Maharashtra Board Class 12 Physics Chapter 13 AC Circuits PDF Download

13. AC Circuits

13.1 Introduction

In school you have learnt that there are two types of supplies of electricity.

First is DC. This is direct current. It has fixed polarity of voltage. The positive and negative ends of the power supply are fixed.

Second is AC. This is alternating current. For AC, the polarity of the voltage keeps changing periodically.

We have studied the generation of AC voltage in the previous chapter. Because of low cost and convenience of transport, the electricity is mostly supplied as AC.

Some of the appliances that we use at home or offices like TV, computer, transistor, radio, etc. convert AC to DC. They use a device like rectifier. You will study this in chapter 16.

However, there are some domestic devices like fan, fridge, air conditioner, induction heater, coil heater, etc., which run directly on AC. Almost all these devices use components like an inductor and a capacitor.

In this chapter we will study the passage of AC through resistors, inductors and capacitors.

Teacher's Note

AC electricity is used in our homes. Most of our appliances work on AC supply. We use AC because it is cheaper to send AC electricity over long distances than DC electricity.

Exam Trick

Remember: AC means the electricity goes back and forth. Just like the current in rivers changes direction with tides. DC is like a straight river that always flows in one direction.

Points to Remember

AC supply changes its polarity again and again.

DC supply has fixed polarity.

Most homes use AC electricity.

AC is cheaper to transport than DC over long distances.

13.2 AC Generator

In the last chapter we have studied that the source of AC (generator) produces a time dependent emf (e) given by:

\(e = e_0 \sin \omega t\) --- (13.1)

Here \(e_0\) is the peak value of emf and \(\omega\) is the angular frequency of rotation of the coil in the AC generator.

As the time variation of current is similar to that of emf, the current in a circuit connected to this generator will be of the form:

\(i = i_0 \sin(\omega t + \phi)\)

Here \(\phi\) represents the phase difference between the current (i) and the emf. And \(i_0\) is the peak value of current.

From Eq (13.1) it can be seen that the induced emf varies sinusoidally with time.

From the graph it can be seen that the direction of the emf is reversed after every half revolution of the coil. This type of emf is called the alternating emf. The corresponding current is called alternating current.

Teacher's Note

An AC generator works like a bicycle dynamo. When you rotate the coil in a magnetic field, it produces AC electricity. This is how power stations generate electricity.

Exam Trick

Remember: Peak value is the maximum value. RMS value is the value used in calculations. RMS is always smaller than peak value. Think: Peak is the top of a mountain, RMS is the average height.

Points to Remember

AC emf changes with time as \(e = e_0 \sin \omega t\).

Peak value is the maximum value of AC.

AC changes its direction every half cycle.

Angular frequency \(\omega\) tells how fast AC is changing.

13.3 Average And RMS Values

Alternating voltages and current go through all values between zero and the peak value in one cycle.

We define some specific values which would be convenient for comparing two voltage or current waveforms.

a) Average Or Mean Value Of AC

This is the average of all values of the voltage (or current) over one half cycle.

As can be seen in a graph, the average over a full cycle is always zero. This is because the average value of sin ωt over a cycle is zero.

So the mean value of AC over a cycle has no significance. The mean value of AC is defined as the average over half cycle.

The average value of current or emf is 0.637 times their peak value.

\(i_{av} = 0.637 \, i_0\) and \(e_{av} = 0.637 \, e_0\)

Here \(i_{av}\) and \(e_{av}\) are the average values of alternating current and emf (voltage) respectively.

b) Root-Mean-Square (Or RMS) Value

A moving coil ammeter and voltmeter measure the average value of current and voltage applied across it respectively.

It is obvious therefore that the moving coil instruments cannot be used to measure the alternating current and voltages.

Hence in order to measure there quantities it is necessary to make use of a property which does not depend upon the changes in direction of alternating current or voltage.

Heating effect depends upon the square of the current. The square of the current is always positive. Hence it does not depend upon the direction of flow of current.

Consider an alternating current of peak value \(i_0\), flowing through a resistance R. Let H be the heat produced in time t.

Now the same quantity of heat (H) can be produced in the same resistance (R) in the same time (t) by passing a steady current of constant magnitude through it.

The value of such steady current is called the effective value or virtual value or rms value of the given alternating current. It is denoted by \(i_{rms}\).

The relation between the rms value and peak value of alternating current is:

\(i_{rms} = \frac{i_0}{\sqrt{2}} = 0.707 \, i_0\)

Similarly it can be shown that:

\(e_{rms} = \frac{e_0}{\sqrt{2}} = 0.707 \, e_0\)

The heat produced by a sinusoidally varying AC over a complete cycle will be:

\(H = \frac{2\pi}{\omega} R(i_{rms})^2\)

It is the same as the heat produced by a DC current of magnitude \(i_{rms}\) for time \(t = \frac{2\pi}{\omega}\).

Teacher's Note

RMS value is very important. When we say 220 V AC supply, we mean 220 V RMS. The peak value is 220 times 1.414 which is 311 V. But AC shock is more dangerous than DC because AC attracts and repels the heart muscles.

Exam Trick

Remember: RMS = peak divided by root 2. Always use RMS value for calculations in exams. Peak value is only for the graph.

Points to Remember

RMS value is 0.707 times the peak value.

Average value is 0.637 times the peak value.

RMS value is used in all electrical calculations.

Moving coil meters read RMS values, not peak values.

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