Alternating Current | Physics Optional Notes for UPSC PDF Download

 Page 1


 
ALTERNATING CURRENT 
ALTERNATING CURRENT AND VOLTAGE 
When we say voltage or current is alternating, it means that it keeps changing in strength 
continuously and switches direction regularly over time. This kind of change can be shown 
using a sine curve or cosine curve, which are wavy lines that go up and down in a smooth 
pattern. So, an alternating voltage or current goes from high to low and then back again, 
repeating this cycle over and over in a predictable way. 
?? = ?? 0
s i n ? ???? ? or ? ?? = ?? 0
c os ? ???? 
Where ? I = Instantaneous value of current at time t , ? I
0
= Amplitude or peak value 
?? = Angular frequency ?? =
2 ?? ?? = 2 ???? ? ?? = time period ? f = frequency  
 
1 as a sine function of ?? 
 
I as a cosine function of ?? 
AMPLITUDE OF AC: 
The maximum value of current in either direction is called peak value or the amplitude of 
current. It is represented by ?? 0
. Peak to peak value = 2 ?? 0
 
PERIODIC TIME: 
The time taken by alternating current to complete one cycle of variation is called periodic 
time or time period of the current. 
FREQUENCY 
Page 2


 
ALTERNATING CURRENT 
ALTERNATING CURRENT AND VOLTAGE 
When we say voltage or current is alternating, it means that it keeps changing in strength 
continuously and switches direction regularly over time. This kind of change can be shown 
using a sine curve or cosine curve, which are wavy lines that go up and down in a smooth 
pattern. So, an alternating voltage or current goes from high to low and then back again, 
repeating this cycle over and over in a predictable way. 
?? = ?? 0
s i n ? ???? ? or ? ?? = ?? 0
c os ? ???? 
Where ? I = Instantaneous value of current at time t , ? I
0
= Amplitude or peak value 
?? = Angular frequency ?? =
2 ?? ?? = 2 ???? ? ?? = time period ? f = frequency  
 
1 as a sine function of ?? 
 
I as a cosine function of ?? 
AMPLITUDE OF AC: 
The maximum value of current in either direction is called peak value or the amplitude of 
current. It is represented by ?? 0
. Peak to peak value = 2 ?? 0
 
PERIODIC TIME: 
The time taken by alternating current to complete one cycle of variation is called periodic 
time or time period of the current. 
FREQUENCY 
The number of cycles completed by an alternating current in one second is called the 
frequency of the current. 
UNIT: cycle/s; (Hz) 
In India: ? ?? = 50 Hz, supply voltage = 220 volt 
In USA: ? f = 60 Hz, supply voltage = 110 volt 
CONDITION REQUIRED FOR CURRENT/ VOLTAGE TO BE ALTERNATING 
? Amplitude is constant 
? Alternate half cycle is positive and half negative 
The alternating current continuously varies in magnitude and periodically reverses its 
direction. 
 
 
 
 
 
AVERAGE VALUE OF A FUNCTION 
? ?? ? =
? ? ?? ?? ?? ? ? ????
 
? ?? ? = Average value of ?? 
Page 3


 
ALTERNATING CURRENT 
ALTERNATING CURRENT AND VOLTAGE 
When we say voltage or current is alternating, it means that it keeps changing in strength 
continuously and switches direction regularly over time. This kind of change can be shown 
using a sine curve or cosine curve, which are wavy lines that go up and down in a smooth 
pattern. So, an alternating voltage or current goes from high to low and then back again, 
repeating this cycle over and over in a predictable way. 
?? = ?? 0
s i n ? ???? ? or ? ?? = ?? 0
c os ? ???? 
Where ? I = Instantaneous value of current at time t , ? I
0
= Amplitude or peak value 
?? = Angular frequency ?? =
2 ?? ?? = 2 ???? ? ?? = time period ? f = frequency  
 
1 as a sine function of ?? 
 
