DC &AC Circuit Analysis

8/9/2019 DC &AC Circuit Analysis

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UNIT-1 DC & AC CIRCUIT ANALYSIS

Introduction to circuit elements:

There are five basic circuit elements based on which the circuits are designed. How

much ever the complexity of the circuit be, the five fundamental elements are the bricks for

that circuit. Those elements are:

1. Resistor2. Capacitor3. Inductor4. Voltage source5. Current source1. Resistor: it opposes the flow of current through it. Resistance is called the physical

property of a resistor by the virtue of which it gives an opposition to the current flow.

Resistance is measured inohms.2. Capacitor: capacitor is a device that stores energy. it has two simple parallel plates with dielectric in between in its construction, each

plate is connected to one end of the supply. Once the supply is turned on, charges of

opposite polarity appear on each plate and thus it stores energy.

Capacitor has a special property that it doesnt allow any sudden changes in voltageacross it.

Capacitance is measured in farads.3. Inductor: An inductor in a simple curl of windings that doesnt allow sudden

changes in current.

Inductance of an inductor is measured in henry.

The above mentioned three elementsresistor , inductorand

capacitor are called passive elements.

4. Voltage source: this serves as a source to create potential difference between two

points. It has two types of sources:


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1. Dc voltage source

2. Ac voltage source

5.Current source:a current source is a source that supplies the current directly.

Current source andvoltage source are called active elements.

Now that we are introduced to the above elements , a combination of the elements

is called a CIRCUIT.

The following parameters form the basic parts of a circuit:

1. Node :a point in a circuit from which two or more circuit elements diverge.Two points between which there is no circuit element cannot be a node.

2. Path : the distance traversed between two nodes is called a path 3. Loop :a closed path is called a loop.4. Branch : a path between two nodes is called the branch of the network.

We have two laws describing the current and voltage across any particular branch.

1. Kirchhoff s current law (KCL) 2. Kirchhoff s voltage law (KVL)


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1. Kirchhoff s current law (KCL):KCL states that the algebraic sum of currents entering / leaving a node is zero. This

law can be explained as follows.

Consider a four way junction. The sum of vehicles entering the signal and leaving the signal

is zero.Because whatever is the number of vehicals entering the signal,all those vehicals will

defnitely leave the junction stapvatingthere.Hence the no.ofvehicals entering is equal to the

number of vehicals leaving.

Simillarly in the network, and are the currents entering, and are leavingcurrents.

+ = +

=1= 0

KIRCHOFFS VOLTAGE LAW:

KVL states that the algebriac sum of voltages arround every

closed path is zero.

For example consider the circuit given below:

let a charge of I be cariied from A to B.there are two different paths from A to B.One isdone from A through element 1 from A to B.The work done is 1 volts.

If other path i.e A element to B is followed,there the work done willbe is

2-

3volts.


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The workdone is however independent of the path

1 = 2 = 3 While applying kvl, simple way of writing down the equation is,while moving

mentally in clockwise direction,write the voltage of each element whose (+1) isentered and write the (-ve) o every voltage first met at theve sign.

is 1 + 2 3 = 01 = 2 3

ALTERNATING CURRENT WAVEFORMS:

The waveform of the most common supply in one domestic use is alternating current and itsshape is sinisoidal.

This function describes the voltage available at household

electrical sockets as well as the voltage of power lines connected to residental and industrial

areas.

The time period of the waveform is given by

WT=2

= 2f


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LAGGING AND LEADING WAVEFORMS:

In some cases the actual waveforms either falls back the original waveform or same times

leads the original waveform by some angle,say .

If the waveform falls back the original waveform,we say that it is

lagging.If the waveform leads the original waveform we say that it is leading.

If leading,we represent if as sin(t-)and if lagging,we represent it as sin(t+).

RMS VALUE:

Let us define the term RMS for a general periodic waveform.

We usually say that the supply available at one domestic level is 230V and

50Hz.what is this 230V? It is not the instantaneous value,since the voltage is not constant.It is

also not the average value,because when the waveform is observed on an oscilloscope,the

average value is also found to be more than 230V.

Hence we define 230V as the effective value which is a measure of the

effectiveness of a voltage source in elelivering the power to a sensitive load.

CALCULATION:The effective values of any periodic current is equal to the value of direct

current flowing through an R-ohm resistor,devices same averge power to the resistor as doesthe periodic current.

P=1

20 R.dt=

20 dt

=1 20

The above expression is a generalized expression.Now let us consider for a sinusiodalwaveform.


