Frequency Response

Some amplifier circuits employing either BJT or FET were analyzed in Chapter 2. Almost all the circuits contained blocking and bypass capacitors in addition to the biasing components, load resistance, and so on. However, these external capacitors and the internal parasitic capacitors of the transistor were considered absent during the analysis because either dc analysis was done or for the simplification in the analysis. There was no mention of the practical limit on the device parameters or on the components used. It does not mean that the obtained expressions for the voltage gain, current gain, and input and output resistance were wrong. These expressions are very important and relevant, and for most of the operations of amplifiers, these expressions are to be employed. However, there is something more, which is also very important, and that remains to be studied.

When the internal device capacitances or the intentionally connected capacitance or load impedance having reactive components are considered, the amplifier gain becomes a complex number, A–θ, instead of a real number. A significant point is that both the gain A and phase angle –θ depend on the input signal frequency, and the gain magnitude decreases at low and at comparatively high signal frequencies; amplification remains almost constant in the mid-frequency range. The frequency response characteristics of an amplifier are the plot of gain and phase with frequency.

Learning Outcomes

After reading this chapter, the reader will be able to

• State the assumptions of kinetic theory of gases (KTG)

• Explain the concept of pressure and calculate the expression for it

• Demonstrate mathematically the gas laws using the expression for pressure derived from KTG

• Present the kinetic interpretation of temperature

• Derive the expression for specific heat at constant volume 𝐶𝑉 and constant pressure 𝐶𝑃

• Explain the concept of degree of freedom

• Solve numerical problems and multiple choice questions on KTG

2.1 Introduction

The kinetic theory of gases (KTG) is a theoretical model that describes the physical properties of a gaseous system in terms of a large number of submicroscopic particles, such as atoms, molecules, and small particles. These constituent elements are in random motion and collide constantly with each other and also with the walls of the container. Considering the molecular composition and characteristic features of such random motion of the molecules, various macroscopic properties of the gaseous system, such as pressure, temperature, viscosity, thermal conductivity, and mass diffusivity can be explained with the help of KTG. In this theory, it is postulated that the pressure exerted by a gas is due to the collision of atoms or molecules moving at different velocities on the walls of a container. It basically attempts to explain the macroscopic properties that are related to the microscopic phenomenon. The physical properties of solids and liquids, in general, are described by their shape, size, mass, volume, etc. Gases, however, have no definite shape, and size. Furthermore, their mass and volume are not directly measurable. In such cases, the KTG can be successfully applied to extract the physical properties of the gaseous system.

A document is an object which is primarily meant forhuman reading. It is not limited to only text.Besides text, which is primarily for reading, it mayalso contain figures, diagrams, photographs, tables,charts, etc. Many of these auxiliary componentsfacilitate the reading experience. Document imageprocessing involves processing of images ofdocuments. Examples of documents comprise of scannedimages of printed or handwritten pages, photographsof documents, etc. In general, images that containdifferent kinds of reading materials are consideredas images of documents. In this context, the imageof a text displayed in the environment is also anexample of document, e.g., an image of a signboard.Such kinds of texts are referred to as scenetexts.

A few examples of images of different kinds ofdocuments are shown in Fig. 16.1. The document inFig. 16.1 (a) contains printed text and graphics.Fig. 16.1 (b) is an example of an official document,which is a typical format of an official purchaseorder. Every organization usually has some standardtemplate or format of each official form used inregular administrative routines, which belongs tothis particular category of documents.

A page of magazine is shown in Fig. 16.1 (c), where thegraphics and text are overlaid in a specific layout.In Fig. 16.1 (d), an example of scene text is shown,which is a photograph of a stone tablet describing ahistorical monument. The image shown in Fig. 16.1(e) is a scanned document of a handwritten page, anexample of writing of the famous Bengali poet andNobel laureate, Rabindranath Tagore (1861–1941).

7.1 INTRODUCTION [LO1]

The problems of stresses and deformations in disks rotating at high speeds are important in the design of both gas and steam turbines, generators and many such rotating machinery in industry. As discussed in earlier chapters, this is another example of axisymmetric problems in polar coordinates. Although the theoretical treatment of a flat disk is simpler, in many industrial applications, disks are tapered. They are usually thicker near the hub, and their theoretical analysis is slightly more involved. We shall first take up the analysis for flat disks.

In the case of rotating disks with centrifugal force as body force, the equation of equilibrium reduces to as in equation (6.1.3).

