Electrostatic field β’Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charges. If two objects have different electrical charges, then an electrostatic field exists between the two objects. An electrostatic field also forms around a single body/object which is electrically charged concerning its surroundings. β’A body is negatively charged (-) when it has an excess of electrons related to its surroundings. A body is positively charged (+) when it has insufficient electrons concerning its surroundings. An electrostatic field is formed by a static charge. β’The mathematical definition of the electrostatic field is derived from Coulomb’s law which defines the vector force between two point charges. Coulomb’s Law Given point charges [q1 , q2 (units=C)] in air located by vectors R1 and R2 , respectively, the vector force acting on charge q2 due to q1 [F 12 (units=N)] is defined by Coulomb’s law as πΉ12 = π1 π2 4ππ0 π 2 −π 1 2 πΰ· 12 - - - - - - (1) where π ΰ· 12 is a unit vector pointing from q1 to q2 and π0 is the free-space permittivity [π0 = 8.854×10-12 F/m]. The permittivity of air is approximately equal to that of free space (vacuum). Note that, according to Coulomb’s law, the force between the point charges is directly proportional to the product of the charges and inversely proportional to the square of the separation distance between the charges. The unit vector pointing from q 1 to q2 can be written as πΰ· 12 = π 2 −π 1 π 2 −π 1 - - - - - - (2) Substituting equation2 in equation 1 π π π −π πΉ12 = 1 2 2 13 - - - - - - (3) 4ππ0 π 2 −π 1 Fig: Coulomb’s Law Electric field An electric field is the physical field that surrounds electrically charged particles and exerts a force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and timevarying electric currents. According to Coulomb’s law, the vector force between two point charges is directly proportional to the product of the two charges. Alternatively, we may view each point charge as producing a force field around it (electric field) that acts on any charge in its vicinity. Since a point charge will repel a charge of like sign, but attract a charge of an unlike sign, we must adopt a convention as to the sign on the electric field force. Electric field, an electric property associated with each point in space when the charge is present in any form. The magnitude and direction of the electric field are expressed by the value of E, called electric field strength or electric field intensity or simply the electric field. Fig: Representation of an Electric Field The space around an electric charge in which its influence can be felt is known as the electric field. The electric field intensity at a point is the force experienced by a unit of positive charge placed at that point. •Electric Field Intensity is a vector quantity. •It is denoted by ‘E’. •Formula: Electric Field = F/q. •Unit of E is NC-1 or Vm-1. The electric field intensity due to a positive charge is always directed away from the charge and the intensity due to a negative charge is always directed toward the charge. Due to a point charge q, the intensity of the electric field at a point d units away from it is given by the expression: π πΈππππ‘πππ πΉππππ πΌππ‘πππ ππ‘π¦ πΈ = 4πππ2 NC-1 The intensity of the electric field at any point due to a number of charges is equal to the vector sum of the intensities produced by the separate charges. The strength of an electric field E at any point may be defined as the electric, or Coulomb, force F exerted per unit positive electric charge q at that point, or simply E = F/q. Permittivity in dielectrics: The dielectric constant of a material, also called the permittivity of a material, represents the ability of a material to concentrate electrostatic lines of flux. In more practical terms, it represents the ability of a material to store electrical energy in the presence of an electric field. Dielectric permittivity (ε) is defined as the ratio between the electric field (πΈ) within a material and the corresponding electric displacement (π·): π· = επΈ When exposed to an electric field, bounded electrical charges of opposing sign will try to separate from one another. For example, the electron clouds of atoms will shift in position relative to their nuclei. The extent of the separation of the electrical charges within a material is represented by the electric polarization (π). The electric field, electric displacement and electric polarization are related by the following expression π· = ε0 πΈ+ π where the permittivity of free-space (ε0=8.8541878176×10−12 F/m) defines the relationship between π· and πΈ if the material is non-polarizable. Therefore, the dielectric permittivity and the electric displacement define how strongly a material becomes electrically polarized under the influence of an electric field. Relative Permittivity The dielectric properties of materials are generally expressed using the relative permittivity (εr). The relative permittivity defines the dielectric properties of a material relative to that of free-space: επ = π ε0 The relative permittivity is both positive and ≥1. Capacitor: A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. Most capacitors contain at