NCERT Textbook - Alternating Current Class 12 Notes | EduRev

Physics Class 12

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Class 12 : NCERT Textbook - Alternating Current Class 12 Notes | EduRev

 Page 1


7.1  INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources. These currents do not change direction with time. But voltages
and currents that vary with time are very common. The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time. Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)*. Today, most of the electrical devices we use require ac voltage.
This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current. The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers. Further, electrical energy can also be transmitted
economically over long distances. AC circuits exhibit characteristics which
are exploited in many devices of daily use. For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter.
Chapter Seven
ALTERNATING
CURRENT
* The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current. Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use. Further, voltage – another phrase commonly
used means potential difference between two points.
Page 2


7.1  INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources. These currents do not change direction with time. But voltages
and currents that vary with time are very common. The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time. Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)*. Today, most of the electrical devices we use require ac voltage.
This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current. The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers. Further, electrical energy can also be transmitted
economically over long distances. AC circuits exhibit characteristics which
are exploited in many devices of daily use. For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter.
Chapter Seven
ALTERNATING
CURRENT
* The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current. Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use. Further, voltage – another phrase commonly
used means potential difference between two points.
Physics
234
NICOLA TESLA (1836 – 1943)
Nicola Tesla  (1836 –
1943) Yugoslov scientist,
inventor and genius. He
conceived the idea of the
rotating magnetic field,
which is the basis of
practically all alternating
current machinery, and
which helped usher in the
age of electric power. He
also invented among other
things the induction motor,
the polyphase system of ac
power, and the high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment.
The SI unit of magnetic field
is named in his honour.
7.2  AC VOLTAGE APPLIED TO A RESISTOR
Figure 7.1 shows a resistor connected to a source e of
ac voltage. The symbol for an ac source in a circuit
diagram is 
~
. We consider a source which produces
sinusoidally varying potential difference across its
terminals. Let this potential difference, also called ac
voltage, be given by
sin
m
v v t ? = (7.1)
where v
m
 is the amplitude of the oscillating potential
difference and ? is its angular frequency.
To find the value of current through the resistor, we
apply Kirchhoff’s loop rule ( ) 0 e =
?
t , to the circuit
shown in Fig. 7.1 to get
= sin
m
v t i R ?
or   
sin
m
v
i t
R
? =
Since R is a constant, we can write this equation as
sin
m
i i t ? = (7.2)
where the current amplitude i
m
 is given by
m
m
v
i
R
=
(7.3)
Equation (7.3) is just Ohm’s law which for resistors works
equally well for both ac and dc voltages. The voltage across a
pure resistor and the current through it, given by Eqs. (7.1) and
(7.2) are plotted as a function of time in Fig. 7.2. Note, in
particular that both v and i reach zero, minimum and maximum
values at the same time. Clearly, the voltage and current are in
phase with each other.
We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle. Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero. The fact that the average current is zero, however, does
FIGURE 7.1  AC voltage applied to a resistor.
FIGURE 7.2 In a pure
resistor, the voltage and
current are in phase. The
minima, zero and maxima
occur at the same
respective times.
Page 3


7.1  INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources. These currents do not change direction with time. But voltages
and currents that vary with time are very common. The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time. Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)*. Today, most of the electrical devices we use require ac voltage.
This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current. The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers. Further, electrical energy can also be transmitted
economically over long distances. AC circuits exhibit characteristics which
are exploited in many devices of daily use. For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter.
Chapter Seven
ALTERNATING
CURRENT
* The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current. Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use. Further, voltage – another phrase commonly
used means potential difference between two points.
Physics
234
NICOLA TESLA (1836 – 1943)
Nicola Tesla  (1836 –
1943) Yugoslov scientist,
inventor and genius. He
conceived the idea of the
rotating magnetic field,
which is the basis of
practically all alternating
current machinery, and
which helped usher in the
age of electric power. He
also invented among other
things the induction motor,
the polyphase system of ac
power, and the high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment.
The SI unit of magnetic field
is named in his honour.
7.2  AC VOLTAGE APPLIED TO A RESISTOR
Figure 7.1 shows a resistor connected to a source e of
ac voltage. The symbol for an ac source in a circuit
diagram is 
~
. We consider a source which produces
sinusoidally varying potential difference across its
terminals. Let this potential difference, also called ac
voltage, be given by
sin
m
v v t ? = (7.1)
where v
m
 is the amplitude of the oscillating potential
difference and ? is its angular frequency.
