FIBER ATTENUATION

FIBER ATTENUATION
The maximum transmission distance between a transmitter and a receiver of an
optical fiber is known as the attenuation of the fiber
[16]
. The attenuation is usually
expressed in decibels per unit length (dB km
-1
) and can be determined by:
α dbL = 10 log10 (Pi / Po )
Where α db is the signal attenuations per unit length in decibels, L is the fiber length, Pi is
the input (transmitted) optical power into the fiber and Po is the output (received) optical
power
[17]
. Such transmission losses in typical fibers used today are less than 5 dB km
-1
verses the metallic wires used in the past with transmissions losses with significantly
higher losses.
In reducing the attenuation of the fiber, it cuts down on the costs since fewer
repeaters are required to restore the signal
[18]
. With this in mind two very important
techniques are considered when manufacturing an optical fiber with a specific
attenuation. The first technique involves purifying the material composition, which
reduces material absorption and Rayleigh scattering of the light rays within the fiber
[19]
.
The second is the preparation method of the fiber that must be done in a controlled
manner such as fiber drawing otherwise microscopic variation in the material density and
compositional fluctuations will result in light scattering in an optical fiber

Optical Fiber Snell's Law

When light is directed into an optical fiber the effectiveness of the wire depends
on its ability to guide the light ray far distances with little scattering or absorption of the
light as possible. Doing so means that the optical fiber must exhibit total internal
reflection within the wire. Thus when considering the propagation of light for an optical
fiber the refractive index of the dielectric medium needs to be accounted for. As light
rays become incident on an interface between two dielectrics with different index of
refractions, refraction occurs between the two mediums. This can be best described by
using Snell’s Law of Refraction which states:
n1 sin φ1 = n2 sin φ2
This equation shows that at certain angles partial internal reflection will arise, as well at
other angles total internal reflection will occur.

This relationship can then be used to find the critical angle φc which serves as the
limiting case of refraction and the angle of incidence
[5]
. By launching the light ray at an
angle φ > φc as seen in figure #2, it is reflected at the same angle to the normal, leading
to total internal reflection within the optical fiber. A typical optical fiber with two
dielectric mediums is shown in figure #2, with the silica core having the index refraction
of n1 and the silica cladding with a lower index of refraction of n2 . With this setup it is
possible to send packets of information through light rays which can propagate through
an optical fiber with very little loss or distortion. However other factors will influence the
effectiveness of the optical fiber due such things like impurities but this will be discussed
in detail in later sections of this paper.
















OPTICAL FIBER TYPES
There are 3 basic types of optical fibers: multimode graded-index fiber,
multimode step-index fiber and single-mode step-index fibers.
A multimode fiber can propagate hundreds of light modes at one time while single-mode
fibers only propagate one mode
[12]
as shown in figure #3.
Figure #3 Optical Fiber Modes
13
The difference between graded-index and step-index fibers is that in a gradedindex
fiber it has a core whose refractive index varies with the distance from the fiber
axis, while the step-index has core with the same refractive index throughout the fiber
[14]
.
Since the single-mode fibers propagate light in one clearly defined path,
intermodal dispersion effects is not present, allowing the fiber to operate at larger
bandwidths than a multimode fiber
[15]
. On the other hand, multimode fibers have large
intermodal dispersion effects due to the many light modes of propagations it handles at
one time. Because of this multimode fibers operate at lower bandwidths, however they
are typically used for enterprise systems such as offices, buildings, universities since they
are more cost effective than single mode ones.

Advantages of Optical Fibre

Let us see the advantages of optical fiber communication over conventional communication system.

1. Enormous Bandwidths

                             The information carrying capacity of a transmission system is directly proportional to the carrier frequency of the transmitted signals. The optical carrier frequency is in the range of 10¹⁴ Hz while the radio frequency is about 10⁶  Hz. Thus the optical fibres have enormous transmission bandwidths and high data rate. Using wavelength division multiplexing operation, the data rate or information carrying capacity of optical fibres is enhanced to many orders of magnitude.

 

2. Low transmission loss

                             

                               Due to the usage of ultra low loss fibres and the erbium doped silica fibres as optical amplifiers, one can achieve almost loss less transmission. Hence for long distance communication fibres of 0.002 dB/km are used. Thus the repeater spacing is more than 100 km.

 

3. Immunity to cross talk

 

                               Since optical fibres are dielectric wave guides, they are free from any electromagnetic interference (EMI) and radio frequency interference (RFI). Since optical interference among different fibres is not possible, cross talk is negligble even many fibres are cabled together.

 

4. Electrical Isolation

 

                              Optical fibres are made from silica which is an electrical insulator. Therefore they do not pick up any electromagnetic wave or any high current lightening. It is also suitable in explosive environment.

 

5. Small size and weight

 

                               The size of the fiber ranges from 10 micrometres to 50 micrometres which is very very small. The space occupied by the fiber cable is negligibly small compared to conventional electrical cables. Optical fibers are light in weight. These advantages make them to use in aircrafts and satellites more effectively.

 

6. Signal security

 

                               The transmitted signal through the fibre does not radiate. Unlike in copper cables, a transmitted signal cannot be drawn from a fiber without tampering it. Thus, the optical fiber communication provides 100%  signal security.

 

7. Ruggedness and flexibility

 

                               The fibre cable can be easily bend or twisted without damaging it. Further the fiber cables are superior than the copper cables in terms of handling, installation, stroage, transportation, maintenance, strength and durability.

 

8. Low cost and availability

 

                               Since the fibres are made of silica which is available in abundance. Hence, there is no shortage of material and optical fibers offer the potential for low cost communication.

 

9. Reliability

 

                              The optical fibres are made from silicon glass which does not undergo any chemical reaction or corrosion. Its quality is not affected by external radiation. Further due to its negligible attenuation and dispersion, optical fiber communication has high reliability. All the above factors also tend to reduce the expenditure on its maintenance.

Optical Fiber

An optical fiber is a thin, flexible, transparent fiber that acts as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of optical fibers is known as fiber optics. Optical fibers are widely used in fiber-optic communications, which permits transmission over longer distances and at higher bandwidths (data rates) than other forms of communication. Fibers are used instead of metal wires because signals travel along them with less loss and are also immune to electromagnetic interference. Fibers are also used for illumination, and are wrapped in bundles so they can be used to carry images, thus allowing viewing in tight spaces. Specially designed fibers are used for a variety of other applications, including sensors and fiber lasers.
Optical fiber typically consists of a transparent core surrounded by a transparent cladding material with a lower index of refraction. Light is kept in the core by total internal reflection. This causes the fiber to act as a waveguide. Fibers which support many propagation paths or transverse modes are called multi-mode fibers (MMF), while those which can only support a single mode are called single-mode fibers (SMF). Multi-mode fibers generally have a larger core diameter, and are used for short-distance communication links and for applications where high power must be transmitted. Single-mode fibers are used for most communication links longer than 1,050 meters (3,440 ft)

Total internal reflection

Main article: Total internal reflection

When light traveling in a dense medium hits a boundary at a steep angle (larger than the "critical angle" for the boundary), the light will be completely reflected. This effect is used in optical fibers to confine light in the core. Light travels along the fiber bouncing back and forth off of the boundary. Because the light must strike the boundary with an angle greater than the critical angle, only light that enters the fiber within a certain range of angles can travel down the fiber without leaking out. This range of angles is called the acceptance cone of the fiber. The size of this acceptance cone is a function of the refractive index difference between the fiber's core and cladding.
In simpler terms, there is a maximum angle from the fiber axis at which light may enter the fiber so that it will propagate, or travel, in the core of the fiber. The sine of this maximum angle is the numerical aperture (NA) of the fiber. Fiber with a larger NA requires less precision to splice and work with than fiber with a smaller NA. Single-mode fiber has a small NA.

