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Unit I

p-n Junction Diode

1. Introduction

Nature has given us three kinds of materials called insulators, conductors andsemiconductors. An insulator is a very poor conductor of electricity. Conductors are mostly

metals, which are excellent conductors of electricity. There are some substances whose

conductivity lies between these two extremes. They are called semiconductors.

Semiconductors are originally non-conductors but start behaving as conductors under certain

operating conditions. It is this beauty of dual behavior that makes them very useful in

electronics. Semiconductors like silicon and germanium belong to the fourth group of the

periodic table of elements. A material belongs to one of the above 3 categories depending

upon its energy-band structure.

1.1 The Energy-Band Concept

Most of the natural elements are crystalline in structure. The structure of a crystal

consists of a spaced array of atoms in three dimensions with some kind of regularity and

repetition. Unlike the electrons in the inner shells of an atom, the outer shell electrons are

influenced by their neighboring atoms. They satisfy their stable valance configuration by

sharing electrons from the neighborhood. Due to this sharing (coupling) of outermost

electrons, the energy level of an isolated electron can not be accurately specified. The energy

levels of a group of electrons is spread into bands of closed and spaced energy states.

1.2 Insulators, Conductors and Semiconductors

Insulators

The energy band structure of diamond (a known insulator) at its normal lattice spacing

is indicated in fig.1.1(a). The band structure is split into two distinct energy bands called

conduction band and valence band with a wide gap between them called Forbidden Gap of

about 6eV. For insulators, the conduction band is almost empty and the valence band is full.

A material cannot conduct electricity unless there are enough number of electrons occupying

energy states in the conduction band. If the electrons in the valence band, which is full, have

to escape to the condition band, they require an energy of about 6eV. The average energy

possessed by an electron at room temperature ( )0300 K is only 26 milli-electron volts.Therefore, an insulator like diamond with very few electrons in the conduction band is an

extremely good insulator. However, at higher temperatures, the number of electrons with

enough energy to cross the forbidden gap increases and makes them available for conduction.Thus the conductivity of a good insulator, such as diamond, increases with temperature.

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Fig. 1.1 Energy band structure of (a) an insulator, (b) a conductor, (c) a semiconductor

crystal at their normal lattice spacing

2

2

ForbiddenGap

GE 6eV=

(Full)

Valence band

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Conductors

In conductors, the outer shell electrons are loosely bound to their parent nuclei and

possess enough energies. There is no forbidden gap. The conduction and valence bands

overlap as shown in fig. 1.1(b). Therefore with a little energy provided externally, they

become capable of conducting electricity. Common metals like gold, silver, copper andaluminum are good conductors.

Semi-conductors

The semi-conductor materials have a relatively small forbidden energy gap of about

1eV as shown in fig.1.1(c). The most important semiconductor materials are Silicon (Si) and

Germanium (Ge) with forbidden energy gaps of 1.1eV and 0.72eV respectively at 0300 K .

Energies of this order cannot be acquired by valence electrons at room temperature.

Germanium and silicon behave as insulators at low temperatures. However, their conductivity

increases with temperature.

It is important to know what energies are possessed by the electrons, whenever we

want to make them available for conduction. The energy distribution function gives this

information. Let Edn represent the number of free electrons per cubic meter whose energy lies

in an interval dE. If E is the density of electrons in this energy interval, we can write

E Edn dE= -- (1.1)

Here it is assumed that the density of electrons is a particular material (electrons per

cubic meter) is constant. However, within each unit volume of a material, electrons may have

all possible energies. It is this distribution of energy E which is of interest to us. Thefunction E can be expressed as

( ) ( )E f E N E = -- (1.2)Where N(E) is the density of the energy states (number of states per electron volt per cubic

meter) in the conduction band and f(E) is the probability that the state with energy E is

occupied by an electron. F(E) is called the Fermi-Dirac Probability function and specifies

the fraction of all states at energy E (electrons volts) occupied, under conditions of thermal

equilibrium. It has been statistically found that

( ) ( )FE EKT

1f E

1 e

=

+

-- (1.3)

Where k=Boltzman constant 5 08.620 10 eV / K = T=Absolute temperature 0 K

EF=Fermi level for the crystal in eV

1.3 Electrons and holes in an Intrinsic Semiconductor

An intrinsic conductor is a purified version of the semiconductor. Germanium and

silicon are mostly used in the fabrication of electronic devices. They have a tetrahedral crystal

structure with an atom at each vertex. A two dimensional picture of a Germanium crystal is

shown in fig. 1.2. Each atom of Germanium has four valence electrons. For achieving a

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chemically stable configuration of 8 outermost electrons, each atom shares one electron each

from four of its neighboring atoms.

