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**Alternating Current:**

**1. Alternating Current:**

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.

Fig: Direct current Fig: 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 (Ï‰t + Ï†)

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 2Ï€ times of frequency f.

Ï‰ = 2Ï€f

Fig: AC wave Fig: 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 (Ï‰t + Ï†)

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 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.

**2. AVERAGE AND RMS VALUE OF ALTERNATING CURRENT:**

**2.1 Average current (Mean current):**

As we know an alternating current is given by

*i* = *i*_{0} sin (Ï‰t + 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 Ï‰T = 2Ï€)

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 Ï‰t

So, *i*_{eff} = =

i.e., =

but as

= 0 and

So, *i*_{eff} =

**2.2 R.M.S Value of alternating current:**

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} = Â» 0.707 *i*_{0}

Similarly the rms voltage is given by

V_{rms} = Â» 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} = R*i* = (*i*_{0}R) sin (Ï‰t + Ï†)

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}(Ï‰t + Ï†) 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 - **Ï†

**Ans: **(a) For ac voltage,

V = V_{0} sin (Ï‰t - Ï†)

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 Ï‰ = 2Ï€f, 2Ï‰f = 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} =

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