[edit] Multi-mode fiber

The propagation of light through a multi-mode optical fiber.

A laser bouncing down an acrylic rod, illustrating the total internal reflection of light in a multi-mode optical fiber.

Main article: Multi-mode optical fiber

Fiber with large core diameter (greater than 10 micrometers) may be analyzed by geometrical optics. Such fiber is called multi-mode fiber, from the electromagnetic analysis (see below). In a step-index multi-mode fiber, rays of light are guided along the fiber core by total internal reflection. Rays that meet the core-cladding boundary at a high angle (measured relative to a line normal to the boundary), greater than the critical angle for this boundary, are completely reflected. The critical angle (minimum angle for total internal reflection) is determined by the difference in index of refraction between the core and cladding materials. Rays that meet the boundary at a low angle are refracted from the core into the cladding, and do not convey light and hence information along the fiber. The critical angle determines the acceptance angle of the fiber, often reported as a numerical aperture. A high numerical aperture allows light to propagate down the fiber in rays both close to the axis and at various angles, allowing efficient coupling of light into the fiber. However, this high numerical aperture increases the amount of dispersion as rays at different angles have different path lengths and therefore take different times to traverse the fiber.

Optical fiber types.

In graded-index fiber, the index of refraction in the core decreases continuously between the axis and the cladding. This causes light rays to bend smoothly as they approach the cladding, rather than reflecting abruptly from the core-cladding boundary. The resulting curved paths reduce multi-path dispersion because high angle rays pass more through the lower-index periphery of the core, rather than the high-index center. The index profile is chosen to minimize the difference in axial propagation speeds of the various rays in the fiber. This ideal index profile is very close to a parabolic relationship between the index and the distance from the axis.

[edit] Single-mode fiber

The structure of a typical single-mode fiber.
1. Core: 8 µm diameter
2. Cladding: 125 µm dia.
3. Buffer: 250 µm dia.
4. Jacket: 400 µm dia.

Main article: Single-mode optical fiber

Fiber with a core diameter less than about ten times the wavelength of the propagating light cannot be modeled using geometric optics. Instead, it must be analyzed as an electromagnetic structure, by solution of Maxwell's equations as reduced to the electromagnetic wave equation. The electromagnetic analysis may also be required to understand behaviors such as speckle that occur when coherent light propagates in multi-mode fiber. As an optical waveguide, the fiber supports one or more confined transverse modes by which light can propagate along the fiber. Fiber supporting only one mode is called single-mode or mono-mode fiber. The behavior of larger-core multi-mode fiber can also be modeled using the wave equation, which shows that such fiber supports more than one mode of propagation (hence the name). The results of such modeling of multi-mode fiber approximately agree with the predictions of geometric optics, if the fiber core is large enough to support more than a few modes.
The waveguide analysis shows that the light energy in the fiber is not completely confined in the core. Instead, especially in single-mode fibers, a significant fraction of the energy in the bound mode travels in the cladding as an evanescent wave.
The most common type of single-mode fiber has a core diameter of 8–10 micrometers and is designed for use in the near infrared. The mode structure depends on the wavelength of the light used, so that this fiber actually supports a small number of additional modes at visible wavelengths. Multi-mode fiber, by comparison, is manufactured with core diameters as small as 50 micrometers and as large as hundreds of micrometers. The normalized frequency V for this fiber should be less than the first zero of the Bessel function J₀ (approximately 2.405).

Fresnel's biprism

Interference by the Division of
the Wavefront
4.1 Theory
There are two methods for producing two coherent sources. One method
involves the division of the wavefront as in the Young’s double slit experiment
and the second method involves the division of the amplitude as in the
Micheslson interferometer. In this experiemnt we study two arrangements
for producing inteference by wavefront division and use these to determine
the wavelength of light. Refer to Chapter 13 of Jenkins and White or Section
9.3 of Hecht for a discussion of Lloyd’s Mirror and Fresnel’s Biprism.



Fresnel’s Biprism
The arrangement from Fresnel biprism is shown in Fig. (4.2). The biprism
consists of two small angle prisms joined together at their bases. When a
wavefront from the source S is incident, the upper portion of the wavefront is
refracted downward and the lower portion is refracted upward. The refracted
wavefronts appear to come from virtual sources S1 and S2. They overlap
in the shaded region and produce inteference fringes. If a be the separation
beteween S1 and S2 and d+D be the source to screen separation, the location
of the m-th bright fringe is given by
ym = mλ D + d
a , (4.8)
where y is measured from the center of the interference pattern. Just as in
the Lloyd’s mirror experiment, the wavelength of light can be determined
from measurements of fringe separation in the interference pattern.

Fringe Width

YOUNG’S DOUBLE SLIT EXPERIMENT A train of plane light waves is incident on a barrier containing two narrow slits separated by a distance’d’. The widths of the slits are small compared with wavelength of the light used, so that interference occurs in the region where the light from S1 overlaps that from S2. A series of alternately bright and dark bands can be observed on a screen placed in this region of overlap. The variation in light intensity along the screen near the centre O shown in the figure Now consider a point P on the screen. The phase difference between the waves at P is θ, where θ= 2π/λ ΔPo (where ΔPo is optical path difference, ΔPo=ΔPg; ΔPg  being the geometrical path difference.)   = 2π/λ  [ S2P - S1P ] (here λ = 1 in air) As     As, D >> d, S2P - S1P ≈  λ d sinθ sin θλ ≈ tanθ( = y/D). [for very small θ]  Thus, θ = 2π/λ (dy/D) For constructive interference, θ = 2nλ    (n = 0, 1, 2...) ⇒  2π/λ (dy/D) = 2nπ     ⇒ y = n λD/d Similarly for destructive interference, y = (2n - 1) λD/2d  (n = 1, 2 ...)   Fringe Width W  It is the separation of two consecutive maxima or two consecutive minima. Near the centre O [where θ is very small], W = yn+1 – yn [yn gives the position of nth maxima on screen]  = λD/d   Intensity Variation on Screen If A and Io represent amplitude of each wave and the associated intensity on screen, then, the resultant intensity at a point on the screen corresponding to the angular position θ as in above figure, is given by   I = Io­ + Io + 2√Io2 cosθ, When θ = 2π(dsinθ)/ λ = 4Io cos2 Φ/2 Illustration 1: A beam of light consisting of two wavelengths 6500 oA and 5200 oA is used to obtain interference fringes in YDE. The distance between the slits is 2.0 mm and the distance between the plane of the slits and the screen is 120 cm. (a)    Find the distance of the third bright fringe on the screen from the central maxima for the wavelength 6500 oA. (b)    What is the least distance from the central maxima where the bright fringes due to both the wavelengths coincide?   Solution: (i) y3 = n. Dλ/d = 3 x 1.2m x 6500 x 10-10m / 2 x 10-3m  = 0.12cm Let nth maxima of light with wavelength 6500 Å coincides with that of mth maxima of 5200Å. (ii) m x 6500Ao x D/d = n x 5200Ao x D/d ⇒ m/n = 5200/6500 = 4/5 Least distance = y4 = 4.D (6500Ao)/d = 4 x 6500 x 10-10 x 1.2/ 2 x 10-3m = 0.16cm

Brewster's Law

Brewster's angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist, Sir David Brewster (1781–1868).

