# Electrical and electronics basics

Power Engineering | Telecommunication| Electronics engineering | Fields and waves | Control System Engineering| Differential Equations

Electrical and Electronics engineering is a branch of engineering science. It involves two subject areas related to circuits which control high voltage and circuits which control low voltage. The high voltage study is basically
electrical engineering and the low voltage study is electronic engineering. The above definition may be unconventional but to my view electrical and electronics engineering is all about circuits. Nicola Tesla
greatly contributed to ac machinery development. He is one of the greatest inventors and scientists of all the time. Then the inventor of radio , Guglielmo Marconi must be considered
a pioneer in the development of wireless communication. Electrical engineering heavily relies on the application of mathematics like rest of the engineering disciplines. Subjects of it are now diverse and unique. Energy
is the most important issue now a days and electrical engineers are trying their best to make the power generation and distribution most efficient. Electricity has conquered darkness when Tomas Alva Edison had
invented the first electrically powered lamp. Then other electrical machinery began to be developed. They replaced old fashioned engines in many power systems. Before going into more details, we need to know that
electricity can be of two forms : one is static electricity and the other is moving or dynamic electricity. There is an intermediate form which may be called quasi-static electricity. Many dynamic electrical phenomena can be modeled with
quasi-static electricity postulates. In quasi-static case the time varying part is negligible compared to spatially varying part.

Electrical systems have been an integral part of all the applications of modern engineering science. Just because without electricity no rocket would fly, no industry would run, not even internet. Electrical engineering not only involves developing new machinery but also controlling them. Control system engineering is a sub field of electrical engineering. It gives stability to electrical system like running generator, electrical motor driven machines and many other electrical things. Robots are also controlled machines. Control system designing and analysis must be explained in details to understand how it works. Technological innovation are being made by the help of fruitful research in electrical and electronics engineering. More and more people are pursuing their career in EEE engineering. But we must not forget all the mathematical theorems and physical laws that have brought light to this prestigious branch of engineering. One of the most important theorems is Fourier analysis. Consequently Fourier analysis has become an indispensable mathematical tool in telecommunication system modelling.

# Electronic circuit theory

Electronic circuit theory includes many principles like Ohm's law, Thevenin theorem, Norton theorem,
superposition principle, node analysis, KVL(kirchoff's voltage law), KCL(Kirchoff's current law) and many others.
The most important concepts of circuit theory are the concept of series and parallel circuit. When a number of components are connected in parallel then different current
flows through them. When the connection is series the same current flows through all of them.

The next concept is the voltage divider rule.

The resistances connected in Delta can be converted to equivalent resistances in Star (Y) as follows

A linear circuit is a type of circuit in which a given sinusoidal input produces a sinusoidal output with same frequency. Otherwise the circuit is non linear. Most of these principles only apply to linear circuits. Thevenin's
theorem is frequently used network analysis. It states that any two terminal branch of a linear circuit with multiple voltage sources and resistors can be replaced with an equivalent source and a resistor connected
in series. The voltage of the source will be just the open circuit voltage of the branch and the the value of resistor is the total equivalent resistance looking back at the branch with all the internal voltage sources being
short circuited. The current through the equivalence branch can be found by dividing the voltage of the source(Vth) by resistance (Rth).

The equivalent voltage and resistance r are connected in series as demanded by the Thevenin's theorem. Norton theorem is a complementary form of Thevenin theorem. Norton equivalent circuit is found by replacing the branch with a current source and an equivalent resistor connected in parallel with each other. In this case the resistance is the same as the Thevenin equivalent resistance and current i is found by dividing the open circuit voltage by the resistor as stated before. KVL as known as Kirchoff's voltage law states that the algebraic sum of all the voltage drops around a closed circuit is equal to the sum of all electromotive forces in the loop.

It is another form of energy conservation law which says energy can neither be created nor destroyed. KCL is the a law that says at any particular node of a circuit sum of all the currents entering is equal to the sum of all the current leaving that node.

