## Electric field due to infinite line charge distribution

Consider an infinitely long straight line carrying uniform line charge with density $\rho&space;_{L}&space;C/m$ and lies on Z-axis from $-\infty$ to $+\infty$.

Consider a point P at which Electric field intensity has to be determined which is produced by the line charge distribution.

from the figure let the co-ordinates of P are $(0,\rho&space;,0)$ ( a point on y-axis) and assume $dQ$ is a small differential charge confirmed to a point  M $(0,0,Z)$ as co-ordinates.

$\therefore&space;dQ$ produces a differential field $\overrightarrow{dE}$

$\overrightarrow{dE}=\frac{dQ}{4\pi&space;\epsilon&space;_{o}R^{2}}\widehat{a_{r}}$

the position vector $\overrightarrow{R}=-Z\overrightarrow{a_{z}}+\rho&space;\overrightarrow{a_{\rho&space;}}$ and the corresponding unit vector $\widehat{a_{r}}&space;=\frac{-Z\overrightarrow{a_{z}}+\rho&space;\overrightarrow{a_{\rho&space;}}}{\sqrt{\rho&space;^{2}+Z^{2}}}$

$\therefore&space;\overrightarrow{dE}&space;=\frac{dQ}{4\pi&space;\epsilon&space;_{o}}({\frac{-Z\overrightarrow{a_{z}}+\rho&space;\overrightarrow{a_{\rho&space;}}}{(\rho&space;^{2}+Z^{2})^{\frac{3}{2}}}})$

$therefore&space;\overrightarrow{dE}&space;=\frac{\rho&space;_{L}dZ}{4\pi&space;\epsilon&space;_{o}}({\frac{-Z\overrightarrow{a_{z}}+\rho&space;\overrightarrow{a_{\rho&space;}}}{(\rho&space;^{2}+Z^{2})^{\frac{3}{2}}}})$

then the Electric field strength $\overrightarrow{E}$ produced by the infinite line charge distribution $\rho&space;_{L}$ is

$\overrightarrow{E}&space;=&space;\int&space;\overrightarrow{dE}$

$\overrightarrow{E}&space;=&space;\int_{z=-\infty&space;}^{\infty&space;}\frac{\rho&space;_{L}dZ}{4\pi&space;\epsilon&space;_{o}}({\frac{-Z\overrightarrow{a_{z}}+\rho&space;\overrightarrow{a_{\rho&space;}}}{(\rho&space;^{2}+Z^{2})^{\frac{3}{2}}}})$

to solve this integral  let $Z=&space;\rho&space;\tan&space;\theta&space;\Rightarrow&space;dZ=\rho&space;\sec&space;^{2}\theta&space;d\theta$

as $Z\rightarrow&space;-\infty&space;\Rightarrow&space;\theta&space;\rightarrow&space;\frac{-\pi&space;}{2}$

$Z\rightarrow&space;\infty&space;\Rightarrow&space;\theta&space;\rightarrow&space;\frac{\pi&space;}{2}$

$\therefore&space;\overrightarrow{E}&space;=&space;\int_{\theta&space;=&space;\frac{-\pi&space;}{2}}^{&space;\frac{\pi&space;}{2}}&space;\frac{\rho&space;_{L}}{4\pi\epsilon&space;_{o}}(\frac{-\rho&space;^{2}\\sec&space;^{2}\theta&space;\tan&space;\theta&space;d\theta&space;\overrightarrow{a_{z}}+\rho&space;^{2}\sec&space;^{2}\theta&space;d\theta&space;\overrightarrow{a_{\rho&space;}}}{(\rho&space;^{2}+\rho&space;^{2}\tan&space;^{2}\theta&space;)^{\frac{3}{2}}})$

$\overrightarrow{E}&space;=&space;\int_{\theta&space;=&space;\frac{-\pi&space;}{2}}^{&space;\frac{\pi&space;}{2}}&space;\frac{\rho&space;_{L}}{4\pi\epsilon&space;_{o}}(\frac{-\rho&space;^{2}\\sec&space;^{2}\theta&space;\tan&space;\theta&space;d\theta&space;\overrightarrow{a_{z}}+\rho&space;^{2}\sec&space;^{2}\theta&space;d\theta&space;\overrightarrow{a_{\rho&space;}}}{\rho&space;^{3}\sec&space;^{3}\theta&space;})$