Fig. 1.2 Two dimensional crystal structure of Germanium

Since the valance electrons of each atom are bonded with four of its neighbors and also with

its nucleus, they are not free and cannot contribute to current flow. Such a pure semiconductor

has a very low conductivity and behaves as an insulator at 00 K. However, at room

temperature (say 0300 K ) some of the covalent bonds will be broken due to thermal energies

available and the liberated electrons wander in a random fashion throughout the crystal. One

such free electron is shown in fig.1.3.

The positive charge remaining at the site of the broken covalent bond due to the

liberated electron is given the name hole. A hole is associated with a positive charge, the

energy required to break the covalent bond is GE which is about 0.72eV for Germanium and

1.1eV for Silicon at room temperature.

Fig. 1.3 Free electron and a hole generated by a broken covalent bond

1.4 The Mechanism of Conduction by Holes

We have seen that when a covalent bond is broken, a hole (absence of electron) is

formed. It is easy for a valence electron in a neighboring atom to leave its covalent bond to

fill this holed as shown in fig 1.4.

Fig. 1.4 The mechanism of hole contributing to conduction. Here a circle with a dot

represents a completed bond and an empty circle represents a hole

There is a hole in fig. 1.4 (a) at ion 4. an electron at site 5 can move into the hole at ion 4. As

a result, the hole has moved to site 5 as shown fig. 1.4(b). A new hole is formed while the

earlier hole is filled due to recombination. This process repeats continuously and gives rise to

hole movement in the valence band. In reality, it is electrons that move in opposite directions.As far as the current flow is concerned, the hole behaves like a positive charge equal in

magnitude to the electron charge. We can therefore consider the holes as physical entities

whose movement in the valence band constitutes a flow of current. Due to the charge packed

environment of the valence band, the whole mobility is relatively less compared to the

electron mobility. It acquires a velocity proportional to the field and the proportionality factor

is called mobility.

Velocity (v) = mobility ( ) field intensity (E)

In a pure semiconductor the number of holes and electrons are equal. Due to thermal

agitation, new hole-electron pairs are generated while the earlier electron-hole pairs disappeardue to recombination.

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Conductivity of Semiconductor

Once an electron-hole pair is created, the two particles move in opposite directions

when an electric field E is applied. Through the two particles are of opposite sign, the current

flow is in the same direction. Conventionally the direction of hole movement is the directionof positive current. The current density J through a semiconductor is given by

( ) 2n pJ n p eE E Amp / m= + = --- (1.4)Where n=free electron concentration, 3m

p=hole concentration, 3m

= conductivity (mhos/meter)n

= electron mobility 20.38m / Vsec (for Ge)

p = hole mobility 20.18m / Vsec (for Ge)E=electric field intensity in V/m

e=electronic charge in coulombs ( )19

1.6 10 C

Hence the conductivity of an intrinsic semiconductor is given by

( )n pJ

n p emE

= = + J

for an intrinsic semi-conductor we also have n=p= in

where in is the intrinsic concentration in3m

Properties of some important semi-conductors at 0300 K is given in Appendix-1

Appendix-1Table A1 Properties of Germanium and Silicon at 0

300 K

Appendix-1

Property Ge Si Units

Atomic Number 32 14

Atomic Weight 72.6 28.1

Density3K g / m 35.32 10 32.33 10 3K g / m

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Table A1

Relative Permittivity 16 12

Atoms/m3 4.4 X 1028 5 X 1028

EGOForbidden gap at 0oK 0.785 1.21 eV

EG Forbidden Gap at 300oK 0.72 1.1 eV

Intrinsic concentration ni 2.5 x 1019 1.5 x 1016 m-3

Intrinsic resistivity 47 x 10-2 2300 - m

Electron mobility n 0.39 0.135 m2/V S

Hole mobility p 0.19 0.045 m2/V S

Electron diffusion constant,

Dn at 300oK 99 x 10-4 39 x 10-4 m2/S

Effective density of states

In conduction band Nc 1.04 x 1025 2.8 x 1025 m-3

Valence band, NV 6.0 x 1024 1.04 x 1024 m-3

Table 1.1 some properties of Ge and Si.