least two electrical conductors often in the form of metallic plates or surfaces separated by a dielectric medium. A capacitor is created out of two metal plates and an insulating material called a dielectric. The metal plates are placed very close to each other, in parallel, but the dielectric sits between them to make sure they don't touch. The dielectric can be made out of all sorts of insulating materials: paper, glass, rubber, ceramic, plastic, or anything that will impede the flow of current. The plates are made of a conductive material: aluminum, tantalum, silver, or other metals. They're each connected to a terminal wire, which is what eventually connects to the rest of the circuit. The capacitance of a capacitor -- how many farads it has -- depends on how it's constructed. More capacitance requires a larger capacitor. Plates with more overlapping surface area provide more capacitance, while more distance between the plates means less capacitance. The total capacitance of a capacitor can be calculated with the equation: π΄ πΆ = ππ , where εr is the dielectric's relative permittivity (a constant value determined by the 4ππ dielectric material), A is the amount of area the plates overlap each other, and d is the distance between the plates. An ideal capacitor is characterized by a constant capacitance C, in farads in the SI system of units, defined as the ratio of the positive or negative charge Q on each conductor to the voltage V between π them. πΆ = π A capacitance of one farad (F) means that one coulomb of charge on each conductor causes a voltage of one volt across the device. In practical devices, charge build-up sometimes affects the capacitor mechanically, causing its capacitance to vary. In this case, capacitance is defined in terms of incremental changes: πΆ= ππ ππ Capacitor Composite: When a parallel plate capacitor has two dielectrics or more between the plates, it is said to be composite capacitor. Different types of composite capacitors are existing. Type 1: In this type, number of dielectric having different thicknesses and relative permittivities are filled in between the two parallel plates. The composite capacitor with three different dielectrics with permitivities εr1, εr2 and εr3 and thicknesses t1, t2 and t3 is shown in Fig. The equivalent capacitance across the terminal A-B is πΆππ = π0 π΄ π‘1 π‘2 π‘3 + + ππ1 ππ2 ππ3 In general, for a composite with 'n' dielectrics, πΆππ = π0 π΄ π‘1 π‘ + 2+ ππ1 ππ2 π‘ ……. +π π ππ The voltage across terminal A-B is π = πΈ1 π‘1 + πΈ2 π‘2 + πΈ3 π‘3 Fig: Type-1 Composite capacitor Type 2: In this type, in the same thickness, 't', the two dielectrics are arranged as shown in Fig. Let the relative permittivity values for the two dielectrics be εr1 and εr2. The Thickness for both is the same but the areas are different. It can be seen from the equivalent circuit that there exists two capacitors in parallel due to two different dielectrics. Equivalent capacitance is πΆππ = πΆ1 + πΆ2 = π0 ππ1 π΄1 π‘1 + π0 ππ2 π΄2 π‘2 = π0 π‘ ππ1 π΄1 + ππ2 π΄2 For 'n' dielectrics arranged in same thickness ‘t’, πΆππ = π0 π‘ ππ1 π΄1 + ππ2 π΄2 + … … … + πππ π΄π Fig: Type-2 Composite capacitor Type 3 : In practice we can have the capacitor which is a combination of above two types. One such capacitor is shown in Fig. Basically it is a type 2 capacitor, consisting of a type 1 capacitor. So there are two capacitors in parallel. The C1 is having thickness t1, relative permittivity εr1 and area A1. πΆ1 = π0 ππ1 π΄1 π‘1 Now the capacitor C2 is again a composite capacitor of Type 1 which itself is made up of two capacitors in series. From the result of Type 1 we can write: Fig: Type-3 Composite capacitor πΆ2 = π0 π΄2 π‘1 −π‘2 ππ2 π‘ +π 2 π3 Hence the total capacitance is the parallel combination of C1 and C2, πΆππ = πΆ1 + πΆ2 = π0 ππ1 π΄1 π‘1 + π0 π΄2 π‘1 −π‘2 ππ2 π‘ +π 2 π3 Dielectric capacitors: An electrically insulating material that becomes polarized in an electric field is called a dielectric. All electrically insulating materials are dielectrics, but some are better dielectrics than others. A good dielectric is one whose molecules allow their electrons to shift strongly in an electric field. In other words, an electric field pulls electrons a fair bit away from their atom, but they do not escape completely from their atom. Placing a dielectric in a capacitor before charging it therefore allows more charge and potential energy to be stored in the capacitor. A parallel plate with a dielectric has a capacitance of π΄ π πΆ = ππ0 = ππΆ0 Where k is a dimensionless constant called the dielectric constant. Because k is >1 for dielectrics, the capacitance increases when a dielectric is placed between the capacitor plates. The dielectric constant of several materials is shown in table. Capacitors in series and parallel When capacitors are connected one after another, they are said to be in series. For capacitors in series, the total capacitance can be found by adding the reciprocals of the individual capacitances, and taking the reciprocal of the sum. Therefore, the total capacitance will be lower than the capacitance of any single capacitor in the circuit. 