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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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Alternating Voltages and Currents
The voltage as a function of time is:
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Alternating Voltages and Currents
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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Alternating Voltages and Currents
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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Alternating Voltages and Currents
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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Alternating Voltages and Currents
By calculating the power and finding theaverage, we see that:
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Alternating Voltages and Currents
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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Alternating Voltages and Currents
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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Capacitors in AC Circuits
In analogy with resistance, we write:
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Capacitors in AC Circuits
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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Inductors in AC Circuits
The voltage across an inductor leads the
current by 90.
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Inductors in AC Circuits
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.
Resonance in Electrical Circuits
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Resonance in Electrical Circuits
The rms voltages across the capacitor and
inductor must be the same; therefore, we cancalculate the resonant frequency.
Resonance in Electrical Circuits
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Resonance in Electrical Circuits
In an RLCcircuit with an ac power source, the
impedance is a minimum at the resonantfrequency:
Resonance in Electrical Circuits
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Resonance in Electrical Circuits
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