When light encounters a boundary between two media with different refractive indices, some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the Fresnel equations, and is dependent upon the incoming light's polarization and angle of incidence.
The Fresnel equations predict that light with the p polarization (electric field polarized in the same plane as the incident ray and the surface normal) will not be reflected if the angle of incidence is

where n₁ and n₂ are the refractive indices of the two media. This equation is known as Brewster's law, and the angle defined by it is Brewster's angle.
The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light. One can imagine that light incident on the surface is absorbed, and then reradiated by oscillating electric dipoles at the interface between the two media. The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction of the dipole moment. Consequently, if the direction of the refracted light is perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles cannot create any reflected light.
With simple geometry this condition can be expressed as:

where θ₁ is the angle of incidence and θ₂ is the angle of refraction.
Using Snell's law,

one can calculate the incident angle θ₁ = θ_B at which no light is reflected:

Solving for θ_B gives:

Ruby laser

Ruby laser (Earliest solid-state laser)

Fig. 14 Schematic of a ruby laser. Capacitor bank is a cabinet containing many capacitors in parallel.  The resultant capacitance could be as large as a Farad.

· The first laser invented (in 1960).

         · Pump source: flash lamp.

         · Classic 3-level laser.

         ·  Very low repetition rate ~1 pulse/min.  Since its repetition rate is so slow, nobody wants to use this type of laser these days.

         · Laser wavelength 694.3 nm, pulse width ~ 10^-8 sec.

Characteristics of Lasers

Laser light has three unique characteristics, that make it different than "ordinary" light. It is:

• Monochromatic
• Directional
• Coherent

Monochromatic means that it consists of one single color or wavelength. Even through some lasers can generate more than one wavelength, the light is extremely pure and consists of a very narrow spectral range. Directional means that the beam is well collimated (very parallel) and travels over long distances with very little spread.
Coherent means that all the individual waves of light are moving precisely together through time and space, i.e. they are in phase.

Laser

The term “laser” is an acronym for (L)ight (A)mplification by
(S)timulated (E)mission of (R)adiation.
Lasers are devices that produce intense beams of light which
are monochromatic, coherent, and highly collimated. The wavelength
(color) of laser light is extremely pure (monochromatic) when compared
to other sources of light, and all of the photons (energy) that
make up the laser beam have a fixed phase relationship (coherence)
with respect to one another. Light from a laser typically has very
low divergence. It can travel over great distances or can be focused
to a very small spot with a brightness which exceeds that of the
sun. Because of these properties, lasers are used in a wide variety
of applications in all walks of life.



SPONTANEOUS AND STIMULATED EMISSION






In general, when an electron is in an excited energy state, it must
eventually decay to a lower level, giving off a photon of radiation.
This event is called “spontaneous emission,” and the photon is
emitted in a random direction and a random phase. The average time
it takes for the electron to decay is called the time constant for spontaneous
emission, and is represented by t.
On the other hand, if an electron is in energy state E2, and its
decay path is to E1, but, before it has a chance to spontaneously
decay, a photon happens to pass by whose energy is approximately
E24E1, there is a probability that the passing photon will cause the
electron to decay in such a manner that a photon is emitted at
exactly the same wavelength, in exactly the same direction, and
with exactly the same phase as the passing photon. This process is
called “stimulated emission.” Absorption, spontaneous emission,
and stimulated emission are illustrated in figure 36.2.
Now consider the group of atoms shown in figure 36.3: all begin
in exactly the same excited state, and most are effectively within
the stimulation range of a passing photon. We also will assume
that t is very long, and that the probability for stimulated emission
is 100 percent. The incoming (stimulating) photon interacts with the
first atom, causing stimulated emission of a coherent photon; these
two photons then interact with the next two atoms in line, and the
result is four coherent photons, on down the line. At the end of the
process, we will have eleven coherent photons, all with identical
phases and all traveling in the same direction. In other words, the
initial photon has been “amplified” by a factor of eleven. Note that
the energy to put these atoms in excited states is provided externally
by some energy source which is usually referred to as the
“pump” source.




Of course, in any real population of atoms, the probability
for stimulated emission is quite small. Furthermore, not all of the
atoms are usually in an excited state; in fact, the opposite is true.
Boltzmann’s principle, a fundamental law of thermodynamics,
states that, when a collection of atoms is at thermal equilibrium, the
relative population of any two energy levels is given by
where N2 and N1 are the populations of the upper and lower
energy states, respectively, T is the equilibrium temperature, and k
is Boltzmann’s constant. Substituting hn for E24E1 yields
For a normal population of atoms, there will always be more
atoms in the lower energy levels than in the upper ones. Since the
probability for an individual atom to absorb a photon is the same as
the probability for an excited atom to emit a photon via stimulated
emission, the collection of real atoms will be a net absorber, not a
net emitter, and amplification will not be possible. Consequently,
to make a laser, we have to create a “population inversion.”






POPULATION INVERSION
Atomic energy states are much more complex than indicated
by the description above. There are many more energy levels, and
each one has its own time constants for decay. The four-level energy
diagram shown in figure 36.4 is representative of some real lasers.
The electron is pumped (excited) into an upper level E4 by some
mechanism (for example, a collision with another atom or absorption
of high-energy radiation). It then decays to E3, then to E2, and
finally to the ground state E1. Let us assume that the time it takes
to decay from E2 to E1 is much longer than the time it takes to
decay from E2 to E1. In a large population of such atoms, at equilibrium
and with a continuous pumping process, a population inversion
will occur between the E3 and E2 energy states, and a photon
entering the population will be amplified coherently.