I as a cosine function of ?? 
AMPLITUDE OF AC: 
The maximum value of current in either direction is called peak value or the amplitude of 
current. It is represented by ?? 0
. Peak to peak value = 2 ?? 0
 
PERIODIC TIME: 
The time taken by alternating current to complete one cycle of variation is called periodic 
time or time period of the current. 
FREQUENCY 
The number of cycles completed by an alternating current in one second is called the 
frequency of the current. 
UNIT: cycle/s; (Hz) 
In India: ? ?? = 50 Hz, supply voltage = 220 volt 
In USA: ? f = 60 Hz, supply voltage = 110 volt 
CONDITION REQUIRED FOR CURRENT/ VOLTAGE TO BE ALTERNATING 
? Amplitude is constant 
? Alternate half cycle is positive and half negative 
The alternating current continuously varies in magnitude and periodically reverses its 
direction. 
 
 
 
 
 
AVERAGE VALUE OF A FUNCTION 
? ?? ? =
? ? ?? ?? ?? ? ? ????
 
? ?? ? = Average value of ?? 
 
Example. ?? = t
2
; < ?? > from 0 to 2s ec . 
Solution: ? ?? ? =
?
0
2
? ?? 2
????
?
0
2
? ????
=
2
3
3 × 2
=
4
3
 
Example. ?? = 2 ?? + 3 ; < ?? > from 0 to 4s ec . 
Solution: ? ?? ? =
?
0
4
? ( 2t + 3 ) dt
?
0
4
? dt
=
( t
2
+ 3t )
0
4
4
=
4
2
+ 3 ( 4 )
4
= 4 + 3 = 7 
AVERAGE VALUE OR MEAN VALUE 
The average value of an alternating current (AC) over any half cycle (positive or 
negative) is the direct current (DC) value that would pass through a circuit and deliver the 
same amount of charge in the same time as the AC does during that half cycle. 
average value of current for half cycle ? ?? ? =
?
0
?? / 2
? ???? ?? ?
0
?? / 2
? ????
 
Average value of ?? = ?? 0
s i n ? ???? over the positive half cycle: 
?? ????
=
? ?
?? 2
0
? ?? 0
s i n ? ?????? ?? ? ?
?? 2
0
?????
=
2 ?? 0
????
[ - c os ? ???? ]
0
?? 2
=
2 ?? 0
?? 
? < sin ? ?? > = < sin ? 2 ?? > = 0
? < c os ? ?? > = < c os ? 2 ?? > = 0
? < sin ? ?? c os ? ?? > = 0
? < sin
2
? ?? > = < c os
2
? ?? > =
1
2
 
? ? For symmetric AC, average value over full cycle = 0, 
Average value of sinusoidal ???? 
Full cycle (+ve) half cycle (-ve) half cycle 
0 
2 I
0
?? 
- 2 I
0
?? 
 
Page 4


 
ALTERNATING CURRENT 
ALTERNATING CURRENT AND VOLTAGE 
When we say voltage or current is alternating, it means that it keeps changing in strength 
continuously and switches direction regularly over time. This kind of change can be shown 
using a sine curve or cosine curve, which are wavy lines that go up and down in a smooth 
pattern. So, an alternating voltage or current goes from high to low and then back again, 
repeating this cycle over and over in a predictable way. 
?? = ?? 0
s i n ? ???? ? or ? ?? = ?? 0
c os ? ???? 
Where ? I = Instantaneous value of current at time t , ? I
0
= Amplitude or peak value 
?? = Angular frequency ?? =
2 ?? ?? = 2 ???? ? ?? = time period ? f = frequency  
 
1 as a sine function of ?? 
 