To find the value of current through the resistor, we
apply Kirchhoff’s loop rule ( ) 0 e =
?
t , to the circuit
shown in Fig. 7.1 to get
= sin
m
v t i R ?
or   
sin
m
v
i t
R
? =
Since R is a constant, we can write this equation as
sin
m
i i t ? = (7.2)
where the current amplitude i
m
 is given by
m
m
v
i
R
=
(7.3)
Equation (7.3) is just Ohm’s law which for resistors works
equally well for both ac and dc voltages. The voltage across a
pure resistor and the current through it, given by Eqs. (7.1) and
(7.2) are plotted as a function of time in Fig. 7.2. Note, in
particular that both v and i reach zero, minimum and maximum
values at the same time. Clearly, the voltage and current are in
phase with each other.
We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle. Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero. The fact that the average current is zero, however, does
FIGURE 7.1  AC voltage applied to a resistor.
FIGURE 7.2 In a pure
resistor, the voltage and
current are in phase. The
minima, zero and maxima
occur at the same
respective times.
Alternating Current
235
GEORGE WESTINGHOUSE (1846 – 1914)
George Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct current. Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current. Westinghouse
was convinced that the
technology of alternating
current was the key to
the electrical future.
He founded the famous
Company named after him
and enlisted the services
of Nicola Tesla and
other inventors in the
development of alternating
current motors and
apparatus for the
transmission of high
tension current, pioneering
in large scale lighting.
not mean that the average power consumed is zero and
that there is no dissipation of electrical energy. As you
know, Joule heating is given by i
2
R and depends on i
2
(which is always positive whether i is positive or negative)
and not on i. Thus, there is Joule heating and
dissipation of electrical energy when an
ac current passes through a resistor.
The instantaneous power dissipated in the resistor is
2 2 2
sin
m
p i R i R t ? = = (7.4)
The average value of p over a cycle is*
2 2 2
sin
m
p i R i R t ? = < > = < >  [7.5(a)]
where the bar over a letter(here, p) denotes its average
value and <......> denotes taking average of the quantity
inside the bracket. Since,  i
2
m 
and R are constants,
2 2
sin
m
p i R t ? = < > [7.5(b)]
Using the trigonometric identity, sin
2
 ?t =
1/2 (1– cos 2?t ), we have < sin
2
 ?t > = (1/2) (1– < cos 2?t >)
and since < cos2?t > = 0**, we have,
2
1
sin
2
t ? < > =
Thus,
2
1
2
m
p i R =
[7.5(c)]
To express ac power in the same form as dc power
(P = I
2
R), a special value of current is defined and used.
It is called, root mean square (rms) or effective current
(Fig. 7.3) and is denoted by I
rms
 or I.
* The average value of a function F (t) over a period T is given by d
0
1
( ) ( )
T
F t F t t
T
=
?
**
[ ]
0
0
sin 2 1 1 1
cos2 cos2 sin 2 0 0
2 2
T T
t
t t dt T
T T T
?
? ? ?
? ?
? ?
< > = = = - =
?
? ?
? ?
FIGURE 7.3 The rms current I is related to the
peak  current i
m
 by I = / 2
m
i = 0.707 i
m
.
Page 4


7.1  INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources. These currents do not change direction with time. But voltages
and currents that vary with time are very common. The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time. Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)*. Today, most of the electrical devices we use require ac voltage.
This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current. The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers. Further, electrical energy can also be transmitted
economically over long distances. AC circuits exhibit characteristics which
are exploited in many devices of daily use. For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter.
Chapter Seven
ALTERNATING
CURRENT
* The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current. Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use. Further, voltage – another phrase commonly
used means potential difference between two points.
Physics
234
NICOLA TESLA (1836 – 1943)
Nicola Tesla  (1836 –
1943) Yugoslov scientist,
inventor and genius. He
conceived the idea of the
rotating magnetic field,
which is the basis of
practically all alternating
current machinery, and
which helped usher in the
age of electric power. He
also invented among other
things the induction motor,
the polyphase system of ac
power, and the high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment.
The SI unit of magnetic field
is named in his honour.
7.2  AC VOLTAGE APPLIED TO A RESISTOR
Figure 7.1 shows a resistor connected to a source e of
ac voltage. The symbol for an ac source in a circuit
diagram is 
~
. We consider a source which produces
sinusoidally varying potential difference across its
terminals. Let this potential difference, also called ac
voltage, be given by
sin
m
v v t ? = (7.1)
where v
m
 is the amplitude of the oscillating potential
difference and ? is its angular frequency.
