The post Energy Density in Electrostatic Fields appeared first on ece4uplp.

]]>To determine the Energy present in an assembly of charges (or) group of charges one must first determine the amount of work necessary to assemble them.

It is seen that , when a unit positive charge is moved from infinity to a point in a field, the work is done by the external source and energy is expended.

If the external source is removed then the unit positive charge will be subjected to a force exerted by the field and will be moved in the direction of force.

Thus to hold the charge at a point in an electrostatic field, an external source has to do work , this energy gets stored in the form of Potential Energy when the test charge is hold at a point in a field.

when external source is removed , the Potential Energy gets converted to a Kinetic Energy.

In order to derive the expression for energy stored in electrostatics (i.e, the expression of such a Potential Energy)

Consider an empty space where there is no electric field at all, the Charge is moved from infinity to a point in the space ,let us say the point as , this requires no work to be done to place a charge from infinity to a point in empty space.

i.e, work done = 0 for placing a charge from infinity to a point in empty space.

now another charge has to be placed from infinity to another point . Now there has to do some work to place at because there is an electric field , which is produced by the charge and is required to move against the field of .

Hence the work required to be done is

i.e, .

Work done to position at = .

Now the charge to be moved from infinity to , there are electric fields due to and , Hence total work done is due to potential at due to charge at and Potential at due to charge at .

Work done to position at = .

Similarly , to place a charge at in a field created by (n-1) charges is ,work done to position at

Total Work done

The total work done is nothing but the Potential energy in the system of charges hence denoted as ,

if charges are placed in reverse order (i.e, first and then and then and finally is placed)

work done to place

work done to place

work done to place

Total work done

EQN (I)+EQN(II) gives

let , and are the resultant Potentials due to all the charges except that charge.

i.e, is the resultant potential due to all the charges except .

Joules.

The above expression represents the Potential Energy stored in the system of n point charges.

simillarly,

Joules

Joules

Joules for different types of charge distributions.

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]]>The post Switches-Circuit Switching appeared first on ece4uplp.

]]>**Circuit Switches (or) Structure of Circuit Switch:-**

The Switches used in Circuit Switching are called Circuit-Switches

**Space-Division Switch:-**

- The paths are separated spatially from one switch to other.
- These were originally designed for analog circuits but currently used for both analog and digital Networks.

**Cross-bar Switch:-**

In this type of Switch we connect n inputs and m outputs using micro switches (Transistors) at each cross point to form a cross-bar switch of size n X m.

The number of cross points required = n X m.

As n and m increases, cross points required also increases, for example n=1000 and m=1000 requires n X m= 1000 X 1000 cross points. A cross-bar with these many number of cross points is impractical and statics show that 25% of the cross points are in use at any given time.

**Multi stage Switch:-**

The solution to Cross-bar Switch is Multi stage switching. Multi stage switching is preferred over cross-bar switches to reduce the number of cross points. Here number of cross-bar switches are combined in several stages.

Suppose an N X N cross-bar Switch can be made into 3 stage Multi bar switch as follows.

- N is divided into groups , that is N/n Cross-bars with n-input lines and k-output lines forms n X k cross points.
- The second stage consists of k Cross-bar switches with each cross-bar switch size as (N/n) X (N/n).
- The third stage consists of N/n cross-bar switches with each switch size as k X n.

The total number of cross points = , so the number of cross points required are less than single-stage cross-bar Switch = .

for example k=2 and n=3 and N=9 then a Multi-stage switch looks like as follows.

The problem in Multi-stage switching is Blocking during periods of heavy traffic, the idea behind Multi stage switch is to share intermediate cross-bars. Blocking means times when one input line can not be connected to an output because there is no path available (all possible switches are occupied). Blocking generally occurs in tele phone systems and this blocking is due to intermediate switches.

Clos criteria gives a condition for a non-blocking Multi stage switch

, and Total no.of Cross points .

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]]>The post GATE problems in EMT appeared first on ece4uplp.

]]>Ans:- Given loss less insulator

Cm/Sec

from

from Equations I and II

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]]>The post Solved Example problems in Electro Magnetic Theory appeared first on ece4uplp.

]]>Ans. Given P(1,3,5)

Cylindrical :-

Similarly

Spherical :-

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]]>The post Phase Locked Loop (PLL) appeared first on ece4uplp.

