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    UNIT-1 : D. C. Circuits

    A DC circuit (Direct Current circuit) is an electrical circuit that consists of any combination of

    constant voltage sources, constant current sources, and resistors. In this case, the circuit voltages

    and currents are constant, i.e., independent of time. More technically, a DC circuit has no

    memory. That is, a particular circuit voltage or current does not depend on the past value of anycircuit voltage or current. This implies that the system of equations that represent a DC circuit do

    not involve integrals or derivatives.

    If a capacitor and/or inductor is added to a DC circuit, the resulting circuit is not, strictly

    speaking, a DC circuit. However, most such circuits have a DC solution. This solution gives the

    circuit voltages and currents when the circuit is in DC steady state. More technically, such a

    circuit is represented by a system of differential equations. The solution to these equations

    usually contain a time varying or transient part as well as constant or steady state part. It is this

    steady state part that is the DC solution. There are some circuits that do not have a DC solution.

    Two simple examples are a constant current source connected to a capacitor and a constantvoltage source connected to an inductor.

    It is common to refer to a circuit that is powered by a DC voltage source such as a battery or

    the output of a DC power supply as a DC circuit even though what is meant is that the circuit is

    DC powered.

    Electric Current:

    Electric current means, depending on the context, a flow of electric charge (a phenomenon) or

    the rate of flow of electric charge (a quantity). This flowing electric charge is typically carried by

    moving electrons, in a conductor such as wire; in an electrolyte, it is instead carried by ions, and,

    in a plasma, by both. The SI unit for measuring the rate of flow of electric charge is the ampere,

    which is charge flowing through some surface at the rate of one coulomb per second. Electric

    current is measured using an ammeter.

    Current:

    The flow of charge is called the current and it is the rate at which electric charges pass though a

    conductor. The charged particle can be either positive or negative. In order for a charge to flow,

    it needs a push (a force) and it is supplied by voltage, or potential difference. The charge flows

    from high potential energy to low potential energy.

    Current, I=V/R

    where the symbol I to represent the quantity current.

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    Electro-motive force(E.M.F):

    Electromotive Force is, the voltage produced by an electric battery or generator in an electrical

    circuit or, more precisely, the energy supplied by a source of electric power in driving a unit

    charge around the circuit. The unit is the volt. A difference in charge between two points in a

    material can be created by an external energy source such as a battery. This causes electrons tomove so that there is an excess of electrons at one point and a deficiency of electrons at a second

    point. This difference in charge is stored as electrical potential energy known as emf. It is the

    emf that causes a current to flow through a circuit.

    Voltage:

    Voltage is electric potential energy per unit charge, measured in joules per coulomb ( = volts). It

    is often referred to as "electric potential", which then must be distinguished from electric

    potential energy by noting that the "potential" is a "per-unit-charge" quantity. Like mechanical

    potential energy, the zero of potential can be chosen at any point, so the difference in voltage isthe quantity which is physically meaningful. The difference in voltage measured when moving

    from point A to point B is equal to the work which would have to be done, per unit charge,

    against the electric field to move the charge from A to B.

    Electric potential:

    A gravitational analogy was relied upon to explain the reasoning behind the relationship between

    location and potential energy. Moving a positive test charge against the direction of an electric

    field is like moving a mass upward within Earth's gravitational field. Both movements would be

    like going against nature and would require work by an external force. This work would in turn

    increase the potential energy of the object. On the other hand, the movement of a positive test

    charge in the direction of an electric field would be like a mass falling downward within Earth's

    gravitational field. Both movements would be like going with nature and would occur without

    the need of work by an external force. This motion would result in the loss of potential energy.

    Potential energy is the stored energy of position of an object and it is related to the location of

    the object within a field.

    Potential Difference:

    A quantity related to the amount of energy needed to move an object from one place to another

    against various types of forces. The term is most often used as an abbreviation of "electricalpotential difference", but it also occurs in many other branches of physics. Only changes in

    potential or potential energy (not the absolute values) can be measured.Electrical potential

    difference is the voltage between two points, or the voltage drop transversely over an impedance

    (from one extremity to another). It is related to the energy needed to move a unit of electrical

    charge from one point to the other against the electrostatic field that is present. The unit of

    electrical potential difference is the volt (joule per coulomb). Gravitational potential difference

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    Kirchhoffs laws:

    Kirchoff's Current Law:

    First law (Current law or Point law):

    The sum of the currents flowing towards any junction in an electric circuit is equal to the sum of

    currents flowing away from the junction. Kirchoff's Current law can be stated in other words as

    the sum of all currents flowing into a node is zero. Or conversely, the sum of all currents leaving

    a node must be zero. As the image demonstrates, the sum of currents I1 and I2 must equal the

    sum of currents I3 and I4 . Current flows through wires much like water flows through pipes. If

    you have a definite amount of water entering a closed pipe system, the amount of water that

    enters the system must equal the amount of water that exists the system. The number of

    branching pipes does not change the net volume of water (or current in our case) in the system.