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(i) FOR A CURRENT WAVEFORM:I(t)= cos (t+)

T=2

=1

+ . 0 = 12 + 12 cos2 + 220 = 14 []02=

2

(ii)

FOR VOLTAGE WAVEFORMS:The generalized voltage equation is:

V(t)=sin(t)T=2

=1 2.0 =1

2 220 1cos 2

2 .

=12 .

22

. []02 =1

2 .22

. 2 =2

AVERAGE VALUES OF A PERIODIC WAVEFORM:

=1 0 sin wt=

1

2 0 sin wt=

2 [ cos ]02

=2 [-cos 2-cos 0]

=

2 [1-1] = 0


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The average value of any periodic waveform is always zero for bothcurrent as well as voltage waveforms.

SIMPLE RLC CIRCUITS:

We have seen the basic elements along with two fundamental sources. Now we will seehow to construct simple circuits using these five elements i.e R,L,C, current source and

voltage source. we also simultaneously drive the equivalent impedance, current and

voltage for the given circuit.

1.SERIES RL CIRCUIT:

Let the resistance be represented by R and inductance by L. The series

RL circuit is supplied by a current source of magnitude i.

The impedance (Z) = R + jwL

=2 + ()2


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=I(R- .1 )

=I.2 + ( 1 )2


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i = (1)2 + ( 1 )2 cos(wt-)

5.PARALLEL RC CIRCUIT:

= 1( 1 ) = 1(wrc)

i = (1)2 + ()2 sin(wt + )

6.RLC IN PARALLEL:

= 1( 1

1

)

Y = (1)2 ( 1 )2

I = sinwt

= Y. .sin wt

= (1)2 ( 1 )2 .sin(wt+1(wc- 1 ))


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Why have we considered= 2fL and = Voltage across an inductor is given by

= L

= L

( i sin wt)

= L.i.w.sinwt

=

(cos ) .

( )

Comparing the above equation with ohms law, we see that resistance offered

by an inductance is R = WL = Similarly the voltage across capacitor is given by

=

1

= 1 . (cos ) = . R

= 1 =POWER FACTOR:

Power factor is defined as the ratio of real power to apparent power. let us first define the

real power and apparent power.

Let the voltage equation is given by

V = cos (wt + )And current be given by

I = cos (wt+)

What is the cos component in the above equations? It is called the power factor.

For example if we need to supply a 33kw of power to an industry we need to send anextra power which is above 33kw. Because most of the loads in an industry are inductiveloads and a few loads are capacitive loads. Irrespective of the loads, whether they are

inductive or capacitive they draw the power which is lagging or leading but definetly notin phase with the load voltage. This creates a cos component which reduces the actual

power sent. Thus this necissiates that we send some extra power which is greater than thepower actually demanded so the actual demand is met.

power is given by the product of voltage and current

p =


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= 2 + 2 +

P =1

2 ( )

P =

cos (

)

In the above expression p is the real power and is called the apparent power.cos( ) is called the power factor

=

DCMACHINES

Machines have become a part and parcel of our life. Machines consists of both generators

and motors. So basically we can classify machines in to two types based on the supply:

1. DC Machines and 2.AC MachinesAs the name itself indicates a machine which demands direct current is

called a DC machine. We are going to confine our studies only to DC generators in the unit

and the later half i.e., DC motors will be dealt in the next unit.

DC GENERATORS:

In general, a generator is the one that consumes mechanical input and gives out a

electrical output.

Before we get in to details, let us have an elaborate explanation of the above

statement.

A basic principle or law behind the operation of any machine is FARADAYSLAW

OF ELECTRO MAGNETIC INDUCTION. It states like this:


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whenever a conductor is placed in the magnetic field and there is a relative motion between

the conductor and the magnetic field, then an E.M.F. is induced across the conductor.Thus, in

order to induce an E.M.F. across the generator we need three basic things:

(i) A set of conductors(ii) Magnetic field(iii) Relative motion between a set of conductors and the magnetic field.

In the DC generators , we create a magnetic field which remains stationery and the

conductors are made to rotate inside the field. The force driving the conductors is mechanical

force and thus a relative motion is created between the magnetic field and the set of

conductors. And as per faradays law of electromagnetic induction , an E.M.F. is induced

across the set of conductors.

Thus a generator is said to convert mechanical input in to electrical output. The E.M.F. is

called Dynamically induced E.M.F.

Theoritical construction:

Let us now analyse the principle of operation of a DC generator from the simple theoretical

construction as shown in the figure.


