d$namic or C resistance.

    • This dynamic resistance is designated r

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    4everse bias+0 characteristic for reverse bias

    • 3hen a reverse+bias voltage is applied across a pn junction,

    there is only an e(tremely small reverse current 4  through

    the junction.

    • igure !.? illustrates what happens as the reverse bias

    voltage is increased negatively from ?0.

    • 3ith ?0 across the pn junction, there is no reverse current.

    • As you gradually increase the reverse bias voltage, there is a

    very small reverse current and the voltage across the pn

     junction increases, as shown in figure !.?%a'.

    • 3hen the applied bias voltage is increased to a value where

    the reverse voltage across the pn junction 04  reaches the

     breakdown value 024, the reverse current begins to increase

    rapidly.

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    4everse bias measurements

    Figure 6.)0 e#erse bias measurements

    (a)

    (b)

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    Reverse bias measurements

    • As you continue to increase the bias voltage, the currentcontinues to increase very rapidly, but the voltage across the

     pn junction increases very little above 024 , as illustrated in

    figure !.?%b'.

    • *rea/do+n, +ith eceptions, is not a normal mode o'operation 'or most pn &unction de#ices.

    • f the results of the type of measurements shown in figure

    >.;, are plotted on a graph, you get the +0 characteristic

    curve for a reverse biased pn junction.• A typical curve is shown in figure !.>.

    • As you can see, there is very little reverse current %usually

    µA or nA' until the reverse voltage across the junction

    reaches appro(imately the breakdown value.

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    Temperature effect• After this point, the reverse voltage remains at

    appro(imately 024 , but 4  increases very rapidly resulting in

    overheating and possible damage.

    Temperature effects on the +0 characteristic

    • or a forward bias pn junction, as temperature is increased,

    the forward current increases for a given value of forward

    voltage.

    • Also, for a given value of forward current, the forward

    voltage decreases.

    • This is shown with the +0 characteristic curves in figure

    !..

    • The red curve is at temperature C°# and the blue curve is at

    an elevated temperature %C°# )∆T'.

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    0+ plot

    Figure 6.)) E''ect o' temperature on the v-i  cur#e

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    Rectification

    !he process of obtaining unidirectional

    currents and voltages from alternating

    currents and voltages is called

    rectification" #utomatic switching in

    circuits is carried out by diodes"

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    6alf wave rectification

    $sing a single diode% as shown in &igure '"()%

    half-wave rectification is obtained"

    *hen P is sufficiently positive with respect to +%

    diode , is switched on and current i flows" *henP is negative with respect to +% diode , is

    switched off"

     !ransformer ! isolates the euipment from

    direct connection with the mains supply and

    enables the mains voltage to be changed"

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    6alf wave rectification

    Figure 6.)3

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    ull wave rectification

    • Two diodes may be used as shown in igure !.* toobtain 'ull +a#e recti'ication. A centre+tappedtransformer T is used.

    • 3hen : is sufficiently positive with respect to G,diode @> conducts and current flows %shown by the

     broken line in igure !.*'.

    • 3hen S is positive with respect to G, diode @conducts and current flows %shown by the continuous

    line in igure !.*'.• The current flowing in 4 is in the same direction for

     both half cycles of the input. The output waveform isthus as shown in igure !.*.

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    ull wave rectification

    Figure 6.)

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    2ridge rectifier 

    • our diodes may be used in a bridge recti'ier circuit,as shown in igure !.C to obtain 'ull +a#erecti'ication.

    • As for the rectifier shown in igure !.*, the currentflowing in 4 is in the same direction for both halfcycles of the input giving the output waveformshown.

    • To smooth the output of the rectifiers described

    above, capacitors having a large capacitance may beconnected across the load resistor 4.

    • The effect of this is shown on the output in igure!.!.

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    2ridge rectifier 

    Figure 6.)5

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    2ridge rectifier 

    Figure 6.)6

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    T4A$SSTH4S

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    Transistors

    • The bipolar junction transistor consists of three regions ofsemiconductor material.