1.5 n Type Semi Conductors

The process of adding impurity atoms by displacing some of the Si(Ge) atoms iscalled doping Fig.1.5 shows the crystal structure when a fifth group impurity and displaces a

silicon atom. Four of the five valence electrons from covalent bonds with the surrounding

silicon atoms. The fifth electron is available for conduction. An intrinsic semiconductor when

doped with fifth group impurity atoms is called an n Type (extrinsic) semiconductor.

Fig. 1.5 Representation of n-type doped crystal

This thV group electron is so weekly bound to its nucleus that it can easily be excited

into the conduction band leaving behind the thV group ion (an atom with positive charge)

fixed in the crystal lattice. The energy states of such impurity ions lie just below the bottom

edge of the conduction band as shown in the energy band diagram offig.1.6.

Atoms of thV group impurity element are called donors and they retain a positive

charges as shown in Fig.1.6 Charge neutrality dictates that the total number of positive

charges must equal the total number of negative charges even when the semiconductor is

doped.

Fig. 1.6 Energy band diagram of an n-type semiconductor

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Therefore, D Dn = N P N+ -- (1.1)

Where n and ND are electron and donor concentrations per cubic meter, p is the

intrinsic concentration of holes (minority carriers) which is negligible. Equation 1.1 is

obtained by neglecting the intrinsic hold concentration p. Compared to ND (the positive ionconcentration) which is typically of the order of 10 20 to 1025 atoms per cubic meter (m-3). The

ionization energies (energies required for liberation of extra carriers introduced by dopant

materials) are different for different impurity elements (Table 1.1)

Table : 1.1 The ionization (binding) energies of donor impurities

The electron conductivity can be computed from the equation 1.2 Neglecting the term

p"p e" whose contribution to conductivity is negligible

e

n(n n p p) = + -----------------------------------1.2

We have n D nN e ---------------------------------1.3

In n type semiconductors, electrons are called majority carriers and holes are called

minority carriers.

1.6 p Type semiconductors:

If we add a small amount of one of the elements from the IIIrd group of periodic table

as an impurity wet get a p type Semiconductor. Fig.1.7 shows the crystal structure when

one such impurity atom displaces a silicon atom. As the three valence electrons from covalent

bonds with three among the surrounding silicon atoms, to complete the covalent bond one

electron is attracted from the valence band as shown in fig.1.7. As a result, the IIIrd group

impurity atom becomes a negative acceptor ion, and net positive charge is left in its valence

band. A hole is then created. A neighboring electron from the same valence band can

complete the covalent band occupying this hole and the hole will move to a new site. Thisprocess continues and gives rise to hole conduction in the valence band. Very little energy is

required to an impurity atom to accept an electron; it is called an acceptor impurity. The

energy state of this added acceptor impurity ion is just above the top of the valence band.

The ionization energies for various III rd group acceptor impurity elements are given

in Table 1.2. At 300oK these energies are available and all the acceptors atoms are ionized

contributing one hold each to the conduction process. Boron is the most common acceptor

material used in semiconductor industry.

Table 1.2 The Ionization (binding) energies of acceptor impurities

Element Ionization Energy

Phosphorous 0.0444 eV

Arsenic 0.049 eV

Antimony 0.039 eV

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Element Ionization Energy

Boron 0.045eV

Gallium 0.065eV

Indium 0.16eV

Fig.1.7

Fig.1.8

Changes neutrality requires that = A Ap N n N --------------1.2= +

We are justified in neglecting the intrinsic negative concentration compared to N A

which is typically of the order of 1020 to 1025 atoms/m3. The conductivity can now be solely

due to holes and equation 1.2 reduces to

p A pN e ---------------1.3 =

In p type semiconductors holes are majority carries and electrons are majority

carriers and electrons are minority carries.

Example: 1.2

Compare the electron and hole concentration, in ap type silicon semiconductor with 2

x 1022 boron atoms per m3 Also compute its conductivity .

