PN Junction Diode

PN Junctuon Diode

A PN Junction Diode is one of the most straightforward semiconductor gadgets around, and which has the electrical trait of going flow through itself in one course as it were. Be that as it may, dissimilar to a resistor, a diode doesn't act directly regarding the applied voltage.A PN-junction diode is formed when a p-type semiconductor is fused to an n-type semiconductor creating a potential barrier voltage across the diode junction.

There are two working locales and three potential "biasing" conditions for the standard Junction Diode and these are:

1. Zero Bias - No outer voltage potential is applied to the PN intersection diode.

2. Switch Bias - The voltage potential is associated negative, (- ve) to the P-type material and positive, (+ve) to the N-type material across the diode which expands the PN intersection diode's width.

3. Forward Bias - The voltage potential is associated positive, (+ve) to the P-type material and negative, (- ve) to the N-type material across the diode which diminishes the PN intersection diodes width.

Zero Biased PN Junction Diode

The potential hindrance that currently exists beats the dispersion of any greater larger part transporters across the intersection down. In any case, the potential obstruction helps minority transporters (scarcely any free electrons in the P-area and hardly any openings in the N-district) to float across the intersection.

Then, at that point, an "Balance" or adjust will be laid out when the greater part transporters are equivalent and both moving in inverse headings, so the net outcome is zero current streaming in the circuit. Whenever this happens the intersection is supposed to be in a territory of "Dynamic Equilibrium".

The minority transporters are continually produced because of nuclear power so this condition of harmony can be broken by raising the temperature of the PN intersection causing an increment in the age of minority transporters, consequently bringing about an expansion in spillage flow yet an electric flow can't stream since no circuit has been associated with the PN intersection.

Reverse Biased PN Junction Diode

Whenever a diode is associated in a Reverse Bias condition, a positive voltage is applied to the N-type material and a negative voltage is applied to the P-type material.

The positive voltage applied to the N-type material draws in electrons towards the positive terminal and away from the intersection, while the openings in the P-type end are additionally drawn in away from the intersection towards the negative anode.

The net outcome is that the consumption layer becomes more extensive because of an absence of electrons and openings and presents a high impedance way, just about a separator and a high potential obstruction is made across the intersection in this way keeping current from moving through the semiconductor material.

Forward Biased PN Junction Diode

Whenever a diode is associated in a Forward Bias condition, a negative voltage is applied to the N-type material and a positive voltage is applied to the P-type material. Assuming that this outer voltage becomes more noteworthy than the worth of the possible boundary, approx. 0.7 volts for silicon and 0.3 volts for germanium, the potential hindrances resistance will be survived and current will begin to stream.

This is on the grounds that the negative voltage pushes or repulses electrons towards the intersection giving them the energy to get over and consolidate with the openings being pushed the other way towards the intersection by the positive voltage. This outcomes in a qualities bend of zero current streaming up to this voltage point, called the "knee" on the static bends and afterward a high current move through the diode with little expansion in the outer voltage as displayed beneath

Networks

 Reciprocal networks

A network is said to be reciprocal if the voltage appearing at port 2 due to
a current applied at port 1 is the same as the voltage appearing at port 1 
when the same current is applied to port 2. Exchanging voltage and current 
results in an equivalent definition of reciprocity. A network that consists 
entirely of linear passive components (that is, resistors, capacitors and 
inductors) is usually reciprocal, a notable exception being passive 
circulators and isolators that contain magnetized materials. In general, it 
will not be reciprocal if it contains active components such as generators 
or transistors.
Reciprocity in electrical networks is a property of a circuit that relates 
voltages and currents at two points. The reciprocity theorem states that the 
current at one point in a circuit due to a voltage at a second point is the 
same as the current at the second point due to the same voltage at the first. 
The reciprocity theorem is valid for almost all passive netwonetworks.