1 πΆπππ‘ππ = 1 πΆ1 + 1 πΆ2 + 1 πΆ3 + ……+ 1 πΆπ−1 + 1 πΆπ If only two capacitors in series, you can use the "product-over-sum" method to calculate the total capacitance: πΆπππ‘ππ = πΆ1 πΆ2 πΆ1 +πΆ2 If two equal-valued capacitors are in series, then total capacitance is half of their value. Capacitors in Parallel Capacitors in parallel refer to the capacitors that are connected together in parallel when the connection of both of its terminals takes place to each terminal of another capacitor. Furthermore, the voltage’s ( Vc ) connected across all the capacitors, whose connection is in parallel, is the same. Then, capacitors in parallel across them have a “common voltage” supply. When capacitors are placed in parallel with one another the total capacitance is simply the sum of all capacitances. This is analogous to the way resistors add when in series. πΆπππ‘ππ = πΆ1 + πΆ1 + … … + πΆπ−1 + πΆπ Charging and discharging of capacitors When positive and negative charges coalesce on the capacitor plates, the capacitor becomes charged. A capacitor can retain its electric field -- hold its charge -- because the positive and negative charges on each of the plates attract each other but never reach each other. At some point the capacitor plates will be so full of charges that they just can't accept anymore. There are enough negative charges on one plate that they can repel any others that try to join. This is where the capacitance (farads) of a capacitor comes into play, which tells you the maximum amount of charge the cap can store. If a path in the circuit is created, which allows the charges to find another path to each other, they'll leave the capacitor, and it will discharge. Charging of a Capacitor Consider a circuit having a capacitance C and a resistance R which are joined in series with a battery of emf ε through a Morse key K as shown in the figure. When the key is pressed, the capacitor begins to store charge. If at any time during charging, I is the current through the circuit and Q is the charge on the capacitor, then Potential difference across resistor = IR, and Potential difference between the plates of the capacitor = Q/C Since the sum of both these potentials is equal to ε, RI + Q/C = ε … (1) As the current stops flowing when the capacitor is fully charged, When Q = Q0 (the maximum value of the charge on the capacitor), I = 0 From equation. (1), Q0 / C = ε … (2) From equations. (1) and (2), π πΌ + π πΆ ππ π0 −π = π0 πΆ ππ‘ πΆπ = or π0 −π πΆ πΆπ = ππ ππ‘ Integrating on both sides, when t=0 : Q=0;t= t: Q=Q; π ππ β«Χ¬β¬0 π −π 0 π0 −π π0 π0 −π = π0 ππ π‘ ππ‘ = β«Χ¬β¬0 πΆπ =− π 1 π‘ = β«Χ¬β¬0 ππ‘ πΆπ π‘ πΆπ −π‘ΰ΅ πΆπ π0 − π = π0 π π = π0 1 − π −π‘ΰ΅ πΆπ −π‘Τ π where π=CR The above equation gives us the value of charge on the capacitor at any time during charging. Discharging of a capacitor: When the key K is released, the circuit is broken without introducing any additional resistance. The battery is now out of the circuit and the capacitor will discharge itself through R. If I is the current at any time during discharge, then putting ε = 0 in RI + Q/C = ε, we get π =0 πΆ ππ π Or π = − ππ‘ πΆ ππ ππ‘ =− π πΆπ π πΌ + When t=0: Q= π0 and when t=t: Q=Q Integrating on both sides π ππ β«π πΧ¬β¬ 0 ππ π π0 =− =− π = π0 π π‘ ππ‘ β«Χ¬β¬0 πΆπ −1 π‘ = β«Χ¬β¬0 ππ‘ πΆπ π‘ πΆπ −π‘ΰ΅ π πΆ Fig: Charging and Discharging of a capacitor = π0 π −π‘Τ π where π=CR The above equation gives the value of charge on the capacitor at any time during discharge. Energy stored in a capacitor: The energy stored in a capacitor is nothing but the electric potential energy and is related to the voltage and charge on the capacitor. If the capacitance of a conductor is C, then it is initially uncharged and it acquires a potential difference V when connected to a battery. If q is the charge on the plate at that time, then π=πΆπ The work done is equal to the product of the potential and charge. Hence, π=ππ If the battery delivers a small amount of charge dQ at a constant potential V, then the work done is ππ = π ππ = π ππ πΆ Now, the total work done in delivering a charge of an amount q to the capacitor is given by π= ππ β«Χ¬β¬0 πΆ ππ = 1 π2 πΆ 2 = 1 π2 2 πΆ Therefore the energy stored in a capacitor is given by 1 π2 π= 2πΆ Substituting π = πΆ π 1 π = πΆπ2 2 The energy stored in a capacitor is given by the above equation Electricity and magnetism: Electricity is often described as being either static or dynamic. The difference between the two is based simply on whether the electrons are at rest (static) or in motion (dynamic). Static electricity is a build up of an electrical charge on the surface of an object. It is considered “static” due to the fact that there is no current flowing as in AC or DC electricity. Static electricity is usually caused when non-conductive materials such as rubber, plastic or glass are rubbed together, causing a transfer of electrons, which then results in an imbalance of charges between the two materials. The fact that there is an imbalance of charges between the two materials means that the objects will exhibit an attractive or repulsive force. The valence of an atom determines its ability to gain or lose an electron, which ultimately determines the chemical and electrical properties of