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Superposition principle is much like linearity property of any system. It can be explained like: the net response of multiple inputs of stimuli of a linear circuit is the sum of the responses that the circuit would have if each input or stimuli were present independently. Another theorem is called the reciprocity theorem. It states that if we swap or alter the position of voltage source and ammeter in a circuit , the ammeter reading will remain the same . In mesh analysis , a complex circuit is subdivided into simpler sub circuits and KCL is applied in each branch to simplify the calculation. In these way current through each loop can be found by solving simultaneous equations of several variables(current i). So linear algebra is needed. Linear algebra is used to study matrix, vector space and solve linear equations. System of simultaneous linear equations can be solved easily using matrix which is a collection of numbers arranged in rows and columns.

# Applications

Electrical and electronics directly relates to computer science, specifically the development of the hardware of Computers. Computers are a modern marvel.
The description of computers requires a vast amount of time and writing, which is best left for discussion about computer science . From microprocessor to RAM(random access memory) and all internal chips are some great developments
of modern nanotechnology. VLSI has inaugurated the development of chips and integrated circuits. Moore's law tells that every two years the number of electronic chips on a single circuit gets doubled. We would not to able to hold any electronic device,
such as smart phone, PDA on our palm without these latest technologies developed by electrical engineers and scientists. All these electronic devices rely heavily on the application of digital signal processing(DSP). DSP
involves the study of discrete signal as opposed to analog system which work with analog signal. Discrete mathematics is used in analyzing such discrete systems. Modern electrical and electronics engineering
are usually subdivided into two main fields in almost all universities all around the world. One is telecommunication and other is power engineering.
Then these two fields has other sub-branch like signals and systems, digital logic design, power station, high voltage study, switch gear and protection, microwave and antenna theory and many other. Some of the Mathematical
courses that are necessary for electrical engineers are complex number analysis, differential equations and some concepts of linear algebra. Filter designing is another aspect of electronics which, involves the act
of removing certain portion of frequency of the signal or make a system susceptible to certain range of frequencies. Filters are highly used in telecommunication system and amplifier or in ac to dc converters. There are basically
four types of filter: first one is low pass filter, the second is high pass filter, third is band pass filter and the fourth is band stop filter.

Three phase electrical system was introduced by Nicola Tesla in the late 80s. Three phase power utilizes three out-of-phase electrical voltage or current. Three phase power has advantages over conventional single phase system. There is
a mathematical relationship between three phase line to line voltage to its phase voltage. To be precise, it is root three times the phase voltage. Phase is the initial angle of a sinusoidal wave at the origin of time t=0. So it is some
kind of angle when we compare two wave forms at a certain time. Phasor algebra plays a pivotal part in ac circuit analysis. In an ac or dc circuit when a voltage is applied , current flows through each branch of the circuit.
In some branch(inductive ) the voltage lead the current and in some (capacitive) current lead the voltage by some angle. The line representing the voltage or current in the argand plane is called phasor (vector).
Circuit impedance is normally complex quantity if the circuit contains capacitive or inductive reactance. So the impedance is also a phasor quantity which can be represented by a line in argand plane. This is a phasor
because it is dependent upon some frequency of current or voltage signal. In inductive branch the inductor stores energy in the form of magnetic field. When current tries to flow through the inductor it initially opposes
the flow because of lenz law. When current across inductor changes rapidly the magnetic flux created in the inductor coil begins to vary rapidly and this
produces back emf in the in the inductor. This back emf cause current to lag behind applied voltage. When there is no change of flux in the inductor coil , the current becomes maximum(minimum back emf). This situation is simply
the statement contained in the voltage- current relationship of inductor.

** V = L ∂i / ∂t **

On the other hand, the reverse situation happens in capacitor branch. When a voltage is applied across the capacitor the charges builds up across the capacitor plates. Energy is stored in the capacitor in the
form of electric field. This electric field begins to oppose the flow of current. When a dc voltage is applied the capacitor acts as an open circuit. In case of ac voltage the capacitor current leads the voltage by an
angle 90(degree) when plotted in a phasor form. Similar equation can be derived for current and voltage across the capacitor.

