Chapter – 7

ALTERNATING CURRENT AND ELECTRICAL MACHINES

ALTERNATING CURRENT
An alternating current is that current whose magnitude changes continuously with time and
direction reverses periodically.
DIRECT CURRENT
Direct current is that current which flows with a constant magnitude in the same direction.

Alternating emf : Ɛ = Ɛ0 sin 𝜔𝑡

Alternating current : 𝐼 = 𝐼0 sin 𝜔𝑡
Ɛ0 = 𝑁𝐵𝐴𝜔 = 𝑝𝑒𝑎𝑘 𝑜𝑟 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑛𝑔 𝐸𝑀𝐹
Ɛ0
𝐼0 = = 𝑝𝑒𝑎𝑘 𝑜𝑟 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎. 𝑐. 𝑎𝑛𝑑 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
𝑅
Time period: The time taken by an alternating current to complete one cycle of its variations is called
its time period.
Frequency: The number of cycles completed per second by an alternating current is called its frequency
and is denoted by f.

AVERAGE (MEAN) VALUE OF A.C. OVER ONE COMPLETE CYCLE

𝑞
𝐼𝑎𝑣 = =0
𝑇
Because the net displacement of the charges in one complete cycle is zero.
AVERAGE (MEAN) VALUE OF A.C. OVER ONE HALF CYCLE
(Refer Notebook for the derivation)
It is defined as that value of direct current which sends the same charge in a circuit in the same
time as is sent by the given alternating current in its half time period.
2
𝐼𝑚 = 𝐼𝑎𝑣 = 𝐼 = 0.637 𝐼0
𝜋 0
2
Ɛ𝑚 = Ɛ𝑎𝑣 = Ɛ0 = 0.637Ɛ0
𝜋
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ROOT MEAN SQUARE (RMS) OR VIRTUAL OR EFFECTIVE VALUE OF A.C.
(Refer Notebook for the derivation)
It is defined as that value of direct current which produces the same heating effect in a given
resistor as is produced by the given alternating current when passed for the same time.
1
𝐼𝑣𝑖𝑟 = 𝐼𝑒𝑓𝑓 = 𝐼𝑟𝑚𝑠 = 𝐼0 = 0.707 𝐼0
√2
1
Ɛ𝑣𝑖𝑟 = Ɛ𝑒𝑓𝑓 = Ɛ𝑟𝑚𝑠 = Ɛ0 = 0.707 Ɛ0
√2

PHASORS AND PHASOR DIAGRAMS

A rotating vector that represents a simultaneously varying quantity is called a Phasor. This vector
is rotating with an angular velocity equal to the angular frequency of that quantity.
A diagram that represents alternating current and voltage of the same frequency as rotating
vectors (phasors) along with proper phase angle between them is called a phasor diagram or Argand
diagram.

A.C. CIRCUIT CONTAINING ONLY A RESISTOR (Refer Notebook for the derivation)

Ɛ = Ɛ0 sin 𝜔𝑡
𝐼 = 𝐼0 sin 𝜔𝑡
Hence the alternating emf Ɛ and the alternating current I are in same phase. 1e;  = 0

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A.C. CIRCUIT CONTAINING ONLY AN INDUCTOR (Refer Notebook for the derivation)

Ɛ = Ɛ0 sin 𝜔𝑡
𝐼 = 𝐼0 sin(𝜔𝑡 − 𝜋⁄2)
The alternating emf Ɛ is ahead of the alternating current I in phase by π/2 radian or the current lags
behind the voltage in phase by π/2 radian.

Inductive reactance XL: It is a measure of the effective resistance or opposition offered by the inductor
to the flow of a.c. through it.
𝑋𝐿 = 𝜔𝐿 = 2𝜋𝑓𝐿
Variation of XL with frequency:

In a.c circuit, 𝑋𝐿 ∝ 𝑓
In d.c circuit, the frequency is zero. Then 𝑋𝐿 = 0. ie; the inductor does not oppose d.c.

