PHYSICS A LEVEL(FORM SIX) NOTES – AC THEORY(1) - EcoleBooks (2024)

AC THEORY

When a battery is connected to a circuit the current flows steadily in one direction, this is called a Direct current (d.c).

The use of Direct currents is limited to a few applications e.g charging of batteries, electroplating etc.

Most of electrical energy is generated and used in the form of alternating current due to many reasons including,

i) Alternating voltages can be changed in value very easily by means of transformers.

ii) A.c motors are simpler in construction and cheaper than d.c motors.

ALTERNATING VOLTAGE AND CURRENT

i) Alternating voltage

An alternating voltage is one whose magnitude changes with time and direction reverses periodically. The instantaneous value (i.e value at any time t) of an alternating voltage is given by,

where,

E = Value of the Alternating voltage at time t

E0 = Maximum value of the Alternating voltage

ω = Angular frequency of supply
From ,where f is the frequency of the alternating voltage, If T is the time period of alternating voltage then

The voltage varies from zero to a positive peak (+E0) then back via zero to negative peak (-E0) and so on.
In time period T, the wave completely cycle.

ii) (ii)Alternating current

This is one whose magnitude changes with time and direction reverses periodically.

The Instantaneous value ( value at any time t) of sinusoidally varying alternating current is given by

where

I = value of alternating current at time t

I0 = maximum value (Amplitude) of alternating current.

ω = Angular frequency of supply.

Figure below shows the waveform of alternating current.

Current varies sinusoidally with time. The current increases gradually from zero to a positive peak (+Io), then back via zero to a negative peak (–Io) and so on.

In time period T, the wave completes a cycle.

An Alternating voltage and current also be represented as a cosine function of time.

Both these representations give the same result as is given by the one containing sine functions.

MEASUREMENT OF ALTERNATING CURRENT

Since the average value of sinusoidal alternating current is zero, an ordinary (DC) ammeter or galvanometer will not show any deflection when connected in an AC circuit.

Due to inertia it will not be possible for the needle to oscillate with the frequency of the current.

Therefore to measure AC we use hot wire instruments because the heating effect of current is independent of the direction of current.

MEAN OR AVERAGE VALUE OF ALTERNATING CURRENT

The mean value or average value of alternating current over one complete circle is zero.

It is because the area of positive half cycle is exactly equal to the area of the negative half cycle.

However, we can find the average or mean value of alternating current over any half cycle.

Half Cycle Average Value of a.c
This is that value of steady current (d.c) which would send the same amount of charge through a circuit for half the time period of a.c as it sent by the a.c through the same circuit in the same time.It is represented by Im or
The instantaneous value of alternating current is given by,
Suppose current I remains constant for a small time. Then small amount of charge sent by alternating current in a small time is given by

If Q is the total charge sent by the positive half cycle of a.c (

If the Im is the half cycle average means values of positive half cycle of a.c then by definition

From equation (i) and (ii)

Hence half cycle average value of a.c is 0.637 times the peak value of a.c
For positive ½ cycle Im +0.637 I0

For negative half cycle Im -0.637 I0

Obviously, average value of a.c over a complete cycle is zero.

MEAN OR AVERAGE VALUE OF ALTERNATIVE VOLTAGE
Half Cycle Average value of alternating

This is that value of steady, (which would send the same amount charge through a circuit for half the time period of alternating as is sent by the alternating through the same circuit in the same time.It is denoted by The instantaneous value of alternating e. m. f is given by
Suppose this alternating is applied to a circuit of resistance R. Then by ohm’s law the instantaneous value of alternating current is

If this current remains constant a small time, then small amount of charge send by alternating in small time is given by

If Q is the total charge sent by positive half cycle of a alternating then

If is the half cycle average or mean value of the positive half cycle of alternating then by definition

From equation (i) and equation (ii)

Therefore, half cycle average value of alternating is 0.637 times the peak value of alternating .
For positive half cycle

For negative half cycle

A d.c voltmeter or ammeter reads average (or d.c) value. Therefore, they can be used to measure alternating voltage on current.

