Chapter 2 AC Circuits

7/25/2019 Chapter 2 AC Circuits

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Chapter 2

Alternating-Current Circuits


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Definition of Alternating Quantity

An alternating quantity changes

continuously in magnitude and alternates in

direction at regular intervals of time.


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Advantages of AC System Over DCSystem

1. AC voltages can be efficiently stepped up/down

using transformer.2. AC motors are cheaper and simpler in

construction than DC motors.

3. Switchgear for AC system is simpler than DCsystem.


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Generation of Single Phase EMF

Consider a rectangular coil of N turns placed in a uniform magnetic

field as shown in the figure. The coil is rotating in the anticlockwise

direction at an uniform angular velocity of rad/sec.


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The maximum flux linking the coil is in the downward direction as shown

in the figure. This flux can be divided into two components, one

component acting along the plane of the coil maxsintand another

component acting perpendicular to the plane of the coil maxcost.


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The component of flux acting along the plane of the coil does not

induce any flux in the coil. Only the component acting perpendicular

to the plane of the coil i.e. maxcostinduces an emf in the coil

Hence the emf induced in the coil is a sinusoidal emf. This

will induce a sinusoidal current in the circuit given by


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Average value of a sine wave

average value over one (or more) cycles is

clearly zero

however, it is often useful to know the

average magnitude of the waveformindependent of its polarity

we can think of this as

the average value over

half a cycle or as the average value

of the rectified signal

pp

p

pav

VV

V

VV

637.02

cos

dsin1

0

0


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Average value of a sine wave


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r.m.s. value of a sine wave

the instantaneous power (p) in a resistor is

given by

therefore the average power is given by

where is the mean-square voltage

R

vp

2

2v

R

v

R

v

avP

2]ofmean)(oraverage[ 2


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While the mean-square voltage is useful,

more often we use the square root of this

quantity, namely the root-mean-square

voltage Vrms

where Vrms =

we can also define Irms=

it is relatively easy to show that (see text for

analysis)

2v

2i

pp

rms VVV 707.02

1p

prms III 707.0

2

1


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r.m.s. values are useful because their

relationship to average power is similar to

the corresponding DC values

rmsrmsavIVP

RIPrmsav

2

R

VP rms

av

2


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Form factor

for any waveformthe form factor is defined

as

for a sine wavethis gives

valueaveragevaluer.m.s.factorForm

11.10.637

0.707factorForm

pVpV


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Peak factor

for any waveformthe peak factor is defined

as

for a sine wavethis gives

valuer.m.s.valuepeakfactorPeak

414.10.707

factorPeak pV

pV


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Alternating Voltages and Currents

Wall sockets provide current and voltage that

vary sinusoidally with time.

Here is a simple ac circuit:


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The voltage as a function of time is:


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Since this circuit has only a resistor, the

current is given by:

Here, the current andvoltage have peaks

at the same time

they are in phase.


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In order to visualize the phase relationships

between the current and voltage in ac circuits,

we define phasors vectors whose length is the

maximum voltage or current, and which rotate

around an origin with the angular speed of the

oscillating current.The instantaneous

value of the voltage or

current represented

by the phasor is its

projection on the y

axis.


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The voltage and current in an ac circuit both

average to zero, making the average useless indescribing their behavior.

We use instead the root mean square (rms); we

square the value, find the mean value, and then

take the square root:

120 volts is the rms value of household ac.


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By calculating the power and finding theaverage, we see that:


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Electrical fires can be started by improper or

damaged wiring because of the heat caused by a

too-large current or resistance.

A fuse is designed to be the hottest point in the

circuit if the current is too high, the fuse melts.

A circuit breaker is similar, except that it is a

bimetallic strip that bends enough to break the

connection when it becomes too hot. When itcools, it can be reset.


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A ground fault circuit interrupter can cut off thecurrent in a short circuit within a millisecond.


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Capacitors in AC Circuits

How is the rms current in the capacitor

related to its capacitance and to the

frequency? The answer, which requirescalculus to derive:


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In analogy with resistance, we write:


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The voltage andcurrent in a capacitor

are not in phase. The

voltage lags by 90.


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RCCircuits

In an RCcircuit, the current across the resistor

and the current across the capacitor are not in

phase. This means that the maximum current isnot the sum of the maximum resistor current

and the maximum capacitor current; they do

not peak at the same time.


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RCCircuits

This phasor diagram

illustrates the phaserelationships. The

voltages across the

capacitor and across the

resistor are at 90in thediagram; if they are

added as vectors, we

find the maximum.