A.C. CIRCUIT CONTAINING ONLY A CAPACITOR (Refer Notebook for the derivation)

Ɛ = Ɛ0 sin 𝜔𝑡
𝐼 = 𝐼0 sin(𝜔𝑡 + 𝜋⁄2)

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The alternating current I is ahead of the alternating emf Ɛ in phase by π/2 radian or the current lags
behind the emf in phase by π/2 radian.

Capacitive reactance: It is the effective resistance or opposition offered by the capacitor to the flow of
a.c. through it.
1 1
𝑋𝐿 = =
𝜔𝐶 2𝜋 𝑓𝐶
Variation of capacitive reactance with frequency:

1
In a.c circuit, 𝑋𝐿 ∝ 𝑓
In d.c circuit, the frequency is zero. Then 𝑋𝐿 = ∞. ie; the capacitor blocks d.c.

A.C. CIRCUIT CONTAINING SERIES LCR-CIRCUIT (Refer Notebook for the derivation)

(i) Inductive LCR (ii) Capacitive LCR

Ɛ0
𝐼0 = , 𝐼𝑓 𝑋𝐿 > 𝑋𝐶
√𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2

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IMPEDANCE (Z)
It is the effective resistance of the series LCR circuit which opposes or impedes the flow of
current through it and is called its impedance. SI unit is ohm.

1 2
𝑍= √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2 = √𝑅 2 + (𝜔𝐿 − ) , 𝐼𝑓 𝑋𝐿 > 𝑋𝐶
𝜔𝐶
Special Cases:
(i) Inductive LCR
𝑋𝐿 > 𝑋𝐶 & 𝑉𝐿 > 𝑉𝐶
𝑍 = √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2
Ɛ = Ɛ0 𝑠𝑖𝑛 𝜔𝑡
𝐼 = 𝐼0 𝑠𝑖𝑛(𝜔𝑡 + )
The alternating emf Ɛ is ahead of the alternating current I in phase by 
𝑅
𝑐𝑜𝑠  = 𝑍
𝑋𝐿 −𝑋𝐶
𝑡𝑎𝑛  = 𝑍

(ii) Capacitive LCR

𝑋𝐿 < 𝑋𝐶 & 𝑉𝐿 < 𝑉𝐶
𝑍 = √𝑅 2 + (𝑋𝐶 − 𝑋𝐿 )2
Ɛ = Ɛ0 𝑠𝑖𝑛 𝜔𝑡
𝐼 = 𝐼0 𝑠𝑖𝑛(𝜔𝑡 − )
The alternating current I is ahead of the alternating emf Ɛ in phase by 
𝑅
𝑐𝑜𝑠  = 𝑍
𝑋𝐶 −𝑋𝐿
𝑡𝑎𝑛  = 𝑍

(iii) Purely resistive LCR (At Resonance)

𝑋𝐿 = 𝑋𝐶 & 𝑉𝐿 = 𝑉𝐶
𝑍 = √𝑅 2 + (𝑋𝐶 − 𝑋𝐿 )2 = 𝑅
Ɛ = Ɛ0 𝑠𝑖𝑛 𝜔𝑡
𝐼 = 𝐼0 𝑠𝑖𝑛 𝜔𝑡
The alternating emf Ɛ and alternating current I are in phase.ie; 𝜑 = 0
𝑅 𝑅
𝑐𝑜𝑠  = 𝑍 = 𝑅 = 1
𝑋𝐿 −𝑋𝐶
𝑡𝑎𝑛  = =0
𝑍

RESONANCE CONDITION
A series LCR-circuit is said to be in the resonance condition when the current through it has its
maximum value (ie; the impendence has its minimum value).
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 𝑖𝑒; 𝑋𝐿 = 𝑋𝐶