I as a cosine function of ?? 
AMPLITUDE OF AC: 
The maximum value of current in either direction is called peak value or the amplitude of 
current. It is represented by ?? 0
. Peak to peak value = 2 ?? 0
 
PERIODIC TIME: 
The time taken by alternating current to complete one cycle of variation is called periodic 
time or time period of the current. 
FREQUENCY 
The number of cycles completed by an alternating current in one second is called the 
frequency of the current. 
UNIT: cycle/s; (Hz) 
In India: ? ?? = 50 Hz, supply voltage = 220 volt 
In USA: ? f = 60 Hz, supply voltage = 110 volt 
CONDITION REQUIRED FOR CURRENT/ VOLTAGE TO BE ALTERNATING 
? Amplitude is constant 
? Alternate half cycle is positive and half negative 
The alternating current continuously varies in magnitude and periodically reverses its 
direction. 
 
 
 
 
 
AVERAGE VALUE OF A FUNCTION 
? ?? ? =
? ? ?? ?? ?? ? ? ????
 
? ?? ? = Average value of ?? 
 
Example. ?? = t
2
; < ?? > from 0 to 2s ec . 
Solution: ? ?? ? =
?
0
2
? ?? 2
????
?
0
2
? ????
=
2
3
3 × 2
=
4
3
 
Example. ?? = 2 ?? + 3 ; < ?? > from 0 to 4s ec . 
Solution: ? ?? ? =
?
0
4
? ( 2t + 3 ) dt
?
0
4
? dt
=
( t
2
+ 3t )
0
4
4
=
4
2
+ 3 ( 4 )
4
= 4 + 3 = 7 
AVERAGE VALUE OR MEAN VALUE 
The average value of an alternating current (AC) over any half cycle (positive or 
negative) is the direct current (DC) value that would pass through a circuit and deliver the 
same amount of charge in the same time as the AC does during that half cycle. 
average value of current for half cycle ? ?? ? =
?
0
?? / 2
? ???? ?? ?
0
?? / 2
? ????
 
Average value of ?? = ?? 0
s i n ? ???? over the positive half cycle: 
?? ????
=
? ?
?? 2
0
? ?? 0
s i n ? ?????? ?? ? ?
?? 2
0
?????
=
2 ?? 0
????
[ - c os ? ???? ]
0
?? 2
=
2 ?? 0
?? 
? < sin ? ?? > = < sin ? 2 ?? > = 0
? < c os ? ?? > = < c os ? 2 ?? > = 0
? < sin ? ?? c os ? ?? > = 0
? < sin
2
? ?? > = < c os
2
? ?? > =
1
2
 
? ? For symmetric AC, average value over full cycle = 0, 
Average value of sinusoidal ???? 
Full cycle (+ve) half cycle (-ve) half cycle 
0 
2 I
0
?? 
- 2 I
0
?? 
 
As the average value of AC over a complete cycle is zero, it is always defined over a half 
cycle which must be either positive or negative 
MAXIMUM VALUE 
? ?? = ?? sin ? ?? 
? ? I
Max. 
= a 
? ?? = ?? + ?? s i n ? ?? 
? ? ?? Max 
= ?? + ?? ( if ?? and ?? > 0 ) 
? ?? = ?? sin ? ?? + ?? c o s ? ?? 
? ? ?? Max. 
= v ?? 2
+ ?? 2
 
E 
? I
Max 
= a ( a > 0 ) 
Average value over ?? complete cycles or long time: 
(i) ?sin ? ( ???? + ?? ) ? = 0 
(ii) ? c os ? ( ???? + ?? ) ? = 0 
(iii) ? sin
2
? ( ???? + ?? ) ? =
1
2
 
(iv) ? c os
2
? ( ???? + ?? ) ? =
1
2
 
(v) ?? = C ? C = c onstant ? ? ? < y ? = C 
(vi) ? ?? = ?? ?? 1
? ?? = Constant ? ? ? < ?? > = ?? < ?? 1
? 
(vii) ?? = ?? 1
+ ?? 2
? ? ? ?? ? = ? ?? 1
? + ? ?? 2
? 
RMS (Root Mean square) Value of a function: 
y
ms
= v ? y
2
? 
Examples: Find the rms value of ?? . 
(1) ?? = ?? 
?? mi n
 From 0 to 2s ec . 
y
min 
= v ? y
2
? = v ? t
2
? =
v
4
3
 