Cyclotron

In the cyclotron, a high-frequency alternating voltage applied across the "D" electrodes (also called "dees") alternately attracts and repels charged particles. The particles, injected near the center of the magnetic field, accelerate only when passing through the gap between the electrodes. The perpendicular magnetic field (passing vertically through the "D" electrodes), combined with the increasing energy of the particles forces the particles to travel in a spiral path.
With no change in energy the charged particles in a magnetic field will follow a circular path. In the cyclotron, energy is applied to the particles as they cross the gap between the dees and so they are accelerated (at the typical sub-relativistic speeds used) and will increase in mass as they approach the speed of light. Either of these effects (increased velocity or increased mass) will increase the radius of the circle and so the path will be a spiral.
(The particles move in a spiral, because a current of electrons or ions, flowing perpendicular to a magnetic field, experiences a force perpendicular to its direction of motion. The charged particles move freely in a vacuum, so the particles follow a spiral path.)
The radius will increase until the particles hit a target at the perimeter of the vacuum chamber. Various materials may be used for the target, and the collisions will create secondary particles which may be guided outside of the cyclotron and into instruments for analysis. The results will enable the calculation of various properties, such as the mean spacing between atoms and the creation of various collision products. Subsequent chemical and particle analysis of the target material may give insight into nuclear transmutation of the elements used in the target.

Mathematics of the cyclotron

[edit] Non-relativistic

The centripetal force is provided by the transverse magnetic field B, and the force on a particle travelling in a magnetic field (which causes it to be angularly displaced, i.e. spiral) is equal to Bqv. So,

(Where m is the mass of the particle, q is its charge, B the magnetic field strength, v is its velocity and r is the radius of its path.)
The speed at which the particles enter the cyclotron due to a potential difference, V.

Therefore,

v/r is equal to angular velocity, ω, so

And since the angular frequency is

ω = 2πf

Therefore,

The frequency of the driving voltage is simply the inverse of this frequency so that the particle crosses between the dees at the same point in the voltage cycle.

A pair of "dee" electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science. The particle exit point may be seen at the top of the upper dee, where the target would be positioned

This shows that for a particle of constant mass, the frequency does not depend upon the radius of the particle's orbit. As the beam spirals out, its frequency does not decrease, and it must continue to accelerate, as it is travelling more distance in the same time. As particles approach the speed of light, they acquire additional mass, requiring modifications to the frequency, or the magnetic field during the acceleration. This is accomplished in the synchrocyclotron.

Limitations of the cyclotron

The spiral path of the cyclotron beam can only "sync up" with klystron-type (constant frequency) voltage sources if the accelerated particles are approximately obeying Newton's Laws of Motion. If the particles become fast enough that relativistic effects become important, the beam gets out of phase with the oscillating electric field, and cannot receive any additional acceleration. The cyclotron is therefore only capable of accelerating particles up to a few percent of the speed of light. To accommodate increased mass the magnetic field may be modified by appropriately shaping the pole pieces as in the isochronous cyclotrons, operating in a pulsed mode and changing the frequency applied to the dees as in the synchrocyclotrons, either of which is limited by the diminishing cost effectiveness of making larger machines. Cost limitations have been overcome by employing the more complex synchrotron or linear accelerator, both of which have the advantage of scalability, offering more power within an improved cost structure as the machines are made larger.

Advantages of the cyclotron

• Cyclotrons have a single electrical driver, which saves both money and power, since more expense may be allocated to increasing efficiency.
• Cyclotrons produce a continuous stream of particles at the target, so the average power is relatively high.
• The compactness of the device reduces other costs, such as its foundations, radiation shielding, and the enclosing building.

2008(Jan) Physics-II

                                                 Unit-I

Q1. What is radioactivity ? Explain natural and artificial radioactivity.        2
Q2. Describe the construction of Aston's mass spectrograph with necessary theory and show how it can be     
used   in detection of isotropes.            2
Q3. What do you mean by nuclear reactor? Give the sketch of it. What is the function of control rod ? Write two factors which should be considered while selecting the site for nuclear reactor.    7
Q4. Write a short note on the structure of atomic nuclei(any four properties). Assuming that protons and neutrons possess equal masses, calculate how many times nuclear matter is denses than water if nuclear redius is given by 1.2 * 10^-15 A m where A is the mass number.

                                                  Unit-II

Q1. Write one difference between a hologram and an ordinary photograph.  2
Q2. Explain the terms : (i)Stimulated Emission   (ii)Population Inversion  (iii)Metastable state.     7
Q3. Give the principle of propagation of light through optical fiber.  Derive an expression for acceptance angle.   7
Q4. (i) A step index fiber is made with a core of index 1.52, a diameter of 29 μm and a fractional index difference of 0.0007. It is operated at a wavelength of 1.3 μm. Find V-number.
Q4. (ii) An optical filter has a line width of 1.5nm and a mean wavelength 550 nm.
With white light incident on the filter calculate coherence length and number of wavelengths in the wave tram.
 
                                                  Unit-III
Q1. What are crystalline and amorphous solids ?     2
Q2. Define unit cell.  Express coordination number and atomic packing densities for Simple Cubic Crystal(SCC), Face Centred Cubic Crystal(FCC), and Body Centred Cubic Crystal(BCC).    7
Q3. Define energy levels and energy bands. Explain with proper diagram,  how on the basis of band theory, solids are classified as conductors, insulators and semiconductors ?   7
Q4. (i)  Find the mobility of electrons in copper assuming that each atom contributes one free electron for conductor. Resistivity of Cu= 1.7 * 10^-6 ohm-cm, atomic wt. = 63.54,  density = 8.96 gm/cc.  7.
(ii) Lead is a face centred cubic with an atomic radius 1.746A. Find spacing for 220 planes.

  
                                               Unit-VI
Q1. Define Hall effect.          2
Q2. Draw a neat and labelled energy diagram of P-N junction at equilibrium. Derive the built in potential barrier:
         V = Vln ND
                      Ni2
Q3. What is superconductivity ? Explain Meissner effect. Derive Type - I and Type -II superconductors.
Q4. (i) Calculate the value of Hall angle '0' for a semiconductor on the basis of the following data :
           Rh = 3.66*10^-4m3/C (Hall coefficient resistivity)
           p = 8.33* 10^-3 ohm-m Bz=0.5 wb/m2(magnetic field)
(ii). In a transistor of common base connection current amplification factor is 0.9. If the emitter current is 1mA,  determine the value of base current


                                     Unit-V
Q1. Write one difference between polar and non-polar dielectrics.  2
Q2. What do you understand by dielectric constant ? Define dielectric susceptibility. Derive a relation between dielectric constant Er and dielectirc susceptibility.
Q3. What are hard and soft magnetic materials ? Indicate the properties sough in each case. Give their application.
Q4. There are 10^27 HCI molecule per cubic meter in a vapour. Determine the orientation polarization at room temperature i.e. 27 C. If vapour is subjected to an electric field of 10^6 V/m The permanent dipole moment of HCI molecule being 1.04 bye unit show that at this temperature and for such a high field the value of alph= μE               is very much less than unity debye unit =3.33*10^-30Cm.