It is because the average value of alternating voltage or current over a complete cycle is zero.

We use a.c meters to measure alternating voltage/current.

ROOT MEAN SQUARE VALUE OF ALTERNATING CURRENT

The average value cannot be used to specify an alternating current (or voltage).

It is because its value is zero over one cycle and cannot be used for power calculations.

Therefore we must search for more suitable criterion to measure the effectiveness of an alternating current or voltage.

The obvious choice would be to measure it in terms of direct current that would do work (or produce heat) at the same average rate as a.c under similar conditions.

This equivalent direct current is called the root mean square () or effective value of alternating current.

Effective or value of Alternating Current

The root mean square (r.m.s) of alternating current is that steady current (d.c) which when flowing through a given resistance for given time produces the same amount of heat as produced by the alternating current when flowing through the same resistance for the same time.

– It is also called virtual value of a.c

– It is denoted by called

For example, when we say that or effective value of an alternating current is 5A, it means that the alternating current will do the work (or produce heat) at the same rate as 5A direct current under similar conditions.

RELATION BETWEEN R.M.S. VALUE AND PEAK VALUE OF A.C

Let the alternating current be represented by

I

If this alternating current flows through a resistance R for a small time, then small amount of heat produced is given.

In one complete cycle for time 0 to T of alternating current the total amount of heat produced in R is given by

If Irms is the virtual or value of the alternating current, then heat produced in R in the same time (0 to T) is given by

From equation (i) and (ii), we have

Hence the value of effective value or virtual value of alternating current is 0.707 times the peak value of alternating current.

The value is the same whether calculated for
half cycle or full cycle.

Alternative Method
Let the alternating current be represented by

I

If this current is passed through a resistance R, then power delivered at any instant is given by

Because the current is squared, power is always positive since the value of varies

between 0 and 1

Average power delivered

If is the virtual or value of the alternating current then by definition

Power delivery P

From equation (I) and (ii)

ROOT MEAN SQUARE VALUE OF ALTERNATING E.M.F

The root mean square (r.m.s) value of alternating voltage is that steady voltage (d.c voltage when applied to a given resistance for a given time produces the same amount of heat as is produced by the alternative when applied to the same resistance for the same time.

It is also called virtual value of alternating e.m.f and is donated by of or
The instantaneous value of alternating is given by

E

When this alternating voltage is applied to a resistance R, small amount of heat produced in a small time is

But

Then,

In one complete cycle for time 0 to T the total amount of heat produced in resistance R is

If Erms is the value of the alternating, then heat produced in the same resistance R for the same time T is

From the equation (i) and equation (ii)

Hence the value of alternating is 0.707 times the peak value of the alternating

Therefore, value is the same whether calculated for half cycle or full cycle.

Importance of r.m.s values

An alternating voltage or current always specified in terms of values.

T
hus an alternating current of 10A is the one which has the same heat effect as 10A d.c under similar conditions. The following points may be noted carefully.

i) The domestic a.c supply is 230V, 50H. It is the or effectively value. It means that the alternative voltage available has the same effect as 230 V d.c under similar conditions.

The equation of this alternative voltage is

ii) When we say that alternating current in a circuit is 5A, we specifying the value.

– It means that the alternating current flowing in the circuit has the same heating effect as 5A d.c under similar conditions.

iii) A.C ammeters and voltmeters record values of current and voltage respectively.

The alternating voltage/current can be measured by utilizing the heating effect of electric current.

Such Instruments are called hot wire instruments and measure the value of the voltage/current since r.m.s value is the same for half cycle or complete cycle.

WORKED EXAMPLES

1. An a.c main supply is given to be 220V what is the average during a positive half cycle?

Solution

Average during positive half is given by



2. An alternating current I is given by

Find

(i) The maximum value

(ii) Frequency

(iii) Time period

(iv) The instantaneous value when t= 3ms

Solution

Comparing the given equation of the alternating current with the standard form

(i) Maximum value,

(ii) Frequency f

f = 50 HZ

(iii) Time period T

(iv)

When

3. Calculate the value of the current shown in figure below