Page 5


 
ALTERNATING CURRENT 
ALTERNATING CURRENT AND VOLTAGE 
When we say voltage or current is alternating, it means that it keeps changing in strength 
continuously and switches direction regularly over time. This kind of change can be shown 
using a sine curve or cosine curve, which are wavy lines that go up and down in a smooth 
pattern. So, an alternating voltage or current goes from high to low and then back again, 
repeating this cycle over and over in a predictable way. 
?? = ?? 0
s i n ? ???? ? or ? ?? = ?? 0
c os ? ???? 
Where ? I = Instantaneous value of current at time t , ? I
0
= Amplitude or peak value 
?? = Angular frequency ?? =
2 ?? ?? = 2 ???? ? ?? = time period ? f = frequency  
 
1 as a sine function of ?? 
 
I as a cosine function of ?? 
AMPLITUDE OF AC: 
The maximum value of current in either direction is called peak value or the amplitude of 
current. It is represented by ?? 0
. Peak to peak value = 2 ?? 0
 
PERIODIC TIME: 
The time taken by alternating current to complete one cycle of variation is called periodic 
time or time period of the current. 
FREQUENCY 
The number of cycles completed by an alternating current in one second is called the 
frequency of the current. 
UNIT: cycle/s; (Hz) 
In India: ? ?? = 50 Hz, supply voltage = 220 volt 
In USA: ? f = 60 Hz, supply voltage = 110 volt 
CONDITION REQUIRED FOR CURRENT/ VOLTAGE TO BE ALTERNATING 
? Amplitude is constant 
? Alternate half cycle is positive and half negative 
The alternating current continuously varies in magnitude and periodically reverses its 
direction. 
 
 
 
 
 
AVERAGE VALUE OF A FUNCTION 
? ?? ? =
? ? ?? ?? ?? ? ? ????
 
? ?? ? = Average value of ?? 
 
Example. ?? = t
2
; < ?? > from 0 to 2s ec . 
Solution: ? ?? ? =
?
0
2
? ?? 2
????
?
0
2
? ????
=
2
3
3 × 2
=
4
3
 
Example. ?? = 2 ?? + 3 ; < ?? > from 0 to 4s ec . 
Solution: ? ?? ? =
?
0
4
? ( 2t + 3 ) dt
?
0
4
? dt
=
( t
2
+ 3t )
0
4
4
=
4
2
+ 3 ( 4 )
4
= 4 + 3 = 7 
AVERAGE VALUE OR MEAN VALUE 
The average value of an alternating current (AC) over any half cycle (positive or 
negative) is the direct current (DC) value that would pass through a circuit and deliver the 
same amount of charge in the same time as the AC does during that half cycle. 
average value of current for half cycle ? ?? ? =
?
0
?? / 2
? ???? ?? ?
0
?? / 2
? ????
 
Average value of ?? = ?? 0
s i n ? ???? over the positive half cycle: 
?? ????
=
? ?
?? 2
0
? ?? 0
s i n ? ?????? ?? ? ?
?? 2
0
?????
=
2 ?? 0
????
[ - c os ? ???? ]
0
?? 2
=
2 ?? 0
?? 
? < sin ? ?? > = < sin ? 2 ?? > = 0
? < c os ? ?? > = < c os ? 2 ?? > = 0
? < sin ? ?? c os ? ?? > = 0
? < sin
2
? ?? > = < c os
2
? ?? > =
1
2
 
? ? For symmetric AC, average value over full cycle = 0, 
Average value of sinusoidal ???? 
Full cycle (+ve) half cycle (-ve) half cycle 
0 
2 I
0
?? 
- 2 I
0
?? 
 