              KbT       

Basic requirements for an acoustically good hall

1. Reflection

In completely free space, sounds travel outwards from their source with diminishing intensity until all the energy has been dissipated in the ever-widening wavefront or lost as heat in the air itself. By contrast, sound waves in a concert hall are repeatedly turned back on themselves and bounced in criss-cross patterns throughout the enclosed space. The audience therefore hears not only the direct sound but also a mixture of later—and weaker—sounds. These multiple reflections are delayed in accordance with the extra distance travelled; and they diminish in intensity through normal dissipation and absorption at each boundary reflection.
A near-perfect reflector such as a polished wood floor will reflect almost 100 per cent of the incident energy, but soft furnishings, porous fibre-tiles, or pliant panels will absorb part of that energy. In practice, the different kinds of absorber are frequency-selective, and good acoustic design depends on careful disposition of various absorbers to control reflections evenly at bass, middle, and treble frequencies.

2. Directional effects

Sound sources are described as directional or non-directional depending on whether they are physically large or small compared with the wavelength of the musical notes being radiated (see acoustics, 10). For similar reasons, reflectors of different sizes and shapes may modify the distribution, i.e. diffusion, of sounds throughout a room not only by frequency-selective reflection but also by re-radiating some bands of frequencies uniformly and others in a directional manner. If curved surfaces are essential, a convex shape is generally preferred because it tends to scatter sounds and help produce uniform listening conditions. Concave surfaces focus sounds back along their axis and give rise to local echoes or dead spots.
A domed ceiling is a classic example of the concave shape to be avoided. The Royal Albert Hall in London is perhaps specially unfortunate in having both an oval plan and a high domed ceiling focusing sounds on to parts of the audience area. The effects of marked echoes were complained of for many years until arrays of ‘flying saucers’ were suspended beneath the huge dome. The underside of the saucers is convex, to scatter the upward-travelling sound waves, and their tops carry absorbent material to capture any sounds that missed the saucers on the way up and rebounded from the ceiling. In the same way, recesses or coffering should be of generous proportions so that their scattering effect will be felt through most of the frequency spectrum.

3. Reverberation

Although members of an audience receive the direct sound followed by a wedge or ‘tail’ of countless reflected waves, they are not normally conscious of these as separate entities or echoes. The hearing mechanism (see ear and hearing) works in such a way that sound repetitions arriving within about ¹/₂₀; of a second of each other are run together and heard as one. Note, however, that ‘flutter echoes’ can arise between parallel walls.
The prolongation effect is known as ‘reverberation’. A smooth decay is to be preferred, secured by careful acoustic design to produce evenly diffused sounds. The time taken for sounds to fall to inaudibility is called the ‘reverberation time’ (strictly the time to fall to a millionth of its original value, or to −60 dB). Reverberation time increases in direct proportion to the volume (size) of the enclosure—the greater distances stretching the decay period—but is reduced by the introduction of absorbent materials. An audience also mops up sound energy quite effectively, so rehearsals in an empty hall sound much more reverberant than the actual concert. To reduce this difference, modern concert-hall seating can be designed so that each seat absorbs about the same amount of sound whether occupied or not.

Ads by Google

SoundPLAN

The world leader in noise and

air pollution planning software

www.soundplan.eu

Gustafs

Fire safety and acoustic properties

Scandinavian interior cladding.

www.gustafs.com

BKL Consultants Ltd.

Acoustics, Noise & Vibration

Sound advice since 1966

www.bkl.ca

CBSE Class 6-12 Exam Help

Get 100% Success in CBSE Exams

Sample Papers, Test & Assessment!

TopperLearning.com/Class6_12

4. Designing for good acoustics

For speech, the principal criteria for design are adequate loudness and a high degree of intelligibility. This suggests a short reverberation time; yet too dry an acoustic will lack the reflected energy needed to carry adequate sound levels to listeners at greater distances from the platform. Attention to room shape and seating layout is necessary, and a sloping or raked floor will help to give listeners in the back row a clear view of the speakers and a better chance of hearing properly.
For music, there are additional acoustic requirements, making acoustic design as much art as science. From an examination of existing halls generally rated as having ‘good acoustics’, Leo L. Beranek, for example, listed 18 criteria of quality in his book Music, Acoustics and Architecture. Historically, increasingly large halls have been built, with correspondingly greater reverberation times, as the size of orchestras has grown. Thus Baroque and chamber music are suited to a reverberation time of less than 1.5 seconds, Classical music about 1.7, and Romantic music about 2.2 seconds. A longer decay at low frequencies makes for fullness of tone or warmth, whereas good definition or clarity demands a rise at high frequencies.
Modern concert halls often incorporate some means of varying the reverberation characteristics to suit different musical or non-musical events. A good example is Symphony Hall in Birmingham (opened in 1991 ) where a movable circular canopy over the platform area directs sound towards specific regions of the auditorium, and reverberation chambers round the periphery can be opened to increase reverberation time.
Performing musicians naturally demand a sense of ease and power in producing adequate tone without fatigue. This is helped by strategic placing near the players of reflecting surfaces which also enable them to hear each other clearly. There seems no doubt that composers of all periods consciously or unconsciously wrote in such a way as to suit the environment in which their music would be performed.

5. Problems of small rooms

In the reverberant sound field of a large hall, the random streams of reflected sound waves produce a reasonably consistent diffusion of sound. In small rooms, however, distinct interference patterns are set up by multiple reflections between parallel walls, floor, and ceiling. These ‘standing wave’ resonances, which form a kind of three-dimensional organ-pipe effect, occur at frequencies of which the distances between the parallel surfaces are multiples of a half-wavelength. A harmonic series of these room resonances, or ‘eigentones’, exists for each room dimension, and the uneven boosting of certain frequencies causes coloration of the sound. Selective bass absorption is needed, or a special design using non-parallel walls.

6. Sound reinforcement

It is economically impossible to limit the use of most halls to musical forces of optimum size and acoustic power. The question of amplification then arises for quiet instruments or voices. In many churches and lecture theatres, the building shape or shortcomings in the acoustic distribution call for augmentation of the natural sounds, either overall or selectively in particular areas. The basic components for sound reinforcement or ‘public address’ are a microphone, amplifier, and loudspeaker. The arrangement is inherently unstable, however, as most users can testify, since any amplified sound from the loudspeaker that falls on the microphone is again amplified and sent to the loudspeaker with the possibility of uncontrolled feedback. Directional microphones can ease the problem since their less sensitive side(s) can be directed towards the loudspeaker(s) and so reduce unwanted pickup of the amplified sound. Directional loudspeaker arrays can also beam the sound waves into specific areas to give more efficient reinforcement without feedback.

7. Assisted resonance

A special kind of sound reinforcement, called assisted resonance, is used in some halls to increase the reverberation time within certain frequency bands. A classic example is the Royal Festival Hall in London, where the original 1948 design had called for a reverberation period of 1.7 seconds, rising to 2.5 seconds at low frequencies. When the hall was built, however, the low-frequency reverberation time measured only 1.4 seconds and, while this gave excellent definition, the hall was criticized as lacking fullness of tone. In 1964 matters were improved by assisted resonance using 172 microphones at roof level, amplifiers tuned to narrow frequency bands in the range 58–700 Hz, and arrays of loudspeakers.