Solution:

We are given that

22 3p N 2 10 holes / mA

= =

( )16 3210 31

22 3

1.5 10 mnn 1.25 10 / m

p 2 10 m

= = =

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p A pN e =

= ( 22 x 1022) x (1.6 x 10-19) x ( 0.045)

=0.144X103 - m = 0.144 x 103 mho m

For intrinsic semiconductors, the Fermi level EF lies in the middle of the energy gap.

If the donor type impurity (n-type) is added to the crystal, we can see from fig.1.6 that

at any given temperature all donor atoms are ionized as the first N D energy states in the

conduction band will be filled. It will now be more difficult for the electrons from the valence

band to overcome the energy gap by thermal agitation. As a consequence, the number of

electron hole pairs thermally generated for the given temperature will be reduced. EF, then

must move closer to the conduction band indicate that most of the energy states in that band

are filled by the donor electrons and fewer holes exist in the valence band as shown in

Fig.1.9. In a similar manner it can be deduced that EF must move closer to the valence band

for a p-material.

Fig 1.9

1.7 Carrier Life Time:

As we have already discussed, thermal agitation produces new electron-hole Paris

while other electron-hole pairs disappear as a result of recombination. On an average the

electron or hole will exist for a time n or p seconds before recombination n ( p ) is called

the Mean Life Time of the electron (hole). Carrier life times, range from few nanoseconds tohundreds of micro seconds. n and p are important parameters in semiconductors becausethey indicate the time required to return to their equilibrium concentrations after a change in

their concentrations takes place due to any reason. Also the device designer can obtain desired

carrier lifetimes by introducing metallic impurities like gold in the semiconductor. These

impurities are capable of inducing new recombination centers which provide energy states in

the forbidden gap. Majority and minority carries in a specific region of a given semiconductor

specimen will have the same life time .

Let Pno and nno be the equilibrium concentrations of holes and electrons in the

specimen semiconductor.

Let no nop and n be the equilibrium concentrations during the exposure to light

radiation. Because the hole and electrons concentration in crease by the same amount.

no no no nop = P n - n ----------------- 1.4=

If the light source is turned off, the concentration should again reach their equilibrium

value decreasing exponentially with a time constant p n= = ( )

This result has been experimentally verified. At any given time t, we can write

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( )

( )

p

n

1/

n no no no

t/

n no no no

P P = P - P e ---------------1.5

n - n = n - n e -----------------1.6

From equations (2.20) and (2.21) we can get the expression for the rate of change (decrease)

in concentration of charges for holes, we get

n nonn no

p

p pdp d(p p ) ..................1.7

dt dt

= =

Similarly for electrons

n non

n no

n

n ndn d(n n ) ..................1.8

dt dt

= =

The negative sign indicates that it is decreasing

1.8 Diffusion and the Einstein equation

It is possible to have the concentration of charges vary with distance in

semiconductors. In such cases there is a concentration gradientdp

dxin the density of charge

carriers. At any cross section of the semiconductor material, the density on one side of the

cross section can be larger than that on the other side. In a given time interval there will then

be a net transport of charges from the higher density side to the lower density side. Causing a

current flow. This process is called diffusion. The diffusion current density J (amperes/squaremeter) is proportional to the concentration gradient. For holes, it is given by

dp 2

p p dxJ = -eD amp / m ...........................1.9

Where Dp is called the diffusion constant for holes. For electron diffusion current density.

2dnn n dx

J = -eD amp / m ...........................1.10

Diffusion and mobility are thermo-dynamic processes and they are not independent. The

relation between D and is given by the famous Einstein equation:

p n

T

p n

D DV ....................................................................1.11= =

Where TT

V11,600

=

VT is called the volt equivalent of temperature.

VT = 0.026 V at T = 300o K

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Also at 300 o K, we can establish the relation between and D as = 39 D

1.9 Diffusion Length

When charge carriers are injected into a semiconductor, a quantity called diffusionlength L is defined for the injected charge carriers as: the distance into the semiconductor at

which the carriers concentration falls to 1e of its value at x = 0. LP represents the diffusion

length for injected holes while Ln represents that for injected electrons. The diffusion length is

also interpreted as the average distance traveled by an injecsted carrier before recombining

with an opposite charge. It depends upon the diffusion constant and mobility as given by the

relation.

