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 any circuit 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 constant voltage 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.

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 to move 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 is the 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 "electrical potential 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 between two points on Earth is related to the energy needed to move a unit mass from one point

to the other against the Earth's gravitational field. The unit of gravitational potential differences is joules per kilogram.

Resistance:

Resistance is the ratio of potential difference across a conductor to the current flowing through it. If energy is used in passing electricity through an object, that object has a resistance.

Ohm’s Law:

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them. The mathematical equation that describes this relationship is:

I=V/R

where I is the current through the resistance in units of amperes, V is the potential difference measured across the resistance in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

Resistance:

Resistance is the opposition that a substance offers to the flow of electric current. It is represented by the uppercase letter R. The standard unit of resistance is the ohm, sometimes written out as a word, and sometimes symbolized by the uppercase Greek letter omega. When an electric current of one ampere passes through a component across which a potential difference (voltage) of one volt exists, then the resistance of that component is one ohm.

In general, when the applied voltage is held constant, the current in a direct-current (DC) electrical circuit is inversely proportional to the resistance. If the resistance is doubled, the current is cut in half; if the resistance is halved, the current is doubled. This rule also holds true for most low-frequency alternating-current (AC) systems, such as household utility circuits.

In some AC circuits, especially at high frequencies, the situation is more complex, because some components in these systems can store and release energy, as well as dissipating or converting it.The electrical resistance per unit length, area, or volume of a substance is known as resistivity. Resistivity figures are often specified for copper and aluminum wire, in ohms per kilometre. Opposition to AC, but not to DC, is a property known as reactance. In an AC circuit, the resistance and reactance combine vectorially to yield impedance.

Kirchhoff’s 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 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 10Ω.Find the potential difference between its ends.

Solution:

Current I = 0.5A.

Resistance R = 10Ω

Potential difference V = ?

V = IR

= 0.5 × 10 = 5V.

Problem :2

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

Solution:

Voltage V = 220V

Resistance R = 100Ω

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

Current ,I = 5A,

Power,P = VI = 220 × 5

= 1100W = 1.1 KW.

Problem: 6

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

Solution:

Resistance of the wire = 0.4Ω

Resistance of bulb = 150Ω

Resistance of rheostat = 120Ω

In series,

Total resistance ,R = 0.4 + 150 +120

= 270.4Ω

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 Ω

Voltage ,V = 4 5V

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

Voltage drop across 2Ω resistor V1 = I R1

= 3 × 2 = 6 Volts.

Voltage drop across 6Ω resistor V2 = I R2

= 3 × 6 = 18 volts.

Voltage drop across 7Ω resistor V3 = I R3

= 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 2Ω,3Ω and 5Ω 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 voltage drop across each resistor and (d) the power dissipated in each resistor.

Solution:

Total resistance R = R1 + R2+ R3.

= 2 +3+5 = 10Ω

Voltage = 20V

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

Voltage drop across 2Ω resistor V1 = I R1

= 2× 2 = 4 volts.

Voltage drop across 3Ω resistor V2 = IR2

= 2 × 3 = 6 volts.

Voltage drop across 5Ω resistor V3 = I R3

= 2 ×5 = 10 volts.

Power dissipated in 2Ω resistor is P1 = I2 R1

= 22 × 2 = 8 watts.

Power dissipated in 3 resistor is P2 = I2 R2.

= 22 × 3 = 12 watts.

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 Ω

Resistance connected in series with lamp = r.

Supply voltage = 200 volt.

Circuit current I = 2A

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

Rt = R + r

100 = 25 + r

r = 75Ω

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 South Pole. 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 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 (ɸ):

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).

aB

φ=

Wb/m2 or T

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

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):

Reluctance is the property of a magnetic material by virtue of which, it opposes the creation of magnetic flux in it.

R= M.M.F/Magnetic Flux =N*I/Ф 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 (µ):

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 (µr):

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 µr = 1, for free space or air.

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.

µr’ 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

H

B=µ

or B

B=µ

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

The unit is Ampere turns per metre.

Faraday’s 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,Ф=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 an electrical 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”.

l

NIH =

dt

dΦ

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

E= - N

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.

Fleming’s 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 (for field), 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 ‘ɸ’ which links the coil. Hence, an e.m.f is induced in the coil, which is given by the equation.

dt

dΦ

dt

dNe

Φ−=

This induced e.m.f. opposes its own cause. Thus induced e.m.f opposes the applied voltage, 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 as statically 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 ‘Ɩ’ meters moving at right angles. A uniform magnetic field of ‘B’ wb/m2 Velocity of ‘ϑ’ m/s. Suppose the conductor moves through a small distance ‘dx’ in ‘dt’ seconds.

Then the area swept by the conductor = ‘Ɩ dx’

Flux cut = dɸ = Flux density × Area swept

= B X Ɩ dx weber

According to faraday’s laws of electromagnetic induction emf ‘e’ induced in the conductor is given by

Volts (N=1, dɸ = B* Ɩ *dx) and dx/dt = ϑ (Velocity)

If the conductor moves at an angle of ‘θ’ to the magnetic field then the velocity at which the conductor moves across the field is ‘ϑ sinθ’

e = B Ɩ ϑ sinθ

where e = induced e.m.f B = magnetic field

ϑ = velocity of conductor

Statically Induced E.M.F:

When both magnetic field and the conductor are stationary, emf induced in a conductor or coil due 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.

υφBl

dt

dxBl

dt

dNe ===

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 emf’s. 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, ɸ = 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

φ−=

dt

diL

di

di

dt

dN

dt

dNeL −=×−=−= φφ

IN

di

dNL

φφ ==

Idi

d φφ =

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

iφα

al

N

IR

NIN

I

NL

rµµφ

0

2

/

. ===

l

aNL r

20µµ=

diLidtidt

diLdtVidw ===

∫∫ ==It

diLidwW00

== ∫ 2

2

0

iLdiiLW

I

2

2IL=

JoulesI

LE2

2

=

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 as self 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, is known as mutually induced e.m.f.

In figure flux ɸ1, links coil 1 and hence an e.m.f. ‘e1’ is induced in it.

This flux ɸ12 which links both coil 1 and coil 2, is called as the mutual flux between the two coils.

ɸ1 = ɸ11 + ɸ12

The mutual flux ɸ12 linking coil 2, induces an e.m.f. el2 in that coil. This e.m.f. is known as the mutually induced e.m.f .

The equation for e12 is also written as

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

dt

dNe 1

11

ϕ−=

dt

dNe 21

221

ϕ−=

dt

diMe 1

2121 −=

M12 is known as the mutual inductance between coil 1 and coil 2, the equation for the mutual inductance MI2 may be written as

Similarly equations can be written, when coil 2 is energised by an alternating current i2, producing a total flux ɸ2 in it

ɸ2= ɸ22 + ɸ21

ɸ2 = total flux produced in coil 2.

ɸ22 = flux that links only coi1.2

ɸ21 = 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 M21 is 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

ϕ−=

dt

dNe 2

22

ϕ−=

dt

diM

dt

dNe 2

2121

121 −=−= ϕ

2

21121 di

dNM

ϕ−=

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 that the two coils are magnetically isolated. Suppose the two coils have self inductances L1 and L2 and M is the mutual inductance, then the Coefficient of Coupling is

1

1212 φ

φ=K and2

2121 φ

φ=K

2112 MMM ==

2

211

1

122 di

dN

di

dNM

φφ ×=

2

2

1

121

)()(

di

Kd

di

KdNN

φφ ×=

2

22

1

11

2

di

dN

di

dNK

φφ ×=

21LL

MK =