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RCCircuits

This has the exact same form as V= I Rif wedefine the impedance, Z:


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RCCircuits

There is a phase anglebetween the voltage and

the current, as seen in the

diagram.


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RCCircuits

The power in the circuit is given by:

Because of this, the factor cos is calledthe power factor.


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Inductors in AC Circuits

Just as with capacitance, we can define

inductive reactance:


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The voltage across an inductor leads the

current by 90.


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The power factor for an RL circuit is:

Currents in resistors,

capacitors, and

inductors as afunction of

frequency:


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RLCCircuits

A phasor diagram is a useful way to analyze an

RLCcircuit.


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RLCCircuits

The phase angle for an RLCcircuit is:

If XL=XC, the phase angle is zero, and the

voltage and current are in phase.

The power factor:


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RLCCircuits

At high frequencies, the capacitive reactance is

very small, while the inductive reactance is verylarge. The opposite is true at low frequencies.

Resonance in Electrical Circuits


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Resonance in Electrical CircuitsIf a charged capacitor is connected across an

inductor, the system will oscillate indefinitely in

the absence of resistance.



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The rms voltages across the capacitor and

inductor must be the same; therefore, we cancalculate the resonant frequency.



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In an RLCcircuit with an ac power source, the

impedance is a minimum at the resonantfrequency:



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The smaller the resistance, the larger the

resonant current:


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THREE PHASE AC CIRCUITS

A three phase supply is a set of three alternating quantities displaced from each other by an

angle of120. A three phase voltage is shown in the figure. It consists of three phases-

phase A, phase B and phase C. Phase A waveform starts at 0. Phase B waveform stars at

120and phase C waveform at240.


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The three phase voltage can be represented by a set of three equations as shown below.

The sum of the three phase voltages at any instant is equal

to zero.

The phasor representation of three phase voltages is as

shown.

The phase A voltage is taken as the

reference and is drawn along the x-axis.

The phase B voltage lags behind the

phase A voltage by 120

. The phase C

voltage lags behind the phase A voltage

by240

and phase B voltage by 120

.


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Generation of Three Phase

Voltage

Three Phase voltage can be generated by placing three rectangular coils

displaced in space by 120

in a uniform magnetic field. When these coilsrotate with a uniform angular velocity of rad/sec, a sinusoidal emf

displaced by 120 is induced in these coils.


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Necessity and advantages of three phase

systems:-

3 power has a constant magnitude whereas 1 power pulsates from zero to peak

value at twice the supply frequency

A 3 system can set up a rotating magnetic field in stationary windings. This is not

possible with a 1 supply.

For the same rating 3 machines are smaller, simpler in construction and have better

operating characteristics than 1 machines

To transmit the same amount of power over a fixed distance at a given voltage, the

3 system requires only 3/4th the weight of copper that is required by the 1 system

The voltage regulation of a 3 transmission line is better than that of 1 line


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Star Connected Load

A balanced star connected load is shown in the figure. A phase voltage is defined as

voltage across any phase of the three phase load. The phase voltages shown in figure areEA, EBand EC. A line voltage is defined as the voltage between any two lines. The line

voltages shown in the figure are EAB, EBC and ECA. The line currents are IA, IBand IC. For

a star connected load, the phase currents are same as the line currents.


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Using Kirchhoffs voltage law, the line voltages can be written in terms of the phase

voltages as shown below.

The phasor diagram shows the three phase voltages and the line voltage EABdrawn from EAand EB phasors. The phasor for current IAis also shown. It is

assumed that the load is inductive.


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From the phasor diagram we see that the line voltage EABleads the phase voltage EAby

30. The magnitude of the two voltages can be related as follows.

Hence for a balanced star connected load we can make the following

conclusions.

Line voltage leads phase voltage by 30


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Delta Connected Load

A balanced delta connected load is shown in the figure. The phase currents IAB, IBC and ICA.The line currents are IA, IBand IC. For a delta connected load, the phase voltages are same

as the line voltages given by EAB, EBC and ECA.


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Using Kirchhoffs current law, the line currents can be written in terms of the phase

currents as shown below.

From the phasor diagram we see that theline current IAlags behind the phase phase

current IAB by 30. The magnitude of the

two currents can be related as follows.

Hence for a balanced delta connected load we can make the following

conclusions.

Line current lags behind phase current

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Chapter 2 AC Circuits