** I = C ∂v / ∂t **

C is the constant called capacitance. It's value is usually denoted by units of f (faraday).

# Resonance

Resonance is a physical phenomena which occurs when a system drives another oscillatory system with maximum amplitude. It is the condition when a system's energy transfer rises to its peak value. There are various
kinds of resonance that occurs in nature. Some of those are magnetic resonance, mechanical resonance and others. Resonance can be very destructive sometimes. A pendulum swinging at its natural frequency
is an example of resonance. When we talk or sing we create resonance in the air inside the vocal tract. the Electrical resonance occurs in electrical circuits when there are inductors and capacitors in the
circuits. The frequency at which the resonance occurs is called resonance frequency. I recommend to watch the video below.

An example of electrical resonance is the time when we tune a channel in electronic radio. When we tune a specific channel the channel's frequency act as a resonance frequency of radio. The resonance in electrical circuit can be
created by both series and parallel connection of inductor and capcitor. In the video above an example of series resonance is explained. The resonance frequency is completely determined by the value of the inductance and capacitance. The graph of average power versus the frequency w can help visualize the resonance phenomena which occurs at the frequency W(nought)

### Transducer

Transducer is an electrical material which shows piezoelectricity. Transducers produce electricity in response to physical quantities like pressure or brightness.

# Phasors

The analysis of three phase or poly-phase system requires some understanding of phasor algebra. Phasor algebra is the algebra of complex numbers on argand plane. Argand plane is a graph where position of a point is fixed by two parameters named theta(θ) and distance from origin(r). The horizontal and vertical axis -es are the real and imaginary axis respectively. We can analyze sine or cosine function as a vector in complex plane. Because a signal has both an amplitude and some phase angle, we can construct an exponential function as a multiplication of a variable and an exponential (e(x)) term, from which we can find the magnitude of any such function and it direction (theta in the complex plane) using Euler's famous identity. The use of complex number on argand plane greatly simplifies calculations of mathematical equations in electrical engineering. Every sine or cosine function is a periodic phenomena.They gradually increase and decrease to same value alternatively. So they are kind of wave forms too. Two sine or cosine waves can vary according to their amplitude and frequency that they have. Phasor gives the total impedance of a R-L-C circuit when we divide voltage phasor by current phasor. Power factor is the cosine of angle between current and voltage of the circuit. It points out important relationship between real and imaginary power. Apparent power is the vector sum of real and apparent power. In fact it is the hypotenuse of a right triangle which base and perpendicular are real and imaginary power. If the apparent power is P and current is I , the real power is V (voltage)X I (current){COS(Theta)} and imaginary power is V(voltage) X I (current) (SIN(theta)).

**Fig: Power triangle**

The imaginary power is not used by the system to do useful work. That is why it is called imaginary power. The real power is what power system utilizes to useful work. The real power is volt-amp whereas the imaginary power is expressed as var or kvar. Inductors and capacitors store energy from the system and they return it to the system periodically. Hence the name imaginary power is coined for the electrical system. It is the power that is not consumed by the load and the electric power company will not charge anything for that.

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# Voltage and Current source

Voltage source is an element of electronic circuit, which provides energy to the other circuit's passive elements. An ideal voltage source maintain an uniform voltage as long as it transfers power to other components in circuit. That means it has almost zero internal resistance. But in practice such voltage source is not possible. So every voltage source has some internal resistance. For example a battery has a very small internal resistance. Its voltage gradually decrease as the load increase. We can use Ohm's law to calculate actual voltage supplied by the voltage source.

**Fig: voltage and current source**

Similarly current source provides or inject current to other elements in electronic circuit. An ideal current source has infinite resistance. So it can maintain constant current to the circuit. But practical current source has some internal resistance also, which is usually high to maintain a current close to actual value of the current source. As the load resistance increase the current value decreases. We can calculate current through the load by using Ohm's law.