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In the above figure the required magnetic field is obtained by employing magnets and the

field is always directed from North pole to South pole. A set of conductors is made to cut

the field with the help of prime mover.

When the conductor AB is under North pole and CD is under South pole, the direction of

current is as shown in the figure. This direction of current is obtained by flemings r ight hand

rule. Now the direction of current, through the load is from L to M.

Now consider the figure shown below:

When the armature is rotated in the clock wise direction conductor CD comes under north

pole and AB comes under south pole. But the current direction through the conductor under

north pole and south pole doesnt change. Hence the current through the load remains

unchanged. The current direction is clearly shown in the figure.

One Important point is to be noted here. The E.M.F. induced across the

conductors varies alternatively but the final current through the load remains unchanged. That

is the E.M.F. induced is always A.C., but with the help of commutator is converted to D.C.

This is illustrated in the graph shown below.


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Whatever the simple construction which we have seen theoretically, practically its not that

simple. Let us now see the constructional details.

CONSTRUCTIONAL DETAILS:

1. Magnetic Frame (or) Yoke2. Pole core and pole shoe3. Armature core4. Armature windings5. Commutator6. Brushes

1. MAGNETIC FRAME (or) YOKE:

- Yoke is an outer covering for all the machine parts.

it is usually made of cast Iron for small machines

it is made with cast steel for larger machines.

it is laminated to reduce eddy current losses

Function of yoke:

it acts as a protecting cover for entire machine

Gives mechanical support for field poles. it provides the return path for the magnetic flux.

2.POLE CORE AND POLE SHOES:

Both pole core and pole shoes are made with cast steel or silicon steel. No laminations

are required. But for large machines they are laminated to reduce the eddy current losses.


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Functions of pole core:

1. it accommodates field windings, and behaves as a magnetic pole when the field isexcited with a D.C.

Functions of pole shoe:

1. gives the mechanical support for field winding.2. Reduces the magnetic reluctance. Hence we can get more flux for a given field

current.

3. It also helps in distributing the flux uniformly in the airgap.3.ARMATURE CORE:

Armature core is preferred with silicon steel, laminated to reduce the eddy current losses.

Functioning of armature loss:

1. they accommodate armature winding.2. It offers low reluctance for the magnetic flux.3. 3. In d.c machines armature slots are sleewed in order to reduce vibrations.

4.ARMATURE WINDING:

Depending upon the application we employ two types of windings:(i) lap winding(ii) wave winding

Lap winding is employed for the applications that require less voltage and large current. It

is as shown below.


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The finishing end of one coil is connected to the beginning of the other coil under the

same pole.

WAVE WINDING:

the finishing end of one coil is connected to the beginning end of the other coil under the

adjacent similar pole.

the winding goes on like a wave as shown below.

This type of winding is employed for low current and large voltage applications.

5.COMMUTATOR:

Commutator drawn with hard drawn copper to reduce the wear and tear

the number of commutator segments is always equal to the number of coils.


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6.BRUSHES:

Function:

The major function of the brushers is to collect the armature current or give the armature

current through the commutator segments to the conductors

- the brushes are usually made with copper or Carbon for small machines.

they are made up of electro graphite for normal ratings and large machines.

they are made up of copper graphite for large currents and low voltage machines.

for small machines carbon brushes are preferred because it has more resistance , therefore

improve the commutation.

Carbon has more thermal stability and hence it can even withstand high temperatures.

Carbon is self lubricating in property. Hence less friction between commutator and

bushes; thus reducing the brush.

E.M.F. EQUATION OF A GENERATOR:

flux per pole in webers

zTotal number of Armature conductors.

pNumber of poles.

NSpeed of the armature

ANo. of parallel paths.

Before getting in to derivation it is necessary to know what a parallel path is.

Parallel path: If the number of conductors available in one path / under one pole carry

current in series, so with respect to the conductors under same pole they are said to be in

series and with respect to the conductors under other poles they are said to be in parallel.

No. of parallel paths depends upon the winding type: i.e.,

For lap winding: No. of poles = Number of parallel paths


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And for wave winding: number of parallel paths=2

Now with all this basic information we will try to derive an expression for the E.M.F. of a

generator.

The E.M.F. induced across the armature conductors is always directly proportional to the

rate of change of flux.

In one rotation each conductor cuts the magnetic flux p times.

flux cut by one conductor in One revolution = P x webersBut it is given that the armature is going to be rotated at a speed of N revolutions per

minute. Hence the number of revolutions made per second = N/60 r.p.s.

flux cut by the conductor per second = flux per revolution x No. of revolutions= p x N/60

E= PN/60 volts

But the number of conductors available For one parallel path = Z/A

E=60

SAMPLE PROBLEM:

1. Calculate the voltage induced in the armature winding of a 4 pole lap wound DCmachine having 728 active conductors and running at 1800 rpm. The flux per pole is

30mwb.