P p P

n n n

L D

and

L D

=

=

Example: 1.3

An n type Germanium semiconductor bar has a mobility of 0.39 m 2/V-sec and

0.19m2/V-Sec for electrons and holes at room temperature. What are the values of diffusion

constants for electrons and holes and what are its units?

Solution:

From Einstein relation of equation (1.11)

P nT

p n

D DV= =

at room temperature (3000 K)

= 0.026 volts

Dn = n TV= 0.39 x 0.026

=0.01 m2/sec.

And

DP = p TV=0.19 x 0.026

=0.005m2/sec

1.10 Hall Effect:

Hall effect enables to determine whether a particular semiconductor piece is of n type or

p type.

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When a metal or semi conductor specimen carrying a current I is placed in a

transverse magnetic field B, then electric field E is induced in the specimen in a direction

perpendicular to both I and B. This phenomena is called Hall effect.

Fig.1.10

Determination of the nature of a semiconductor:

Let a semiconductor bar carry currentIin the positiveXdirection. Let a magnetic field

B be applied in the positiveZdirection. Then according to Hall effect, a force gets exerted on

the charge carriers in the negative Y direction. Hence irrespective of the nature of charge

carriers (Whether holes or electrons), the charges get pressed downwards i.e., towards face 1

of the specimen.

In an n type specimen, current is carried almost fully by electrons. These electrons

as a result of Hall effect accumulate on face 1, which gets negatively charged relative to face2. A potential difference develops between face 1 and face 2 called Hall voltage. In a p-type

specimen the Hall voltage is positive at face 1. Hall effect can also be used to determine the

mobility of charge carriers.

Fig.1.11

Under equilibrium conditions, the force on the carrier due to electric field is equal to

force exerted on it by magnetic field.

eE= eBv--------------------------------------------------------------1.15

But

H

H

VE

d

V E d

=

= (d is the distance between two faces 1 and 2)

Current density1 1

J v -------------------------1.16A Wd

= = =

( is charge density, Wis the width of the specimen along the magnetic field.

2.P N Junction:

A p n junction is formed when a p type semiconductor is brought in contact with

n type semi conductor through a fabrication process. There are a number of ways in which a

p n junction can be fabricated. Four of the methods are briefly described belowl Generally a

crystal pulling process is used. Impurities of P and N type are alternatively added to the

molten semiconductor material and pulled. Ion this method, the mother intrinsic

semiconductor is amalgamated firstly with trivalent (p type) and then with Pentavalent

impurities (n type). Then the mother crystal is sliced at right angle to the plane of junctions.

The junction thus formed is called a grown junction. Each such slice is called a p-n diode.

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Pn junction diode is symbolically represented as follows with the arrow indicating the

direction of conventional current flow.

P N

Fig. 1.12 symbol of a diode. The arrow indicates the4 direction of conventional current flow.

2.1 Qualitative theory of p-n Junction diode under open circuit conditions

Let us have a look sat fig. 1.7.

As soon as a p-n junction is formed, the following events take place.

a. Holes from p-region diffuse into the n-region. They combine with the free

electrons in the n-region.

b. Free electrons from the n-region diffuse into the p0region. These electrons

combine with the holes.

c. This diffusion takes place because there is a difference in their concentrations in

the two regions and also because they move haphazardly due to thermal energy.

d. The diffusion of holes and free electrons across the junction occurs for a very short

time. After a few crossings recombination of holes and electrons in the immediate

neighborhood of the junction takes place, a force of restraint is set up

automatically.

e. Each recombination eliminates a hole and a free electron. After this sweep of

majority carriers, few negative acceptor ions in the p-region and few positive

donor ions in the n-region in the immediate neighborhood of the junction are left

uncovered (uncompensated). Additional holes trying to diffuse into n-region are

repelled by the uncompensated positive charge of the donor ions. The electrons

trying to diffuse into the p region are repelled by the uncompensated negative

charges of the acceptor ions. As a result recombination of holes and electrons

cannot occur totally.

f. The region containing uncompensated acceptor and donor ions is called depletion

region i.e., there is a depletion of mobile charges in this region. Since this region

has immobile ions which are electrically charged, it is also referred to as the space-charge region. The electric field between the acceptor and the donor ions is called

a barrier. The physical distance from the starting of one side of the barrier to the

ending of the other side is referred to as the width of the barrier. The difference of

potential from one side of the barrier to the other side is called barrier potential.

