Alternating Current

An electric current whose magnitude changes continuously with time and changes its
direction periodically, is called an alternating current.

The instantaneous value of alternating current at any instant of time t is given by

I = Io sin ωt

where, 10 = peak value of alternating current.

The variation of alternating current with time is shown in graph given below

Mean or average value of alternating current for first half cycle

Im = 2Io / π = 0.637 Io

Mean or average value of alternating current for next half cycle

I’m = – 2Io / π = – 0.637 Io

Mean or average value of alternating current for one complete cycle = O.

Root mean square value of alternating current

Iv = Irms = Io / √2 = 0.707 Io

Where, Io = peak value of alternating current.

Root mean square value of alternating voltage

Vrms = Vo / √2 = 0.707 Io = 0.707 Vo

Reactance

The opposition offered by an inductor or by a capacitor in the path of flow of

Reactance is of two types

(i) Inductive Reactance (XL) Inductive reactance is the resistance offered by an

Inductive reactance (XL) = Lω = L2πf = L2π / T

Its unit is ohm. XL ∝ f

For direct current, XL = 0 (f = 0)

(ii) Capacitive Reactance (Xc) Capacitive reactance is the resistance offered by an

Capacitive reactance,

Xc = 1 / Cω = 1 / C2πf = T / C 2π

Its unit is ohm Xc ∝ 1 / f

For direct current, Xc = ∞ (f = 0)

The opposition offered by an AC circuit containing more than one out of three
components L, C and R, is called impedance (Z) of the circuit.

Impedance of an AC circuit, Z = √R2 + (XL – XC)2

Its SI unit is ohm.

Power in an AC Circuit

The power is defined as the rate at which work is being in the circuit.

The average power in an AC circuit,

Pav = Vrms irms cos θ

= V / √2 i / √2 cos θ = Vi / √2 cos θ

where, cos θ = Resistance(R) / Impedance (Z) is called the power factor 0f AC

Current and Potential Relations

Here, we will discuss current and potential relations for different AC circuits.

(i) Pure Resistive Circuit (R circuit)

(a) Alternating emf, E = Eo sin ωt

(b) Alternating current, I = Io sin ωt

(c) Alternating emf and alternating current both are in the same phase.

(d) Average power decay, (P) = Ev . Iv

(e) Power factor, cos θ = 1

(ii) Pure Inductive Circuit (L Circuit)

(a) Alternating emf, E = Eo sin ωt

(b) Alternating current, I = Io sin (ωt – π / 2)

(c) Alternating current lags behind alternating emf by π / 2.

(d) Inductive reactance, XL = Lω = L2πf

(e) Average power decay, (P) = 0

(f) Power factor, cos θ = cos 90° = 0

(iii) Pure Capacitive Circuit

(a) Alternating emf, E = Eo sin ωt

(b) Alternating current, I = Io sin (ωt + π / 2)

(d) Inductive reactance, XL = Cω = C2πf

(e) Average power decay, (P) = 0

(f) Power factor, cos θ = cos 90° = 0

(iv) R – C Circuit

E = Eo sin ωt

I = Eo / 2 sin (ωt – φ)

Z = √R2 + (1 / ωC)2

tan φ = – 1 / ωC / R

Current leading the voltage by φ

V2 = V2R = V2C

(v) L – C Circuit

(vi) L – C – R Circuit

(a) Alternating emf, E = Eo sin Ωt

(b) Alternating current, I = Io sin (Ωt ± θ)

(c) Alternating current lags leads behind alternating emf by ω.

(d) Resultant voltage, V = √V2R + (VL – VC)2

(e) Impedance, Z = √R2 + (XL – XC)2

(f) Power factor, cos θ = R / Z = R / √√R2 + (XL – XC)2

(g) Average power decay, (P)= EVIV cos θ

Resonance in AC Circuit

The condition in which current is maximum or impedance is minimum in an AC circuit,

(i) Series Resonance Circuit

In this circuit components L, C and R are connected in series.

At resonance = XL = XC

Resonance frequency f = 1 / 2π√LC

A series resonance circuit is also known as acception circuit.

(ii) Parallel Resonance Circuit

In this circuit L and C are connected in parallel with each other.

Impedance (Z) of the circuit is maximum.

Current in the circuit is minimum.

Wattless Current

Average power is given by

Pav = Erms = Irms cos θ

Here the Irms cos φ contributes for power dissipation. Therefore, it is called
wattless current.

AC Generator or Dynamo

It is a device which converts mechanical energy into alternating current energy.

Its working is based on electromagnetic induction.

The induced emf produced by the AC generator is given by

e = NBAω sin ωt = eo = sin ωt

There are four main parts of an AC generator

(i) Armature It is rectangular coil of insulated copper wire having a large number
of turns.

(ii) Field Magnets These are two pole pieces of a strong electromagnet.

(iii) Slip Rings These are two hollow metallic rings.

(iv) Brushes These are two flexible metals or carbon rods, which remains slightly
in contact with slip rings .

Note An DC generator or dynamo contains split rings or commutator inspite of slip

DC Motor

It is a device which converts electrical energy into mechanical energy.

Its working is based on the fact that when a current carrying coil is placed in
uniform magnetic field a torque acts on it.

Torque acting on a current carrying coil placed in uniform magnetic field

τ = NBIA sin θ

When armature coil rotates a back emf is produced in the coil.

Efficiency of a motor,

η = Back emf / Applied emf = E / V

Transformer
It is a device which can change a low voltage of high current into a high voltage
of low current and vice-versa.

Its working is based on mutual induction.

There are two types of transformers.

(i) Step-up Transformers It converts a low voltage of high current into a high
voltage of low current.

In this transformer,

Ns > NP, Es > EP

and IP > IS

(ii) Step-down Transformer It converts a high voltage of low current into a low
voltage of high current.

In this transformer,

NP > NS, EP > ES and IP < IS

Transformation Ratio

Transformation ratio,

K = NS / NP = ES / EP = IP / IS

For step-up transformer, K > 1

For step-down transformer, K < 1

Energy Losses in a Transformer

The main energy losses in a transformer are given below

Iron loss
Copper loss
Flux loss
Hysteresis loss
Humming loss
Important Points

Transformer does not operate on direct current. It operates only on alternating

A choke coil is a pure inductor. Average power consumed per cycle is zero in a
choke coil.
A DC motor connects DC energy from a battery into mechanical energy of rotation.
An AC dynamo/generator produces are energy from mechanical energy of rotation of a
coil.
An induction coil generates high voltages of the order of 1OS V from a battery.
It is based on the phenomenon of mutual induction