a. Alternating Current
The current which changes in direction and magnitude with time is called alternating current. For e.g. the electric
mains supply in our homes which varies sinusoidally with time. AC voltage is represented as-
@
~
t
o
Average value of a function
The average value of any function y = f(x) over an interval from x1 to x2 can be calculated as-
-
The following are some results for average value of trigonometric functions which should be memorized-
1)
2)
For a sinusoidal wave, the average value in one time period will be 0 as equal current flows in forward and backward
direction. Therefore, the average value is calculated over half a time period-
Root mean squared (RMS) value/ Effective value
RMS value for an ac current is that value of dc current which when applied to the same resistor for the same amount
of time, produces the same amount of heat as ac.
Mathematically,
Note: - ‘An ac current of 4A’ implies that the RMS value is 4A
Let current be I = i˳ Sin(ωt), where i˳ is the peak value. The RMS value is related to peak value as-
IBE
rn s
,Similarly, the rms voltage or effective voltage-
AC voltage applied to a Resistor
Consider a resistor of resistance R is connected to an alternating voltage source v= v˳Sinωt, where v˳ is the amplitude
of the oscillating potential difference and ω is the angular frequency. Using Kirchhoff’s loop law-
•
t t
-
-
Where-
In the graph, both v and i reach minimum and maximum values
Simultaneously. Therefore, for a resistor voltage and current are
In phase with each other i.e. phase difference = 0.
Power consumed
In complete cycle, since equal current flows in both directions, the avg current over a complete cycle is 0 but the
average power consumed is not zero. This is because the power consumed is given by i2R (according to Joule’s law)
which depends on i2 and not i and i2 is always positive whether i is positive or negative. Thus, there is Joule heating
and dissipation of electrical energy when ac current passes through a resistor.
Instantaneous power dissipated in the circuit-
Therefore, average power dissipated-
Also-
From 1
Note: The instantaneous values of ac current/ voltage obey dc laws like Kirchhoff’s loop and junction law.
, Representation of AC voltage and current by Rotating vectors- Phasors
A phasor is a vector which rotates about the origin with angular speed ω. The orthogonal components of phasors v
and I give the instantaneous values v and i and their magnitude represents the amplitude v˳ and i˳
W
O
=
wt
•
Note: Though the voltage and current in an ac circuit are represented by vectors they are not vectors themselves.
They are scalar quantities.
AC voltage applied to an inductor (PYQ 2011)
Consider an inductor of self inductor L and negligible resistance in its windings. Thus, the circuit is purely inductive.
Let the voltage be v= v˳ Sinωt. Using Kirchhoff’s loop law-
t
+
The negative sign is in accordance with Lenz’ law
-
-
The integration constant has dimensions of current and is time independent. Since the source emf and the current
oscillate symmetrically about zero, the integration constant will be 0
Inductive reactance (XL)
The quantity ωL is analogous to resistance and is called Inductive reactance.
L
- Dimensions are same as Resistance | SI Unit- Ohm (Ω)
W
Phasor
Comparing the expression for current and voltage in a purely inductive circuit, we see that the current lags behind
voltage by π/2 or 1/4 th of the cycle.
Vo
Io
The current which changes in direction and magnitude with time is called alternating current. For e.g. the electric
mains supply in our homes which varies sinusoidally with time. AC voltage is represented as-
@
~
t
o
Average value of a function
The average value of any function y = f(x) over an interval from x1 to x2 can be calculated as-
-
The following are some results for average value of trigonometric functions which should be memorized-
1)
2)
For a sinusoidal wave, the average value in one time period will be 0 as equal current flows in forward and backward
direction. Therefore, the average value is calculated over half a time period-
Root mean squared (RMS) value/ Effective value
RMS value for an ac current is that value of dc current which when applied to the same resistor for the same amount
of time, produces the same amount of heat as ac.
Mathematically,
Note: - ‘An ac current of 4A’ implies that the RMS value is 4A
Let current be I = i˳ Sin(ωt), where i˳ is the peak value. The RMS value is related to peak value as-
IBE
rn s
,Similarly, the rms voltage or effective voltage-
AC voltage applied to a Resistor
Consider a resistor of resistance R is connected to an alternating voltage source v= v˳Sinωt, where v˳ is the amplitude
of the oscillating potential difference and ω is the angular frequency. Using Kirchhoff’s loop law-
•
t t
-
-
Where-
In the graph, both v and i reach minimum and maximum values
Simultaneously. Therefore, for a resistor voltage and current are
In phase with each other i.e. phase difference = 0.
Power consumed
In complete cycle, since equal current flows in both directions, the avg current over a complete cycle is 0 but the
average power consumed is not zero. This is because the power consumed is given by i2R (according to Joule’s law)
which depends on i2 and not i and i2 is always positive whether i is positive or negative. Thus, there is Joule heating
and dissipation of electrical energy when ac current passes through a resistor.
Instantaneous power dissipated in the circuit-
Therefore, average power dissipated-
Also-
From 1
Note: The instantaneous values of ac current/ voltage obey dc laws like Kirchhoff’s loop and junction law.
, Representation of AC voltage and current by Rotating vectors- Phasors
A phasor is a vector which rotates about the origin with angular speed ω. The orthogonal components of phasors v
and I give the instantaneous values v and i and their magnitude represents the amplitude v˳ and i˳
W
O
=
wt
•
Note: Though the voltage and current in an ac circuit are represented by vectors they are not vectors themselves.
They are scalar quantities.
AC voltage applied to an inductor (PYQ 2011)
Consider an inductor of self inductor L and negligible resistance in its windings. Thus, the circuit is purely inductive.
Let the voltage be v= v˳ Sinωt. Using Kirchhoff’s loop law-
t
+
The negative sign is in accordance with Lenz’ law
-
-
The integration constant has dimensions of current and is time independent. Since the source emf and the current
oscillate symmetrically about zero, the integration constant will be 0
Inductive reactance (XL)
The quantity ωL is analogous to resistance and is called Inductive reactance.
L
- Dimensions are same as Resistance | SI Unit- Ohm (Ω)
W
Phasor
Comparing the expression for current and voltage in a purely inductive circuit, we see that the current lags behind
voltage by π/2 or 1/4 th of the cycle.
Vo
Io
