Alternating current

 1. Alternating Current

Alternating current is the one which changes in magnitude continuously and in direction periodically. The maximum value of current is called current-amplitude or peak value of current.

It is expressed as

I = I0 sin ωt

Similarly alternating voltage (or emf) is

V = V0 sin ωt

2. Mean and RMS Value of Alternating Currents

The mean value of alternating current over complete cycle is zero

(Imean)full cycle =0

While for half cycle it is

(Imean)half cycle =2I0π=0.636I0

Vav=2V0π=0.636V0

An electrical device reads root mean square value as

Irms=(I2)mean−−−−−−−√=I02√=0.707I0;Vrms=V02√=0.707V0

3. Phase Difference between Voltage and Current

In a circuit having a reactive component, there is always a phase difference between applied voltage and the alternating current.

If E = E0 sin ωt

Current is I = I0 sin (ωt+φ)

where φ is the phase difference between voltage and current.

4. Impedance and Reactance

Impedance: The opposition offered by a reactive component and a reactive component to an alternating current is called Impedance. It is denoted as Z. Its unit is ohm.

Z=VI=V0I0=VrmsIrms

Reactance : The opposition offered by inductance and capacitance or both in ac circuit is called reactance. It is denoted by XC or XL.

The opposition due to inductor alone is called the inductive reactance while that due to capacitance alone is called the capacitive reactance.

Inductive reactance, XL=ωL

Capacitive reactance, XC=1ωC

5. Purely Resistive Circuit

If a circuit contains pure resistance, then phase difference φ=0 i.e., current and voltage are in the same phase.

Impedance, Z = R

6. Purely Inductive Circuit

If a circuit contains pure inductance, then φ=–π2, i.e., current lags behind the applied voltage by an angle π2.

i.e., If V = V0 sin ωt

I=I0sin(ωt–π2)

In this case inductive reactance, XL= ωL

The inductive reactance increases with the increase of frequency of AC linearly (fig. b).

7. Purely Capacitive Circuit

If circuit contains pure capacitance, then φ=π2, i.e., current leads the applied voltage by angle π2.i.e.,

V=V0sinωt,I=I0sin(ωt+π2)

Capacitance reactance, XC=1ωC

Clearly capacitance reactance (XC) is inversely proportional to the frequency ν (fig. b).

8. LC Oscillations

A circuit containing inductance L and capacitance C is called an LC circuit. If capacitor is charged initially and ac source is removed, then electrostatic energy of capacitor (q20/2C) is converted into magnetic energy of inductor (12LI2) and vice versa periodically; such oscillations of energy are called LC oscillations. The frequency is given by

ω=1LC√⇒2πν=1LC√

9. Series LCR Circuit

If a circuit contains inductance L, capacitance C and resistance R, connected in series to an alternating voltage V = V0 sin ωt

then impedance Z=R2+(XC–XL)2−−−−−−−−−−−−−−√

and phase φ=tan–1XC–XLR

Net voltage V=V2R+(VC–VL)2−−−−−−−−−−−−−√

10. Resonant Circuits

Series LCR circuit : In series LCR circuit, when phase (φ) between current and voltage is zero, the circuit is said to be resonant circuit. In resonant circuit XC=XLor1ωC=ωL

⇒ω=1LC√

Resonant angular frequency ωr=1LC√

(linear) frequency, νr=ωr2π=12πLC√

At resonant frequency φ =0, V = VR

Quality factor (Q)

The quality factor (Q) of series LCR circuit is defined as the ratio of the resonant frequency to frequency band width of the resonant curve.

Q=ωrω2–ω1=ωrLR

Clearly, smaller the value of R, larger is the quality factor and sharper the resonance. Thus quality factor determines the nature of sharpness of resonance. It has no unit.

11. Power Dissipation in AC Circuit is

P=VrmsIrmscosφ=12V0I0cosφ

where cosφ=RZ is the power factor.

For maximum power

cos φ =1 or Z = R

i.e., circuit is purely resistive.

For minimum power

cos φ =0 or R = 0

i.e., circuit should be free from ohmic resistance.

Power loss, P = i2R

12. Wattless Current

In purely inductive or purely capacitive circuit, power loss is zero. In such a circuit, current flowing is called wattless current.

Iwattless=Isinφ=I(XCZ)=I(XLZ)

13. AC Generator

It is a device used to convert mechanical energy into electrical energy and is based on the phenomenon of electromagnetic induction. If a coil of N turns, area A is rotated at frequency f in uniform magnetic field of induction B, then motional emf in coil (if initially it is perpendicular to field) is

ε=NBA ω sin ωt with ω = 2πν

Peak emf, ε0= NBA ω

14. Transformer

A transformer is a device which converts low ac voltage into high ac voltage and vice versa. It works on the principle of mutual induction. If Np and NS are the number of turns in primary and secondary coils, VP and IP are voltage and current in primary coil, then voltage (VS) and current (IS) in secondary coil will be

VS=(NSNP)VPandIS=(NPNS)IP

Step up transformer increases the voltage while step down transformer decreases the voltage.

In step up transformer VS > VP so NS > NP

In step down transformer VS < VP so NS < NP

Energy Losses and Efficiency of a Transformer

(i) Copper Losses: When current flows in primary and secondary coils, heat is produced. The power loss due to Joule heating in coils will be i2R where R is resistance and i is the current.

(ii) Iron Losses (Eddy currents): The varying magnetic flux produces eddy currents in iron-core, which leads to dissipation of energy in the core of transformer. This is minimised by using a laminated iron core or by cutting slots in the plate.

(iii) Flux Leakage: In actual transformer, the coupling of primary and secondary coils is never perfect, i.e., the whole of magnetic flux generated in primary coil is never linked up with the secondary coil. This causes loss of energy.

(iv) Hysteresis Loss: The alternating current flowing through the coils magnetises and demagnetises the iron core repeatedly. The complete cycle of magnetisation and demagnetisation is termed as hysteresis. During each cycle some energy is dissipated. However, this loss of energy is minimised by choosing silicon-iron core having a thin hysteresis loop.

(v) Humming Losses: Due to the passage of alternating current, the core of transformer starts vibrating and produces humming sound. Due to this a feeble part of electrical energy is lost in the form of humming sound.

On account of these losses the output power obtained across secondary coil is less than input power given to primary. Therefore, the efficiency of a practical transformer is always less than 100%.

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