𝑍 = √𝑅 2 + (𝑋𝐶 − 𝑋𝐿 )2 = 𝑅 (𝑍 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑖𝑒; 𝐿𝐶𝑅 𝑖𝑠 𝑝𝑢𝑟𝑒𝑦 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑒)

1
Angular Resonant frequency, 𝑟 =
√𝐿𝐶
1
Linear Resonant frequency, 𝑓𝑟 = 2
√𝐿𝐶

Characteristics of LCR resonant circuit:

1. X L = X C
2. The impedance is minimum and purely resistive.
Ɛ0
3. The current is maximum, 𝑅
Ɛ2𝑟𝑚𝑠
4. Power dissipation is maximum, 𝑅
5. Series resonance can occur at all values of R.
6. The current is in phase with the voltage
7. Power factor is unity
8. The series resonant circuit is called an acceptor circuit. When a number of frequencies are fed to
it, it accepts only one frequency 𝑓𝑟 and rejects the other frequencies.
Sharpness of resonance (Q – factor):

The Q – factor of a series resonant circuit is defined as the ratio of the resonant frequency to the
difference in two frequencies taken on both sides of the resonant frequency such that at
1
each frequency, the current amplitude becomes times the value of the resonant
√2
frequency.
r r
Q= = 𝑊ℎ𝑒𝑟𝑒 2  = 𝐵𝑎𝑛𝑑 𝑤𝑖𝑑𝑡ℎ
2 − 1 2 
The Q-factor of a series LCR-circuit may also be defined as the ratio of the voltage drop across the
inductance (or capacitance) at resonance to the applied voltage.

r 𝐿 1 1 𝐿
Q= = = √
𝑅 r 𝐶 𝑅 𝑅 𝐶

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AVERAGE POWER IN AN A.C CIRCUIT (Refer Notebook for the derivation)
If the instantons value of alternating EMF is, Ɛ = Ɛ0 𝑠𝑖𝑛 𝜔𝑡 and
the instantons value of alternating current is, 𝐼 = 𝐼0 𝑠𝑖𝑛(𝜔𝑡 + )
Then the average power dissipated, 𝑃𝑎𝑣 = 𝑟𝑚𝑠 𝐼𝑟𝑚𝑠 cos 
Special Cases:
(i) Power dissipated in a Pure Inductive circuit
𝜋
= , cos  = 0
2
𝑃𝑎𝑣 = 0
Thus, the average poser consumed in an inductive circuit over a complete cycle is zero
(ii) Power dissipated in a Pure Capacitive circuit
𝜋
= , cos  = 0
2
𝑃𝑎𝑣 = 0
Thus, the average poser consumed in a capacitive circuit over a complete cycle is zero
(iii) Power dissipated in a Pure Capacitive circuit
 = 0, cos  = 1
Ɛ2𝑟𝑚𝑠
𝑃𝑎𝑣 = 𝑟𝑚𝑠 𝐼𝑟𝑚𝑠 =
𝑅
(iv) Power dissipated in an LCR circuit at resonance (Pure resistive)
 = 0, cos  = 1
Ɛ2𝑟𝑚𝑠
𝑃𝑎𝑣 = 𝑟𝑚𝑠 𝐼𝑟𝑚𝑠 =
𝑅
POWER FACTOR (𝐜𝐨𝐬 )
𝑃𝑎𝑣 𝑇𝑟𝑢𝑒 𝑃𝑜𝑤𝑒𝑟
cos  = =
𝑟𝑚𝑠 𝐼𝑟𝑚𝑠 𝐴𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑃𝑜𝑤𝑒𝑟
Thus, the power factor may be defined as the ratio of the true power to the apparent power of an
a.c. circuit.
WATTLES CURRENT
The current in a.c. circuit is said to be wattles if the average power consumed in the circuit is
zero.
Energy produced in a Resistor: 𝐻 = 𝐼 2 𝑅𝑡
1
Energy stored in a Capacitor: 𝑈= 𝐶𝑉 2
2
1
Energy stored in an inductor: 𝑈= 𝐿𝐼02
2

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ELECTRICAL MACHINES

(1) TRANSFORMER
A transformer is an electrical device for converting an alternating current at low voltage into
that at high voltage or vice versa. If it increases the input voltage, it is called step up transformer and
if it decreases the input voltage, it is called step down transformer.