]]>Let the input to PLL is an FM signal

let

Now the signal at the output of VCO is FM signal (another FM signal, which is different from input FM signal) Since Voltage Controlled Oscillator is an FM generator.

the corresponding phase

It is observed that S(t) and b(t) are out of phase by . Now these signals are applied to a phase detector , which is basically a multiplier

the error signal

on further simplification , the product yields a higher frequency term (Sum) and a lower frequency term (difference)

This product e(t) is given to a loop filter , Since the loop filter is a LPF it allows the difference and term and rejects the higher frequency term.

the over all output of a loop filter is

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]]>The post Frequency domain representation of a Wide Band FM appeared first on ece4uplp.

]]>i.e, Single-tone FM signal is

Now by expressing the above signal in terms of Phasor notation ( , None of the terms can be neglected)

Let is the complex envelope of FM signal.

is a periodic function with period . This can be expressed in it’s Complex Fourier Series expansion.

i.e, this approximation is valid over . Now the Fourier Coefficient

let implies

as and

let as order Bessel Function of first kind then .

Continuous Fourier Series expansion of

Now substituting this in the Equation (I)

The Frequency spectrum can be obtained by taking Fourier Transform

n value | wide Band FM signal |

0 | |

1 | |

-1 | |

… | …. |

From the above Equation it is clear that

- FM signal has infinite number of side bands at frequencies for n values changing from to .
- The relative amplitudes of all the side bands depends on the value of .
- The number of significant side bands depends on the modulation index .
- The average power of FM wave is Watts.

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]]>The post Mutual Information I(X ; Y) Properties appeared first on ece4uplp.

]]>Mutual Information is given by equation

we know that

Substitute Equation (II) in Equation (I)

The above Equation can be written as

we knew that

This result can be applied to Mutual Information , If and be , Both and are two probability distributions on same alphabet , then Equation (III) becomes

i.e, , Which implies that Mutual Information is always Non-negative (Positive).

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]]>The post Example Problems in Electro Magnetic Theory Wave propagation appeared first on ece4uplp.

]]>Ans. Given , and

and

Find the Loss tangent

So given medium is a Conductor (Copper)

then

, .

.

.

2. If for a medium in which a wave with a frequency of is propagating . Determine the propagation constant and intrinsic impedance of the medium when

Ans: Given , , and .

Since , the given medium is a lossless Di-electric.

which implies

.

Ω.

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]]>The post Delta modulation and Demodulation appeared first on ece4uplp.

]]>i.e, Delta Modulation (DM) is a Modulation scheme in which an incoming message signal is over sampled (i.e, at a rate much higher than the Nyquist rate ) to purposely increase the correlation between adjacent samples of the signal. Over sampling is done to permit the use of a sample Quantizing strategy for constructing the encoded signal.

Signaling rate and Transmission Band Width are quite large in PCM. DM is used to overcome these problems in PCM .

DM transmits one bit per sample.

The process of approximation in Delta Modulation is as follows:-

The difference between the input () and the approximation () is quantized into only two levels corresponding to Positive and negative differences.

i.e, If the approximation () falls below the signal ()at any sampling epoch(the beginning of a period)output signal level is increased by .

On the other hand the approximation () lies above the signal () , output signal level is diminished by provided that the input signal does not change too rapidly from sample to sample.

it is observed that the change in stair case approximation lies with in .

This process can be illustrated in the following figure

**Delta Modulated System:- **The DM system consists of Delta Modulator and Delta Demodulator.

**Delta Modulator:- **

Mathematical equations involved in DM Transmitter are

error signal:

Present sample of the (input) sampled signal:

last sample approximation of stair case signal:

Quantized error signal( output of one-bit Quantizer):

if .

and .

encoding has to be done on the after Quantization that is when the output level is increased by from its previous quantized level, bit ‘1’ is transmitted .

similarly when output is diminished by from the previous level a ‘0’ is transmitted.

from the accumulator

where is the Quantization error.

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]]>The post Fourier Series and it’s applications appeared first on ece4uplp.

]]>The alternative representation if a set of complex exponentials are used,

The resulting representations are known as Fourier Series in Continuous-Time . Here we focus on representation of Continuous-Time and Discrete-Time periodic signals in terms of basic signals as Fourier Series and extend the analysis to the Fourier Transform representation of broad classes of aperiodic, finite energy signals.

These Fourier Series & Fourier Transform representations are most powerful tools used

- In the analyzation of signals and LTI systems.
- Designing of Signals & Systems.
- Gives insight to S&S.

The development of Fourier series analysis has a long history involving a great many individuals and the investigation of many different physical phenomena.

The concept of using “Trigonometric Sums”, that is sum of harmonically related sines and cosines (or) periodic complex exponentials are used to predict astronomical events.