To find the value of current through the resistor, we
apply Kirchhoff’s loop rule ( ) 0 e =
?
t , to the circuit
shown in Fig. 7.1 to get
= sin
m
v t i R ?
or   
sin
m
v
i t
R
? =
Since R is a constant, we can write this equation as
sin
m
i i t ? = (7.2)
where the current amplitude i
m
 is given by
m
m
v
i
R
=
(7.3)
Equation (7.3) is just Ohm’s law which for resistors works
equally well for both ac and dc voltages. The voltage across a
pure resistor and the current through it, given by Eqs. (7.1) and
(7.2) are plotted as a function of time in Fig. 7.2. Note, in
particular that both v and i reach zero, minimum and maximum
values at the same time. Clearly, the voltage and current are in
phase with each other.
We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle. Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero. The fact that the average current is zero, however, does
FIGURE 7.1  AC voltage applied to a resistor.
FIGURE 7.2 In a pure
resistor, the voltage and
current are in phase. The
minima, zero and maxima
occur at the same
respective times.
Alternating Current
235
GEORGE WESTINGHOUSE (1846 – 1914)
George Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct current. Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current. Westinghouse
was convinced that the
technology of alternating
current was the key to
the electrical future.
He founded the famous
Company named after him
and enlisted the services
of Nicola Tesla and
other inventors in the
development of alternating
current motors and
apparatus for the
transmission of high
tension current, pioneering
in large scale lighting.
not mean that the average power consumed is zero and
that there is no dissipation of electrical energy. As you
know, Joule heating is given by i
2
R and depends on i
2
(which is always positive whether i is positive or negative)
and not on i. Thus, there is Joule heating and
dissipation of electrical energy when an
ac current passes through a resistor.
The instantaneous power dissipated in the resistor is
2 2 2
sin
m
p i R i R t ? = = (7.4)
The average value of p over a cycle is*
2 2 2
sin
m
p i R i R t ? = < > = < >  [7.5(a)]
where the bar over a letter(here, p) denotes its average
value and <......> denotes taking average of the quantity
inside the bracket. Since,  i
2
m 
and R are constants,
2 2
sin
m
p i R t ? = < > [7.5(b)]
Using the trigonometric identity, sin
2
 ?t =
1/2 (1– cos 2?t ), we have < sin
2
 ?t > = (1/2) (1– < cos 2?t >)
and since < cos2?t > = 0**, we have,
2
1
sin
2
t ? < > =
Thus,
2
1
2
m
p i R =
[7.5(c)]
To express ac power in the same form as dc power
(P = I
2
R), a special value of current is defined and used.
It is called, root mean square (rms) or effective current
(Fig. 7.3) and is denoted by I
rms
 or I.
* The average value of a function F (t) over a period T is given by d
0
1
( ) ( )
T
F t F t t
T
=
?
**
[ ]
0
0
sin 2 1 1 1
cos2 cos2 sin 2 0 0
2 2
T T
t
t t dt T
T T T
?
? ? ?
? ?
? ?
< > = = = - =
?
? ?
? ?
FIGURE 7.3 The rms current I is related to the
peak  current i
m
 by I = / 2
m
i = 0.707 i
m
.
Physics
236
It is defined by
2 2
1
2 2
m
m
i
I i i = = =
= 0.707 i
m
(7.6)
In terms of I, the average power, denoted by P is
2 2
1
2
m
p
P i R I R = = =
(7.7)
Similarly, we define the rms voltage or effective voltage by
V = 
2
m
v
 = 0.707 v
m
(7.8)
From Eq. (7.3), we have
v
m
 = i
m
R
or,  
2 2
m m
v i
R =
or,  V = IR (7.9)
Equation (7.9) gives the relation between ac current and ac voltage
and is similar to that in the dc case. This shows the advantage of
introducing the concept of rms values. In terms of rms values, the equation
for power [Eq. (7.7)] and relation between current and voltage in ac circuits
are essentially the same as those for the dc case.
It is customary to measure and specify rms values for ac quantities. For
example, the household line voltage of 220 V is an rms value with a peak
voltage of
v
m
 = 
2
 V =  (1.414)(220 V) = 311 V
In fact, the I or rms current is the equivalent dc current that would
produce the same average power loss as the alternating current. Equation
(7.7) can also be written as
P = V
2
 / R = I V    (since V = I R)
Example 7.1 A light  bulb is rated at 100W for a 220 V supply. Find
(a) the resistance of the bulb; (b) the peak voltage of the source; and
(c) the rms current through the bulb.