8. Sound insulation

A requirement in every type of auditorium is for the lowest practical level of extraneous noise, whether airborne or transmitted through the structure of the building. A first step in planning is to choose a quiet site—not very practicable in a large city—and to design the building with as many layers or shells as possible on the side nearest to railway lines or other identifiable sources of noise. Aircraft noise is an increasing problem requiring the use of massive roofs on insulating supports, with suspended ceilings, floating floors, and multi-layered exterior walls. The Bridgewater Hall in Manchester (opened in 1996 ) has achieved almost total exclusion of external noise. Its massive 22,500-tonne weight is suspended on some 300 isolation spring bearings, and its three-layer roof has an outer sheet of steel lined with acoustic panels.

Read more: architectural acoustics - 1. Reflection, 2. Directional effects, 3 Reverberation, 4. Designing for good acoustics http://arts.jrank.org/pages/366/architectural-acoustics.html#ixzz1Eb5uUVg3

Digital Electronics

Digital electronics

Digital electronics envolves initially for performing numerical computations quickly and accurately. The inventions of transistor in 1947 paved the way for fabrication of logic circuits and then digital ICs which are efficient and faster than the analog circuita. In almost of the digital circuits diodes and transistors are used as switches to change from one voltage level to another. Since a switch may be oped or closed, the output states of digital circuits are designed as OFF and ON states. These two states correspond to the 0 and 1 states in positive logic systems. Today digital circuits are used in diverse applications such as CD  player , stereos, TVs, Telephones....

I

Numerical Presentation

The quantities that are to be measured, monitored, recorded, processed and controlled are analog and digital, depending on the type of system used. It is important when dealing with various quantities that we be able to represent their values efficiently and accurately. There are basically two ways of representing the numerical value of quantities: analog and digital.

Advantages

• Easier to design. Exact values of voltage or current are not important, only the range (HIGH or LOW) in which they fall.
• Information storage is easy.
• Accuracy and precision are greater.
• Operations can be programmed. Analog systems can also be programmed, but the available operations variety and complexity is severely limited.
• Digital circuits are less affected by noise, as long as the noise is not large enough to prevent us from distinguishing HIGH from LOW (we discuss this in detail in an advanced digital tutorial section).
• More digital circuitry can be fabricated on IC chips.

Limitations of Digital Techniques

• Most physical quantities in real world are analog in nature, and these quantities are often the inputs and outputs that are being monitored, operated on, and controlled by a system. Thus conversion to digital format and re-conversion to analog format is needed.

Many number systems are in use in digital technology. The most common are the decimal, binary, octal, and hexadecimal systems. The decimal system is clearly the most familiar to us because it is a tool that we use every day. Examining some of its characteristics will help us to better understand the other systems. In the next few pages we shall introduce four numerical representation systems that are used in the digital system. There are other systems, which we will look at briefly.

• Decimal
• Binary
• Octal
• Hexadecimal

   

  Decimal System

The decimal system is composed of 10 numerals or symbols. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these symbols as digits of a number, we can express any quantity. The decimal system is also called the base-10 system because it has 10 digits.

   

103	102	101	100		10-1	10-2	10-3
=1000	=100	=10	=1	.	=0.1	=0.01	=0.001
Most Significant Digit				Decimal point			Least Significant Digit

   

Even though the decimal system has only 10 symbols, any number of any magnitude can be expressed by using our system of positional weighting.

   

  Decimal Examples    

• 3.14₁₀
• 52₁₀
• 1024₁₀
• 64000₁₀

   

  Binary System

In the binary system, there are only two symbols or possible digit values, 0 and 1. This base-2 system can be used to represent any quantity that can be represented in decimal or other base system.

   

23	22	21	20		2-1	2-2	2-3
=8	=4	=2	=1	.	=0.5	=0.25	=0.125
Most Significant Digit				Binary point			Least Significant Digit

   

  Binary Counting

The Binary counting sequence is shown in the table:

   

23	22	21	20	Decimal
0	0	0	0	0
0	0	0	1	1
0	0	1	0	2
0	0	1	1	3
0	1	0	0	4
0	1	0	1	5
0	1	1	0	6
0	1	1	1	7
1	0	0	0	8
1	0	0	1	9
1	0	1	0	10
1	0	1	1	11
1	1	0	0	12
1	1	0	1	13
1	1	1	0	14
1	1	1	1	15

   

   

  Representing Binary Quantities

In digital systems the information that is being processed is usually presented in binary form. Binary quantities can be represented by any device that has only two operating states or possible conditions. E.g.. a switch is only open or closed. We arbitrarily (as we define them) let an open switch represent binary 0 and a closed switch represent binary 1. Thus we can represent any binary number by using series of switches.

   

  Typical Voltage Assignment

Binary 1:Any voltage between 2V to 5V

Binary 0:Any voltage between 0V to 0.8V

Not used:Voltage between 0.8V to 2V in 5 Volt CMOS and TTL Logic, this may cause error in a digital circuit. Today's digital circuits works at 1.8 volts, so this statement may not hold true for all logic circuits.

   

   

We can see another significant difference between digital and analog systems. In digital systems, the exact voltage value is not important; eg, a voltage of 3.6V means the same as a voltage of 4.3V. In analog systems, the exact voltage value is important.

   

The binary number system is the most important one in digital systems, but several others are also important. The decimal system is important because it is universally used to represent quantities outside a digital system. This means that there will be situations where decimal values have to be converted to binary values before they are entered into the digital system.

   

In additional to binary and decimal, two other number systems find wide-spread applications in digital systems. The octal (base-8) and hexadecimal (base-16) number systems are both used for the same purpose- to provide an efficient means for representing large binary system.

   

  Octal System

The octal number system has a base of eight, meaning that it has eight possible digits: 0,1,2,3,4,5,6,7.

   

83	82	81	80		8-1	8-2	8-3
=512	=64	=8	=1	.	=1/8	=1/64	=1/512
Most Significant Digit				Octal point			Least Significant Digit

   

  Octal to Decimal Conversion   

• 237₈ = 2 x (8²) + 3 x (8¹) + 7 x (8⁰) = 159₁₀
• 24.6₈ = 2 x (8¹) + 4 x (8⁰) + 6 x (8^-1) = 20.75₁₀
• 11.1₈ = 1 x (8¹) + 1 x (8⁰) + 1 x (8^-1) = 9.125₁₀
• 12.3₈ = 1 x (8¹) + 2 x (8⁰) + 3 x (8^-1) = 10.375₁₀

   

  Hexadecimal System

The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It uses the digits 0 through 9 plus the letters A, B, C, D, E, and F as the 16 digit symbols.

   

163	162	161	160		16-1	16-2	16-3
=4096	=256	=16	=1	.	=1/16	=1/256	=1/4096
Most Significant Digit				Hexa Decimal point			Least Significant Digit

   

  Hexadecimal to Decimal Conversion   

• 24.6₁₆ = 2 x (16¹) + 4 x (16⁰) + 6 x (16^-1) = 36.375₁₀
• 11.1₁₆ = 1 x (16¹) + 1 x (16⁰) + 1 x (16^-1) = 17.0625₁₀
• 12.3₁₆ = 1 x (16¹) + 2 x (16⁰) + 3 x (16^-1) = 18.1875₁₀

Symbolic Logic

Boolean algebra derives its name from the mathematician George Boole. Symbolic Logic uses values, variables and operations :

   

• True is represented by the value 1.
• False is represented by the value 0.