the atom. These properties can be categorized as being a conductor, semiconductor or insulator, depending on the ability of the material to produce free electrons. Magnetism: Magnetism is the force exerted by magnets when they attract or repel each other. Magnetism is caused Every substance is made up of tiny units called atoms. Each atom has electrons, particles that carry electric charges. Spinning like tops, the electrons circle the nucleus, or core, of an atom. Their movement generates an electric current and causes each electron to act like a microscopic magnet.by the motion of electric charges. In most substances, equal numbers of electrons spin in opposite directions, which cancels out their magnetism. That is why materials such as cloth or paper are said to be weakly magnetic. In substances such as iron, cobalt, and nickel, most of the electrons spin in the same direction. This makes the atoms in these substances strongly magnetic—but they are not yet magnets. To become magnetized, another strongly magnetic substance must enter the magnetic field of an existing magnet. The magnetic field is the area around a magnet that has a magnetic force. All magnets have north and south poles. Opposite poles are attracted to each other, while the same poles repel each other. When you rub a piece of iron along a magnet, the north-seeking poles of the atoms in the iron line up in the same direction. The force generated by the aligned atoms creates a magnetic field. The piece of iron has become a magnet. Some substances can be magnetized by an electric current. When electricity runs through a coil of wire, it produces a magnetic field. The field around the coil will disappear, however, as soon as the electric current is turned off. Magnetic field: Magnetic Field is the region around a magnetic material or a moving electric charge within which the force of magnetism acts. A magnetic field is a vector field in the neighborhood of a magnet, electric current, or changing electric field in which magnetic forces are observable. A magnetic field is produced by moving electric charges and intrinsic magnetic moments of elementary particles associated with a fundamental quantum property known as spin. Magnetic field and electric field are both interrelated and are components of the electromagnetic force, one of the four fundamental forces of nature. Symbol B or H Unit Tesla Base Unit (Newton.Second)/Coul omb A magnetic field can be illustrated in two different ways. •Magnetic Field Vector •Magnetic Field Lines Magnetic Field Vector The magnetic field can be mathematically described as a vector field. The vector field is a set of many vectors that are drawn on a grid. In this case, each vector points in the direction that a compass would point and has a length dependent on the strength of the magnetic force. Fig: Vector Field of a Bar Magnet Magnetic Field Lines Magnetic field lines are a visual tool used to represent magnetic fields. They describe the direction of the magnetic force on a north monopole at any given position. The density of the lines indicates the magnitude of the field. Taking an instance, the magnetic field is stronger and crowded near the poles of a magnet. As we move away from the poles, it is weak, and the lines become less dense. Fig: A magnetic field lines plot for a bar magnet Magnetic Field Intensity Magnetic field strength is also magnetic field intensity or magnetic intensity. It is represented as vector H and is defined as the ratio of the MMF needed to create a certain Flux Density (B) within a particular material per unit length of that material. Magnetic field intensity is measured in units of amperes/metre. It is given by the formula: π» = π΅π − π Where, •B is the magnetic flux density •M is the magnetization •μ is the magnetic permeability The SI unit of magnetic field intensity is Tesla. One tesla (1 T) is defined as the field intensity generating one newton of force per ampere of current per metre of conductor. Faraday’s law: Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators , and solenoids. Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced. If the conductor circuit is closed, a current is induced, which is called induced current. Faraday’s second law of electromagnetic induction states that The induced emf in a coil is equal to the rate of change of flux linkage. The flux linkage is the product of the number of turns in the coil and the flux associated with the coil. The formula of Faraday’s law is given below π = −π β∅ βπ‘ Where ε is the electromotive force, Φ is the magnetic flux, and N is the number of turns. Fig: Magnetic field intensity in a closed loop Derivation: Consider a magnet approaching a coil. Consider two-time instances T1 and T2. Flux linkage with the coil at the time T1 is given by NΦ1. Flux linkage with the coil at the time T2 is given by NΦ2 Change in the flux linkage is given by N(Φ2 – Φ1) Let us consider this change in flux linkage as Φ = Φ2 – Φ1 Hence, the change in flux linkage is given by NΦ The rate of change of flux linkage is given by NΦ/t Taking the derivative of the above equation, we get N dΦ/dt Increasing the speed of the relative motion between the coil and the magnet, results in the increased emf According to Faraday’s second law of electromagnetic induction, we know that the induced emf in a coil is equal to the rate of change of flux linkage. Therefore, πΈ = π π∅ ππ‘ Considering Lenz’s law, which states that “The polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.” πΈ = −π π∅ ππ‘ From the above equation, we can conclude the following •Increase in the number of turns in the coil increases the induced emf •Increasing the magnetic field strength increases the induced emf Inductance: β’Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. L is used to represent the inductance, and Henry is the SI unit of inductance. 