$\overrightarrow{E}&space;=&space;\frac{\rho&space;_{L}}{4\pi\epsilon&space;_{o}\rho&space;}[\int_{\theta&space;=&space;\frac{-\pi&space;}{2}}^{&space;\frac{\pi&space;}{2}}&space;-\sin&space;\theta&space;\overrightarrow{a_{z}&space;}d\theta&space;+\int_{\theta&space;=&space;\frac{-\pi&space;}{2}}^{&space;\frac{\pi&space;}{2}}&space;\cos\theta&space;d\theta&space;\overrightarrow{a_{\rho&space;}}]$

$\overrightarrow{E}=&space;\frac{\rho&space;_{L}}{4\pi\epsilon&space;_{o}\rho&space;}[0+2\overrightarrow{a_{\rho&space;}}]$

$\therefore&space;\overrightarrow{E}=&space;\frac{\rho&space;_{L}}{2\pi\epsilon&space;_{o}\rho&space;}\overrightarrow{a_{\rho&space;}}&space;Newtons/Coulomb$.

$\overrightarrow{E}$ is a function of $\rho$$\rho$   only,tere is no $\overrightarrow{a_{z}}$ component and $\rho$ is the perpendicular distance from the point P to line charge distribution $\rho&space;_{L}$.

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## Compensators-Introduction

Compensators are corrective sub systems to compensate the deficiency in the performance of the plant or system, so given a plant and a set of specifications suitable compensators are to be designed so that the overall system will meet given specifications. Proper selection of performance specifications is the most important step in the design of compensators.

i.e, All the control systems are designed to achieve specific objectives that is the requirements are defined for the control system. A good control system has less error, good accuracy, good speed of response, good relative stability, good damping which will not cause unusual Overshoots etc.

For stationary performance of the system, gain is adjusted first but gain adjustment alone can not provide satisfactory results. When gain increases, Steady-state behavior of system improves but results into poor Transient response (or) even instability.

The desired behavior of a system is specified in terms of

• Transient response.
• Steady-state error($e_{ss}$).

$e_{ss}$ →is usually specified in terms of constants $k_{a},k_{v}$ and $k_{p}$ for $u(t),r(t)$ and $p(t)$ as inputs.

Transient response → it measures relative stability and speed of response which are specified in time or frequency domain.

In time domain the measure of relative stability is in terms of $\xi$ or $M_{p}$, while the speed response is measured in terms of rise time $t_{r}$ , settling time $t_{s}$ or natural frequency $\omega&space;_{n}$. Where as in frequency-domain the measure of relative stability is given by Resonant peak $M_{r}$ or Phase margin $\varphi&space;_{pm}$ and the speed response is measured by Resonant frequency $\omega&space;_{r}$ or band width $\omega&space;_{b}$.

Once a set of performance specifications have been selected, the next step is to chose the appropriate compensator. There exists Electrical,hydraulic,pneumatic and mechanical compensators and in this context we prefer Electrical compenators.

An external device which is used to alter the behavior of the system so as to achieve given specifications is called as compensator [ compensators are added to the original system].

Compensators can be added to the system in series or in parallel or in combination of both.

Series Compensation:-

The flow of signal in series scheme is from lower energy level towards higher energy level. This requires additional amplifiers to increase the gain and also provide necessary isolation. The number of components required in series scheme is more than in parallel scheme.

Parallel Compensation:-

In this compensation technique energy flow is from higher energy level to lower energy level.  As there is no need of any amplifiers additional components required are less.

Series-Parallel Compensation:-

This is a compensation technique which utilizes the advantages of both series and parallel compensation techniques.