As the average value of AC over a complete cycle is zero, it is always defined over a half 
cycle which must be either positive or negative 
MAXIMUM VALUE 
? ?? = ?? sin ? ?? 
? ? I
Max. 
= a 
? ?? = ?? + ?? s i n ? ?? 
? ? ?? Max 
= ?? + ?? ( if ?? and ?? > 0 ) 
? ?? = ?? sin ? ?? + ?? c o s ? ?? 
? ? ?? Max. 
= v ?? 2
+ ?? 2
 
E 
? I
Max 
= a ( a > 0 ) 
Average value over ?? complete cycles or long time: 
(i) ?sin ? ( ???? + ?? ) ? = 0 
(ii) ? c os ? ( ???? + ?? ) ? = 0 
(iii) ? sin
2
? ( ???? + ?? ) ? =
1
2
 
(iv) ? c os
2
? ( ???? + ?? ) ? =
1
2
 
(v) ?? = C ? C = c onstant ? ? ? < y ? = C 
(vi) ? ?? = ?? ?? 1
? ?? = Constant ? ? ? < ?? > = ?? < ?? 1
? 
(vii) ?? = ?? 1
+ ?? 2
? ? ? ?? ? = ? ?? 1
? + ? ?? 2
? 
RMS (Root Mean square) Value of a function: 
y
ms
= v ? y
2
? 
Examples: Find the rms value of ?? . 
(1) ?? = ?? 
?? mi n
 From 0 to 2s ec . 
y
min 
= v ? y
2
? = v ? t
2
? =
v
4
3
 
(2) ?? = ?? ? ?? ?? ,
= ?? 
(3) ?? = Cy
1
? ?? rm , s
= C ( y
1
)
mm
 
(4) ?? = 4 + 6s i n ? ???? 
Sol. ?? 2
= 2
2
( 2 + 3s i n ? ???? )
2
? ? ? ?? 2
= 4 ( 4 + 9 sin
2
? ???? + 12s i n ? ???? ) ? ?? 2
= 16 + 36 s i n
2
? ???? +
48s i n ? ???? 
? y
2
? = 16 +
36
2
= 16 + 18 ? y
r ms
= v ? y
2
? = v 34 
ROOT MEAN SQUARE (rms) VALUE 
The RMS (Root Mean Square) value of an alternating current (AC) is the value of direct 
current (DC) that would generate the same amount of heat in a given resistance over a 
specific time period as the AC does when passed through the same resistance for the 
same duration. 
?? r ms
= v
? ?
?? 0
? ?? 2
dt
? ?
?? 0
? dt
? rms value = virtual value = apparent value  
R.M.S value of ?? = ?? 0
s i n ? ???? : 
?? r m s
= v
? ?
?? 0
? ( ?? 0
s i n ? ???? )
2
????
? ?
?? 0
?????
= v
?? 0
2
?? ? ?
?? 0
? sin
2
? ?????? ?? = ?? 0
v
1
?? ? ?
?? 0
? [
1 - c os ? 2 ????
2
] ????
= ?? 0
v
1
?? [
?? 2
-
sin ? 2 ????
2 × 2 ?? ]
0
?? =
?? 0
v 2
 
? If nothing is mentioned then values printed in a.c circuit on electrical appliances, 
any given or unknown values, reading of AC meters are assumed to be RMS. 
Current Average Peak RMS Angular frequency 
?? 1
= ?? 0
s i n ? ( ???? + ?? ) 0 ?? 0
 
?? 0
v 2
 ?? 
?? 2
= ?? 0
s i n ? ???? c os ? ???? =
?? 0
2
sin ? 2 ???? 0 
?? 0
2
 
?? 0
2 v 2
 2 ?? 
?? 3
= ?? 0
s i n ? ???? + ?? 0
c os ? ???? 0 
v 2 ?? 0
 ?? 0
 ??