Electricity (Lec-7& 8)

Growth and decay current in L-R circuit, AC and DC,

AC through inductor, capacitor, resistor, LCR in series
and parallel circuit, Exercise.
1. Growth and decay of current in L-R circuit

See word file Lecture 6

Q2: What is alternative current and its details?

Alternating Currents
An alternating current is a current that a periodic function of time. Or
an alternating current is one which passes through a cycle of changes
at regular intervals. Alternating current (AC), is an electric current in
which the flow of electric charge periodically reverses direction,
whereas in direct current (DC, also dc), the flow of electric charge is
only in one direction. The value of current at any instant t is given by,
sinusoidal wave.
Comparison Chart

PARAMETERS AC (ALTERNATING CURRENT) DC (DIRECT CURRENT)

Definition AC is the type of electric current which varies DC is the type of electric current which
instantaneously with time. remains constant with time.

Long Distance Suitable for long distance transmission as power Not suitable as power loss is directly
Transmission losses are minimum. proportional to distance.
Flow of Electrons Bi-directional flow of electrons Uni-directional flow of electrons

Freqency Between 50 Hz to 60 Hz, different in different Frequency of DC is zero.

countries
Power Factor It lies between 0 and 1. In DC, it will be always 1.
Graphical Sinusoidal Wave Constant line
Representation
Generation By placing current carrying coil in rotating By placing current carrying coil in steady
Mechanism magnetic field. magnetic field.
Generating Devices Alternators Cells or batteries

Type of Load It can be resistive, inductive or capacitive. Only resistive

Capacitive Impedance Capacitor allows DC to pass thorugh it, thus Capacitor blocks DC, thus capacitive
capacitive impedance will be low. impedance will be infinite.

Applications High Voltage application such as home Low Voltage applications in electronic
appliances, office equipments. circuits
Circuit containing pure Resistance only:
Let an AC source of emf E be connected to a pure resistance
R. The instantaneous emf from the source is given by
E = Eo sin wt
From Ohm's law
I = E/R = Eo/R . sin wt
I = Io sin wt
Circuit containing pure inductor only:

Inductors do not behave the same as resistors. Whereas resistors

simply oppose the flow of electrons through them (by dropping a
voltage directly proportional to the current), inductors oppose changes
in current through them, by dropping a voltage directly proportional
to the rate of change of current.
Let an AC source be connected across a pure inductive element. If the
alternating current I = Io sin wt flows through it. Then
According to Kirchhoff's Law

E = LIocost

E = Eo cost
Circuit containing pure capacitance only:

The instantaneous power changes at twice the frequency that is applied.

The average power will be zero because the average of a sine

function is zero
Let an AC source be connected across pure inductive element.

If E = Eosinwt then,
q = EC = CEo sin wt

I = Io cos t

Here Io = Eo C = EoC
Eo / Io = 1/C
Flow of A.C through inductor, capacitor and
resistance in series:
Series RLC circuits contain two energy storage elements, an
inductance L and a capacitance C. Consider the RLC circuit
below.
 VR = IR
VL = IjwL
1
VC = I
jwC

Thus in vector form the emf of the circuit is

1
[ R + j ( wL − )]I = 0e jwt
wC
Where I represents the instantaneous current in circuit,
is given by
  0e jwt

I=
1
R + j ( wL − )
wC
1
R − j ( wL −
)
1 wC
=
1 1 2
R + j ( wL − ) R 2 + ( wL − )
wC wC
=  − j
where
R
=
1 2
R 2 + ( wL − )
wC
1
wL −
wC
=
1 2
R 2 + [ wL − ]
wC
 And If we put
1
wL −
wC
tan  =
R
R
cos =
1 2
R 2 + ( wL − )
wC
1
wL −
wC
sin =
1 2
R 2 + ( wL − )
wC
 Hence
1 1
= (cos − j sin  )
1 1 2
R + j ( wL − ) R 2 + ( wL − )
wC wC
1
= e − j
1 2
R 2 + ( wL −)
wC
Then the exp resion for current
0 j ( wt − )
I= e
1 2
R 2 + ( wL − )
wC
or
I = I o e j ( wt− )
where
0
I0 =
1 2
R 2 + ( wL − )
wC
1 2
R 2 + ( wL − ) =Z
I0 is the peak value of the current. And Z is the impedance wC
The equation shows that the current lags the applied voltage in phase
by an angle  given by
1
wL −
wC
 = tan −1
R

Special Cases
1
1. wL  The circuit behaves as an inductive circuit
wC
1
2. wL  The circuit behaves as capacitative circuit
wC
1
3. wL = The circuit behaves as purely resistive circuit
wC
• The Parallel Resonance Circuit (Home W)

A parallel circuit containing a resistance, R, an inductance, L

and a capacitance, C will produce a parallel resonance (also
called anti-resonance) circuit when the resultant current
through the parallel combination is in phase with the supply
voltage. At resonance there will be a large circulating current
between the inductor and the capacitor due to the energy of the
oscillations, then parallel circuits produce current resonance.
 ⇒ I = V [1/R + j⟮C−1/L⟯

Resonant Frequency
We know that the resonant frequency, fr is the frequency at which, resonance occurs. In
parallel RLC circuit resonance occurs, when the imaginary term of admittance, Y [1/R
+ j⟮C−1/L⟯ is zero. i.e., the value of L=1/C then Y= 1/R

Therefore, the voltage across all the elements of parallel RLC circuit at resonance is V =
IR.
At resonance, the admittance of parallel RLC circuit reaches to minimum value.
Hence, maximum voltage is present across each element of this circuit at resonance.
The supply voltage magnitude:
I
V=
1 2
R 2 + ( wL − )
wC
• At resonance, ω = ωr,
V=R

• Current through the resistance at

ωr: V V IR
IR = R
= = =I
R R R
Differences between series and the Parallel Resonance Circuit
THANKS TO ALL