Symmetrical networks

A network is symmetrical if its input impedance is equal to its output impedance. Most often, but not necessarily, 

symmetrical networks are also physically symmetrical. Sometimes also antimetrical networks are of interest. These 

are networks where the input and output impedances are the duals of each othnetwork

Lossless network

A lossless network is one, that contains no resistors or other dissipative elements.

Two-port network

 Two-port network

A two-port network (a kind of four-terminal network or quadripole) is an 

electrical network (circuit) or device with two pairs of terminals to connect to 

external circuits. Two terminals constitute a port if the currents applied to them 

satisfy the essential requirement known as the port condition: the electric current 

entering one terminal must equal the current emerging from the other terminal on 

the same port. The ports constitute interfaces where the network connects to other 

networks, the points where signals are applied or outputs are taken. In a two-port 

network, often port 1 is considered the input port, and port 2 is considered the 

output port.

The two-port network model is used in mathematical circuit analysis techniques 

to isolate portions of larger circuits. A two-port network is regarded as a "black 

box" with its properties specified by a matrix of numbers. This allows the response 

of the network to signals applied to the ports to be calculated easily, without solving for all the internal voltages 

and currents in the network. It also allows similar circuits or devices to be compared easily. For example, transistors 

are often regarded as two-ports, characterized by their h-parameters (see below) which are listed by the 

manufacturer. Any linear circuit with four terminals can be regarded as a two-port network provided that it does not 

contain an independent source and satisfies the port conditions.

Examples of circuits analyzed as two-ports are filters, matching networks, transmission lines, transformers, and 

small-signal models for transistors (such as the hybrid-pi model). The analysis of passive two-port networks is an 

outgrowth of reciprocity theorems first derived by Lorentz.

In two-port mathematical models, the network is described by a 2 by 2 square matrix of complex numbers. The 

common models that are used are referred to as z-parameters, y-parameters, h-parameters, g-parameters, and 

ABCD-parameters, each described individually below. These are all limited to linear networks since an underlying 

assumption of their derivation is that any given circuit condition is a linear superposition of various short-circuit 

and open-circuit conditions. They are usually expressed in matrix notation, and they establish relations between the 

variables

 V1 , voltage across port 1

 I1, current into port 1

 V2 , voltage across port 2

 I2, current into port 2

which are shown in figure 1. The difference between the various models lies in which of these variables are regarded 

as the independent variables. These current and voltage variables are most useful at low-to-moderate frequencies. 

At high frequencies (e.g., microwave frequencies), the use of power and energy variables is more appropriate, and 

the two-port current-voltage approach is replaced by an approach based upon scattering parameters.

The port conditions

The port condition is that a pair of poles of a circuit is considered a port if and only if the current flowing into one 

pole from outside the circuit is equal to the current flowing out of the other pole into the external circuit. 

Equivalently, the algebraic sum of the currents flowing into the two poles from the external circuit must be zero

Filter Circuit

Filter Circuit:

  Filters are electrical networks used to separate alternating from direct current components or to separate a group of 

A.C. components included within a particular frequency range from those lying outside this range. So a filter can 

be defined as a network that in its ideal form has at least one range of frequency in which the attenuation is zero 

(pass hand) and at least one range of frequency in which the attenuation is infinite (attenuation band). The 

frequencies which separate a pass band and attenuation hand are called cut-off frequencies. To achieve the desired 

effect, the filter is designed to provide a low attenuation for frequency components within a particular pass band  

range and a high attenuation at frequencies within other stop band ranges. The networks provide a uniform response 

over a wide range of frequencies than that obtained with resonant circuits. Filters are commonly classified in

accordance with their selectivity characteristics as below :

(a) A low pass filter. It transmits all frequencies below a limiting frequency 𝑓𝑐

. known as cut-off frequency. and 

stops all these above this frequency.

(b) A high pass filter. It passes frequencies above the cut-off freque

ncy and stops all those below this frequency. 

(c) A band pass filter. It passes frequencies in a particular hand between two cut-off frequencies and stops those 

above and below this band limit. 

(d) A band elimination filter. It stops frequencies within a specified band and passes those above and below the 

units of this hand.