1 Henry is defined as the amount of inductance required to produce an emf of 1 volt in a conductor when the current change in the conductor is at the rate of 1 Ampere per second. β’An electric current flowing through a conductor creates a magnetic field around it. The strength of the field depends upon the magnitude of the current. The generated magnetic field follows any changes in the current, and from Faraday’s law of induction, we know that changing the magnetic field induces an electromotive force in the conductor. β’Considering this principle, inductance is defined as the ratio of the induced voltage to the rate of change of current causing it. The electronic component designed to add inductance to a circuit is an inductor. β’Inductance formula is given by πΏ = ππ 2 π΄ π Where, L = Inductance (H), μ = Permeability (Wb/Am), N = The coil’s number of turns, A = The coil’s cross sectional area, l = Length of coil (m). Types of inductance: Inductance is classified into two types: Self Inductance Mutual Inductance Self-inductance: When there is a change in the current or magnetic flux of the coil, an electromotive force is induced. This phenomenon is termed Self Inductance. When the current starts flowing through the coil at any instant, it is found that, that the magnetic flux becomes directly proportional to the current passing through the circuit. The relation is given as: ∅=πΏ×πΌ Where L is termed as the self-inductance of the coil or the coefficient of self-inductance, the self-inductance depends on the cross-sectional area, the permeability of the material, and the number of turns in the coil. The rate of change of magnetic flux in the coil is given as, π=− π∅ ππ‘ =− π πΏπΌ ππ‘ = −πΏ ππΌ ππ‘ Where, L is the self inductance in Henries, N is the number of turns,Φ is the magnetic flux, and I is the current in amperes Mutual Inductance: Consider two coils: P – coil (Primary coil) and S – coil (Secondary coil). A battery and a key are connected to the P-coil, whereas a galvanometer is connected across the S-coil. When there is a change in the current or magnetic flux linked with the two coils, an opposing electromotive force is produced across each coil, and this phenomenon is termed Mutual Inductance. This phenomenon is given by the relation: ∅= π×πΌ Where M is termed as the mutual inductance of the two coils or the coefficient of the mutual inductance of the two coils. The rate of change of magnetic flux in the coil is given as, π=− π∅ ππ‘ =− π ππΌ ππ‘ = −π ππΌ ππ‘ Mutual inductance formula is given by π= π0 ππ π1 π2 π΄ π Where, μ0 is the permeability of free space, μr is the relative permeability of the soft iron core, N is the number of turns in coil, A is the cross-sectional area in m2, l is the length of the coil in m Ampere’s Law: According to Ampere’s law, magnetic fields are related to the electric current produced in them. The law specifies the magnetic field that is associated with a given current or viceversa, provided that the electric field doesn’t change with time. “The magnetic field created by an electric current is proportional to the size of that electric current with a constant of proportionality equal to the permeability of free space.” Ampere’s circuital law can be written as the line integral of the magnetic field surrounding closed-loop equals the number of times the algebraic sum of currents passing through the loop. β«π» Χ―β¬. ππΏ = πΌπππ Suppose a conductor carries a current I, then this current flow generates a magnetic field that surrounds the wire. The equation’s left side describes that if an imaginary path encircles the wire and the magnetic field is added at every point, then it is numerically equal to the current encircled by this route, indicated by Ienc. Suppose you have a long enough wire that carries a constant current I in amps. In the shown figure a long wire exists that carries current in Amps. We need to find out how much is the magnetic field at a distance r. Therefore, we sketch an imaginary route around the wire indicated by dotted blue toward the right in the figure. According to the second equation, if the magnetic field is integrated along the blue path, then it has to be equal to the current enclosed, I. The magnetic field doesn’t vary at a distance r due to symmetry. The path length (in blue) in figure 1 is equal to the circumference of a circle,2πr. When a constant value H is added to the magnetic field, the equation’s left side looks like this: β«π» Χ―β¬. ππΏ = 2ππ π» = πΌπππ π»= πΌπππ 2ππ Magnetic circuit: A magnetic