The p-n junction, thus formed is shown in fig.1.7(a)

For a silicon p-n junction, the barrier potential is about 0.7V, where as for germanium

it is about 0.3V.

g. The minority carriers are constantly generated due to thermal energy. The electric

field built up across the junction pointing from n-region to p-region tends to send adrift current (due to minority carriers) across the junction. The drift current exactly

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counter balances the diffusion current. In the absence of an applied bias voltage,

the net flow of charge in any one direction for a semiconductor diode is zero.

Fig 1.13 (b) shows the general shape of the charge density e which depends upon how

the diode is doped. Since the region of the junction does not have any mobile chares, it is

called the charge depletion region or space charge region or transition region. Thethickness of this region is about 0.5 micron (or micro meters).

2.1.1 Electric Field Intensity: the charge density is zero at the junction. It is positive to the

right of the junction and negative to the left. When equilibrium is attained there is no

movement of charge across the junction. The field intensity variation is proportional to

the integral of the charge density variation.

From poisons equation

2

d v

dx

2 =

------------- (1.21)

Where 0 r= is the permittivity of the semiconductor material. r is the relativepermittivity and 0 is the permittivity of free space.

The field intensity is obtained by integrating equation (1.19)

dvE dx

dx

= =

-----------(1.22)The electric field intensity variation is shown fig.1.7(c)

2.1.2 Potential: As shown in fig.1.13 (d), the potential variation in the depletion region is

the ve integral of the field intensity variation shown in fig. 1.13(c). This variation

indicates the formation of a potential energy barrier against the further diffusion of

holes into the n-region. Similarly potential energy barrier for electrons is shown infig.1.13(e).

The electric potential v is given by

V F.dx= ---------(1.23)

Figures

Fig. 1.13 Schematic picture of a p-n junction diode indicating (a) structure, (b) chargedensity distribution, (c) field intensity distribution, (d) potential energy barrier for the

hole, (e) potential energy barrier for the electron.

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2.2

2. p-n Junction Diode in Reverse Bias

The p-n junction is reverse biased when a battery of voltage V volts is connected

across the terminals, with negative terminal connected to pside and positive terminal to n-

side. This polarity of the battery connection causes both the holes in p-region and electrons inn-region to move away from the junction. Consequently, the width of the depletion region

increases on either side of the junction. But there are very few holes in n-region and very few

free electrons in the p-region, thermally generated that cross the junction. An extremely small

current flows due to these thermally generated holes and electrons. This current is called the

diode reverse saturation current and its magnitude is generally represented by 0I . The

magnitude of 0I is independent of the reverse bias but increases with increase in temperature.

It is in the order of A for Ge and nA for Si.

Fig. 1.14.

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UNIT II

RECTIFIERS AND FILTERS

2.0 Introduction

Almost all electronic circuits operate on d.c. power supply. Let us consider a transistor

receiver which operates on 9V d.c. to be provided by an eliminator. The eliminator is a circuit

that eliminates the varying components of the a.c. power available from our supply mains and

provides a steady d.c. at the required voltage. Through we can use dry cells or wet batteries,

they dont always serve the purpose due to their low supply voltages, initial cost, running

costs and maintenance. So it is better to depend on our a.c. mains which is always available

and economical to get the required d.c. voltages.

The process of extracting the required d.c. power from a.c. mains is shown in the block

diagram of fig. 2.1.

2.1 p.n Junction as a Rectifier:

As seen from chapter 1, there is a low resistance conducting path for can cut when the

diode is forward biased. Under reverse bias there is hardly any current (except the reverse

saturation current which is extremely low compared to the current . forward bias).

This properly of the PN junction diode utilized is a circuit called rectifier. If an alternating

voltage is applied to a PN junction diode, current flows only during the positive ------ half

cycle of the input waveform. No or current flows during the negative half cycle because the

diode gets reverse biased. The current through the diode will be unilateral (only in +ve d)

or simply rectified current. The voltage developed across a reason due to this current flow

will.. be unilateral or rectified.