Some Applications of Faraday’s Law

 The Ground Fault Interrupter (GFI of GFCI)

The ground fault interrupter is an interesting safety device that protects users of electrical appliances against electric shock. Its operation makes use of Faraday’s law. In the GFI shown in Figure 3, wire 1 leads from the wall outlet to the appliance to be protected, and wire 2 leads from the appliance back to the wall outlet. An iron ring surrounds the two wires, and a sensing coil is wrapped around part of the ring. Because the currents in the wires are in opposite directions, the net magnetic flux through the sensing coil due to the currents is zero. However, if the return current in wire 2 changes, the net magnetic flux through the sensing coil is no longer zero. (This can happen, for example, if the appliance gets wet, enabling current to leak to ground.) Because household current is alternating (meaning that its direction keeps reversing), the magnetic flux through the sensing coil changes with time, inducing an emf in the coil. This induced emf is used to trigger a circuit breaker, which stops the current before it is able to reach a harmful level.

Production of Sound in an Electric Guitar

Another interesting application of Faraday’s law is the production of sound in an electric guitar (Fig. 4). The coil in this case, called the pickup coil, is placed near the vibrating guitar string, which is made of a metal that can be magnetized. A permanent magnet inside the coil magnetizes the portion of the string nearest Lenz’s law the coil. When the string vibrates at some frequency, its magnetized segment produces a changing magnetic flux through the coil. The changing flux induces an emf in the coil that is fed to an amplifier. The output of the amplifier is sent to the loudspeakers, which produce the sound waves we hear

Induction Heater

This electric range cooks food on the basis of the principle of induction. An oscillating current is passed through a coil placed below the cooking surface, which is made of a special glass. The current produces an oscillating magnetic field, which induces a current in the cooking utensil. Because the cooking utensil has some electrical resistance, the electrical energy associated with the induced current is transformed to internal energy, causing the utensil and its contents to become hot.

Faraday’s law of electromagnetic induction

 Faraday’s law of electromagnetic induction

To see how an emf can be induced by a changing magnetic field, let us consider a loop of wire connected to a galvanometer, as illustrated in Figure 1. When a magnet is moved toward the loop, the galvanometer needle deflects in one direction, arbitrarily shown to the right in Figure 1a. When the magnet is moved away from the loop, the needle deflects in the opposite direction, as shown in Figure 1c. When the magnet is held stationary relative to the loop (Fig. 1b), no deflection is observed. Finally, if the magnet is held stationary and the loop is moved either toward or away from it, the needle deflects. From these observations, we conclude that the loop “knows” that the magnet is moving relative to it because it experiences a change in magnetic field. Thus, it seems that a relationship exists between current and changing magnetic field. These results are quite remarkable in view of the fact that a current is set up even though no batteries are present in the circuit! We call such a current an induced current and say that it is produced by an induced emf.

Now let us describe an experiment conducted by Faraday and illustrated in Figure 2. A primary coil is connected to a switch and a battery. The coil is wrapped around a ring, and a current in the coil produces a magnetic field when the switch is closed. A secondary coil also is wrapped around the ring and is connected to a galvanometer. No battery is present in the secondary circuit, and the secondary coil is not connected to the primary coil. Any current detected in the secondary circuit must be induced by some external agent. Initially, you might guess that no current is ever detected in the secondary circuit. However, something quite amazing happens when the switch in the primary circuit is either suddenly closed or suddenly opened. At the instant the switch is closed, the galvanometer needle deflects in one direction and then returns to zero. At the instant the switch is opened, the needle deflects in the opposite direction and again returns to zero. Finally, the galvanometer reads zero when there is either a steady current or no current in the primary circuit.

The key to understanding what happens in this experiment is to first note that when the switch is closed, the current in the primary circuit produces a magnetic field in the region of the circuit, and it is this magnetic field that penetrates the secondary circuit. Furthermore, when the switch is closed, the magnetic field produced by the current in the primary circuit changes from zero to some value over some finite time, and it is this changing field that induces a current in the secondary circuit.

