Alternating Current: Average & RMS Value | Physics Class 12 - NEET PDF Download

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.

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 = i₀ sin (ωt + φ)

Here i is instantaneous value of current i.e., the magnitude of current at any instant of time and i₀ 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 = V₀ sin (ωt + φ)

It also varies alternatively as shown in the figure (b), where V is instantaneous voltage and V₀ 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

Average and RMS value of Alternating Current

Average current (Mean current)

As we know an alternating current is given by

i = i₀ 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 i₀

Similarly V_avg =  0.637 V₀

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 =

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.,

i²_avg =  

=  =

=  = 

 =

i_rms =  » 0.707 i₀

Similarly the rms voltage is given by

V_rms =  » 0.707 V₀

The significance of rms current and rms voltage may be shown by considering a resistance R carrying a current i = i₀ sin (wt + f)

The voltage across the resistor will be

V_R = Ri = (i₀R) sin (ωt + φ)

The thermal energy developed in the resistor during the time t to t + dt is

i² R dt = i₀²R sin²(ωt + φ) dt

The thermal energy developed in one time period is

U =  =  = RT  = i²_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 = V₀ sin (ωt - φ)

The peak value of voltage

V₀ =  = 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 = i₀ (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 =

Power in an AC circuit

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 = V₀ sinωt

and i = i₀ sin(ωt + φ)

dW = V₀i₀ sin ωt sin (ωt + φ) dt

= V₀ i₀ (sin² ωt cos f + sinωt cos ωt sin φ) dt

The total work done in a complete cycle is

W = V₀i₀ cos + V₀i₀sin

+  =

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

• Here, the term cos φ is known as the power factor.
• It is said to be leading if current leads voltage, lagging if current lags voltage. Thus, a power factor of 0.5 lagging means current lags the voltage by 60° (as cos^-10.5 = 60°). The product of V_rms and i_rms gives the apparent power. While the true power is obtained by multiplying the apparent power by the power factor cos φ. Thus,
• and apparent power = V_rms × i_rms
• True power = apparent power × power factor
• For φ= 0°, the current and voltage are in phase. The power is thus, maximum (V_rms× i_rms). For
• φ = 90°, the power is zero. The current is then stated as wattless. Such a case will arise when resistance in the circuit is zero. The circuit is purely inductive or capacitive.