Combining this with displacement equations, we have, as in equation (6.1.5), a general equation for determining the stress distribution in axisymmetric problems. This is given as

This is a nonhomogeneous differential equation. The associated homogeneous equation (complementary equation) is

The solution of this equation is Lame's equation as discussed in Chapter 6, equation (6.2.3), and taking into consideration the particular solution, the solution to equation (7.1.2) turns out to be

We may also determine the radial displacement from equation (6.2.11), and this is given as

We may therefore write the stresses and displacement for the rotating disk under one bracket as

With these introductory basic equations, we shall now set out to discuss the stress distribution and displacement in rotating disks.

Specialization characterizes all economies to some degree, but its variation is profound, and an objective of economic theory has been to explain its development. Since Adam Smith, economic specialization has been a focus of social scientific inquiry into the evolution of sociopolitical-economic complexity. In the words of Henrich and Boyd (2008:715), “Anthropologists and sociologists … have defended a wide variety of theories that link economic specialization, a division of labor, and the emergence of socially stratified inequality since the birth of their discipline at the end of the 19th century.” Archaeological inquiry, however, compels us to rethink this simple correlation. As the flip-side to self-sufficiency (Chapter 3), we examine variation in economic specialization found in thirteen ancient, premodern, or small-scale economies across Africa, Eurasia, and the Americas. Our analysis looks at the nature, type, and scale of specialization found in societies of different sizes and internal complexity. This is followed by a discussion of production, distribution, and infrastructural/service specializations, and where they occur within the thirteen societies examined. Although specialization apparently has different causes related to efficiency, it links strongly to developing markets with their expanded access to demand.

6.1 INTRODUCTION [LO1]

In earlier chapters, we have discussed axisymmetric problems in two-dimensional (2D) polar coordinate systems. Thick cylinders fall into this class of problems. Cylindrical pressure vessels, hydraulic cylinders, gun-barrels, and pipes carrying fluids at high pressure develop radial and tangential stresses (circumferential). Longitudinal stresses can also be developed if the ends are closed. Therefore, ideally, this is a triaxial stress system as shown in Figure 6.1.

• (a) Circumferential or hoop stress (σθ)

• (b) Longitudinal stress (σz)

• (c) Radial stress (σr)

If the wall thickness of a hollow cylinder is less than about 10% of its radius, it may be treated as a thin cylinder. Cylinders with higher wall thickness are considered to be thick cylinders. Before analyzing the stress in a thick cylinder, we should briefly consider the stress state in thin cylinders, where radial stress is small compared to the other stresses, and this can be neglected. Stress variation across the thin wall is also negligible. Analysis of thin-walled pressure vessels may therefore be carried out on the basis of biaxial stress system. Since the presence of shear stress at the cut section would lead to incompatible distortion, the longitudinal and circumferential stresses in this case are both principal stresses. We now take another section of the cut section as shown in Figure 6.2 (a) to consider the equilibrium of the section, and this is shown in Figure 6.2 (b).

The section is acted upon by internal pressure p and the circumferential stress developed at the cut section is σθ. Force on an infinitesimal small area subtended by angle dθ at θ inclination from the horizontal axis is pridθ.

Introduction

Communication through optical fiber is one of the remarkable discoveries of the twenty-first century that have brought a revolution in modern times. The transfer of information over long distances was earlier performed through copper wires and coaxial cables. The limitation of these devices, such as limited bandwidth, could not fulfill modern needs and hence were replaced by glass fiber. It was the effort of Alexander Graham Bell, who in 1880 used the light as a carrier of signal. Since the attenuation in the optical fibers was quite high, an attempt to minimize it was done to improve it, and today its features are so fantastic that optical communication through glass fiber with low loss has become a reality. A large number of advantages of optical fibers over the traditional wires and coaxial cables are not hidden now and have been accepted over the entire globe. Apart from their use in communication, optical fibers are widely used in other areas also. In a nutshell, we can therefore say that fiber optics is a backbone of communication infrastructure.

The optical fiber is a cylindrical waveguide system operating at optical frequency. It consists of a core at the center and a cladding outside the core. The core is generally a cylindrical dielectric glass, and cladding is the second dielectric cover usually of glass with a lower refractive index n2, as shown in Figure 13.1.