circuit, is a closed path to which a magnetic field, represented as lines of magnetic flux, is confined. In contrast to an electric circuit through which electric charge flows, nothing actually flows in a magnetic circuit. In a ring-shaped electromagnet with a small air gap, the magnetic field or flux is almost entirely confined to the metal core and the air gap, which together form the magnetic circuit. The reluctance r of a magnetic circuit is analogous to the resistance of an electric circuit. Reluctance depends on the geometrical and material properties of the circuit that offer opposition to the presence of magnetic flux. Reluctance of a given part of a magnetic circuit is proportional to its length and inversely proportional to its cross-sectional area and a magnetic property of the given material called its permeability. In a magnetic circuit, in summary, the magnetic flux is quantitatively equal to the magnetomotive force divided by the reluctance; Φ = f/r. Fig: Electro Magnet with Air Gap Transformer: A transformer is a device used to transmit electric energy. The transmission current is AC. It is commonly used to increase or decrease the supply voltage without a change in the frequency of AC between circuits. The transformer works on the principle of Faraday’s law of electromagnetic induction and mutual induction. There are usually two coils primary coil and secondary coil on the transformer core. The core laminations are joined in the form of strips. The two coils have high mutual inductance. When an alternating current pass through the primary coil it creates a varying magnetic flux. As per faraday’s law of electromagnetic induction, this change in magnetic flux induces an emf (electromotive force) in the secondary coil which is linked to the core having a primary coil. This is mutual induction. Overall, a transformer carries the below operations: 1.Transfer of electrical energy from circuit to another 2.Transfer of electrical power through electromagnetic induction 3.Electric power transfer without any change in frequency 4.Two circuits are linked with mutual induction Fig: Magnetic field intensity in a closed loop Fig: Formation of magnetic flux lines around a current-carrying wire Fig: formation of varying magnetic flux lines around a wire-wound. Classification of transformers: Based on application: β Step-Up Transformer β Step-Down Transformer Based on construction β Core-type transformer β Shell-type transformer Based on the number of phases β Single-phase transformer β Three-phase transformer Based on the location β Indoor type transformer β Outdoor type transformer β Station transformer Single-Phase Transformer: A single phase transformer is a type of transformer which operates on single-phase power. A transformer is a passive electrical device that transfers electrical energy from one circuit to another through the process of electromagnetic induction. It is most commonly used to increase (‘step up’) or decrease (‘step down’) voltage levels between circuits. A single phase transformer consists of a magnetic iron core serving as a magnetic transformer part and transformer cooper winding serving as an electrical part. A single phase transformer is a high-efficiency piece of electrical equipment, and its losses are very low because there isn’t any mechanical friction involved in its operation. Principle of operation: The single-phase transformer works on the principle of Faraday’s Law of Electromagnetic Induction. Typically, mutual induction between primary and secondary windings is responsible for the transformer operation in an electrical transformer. When the primary of a transformer is connected to an AC supply, the current flows in the coil and the magnetic field build-up. This condition is known as mutual inductance and the flow of current is as per the Faraday’s Law of electromagnetic induction. As the current increases from zero to its maximum value, the magnetic field strengthens and is given by dΙΈ/dt. This electromagnet forms the magnetic lines of force and expands outward from the coil forming a path of magnetic flux. The turns of both windings get linked by this magnetic flux. The strength of a magnetic field generated in the core depends on the number of turns in the winding and the amount of current. The magnetic flux and current are directly proportional to each other. As the magnetic lines of flux flow around the core, it passes through the secondary winding, inducing voltage across it. The Faraday’s Law is used to determine the voltage induced across the secondary coil and it is given by: N. dΙΈ/dt where,‘N’ is the number of coil turns The frequency is the same in primary and secondary windings. Thus, we can say that the voltage induced is the same in both the windings as the same magnetic flux links both the coils together. Also, the total voltage induced is directly proportional to the number of turns in the coil. Let us assume that the primary and secondary windings of the transformer have single turns on each. Assuming no losses, the current flows through the coil to produce magnetic flux and induce voltage of one volt across the secondary. Due to AC supply, magnetic flux varies sinusoidally