    Kirchoff's Voltage Law:

    Second law (voltage law or Mesh law):

    In any closed circuit or mesh, the algebraic sum of all the electromotive forces and the voltage

    drops is equal to zero.Kirchoff's voltage law can be stated in words as the sum of all voltage

    drops and rises in a closed loop equals zero. As the image below demonstrates, loop 1 and loop 2

    are both closed loops within the circuit. The sum of all voltage drops and rises around loop 1

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    equals zero, and the sum of all voltage drops and rises in loop 2 must also equal zero. A closed

    loop can be defined as any path in which the originating point in the loop is also the ending point

    for the loop. No matter how the loopis defined or drawn, the sum of the voltages in the loop must

    be zero.

    Problem 1:

    A current of 0.5 A is flowing through the resistance of 10.Find the potential difference between

    its ends.

    Solution:

    Current I = 0.5A.

    Resistance R = 10

    Potential difference V = ?

    V = IR

    = 0.5 10 = 5V.

    Problem :2

    A supply voltage of 220V is applied to a 100 resistor.Find the current flowing through it.

    Solution:

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    Voltage V = 220V

    Resistance R = 100

    Current I = V = 220 = 2.2 A.

    R 100

    Problem : 3

    Calculate the resistance of the conductor if a current of 2A flows through it when the potential

    difference across its ends is 6V.

    Solution:

    Current I = 2A.

    Potential difference = V = 6.

    Resistance R = V/I

    = 6 /2

    = 3 ohm.

    Problem: 4

    Calculate the current and resistance of a 100 W ,200V electric bulb.

    Solution:

    Power,P = 100W

    Voltage,V = 200V

    Power p = VI

    Current I = P/V = 100/200 = 0.5A

    Resistance R = V /I = 200/0.5 = 400W.

    Problem: 5

    Calculate the power rating of the heater coil when used on 220V supply taking 5 Amps.

    Solution:

    Voltage ,V = 220V

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    Current ,I = 5A,

    Power,P = VI = 220 5

    = 1100W = 1.1 KW.

    Problem: 6

    A circuit is made of 0.4 wire, a 150 bulb and a 120 rheostat connected in series. Determine

    the total resistance of the resistance of the circuit.

    Solution:

    Resistance of the wire = 0.4

    Resistance of bulb = 150

    Resistance of rheostat = 120

    In series,

    Total resistance ,R = 0.4 + 150 +120

    = 270.4

    Problem :7

    In the circuit shown in fig .find the current, voltage drop across each resistor and the power

    dissipated in each resistor.

    Solution:

    Total resistance of the circuit = 2 + 6 +7

    R = 15

    Voltage ,V = 4 5V

    Circuit current ,I = V /R = 45 /15 = 3A

    Voltage drop across 2 resistor V1 = I R1

    = 3 2 = 6 Volts.

    Voltage drop across 6 resistor V2 = I R2

    = 3 6 = 18 volts.

    Voltage drop across 7 resistor V3 = I R3

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    = 3 7 = 21 volts.

    Power dissipated in R1 is P1 = P R1

    = 32 2 = 18 watts.

    Power dissipated in R2 is P2 = I2 R2.

    = 32 6 = 54 watts.

    Power dissipated in R3 is P3 = I2 R3.

    = 32 7 = 63 watts.

    Problem : 8

    Three resistances of values 2,3 and 5 are connected in series across 20 V,D.C supply

    .Calculate (a) equivalent resistance of the circuit (b) the total current of the circuit (c) the voltagedrop across each resistor and (d) the power dissipated in each resistor.

    Solution:

    Total resistance R = R1 + R2+ R3.

    = 2 +3+5 = 10

    Voltage = 20V

    Total current I = V/R = 20/10 = 2A.

    Voltage drop across 2 resistor V1 = I R1

    = 2 2 = 4 volts.

    Voltage drop across 3 resistor V2 = IR2

    = 2 3 = 6 volts.

    Voltage drop across 5 resistor V3 = I R3

    = 2 5 = 10 volts.

    Power dissipated in 2 resistor is P1 = I2 R1

    = 22 2 = 8 watts.

    Power dissipated in 3 resistor is P2 = I2 R2.

    = 22 3 = 12 watts.