Solution
(a) We are given P = 100 W and V = 220 V. The resistance of the
bulb is
( )
2
2
220 V
484
100 W
V
R
P
= = = O
(b) The peak voltage of the source is
V 2 311
m
v V = =
(c) Since, P = I V
100 W
0.450A
220 V
P
I
V
= = =
 EXAMPLE 7.1
Page 5


7.1  INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources. These currents do not change direction with time. But voltages
and currents that vary with time are very common. The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time. Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)*. Today, most of the electrical devices we use require ac voltage.
This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current. The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers. Further, electrical energy can also be transmitted
economically over long distances. AC circuits exhibit characteristics which
are exploited in many devices of daily use. For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter.
Chapter Seven
ALTERNATING
CURRENT
* The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current. Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use. Further, voltage – another phrase commonly
used means potential difference between two points.
Physics
234
NICOLA TESLA (1836 – 1943)
Nicola Tesla  (1836 –
1943) Yugoslov scientist,
inventor and genius. He
conceived the idea of the
rotating magnetic field,
which is the basis of
practically all alternating
current machinery, and
which helped usher in the
age of electric power. He
also invented among other
things the induction motor,
the polyphase system of ac
power, and the high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment.
The SI unit of magnetic field
is named in his honour.
7.2  AC VOLTAGE APPLIED TO A RESISTOR
Figure 7.1 shows a resistor connected to a source e of
ac voltage. The symbol for an ac source in a circuit
diagram is 
~
. We consider a source which produces
sinusoidally varying potential difference across its
terminals. Let this potential difference, also called ac
voltage, be given by
sin
m
v v t ? = (7.1)
where v
m
 is the amplitude of the oscillating potential
difference and ? is its angular frequency.
To find the value of current through the resistor, we
apply Kirchhoff’s loop rule ( ) 0 e =
?
t , to the circuit
shown in Fig. 7.1 to get
= sin
m
v t i R ?
or   
sin
m
v
i t
R
? =
Since R is a constant, we can write this equation as
sin
m
i i t ? = (7.2)
where the current amplitude i
m
 is given by
m
m
v
i
R
=
(7.3)
Equation (7.3) is just Ohm’s law which for resistors works
equally well for both ac and dc voltages. The voltage across a
pure resistor and the current through it, given by Eqs. (7.1) and
(7.2) are plotted as a function of time in Fig. 7.2. Note, in
particular that both v and i reach zero, minimum and maximum
values at the same time. Clearly, the voltage and current are in
phase with each other.
We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle. Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero. The fact that the average current is zero, however, does
FIGURE 7.1  AC voltage applied to a resistor.
FIGURE 7.2 In a pure
resistor, the voltage and
current are in phase. The
minima, zero and maxima
occur at the same
respective times.
Alternating Current
235
GEORGE WESTINGHOUSE (1846 – 1914)
George Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct current. Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current. Westinghouse
was convinced that the
technology of alternating
current was the key to
the electrical future.
He founded the famous
Company named after him
and enlisted the services
of Nicola Tesla and
other inventors in the
development of alternating
current motors and
apparatus for the
transmission of high
tension current, pioneering
in large scale lighting.
not mean that the average power consumed is zero and
that there is no dissipation of electrical energy. As you
know, Joule heating is given by i
2
R and depends on i
2
(which is always positive whether i is positive or negative)
and not on i. Thus, there is Joule heating and
dissipation of electrical energy when an
ac current passes through a resistor.
The instantaneous power dissipated in the resistor is
2 2 2
sin
m
p i R i R t ? = = (7.4)
The average value of p over a cycle is*
2 2 2
sin
m
p i R i R t ? = < > = < >  [7.5(a)]
where the bar over a letter(here, p) denotes its average
value and <......> denotes taking average of the quantity
inside the bracket. Since,  i
2
m 
and R are constants,
2 2
sin
m
p i R t ? = < > [7.5(b)]
Using the trigonometric identity, sin
2
 ?t =
1/2 (1– cos 2?t ), we have < sin
2
 ?t > = (1/2) (1– < cos 2?t >)
and since < cos2?t > = 0**, we have,
2
1
sin
2
t ? < > =
Thus,
2
1
2
m
p i R =
[7.5(c)]
To express ac power in the same form as dc power
(P = I
2
R), a special value of current is defined and used.
It is called, root mean square (rms) or effective current
(Fig. 7.3) and is denoted by I
rms
 or I.
* The average value of a function F (t) over a period T is given by d
0
1
( ) ( )
T
F t F t t
T
=
?
**
[ ]
0
0
sin 2 1 1 1
cos2 cos2 sin 2 0 0
2 2
T T
t
t t dt T
T T T
?