As a result of these observations, Faraday concluded that an electric current can be induced in a circuit (the secondary circuit in our setup) by a changing magnetic field. The induced current exists for only a short time while the magnetic field through the secondary coil is changing. Once the magnetic field reaches a steady value, the current in the secondary coil disappears. In effect, the secondary circuit behaves as though a source of emf were connected to it for a short time. It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field.

The experiments shown in Figures 1 and 2 have one thing in common: In each case, an emf is induced in the circuit when the magnetic flux through the circuit changes with time.

Atoms and the Bohr Model

 INTRODUCTION

The concept of the atom was introduced by John Dalton in 1803 to explain the

chemical combination of elements to form compounds. The idea got further

confirmation when kinetic theory was developed to explain the behaviour of

gases. However, real understanding of the structure of the atom became possible

after the discovery of the electron by J.J. Thomson in 1897 and the realization

that all atoms contain electrons. The electron is a negatively-charged particle

having mass which is very small compared to the mass of an atom. Therefore,

the atom must also contain positively-charged matter, having mass almost equal

to the mass of the whole atom. Thomson suggested the plum-pudding model

of the atom, according to which the electrons are embedded in a uniform sphere

of positively-charged matter so that the atom as a whole is neutral.

Alpha-Scattering Experiment

In order to test the Thomson model, Geiger and Marsden carried out the following

experiment in 1908 under the guidance of Rutherford. Alpha-particles from a radioactive source were collimated into a narrow beam and then allowed to fall

on thin metal foils. The a-particles scattered in different directions were

detected. It was found that (a) most of the a-particles passed through the gold

foil without appreciable deflection, and (b) some of the a-particles suffered

fairly large deflections—in fact an unexpectedly large number even retraced

their path.

The Rutherford Nuclear Model

It is obvious that the above results cannot be explained on the basis of

Thomson’s model. Observation (a) requires that most of the space in the metal

foil must be empty. Observation (b) requires that the positively-charged matter

in an atom cannot be uniformly distributed but must be concentrated in a small

volume. Based on these facts, Rutherford proposed a new model known as the

nuclear model or the planetary model. According to this model, the whole of

the positive charge, which carries almost the entire mass of the atom, is concentrated

in a tiny central core called the nucleus. The electrons revolve around

the nucleus in orbits, leaving most of the volume of the atom unoccupied.

Difficulties with the Rutherford Model

The nuclear atom proposed by Rutherford could not be accepted due to the

following problems. An electron moving in a circle is continuously accelerated

towards the nucleus. According to classical electromagnetic theory, an

accelerated charge radiates electromagnetic energy. As such, the energy of the

electron would continuously decrease, its orbit would become smaller and

smaller and ultimately it would spiral into the nucleus. However, we know that

this does not happen and atoms are stable. Further, according to the classical

theory, the frequency of the radiation emitted by the electron is equal to the

frequency of revolution. Therefore, the spiralling electron would emit radiation

of continuously increasing frequency till it falls into the nucleus. However,

atoms do not radiate unless excited, in which case they radiate discrete, rather

than continuous, frequencies. We discuss this in more detail below.

Geometric Representation

Spacetime:

 Spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum (a continuous sequence in which adjacent elements are not perceptibly different from each other). Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur.

Minkowski Space: Minkowski space (or Minkowski spacetime) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.

Minkowski Space:

Minkowski space is closely associated with Einstein's theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frame of references will agree on the total distance in spacetime between events. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.

Space-Time Diagrams:

According to classical physics, the time coordinate is unaffected by a transformation from one inertial frame to another i.e. the time coordinate, t', of one inertial system does not depend on the space coordinates, x, y, z of another inertial system, the transformation equation being t' = t.

