Until now, we have studied only circuits with direct current (dc) which flows only in one direction. The primary source of emf in such circuit is a battery. When a resistance is connected across the terminals of the battery, a current is established in the circuits, which flows in a unique direction from the positive terminal to the negative terminal via the external resistance.
Direct current Alternating current
But most of the electric power generated and used in the world is in the form of alternating current (ac), the magnitude of which changes continuously with time and direction is reversed periodically (as shown in figure III & IV) and it is given by
i = i_{0} sin (ωt + φ)
Here i is instantaneous value of current i.e., the magnitude of current at any instant of time and i_{0} is the maximum value of current which is called peak current or the current amplitude and the current repeats its value after each time interval T = as shown in figure. This time interval is called the time period and w is angular frequency which is equal to 2π times of frequency f.
ω = 2πf
AC wave Triangular wave
The current is positive for half the time period and negative for remaining half period. It means that the direction of current is reversed after each half time period. The frequency of ac in India is 50 Hz.
An alternating voltage is given by:
V = V_{0} sin (ωt + φ)
It also varies alternatively as shown in the figure (b), where V is instantaneous voltage and V_{0} is peak voltage. It is produced by ac generator also called as ac dynamo.
AC Voltage
AC Circuit: An ac circuit consists of circuit element i.e., resistor, capacitor, inductor or any combination of these and a generator that provides the alternating current as shown in figure. The ac source is represented by symbol in the circuit.
AC Circuit
As we know an alternating current is given by
i = i_{0} sin (ωt + f) ...(1)
The mean or the average value of ac over any time T is given by
Using equation (1)
In one complete cycle, the average current
=  = = 0 (as ωT = 2π)
Since ac is positive during the first half cycle and negative during the other half cycle so i_{avg} will be zero for long time also. Hence the dc instrument will indicate zero deflection when connected to a branch carrying ac current. So it is defined for either positive half cycle or negative half cycle.
_{ = } 0.637 i_{0}
Similarly V_{avg} = 0.637 V_{0}
Ex. 1 If a direct current of value a ampere is superimposed on an alternating current i = b sin wt flowing through a wire, what is the effective value of the resulting current in the circuit ?
Ans: As current at any instant in the circuit will be,
i = i_{dc} + i_{ac} = a + b sin ωt
So, i_{eff} = =
i.e., =
but as
= 0 and
So, i_{eff} =
The notation rms refers to root mean square, which is given by square root of mean of square current.
i.e.,
i^{2}_{avg} =
= =
= =
=
i_{rms} = » 0.707 i_{0}
Similarly the rms voltage is given by
V_{rms} = » 0.707 V_{0}
The significance of rms current and rms voltage may be shown by considering a resistance R carrying a current i = i_{0} sin (wt + f)
The voltage across the resistor will be
V_{R} = Ri = (i_{0}R) sin (ωt + φ)
The thermal energy developed in the resistor during the time t to t + dt is
i^{2} R dt = i_{0}^{2}R sin^{2}(ωt + φ) dt
The thermal energy developed in one time period is
U = = = RT = i^{2}_{rms}^{ }RT
It means the root mean square value of ac is that value of steady current, which would generated the same amount of heat in a given resistance in a given time.
So in ac circuits, current and ac voltage are measured in terms of their rms values. Likes when we say that the house hold supply is 220 V ac it means the rms value is 220 V and peak value is = 311 V.
Ex. 2 If the voltage in an ac circuit is represented by the equation, V = sin (314t  φ),calculate (a) peak and rms value of the voltage, (b) average voltage, (c) frequency of ac.
Ans: (a) For ac voltage,
V = V_{0} sin (ωt  φ)
The peak value of voltage
V_{0} = = 311 V
The rms value of voltage
(b) Average voltage in full cycle is zero. Average voltage in half cycle is
V_{avg} = = = 198.17 V
(c) As ω = 2πf, 2ωf = 314
i.e., f = = 50 Hz
Ex. 3 The electric current in a circuit is given by i = i_{0} (t/T) for some time. Calculate the rms current for the period t = 0 to t = T.
Ans: The mean square current is
=
Thus, the rms current is
i_{rms} =
In case of a steady current the rate of doing work is given by,
P = Vi
In an alternating circuit, current and voltage both vary with time, so the work done by the source in time interval dt is given by
dW= Vidt
Suppose in an ac, the current is leading the voltage by an angle φ. Then we can write,
V = V_{0} sinωt
and i = i_{0} sin(ωt + φ)
dW = V_{0}i_{0} sin ωt sin (ωt + φ) dt
= V_{0} i_{0} (sin^{2} ωt cos f + sinωt cos ωt sin φ) dt
The total work done in a complete cycle is
W = V_{0}i_{0} cos + V_{0}i_{0}sin
+ =
The average power delivered by the source is, therefore,
P = = =
= V_{rms} i_{rms} cos φ
or P_{one cycle} = V_{rms} i_{rms} cos φ
105 videos425 docs114 tests

1. What is the relationship between the average and RMS value of an Alternating Current in a circuit? 
2. How is the average value of an Alternating Current calculated in a circuit? 
3. Why is the RMS value of an Alternating Current more commonly used than the average value in calculations? 
4. How does the RMS value of an Alternating Current affect the power consumption in a circuit? 
5. Can the average and RMS value of an Alternating Current be the same for any waveform other than a sinusoidal wave? 

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