Fig. 2.1 Block diagram of a regulated power supply

It involves three steps.

i. Rectification: Which converts the a.c. into unidirectional current/voltage. Half wave

rectifiers, full wave rectifiers and bridge rectifiers are some of the circuits used forrectification.

ii. Filtering: which smoothens the ripples in the uni-directional current/ voltage provided

by a rectifier. Simple inductor and capacitor filters, L section, section, multiple L and section filters help us in achieving the required smoothening.

iii. Regulation: regulation is the process of maintaining the output voltage/current at the

output of filters at a steady value irrespective of changes in the mains supply voltage,

component tolerances and the varying demands of the load (appliance). Simple Zener diode

regulator, series and shunt regulators are some of the circuits that regulate d.c. power obtained

in steps (i) and (ii)

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--------------------

2.2 Half wave rectifier (HWR)

A rectifieris an electrical circuit which offers a low resistance to the current in one

direction and offers high resistance in the direction opposite to it.

The domestic supply waveform is sinusoidal in nature and the average value of a

sinusoidal wave is zero. The rectifier is capable of converting an input sinusoidal

waveform into a unidirectional waveform with a non-zero average value.

A half wave rectifier circuit uses a single diode in achieving the rectification. The half

wave rectifier circuit, its input waveform and the output waveform are shown in fig.

2.2.

Rectifiers, Filters and Regulators

Fig. 2.2. The HWR circuit

The basic HWR circuit consists of an input transformer, a rectifying device and a

resistance LR to act as a load.

The transformer can be step-up or step-down depending on the need. It also provides

isolation from the mains-supply which is an important safety aspect for the users. The

transformer is internally well shielded to prevent unwanted electrical noise from

entering the dc power supply circuit connected to the secondary.

The rectifying device is a p-n diode which has a piece-wise linear approximation as

shown in fig. 2.3. the device has infinite resistance in the reverse direction and a small

and constant resistance fR in the forward direction (for yV V> ). A p-n diode isnormally used for low voltages. For high voltages vacuum tube diodes are used.

Analysis

Assuming the ideal diode characteristic of fig. 2.3, the current I through the

diode and LR is

Fig. 2.3 Piecewise linear (ideal) approximation of the p-n diode characteristic

I = ( )0 0mI sin t, for 0 t 0 to180

( )0 00, for t 2 180 to360= ------ (2.1)

As shown in the figure 3.2(c)

The peak input currentm

max

f L

VI

R R=

+ ------ (2.2)

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When the diode is non conducting during to 2 , the transformer secondary voltage

iv will appear across the load resistance.

Mathematically,

f m fV i.R I R sin t= =

mV sin t= ( )t 2 (2.7)Fig. 3.4 shows the plot of the voltage across the diode. The voltmeter reads dcV , which

is given by2

dc m f m0

1V I R sin t d t V sin t d t

2

= +

[ ]m f m1

I R V=

( )m f m f L1

I R I R R = + from equation (2.2)

( ) m LDC dc LI R

V HWR I R

= = --- (2.8)

Fig. 2.4 voltage across the diode of the HWR circuit

From the above discussion 3 points are to be noted

i) The dc voltage dcV is dc LI R , which is negative the ve sign is justifiedbecause of the fact around the complete loop of a circuit must always add

up to zero.

ii) ( ) dc fdc HWRV I .R

Because the P-N diode is a non-linear device whose resistance is not the

same during the entire cycle.

The diode conducts only for 0 to and offers a low resistance where as itoffers high resistance during to 2 )

iii) The voltage across LR is dc LI .R because the load resistance LR is a linear

component and its value is constant thorough out the cycle.

The a.c power input to the HWR

The average value of the instantaneous power supplied by the mains is computed as2

i0

1Pi v .id t

2

= ---(2.9)

For the HWR, ( )i f S Lv i R R R = + + for 0 t

( SR = resistance of the transformer secondary)

( ) ( )Pi 2

f S LHWR 0

1i R R R d t

2

= + +

( )2 2m f S L0

1I sin t R R R d t

2

= + +

( )2m

f S L

IR R R

4= + + from (2.4)

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( )2rms f L SI R R R = + + ---- (2.10)

( )2

mL f S L

IR if R R R

2

= +

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from equation (2.15) and (2.16), we get

md.c L

f S L

V 1V . R

R R R

= + +

S f L f Sm

f S L S f L

R R R R R V

R R R R R R

+ + +=

+ + + + f sm m

S f L

R RV V.R R R

+=

+ +

( )m d.c f SV

I R R= +

---- (2.17)

Equation 2.17 is an equation of a straight line with an intercept equal to mV

and slope given

by ( )f SR R+ . The regulation characteristics is shown in fig. 3.5.