In relativity, however, space and time are interdependent. The time coordinate of one inertial system depends on both the time and the space coordinates of another inertial system, the transformation equation being.

t′= {𝑡−(𝑣𝑥/𝑐2)}/ √(1−𝑣2𝑐2)

Hence, instead of treating space and time separately, as is quite properly done in classical theory, it is natural in relativity to treat them together. H. Minkowski was first to show clearly how this could be done. In what follows, we shall consider only one space axis, the x-axis, and shall ignore the y and z axes. We lose no generality by this algebraic simplification and this procedure will enable us to focus more clearly on the interdependence of space and time and its geometric representation. The coordinates of an event are given then by x and t. All possible space-time coordinates can be represented on a space-time diagram in which the space axis is horizontal and the time axis is vertical. It is convenient to keep the dimensions of the coordinates the same; this is easily done by multiplying the time t by the universal constant c, the velocity of light. Let ct be represented by the symbol w. Then, the Lorentz transformation equations can be written as follows.

Notice the symmetry in this form of the equations. To represent the situation geometrically, we begin by drawing the x and w axes of frame S orthogonal (perpendicular) to one another

Fig-1

If we wanted to represent the motion of a particle in this frame, we would draw a curve, called a world line, which gives the loci of space-time points corresponding to the motion. Minkowski referred to space-time as "the world." Hence, events are world points and a collection of events giving the history of a particle is a world line. Physical laws on the interaction of particles can be thought of as the geometric relations between their world lines. In this sense, Minkowski may be said to have geometrized physics.

The tangent to the world line at any point, being 𝑑𝑥𝑑𝑤=1𝑐𝑑𝑥𝑑𝑡, is always inclined at an angle less than 45° with the time axis. For this angle (see Fig. 1) is given by tan𝜃 = 𝑑𝑥𝑑𝑤 = 𝑢𝑐 and we must have u < c for a material particle. The world line of a light wave, for which u = c, is a straight line making a 45° angle with the axes

Gaussian Surface

 A Gaussian surface is a closed imaginary surface in three-dimensional space through which the flux of a vector field is calculated. It enclosed all the charges for which flux is to be calculated.

Gauss's Law:

The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1813. It is one of Maxwell's four equations, which form the basis of classical electrodynamics.

The net electric flux through any hypothetical closed surface (Gaussian Surface) is equal to ...... times the net electric charge within that closed surface.

Integral Form of Gauss's Law

Gauss's law may be expressed as:

Where ΦE is the electric flux through a closed surface S enclosing any volume V, Q is the total charge enclosed within V, and ε0 is the electric permittivity. The electric flux ΦE is defined as a surface integral of the electric field:

Where E is the electric field, dA is a vector representing an infinitesimal element of area of the surface, and · represents the dot product of two vectors.

Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form.

Electric field

 An electric field (sometimes abbreviated as E-field) is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them. Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge by an infinitesimal test charge at that point. The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m).

Electric fields are created by electric charges, or by time-varying magnetic fields. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces (or interactions) of nature.

Electric Lines of Force:

An electric line of force is an imaginary continuous line or curve drawn in an electric field such that tangent to it at any point gives the direction of the electric force at that point. The direction of a line of force is the direction along which a small free positive charge will move along the line. Field lines directed into a closed surface are considered negative; those directed out of a closed surface are positive. If there is no net charge within a closed surface, every field line directed into the surface continues through the interior and is directed outward elsewhere on the surface.

Electric Flux:

Electric flux is a property of an electric field that may be thought of as the number of electric lines of force (or electric field lines) that intersect a given area. Electric field lines are considered to originate on positive electric charges and to terminate on negative charges. If a net charge is contained inside a closed surface, the total flux through the surface is proportional to the enclosed charge, positive if it is positive, negative if it is negative.

The mathematical relation between electric flux and enclosed charge is known as Gaussfs law for the electric field, one of the fundamental laws of electromagnetism.

Where E is the electric field (having units of V/m), E is its magnitude, S is the area of the surface, and .. is the angle between the electric field lines and the normal (perpendicular) to S.

For a non-uniform electric field, the electric flux dƒ³E through a small surface area dS is given by

Cut of Frequency

 The cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced (attenuated or reflected) rather than passing through.

