Topics to be discussed1. CURRENT2. TYPES OF CURRENT3. A.C4. D.C5. P.D.C6. TERMS USED IN A.C7. RECTANCE8. IMPEDENCE9. LCR circuit

CURRENTThe rate of flow of charge through any cross-section of a substance is

known as CURRENT.

CURRENT =

I=

S.I UNITS : ampere(A)

1A=1CS-1

TOTAL CHARGE FLOWING(Q) TIME TAKEN(T)

Q T

TYPES OF CURRENT

ALTERNATING CURRENT

• THE CURRENT WHICH CHANGES ITS MAGNITUDE CONTINUOUSLY WITH TIME AND REVERSES ITS DIRECTION PERIODICALLY.

DIRECT CURRENT

• THE CURRENT WHICH NEITHER CHANGES ITS MAGNITUDE NOR ITS DIRECTION WITH TIME.

PULSATING D.C

• THE CURRENT WHICH CHANGES ITS MAGNITUDE CONTINUOUSLY WITH TIME BUT KEEPS ITS DIRECTION SAME THROUGH OUT THE TIME .

TERMS USED IN ALTERNATING

CURRENT/EMF• PEAK VALUE:- the maximum value of alternating current /emf in the positive or –ve

direction is called amplitude or peak value of alternating current/emf.

• It is denoted by I0 or (E0)

• ROOT MEAN SQUARE OF ALTERNATING CURRENT:-Root mean square or effective or virtual value of alternating current is defined as that steady current

which when passed through any resistance for any given time would produce the same amount of heat as is produced by the alternating current when passed through

the same resistance for the same time.

• Irms=IV=I0/ √2 = 0.707I0

Erms =EV =E0/√2 =0.707 E0 The rms value of alternating current/emf is 0.707 times its peak value.

• FREQUENCY OF ALTERNATING CURRENT/EMF :- it is defined as the number of cycles completed by alternating current/emf in one second.

MATHEMATICALLY, = 1 time period (T)

BUT T= 2∏/ ω = ω/ 2∏

UNITS : hertz (HZ)

ANGULAR FREQUENCY:- It is given by ω=2∏

it is expressed in radians per second.

•SIGNIFICANCE OF OPERATOR j

consider a vector B represented by OP1. If we rotate the vector B through 1800

we get the vector –B represented by OP2 .Therefore , if OP1 = B

then OP2 = -B Hence , If we multiply a vector by -1 , it is turned through 1800 in the

anticlockwise direction . But -1 = √-1 * √-1

So in order to rotate a vector through 1800 , we multiply it twice by √-1.Hence in order to rotate it through 900 in anticlockwise direction it has to be

multiplied by √-1 as shown in fig.

The factor √-1 is denoted by j

it is called j operator . It is not real number but it is an imaginary number.

REACTANCEIt is the opposition offered by pure inductance or pure capacitance to the flow of

electricity in a circuit.IT IS MEASURED IN ohm.

REACTANCE

INDUCTIVE REACTANCE(XL)

CAPACITIVE REACTANCE(XC

)

IT IS ASSOCIATED

WITH THE MAGNETIC

FIELD.

IT IS ASSOCIATED WITH THE

CHANGING ELECTRIC FIELD B/W TWO CONDUCTING

SURFACES.

REACTANCE DUE TO INDUCTANCE (L) IS

XL= ωL=2∏ L

REACTANCE DUE TO CAPACITANCE (C) IS

XC=1/ωc=1/2∏ C

FOR D.C = 0

so XL=0

FOR A.C XL α

FOR D.C =0 so XC=∞

FOR A.C

XC α 1/

IMPEDENCE(z)

It is the opposition offered by an electronic device (L,C,R) to

the flow of electricity in a circuit.

It is measured in ohm

RESISTIVE , INDUCTIVE , CAPACITIVE A.C CIRCUITS

THESE CIRCUITS CONTAINS ONLY resistance OR ONLY INDUCTANCE OR ONLY CAPACITANCE WHICH ARE FURTHER CONNECTED TO A.C SUPPLY

i.e E = E0sinωt = E0ejωt

1. RESISITIVE A.C CIRCUITS :

The circuits contains only resistance

E=E0ejωt

Let the applied instantaneous emf beE=E0ejωt &

Instantaneous current be I=(E0/R) ejωt = I=I0ejωt

There is no phase difference b/w emf & current

IB E

O A

2. INDUCTIVE A.C CIRCUITSThe circuit contains only inductance

E=E0ejωt

let the applied instantaneous emf be E=E0ejωt &

instantaneous circuit current be

I=(E0/XL) e(jωt-∏/2) = I = I0e(jωt-∏/2)

In this circuit current lags behind emf by ∏/2

3. CAPACITIVE A.C CIRCUITSThe circuit contains only capacitance

E=E0ejωt

let the applied instantaneous emf be E=E0ejωt &

instantaneous current be I = (E0/XC)e(jωt+∏/2) I = I0e(jωt+∏/2)

In this circuit current leads emf by ∏/2

A.C circuit containing INDUCTANCE , CAPACITANCE & resistance

Let there be an A.C circuit containing Inductance(L) , Capacitance(C) , resistance(R) connected in series as shown in figure .