# Hysteresis

Hysteresis is a phenomena which occurs in many materials with magnetic properties. It is a curve that shows relationship between magnetic field strength (H) and magnetic field(B) .**Fig: Hysteresis curve**

It is actually a loop that has a strange characteristic. It is the loop that defines a history of B and H. Once a complete loop is formed by traversing the the H axis both ways , it can retain some magnetic field(B) in absence of the effect causing it. Thus it shows information storing behavior. It has a kind of ability of remembering the past. The graph shows the field (B) can have either positive or negative values depending on which way the loop is traversed. These values are called residual magnetism. Hysteresis phenomena has a wide ranging applications. It is this concept that is applied in hard drive and other kinds of memory devices.

### Earth Resistance and conductivity

Earth is a electrical conductor. It is not very good conductor but is still a good replace for wires. The resistance of earth is dependable on the contact surface and distance from the contact surface. Theoretically the resistance of the whole earth is zero as the surface area is enormous. The formula for finding resistance of any material is :

This is the formula to find dc resistance of any material of length l and surface area A. Resistance is of two kinds : one is dc and other is ac resistance. Ac resistance arise due to ac currents and is somewhat complicated concept.

The formula for finding earth restance is

So what is the application of earth's resistance in electrical engineering. We have to take into various factors including earth's resistance when gounding electrical appliances and machineries. This brings us to the concept "grounding". Gounding means any physical connecting to earth. As the earth is an infinite sink of elctrical charge , its potential does not change when electricty goes into it. Its potential can be taken as zero as a reference point.

Earth's resistance plays a crucial role in cable fault testing. Now a days underground power cables are used. Cables are buried underneath the earth and are used to transmit power from one place to another place. But insulation of cables is sometimes damaged and power leakage occurs. The electricity flows from the fault location towards all direction inside the earth. This poses a severe leakage current. So fault needs to be located and repai red. There creates an electric field which help the instrument locate the underground cable fault from above the ground.

There is no doubt that from the fault location electricity flows in all directions. Earth acts like a conductor in this case. As the distance from the fault location increase the resistance increase and intensity of electric current decreases.

## Magnetostriction

Magnetostriction is a physical phenomena that occurs in electrical transformers. More specifically it is a property of ferromagnetic matter , which causes them expands and contracts in response to an external magnetic field B. This effect shows this kind of magnetostrictive material can converts electromagnetic energy into mechanical energy. The effect of magnetostriction is created by the devopment of strain. We know that strain is a kind of force that try to deform an object. Mechanical strain creates the contraction of the core of a transformer when the magnetic field is applied. Under operation of a transformer a humming sound can be heard from the transformer. This humming sound is the result of magnetostriction effect. Transformer core vibrates a frequency that is double of the system frequency.So magnetostriction is actually harmful for transformer's operation.

## Insulation Resistance of a cylindrical wire

Insulation of a conductor is very important when designing a conductor. If the conductor is cylindrical then the method of finding insulation is easy.A cross section of a typical insulated conductor is

Let d = Diameter of conductor or core

r = d / 2

D = Diameter with sheath

R = D / 2

As the leakage current flows radially outwards, the length along which the current flows in an elementary ring is dx. While the cross sectional area perpendicular to the flow of current depends on the length l of the cable.

Cross section area = Surface area for length l of cable = (2 π x) × l

Therefore the resistance of this elementary cylindrical shell is,

dRi = ρ dx / (2 π x l)

Where ρ = Resistivity of the insulating material

The total or net insulation resistance of the cable can be obtained by integrating the resistance of an elementary ring from inner radius upto the outer radius i.e. r to R.

he value of is always high. The expression proves that the insulation resistance is inversely proportional to its length. So as the cable length increases, the insulation resistance decreases.

## Engineering mathematics

Inverse Laplace Transform using Partial Fraction Method is :If degree of numerator polynomial N(s) is higher than the degree of D(s) , than we should divide \displaystyle N(s) to obtain quoitent and remainder term.