Typically, in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response where a transition band and passband meet, for example, as defined by a half-power point (a frequency for which the output of the circuit is -3 dB of the nominal passband value). Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.

In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.

Two-port network

 A two-port network (a kind of four-terminal network or quadripole) is an electrical network (circuit) or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the electric current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port, and port 2 is considered the output port.

The two-port network model is used in mathematical circuit analysis techniques to isolate portions of larger circuits. A two-port network is regarded as a "black box" with its properties specified by a matrix of numbers. This allows the response of the network to signals applied to the ports to be calculated easily, without solving for all the internal voltages and currents in the network. It also allows similar circuits or devices to be compared easily. For example, transistors are often regarded as two-ports, characterized by their h-parameters (see below) which are listed by the manufacturer. Any linear circuit with four terminals can be regarded as a two-port network provided that it does not contain an independent source and satisfies the port conditions.

Examples of circuits analyzed as two-ports are filters, matching networks, transmission lines, transformers, and small-signal models for transistors (such as the hybrid-pi model). The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz.

In two-port mathematical models, the network is described by a 2 by 2 square matrix of complex numbers. The common models that are used are referred to as z-parameters, y-parameters, h-parameters, g-parameters, and ABCD-parameters, each described individually below. These are all limited to linear networks since an underlying assumption of their derivation is that any given circuit condition is a linear superposition of various short-circuit and open-circuit conditions. They are usually expressed in matrix notation, and they establish relations between the variables

V1 , voltage across port 1

I1, current into port 1

V2 , voltage across port 2

I2, current into port 2

which are shown in figure 1. The difference between the various models lies in which of these variables are regarded as the independent variables. These current and voltage variables are most useful at low-to-moderate frequencies. At high frequencies (e.g., microwave frequencies), the use of power and energy variables is more appropriate, and the two-port current-voltage approach is replaced by an approach based upon scattering parameters.

 

Vibration:

A vibration is a kind of motion that changes the direction of sense and after a regular interval of time. It is also a to and fro motion.

 

Free Vibration:

A closed vibratory system is given and initial excitation and is allowed to vibrate without further influence executes free vibration.

 

Criteria of Vibration

a) It has restoring force

b) It has inertia of motion

c) It has also initial excitation

 

Simple Harmonic Motion:

When a body moves such that its acceleration is proportional to its displacement from its equilibrium position or any other fixed point at its path and be always directed towards that point then the motion of the body is called simple harmonic motion. 

Characteristics of S.H.M. / Criteria of S.H.M.

a) The motion is periodic motion in a particular case

b) the acceleration of the body is proportional to its displacement from its equilibrium position

c) The acceleration of the body is directed towards a certain fixed point

d) The motion is isochronous because the expression for the time period  is independent of the amplitude of the motion

e) The equation does not conclude the motion of the particle along the circle

f) Acceleration varies directly as its distance from the equilibrium point.

 

Stationary Wave

 

When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions superimpose on each other, stationary waves are formed. 

Analytical method:

   Let us consider a progressive wave of amplitude a and wavelength λ travelling in the direction of X axis. 

y₁ = a sin 2π [t/T – x/λ]         …... (1)

This wave is reflected from a free end and it travels in the negative direction of X axis, then 

y₂ = a sin 2π [t/T + x/λ]         …... (2)

According to principle of superposition, the resultant displacement is,    

y = y₁+y₂   = a [sin 2π (t/T – x/λ) + sin 2π (t/T + x/λ)]

                                                       = a [2sin (2πt/T) cos  (2πx/λ)]

So, y = 2a cos (2πx/λ) sin (2πt/T)         …... (3)

Characteristics of stationary waves:

   The waveform remains stationary.   Nodes and antinodes are formed alternately.   The points where displacement is zero are called nodes and the points where the displacement is maximum are called antinodes.   Pressure changes are maximum at nodes and minimum at antinodes.  