and it is given by, ∅ = ∅πππ₯ sin ππ‘ The relationship between the induced emf, E in the coil windings of N turns is given by, π¬=π΅ π ∅ π π π¬ = π΅ × π × ∅πππ cos ππ∅ π¬πππ = π΅π∅πππ π¬πππ = π΅π π∅πππ π¬πππ = π. ππππ΅∅πππ Where, ‘f’ is the frequency in Hertz, given by ω/2π, ‘N’ is the number of coil windings, ‘ΙΈ’ is s the amount of flux in Webers. The above equation is the Transformer EMF Equation. For emf of a primary winding of a transformer E, N will be the number of primary turns (NP), while for the emf, E of a secondary winding of a transformer, the number of turns, N will be (NS). Voltage transformation ratio: If N1 – number of turns in primary, N2 – number of turns in secondary, Φm – maximum flux in weber (Wb),T – time period. Time is taken for 1 cycle. The flux formed is a sinusoidal wave. It rises to a maximum value Φm and decreases to negative maximum Φm. So, flux reaches a maximum in one-quarter of a cycle. The time taken is equal to T/4. Average rate of change of flux = Φm/(T/4) = 4fΦm Where f = frequency T = 1/f Induced emf per turn = rate of change of flux per turn Form factor = rms value / average value Rms value = 1.11 (4fΦm) = 4.44 fΦm [form factor of sine wave is 1.11] RMS value of emf induced in winding = RMS value of emf per turn x no of turns Primary Winding: Rms value of induced emf = E1 = 4.44 fΦm * N1 Secondary winding: Rms value of induced emf = E2 = 4.44 fΦm * N2 πΈ1 π1 = πΈ2 π2 = 4.44 π ∅π This is the emf equation of the transformer. For an ideal transformer at no load condition, E1 = supply voltage on the primary winding. E2 = terminal voltage (theoretical or calculated) on the secondary winding. πΈ1 π1 = πΈ2 π2 =π K is called the voltage transformation ratio, which is a constant. Case1: if N2 > N1, K>1 it is called a step-up transformer. Case 2: if N2< N1, K<1 it is called a step-down transformer. Current ratio: Transformer transfers electrical power from one circuit to another circuit very efficiently with negligible power loss. πππ€ππ ππππ’π‘ = πππ€ππ ππ’π‘ππ’π‘ Since π = ππΌ cos ∅ Thus,π1 πΌ1 cos ∅1 = π2 πΌ2 cos ∅2 Where πΌ1 and πΌ2 are primary and secondary currents respectively, and cos ∅1 , cos ∅2 are the corresponding power factors. For a transformer particulary at full load, power factors of primary and secondary are nearly equal. π1 πΌ1 = π2 πΌ2 πΌ1 πΌ2 = π2 π1 π2 π1 = πΌ1 = π2 =K πΌ 2 π 1 From the above equation it is clear that primary and secondary currnets of a transformer are inverlsy proportional to their respective turns of voltages KVA rating of a transformer, and regulation of a transformer: kVA stands for Kilovolt-Ampere and is the rating normally used to rate a transformer. The size of a transformer is determined by the kVA of the load. In many circumstances the power required by the load is equivalent to the rating of the transformer expressed in either VA or kVA. For example a 1KW (1000 Watts) load would require a 1kVA transformer @ unity power factor. The Current that passes through transformer windings will determine the Copper Losses, whereas Iron Losses, Core Losses or Insulation Losses depends on voltage. Regulation: The term voltage regulation identifies the characteristic of the voltage change in the transformer with loading. The voltage regulation of the transformer is defined as the arithmetical difference in the secondary terminal voltage between no-load (I2=0) and full rated load (I2 = I2fl) at a given power factor with the same value of primary voltage for both rated load and no-load. The numerical difference between no-load and full-load voltage is called inherent voltage regulation. Inherent voltage regulation= π½πππ − π½πππ Where V2fl = rated secondary terminal voltage at rated load. V2nl = no load secondary terminal voltage with the same value of primary voltage for both rated load and no load. Per unit voltage regulation at full load is = Percent voltage regulation at full load = π½πππ − π½πππ π½πππ π½πππ − π½πππ π½πππ π½π =ππππ π‘πππ‘ × 100 π½π =ππππ π‘πππ‘ Voltage Regulation in terms of primary values: Per unit voltage regulation = Where π½πππ = π½πππ − π½πππ π½πππ π1 π Then per unit voltage regulation is = = π1 π − π½πππ π½πππ Transformer Efficiency πΈπππππππππ¦(η) = πΈπππππππππ¦(η) = ππ’π‘ππ’π‘ πππ€ππ ππππ’π‘ πππ€ππ πππ’π‘ πππ’π‘ +ππππ π ππ × 100 × 100 Therefore the power efficiency at load current I2 and power factor cos ∅2 will be η= π2 πΌ2 cos ∅2 π2 πΌ2 cos ∅2 +πΌ22 π π2 +ππ ηππ = π2 πΌ2ππ cos ∅2 2 π +π π2 πΌ2 cos ∅2 +πΌ2ππ π2 π If π2ππ = ππ΄ Then ηππ = 2ππ = π2 πΌ2ππ = full- load VA = rated VA π2 cos ∅2 2 π +π π2 cos ∅2 +πΌ2ππ π2 π Electromechanical energy conversion A device which converts electrical energy into mechanical energy or mechanical energy into electrical energy is known as electromechanical energy conversion device. The electromechanical energy conversion takes place through the medium of a magnetic field. The magnetic field is used as a coupling medium between electrical and mechanical systems. It is because the energy storing capacity of a magnetic field is very high. Therefore, an electromechanical energy converter has three main parts • Mechanical system • Coupling medium • Electrical system The electromechanical energy converters are of