    •  Hne type is called a p+n+p transistor, in which two regions of 

      p+type material sandwich a very thin layer of n+type material.

    • A second type is called an n+p+n transistor, in which tworegions of n+type material sandwich a very thin layer of p+typematerial.

    • 2oth of these types of transistors consist of two p+n junctions placed very close to one another in a back+to+backarrangement on a single piece of semiconductor material.

    • @iagrams depicting these two types of transistors are shown inigure !.8

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    Transistors

    • The two p+type material regions of the p+n+p transistor arecalled the emitter and collector and the n+type material iscalled the base.

    • Similarly, the two n+type material regions of the n+p+ntransistor are called the emitter and collector and the p+type

    material region is called the base, as shown in igure !.8.

    Figure 6.)

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    !ransistors

    • Transistors have three connecting leads and inoperation an electrical input to one pair ofconnections, say the emitter and base connections cancontrol the output from another pair, say the collectorand emitter connections.

    • This type of operation is achieved by appropriately biasing the two internal p+n junctions.

    • 3hen batteries and resistors are connected to a p+n+ptransistor, as shown in igure !.9%a', the base+emitter junction is 'or+ard biased and the base+collector junction is re#erse biased.

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    !ransistors

    • Similarly, an n+p+n transistor has its base+emitter junction

    forward biased and its base+collector junction reverse biased

    when the batteries are connected as shown in igure !.9%b'.

    Figure 6.)"

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    !ransistors

    • or a silicon p+n+p transistor, biased as shown in

    igure !.9%a', if the base+emitter junction is

    considered on its own, it is forward biased and a

    current flows.• This is depicted in igure !.;%a'. or e(ample, if 4 E 

    is >??? ohm, the battery is *.C0 and the voltage drop

    across the junction is taken as ?.8 0, the current

    flowing is given by *.C+ ?.8D>??? I ".9mA"

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    !ransistors

    • 3hen the base+collector junction is considered on its

    own, as shown in igure !.;%b', it is reverse biased

    and the collector current is something less than > JA.

    • 6owever, when both e(ternal circuits are connectedto the transistor, most of the ".9mA of current flowing

    in the emitter, which previously flowed from the base

    connection, now flows out through the collector

    connection due to transistor action.

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    !ransistors

    Figure 6.)%

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    Transistors operation

    • n a p-n-p transistor, connected as shown in igure!.9%a', transistor action is accounted for as followsK

    • %a' The majority carriers in the emitter p+typematerial are holes

    • %b' The base+emitter junction is forward biased to themajority carriers and the holes cross the junction andappear in the base region

    • %c' The base region is very thin and is only lightly

    doped with electrons so although some electron+hole pairs are formed, many holes are left in the baseregion

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    Transistors operation

    • %d' The base+collector junction is reverse biased to

    electrons in the base region and holes in the collector

    region, but forward biased to holes in the base region1

    these holes are attracted by the negative potential atthe collector terminal

    • %e' A large proportion of the holes in the base region

    cross the base collector junction into the collector

    region, creating a collector current1 conventional

    current flow is in the direction of hole movement.

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    Transistors operation

    •The transistor action is shown diagrammatically

    in igure !."?.• or transistors having very thin base regions, up

    to ;;.CL of the holes leaving the emitter cross the

     base collector junction.

    Figure 6.30

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    Transistors operation

    • n an n-p-n transistor, connected as shown in

    igure !.9%b', transistor action is accounted

    for as followsK

    • %a'The majority carriers in the n+type emitter

    material are electrons

    • %b' The base+emitter junction is forward biased

    to these majority carriers and electrons cross

    the junction and appear in the base region

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    Transistors operation

    .c !he base region is very thin and only lightly

    doped with holes% so some recombination with

    holes occurs but many electrons are left in the

    base region .d !he base-collector 0unction is reverse biased to

    holes in the base region and electrons in the

    collector region% but is forward biased to electrons

    in the base region1 these electrons are attracted bythe positive potential at the collector terminal

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    Transistors operation

    • %e' A large proportion of the electrons in the base

    region cross the base collector junction into the

    collector region, creating a collector current.