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    Power dissipated in 5 resistor is P3 = I2 R3

    = 22 5 = 20 watts.

    Problem: 9

    A lamp can work on a 50 volt mains taking 2 amps.What value of the resistance must be

    connected in series with it so that it can be operated from 200 volt mains giving the same power.

    Solution:

    Lamp voltage ,V = 50V

    Current ,I = 2 amps.

    Resistance of the lamp = V/I = 50/25 = 25

    Resistance connected in series with lamp = r.

    Supply voltage = 200 volt.

    Circuit current I = 2A

    Total resistance Rt= V/I = 200/2 = 100

    Rt = R + r

    100 = 25 + r

    r = 75

    2) Electromagnetism

    Electromagnetism describes the relationship between electricity and magnetism. We use

    electromagnets to generate electricity, store memory on our computers, generate pictures on a

    television screen. Electromagnetism works on the principle that an electric current through a wire

    generates a magnetic field. In a bar magnet, the magnetic field runs from the North to the SouthPole. In a wire, the magnetic field forms around the wire. An electromagnet can be created by

    wrap wire around a metal object.

    Magnetic field:

    When DC electricity is passed through a wire, a magnetic field rotates around the wire in a

    specific direction. When bars of magnetic materials Iron, Cobalt, Nickel are wound with a coil

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    and current is passed though them, they become electromagnets. The strength of such an

    electromagnet depends on the number of turns in the coil and the magnitude of the current

    passing through it .The region or space around a magnet in which the magnetic effects are felt is

    known as the magnetic field.The magnetic field is represented by magnetic lines of force, which

    start from the North Pole and go into the South Pole. The magnetic lines of force are always closed

    lines. The magnetic lines of force are purely imaginary lines. They do not intersect each other.

    They are like elastic bands which always try to shorten themselves.

    Magnetic Flux ():

    The entire magnetic lines of force representing a magnetic field is known as the magnetic flux .Its unit is

    weber, abbreviated as Wb, named after Wilhelm Eduard Weber (1804-91), a German Physicist.

    The magnetic flux per unit area, the area being normal to the lines of flux is known as the flux

    density. The unit is Weber per square metre (Wb/m2) or Tesla (T).

    Wb/m2or T

    Magneto-Motive Force or M.M.F. :

    aB

    =

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    M.M.F. is defined as the magnetic force, which creates magnetic flux is a magnetic material.

    M.M.F = N * I ampere turns (AT), Where, N = Number of turns in the coil, I = Current through the coil

    Reluctance (R):

    Reluctanceis the property of a magnetic material by virtue of which, it opposes the creation of magneticflux in it.

    R= M.M.F/Magnetic Flux =N*I/ ampere turns per weber (AT/Wb).

    Permeability:

    Permeability is basically the property of the magnetic material by virtue of which the magnetic flux can

    be easily created in it. For any magnetic material, there are two permeabilities:

    (i) Absolute permeability (ii) Relative permeability

    Absolute Permeability ():

    The absolute permeability of a magnetic material is defined as the flux induced in the magnetic material

    per unit magnetising force.

    where, H = magnetising force

    Relative Permeability (r):

    For defining the relative permeability of a magnetic material, the permeability of free space or air is taken

    as reference. The relative permeability of free space or air is taken as unity. i.e r= 1,for free space orair.

    The relative permeability is defined as the ratio of the flux density induced in the magnetic to the flux

    density induced in free space or air when the same magnetising force is applied.

    r is dimensionless ,If the permeability of iron is 500, it means that, iron is 500

    times more magnetic than free space or air.

    Magnetising Force (H): This is also referred as magnetic field strength or magnetic field

    intensity.The magnetic field intensity at any point in a magnetic field is defined as the force

    experienced by a unit North Pole placed at that point, both in magnitude and direction. The

    magnetising force may also be defined as the number of ampere turns produced per unit length

    AT/m

    H

    B=

    o

    rB

    B=

    l

    NIH=

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    The unit isAmpere turns per metre.

    Faradays Laws:

    First Law: Whenever the flux linking with a conductor changes, an emf is induced in it.

    Second Law: The magnitude of the emf induced in a conductor or coil is equal to the rate of

    change of flux linkages of it.

    Faraday's law of electromagnetic induction states that:

    Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be

    "induced" in the coil. The change could be produced by changing the magnetic field strength,

    moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field,

    rotating the coil relative to the magnet, etc.

    E= -N

    where E= electromotive force in volts,N= No.of turns of wire,=Magnetic flux density in

    webers

    The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the

    number of turns in the coil. It involves the interaction of charge with magnetic field.