## Table of Z-Transforms for some standard signals

 Signal Z-Transform Region of Convergence (ROC) $\delta&space;(n)$ $1$ entire Z-plane $u[n]$ $\frac{1}{1-z^{-1}}&space;or\frac{z}{z-1}$ $\left&space;|&space;z&space;\right&space;|>&space;1$ $u[-n-1]$ $\frac{-1}{1-z^{-1}}&space;or\frac{-z}{z-1}$ $\left&space;|&space;z&space;\right&space;|<&space;1$ $a^{n}u[n]$ $\frac{1}{1-az^{-1}}&space;or\frac{z}{z-a}$ $\left&space;|&space;z&space;\right&space;|>&space;a$ $-a^{n}u[-n-1]$ $\frac{1}{1-az^{-1}}&space;or\frac{z}{z-a}$ $\left&space;|&space;z&space;\right&space;|<&space;a$ $na^{n}u[n]$ $\frac{az^{-1}}{(1-az^{-1})^{2}}&space;(or&space;)\frac{az}{(z-a)^{2}}$ $\left&space;|&space;z&space;\right&space;|>&space;a$ $cos&space;(\omega&space;_{o}n)u[n]$ $\frac{1-cos(\omega&space;_{o})z^{-1}}{1-2cos(\omega&space;_{o})z^{-1}+z^{-2}}$ $\left&space;|&space;z&space;\right&space;|>&space;1$ $sin&space;(\omega&space;_{o}n)u[n]$ $\frac{sin(\omega&space;_{o})z^{-1}}{1-2cos(\omega&space;_{o})z^{-1}+z^{-2}}$ $\left&space;|&space;z&space;\right&space;|>&space;1$ $r^{n}cos&space;(\omega&space;_{o}n)u[n]$ $\frac{1-rz^{-1}cos(\omega&space;_{o})}{1-2rz^{-1}cos(\omega&space;_{o})+z^{-2}r^{2}}$ $\left&space;|&space;z&space;\right&space;|>&space;\left&space;|&space;r&space;\right&space;|$ $r^{n}sin&space;(\omega&space;_{o}n)u[n]$ $\frac{rz^{-1}sin(\omega&space;_{o})}{1-2rz^{-1}cos(\omega&space;_{o})+z^{-2}r^{2}}$ $\left&space;|&space;z&space;\right&space;|>&space;\left&space;|&space;r&space;\right&space;|$ $\frac{1}{n},n>&space;0$ $-\ln&space;[1-z^{-1}]$ $\left&space;|&space;z&space;\right&space;|>&space;1$ $a^{\left&space;|&space;n&space;\right&space;|}&space;\forall&space;n$ $\frac{(1-a^{2})}{(1-az)(1-az^{-1})}$ $\left&space;|&space;a&space;\right&space;|<&space;\left&space;|&space;z&space;\right&space;|<&space;\frac{1}{\left&space;|&space;a&space;\right&space;|}$ $-na^{n}u[-n-1]$ $\frac{az^{-1}}{(1-az^{-1})^{2}}&space;(or)\frac{az}{(z-a)^{2}}$ $\left&space;|&space;z&space;\right&space;|>&space;\left&space;|&space;a&space;\right&space;|$ $(n+1)a^{n}u[n]$ $\frac{1}{(1-az^{-1})^{2}}&space;(or)\frac{z^{2}}{(z-a)^{2}}$ $\left&space;|&space;z&space;\right&space;|>&space;\left&space;|&space;a&space;\right&space;|$
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## Choice Of Intermediate Frequency (or) IF Amplifier in a Radio Receiver

Choice of Intermediate Frequency of a receiving system is usually a compromise , since there are reasons why it is neither low nor high, nor in a certain range between the two.

The following are the major factors influencing the choice of the Intermediate Frequency in any particular system.

1. If the IF is too high poor selectivity and poor adjacent channel rejection results unless sharp cut-off filters(crystal/mechanical filters) are used in the IF stage.
2. A high value of Intermediate Frequency(IF) increases tracking difficulties.
3. If we chose IF as low frequency, image frequency rejection becomes poorer. i.e, if $\frac{f_{si}}{f_{s}}$ is more IFRR(image Frequency Rejection Ratio) has been improved, which requires a high Intermediate Frequency($f_{si}$). Similarly when $f_{s}$ is more IFRR becomes worst.
4. Average Intermediate Frequency(IF) can make the selectivity too sharp cutting of the side bands.This problem arises because the Q must be low when the IF is low, unless crystal or mechanical filters are used and hence gain per stage is low. Thus a designer is more likely to raise Q rather than increasing the number of IF amplifiers.
5. If IF is very low , the frequency stability of local oscillator must be made correspondingly high.
6. IF must not fall in the tuning range of the receiver or else instability occurs and hetero dyne whistles (noise) will be heard.