Similarly, if we consider the vertical deflection of the string at time t and at a distance x along the string then for any fixed instant of time, the normal modes are harmonically related sinusoidal functions of x.

The scientist Fourier’s work, which motivated him physically was the phenomenon of heat propagation and diffusion. So he found that the temperature distribution through a body can be represented by using harmonically related sinusoidal signals.

In addition to this he said that any periodic signal could be represented by such a series.

Fourier obtained a representation for aperiodic (or) non-periodic signals not as weighted sum of harmonically related sinusoidals but as weighted integrals of Sinusoids that are not harmonically related, which is known as Fourier Integral (or) Fourier Transform.

In mathematics, we use the analysis of Fourier Series and Integrals in

- The theory of Integration.
- Point-set topology.
- and in the eigen function expansions.

In addition to the original studies of vibration and heat diffusion, there are numerous other problems in science and Engineering in which sinusoidal signals arise naturally, and therefore Fourier Series and Fourier T/F’s plays an important role.

for example, Sine signals arise naturally in describing the motion of the planets and the periodic behavior of the earth’s climate.

A.C current sources generate sinusoidal signals as voltages and currents. As we will see the tools of Fourier analysis enable us to analyze the response of an LTI system such as a circuit to such Sine inputs.

Waves in the ocean consists of the linear combination of sine waves with different spatial periods (or) wave lengths.

Signals transmitted by radio and T.V stations are sinusoidal in nature as well.

The problems of mathematical physics focus on phenomena in Continuous Time, the tools of Fourier analysis for DT signals and systems have their own distinct historical roots and equally rich set of applications.

In particular, DT concepts and methods are fundamental to the discipline of numerical analysis , formulas for the processing of discrete sets of data points to produce numerical approximations for interpolation and differentiation were being investigated.

FFT known as Fast Fourier Transform algorithm was developed, which suited perfectly for efficient digital implementation and it reduced the time required to compute transform by orders of magnitude (which utilizes the DTFS and DTFT practically).

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]]>The post Block Diagram of Digital Communication System/Elements of DCS appeared first on ece4uplp.

]]>Our basic aim is to understand the various modules and sub systems in the system. If we are trying to understand the design and various features of DCS , it is plus imperative that we have to understand how we should design a transmitter and we must understand how to design a very good quality Receiver. Therefore one must know the features of the channel to design a good Transmitter as well as receiver that is the channel and it’s contribution will come repeatedly in digital Communications.

**Source:-** the primary block (or) the starting point of a DCS is an information source, it may be an analog/digital source , for example the signal considered is analog in nature, then the signal generated by the source is some kind of electrical signal which is random in nature. if the signal is a speech signal (not an electrical signal) that has to be converted into electrical signal by means of a Transducer, which can be considered as a part of source itself.

**Sampling & Quantization:-** the secondary block involves the conversion of analog to discrete signal

this involves the following steps

**Sampling:- **it is the process that involves in the conversion of Continuous Amplitude Continuous Time (CACT) signal into Continuous Amplitude Discrete Time (CADT) signal.

**Quantization:- **it is the process that involves in the conversion of Continuous Amplitude Discrete Time (CADT) signal into Discrete Amplitude Discrete Time (DADT) signal.

**Source Encoder:- ** An important problem in Digital Communications is the efficient representation of data generated by a Discrete Source, this is accomplished by source encoder.

” The process of representation of incoming data from a Discrete source into a more suitable form required for Transmission is known as source encoding”

Note:-The blocks Sampler, Quantizer followed by an Encoder constructs ADC (Analog to Digital Converter).

∴ the output of Source encoder is a Digital Signal, the advantages of Source coding are

- It reduces the Redundancy.
- Minimizes the average bit rate.

**Channel encoder:-**Channel coding is also known as error control coding. Channel coding is a technique which reduces the probability of error by reducing Signal to Noise Ratio at the expense of Transmission Band Width.The device that performs the channel coding is known as Channel encoder.

Channel encoding increases the redundancy of incoming data , this also involves error detection and error correction along with the channel decoder at the receiver.

**Spreading Techniques:-** Spread Spectrum techniques are the methods by which a signal generated with a particular Band Width is deliberately spread in the frequency domain, resulting in a signal with a wider Band width.

There are two types of spreading techniques available

- Direct Sequence Spread Spectrum Techniques.
- Frequency Hopping Spread Spectrum Techniques.

The output of a spreaded signal is very much larger than incoming sequence. Spreading increases the BW required for transmission, which is a disadvantage even though spreading is done for high security of data.

SS techniques are used in Military applications.