Variables are represented by letters and can have one of two values, either 0 or 1. Operations are functions of one or more variables.

• AND is represented by X.Y
• OR is represented by X + Y
• NOT is represented by X' . Throughout this tutorial the X' form will be used and sometime !X will be used.

These basic operations can be combined to give expressions.

   

Example :

   

• X
• X.Y
• W.X.Y + Z

   

  Precedence

As with any other branch of mathematics, these operators have an order of precedence. NOT operations have the highest precedence, followed by AND operations, followed by OR operations. Brackets can be used as with other forms of algebra. e.g.

   

X.Y + Z and X.(Y + Z) are not the same function.

   

  Function Definitions

The logic operations given previously are defined as follows :

   

Define f(X,Y) to be some function of the variables X and Y.

   

f(X,Y) = X.Y

• 1 if X = 1 and Y = 1
• 0 Otherwise

   

f(X,Y) = X + Y

• 1 if X = 1 or Y = 1
• 0 Otherwise

   

f(X) = X'

• 1 if X = 0
• 0 Otherwise

   

  Truth Tables

Truth tables are a means of representing the results of a logic function using a table. They are constructed by defining all possible combinations of the inputs to a function, and then calculating the output for each combination in turn. For the three functions we have just defined, the truth tables are as follows.

   

AND

X	Y	F(X,Y)
0	0	0
0	1	0
1	0	0
1	1	1

   

OR

X	Y	F(X,Y)
0	0	0
0	1	1
1	0	1
1	1	1

   

NOT

X	F(X)
0	1
1	0

   

Truth tables may contain as many input variables as desired

   

F(X,Y,Z) = X.Y + Z

X	Y	Z	F(X,Y,Z)
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

   

  Boolean Switching Algebras

A Boolean Switching Algebra is one which deals only with two-valued variables. Boole's general theory covers algebras which deal with variables which can hold n values.

   

  Axioms

Consider a set S = { 0. 1}

Consider two binary operations, + and . , and one unary operation, -- , that act on these elements. [S, ., +, --, 0, 1] is called a switching algebra that satisfies the following axioms S

   

  Closure    

If X S and Y S then X.Y S

If X S and Y S then X+Y S

   

  Identity    

an identity 0 for + such that X + 0 = X

an identity 1 for . such that X . 1 = X

   

  Commutative Laws    

X + Y = Y + X

X . Y = Y . X

   

  Distributive Laws    

X.(Y + Z ) = X.Y + X.Z

X + Y.Z = (X + Y) . (X + Z)

   

  Complement    

X S a complement X'such that

X + X' = 1

X . X' = 0

The complement X' is unique.

   

   

  Theorems   

A number of theorems may be proved for switching algebras

   

  Idempotent Law   

X + X = X

X . X = X

   

  DeMorgan's Law   

(X + Y)' = X' . Y', These can be proved by the use of truth tables.

   

Proof of (X + Y)' = X' . Y'

   

X	Y	X+Y	(X+Y)'
0	0	0	1
0	1	1	0
1	0	1	0
1	1	1	0

   

X	Y	X'	Y'	X'.Y'
0	0	1	1	1
0	1	1	0	0
1	0	0	1	0
1	1	0	0	0

   

The two truth tables are identical, and so the two expressions are identical.

   

(X.Y) = X' + Y', These can be proved by the use of truth tables.

   

Proof of (X.Y) = X' + Y'

   

X	Y	X.Y	(X.Y)'
0	0	0	1
0	1	0	1
1	0	0	1
1	1	1	0

   

X	Y	X'	Y'	X'+Y'
0	0	1	1	1
0	1	1	0	1
1	0	0	1	1
1	1	0	0	0

   

Note :DeMorgans Laws are applicable for any number of variables.

   

  Boundedness Law   

X + 1 = 1

X . 0 = 0

   

  Absorption Law   

X + (X . Y) = X

X . (X + Y ) = X

   

  Elimination Law   

X + (X' . Y) = X + Y

X.(X' + Y) = X.Y

   

  Unique Complement theorem   

If X + Y = 1 and X.Y = 0 then X = Y'

   

  Involution theorem   

X'' = X

0' = 1

   

  Associative Properties   

X + (Y + Z) = (X + Y) + Z

X . ( Y . Z ) = ( X . Y ) . Z

   

  Duality Principle

In Boolean algebras the duality Principle can be is obtained by interchanging AND and OR operators and replacing 0's by 1's and 1's by 0's. Compare the identities on the left side with the identities on the right.

   

Example

   

X.Y+Z' = (X'+Y').Z

   

  Consensus theorem   

X.Y + X'.Z + Y.Z = X.Y + X'.Z

or dual form as below

(X + Y).(X' + Z).(Y + Z) = (X + Y).(X' + Z)

   

Proof of X.Y + X'.Z + Y.Z = X.Y + X'.Z:

   

X.Y + X'.Z + Y.Z	= X.Y + X'.Z
X.Y + X'.Z + (X+X').Y.Z	= X.Y + X'.Z
X.Y.(1+Z) + X'.Z.(1+Y)	= X.Y + X'.Z
X.Y + X'.Z	= X.Y + X'.Z

   

(X.Y'+Z).(X+Y).Z = X.Z+Y.Z instead of X.Z+Y'.Z

X.Y'Z+X.Z+Y.Z

(X.Y'+X+Y).Z

(X+Y).Z

X.Z+Y.Z

   

The term which is left out is called the consensus term.

   

Given a pair of terms for which a variable appears in one term, and its complement in the other, then the consensus term is formed by ANDing the original terms together, leaving out the selected variable and its complement.

   

Example :

The consensus of X.Y and X'.Z is Y.Z

   

The consensus of X.Y.Z and Y'.Z'.W' is (X.Z).(Z.W')

   

  Shannon Expansion Theorem

The Shannon Expansion Theorem is used to expand a Boolean logic function (F) in terms of (or with respect to) a Boolean variable (X), as in the following forms.

   

F = X . F (X = 1) + X' . F (X = 0)

   

where F (X = 1) represents the function F evaluated with X set equal to 1; F (X = 0) represents the function F evaluated with X set equal to 0.

   

Also the following function F can be expanded with respect to X,

   

F = X' . Y + X . Y . Z' + X' . Y' . Z

   

= X . (Y . Z') + X' . (Y + Y' . Z)

   

Thus, the function F can be split into two smaller functions.

   

F (X = '1') = Y . Z'

   

This is known as the cofactor of F with respect to X in the previous logic equation. The cofactor of F with respect to X may also be represented as F X (the cofactor of F with respect to X' is F X' ). Using the Shannon Expansion Theorem, a Boolean function may be expanded with respect to any of its variables. For example, if we expand F with respect to Y instead of X,

   

F = X' . Y + X . Y . Z' + X' . Y' . Z

   

= Y . (X' + X . Z') + Y' . (X' . Z)

   

A function may be expanded as many times as the number of variables it contains until the canonical form is reached. The canonical form is a unique representation for any Boolean function that uses only minterms. A minterm is a product term that contains all the variables of F¿such as X . Y' . Z).

