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Topics to be discussed
1. CURRENT
2. TYPES OF CURRENT
3. A.C
4. D.C
5. P.D.C
6. TERMS USED IN A.C
7. RECTANCE
8. IMPEDENCE
9. LCR circuit
CURRENT
The rate of flow of charge through any
cross-section of a substance is known as
CURRENT.
CURRENT =
TOTAL CHARGE FLOWING(Q)
TIME TAKEN(T)
Q
I= T
S.I UNITS : ampere(A)
1A=1CS-1
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.
REACTANCE
It is the opposition offered by pure inductance or pure capacitance to the flow of
electricity in a circuit.
IT IS MEASURED IN ohm.
FOR D.C
=0
FOR D.C
=0
REACTANCE
so X =∞
so
C
XL=0
FOR A.C
FOR A.C
XL α
XC α 1/
INDUCTIVE
REACTANCE(XL)
IT IS
ASSOCIATED
WITH THE
MAGNETIC
FIELD.
REACTANCE DUE TO
INDUCTANCE (L) IS
XL= ωL=2∏ L
CAPACITIVE
REACTANCE(XC)
IT IS ASSOCIATED
WITH THE
CHANGING
ELECTRIC FIELD B/W
TWO CONDUCTING
SURFACES.
REACTANCE DUE TO
CAPACITANCE (C) IS
XC=1/ωc=1/2∏
C
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 be
E=E0ejωt &
Instantaneous current be I=(E0/R) ejωt = I=I0ejωt
I
There is no phase difference b/w emf & current
O
B
E
A
2. INDUCTIVE A.C CIRCUITS
The 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 CIRCUITS
The 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 --------------------- 1
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----------- 2
Potential drop across inductor , ELO=IOXL
Current lags = ELO = jI0XL---------------- 3
Potential drop across capacitance , ECO=I0XC
Current leads = ECO = -jI0Xc------------- 4
E = ERO + ELO + ECO
USING EQNS. 2 , 3 & 4 IN ABOVE EQNS.
E = I0[R + j (XL-XC)]----------- 5
E0 = I0Z
where, Z = R + j(XL-XC)
z is known as complex impedence of circuit.
Z = IZI e jᶲ----------------- 6
where, IZI = √R2+(XL-XC)2--------------- 7
&
tan φ = (XL-XC)/R------------ 8
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) ---------------- 9
COMPARING EQ. 1 & 9 , IT FOLLOWS THAT THERE IS A PHASE DIFFERENCE OF φ B/W
EMF & CURRENT.
CASE -1
XL >XC
THEN tan φ = XL –XC / R
φ is positive
is positive
so current lags behind emf by angle
ᵩ
CASE -2
XL <XC
THEN tan φ = XL –XC / R
is negative
φ is negative
so current leads emf by angle φ
Therefre, I = I0e j (ωt + φ )
THEREFORE ,
CASE -3
XL = XC
tan φ = XL –XC / R = 0
φ =0
so current & emf are in phase , impedence IZI = R
IZI is minimum
hence , (I) in circuit is maximum
This is the case of RESONANCE.
SERIES RESONANT CIRCUIT
The 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 CIRCUIT
A parallel resonant circuit consists of an inductance L & capacitance C
connected in parallel to the alternating emf.
RESONANT FREQUENCY
2 2
0 = ω0/2∏=1/2∏ √1 / LC - R /L
The impedence of the circuit is maximum & hence the current (I)
will be minimum hence resonance circuit is called
REJECTOR circuit
SHARPNESS OF RESONANCE
Sharpness 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 CIRCUIT
OR
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=
ω0L
R
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.
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.