**Modulator:-** Spreaded sequence is modulated by using digital modulation schemes like ASK, PSK, FSK etc depending up on the requirement, now the transmitting antenna transmits the modulated data into the channel.

**Receiver:-** Once you understood the process involved in transmitter Block. One should perform reverse operations in the receiver block.

i.e the input of the demodulator is demodulated after that de-spreaded and then the channel decoder removes the redundancy added by the channel encoder ,the output of channel decoder is then source decoded and is given to Destination.

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]]>The post Why Digital Communication is preferred over Analog Communication? appeared first on ece4uplp.

]]>Communication is the process of establishing Connection (or) link between two points (which are separated by some distance) and transporting information between those two points. The electronic equipment used for communication purpose is called Communication equipment. The equipment when assembled together forms a communication system.

Examples of different types of communications are

- Line Telephony & Telegraphy.
- Radio Broadcasting.
- Point-to-Point Communication.
- Mobile Communication.
- TV Broadcasting.
- Radar and Satellite Communications etc.

**Why Digital?**

A General Communication system has two devices and a medium (channel) connecting those two devices. This can be understood that a Transmitter and Receiver are separated by a medium called as Communication channel. To transport an information-bearing signal from one point to another point over a communication channel either Analog or digital modulation techniques are used.

Now Coming to the point, Why Digital communication is preferred over analog Communication?

Why are communication systems, military and commercial alike, going digital?

- There are many reasons; the primary advantage is the ease with which digital signals compared with analog signals are generated. That is the generation of digital signals is much easier compared to analog signals.
**Propagation of Digital pulse through a Transmission line:-**

When an ideal binary digital pulse propagating along a Transmission line. The shape of the waveform is affected by two mechanisms

- Distortion caused on the ideal pulse because all Transmission lines and Circuits have some Non-ideal frequency Transfer function.
- Unwanted electrical noise (or) other interference further distorts the pulse wave form.

Both of these mechanisms cause the pulse shape to degrade as a function of line length. During the time that the transmitted pulse can still be reliably identified (i.e. before it is degraded to an ambiguous state). The pulse is amplified by a digital amplifier that recovers its original ideal shape. The pulse is thus “re-born” (or) regenerated.

Circuits that perform this function at regular intervals along Transmission system are called “regenerative repeaters’. This is one of the reasons why Digital is preferred over Analog.

**3.Digital Circuits Vs Analog Circuits:-**

Digital Circuits are less subject to distortion and Interference than are analog circuits because binary digital circuits operate in one of two states FULLY ON (or) FULLY OFF to be meaningful, a disturbance must be large enough to change the circuit operating point from one state to another. Such two state operation facilitates signal representation and thus prevents noise and other disturbances from accumulating in transmission.

However, analog signals are not two-state signals, they can take an infinite variety of shapes with analog circuits and even a small disturbance can render the reproduced wave form unacceptably distorted. Once the analog signal is distorted, the distortion cannot be removed by amplification because accumulated noise is irrecoverably bound to analog signals, they cannot be perfectly generated.

4. With digital techniques, extremely low error rates and high signal fidelity is possible through error detection and correction but similar procedures are not available with analog techniques.

5. Digital circuits are more reliable and can be produced at a lower cost than analog circuits also; digital hardware lends itself to more flexible implementation than analog hardware.

Ex: – Microprocessors, Digital switching and large scale Integrated circuits.

6. The combining of Digital signals using Time Division Multiplexing (TDM) is simpler than the combining of analog signals using Frequency Division Multiplexing (FDM).

7. Digital techniques lend themselves naturally to signal processing functions that protect against interference and jamming (or) that provide encryption and privacy and also much data communication is from computer to computer (or) from digital instruments (or) terminal to computer, such digital terminations are normally best served by Digital Communication links.

8. Digital systems tend to be very signal-processing intensive compared with analog systems.

Apart from pros there exists a con in Digital Communications that is non-graceful degradation when the SNR drops below a certain threshold, the quality of service can change suddenly from very good to very poor. In contrast most analog Communication Systems degrade more gracefully.

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]]>The post Maxwell’s Equations in Point (or Differential form) and Integral form appeared first on ece4uplp.

]]>- .
- .
- .
- .

The 4 Equations above are known as Maxwell’s Equations. Since Maxwell contributed to their development and establishes them as a self-consistent set. Each differential Equation has its integral part. One form may be derived from the other with the help of Stoke’s theorem (or) Divergence theorem.

A word statement of the field Equations is readily obtained from their mathematical statement in the integral form.

1. .

i.e, The magneto motive force ( is m.m.f)around a closed path is equal to the conduction current plus the time derivative of the electric displacement through any surface bounded by the path.