    • The transistor action is shown diagrammatically inigure !.">. As stated in earlier section conventional

    current flow is taken to be in the direction of hole

    flow, that is, in the opposite direction to electron flow,

    hence the directions of the conventional current floware as shown in igure !.">.

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    Transistors operation

    Figure 6.31

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    Transistors operation

    •or a p+n+p transistor, the base+collector junction isreverse biased for majority carriers.

    • 6owever, a small leakage current, #2H flows from the base to the collector due to thermally generated minority

    carriers %electrons in the collector and holes in the base', being present. The base+collector junction is forward biased to these minority carriers.

    • f a proportion, M, %having a value of up to ?.;;C in

    modern transistors', of the holes passing into the basefrom the emitter, pass through the base collector junction,then the various currents flowing in a p+n+p transistor areas shown in igure !."%a'.

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    Transistors operation

    • Similarly, for an n+p+n transistor, the base+collector junction is reversed biased for majority carriers, but asmall leakage current, #2H flows from the collector tothe base due to thermally generated minority carriers

    %holes in the collector and electrons in the base', being present.

    •  The base+collector junction is forward biased to theseminority carriers.

    • f a proportion, M, of the electrons passing through the base+emitter junction also pass through the base+collector junction then the currents flowing in an n+p+n transistor are as shown in igure !."%b'.

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    Transistors operation

    Figure 6.3)

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    Transistor symbols

    • Symbols are used to represent p+n+p and n+p+ntransistors in circuit diagrams and are as shown inigure !."".

    • The arrow head drawn on the emitter of the symbol isin the direction of conventional emitter current %holeflow'.

    • The potentials marked at the collector, base and

    emitter are typical values for a silicon transistorhaving a potential difference of !0 between itscollector and its emitter.

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    Transistor symbols

    Figure 6.33

    ! i b l

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    !ransistor symbols

    • The voltage of ?.!0 across the base and emitteris that re5uired to reduce the potential barrierand if it is raised slightly to, say, ?.! 0, it is

    likely that the collector current will double toabout mA.

    • Thus a small change of voltage between theemitter and the base can give a relatively largechange of current in the emitter circuit1 becauseof this, transistors can be used as amplifiers.

    i i

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    Transistor connections

    • There are three ways of connecting a transistor, dependingon the use to which it is being put.

    • The ways are classified by the electrode that is commonto both the input and the output. They are calledK

    •%a' common+base configuration, shown in igure !."*%a'• %b' common+emitter configuration, shown in igure!."*%b'

    • %c' common+collector configuration, shown in igure!."*%c'

    • These configurations are for an n+p+n transistor.• The current flows shown are all reversed for a p+n+p

    transistor .

    i i

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    Transistor connections

    Figure 6.3

    Relation between I I 2 I

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    Relation between IE%IC 2 IB

    • The various current components that flow within a transistor areK• The emitter current E.

    • The base current 2.

    • The collector current #.

     E I 2 )#  • Emitter to collector current gain Nd.c is the ratio of collector

    current to emitter current

     E 

    cd 

     I 

     I =

    .α 

     

    .'")

    4 l ti b t 7

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    4elation between E,# 7 2

    • Substituting for EI#)2 in e5uation %!."'

      we get

    cd .β 

     B

    cd 

     I 

     I +

    =

    >

    >.α 

     BC 

    cd 

     I  I 

     I 

    +

    =.

    α 

    %!.C'

    • 3here is the base to collector gain,

    cd 

    cd 

    .

    .>

    >

    >

    β 

    α 

    +

    = %!.!'

    %!.*'

     B

    cd 

     I 

     I =

    .

    β 

    4 l ti b t 7

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    4elation between E,# 7 2

    Rearranging euation .'"' we get

    cd 

    cd 

    cd 

    .

    ..

    >   α 

    α β 

    −= %!.8'

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