Principle: It works on the principle of mutual induction, i.e., when a changing current is passed through
one of the two inductively coupled coils, an induced emf is set up in the other coil.
Construction: Refer NCERT
Theory: (Refer Notebook for the derivation)
Ɛ2 𝑁2
=
Ɛ1 𝑁1
𝑁2
The ratio is called turns ratio of the transformer. It is also called transformation ratio.
𝑁1
𝐼1 Ɛ2 𝑁2
= =
𝐼2 Ɛ1 𝑁1
In a step-up transformer, 𝑁1 < 𝑁2 , then Ɛ1 < Ɛ2 and 𝐼1 > 𝐼2
In a step-down transformer, 𝑁1 > 𝑁2 , then Ɛ1 > Ɛ2 and 𝐼1 < 𝐼2

Assumptions used:
1. The primary resistance and the current are low
2. There is no flux leakage
3. The terminals of the secondary are open
Energy losses in transformers: (Refer Book for the explanations)
1. Copper loss
2. Eddy current loss
3. Hysteresis loss
4. Flux leakage
5. Humming loss

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Uses of transformers:
1. Small transformers are used in radio receivers, telephones, loud speakers, etc.
2. In voltage regulators for TV, refrigerators, air-conditioners, computers, etc.
3. In stabilized power supplies.
4. A step-down transformer is used for obtaining large current for electric welding.
5. A step-down transformer is used in induction furnace for melting metals.
6. A step-up transformer is used for the production of X-rays.

(2) A.C. GENERATOR

A generator or dynamo is a device which converts mechanical energy into electrical
energy.
Principle: The working of an a.c. generator is based on the principle of electromagnetic induction.
Whenever the magnetic flux linked with a closed circuit changes, an emf (and hence a current) is induced
in it.
Construction: Refer NCERT
1. Field magnet
2. Armature
3. Slip rings
4. Brushes
5. Source of energy

Working: (Refer NCERT)

Theory: (Refer Notebook, Chapter 6)
Ɛ = Ɛ0 sin 𝜔𝑡
𝐼 = 𝐼0 sin 𝜔𝑡
Ɛ0 = 𝑁𝐵𝐴𝜔 (𝑃𝑒𝑎𝑘 𝑜𝑟 𝑎𝑚𝑝𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑖𝑛𝑑𝑢𝑐𝑒𝑑 𝐸𝑀𝐹)
Ɛ0 𝑁𝐵𝐴𝜔
𝐼0 = = (𝑃𝑒𝑎𝑘 𝑜𝑟 𝑎𝑚𝑝𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑖𝑛𝑑𝑢𝑐𝑒𝑑 𝑐𝑢𝑟𝑟𝑒𝑛𝑡)
𝑅 𝑅

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Advantages of a.c. over d.c.

1. The generation of a.c. is more economical than d.c.

2. The alternating voltage can be easily stepped up or stepped down by using a transformer.
3. The alternating currents can be reduced by using by a choke coil without any significant wastage
of energy.
4. The alternating currents can be transmitted to distant places without any significant line loss.
5. Also a.c. can be easily converted into d.c. by using rectifiers.
6. A.C. machines are simple and robust and do not require much attention during their use.

Disadvantages of a.c. over d.c.

1. Peak value of a.c. is high (𝐼0 = √2 𝐼𝑟𝑚𝑠 ). It is dangerous to work with a.c.
2. In phenomena like electroplating, electro-refining, electrotyping, etc; a.c. cannot be used.
3. A.C. is transmitted more from the surface of conductor than from inside. This is called skin
effect. Therefore, several fine insulated wires (and not a single thick wire) are required for the
transmission of a.c.

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