    Electromagnetic induction is the production of voltage across a conductor situated in a changing

    magnetic field or a conductor moving through a stationary magnetic field. Faraday found that the

    electromotive force (EMF) produced around a closed path is proportional to the rate of change of

    the magnetic flux through any surface bounded by that path. In practice, this means that anelectrical current will be induced in any closed circuit when the magnetic flux through a surface

    bounded by the conductor changes. Electromagnetic induction underlies the operation of

    generators, all electric motors, transformers, induction motors, synchronous motors, solenoids,

    and most other electrical machines.

    Lenz Law:

    The emf induced in an electric circuit always acts in such a direction that the current it drives

    around the circuit opposes the change in magnetic flux which produces the emf.

    The direction mentioned in Lenz's law can be thought of as the result of the minus sign in the

    below equation

    E= - N

    dt

    d

    dt

    d

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    When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity

    of the induced emf is such that it produces a current whose magnetic field opposes the change

    which produces it. The induced magnetic field inside any loop of wire always acts to keep the

    magnetic flux in the loop constant.

    Flemings Rules:

    The relative directions of the magnetic field, current, and motion in an electric generator or

    motor, can be represented using one's fingers. The three directions are represented by the thumb

    (for motion),forefinger(forfield), and middle finger(for conventional current), all held at right

    angles to each other. The right hand is used for generators and the left for motors. The rules were

    devised by the English physicist John Fleming.

    Fleming's Right hand rule:

    The Right hand is held with the thumb index finger and middle finger mutually at right angles.

    The Fore finger represents the direction of the Magnetic Field. The Middle finger represents the

    direction of the Induced E.M.F or Current. The Thumb represents the direction of the Motion of

    the Conductor.

    An emf is induced in a coil or conductor whenever there is change in the flux linkages.

    Depending on the way in which the changes are brought about, there are two types:

    Statically induced E.M.F and

    Dynamically induced E.M.F

    Consider a coil of N turns as shown in Figure to which an alternating voltage V is applied,

    due to which an alternating current I flows through the coil. This alternating current, produces

    an alternating flux which links the coil. Hence, an e.m.f is induced in the coil, which is given

    by the equation.

    dt

    dNe

    =

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    This induced e.m.f. opposes its own cause. Thus induced e.m.f opposes the appliedvoltage, which is the very cause of it.

    The change of flux linking an electric circuit can take place in two ways. When a conductor cuts

    across a magnetic field of constant flux density, the flux changes and an e.m.f. is induced in the

    conductor.This type of e.m.f. induced is known as dynamically induced e.m.f.When the electric

    circuit is in the form of a stationary coil , produces an alternating flux ,due to current passing

    through coil which links the coil. An e.m.f. is induced in the stationary coil. This type of e.m.f

    induced is known asstatically induced e.m.f.

    Dynamically Induced E.M.F:

    The emf induced in a conductor due to the relative motion between the magnetic field and the

    conductor is called the dynamically induced emf.

    Dynamically induced emf is obtained either,

    1. Keeping the magnetic field stationary and moving the conductor in the magnetic field.

    2. Keeping the conductor stationary and moving the field over the conductor.

    Expression for Dynamically Induced E.M.F:

    Consider a single conductor of length meters moving at right angles. A uniform magnetic

    field of B wb/m2 Velocity of m/s. Suppose the conductor moves through a small distance

    dx in dt seconds.

    Then the area swept by the conductor = dx

    Flux cut = d = Flux density Area swept

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    = B X dx weber

    According to faradays laws of electromagnetic induction emf e induced in the conductor is

    given by

    Volts (N=1, d = B* *dx) and dx/dt = (Velocity)

    If the conductor moves at an angle of to the magnetic field then the velocity at which the

    conductor moves across the field is sin

    e = B sin

    where e = induced e.m.f

    B = magnetic field

    = velocity of conductor

    Statically Induced E.M.F:

    When both magnetic field and the conductor are stationary, emf induced in a conductor or coildue to the variation of flux linking with the conductor is called statistically induced emf. In this

    neither the conductor nor the magnetic field moves, but the strength of the magnetic field varies.

    Bldt

    dxBl

    dt

    dNe ===

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    Applications:

    Dynamically induced emf principle is used in D.C generators, alternators, in cycle dynamo etc.

    Statically induced emf principle is used in all types of transformers like power transformers,

    distribution transformers and chokes.