Learning Outcomes After reading this chapter, the reader will be able to

• Know various types of thermodynamic systems such as open, closed, and isolated, and the surroundings

• Classify between intensive and extensive thermodynamic variables

• Understand various types of equilibrium conditions satisfied by a thermodynamic system

• State the zeroth law of thermodynamics and highlight its physical significance

• Comprehend the idea of temperature from the zeroth law of thermodynamics

• Solve numerical problems and multiple choice questions on thermodynamic equilibrium and the zeroth law of thermodynamics

7.1 Introduction

Heat is a form of energy. It can be transformed from one form to another as well as can be transferred between various objects maintained at suitable temperatures. For example, in an electric motor, heat is transformed into mechanical energy by the turbine to power the motor. This mechanical energy is then transformed into electrical energy by the engine to illuminate light bulbs. “Thermodynamics” is a branch of physics that deals with heat and the transformation of heat from one form to another, work, temperature, and their relation to energy, entropy, and other physical properties of matter and radiation. It establishes the relation between heat and various forms of energy and describes the transformations that occur in thermal energy from one energy state to another and how this transformation affects matter. A thermodynamic system is described within a framework based on the four laws of thermodynamics that facilitate a quantitative description of the average macroscopic properties of the system in equilibrium. Macroscopic matter refers to large objects that consist of many atoms and molecules. The average properties of such macroscopic systems are determined by the physical quantities such as volume, pressure, and temperature that do not depend upon the detailed microscopic positions and velocities of the atoms and the molecules comprising the macroscopic system. In the equilibrium state of a thermodynamic system, these average properties also do not change with time. These physical quantities are called thermodynamic coordinates, variables, or parameters. If a subset of these properties are experimentally measured, the rest of them can be calculated using thermodynamic relations. Thermodynamics not only gives the exact description of the state of equilibrium but also provides an approximate description (to a very high degree of precision!) of relatively slow processes. This branch of physics can be successfully applied to a wide variety of topics in science, such as physics, physical chemistry, biochemistry, chemical engineering, and mechanical engineering, but also in other complex fields, such as meteorology.

5.1 INTRODUCTION [LO1]

In any three-dimensional (3D) elasticity problem, there are 15 unknown parameters: 6 stress components, 6 strain components, and 3 displacements. There are 15 related equations: 3 equations of equilibrium, 6 compatibility equations, and 6 constitutive equations. Solutions to a particular elasticity problem require evaluation of these 15 unknown parameters using 15 equations, satisfying all the boundary conditions. As discussed in the earlier chapters, there may be displacement or stress, or mixed boundary conditions. In many cases, solutions to 3D problems are not easy analytically. Even numerical solutions may be difficult.

There are mainly three methods of simplification of solution techniques:

• (a) If the boundary conditions are in terms of stresses, stress function approach may be made as discussed in the earlier chapter. This makes the solution simpler.

• (b) Assumptions of plane stress and plane strain reduce 3D problems to two-dimensional (2D) ones and this also makes the solution simpler.

• (c) Use of St. Venant's principle and superposition principle also makes the solution of elasticity problems simpler.

An introduction to stress function approach has been discussed in Chapter 4. We therefore start our discussion on plane stress and plane strain approaches.

5.2 PLANE STRESS AND PLANE STRAIN PROBLEMS [LO2]

The idealizations of both plane stress and plane strain states are suitable for certain classes of problems that are made to reduce the complexity of solutions. We shall consider the plane stress state first.

Learning Outcomes

After reading this chapter, the reader will be able to

• Learn the basic concept of the theory of probability

• List the assumptions used in the derivation of Maxwell's speed distribution law

• Derive Maxwell's speed distribution law and test its validity experimentally

• Calculate average, root mean square and most probable speed, energy, and momentum in one, two, and three dimensions, respectively

• State and prove the law of equipartition of energy

• Calculate the specific heat of gases

• Solve numerical problems and multiple choice questions on the distribution of molecular speed, energy, and momentum

3.1 Introduction

In Chapter 2, various characteristic features of a gaseous system based on the model of the kinetic theory of gases (KTG) have been discussed elaborately. Macroscopic properties and various relations among the thermodynamic variables have been explained in terms of this kinetic model.

According to the assumptions used in this model, a gaseous system is composed of a large number of particles (atoms or molecules) with practically no volume occupied by them. Most of the times, these molecules move randomly through empty space at temperatures above absolute zero, and such motions remain unaffected by the presence of other particles. This motion of the molecules is extremely chaotic and is characterized by straight-line trajectories interrupted by collisions with other molecules or with a physical boundary. In such a collision, the transfer of kinetic energy with a change in direction takes place depending on the nature of the relative kinetic energies of the particles. Any individual molecule collides with others at a huge rate, typically of the order of a billion times per second. This chapter is focused to present a comprehensive and quantitative discussion on the distributions of velocities, energies, and momenta of these molecules in various dimensions.