two types − • Gross-motion devices − Such as electrical motors or generators. • Incremental motion devices − Such as microphones, loudspeakers, electromagnetic relays and electrical measuring instruments, etc. When the electromechanical energy conversion takes place from electrical energy to mechanical energy, the converter is known as motor. Whereas, when the conversion takes place from mechanical energy to electrical energy, the device is known as generator. Fig: Electromechanical energy conversion Principle of Conservation of Energy The principle of conservation of energy states that “the energy can neither be created not destroyed. It can only be converted from one form to another”. In an electromechanical energy conversion device, the total input energy is equal to the sum of following three components − •Energy dissipated, •Energy stored, and •Useful output energy. Hence, the principle of electromechanical energy conversion is based on the following two equations The energy balance equation or energy transfer equation for motoring action can be written as πΈππππ‘πππππ ππππππ¦ ππππ’π‘ = πΈπππππ¦ πππ π ππππ‘ππ ππ πππππ‘πππππ πππ π ππ + πΈπππππ¦ π π‘ππππ ππ πππ’πππππ πππππ’π + πππβππππππ ππππππ¦ ππ’π‘ππ’π‘ The energy balance equation or energy transfer equation for generating action can be written as πππβππππππ ππππππ¦ ππππ’π‘ = πΈππππ‘πππππ ππππππ¦ ππ’π‘ππ’π‘ + πΈπππππ¦ π π‘ππππ ππ πππ’πππππ πππππ’π + πΈπππππ¦ πππ π ππππ‘ππ DC motor: A DC motor is defined as a class of electrical motors that convert direct current electrical energy into mechanical energy. Working Principle of DC motor: When kept in a magnetic field, a current-carrying conductor gains torque and develops a tendency to move. In short, when electric fields and magnetic fields interact, a mechanical force arises. This is the principle on which the DC motors work. Construction parts of a DC motor: Armature or Rotor: The armature of a DC motor is a cylinder of magnetic laminations that are insulated from one another. The armature is perpendicular to the axis of the cylinder. The armature is a rotating part that rotates on its axis and is separated from the field coil by an air gap. Fig: DC motor construction parts Field Coil or Stator: A DC motor field coil is a non-moving part on which winding is wound to produce a magnetic field. This electro-magnet has a cylindrical cavity between its poles. Commutator and Brushes: The commutator of a DC motor is a cylindrical structure that is made of copper segments stacked together but insulated from each other using mica. The primary function of a commutator is to supply electrical current to the armature winding. The brushes of a DC motor are made with graphite and carbon structure. These brushes conduct electric current from the external circuit to the rotating commutator. Hence, we come to understand that the commutator and the brush unit are concerned with transmitting the power from the static electrical circuit to the mechanically rotating region or the rotor. Fig: Production of torque in a DC motor Types of DC motors: AC motors: An AC motor is a motor that converts alternating current into mechanical power. The stator and the rotor are important parts of AC motors. The stator is the stationary part of the motor, and the rotor is the rotating part of the motor. The AC motor may be single-phase or three-phase. An AC motor works on the principle of electromagnetic induction. Construction: An alternating current drives an AC motor. The stationary stator and the rotating rotor are important parts of AC motors. Stator: The stator is the stationary part of the motor that delivers a rotating magnetic field to interact with the rotor. Stator Core: The stator core is made of thin metal sheets known as laminations. Laminations are used to reduce energy loss. Stator Windings: Stator windings are stacked together, forming a hollow cylinder. The slots of the stator core coils of insulated wires are insulated. When the assembled motor operates, the stator windings are connected to a power source. Each group of coils, along with the steel core, becomes an electromagnet when the current is applied. Fig: Stator Rotor: A rotor is a central component of a motor that is fixed to the shaft. The most common type of rotor used in an AC motor is the squirrel cage rotor. A squirrel-cage rotor is cylindrical and is made by stacking thin steel laminations. Instead of inserting wire coils between the slots, conductor bars are die-cast into the evenly spaced slots around the cylinder. Once the conductor bars are die-casted, they are electrically and mechanically connected to the end rings. Motor Shaft: The rotor is pressed onto a steel shaft to form a rotor assembly. The shaft extends outside the motor casing allowing connection to an external system to transmit the rotational power. Bearings: Bearings hold the motor shaft in place. The bearings minimize the shaft’s friction connected to the casing, which increases the motor’s efficiency. Enclosure: The enclosure protects the internal parts of the motor from water and other environmental elements. The enclosure consists of a frame and two end brackets. Fig: Squirrel cage rotor Types of AC motor:
