ALTERNATING CURRENTS

1. Alternating Current and Alternating EMF

An alternating current is one whose magnitude changes continuously

with time between zero and a maximum value and whose direction

reverses periodically.

The simplest type of alternating current is one which varies with time

simple harmonically. It is represented by

0i i sin t (i)

or 0i i cos t, (ii)

where i is the instantaneous value of the current at time t and i0 is the

maximum (or peak) value of the current and is called 'current amplitude'.

is called the 'angular frequency' of alternating current (a.c.) and is

given by

2

2 fT

where T is the 'time period' and f is the frequency of a.c.

Eq. (i) and (ii) represent alternating current i as a sine function of time t

and as a cosine function of time t respectively. Both representations lead

to the same result.

Fig. 1 (a) and (b) are graphical representations of i as sine function of t

and as cosine function of t respectively. The complete set of variations of

the current in one time-period T is called a 'cycle.'

The emf (or voltage) which produces alternating current in some circuit

is called 'alternating emf' (or voltage). Thus, the emf (or voltage) whose

magnitude changes continuously with time between zero and a

maximum value and whose direction reverses periodically is known as

alternating emf (or voltage).

The instantaneous value of alternating emf may be represented by

E = E0 sin t ...(iii)

or E = E0 cos t (iv)

The graphical representations of E as sine and cosine functions of t are of

the same form as those of i (Fig. la and b).

2. Amplitude, Periodic Time and Frequency of Alternating Current

Amplitude :

The alternating current varies in magnitude and reverses in direction

periodically. The maximum value of the current in either direction is

called the 'peak value' or the 'amplitude' of the current. It is represented

by i0.

Periodic Time :

The time taken by the alternating current to complete one cycle of

variations is called the 'periodic-time' of the current. If, in eq. (i) or (ii), t

be increased by

2, the value of i remains unchanged. Hence the

periodic-time T of the alternating current is given by

T =

2.

Frequency :

The number of cycles completed by an alternating current in one second

is called the 'frequency' of the current. If the periodic-time of the

alternating current be T, then its frequency is

f = 1

T

But T =

2.

f .2

The unit of frequency is 'cycles/second' (c/s) or hertz (Hz). The frequency

of the domestic alternating current is 50 cycles/second or 50 Hz.

3. Mean (or Average) Value and Root-Mean-Square Value of Alternating

Current

Mean (or Average) Value of Alternating Current :

An alternating current flows during one half-cycle in one direction and

during the other half-cycle in the opposite direction. Hence, for one

complete cycle the mean value of alternating current is zero. This is

why there is no deflection in a moving-coil galvanometer when an

alternating current passes through it.

However, the mean value of alternating current over half a cycle is a

finite quantity and, in fact, it is this quantity which is defined as the

'mean value' of alternating current. It is given by

imean =

T

2

0

1i dt,

T

2

where i is instantaneous value of the current. Now

i = i0 sin t,

where i0 is the peak value of the current and

T =

2.

mean 0

2i i

= 0.637 i0.

Thus, the mean (or average) value of a.c. for a half-cycle (t = 0 to t = T

2)

is 0.637 times, or 63.7%, of the peak value.

Similarly, the mean value of a.c. for the other half-cycle (t = T

2 to t = T)

will be 0.637 i0. Obviously, it is zero for the full cycle.

Root-mean-square Value of Alternating Current:

The root-mean- square (rms) value of an alternating current is defined

as the square-root of the average of i2 during a complete cycle, where i

is the instantaneous value of the alternating current.

Now, the average value of i2 over a complete cycle is given by

T

2 2

0

1i i dt.

T

Putting i = i0 sin t

and T =

2,

we get

2

2 0ii .2

Hence the root-mean-square value of the alternating current is given by

2rmsi i

= 0 0i

0.707 i .2

Thus, the root-mean-square value of an alternating current is 0.707

times, or 70.7%, of the peak value.

In a similar way, the root-mean-square value of an alternating emf

represented by E = E0 sin t is given by

0rmsE

E2

= 0.707 E0.

The ammeter measuring alternating current gives directly the rms value

of the current. Similarly, the voltmeter measuring alternating voltage

gives the rms value of the voltage. If an ammeter connected in an

alternating-current circuit reads 5 A, it means that the rms value of the

current is 5 A. In our houses, the alternating current is supplied at 220 V.

It actually means that the rms value of the alternating voltage is 220 V,

The peak voltage of the current is

V0 = 2 rms value of the voltage

= 2 220

= 311 V.

Hence the voltage in domestic supply varies from + 311 V to 311 V in

each cycle. It is due to this reason that 220 V a.c. is more dangerous than

220 V d.c.

The root-mean-square value of alternating current has special

significance. If an alternating current given by i = i0 sin t passes through

a resistance R, the instantaneous rate of production of heat is

P = i2 R.

Since the magnitude of the current i is changing, the rate of production

of heat P will also be changing. The average' rate of production of heat

over one complete cycle of current is

2P i R,

where 2i is the mean of the square of the current (i2) over one complete

cycle, i.e. 2i = (irms)2. Thus

2

rmsP i R.

If we pass a direct current of strength irms in a resistance R, then also the

rate of production of heat will be (irms)2 R. Thus, the root-mean-square

value of an alternating current is equal to that direct-current which

would produce heat in a given resistance at the same rate as the

alternating current. Hence, the root-mean-square value of an

alternating-current (irms) is also called the 'effective value' or 'virtual

value' of the current.

ivirtual = irms

= 0i

.2

Similarly, the root-mean-square value of an alternating emf is equal to

that direct emf which would produce heat in a given resistance at the

same rate as the alternating emf. Hence, the root-mean-square value of

an alternating emf (Erms) is also called the 'effective value' or 'virtual

value' of the emf.

Evirtual = Erms

= 0E

.2

4. Measurement of Alternating Current and Voltage

If we pass an alternating current through an ordinary ammeter, then

there will be no deflection in its needle. The reason is that the direction

of alternating current changes very rapidly. Therefore, the direction of

the magnetic field produced due to the current in the ammeter-coil and

hence the direction of the deflection of the needle will also change with

equal rapidity. But, because of its inertia, the needle cannot change its

direction with that rapidity. So, it remains stationary in its mean position.

Similarly, on passing an alternating current through a voltameter, the

polarity of the plates of the voltameter will change very rapidly and, as a

result, there will be no electrolysis.

Clearly, to measure alternating current, some property of the current

which is independent of the direction of the current, will have to be

used. An example is the 'heating effect' of current. We know that when

an electric current is passing through a resistance wire, the wire is

heated and its length is increased, whatever be the direction of current.

Hence it can be used to measure the alternating current. Ammeters and

voltmeters used to measure alternating currents and alternating

voltages are based on this principle. These are called 'hot-wire ammeter'

and 'hot-wire voltmeter'.

Hot-wire Ammeter :

It consists of a resistance-wire AB (Fig. 2) of high coefficient of linear

expansion and high specific resistance which is stretched between two

screws. This wire is made of platinum-iridium alloy which is not easily

oxidised. A silk-thread tied to the middle point C of the wire AB is wound

round a cylinder D and then connected to a spring M. The cylinder D can

rotate about its axis. At its centre is attached a pointer which moves over

a scale graduated in ampere. A and B are connected to two binding-

screws P1 and P2 fitted at the base of the instrument. A thick wire called

the 'shunt' is connected between P1 and P2 in parallel with AB.

Working :

When alternating current is passed through the wire AB, the length of

the wire increases and it becomes loose. As a result, the spring M pulls

the silk thread downward and the cylinder D rotates in the clockwise

direction and the pointer moves towards right on the scale. Whatever be

the direction of current in the wire, the pointer will always move

towards right. (That is why no positive and negative signs are marked on

the scale). The deflection of the pointer is proportional to the increase in

the length of the wire and this increase in length is proportional to the

heat produced. According to Joule's law, the heat produced in the wire is

proportional to the square of the strength of the current in the wire.

Thus, the deflection 6 of the pointer is proportional to the square of the

current (i2) through the wire, i.e.,

2i .

But the scale is so graduated that it reads directly in ampere. This is why,

the marks on the scale are not equidistant, and the distance between

them goes on increasing.

Hot-Wire Voltmeter:

The construction of the hot-wire voltmeter is just like the hot-wire

ammeter except that instead of shunt, a high-resistance wire is

connected in series with AB.

Here, the deflection 0 of the pointer is proportional to the square of the

potential difference (V2) across the ends of the wire, i.e.,

2V

But the scale is so graduated that it reads directly in volt. This is why the

scale is not marked at equal distances. In the beginning the marks are

closer but gradually the distance between them increases.

5. Phasors and Phasor Diagrams

In an alternating-current circuit, although the frequency of alternating

current and alternating emf is same but it is not necessary that the

alternating current and alternating emf be in the same phase. Usually,

when the current in the circuit is maximum, the emf is not maximum.

The phase difference between the two depends upon the type of the

circuit. In certain circuits, the current reaches its maximum value after

the emf becomes maximum. Then the current is said to 'lag' behind the

emf. In certain other circuits, the current reaches its maximum value

before the emf has become maximum. In such cases the current is said

to 'lead' the emf.

The study of a.c. circuits is much simplified if we treat alternating current

and alternating emf as vectors with the angle between the vectors equal

to the phase difference between the current and the emf. The current

and emf vectors are more appropriately called 'phasors'. A diagram

representing alternating current and alternating voltage (of same

frequency) as vectors (phasors) with the phase angle between them is

called a 'phasor diagram'.

6. Different Types of A.C. Circuits

In an alternating-current circuit the phase difference between the

current and the emf depends upon the type of the circuit. We shall first

study simple circuits containing only one element (resistor, inductor or

capacitor) and then circuits containing combinations of these basic

circuit elements.

(i) Circuit containing Resistance only:

Let an alternating emf given by

E = E0 sin t, (i)

be applied across a pure (non-inductive) resistance R Let i be the

instantaneous current in the circuit. By Ohm's law, the applied emf at

any instant, E, must be equal to iR, the p.d. across the resistance at that

instant. That is

E = iR

or E0 sin t = iR

or i = 0E

sin t.R

But 0 0E

i ,R

the peak value of the current in the circuit.

0i i sin t. (ii)

A comparison of eq. (ii) with eq. (i) shows that in a pure resistor the

current is always in phase with the applied emf. This phase relationship

graphically.

(ii) Circuit containing Inductance only :

Let an alternating emf given by

E = E0 sin t,

be applied across a pure (zero resistance) coil of inductance L . As the

current i in the coil varies continuously, an opposing (back) emf is

induced in the coil whose magnitude is di

L ,dt

where di

dt is the rate of change of current. The net instantaneous emf is

thus E0 sin Ldi

.dt

But this should be zero because there is no

resistance in the circuit. Thus

0

diE sin t L 0

dt

or 0di

L E sin tdt

or 0E

di sin t dt.L

Integrating both sides,

we have

0Edi sin t dt

L

or

0E cos ti

L

=

0E cos tL

or i =

0E sin t .L 2

The maximum value of sin

t

2 is 1. Therefore,

0E

L is the

maximum current is the circuit. Thus

i =

0i sin t ,

2

where

00

Ei

Lis the peak value of current. A comparison of this

equation with the emf equation E = E0 sin t shows that in a pure

inductor the current lags behind the emf by a phase angle of

2 or 90

(or the emf leads the current by a phase angle of

2 or 90). This means

that when the emf is maximum, the current is zero and vice-versa. Fig. 7

(a) shows this phase relationship graphically. Fig. 7 (b) is the

corresponding phasor diagram.

Inductive Reactance :

The peak value of current in the coil is

00

Ei .

L

Applying Ohm's law, we find that the product L has the dimensions of

resistance. It represents the 'effective opposition' of the coil to the flow

of alternating current. It is known as the 'reactance of the coil' or

'inductive reactance' and is denoted by XL. Thus

XL = L = 2 fL,

Where f is the frequency of the alternating current. Thus, the inductive

reactance increases with increasing frequency of the current. This is

easy to understand.

The inductive reactance XL (= 2 f L) is zero for d.c. for which f = 0.

(iii) Circuit containing Capacitance only: Let an alternating emf given by

E = E0 sin t be applied across the plates of a perfect capacitor of

capacitance C. As the emf is alternating, the charge on the capacitor

plates varies continuously and correspondingly current flows in the

connecting leads. Let q be the charge on the capacitor plates and i the

current in the circuit at any instant. Since there is no resistance in the

circuit, the instantaneous p.d. q

C across the capacitor plates must be

equal to the (alternating) applied e.m.f. This is

0q

E sin t.C

The instantaneous current i in the circuit is, therefore, given by

dq

idt

= 0d

E sin tdt

= 0E cos t

=

0E sin t1 2

C

The maximum value of sin

t is 1.

2 Therefore,

0E

1

C

is the

maximum current in the circuit.Thus

0i i sin t .

2

where

00

Ei

1

C

is the peak value of current. A comparison of this

equation with the emf equation E = E0 sin t shows that in a perfect

capacitor the current leads the emf by a phase angle of

2 or 90 (or

the emf lags behind the current by a phase angle of

2 or 90). This

means that when the emf is zero, the current is maximum and vice-

versa. shows this phase relationship graphically. The corresponding

phasor diagram.

Capacitive Reactance:

The peak value of current in the capacitor circuit is

00

Ei .

1

C

Applying Ohm's law, we find that the quantity

1

C has the dimensions

of resistance. It represents the 'effective opposition' of the capacitor to

the flow of alternating current. It is known as the 'reactance of the

capacitor' or 'capacitve reactance' and is denoted by XC. Thus

C

1 1X ,

C 2 f C

where f is the frequency of the alternating current. Thus, the capacitive

reactance decreases with increasing frequency of current.

When C is in farad and f in hertz (cycles/sec), then XC

1

2 f C is in

ohm.

The capacitive reactance

C

1X

2 f C in infinite for d.c. for which f =

0.

The reciprocal of reactance (inductive or capacitive) is called

'susceptance' of the a.c. circuit. It is measured in 'mho' or 'ohm1 (1) or

'siemen' (S).

(iv) Circuit containing Inductance and Resistance in Series (LR Series

Circuit):

Let an alternating emf E = E0 sin t be applied to a circuit containing an

inductance L and a non-inductive resistance R in series. The same current

will flow both in L and R.

Let i be the current in the circuit at any instant and VL and VR the p.d.'s

across L and R respectively at that instant. Then

VL = iXL

and VR = iR,

where XL is the inductive reactance. Now, VR is in phase with the current

i, while VL leads i by 90. Thus VR and VL are mutually at right angles.

(Their sum is not equal to the impressed emf E.)

The vector OA represents VR (which is in phase with i), while OB

represents VL (which leads i by 90). The vector OD represents the

resultant of VR and VL, which is the applied emf E. Thus

2 2 2R LE V V

or 2 2 2 2LE i R X

or 2 2L

Ei .

R X

Applying Ohm's law, we see that 2 2LR X is the effective resistance of

the circuit. It is called the 'impedance' of the circuit and is represented

by Z. Thus, in LR circuit,

we have

Z = 2 2LR X

But XL = L.

22Z R L .

The quantity Z, measured in ohm, is called impedance because it

impedes the flow of alternating current in the circuit.

The reciprocal of impedance is called 'admittance' of the a.c. circuit. It is

measured in 'mho' or 'ohm1'

(1) or 'siemen' (S).

(v) Circuit containing Capacitance and Resistance in series (CR Series

Circuit):

In this case (Fig. 11 a), the instantaneous p.d.'s across C and K are given

by

VC = i XC

and VR = i R,

where XC is the capacitive reactance and i is the instantaneous current.

Now VR is in phase with i, while VC lags behind i by 90. The phasor

diagram is drawn, in which the vector OA represents VR (in phase with i)

and the vector OB represents VC (lagging behind i by 90). The vector OD

represents the resultant of VR and VC which is the applied emf E. Thus

2 2 2R CE V V

or 2 2 2 2CE i R X

or i = 2 2C

E.

R X

Appling Ohm's law 2 2CR X is the effective resistance of the circuit

and is called the 'impedance' Z of the circuit. Thus, in CR circuit,

We have

2 2CZ R X .

But XC =

1.

C

2

2 1Z R .C

The phasor diagram (Fig. 11 b) shows that in CR circuit the aplied emf E

lags behind the current i (or the current leads the emf E) by a phase

angle , given by

C C

R

1V X Ctan .V R R

(vi) Circuit containing Inductance and Capacitance (LC Circuit):

In this case, the potential difference VL across L will lead the current i in

phase by 90, while the potential difference VC across C will lag behind

the current i in phase by 90. Thus the phase difference between VL and

VC will be 180, i.e. they will be in opposite phase to each other. Hence,

the resultant potential difference in LC circuit is

E = VL ~ VC

and the impedance of the circuit is

Z = XL ~ XC.

If, in the circuit, XL, = XC, then the impedance Z = 0. In this situation the

amplitude of current in the circuit would be infinite. It will be the

condition of 'electrical resonance'. Thus, in the condition of electrical

resonance, we have

XL = XC

or

1L

C

or

12 f L

2 f C

or

2

2

1f

4 LC

or

1 1f .

2 LC

This is called the 'resonant frequency' of the circuit.

(vii) Circuit containing Inductance, Capacitance and Resistance in Series (L

CR Series Circuit):

Let an alternating emf E = E0 sin t be applied to a circuit containing an

inductance L, a capacitance C and a resistance R all joined in series (Fig.

12a). The same current will flow in all the three and the vector sum of

the p.d.'s across them will be equal to the applied emf.

Let i be the current in the circuit at any instant of time and VL, VC and VR

the p.d.'s across L, C and R respectively at that instant. Then

VL = i XL,

VC = i XC

and VK = i R,

where XL and XC are the, inductive and capacitive reactances

respectively.

Now, VR is in phase with i but VL leads i by 90 while VC lags behind i by

90. The phasor diagram is drawn. In this diagram, the vector OA

represents VR (which is in phase with i), the vector OB represents VL

(which leads i by 90) and the vector OC represents VC (which lags behind

i by 90). VL and VC are opposite to each other. If VL > VC (as shown in

Fig.), then their resultant will be (VL VC) which is represented by the

vector OD. Finally, the vector OF represents the resultant of VR and (VL

VC), that is, the resultant of all the three, which is the applied emf E. Thus

22 2

R L CE V V V

or

22 2 2

L CE i R X X

or i =

22

L C

E.

R X X

Applying Ohm's law, we see that 22

L CR X X is the effective

resistance of the circuit and is called the 'impedance' Z of the circuit.

Thus, in LCR circuit,

we have

Z = 22

L CR X X

But XL = L

and XC =

1.

C

2

2 1Z R L .C

The phasor diagram (Fig. 12 b) shows that in LCR circuit the applied

emf E leads the current i by a phase angle , given by

L C

R

V Vtan

V

= L CX X

R

=

1L

C .R

The following three cases arise :

(i) When L >

1,

C then tan is positive i.e. is positive. In this

case, the emf E leads the current i.

(ii) When L <

1,

C then tan is negative i.e. is negative. In this

case, the emf E lags behind the current i.

(iii) When

1L ,

C then tan = 0 i.e. = 0. In this case, the emf

E and the current i are in phase.

Again, when L =

1

C,

we have

2

2 1Z R L R,C

which is the minimum value Z can have. Thus, in this case, the

impedance is minimum (and is purely resistive) and hence the

current is maximum. This is the case of 'electrical resonance'.

Hence at resonance

1L

C

(or XL = XC)

or 1

.LC

But = 2f, where f is the frequency of the applied emf.

Therefore

1f

2 LC

= f0 (say),

where

0

1f

2 LC is the natural frequency of the circuit when the

resistance is zero (or small). Thus, the condition for resonance is that the

frequency of the applied emf should be equal to the natural frequency of

the circuit when the resistance of the circuit is zero.

Impedance Triangle :

The impedance of an LCR a.c. circuit is given by

22

L CZ R X X

and the phase relationship given by

L CX X

tan ,R

in terms of the resistance and the reactances of the circuit elements may

be expressed by means of a right-angled triangle, as shown in Fig. 13.

This triangle is called as 'impedance triangle'.

7. Series Resonant Circuit

Series Resonant Circuit :

Let us consider an alternating-current circuit having inductance L,

capacitance C and resistance R connected in series. The impedance of

the circuit is

22

L CZ R X X

=

2

2 1R L .C

If the phase difference between the applied alternating voltage and the

resulting current be , then

L CX X

tanR

=

1L

C .R

the inductive reactance XL and the capacitive reactance XC of the circuit

be equal, then the voltage and the current will be in the same phase ( =

0). In this condition, the impedance Z of the circuit will be minimum (= R)

and the current i will be maximum. This is the condition of resonance.

Thus, for resonance,

we have

L CX X

1L

C

1

LC

f =

2

=

1

2 LC

is the frequency of the applied alternating emf. For resonance, it should

be

1.

2 LC

But

1

2 LC is the natural frequency of an LCR circuit (when R is

small). So, in a series resonant circuit, the frequency of the applied

voltage is equal to the natural frequency of the circuit.

A graph between the frequency f of the applied voltage and the current i

in the circuit is shown, in which

1f

2 LC is the natural frequency

of the circuit. For f = f0, the current is maximum (imax). Both for f < f0, and

f > f0, the current is less than its maximum value.

The maximum value of the current, however, depends on the resistance

R in the circuit. Two curves between f and i for circuit resistances R and

2R. The maximum current for 2R is half of that for R. Also, the resonance

is sharper for smaller resistance. The resonant frequency f0 remains,

however, unaffected.

The characteristic of the series resonant circuit is that the potential

differences available across the inductance L and across the capacitor C

may be much more than the applied voltage.

8. Q-factor of an LCR Circuit : Sharpness of Resonance : Half-power

Frequencies

Sharpness of Resonance :

When an alternating emf E0 sin t is applied to an LCR circuit, electrical

oscillations occur in the circuit with the frequency of the applied emf.

The amplitude of these oscillations (current amplitude) in the circuit is

given by

i0 =

0

2

2

E,

1R L

C

where

2

2 1R LC

is the "impedance" Z of the circuit.

The rapidity with which the current falls from its resonant value

0E

R

with change in applied frequency is known as 'sharpness of resonance'.

It is measured by the ratio of the resonant frequency 0 to the difference

of two frequencies 1 and 2 at which the current falls to 1

2 of the

resonant value. That is,

sharpness of resonance =

0

0 1

.

1 and 2 are known as 'half-power frequencies', because at these

frequencies the power in the circuit reduces to half its maximum value.

The difference of half-power frequencies, 2 1, is known as "band-

width". The smaller is the band-width, the sharper is the resonance.

Expression for Band-Width:

At resonant frequency 0, the impedance is R. Therefore, at 1 and 2

it must be 2 R , that is,

2

2 1Z R LC

= 2 R

or

2

2 21R L 2Rc

or

2

21L RC

or

1L R.

C

Thus, if 2 > 1,

we can write

1

1

1L R

C (i)

and

2

2

1L R.

C (ii)

Adding (i) and (ii),

we get

1 21 2

1 2

1L 0

C

or 1 21

.LC

Subtracting (i) from (ii),

we get

2 12 1

1 2

1L 2R

C

or

2 1

1 2

1L 2R

C

or 2 1 L L 2R

1 2

1

LC

or 2 1R

.L

(iii)

This is the expression for the "band-width".

Expression for Half-power Frequencies:

From, we have

2 12 0 2

= 0R

2L

and

2 11 0 2

0R

.2L

Quality (Q) Factor of LCR Circuit:

The Q-factor of an LCR circuit is a dimensionless quantity which

describes quantitatively the sharpness of resonance of the circuit. It is

defined as

Q = resonant frequency

band-width

=

0

2 1

,

where 2 and 1 are the half-power frequencies.

Substituting the value of 2 1 from eq. (iii),

we get

0L

Q .R

Also, 01

.LC

1 L

Q .R C

9. Power in a.c. Circuit

The rate of dissipation of energy in an electrical circuit is called the

'power'. It is equal to the product of the voltage and the current. If the

current be in ampere, the voltage in volt, then the power will be in

'watt'. The power of an alternating-current circuit depends upon the

phase difference between the voltage and the current.

The instantaneous values of the voltage and the current in an a.c. circuit

are given by

0E E sin t

and 0i i sin t

where is the phase difference between the voltage and the current.

Then, the instantaneous power in the circuit is

Pins = E i

= E0 sin t i0 sin (t )

= 0 0E i sin t sin t cos cos t sin

= 20 0E i sin t cos sin t cos t sin

=

2

0 0

1E i sin t cos sin2 t sin .

2

For one complete cycle, 21

sin t and sin 2 t 0.2

Therefore, the average power P in the circuit is given by

0 01

P E i cos2

= 0 0E i

cos2 2

or rms rmsP E i cos .

cos is known as the 'power factor' of the circuit and its value depends

upon the nature of the circuit.

Wattless Current:

If the circuit contains either inductance only or capacitance only

(resistance is zero), then the phase difference between current and

voltage is 90. The average power in such a circuit is

P = Erms irms cos 90 = 0.

[ cos 90 = 0]

Thus, if the resistance in an a.c. circuit is zero, although current flows in

the circuit, yet the average power remains zero, that is, there is no

energy dissipation in the circuit. The current in such a circuit is called

'wattless current'. In practice, however, a wattless current is not a reality

because no circuit can be entirely resistanceless.

It is for this reason that in a.c. circuits, either an inductor or a capacitor is

used for controlling the current.

Choke Coil

In a direct-current circuit, the current is reduced by means of a rheostat.

This results in a loss of electrical energy i2 R per second as heat. The

current in an alternating-current circuit may, however, be reduced by

means of a device which involves little loss of energy. This device is

called 'choke-coil'.

Construction :

It is a high-inductance coil made of thick, insulated copper wire wound

closely in a large number of turns over a soft-iron laminated core. Since

the wire is of copper and is thick, its resistance (R) is almost zero, but due

to the large number of turns and high permeability of the iron-core, its

inductance (L) is quite high. The coil, therefore, offers a large reactance

(L) and contributes to the impedance 2 2 2R L of the circuit.

Thus, it reduces the alternating current appreciably. There is an

arrangement to insert the core into the coil to any desired depth. More

is the length of the core inside the coil; greater is the inductance (L) of

the coil and so greater the impedance. Hence this coil can be used to

vary the current in the alternating-current circuit.

The current in an alternating-current circuit can also be reduced by

inserting a resistance R through a rheostat, but then an energy irms2 R will

be lost as heat in each cycle of current, where irms is the current in the

circuit. On the other hand, the loss of energy in a choke-coil is almost

negligible.

The average power dissipated in the choke coil (LR circuit) is given by

rms rmsP E i cos , (i)

where the power factor is given by

2 2 2R

cos .R L

Since the resistance R of the choke-coil is nearly zero and its inductance L

is very high, so

cos 0 approx.

Thus, according to eq. (i), the average power dissipated in choke-coil will

be nearly zero.

The resistance R of a choke coil is not exactly zero. That is why, in

practice, some electrical energy is lost as heat. In addition to it, energy is

also lost due to hysteresis-loss in the iron-core of the choke-coil. The loss

of energy due to eddy currents is reduced by laminating the iron-core.

The choke-coil can be used only in a.c. circuits, not in d.c. circuits,

because for direct-current ( = 0) the inductive reactance L of the coil

is zero, only the resistance of the coil remains effective which too is

almost zero.

Transformers

A transformer is a device, based on the principle of mutual induction,

which is used for converting large alternating current at low voltage into

small current at high voltage, and vice-versa. The transformers which

convert low voltages into higher ones are called 'step-up' transformers,

while those which convert high voltages into lower ones are called 'step-

down' transformers. Transformers are used only in a.c. (not in d.c.).

Construction :

A simple transformer consists of two coils called the 'primary' and the

'secondary', which are insulated from each other and wound on a

common soft-iron laminated core, One of the two coils has a smaller

number of turns of thick insulated copper wire while the other has a

large number of turns of thin insulated copper wire. In a step-up

transformer the coil of thick copper wire having smaller number of turns

is the primary coil, and the coil of thin wire having larger number of turns

is the secondary coil. In the step-down transformer, the order is

reversed.

Theory :

The given source of emf, say a.c. mains, is always connected to the

primary coil. When alternating current flows through the primary coil,

then in each cycle of current, the core is magnetised once in one

direction and once in the opposite direction. Hence an alternating

magnetic flux is produced in the core. Since the secondary coil is also

wound on the same core, the magnetic flux passing through it changes

continuously due to the repeated magnetisation and demagnetisation of

the core. Therefore, by mutual induction, an alternating emf of the same

frequency is induced in the secondary coil. The induced emf in the

secondary coil depends upon the ratio of the number of turns in the two

coils.

Let Np, and Ns be the number of turns in the primary and the secondary

coils respectively. Let us assume that there is no leakage of magnetic flux

so that the same flux passes through each turn of the primary and the

secondary. Let be the flux linked with each turn of either coil at any

instant. Then, by Faraday's law of electromagnetic induction, the emf

induced in the primary coil is given by

p pe N ,t

and the emf induced in the secondary coil is given by

s se N .t

s s

p P

e N.

e N

If the resistance of the primary circuit be negligible and there be no loss

of energy in it, then the induced emf ep in the primary coil will be nearly

equal to the applied voltage Vp across its ends. Similarly, if the secondary

circuit is open, then the voltage Vs across its ends will be equal to the

emf es induced in it. Under these ideal conditions, we have

s s s

p p p

V e Nr,

V e N

where r is called the 'transformation ratio'. In step-up transformer, r is

more than 1; whereas in step-down transformer, r is less than 1. Thus

voltage obtained across secondary

voltage applied across primary

= no. of turns in secondary

.no. of turns in primary

If ip and is be the currents in the primary and the secondary at any instant

and the energy losses be zero, then

power in the secondary

= power in the primary

i.e. s s p pV i V i

or p s s

s p p

i V Nr.

i V N

Thus, when the voltage is stepped-up, the current is correspondingly

reduced in the same ratio, and vice-versa. Thus, the energy obtained

from the secondary coil is equal to the energy given to the primary coil.

Obviously, a transformer is not a generator of electricity.

Energy Losses in a Transformer:

In practice, the power output of a transformer is less than the power

input because of unavoidable energy losses. These losses are :

(i) Copper Losses :

As the alternating current flows through the primary and the

secondary, heat is developed inside the copper turns. This waste

of energy is known as 'copper losses'.

(ii) Eddy Current Losses :

Eddy currents are set up in the iron core of the transformer and

generate heat, with consequent loss of energy. To minimise

these losses the iron core is laminated by making it of a number

of thin sheets of iron insulated from each other, instead of

making it from one solid piece of iron.

(iii) Flux Losses :

The coupling of the primary and the secondary coils is never

perfect. Therefore, the whole of the magnetic flux generated in

the primary does not pass through the secondary.

On account of these losses, we have

s s p pV i V i .

(iv) Hysteresis Losses :

During each cycle of a.c. the core is taken through a complete

cycle of "magnetisation. The energy expanded in this process is

finally converted into heat and is therefore wasted. This loss is

minimised by using the core of a magnetic alloy for which the

area of the hysteresis loop is a minimum.

Utility of Transformers in Long-distance Power Transmission :

If electric power generated at the power station, say 22,000 watt, be

transmitted over long distances at the same voltage as required by

consumers, say 220 volt, a number of disadvantages will arise :

(i) The current i flowing through the line wires will be very high (=

22,000/220 = 100 amp). Hence a large amount of energy (i2 Rt)

will be lost as heat during transmission, R being the resistance of

the line wires.

(ii) The voltage drop along the line wire (iR) will be considerable.

Hence the voltage at the receiving station will be considerably

lower than the voltage at the generating station.

(iii) The line wires, which are to carry the high current, will have to

be made thick. Such wires will be expensive and require

stronger poles to support them.

If however, the power is transmitted at a high voltage, (say

11,000 volt), all the above disadvantages almost disappear. The

current flowing through the line wires will then be only

22,000/11,000 = 2 amp. This will cause much less heating, much

less voltage-drop along the line wire and will require much

thinner line wires.

Hence, the electric power generated at the power station is

stepped-up to a very high voltage by means step-up transformer

and transmitted to distant places. At the place where the supply

is required, it is again stepped-down by a step-down

transformer.

Only alternating-current is suitable for this transmission because

only alternating-voltage can be stepped-up or stepped-down by

means of transformers.

AC GENERATOR OR DYNAMO

It is used to convert mechanical energy into electrical energy.

Construction :

The main components of ac generator are :

(i) Armature coil:

If consist of large number of turns of insulated copper wire

wound over iron core.

(ii) Magnet :

Strong permanent magnet (for small generator) or an

electromagnet (for large generator) with cylindrical poles in

shape.

(iii) Slip rings :

The two ends of the armature coil are connected to two brass

rings R1 and R2. These rings rotate along with the armature coil.

(iv) Brushes :

Two carbon brushes (B1 and B2), are pressed against the slip

rings. These brushes are connected to the load through which

the output is obtained.

Principle :

It works on the principle of electromagnetic induction. According to it

when a coil is rotated in magnetic field, an emf is induced in the coil. The

coil may be rotated by water energy, steam energy or oil energy. Let at

any instant magnetic flux through armature coil,

NB = NBA cos

= NBA cos t

The induced emf

e =

Bd

dt

= NBA sin t

or e = e0 sin t,

where e0 = NBA .

and induced current

i = e

R

= 0e

sin tR

= i0 sin t

DC GENERATOR

It produces direct current. It is possible by providing split rings or

commutator in place of slip rings. DC generator consists of:

(i) armature coil,

(ii) magnet,

(iii) split rings,

(iv) brushes.

For DC generator

output current

0i i sin t

DC MOTOR

It is an electrical machine which converts electrical energy into

mechanical energy.

Principle:

It is based on the fact that a current carrying coil placed in magnetic field

experiences a torque. Because of this torque the coil starts rotating.

Construction:

It consists of

(i) strong magnet,

(ii) armature,

(iii) split rings,

(iv) brushes.

By experiments,

M12 = M12 = M.

Thus

21di

e Mdt

(1)

and 12di

edt

(2)

The SI unit of M is henry.

The coefficient of mutual induction depends on the shape, size, the

mutual arrangement of the coils, as well as the magnetic permeability of

the medium surrounding the coils.

Reciprocity theorem

Calculations show (and experiments confirm) that in the absence of

material medium between the coils, the coefficients M12 and M21 are

equal:

M12 = M21

This property of mutual inductance is called the reciprocity theorem.

Because of this reason, we do not have to distinguish between M12 and

M21 and can simply speak of the mutual inductance of two circuits.

Example 43.

Two circular loops 1 and 2 whose centres coincide lie in a plane (see fig.

8.82). The radii of the loops are a1 and a2. Current i flows in loop 1. Find

the magnetic flux 2 associated by loop 2, if a1

0 < k < 1.

For tight coupling k = 1,

for loose coupling k < 1

k can be defined as,

k = Magnetic flux linked in secondary

Magnetic flux linked in primary