   All the particles except those at the nodes, execute simple harmonic motions of same period.   Amplitude of each particle is not the same, it is maximum at antinodes decreases gradually and is zero at the nodes.   The velocity of the particles at the nodes is zero. It increases gradually and is maximum at the antinodes.   Distance between any two consecutive nodes or antinodes is equal to ^λ₂, whereas the distance between a node and its adjacent antinode is equal to λ/4.   There is no transfer of energy. All the particles of the medium pass through their mean position simultaneously twice during each vibration.   Particles in the same segment vibrate in the same phase and between the neighboring segments, the particles vibrate in opposite phase.

 

This is the equation of a stationary wave.   (a) At points where x = 0, λ/2, λ, 3λ/2, the values of cos 2πx/λ = ±1   ∴ A = + 2a. At these points the resultant amplitude is maximum. They are called antinodes as shown in figure.   (b) At points where x = λ/4, 3λ/4, 5λ/4..... the values of cos 2πx/λ = 0.   ∴ A = 0. The resultant amplitude is zero at these points. They are called nodes.      The distance between any two successive antinodes or nodes is equal to λ/2 and the distance between an antinode and a node is λ/4.   (c) When t = 0, T/2, T, 3T/2, 2T, then sin 2πt/T = 0, the displacement is zero.   (d) When t = T/4, 3T/4, 5T/4 etc,....sin 2πt/T = ±1,  the displacement is maximum.   

Equivalence Between Voltage Source and Current Source

 Practically, a voltage source is not different from a current source. In fact, a source can either work as a current source or as a voltage source. It merely depends upon its working conditions. If the value of the load impedance is very large compared to the internal impedance of the source, it proves advantageous to treat the source as a voltage source. On the other hand, if the value of the load impedance is very small compared to the internal impedance, it is better to represent the source as a current source. From the circuit point of view, it does not matter at all whether the source is treated as a current source or a voltage source. In fact, it is possible to convert a voltage source into a current source and vice-versa.

Fig.1: A source connected to a load

Concept of Current Source

 Like a constant voltage source, there may be a constant current source – a source that supplies a constant current to a load even its impedance varies. Ideally, the current supplied by it should remain constant, no matter what the load impedance is.

A symbolic representation of such an ideal current source is shown in Fig. 1(a). The arrow inside the circle indicates the direction in which current will flow in the circuit when a load is connected to the source. Fig. 1(b) shows the V-I characteristic of an ideal current source. Let us connect a variable load impedance ZL to a constant current source as shown in Fig. 1(c). As stated above, the current supplied by the source should remain constant at is for all values of load impedance.

Fig.1: (a) Symbol for an ideal current source (b) V-I characteristic of an ideal current source (c) A variable load connected to an ideal current source (d) Symbol for a practical current source

It means even if ZL is made infinity, the current through this should remain IS (same). Now, we must see if any practical current source could satisfy this condition. The load impedance ZL = ∞ means no conducting path, external to the source, exists between the terminals A and B. Hence, it is a physical impossibility for current to flow between terminals A and B. If the source could maintain a current Is through an infinitely large load impedance, there would have been an infinitely large voltage drop across the load. It would then have consumed infinite power from the source. Of course, no practical source could ever supply infinite power.

The maximum voltage that the current source can deliver to the load is called compliance voltage. During the variation in the load the current source work like ideal source, provides the unlimited resistance but, when the voltage value at the output reaches to compliance voltage, then it starts to behave like a real source and provides the limited value of resistance.

A practical current source supplies current IS to a short-circuit (i.e. when ZL= 0). That is why the current IS is called short-circuit current. But, when we increase the load impedance, the current falls below IS. When the load impedance ZL is made infinite (i.e., the terminals A and B are open-circuited), the load current reduces to zero. It means there should be some path (inside the source itself) through which the current IS can flow. When some finite load impedance is connected, only a part of this current IS flows through the load. The remaining current goes through the path inside the source. This inside path has an impedance ZS, and is called the internal impedance. The symbolic representation of such a practical current source is shown in Fig. 1(d).