Frequencies used:-

1. Standard AM broadcast receivers tuned to (540 KHz-1650 KHz) or(6 MHz-18 MHz) and European long wave band (150 KHZ- 350 KHz) uses IF in the range (438 KHz- 465 KHz). 455 KHz is the most popular value used.
2. FM receivers using the standard (88 MHz -108 MHz) band have an IF which is almost always 10.7 MHz.
3. TV Receivers in the  VHF band (54 MHz-223 MHz),UHF band (470 MHz-940 MHz) uses IF between (26 MHz-46 MHz) and the popular values are 36 MHz and 46 MHz.
4. AM-SSB Receviers employed for short-wave reception in the short wave band / VHF band uses IF in the range (1.6 MHz to 2.3 MHz).

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## Working /Operation of NPN and PNP Transistor

NPN Transistor Working:-

For Normal operation of NPN Transistor Emitter junction JE is Forward Biased and Collector junction JC is Reverse Biased.

The applied Forward Biased at Emitter-Base junction injects a large number of electrons into the N-region and these electrons have enough energy to overcome the JE junction and enter into the very thin lightly doped Base region.

Since Base is very lightly doped very few electrons recombine with the holes in the P-type Base region and constitutes a small Base current IB in μA.

The electrons in the Emitter region are more when compared to electrons in the Collector. Only 5% (or) 1% of injected electrons combines with the holes in Base to produce Iand remaining 95% (or0 99% of electrons diffuse into Collector region due to extremely small thickness of Base.

Since Collector junction is Reverse-Biased a strong Electro-static field develops between Base and Collector. The field immediately collects the diffused electrons which enters Collector junction and are collected by the Collector(Positive electrode).

Thus injected electrons from Emitter reaches Collector constituting a current known as $I_{E}=I_{B}+I_{C}$ Thus Emitter current is sum of Base current and Collector current. $I_{B}$ is very small in the Base region.

Current directions are  always from negative to positive and Majority carriers are electrons in NPN Transistor.

NPN Transistor is preferred over PNP since the mobility of electron is more than that of hole that is electron moves faster than holes.

PNP Transistor Working:-

For Normal operation of PNP Transistor Emitter junction JE is forward Biased and Collector junction JC is reverse biased.

The applied FB at Emitter-Base junction injects a large number of holes in the P-type emitter region and these holes have enough energy to enter into very thin lightly doped Base region. Base is very lightly doped N-type region. Therefore very few holes combines with the Base region and constitutes a small Base current IB (in Micro Amperes).

The holes in the Emitter region are more when compared to holes in the collector region.Only 5% or 1% of injected holes from Emitter combines with the electrons in the Base to produce IB and remaining 95% (or) 99% of holes diffuse into Collector region  due to extremely small thickness of Base.

Since Collector junction is Reverse-Biased a strong Electro-static field develops between Base and Collector. The field immediately collects the diffused holes which enters Collector junction and are collected by the Collector(negative electrode).

Thus injected holes from Emitter reaches Collector constituting a current known as $I_{E}=I_{B}+I_{C}$$I_{B}$ is very small in the Base region.

Majority carriers are holes in PNP Transistor.

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## Working of PN-junction Diode under Forward and Reverse Bias Conditions

In order to consider the working of a diode,we shall consider the effect of forward and Reverse Bias across PN-junction.

Forward Bias:-

Forward Bias means the Positive terminal of the Battery has been  connected to P-type and negative terminal to N-type in a PN-junction diode that is when an external voltage is applied to PN-junction in such a way that it cancels the barrier potential and permits the current flow such a bias  is called as Forward-Bais.

Under No Bias voltage condition, Near the junction the holes moves towards the junction and electrons as well forms a region known as Depletion region, the region depleted with immobile ions .

when the applied voltage V establishes an electric field opposite to the potential barrier , as a result the width of the potential barrier is reduced as it is very small

0.3 Volts in Ge diode and

0.7 Volts in Si diode.

∴ a small voltage (V) is sufficient to completely eliminate the barrier that is the barrier is completely eliminated and the resistance at the junction becomes zero and the current flow across the diode can be explained as follows.

Now holes move towards junction simultaneously electrons since holes and electrons were repelled by the opposite terminals of the Battery, As the Battery voltage is sufficiently greater than barrier voltage electrons and holes gets sufficient energy to cross the barrier easily.