2. .

The electro motive force ( is e.m.f)around a closed path is equal to the time derivative of the magnetic displacement through any surface bounded by the path.

3. .

The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume.

4. .

The net magnetic flux emerging through any close surface is zero.

the time-derivative of electric displacement is called displacement current. The term electric current is then to include both conduction current and displacement current. If the time-derivative of electric displacement is called an electric current, similarly is known as magnetic current, e.m.f as electric voltage and m.m.f as magnetic voltage.

the first two Maxwell’s Equations can be stated as

- The magnetic voltage around a closed path is equal to the electric current through the path.
- The electric voltage around a closed path is equal to the magnetic current through the path.

Maxwell’s Equations of static-fields in differential form and integral form are:

- .
- .
- .
- .

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]]>The post Inconsistensy in Ampere’s law (or) Displacement Current density appeared first on ece4uplp.

]]>From Ampere’s Circuital law which is applicable to Steady Magnetic fields

By taking divergence of Ampere’s law the Ampere’s law is not consistent with time-varying fields

,since

the divergence of the curl is identically zero which implies , but from the continuity equation which is not equal to zero, as is an unrealistic limitation(i.e we can not assume as zero) .

to make a compromise between the above two situations we must add an unknown term to Ampere’s Circuital law

i.e,

then by taking the Divergence of the above equation

from Equation(1),Equation(4) becomes

thus

from Maxwell’s first Equation

then Equation (5) becomes

then

This is the equation obtained which does not disagree with the continuity equation. It is also consistent with all other results. This is a second Maxwell’s Equation is time-varying fields so the term has the dimensions of current density Amperes/Square-meter. Since it results from a time-varying electric flux density ( ) , Maxwell termed it as displacement current density .

up to this point three current densities are there , and .

when the medium is Non-conducting medium

the total displacement current crossing any given surface is expressed by the surface integral

from Ampere’s law

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]]>The post Correlation Receiver(Special case of Optimum Receiver) appeared first on ece4uplp.

]]>from the figure output of the filter after sampling at seconds is

at output becomes

Now by substituting

by substituting the Equation(2) in Equation(1) over the limits

Now by replacing with the above equation becomes

The equation (3) suggests that the Optimum Receiver can be implemented as shown in the figure, this form of the Receiver is called as correlation Receiver. This receiver requires the integration operation be ideal with zero initial conditions. Correlation Receiver performs coherent-detection.

in general Correlation Receiver can be approximated with Integrate and dump filter.

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]]>The post Relation between E and V appeared first on ece4uplp.

]]>i.e,

.

This equation implies that the total work done in moving a charge from A to B and then from B to A is zero.

i.e,

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]]>The post Electric Potential (V) appeared first on ece4uplp.

]]>Assume a test charge at A in an Electric field, let points A and B are located at units from the origin O,from Coulomb’s law the force acting on a test charge is

The work done in moving a point charge along a differential length is is given by

so the total work done in moving a point charge from A to B is

the direction of work done is always opposite to the direction of displacement.

where A is the initial point and B is the final point. Dividing the work done by the charge gives the potential energy per unit charge denoted by ,this is also known as potential difference between the two points A and B.

Thus

if we take B as initial point and A as final point , then

To derive the expression for V in terms of charge Q and distance r , we can use the concept of Electric field intensity produced by a charge Q, which is placed at a distance r

i.e,

from Equation(1)

since

similarly,

where and are the scalar potentials at the points A and B respectively. If A is located at with respect to origin ,with zero potential and B is located at a distance r with respect to origin. then the work done in moving a charge from A (infinity) to B is given by

here

volts.

hence the potential at any point is the potential difference between that point and a chosen point at which the potential is zero. In other words assuming Zero potential at infinity .

The potential at a distance r from a point charge is the work done per unit charge by an external agent in transferring a test charge from infinity to that point.

i.e,

So a point charge located at a point P with position vector then the potential at another point Q with a position vector is

As like superposition principle is applicable to V also that is for n point charges located at points with position vectors

then the potential at is

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]]>The post few problems on Auto correlatioon Function(ACF) and Energy Spectral Density(ESD) appeared first on ece4uplp.

]]>Ans. We know that Auto correlation function forms fourier transform pair with Energy Spectral Density function

the Fourier Transform of

here

the Fourier Transform of x(t) is X(f) and is and the Energy Spectral Density

By finding the inverse Fourier Transform of S(f) gives the Auto Correlation Function

the ACF of the given signal is inverse Fourier Transform of S(f) which is .

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]]>