Now, if terminals AB are open-circuited (ZL = ∞) in Fig. 1(d), the terminal voltage does not have to be infinite. It is now a finite value, VT = IS ZS. It means that the source does not have to supply infinite power!

Practical Voltage Source

 An ideal voltage source is not practically possible. There is no source which can attain it terminal voltage constant when its terminals are short-circuited. If it could do so, it would mean that it can supply an infinite amount of power to a short-circuit. This is not possible. Hence, an ideal voltage source does not exist in practice. However, the concept of an ideal voltage source is very helpful in understanding the circuits containing a practical voltage source.

A practical voltage source can be considered to consist of an ideal voltage source in series with an impedance. This impedance is called the internal impedance of the source. The symbolic representation of practical voltage sources are shown in Fig. 1.

Fig. 1: Practical voltage source: (a) DC voltage source (b) AC voltage source

It is not possible to reach any other terminal except A and B. These are the terminals available for making external connections. In the dc source, since the upper terminal of the ideal voltage source is marked positive, the terminal A will be positive with respect to terminal B. In the ac source in Fig.1(b), the upper terminal of the ideal voltage source is marked as positive and lower as negative. The marking of positive and negative on an ac source does not mean the same thing as the markings on a dc source. Here (in ac), it means that the upper terminal (terminal A) of the ideal voltage source is positive with respect to the lower terminal at that particular instant. In the next half-cycle of ac, the lower terminal will be positive and the upper negative. Thus, the positive and negative markings on an ac source indicate the polarities at a given instant of time. In some books you will find the reference polarities marked by, instead of positive and negative signs, an arrow pointing towards the positive terminal.

The question naturally arises: What should be the characteristics of a source so that it may be considered a good enough constant voltage source? An ideal voltage source, of course, must have zero internal impedance. In practice, no source can be an ideal one. Therefore, it is necessary to determine how much the value of the internal impedance ZS should be, so that it can be called a good practical voltage source.

Let us consider an example. A dc source has an open-circuit voltage of 2 V, and internal resistance of only 1 Ù. It is connected to a load resistance RL as shown in Fig. 2(a). The load resistance can assume any value ranging from 1 Ù to 10 Ù. Let us now find the variation in the terminal voltage of the source. When the load resistance RL is 1 Ù the total resistance in the circuit is 1 Ù + 1 Ù = 2 Ù. The current in the circuit is

𝐼𝑇  = 𝑉𝑆 /(𝑅𝑆+𝑅𝐿1) = 2/(1+1) = 1 𝐴

Fig. 2: Voltage sources connected to variable loads

The terminal voltage is then 𝑉𝑇1=𝐼1×𝑅𝐿1=𝑉𝑆/(𝑅𝑆+𝑅𝐿1)×𝑅𝐿1 =21+1×1=1.0 𝑉

When the load resistance becomes 10 Ω, the total resistance in the circuit becomes 10 Ω + 1 Ω =11 Ω. We can again find the terminal voltage as 𝑉𝑇2=𝐼2×𝑅𝐿2=𝑉𝑆/(𝑅𝑆+𝑅𝐿2)×𝑅𝐿2 =21+10×10=1.818 𝑉

Thus, we find that the maximum voltage available across the terminals of the source is 1.818 V. When the load resistance varies between its extreme limits—from 1 Ω to 10 Ω, the terminal voltage varies from 1 V to 1.818 V. This is certainly a large variation. The variation in the terminal voltage is more than 40 % of the maximum voltage.

Let us consider another example. A 600 Ω, 2 V ac source is connected to a variable load, as shown in Fig. 2(b). The load impedance ZL can vary from 50 KΩ to 500 KΩ, again a variation having the same ratio of 1 : 10, as in the case of the first example. We can find the variation in the terminal voltage of the source. When the load impedance is 50 KΩ, the terminal voltage is

𝑉𝑇1=𝐼1×𝑍𝐿1=𝑉𝑆/(𝑍𝑆+𝑍𝐿1)×𝑍𝐿1 =2/(600+50000)×50000=1.976 𝑉