Measurement of the velocities of the molecules at a given time leads to a large distribution of values; some molecules may move very slowly and others very quickly. As these molecules move constantly in different directions, the velocity could be momentarily equal to zero

Learning Outcomes

After reading this chapter, the reader will be able to

• Gather knowledge about state function and its properties

• Understand the meaning of internal energy and its significance in formulating the first law of thermodynamics

• Formulate the first law of thermodynamics and apply it to various thermodynamic processes

• Grasp the idea of various thermodynamic processes and the related work done in these processes

• Find a relation between specific heats at constant volume and constant pressure for ideal and real gases

• Find expression for isothermal compressibility and volume expansion coefficient for ideal and real gases

• Understand how the temperature of air varies with height assuming an adiabatic process

• Solve numerical problems and multiple choice questions on the first law of Thermodynamics

8.1 Introduction

Systems, surroundings, and interactions between them play a vital role in the development of the subject—thermodynamics. To extract out the physical properties of a thermodynamic system, it is essential to have knowledge about the fundamental laws and concepts of thermodynamics. For example, heat and work are two interrelated concepts. Heat is the transfer of thermal energy between two bodies that are at different temperatures and are not equal to thermal energy. Work is the external physical parameter used to transfer energy between a system and its surroundings. Further, work is needed to create heat and to transfer the thermal energy. Thus, work and heat together allow systems to exchange energy. The relationship between these two physical quantities, heat and work, can be analyzed through the laws and concepts of thermodynamics. The interaction between heat and other types of energy is primarily focused on the topic of thermodynamics. To understand the relationship between heat and work, it is required to have an idea about a third linking factor, known as the change in internal energy

A2.1 Mathematical operations

Try the following operations and commands on the command line in the command window of Matlab.

A2.1.1 Defining an array and arranging in ascending and descending order

Introduction

Numerous microscopic events, including atomic stability, blackbody radiation, the photoelectric effect, and atomic spectroscopy, could not be explained by classical physics. When Max Planck presented the idea of the quantum of energy in 1900, it marked the first significant advancement. Only after positing that the energy exchange between radiation and its surroundings occurs in discrete, or quantized, amounts was he able to replicate the experimental findings in his attempts to understand the phenomenon of blackbody radiation. He claimed that an electromagnetic wave of frequency v and matter can only exchange energy in integer multiples of h, or what he termed a quantum's energy, where h is a fundamental constant known as Planck's constant. The concept of quantizing electromagnetic radiation proved to have far-reaching effects.

Blackbody radiation was correctly explained by Planck's hypothesis, which inspired fresh thinking and set off a wave of new findings that provided answers to the most pressing issues of the day.

Planck's quantum idea received a potent reinforcement from Einstein in 1905. Einstein realized that Planck's theory of the quantization of electromagnetic waves must also apply to light when attempting to comprehend the photoelectric effect. So, adopting Planck's methodology, he proposed that light itself is composed of discrete energy units (or minuscule particles) called photons, each of which has energy hv, which corresponds to the light's frequency.

All societies mobilize resources for different purposes. The product of human labor, these goods and services become especially important in the formation and support of multi-scalar organizations that include communities, regional polities, and beyond. The labor and its resources must be mobilized to finance these organizations as they are developed and maintained across time.

Introduction

The word laser is an abbreviation for “light amplification by stimulated emission of radiation.” Historically, the laser is the outgrowth of maser, which means “microwave amplification by stimulated emission of radiation.” If the stimulated radiation lies in optical region, the device is called optical maser or laser. Laser beam is highly monochromatic, highly coherent, intense, and highly collimated with a small diversion.

Using ruby as the active material, T. H. Maiman invented the first laser system in 1960. Thus, it is known as Ruby Laser.

ABSORPTION, SPONTANEOUS EMISSION, AND STIMULATED EMISSION OF RADIATION

Absorption of Radiation:An atom has a number of quantized states. Initially, an atom is in ground state, that is, all its electrons possess lowest possible energy state. When energy is given to an atom, it goes to excited state, that is, its electron jumps to a higher energy state by absorbing a quantum of radiation or photon. This process is called the absorption of radiation shown in Figure 5.1.

If E1 and E2 are the energies of an electron in initial and final states and ν is the frequency of absorbed radiation, then

where h is Plank's constant. Thus, absorption is a stimulated process.

SPONTANEOUS EMISSION

An atom in an excited state remains for only about 10−8 sec. After staying 10−8 sec, the atom jumps automatically to a lower energy state, emitting a radiation. If initially an atom is in excited state, then it spontaneously jumps to ground state, emitting a photon of frequency ν, given by