    There are two types of statically induced emfs. They are :

    Self-induced emf

    Mutually-induced emf

    1.Self-induced E.M.F:

    Self inductance is defined as the induction of a voltage in a current-carrying wire when the

    current in the wire itself is changing. In the case of self-inductance, the magnetic field created by

    a changing current (AC) in the circuit itself induces a voltage in the same circuit. Self induced

    emf will be in opposition to the applied voltage.

    eL =Induced emf & measured in volts.

    where N = Number of turns of the coil, = Flux linking with the coil

    -ve sign indicates induced emf is in the direction opposite to the applied voltage.

    Increasing the number of turns or the rate of change of magnetic flux increases the amount of

    induced voltage .

    Where

    L is a constant known as self inductance of the coil

    dt

    dNeL

    =

    dt

    diL

    di

    di

    dt

    dN

    dt

    dNeL ===

    INdi

    dNL

    ==

    Idi

    d =

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    is a constant because

    Therefore

    Energy Stored in an Inductor:

    An inductor is an inductive coil, which possesses both inductance and a small resistance. If the

    resistance is neglected, it is called as an ideal inductor. A pure inductance does not consume any

    energy and the energy supplied to the coil is stored in the form of an electromagnetic field.The

    induced e.m.f opposes any change in the value of the current flowing through the coil. Hence in

    order to establish a steady current of I amperes in t seconds, work has to be done to overcome

    the opposition due to the induced e.m.f.

    If Induced emf

    Applied voltage

    The work done in dt seconds is given by

    The work done in t seconds is given by

    This work done is stored in the coil in the form of an electromagnetic field. The energy stored in

    a coil of inductance L henrys(H) in the form of an electromagnetic field is given by

    i

    al

    N

    IR

    NIN

    I

    NL

    r

    0

    2

    /

    .===

    laNL r

    2

    0=

    diLidtidt

    diLdtVidw ===

    ==It

    diLidwW00

    == 22

    0

    i

    LdiiLW

    I

    2

    2IL=

    JoulesI

    LE2

    2

    =

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    Mutually-induced E.M.F:

    Two coils, which are placed close to each other are said to be mutually coupled, when a part of

    the alternating flux produced in one coil links the other coil An e.m.f is induced in both the coils.

    The e.m.f. induced in the first coil, where the flux is produced, is called asself induced e.m.f The

    e.m.f. induced in the second coil, which links a part of the flux produced in the first coil, isknown as mutually induced e.m.f.

    In figure flux 1, links coil 1 and hence an e.m.f. e1is induced in it.

    This flux 12which links both coil 1 and coil 2, is called as the mutual fluxbetween the two

    coils.

    1= 11+ 12

    The mutual flux 12 linking coil 2, induces an e.m.f. el2in that coil. This e.m.f. is known as the

    mutually induced e.m.f .

    The equation for e12is also written as

    i.e e.m.f induced in coil 2 due to current flowing in coil 1

    dt

    dNe 111 =

    dtdNe 21221 =

    dt

    diMe 1

    2121 =

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    M12is known as the mutual inductancebetween coil 1 and coil 2, the equation for the mutual

    inductance MI2may be written as

    Similarly equations can be written, when coil 2 is energised by an alternating current i2,

    producing a total flux 2in it

    2= 22+ 21

    2= total flux produced in coil 2.

    22= flux that links only coi1.2

    21= flux that links both coil 2 and coil 1.

    The self induced e.m.f. in coil 2 is given, by

    The mutually induced e.m.f in coil 1 is given by

    Where M21is the mutual inductance.

    As the coupling between the two coils is bilateral M12= M21= M

    The mutual inductance between any two coils, which are placed close to each other, can be

    defined as the ability of one coil to induce an e.m.f in the other coil, when an alternating current

    flows through one of the coils.

    1

    12221

    di

    dNM =

    dt

    dNe 222 =

    dt

    diM

    dt

    dNe 221

    21121 ==

    2

    21121

    di

    dNM =

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    Coefficient of Coupling (K):

    The Coefficient of Coupling (K) between two coils is defined as the fraction of magnetic flux

    produced by the current in one coil that links the other coil. The Co-efficient of coupling is the

    ratio of the mutual flux to the total flux. The Coefficient of Coupling (K) has a maximum value

    of 1 (or 100%) The Coefficient of Coupling has a minimum value of zero which indicates thatthe two coils are magnetically isolated. Suppose the two coils have self inductances L1and L2

    and M is the mutual inductance, then the Coefficient of Coupling is

    1

    1212

    =K and

    2

    2121

    =K

    2112 MMM ==

    2

    211

    1

    122

    di

    dN

    di

    dNM

    =

    2

    2

    1

    121

    )()(

    di

    Kd

    di

    KdNN

    =

    2

    22

    1

    11

    2

    di

    dN

    di

    dNK

    =

    21LL

    M

    K=