Fig. 2.5 regulation characteristic of a HWR

At no load i.e., d.cI 0=

md.c

VV =

and it decreases linearly with the increasing of the factor ( )d.c f SI . R R + . The

greater this ( )f SR R+ , the larger this decrease will be fR can be called as the effective

internal resistance of the power supply using the HWR circuit, if the d.c resistance SR is not

taken into account.

2.10

Example 2.11: Given the value ( )f SR R 15+ = . Find the % regulation of a 9V, 200mAHWR?

Solution:

The output voltage at no load is 9V.

The HWR is required to draw a maximum load current of 200mA. The full load d.c voltage

can be computed as

( )fullloadV 9 0.2 15 6V= =

The % regulation =n o lo ad full load

fullload

V V100

V

9 6100 50%

6

= =

Readers may note that a higher % regulation implies bad regulation. A half wave rectifier is

not a well regulated source of d.c power.

Ripple factor

As soon from fig. 2.2, the conversion from an alternating current into a unidirectional current

in a HWR circuit does not give a steady d.c current. Periodically fluctuating components still

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remain in the output wave. A measure of the fluctuating components is given by the ripple

factor r which is defined as

rms valueof alternatingcomponentsof the waver

average valueof thewave=

Or

' '

rms rms

d.c d.c

I Vr

I V= = --- (2.18)

The terms'

rmsV and

'

rmsI represent the rms values of the a.c components of the voltage/ current

wave at the output of the HWR. Looking at the instantaneous values, we have'

dci i I=

Then ( ) ( )2 2'

rms d.c0

1I i I d t

2

=

( )2

2 2

d.c d.c0

1i 2I .i I d t

2

= +

The first term of the integral which is2 2

0

1 i .d t2

is

2

rmsI

Since2

d.c0

1i d t I

2

= by definition, the second term of the integral is

( ) 2d.c d.c d.c2I I 2I =

The rms ripple current'

rmsI is given by

' 2 2 2

rms rms d.c d.cI I 2 I I= +

2 2

rms dcI I=

Therefore, from equation (2.18), we have22 2

rms dc rms

d.c dc

I I Ir 1

I I

= =

---- (2.19)

Equation (2.19) is independent of the wave shape of the current and applies to all circuits, not

only HWR.

In the case of a HWR.

m

rms

mdc

II 2 1.57

II 2

= = =

And

( ) ( )2

HWRr 1.57 1 1.21= = ---- (2.20)

Equation (2.20) tells that in the case of a HWR circuit, the ripple component exceeds the d.c

output voltage/ current. Hence a HWR is seldom (rarely) used for converting a.c into d.c

voltage/current.

3.12 Ripple Frequency:

The number of half sinusoids per second in the output of a rectifier is often called ripple

frequency. In the case of a HWR there is one half sinusoid per cycle and there are 50 cycles

per second (50Hz) in our domestic supply. Therefore the ripple frequency of the HWR is

50Hz.

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Transformer Utilization factor (TUF)

The transformer utilization factor indicates the extent to which the transformer is utilized.

TUF is defined as the ratio.

d.c powerdelivered to theloadTUF

a.c power ratingof thetransformer= --- (2.22)

For the HWR,m

rms

VV

2= and mrms

II

2= and md.c

II =

2

mL

m m

IR

TUFV I

2 2

=

Neglecting the diode forward resistance and the transformer secondary resistance, we can

write

m m LV I R=

( )

2m

LHWR 2 2

m L

I 2 2TUF .R

I R

;

2

2 20.287=

; ----(2.22)

Form factor

The form factor of a wave is defined as the ratio of the rms value to the average value.

The form factor for half wave rectified wave form is therefore given by

( )

m

HWR

m

I

2form factor 1.57

I 2

= = =

---- (2.23)

Peak factor:

It is defined as the ratio of the peak value to the rms value of the output voltage for a

HWR

mm

Vpeak factor V 2

2= = ---- (2.24)

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