NEET  >  Physics Class 12  >  Alternating Current: Average & RMS Value

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

1 Crore+ students have signed up on EduRev. Have you?

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 currentFig: Direct current Fig: Alternating currentFig: 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 = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET 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

Fig: AC waveFig: AC wave Fig: Triangular waveFig: 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.

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

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 Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET in the circuit.

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET


2. AVERAGE AND RMS VALUE OF ALTERNATING CURRENT:

2.1 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

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

Using equation (1)

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

In one complete cycle, the average current

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = -Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = 0 (as ωT = 2π)

Since ac is positive during the first half cycle and negative during the other half cycle so iavg 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.

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET0.637 i0

Similarly Vavg = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET0.637 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 ?

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

Ans: As current at any instant in the circuit will be,

i = idc + iaca + b sin ωt

So, ieff = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

i.e., = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

but as

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = 0 and Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

So, ieff = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

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., Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

i2avg = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET 

= Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

= Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = 

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

irms = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET » 0.707 i0

Similarly the rms voltage is given by

Vrms = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET » 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

VR = 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 = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = RT Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = i2rms 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 = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = 311 V

The rms value of voltage

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

(b) Average voltage in full cycle is zero. Average voltage in half cycle is

Vavg = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = 198.17 V

(c) As ω = 2πf, 2ωf = 314

i.e., f = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = 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

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

Thus, the rms current is

irms = Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

The document Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET is a part of the NEET Course Physics Class 12.
All you need of NEET at this link: NEET
157 videos|452 docs|213 tests
Use Code STAYHOME200 and get INR 200 additional OFF
Use Coupon Code
157 videos|452 docs|213 tests
Download as PDF

How to Prepare for NEET

Read our guide to prepare for NEET which is created by Toppers & the best Teachers

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

,

study material

,

mock tests for examination

,

past year papers

,

ppt

,

MCQs

,

Exam

,

Viva Questions

,

video lectures

,

Semester Notes

,

Summary

,

Extra Questions

,

Sample Paper

,

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

,

shortcuts and tricks

,

Objective type Questions

,

practice quizzes

,

Important questions

,

Free

,

pdf

,

Previous Year Questions with Solutions

,

Alternating Current: Average & RMS Value - Notes | Study Physics Class 12 - NEET

;