   

Any Boolean function can be implemented using multiplexer blocks by representing it as a series of terms derived using the Shannon Expansion Theorem.

   

  Summary of Laws And Theorms   

Identity	Dual
Operations with 0 and 1
X + 0 = X (identity)	X.1 = X
X + 1 = 1 (null element)	X.0 = 0
Idempotency theorem
X + X = X	X.X = X
Complementarity
X + X' = 1	X.X' = 0
Involution theorem
(X')' = X
Cummutative law
X + Y = Y + X	X.Y = Y X
Associative law
(X + Y) + Z = X + (Y + Z) = X + Y + Z	(XY)Z = X(YZ) = XYZ
Distributive law
X(Y + Z) = XY + XZ	X + (YZ) = (X + Y)(X + Z)
DeMorgan's theorem
(X + Y + Z + ...)' = X'Y'Z'... or { f ( X1,X2,...,Xn,0,1,+,. ) } = { f ( X1',X2',...,Xn',1,0,.,+ ) }	(XYZ...)' = X' + Y' + Z' + ...
Simplification theorems
XY + XY' = X (uniting)	(X + Y)(X + Y') = X
X + XY = X (absorption)	X(X + Y) = X
(X + Y')Y = XY (adsorption)	XY' + Y = X + Y
Consensus theorem
XY + X'Z + YZ = XY + X'Z	(X + Y)(X' + Z)(Y + Z) = (X + Y)(X' + Z)
Duality
(X + Y + Z + ...)D = XYZ... or {f(X1,X2,...,Xn,0,1,+,.)}D = f(X1,X2,...,Xn,1,0,.,+)	(XYZ ...)D = X + Y + Z + ...
Shannon Expansion Theorem
f(X1,...,Xk,...Xn)	Xk * f(X1,..., 1 ,...Xn) + Xk' * f(X1,..., 0 ,...Xn)
f(X1,...,Xk,...Xn)	[Xk + f(X1,..., 0 ,...Xn)] * [Xk' + f(X1,..., 1 ,...Xn)]

Logic Gates

A logic gate is an electronic circuit/device which makes the logical decisions. To arrive at this decisions, the most common logic gates used are OR, AND, NOT, NAND, and NOR gates. The NAND and NOR gates are called universal gates. The exclusive-OR gate is another logic gate which can be constructed using AND, OR and NOT gate.

   

Logic gates have one or more inputs and only one output. The output is active only for certain input combinations. Logic gates are the building blocks of any digital circuit. Logic gates are also called switches. With the advent of integrated circuits, switches have been replaced by TTL (Transistor Transistor Logic) circuits and CMOS circuits. Here I give example circuits on how to construct simples gates.

Symbolic Logic

Boolean algebra derives its name from the mathematician George Boole. Symbolic Logic uses values, variables and operations.

   

  Inversion

A small circle on an input or an output indicates inversion. See the NOT, NAND and NOR gates given below for examples.

   

   

  Multiple Input Gates

Given commutative and associative laws, many logic gates can be implemented with more than two inputs, and for reasons of space in circuits, usually multiple input, complex gates are made. You will encounter such gates in real world (maybe you could analyze an ASIC lib to find this).

   

  Gates Types    

• AND
• OR
• NOT
• BUF
• NAND
• NOR
• XOR
• XNOR

   

   

  AND Gate

The AND gate performs logical multiplication, commonly known as AND function. The AND gate has two or more inputs and single output. The output of AND gate is HIGH only when all its inputs are HIGH (i.e. even if one input is LOW, Output will be LOW).

   

If X and Y are two inputs, then output F can be represented mathematically as F = X.Y, Here dot (.) denotes the AND operation. Truth table and symbol of the AND gate is shown in the figure below.

   

Symbol

   

   

Truth Table

   

X	Y	F=(X.Y)
0	0	0
0	1	0
1	0	0
1	1	1

   

Two input AND gate using "diode-resistor" logic is shown in figure below, where X, Y are inputs and F is the output.

   

Circuit

   

If X = 0 and Y = 0, then both diodes D1 and D2 are forward biased and thus both diodes conduct and pull F low.

   

If X = 0 and Y = 1, D2 is reverse biased, thus does not conduct. But D1 is forward biased, thus conducts and thus pulls F low.

   

If X = 1 and Y = 0, D1 is reverse biased, thus does not conduct. But D2 is forward biased, thus conducts and thus pulls F low.

   

If X = 1 and Y = 1, then both diodes D1 and D2 are reverse biased and thus both the diodes are in cut-off and thus there is no drop in voltage at F. Thus F is HIGH.

   

   

  Switch Representation of AND Gate

In the figure below, X and Y are two switches which have been connected in series (or just cascaded) with the load LED and source battery. When both switches are closed, current flows to LED.

   

   

  Three Input AND gate

Since we have already seen how a AND gate works and I will just list the truth table of a 3 input AND gate. The figure below shows its symbol and truth table.

   

Circuit

   

Truth Table

X	Y	Z	F=X.Y.Z
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

   

  OR Gate

The OR gate performs logical addition, commonly known as OR function. The OR gate has two or more inputs and single output. The output of OR gate is HIGH only when any one of its inputs are HIGH (i.e. even if one input is HIGH, Output will be HIGH).

   

If X and Y are two inputs, then output F can be represented mathematically as F = X+Y. Here plus sign (+) denotes the OR operation. Truth table and symbol of the OR gate is shown in the figure below.

   

Symbol

   

   

Truth Table

   

X	Y	F=(X+Y)
0	0	0
0	1	1
1	0	1
1	1	1

   

Two input OR gate using "diode-resistor" logic is shown in figure below, where X, Y are inputs and F is the output.

   

Circuit

   

If X = 0 and Y = 0, then both diodes D1 and D2 are reverse biased and thus both the diodes are in cut-off and thus F is low.

   

If X = 0 and Y = 1, D1 is reverse biased, thus does not conduct. But D2 is forward biased, thus conducts and thus pulling F to HIGH.

   

If X = 1 and Y = 0, D2 is reverse biased, thus does not conduct. But D1 is forward biased, thus conducts and thus pulling F to HIGH.

   

If X = 1 and Y = 1, then both diodes D1 and D2 are forward biased and thus both the diodes conduct and thus F is HIGH.

   

  Switch Representation of OR Gate

In the figure, X and Y are two switches which have been connected in parallel, and this is connected in series with the load LED and source battery. When both switches are open, current does not flow to LED, but when any switch is closed then current flows.

   

   

  Three Input OR gate

Since we have already seen how an OR gate works, I will just list the truth table of a 3-input OR gate. The figure below shows its circuit and truth table.

   

Circuit

   

Truth Table

X	Y	Z	F=X+Y+Z
0	0	0	0
0	0	1	1
0	1	0	1
0	1	1	1
1	0	0	1
1	0	1	1
1	1	0	1
1	1	1	1