A: = 30mwb

Z=728

N=1800 r.p.m

P=4

No. of parallel paths A = 4


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E =60

=0.03728 1800

60

4

4

= 655.2v

2.Calculate the E.M.F. induced in case of wave wound machine.

A: No. of parallel paths = A= 2

E =0.03728 1800

60

4

2

=1310.4 volts

TYPES OF GENERATORS BASED ON EXCITATION:

We have seen the division of machines based on the supply given. Now let us classify the

generators based on the type of excitation given.

Excitation is nothing but the way of generating the flux. In our machines we usually

employ electromagnets. So the windings given for generating field are called field windingsand current given to these windings to generate field is what is called as excitation.

Basically there are two types of generators.

1. Separately excited D.C. generators2. Self excited D.C. generator

1.SEPARATELY EXCITED DC GENERATOR: A DC generator whose field winding is

excited from a separate DC source, such as a battery, the generator is called a separately

excited DC generator.

The circuit diagram and the relevant equations are as shown.


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Armature current

load current

excitation voltage Generator E.M.F.V Terminal voltage

Let = =

V = Power developed = 4 = .I

Power delivered= =V.ISELF EXCITED DC GENERATOR:

A DC generator whose field winding is excited by the current supplied by generator itself

is called a self excited DC generator.

Usually the voltage of the self excited generators does not start from origin. i.e., even at

zero field currents we have same amount of Voltage is due to residual magnetism. The

magnetism whatever is retained by the material from the previous excitations. It circulates

same current in the field winding during the next start. Hence this develops a slight E.M.F.

and this E.M.F. and this E.M.F. in turn increases the field current. Hence the E.M.F. in a self

excited generator is gradually developed.


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There are three types of self excited generators.

1. series wound generator2. shunt wound generator3. compound wound

1.Series wound Generator:

If the field winding placed in series with the armature, then it is called a series wound

generator.

A series field winding must always carry fuel load current through it. Hence it must offer

very less resistance to the flow of current

thus we give only less number of turns in the series field winding.

= =

V = + = =


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CHARACTERSTICS OF SERIES WOUND GENERATORS:

We have two major characteristics to be studied for every machine.

1.open circuit characteristics

2. load characteristics

1.open circuit characteristics:

open circuit characteristics is a graph between the field current on x axis and E.M.F.

generated on Yaxis

2. Load characteristics:

load characteristics is a graph between load current taken on X axis and E.M.F. induced on

the terminals taken on yaxis

In series wound generators the field current and the load current both are same because

they are connected in series. Hence we will plot a graph on the same axis:

The graph is shown below:

load characteristics is once again analyzed in two different forms.

(a) Internal characteristics: It is a graph between two parameters that are internal to themachine.,i.e, Ia taken on xaxis and induced E.M.F. taken on Y- axis.


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the deviation of internal characteristics from that of ideal characteristics is due to a drop

known as armature reaction drop.

(b) External characteristics: It is a graph between the parameters that are completelyexternal to the circuit. i.e., between load current (x- axis) and terminal voltage(v) on

Yaxis

The further deviation of these external characteristics from the ideal characteristics when

compared to internal characteristics is only due to a drop called as armature drop (or) drop

From the above characteristics we calculate a very important factor called as CRITICAL

LOAD RESISTANCE. Def; it is the load resistance beyond which the generator will fail to

excite. i.e.., if load resistance > critical load resistance the generator will not produce any

e.m.f.

Procedure to determine C.L.R:

1.draw a perfect tangent to the external characteristics.

2.determine the shape that gives a value of resistance. That itself is critical load resistance

Hence care should be taken that the series generator load resistance should always be less

than critical load resistance.

2.shunt wound DC generator:

In this type of generator the field winding is connected in parallel with the armature.


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the effective power of the generator is proportional to current delivered to the external

load circuit. hence the field current must be less.

In order to have less field current, we increase the resistance of the field winding by

increasing the number of turns in it.

The resistance is usually as high as 100. = +

=

= + V = = =

Characteristics of DC shunt generator:

1.open circuit characteristics:

From the above curve we calculate the critical field resistance.

It is the field resistance at which the generator just excites.

The procedure to calculate the value is same as that stated for series generator.


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DC &AC Circuit Analysis

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