The continuous current in external circuit is due to electrons, the current in N-type material is due to movement of free electrons, when these electrons reaches the junction they combine with the holes at the junction and releases a new electron.Similarly, in the P-type region current is due to holes.

i.e, when an electron-hole combination takes place near the junction ,   A co-valent bond near positive terminal of the battery breaks down and it liberates an electron which moves towards positive terminal of the Battery as electron movement is  towards positive terminal of the Battery this can be treated as hole movement in opposite direction.

therefore the constant movement of electrons and holes towards opposite terminals creates a high forward current in the external circuit.

PN-juction Diode in Reverse-Bias:-

When an External voltage V is applied to a PN-junction in such a way(direction) that it increases the Potential barrier is called as Reverse Bias that is Positive terminal of the Battery connected to N-type and negative terminal to P-type.

The applied voltage V acts in the Same direction to that of Potential Barrier.

that is when the PN-junction is Reverse Biased

• The junction Potential Barrier width increases.
• The junction offers higher resistance.
• electrons and holes move away from the junction and a very small current flows through the junction because of  minority carriers known as Reverse saturation current.

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## Comparision table HWR ,FWR and Bridge Rectifier

 Parameter Half-Wave Rectifier Full-wave Rectifier Bridge Rectifier No of diodes 1 2 4 Maximum Efficiency ( η ) 40.6% 81.2% 81.2% $V_{dc}$ $\frac{V_{m}}{\pi&space;}$ $\frac{2V_{m}}{\pi&space;}$ $\frac{2V_{m}}{\pi&space;}$ $I_{dc}$ $\frac{I_{m}}{\pi&space;}$ $\frac{2I_{m}}{\pi&space;}$ $\frac{2I_{m}}{\pi&space;}$ Output RMS voltage $\frac{V_{m}}{2}$ $\frac{V_{m}}{\sqrt{2}}$ $\frac{V_{m}}{\sqrt{2}}$ Average current Idc $I_{dc}$ $\frac{I_{dc}}{{2}}$ $\frac{I_{dc}}{{2}}$ Ripple Factor ($\Gamma$) 1.21 0.48 0.48 Peak Inverse Voltage (PIV) $V_{m}$ $2V_{m}$ $V_{m}$ Output Frequency f 2f 2f TUF(Transformer Utilization Factor) 0.287 0.693 0.812 Form Factor 1.57 1.11 1.11 Peak factor 2 $\sqrt{2}$ $\sqrt{2}$
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## Generation of PWM and PPM using Wave forms

PWM Generator:-

The circuit that generates PWM wave is as follows, Here in this circuit Op-Amp works in comparator mode.It compares two voltages, modulating voltage with Saw-tooth Voltage. Saw-tooth voltage is taken as reference voltage.

 condition Output voltage Vo(t) $m(t)>&space;V_{r}(t)$ Low $V_{r}(t)>m(t)$ High

from the graphs whenever modulating voltage dominates saw-tooth voltage corresponding output is low.

Similarly, when saw-tooth voltage dominates modulating voltage corresponding output is High.

Then the resultant output voltage is a PWM signal.

PPM Generator:-

Now, a PPM signal has been generated by passing the PWM signal through a Mono-stable Multi vibrator . Here the resultant signal is a PPM signal with the pulse starting with respect to trailing edge of PWM signal.

The width and Amplitude of Pulse remains constant only the position of the pulse changes with respect to m(t) .

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## Band Pass Sampling

However when the given signal is a Band Pass signal then a different criterion must be used to sample the signal , the Band Pass signal x(t) whose maximum BW is ‘$2f_{m}/W$‘ Hz can be completely represented and recovered from it’s samples if it is  sampled at the minimum rate of greater than or  equals to twice that of the BW.

then sampling rate $f_{s}\geq&space;2&space;X&space;BW$

i.e, $f_{s}\geq&space;4f_{m}&space;or&space;f_{s}\geq&space;2W$

Any band pass signal in time-domain can be represented in it’s in-phase $x_{I}(t)$ and quadrature phase $x_{Q}(t)$ components as

$x(t)&space;=&space;x_{I}(t)cos&space;2\pi&space;f_{c}t&space;\pm&space;x_{Q}(t)sin&space;2\pi&space;f_{c}t$

after sampling the band pass signal, the signal after reconstruction is

$x(t)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}sinc(2f_{m}t-\frac{n}{2})cos(2\pi&space;f_{c}(t-\frac{n}{4f_{m}}))$

$T_{s}&space;=&space;\frac{1}{4f_{m}}$, where BW of band pass signal is $2f_{m}$  Hz

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