If there are simple and distinct Real Roots

If there are multiple Real Roots

The inverse laplace transform theorem is the opposite operation of laplace transform

A table of inverse laplace transforms' method is given for convenience

If there are complex conjugate roots

## Equation of state and its solution

The output response is dependent or contingent opon the state variables and their initial values. Hence it is necessary to obtain the state vector x(t) which satisfies the state equation at any time t . This is called the solution of state equations, which helps to obtain the output response of a system.Consider the state equation of linear time invariant system as

Premultiplying both sides of the first equation by e^{-At}

Assuming initial time t = t_0 and integrating both sides from t = 0 -> t_0

For the Causal LTI system t_0 = 0

The solution is broken into two parts. The first part is zero input response (ZIR). ZIR depends only on the nature of the system.

x_{ZIR} = e^{At} x(0)

The other part is the part due to input. It is called forced response or zero state response (ZSR). ZSR is contingent on the nature of the system as well as on the input.

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## RLC circuit

R_L-C circuits play a vital role in electrical and electronics. A typical series R-R-C circuit isIn a R-L-C circuit

## Phasor Diagram of RLC Series Circuit

The phasor diagram of the RLC Series Circuit when the circuit is acting as an inductive circuit that means (VL>VC) is depicted below and if (VL< VC) the circuit will behave as a capacitive circuit.Where voltage through capacitance and inductance will lead or lag the current I repsectively. The voltage across resistance will be in phase with the current I. The parameters are (using vector diagram)

Phase angle of the circuit is

Power in RLC Series Circuit The product of voltage and current is defined as power.

p = VICOSφ = I^R

Where cosϕ is the power factor of the circuit and is expressed as Cos φ = V_R/V = R/Z where z is the impedence of the circuit .

Impedance Triangle of RLC Series Circuit

When the quantities of the phasor diagram are broken down by the common factor I then the right angle triangle is attained known as impedance triangle. The impedance triangle of the RL series circuit, when (XL > XC) is shown below

Impedance triangle is shown below when the circuit acts as a RC series circuit (XL< XC)

## two Wattmeter Method of Power Measurement

Two Wattmeter Method can be deployed to measure the power in a 3 phase, three wire star or delta connected the balanced or unbalanced load. In Two wattmeter method the current coils of the wattmeter are attached with any two lines, say R and Y and the potential coil of each wattmeter is joined on the same line, the third line i.e. B as shown below in figure (A).## Superconductor

Superconductivity is the property of soome metallic substance. When some specific materials are cooled down below some low temparature its resistance almost decreases to zero. At that specific temparature the metal becomes a superconductor. When a magnet is placed above the superconductor magnetic levitation occurs.To understand electronics and electrical engineering , quantum mechanics must be understood to some extent. All the stuffs related to electricity can be explained using the principles of quantum mechanics. Let us review some of the basics of weird quantum mechanical stuffs in brief.

Impact parameter is used in Rutherford scattering. It is the minimum perpendicular distance from the center of the target , for which the projectile would be undeflected. That is to say, it would make an impact with the target. It is related to the scattering angle by the equation.

r(min) is the closest distance from the center. Where U(r) is the potential energy of the target , and b is the impact parameter. It can be easily shown that θ = 0 when b <= R and in that case b = R cos(θ/2). V(∞) is the velocity of the projectile far from the center. The condition for potential energy for a hard sphere U(r) = 0 when r > R and U(r) = ∞ when r < R .

We can calculate the motion of electron using differential equation. To do that we have to know a little bit of quantum mechanics.

De-broglie said that the motion of an electron is equivalent to the motion of an wave-packet with group velocity v(g).

Let external electric field ε is applied to influence the electron. Then the gain of energy in time dt is

dE = -eεvdt;

Putting the value of velovity v we get

The formula for effective mass of electron is

More detailed analysis can be made.

## More theories about electron

Most of the theories of electron are related to Schrodinger's equation. Some are the theories of the Bohr's atomic model . The total energy of electron isA short glance at the quantum mechanical theories can be made

## Equation of our universe

First complete understanding of the universe was possible because of Einstein's invention of his general theory of relativity. Various scientists solved his field equations and created different models of our universe. Friedman solved his equation and showed that our universe could have a history. Here is the mathematical model of Friedman's universe.You can provide your comment and response below: