edc material 01.06.10
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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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