? ? ?
? ?
? ?
< > = = = - =
?
? ?
? ?
FIGURE 7.3 The rms current I is related to the
peak  current i
m
 by I = / 2
m
i = 0.707 i
m
.
Physics
236
It is defined by
2 2
1
2 2
m
m
i
I i i = = =
= 0.707 i
m
(7.6)
In terms of I, the average power, denoted by P is
2 2
1
2
m
p
P i R I R = = =
(7.7)
Similarly, we define the rms voltage or effective voltage by
V = 
2
m
v
 = 0.707 v
m
(7.8)
From Eq. (7.3), we have
v
m
 = i
m
R
or,  
2 2
m m
v i
R =
or,  V = IR (7.9)
Equation (7.9) gives the relation between ac current and ac voltage
and is similar to that in the dc case. This shows the advantage of
introducing the concept of rms values. In terms of rms values, the equation
for power [Eq. (7.7)] and relation between current and voltage in ac circuits
are essentially the same as those for the dc case.
It is customary to measure and specify rms values for ac quantities. For
example, the household line voltage of 220 V is an rms value with a peak
voltage of
v
m
 = 
2
 V =  (1.414)(220 V) = 311 V
In fact, the I or rms current is the equivalent dc current that would
produce the same average power loss as the alternating current. Equation
(7.7) can also be written as
P = V
2
 / R = I V    (since V = I R)
Example 7.1 A light  bulb is rated at 100W for a 220 V supply. Find
(a) the resistance of the bulb; (b) the peak voltage of the source; and
(c) the rms current through the bulb.
Solution
(a) We are given P = 100 W and V = 220 V. The resistance of the
bulb is
( )
2
2
220 V
484
100 W
V
R
P
= = = O
(b) The peak voltage of the source is
V 2 311
m
v V = =
(c) Since, P = I V
100 W
0.450A
220 V
P
I
V
= = =
 EXAMPLE 7.1
Alternating Current
237
7.3 REPRESENTATION OF AC CURRENT AND VOLTAGE
BY ROTATING VECTORS — PHASORS
In the previous section, we learnt that the current through  a resistor is
in phase with the ac voltage. But this is not so in the case  of an inductor,
a capacitor or a combination  of these circuit elements. In order to show
phase relationship between voltage and current
in an ac circuit, we use the notion of phasors.
The analysis of an ac circuit is facilitated by the
use of a phasor diagram. A phasor* is a vector
which rotates about the origin with angular
speed ?, as shown in Fig. 7.4. The vertical
components of phasors V and I represent the
sinusoidally varying quantities v and i. The
magnitudes of phasors V and I represent  the
amplitudes or the peak values v
m
 and i
m
 of these
oscillating quantities. Figure 7.4(a) shows the
voltage and current phasors and their
relationship at time t
1
 for the case of an ac source
connected to a resistor i.e., corresponding to the
circuit shown in Fig. 7.1. The projection of
voltage and current phasors on vertical axis, i.e., v
m
 sin?t and i
m
 sin?t,
respectively represent the value of voltage and current at that instant. As
they rotate with frequency ?, curves in Fig. 7.4(b) are generated.
From Fig. 7.4(a) we see that phasors V and I for the case of a resistor are
in the same direction. This is so for all times. This means that the phase
angle between the voltage and the current is zero.
7.4  AC VOLTAGE APPLIED TO AN INDUCTOR
Figure 7.5 shows an ac source connected to an inductor. Usually,
inductors have appreciable resistance in their windings, but we shall
assume that this inductor has negligible resistance.
Thus, the circuit is a purely inductive ac circuit. Let
the voltage across the source be v = v
m
 sin?t. Using
the Kirchhoff’s loop rule, ( ) 0 t e =
?
, and  since there
is no resistor in the circuit,
d
0
d
i
v L
t
- =
(7.10)
where the second term is the self-induced Faraday
emf in the inductor; and L is the self-inductance of
FIGURE 7.4 (a) A phasor diagram for the
circuit in Fig 7.1. (b) Graph of v and
i versus ?t.
FIGURE 7.5  An ac source
connected to an inductor.
* Though voltage and current in ac circuit are represented by phasors – rotating
vectors, they are not vectors themselves. They are scalar quantities. It so happens
that the amplitudes and phases of harmonically varying scalars combine
mathematically in the same way as do the projections of rotating vectors of
corresponding magnitudes and directions. The rotating vectors that represent
harmonically varying scalar quantities are introduced only to provide us with a
simple way of adding these quantities using a rule that we already know.
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