Let the instantaneous emf be E=E0ejωt ---------------------

Where, E0 = peak value of applied emf .

ω= angular frequency of applied emf .Let I0 be the peak value of current in the circuit

Then, potential drop across resistor, ERO =I0R-----------

Potential drop across inductor , ELO=IOXL

Current lags = ELO = jI0XL----------------

Potential drop across capacitance , ECO=I0XC

Current leads = ECO = -jI0Xc-------------

1

2

3

4

E = ERO + ELO + ECO

USING EQNS. 2 , 3 & 4 IN ABOVE EQNS.E = I0[R + j (XL-XC)]-----------

E0 = I0Zwhere, Z = R + j(XL-XC)

z is known as complex impedence of circuit.Z = IZI e jᶲ-----------------

where, IZI = √R2+(XL-XC)2---------------& tan φ = (XL-XC)/R------------

where, φ is the phase difference b/w current & emf so, instantaneous current in the circuit is given by

I = E/Z

USING EQNS. 6 & 1 IN ABOVE EQNS. I =E0ejωt/ IZI e jᶲ

I = I0e j (ωt-ᶲ) ----------------

COMPARING EQ. 1 & 9 , IT FOLLOWS THAT THERE IS A PHASE DIFFERENCE OF φ B/W EMF & CURRENT.

5

6

7

8

9

CASE -1 XL >XC

THEN tan φ = XL –XC / R is positive φ is positive

so current lags behind emf by angle ᵩ

CASE -2XL <XC

THEN tan φ = XL –XC / R is negative φ is negativeso current leads emf by angle φTherefre, I = I0e j (ωt + φ )

CASE -3XL = XC

THEREFORE , tan φ = XL –XC / R = 0

φ = 0

so current & emf are in phase , impedence IZI = RIZI is minimum

hence , (I) in circuit is maximum This is the case of RESONANCE.

SERIES RESONANT CIRCUITThe impedence of an A.C circuit with resistance , inductance, & capacitance in series is given by

IZI = E/√R2+(XL-XC)2 = E/√R2+(ωL-1/ωc)2

IN THE ABOVE EQNS. the inductive reactance is proportional to the frequency of A.C as ωL=2∏ L & the capacitive reactance is inversely prportional to frequency as Xc=1/ 2∏ C

When ωL=1/ωc the impedence of the circuit becomes minimum & is given by IZI=R &

The current in the circuit becomes maximum & is given by I = E/R

The particular frequency 0 at which the impedence of the circuit becomes minimum &

the current becomes maximum is called (hence this circuit is called acceptor circuit ) as RESONANT FREQUENCY OF THE CIRCUIT.

0=1/2∏√LC

PARALLEL RESONANT CIRCUITA parallel resonant circuit consists of an inductance L & capacitance C

connected in parallel to the alternating emf.

RESONANT FREQUENCY 0 = ω0/2∏=1/2∏ √1 / LC - R2/L2

The impedence of the circuit is maximum & hence the current (I) will be minimum hence resonance circuit is

called REJECTOR circuit

SHARPNESS OF RESONANCESharpness is the measure of rapidity with which current falls from its maximum value when frequency is changed

above or below resonance frequency.MATHEMATICALLY, it is defined as ratio of resonance

frequency & bandwidth

S = ω0/ ω

Where, bandwidth ω= ω2 - ω1

ω2 & ω1 are upper & lower cut off frequency at which

current falls to 1/√2 times the maximum current.

QUALITY FACTOR OF A RESONANCE CIRCUITOR

Q--- FACTOR

It is the measure of its ability to discriminate resonance frequency from other frequency's.

It is defined as 2∏ times the ratio of maximum energy stored in the circuit per cycle to energy dissipated per

cycle.Q = 2∏ (max. energy stored per cycle)

Energy dissipated per cycle

FOR SERIES RESONANCE CIRCUIT: QS=XL/R = XC/R

FOR PARALLEL RESONANCE CIRCUIT:QP = XL/R

IMPORTANCE OF Q – FACTOR

Q=

It is obvious that a high Q-circuit has a low resistance & low resistance LCR circuit has a greater sharpness.

Hence, in a series LCR circuit having low resistance ,the current amplitude falls from its resonant value rapidly . The selectivity of

such a circuit is good.

While that of high resistance LCR circuit selectivity is poor & resonance is flat.

So, in the tunings circuit in radio receivers at sharp resonance Q is adjusted to be small. Hence , RESONANT FREQUENCY is maximum.

ω0L R

COMPARISON BETWEEN SERIES & PARALLEL RESONANCE CIRCUIT

SERIES

1. The inductive & capacitive voltages are equal & opposite.

2. It is used to produce the maximum current & minimum impedence.

3. This circuit is called acceptor circuit since it accepts maximum current at resonance.

PARALLEL

1. The inductive & capacitive currents are equal & opposite .

2. It is used to provide maximum voltage & minimum current.

3. This circuit is called rejector circuit since it allows minimum current at resonance.